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This is a survey on Reidemeister torsion for hyperbolic three-manifolds of finite volume.
Torsions are viewed as topological invariants and
also as functions on the variety of representations in $\SL_2(\CC)$. In both cases,
the torsions may also be computed after composing with
finite dimensional representations
of $\SL_2(\CC)$.
In addition the paper deals with the torsion of the adjoint representation as a function on the variety of $\PSL_{n+1}(\CC)$-characters,
using that the first cohomology group with coefficients twisted by the adjoint is the tangent space to the variety of characters.
MSC: 57Q10; 57M27
Keywords: Reidemeister torsion; hyperbolic three-manifold; character variety.
§ INTRODUCTION
The goal of this paper is to survey some results on Reidemeister torsions of orientable, hyperbolic
three-manifolds of finite volume. Torsions are viewed as invariants of hyperbolic manifolds and also
as functions on the variety of
The paper uses combinatorial torsion, though some relevant developments
described here are proved using analytic torsion, which is briefly mentioned.
No details on other kinds of torsion are provided, like $L^2$-torsion.
There are remarkable recent surveys on twisted Alexander polynomials <cit.>
and on abelian torsions <cit.>.
There are also the classical surveys of Milnor <cit.> and Turaev <cit.>, as well as the books by Turaev <cit.>
and Nicolaescu <cit.>.
Besides Section <ref> (devoted to general tools on torsions),
only orientable hyperbolic three-manifolds $M^3$ of finite volume are considered.
The hyperbolic structure is unique, by Mostow-Prasad theorem, so their holonomy is unique up to conjugacy. It lifts to a representation
in $\SL_2(\CC)$. The lift is naturally associated to a spin structure $\sigma$
and it is acyclic, by a theorem of Raghunathan.
Thus this yields a topological invariant of the oriented manifold with a spin
structure (Definition <ref> in the closed case
and Definition <ref> in the non-compact case):
\tau(M^3, \sigma)\in \CC^*.
Its main properties are described, in particular, its behavior by Dehn filling
allows to construct sequences of closed manifolds whose volume stays bounded
the torsion converges to infinity (Corollary <ref>).
This must be compared with Theorem <ref> (due to Bergeron and Venkatesh)
on the asymptotic behavior
of torsions by coverings, that yields sequences of coverings $M^3_ n\to M^3 $
so that
\left|\tau(M^3_ n, \sigma)\right|
grows with the exponential of the volume of $M_n^3$.
For those sequences the injectivity radius converges to infinity. Some results
suggest that short geodesics may play a role in the behavior of this torsion,
for instance the surgery formula,
Proposition <ref>.
Additionally, the paper deals with finite dimensional irreducible
representations of $\SL_2(\CC)$. Their composition with the lift of the
provides a family of invariants of a
closed hyperbolic manifold, oriented and with a spin structure.
A remarkable theorem of Müller relates the asymptotic behavior of those
invariants with the volume (Theorem <ref>).
For simplicity,
finite volume hyperbolic three-manifolds are assumed to have
a single cusp.
I am interested in the torsion as a function on the variety of characters. The
distinguished component
is the component of the variety of $ \SL_2(\CC)$-characters that contains a
lift of the holonomy of the complete structure.
It is easy to see that the torsion is a rational (meromorphic) function on this
curve. It is shown here that for a knot
exterior (more generally, for a manifold with first Betti number 1), this
torsion is a regular function (it has no poles)
on the distinguished component.
The paper also discusses the functions obtained by composing the representation
with the irreducible representations of $\SL_2(\CC)$.
In particular the torsion for the adjoint representation occurs in the volume
conjecture, which is also quickly mentioned but not analyzed.
Finally I describe the torsion of the adjoint representation as a function on
the variety of characters in $\PSL_{n+1}(\CC)$.
I considered the case for $\PSL_{2}(\CC)$ in <cit.> and
Kitayama and Terashima discuss the general case
for $\PSL_{n+1}(\CC)$ in
<cit.>. The relevant fact for this torsion is a result of
André Weil, that identifies the tangent space to the variety of characters
with the first cohomology group with coefficients twisted by the adjoint
representation (under generic hypothesis).
In this way this torsion is related to local parameterizations of the
deformation space, which amounts to choose peripheral curves. In particular,
a formula for the change of curve is provided, and this allows to define a
volume form under some circumstances (not at the holonomy of the complete
but for instance for characters in $\operatorname{SU}(n)$), as done by Witten
for surfaces and Dubois for knot exteriors and $\operatorname{SU}(2)$.
In addition Weil's interpretation allows to compute the torsion for surface
bundles from the tangent map of the monodromy on the deformation space of the
The paper is organized as follows. Section <ref> is devoted to
the preliminaries on combinatorial torsion, including examples
as Seifert fibered manifolds, Witten's theorem on the volume form on
representations of surfaces, and Johnson's construction of an
analog to Casson's invariant. This section concludes with a brief recall of
analytic torsion and Cheeger-Müller theorem on the equivalence of both.
Section <ref> discusses the torsion of a hyperbolic
three-manifold for the lift of the holonomy in $\SL_2(\CC)$. It addresses
first closed manifolds and then the cusped ones, considering both the invariant
and the function on the distinguished component of the
variety of characters.
Section <ref> deals with the analog to the previous section, but
instead of the torsion of representations in $\SL_2(\CC)$,
it considers torsions of compositions with finite dimensional representations
of $\SL_2(\CC)$. In particular it mentions the torsion of the adjoint
representation as part of the volume conjecture.
Section <ref> deals with the the torsion of the adjoint on the variety
of characters in $\PSL_{n+1}(\CC)$, in particular recalling the work
of Kitayama and Terashima and the author.
The paper concludes with two appendices. Appendix <ref> is devoted
to the proof of a technical result: the trivial representation does not lie in
the distinguished component of
the variety of characters if the first Betti number is one. This is used in
Section <ref> to prove that the torsion in $\SL_2(\CC)$ is a
regular function
without poles. Appendix <ref> provides results in cohomology
that are required for the torsions of Sections <ref> and
<ref>. It is based mainly on a vanishing theorem of Raghunathan
and it also discusses bases for the cohomology groups.
I am grateful to Teruaki Kitano, Takahiro Kitayama, and the anonymous referee
for useful remarks.
§ COMBINATORIAL AND ANALYTIC TORSIONS
This section overviews the definition and main properties of combinatorial
as they are used later in some proofs. It also recalls briefly the definition
of analytic torsion, as it is used in many relevant results.
§.§ Combinatorial torsion
For a detailed definition of combinatorial torsion see <cit.>. This section follows mainly <cit.>,
in particular its convention with the power $(-1)^{i+1}$ instead of $(-1)^{i}$
for the alternated product.
(See Remark <ref> on this convention.) The definition is given
for chain and for cochain complexes, in a way that both homology and cohomology
give the
same definition of the torsion of a manifold.
§.§.§ Torsion of a chain complex
Let $F$ be a field and $C_*=(C_*,\partial)$ a chain complex of finite
dimensional $F$-vector spaces:
C_d\xrightarrow{\partial} C_{d-1} \xrightarrow{\partial}\cdots
\xrightarrow{\partial} C_0.
The subspaces of boundaries and cycles are denoted by
\xrightarrow{\partial} C_i)$ and $Z_i=\ker(C_{i}
\xrightarrow{\partial} C_{i-1} )$ respectively, the homology is denoted by by
is an $F$-basis for $C_i$ and
is a basis for $H_i$, if nonzero. For the definition of torsion, a basis
$$b_i=\{b_{i,1},\ldots,b_{i,r_i} \}$$
for $B_i$ is also required. Using the
exact sequences:
\begin{gather}
\label{eqn:cycle} 0\to Z_i\to C_i \xrightarrow{\partial} B_{i-1}\to 0 \\
\label{eqn:homology} 0\to B_i\to Z_i\to H_i\to 0,
\end{gather}
lift $b_{i-1}$ to $\tilde b_{i-1}\subset C_i$ in (<ref>) and $h_i$ to
$\tilde h_i\subset Z_i\subset C_ i$
in (<ref>) and construct a new basis for $C_i$:
\begin{equation}
b_i\sqcup \tilde b_{i-1}\sqcup \tilde h_i ,
\end{equation}
where $\sqcup$ denotes the disjoint union.
The bases are compared with
the determinant of the corresponding matrix.
Given two bases $\alpha=\{\alpha_1,\ldots,\alpha_r\}$ and
$\beta=\{\beta_1,\ldots,\beta_r\}$ for $F^r$, if $(\eta_{ij})\in
\operatorname{M}_n(F)$
is the matrix that relates the bases, i.e. if $\alpha_i= \sum_j
\eta_{ij}\beta_j$, set
\begin{equation}
\label{eqn:det}
[\alpha,\beta]=\det(\eta_{ij})\in F^*.
\end{equation}
The torsion of the chain complex $C_*$ with bases $\{c_i\}$ and bases
for $H_i$ is:
\begin{equation}
\label{eqn:torcplx}
\tor(C_*,\{c_i\},\{h_i\})= \prod_{i=0}^{d} [ b^i\sqcup \tilde b_{i-1}\sqcup
\tilde h_i, c_i] ^{(-1)^{i+1}}\in F^*/\{\pm 1\}.
\end{equation}
As it is an alternated product, it is easy to see that it does not depend on the
choice of $b_i$, the basis for $B_i$, and it is also straightforward
that it does not depend on the lifts $\tilde b_{i-1}$ and $\tilde h_{i}$.
In the alternated product defining the torsion of a complex (<ref>),
may authors use ${(-1)^{i}}$
instead of ${(-1)^{i+1}}$.
Definition <ref> follows the convention of
<cit.> for instance,
but opposite to <cit.> and all papers on analytic torsion
and quantum invariants <cit.>.
I followed this convention in <cit.> but not in <cit.>.
This convention is better suited for polynomials, also for functions on the
deformation space; the opposite one
is more useful for the interpretation of the torsion
as a volume form.
We follow a different definition for the torsion of a cocomplex, which gives a
nonstandard statement of
Lemma <ref> (in the standard version it is corrected by a power
There is a formula of change of bases.
For different choices $c'_i$ and $h'_i$ of bases for $C_i$ and $H_i$ we have
\begin{equation}
\label{eqn:change}
\frac{ \tor(C_*,\{c_i'\},\{h_i'\} )}{ \tor(C_*,\{c_i\},\{h_i\}) } =
\prod_{i=0}^{d} \left(\frac{[c'_i,c_i ]}{ [h'_i,h_i ]}\right)^{(-1)^i}.
\end{equation}
The proof is straightforward, see
<cit.> for details.
Let $C^*=(C^*,\delta)$ be a cochain complex with bases $\{c^i\}$, and
cohomology bases $\{h^i\}$, by constructing the $b^i$ in a similar way,
its torsion is defined as
\begin{equation}
\label{eqn:cohom}
\tor(C^*,\{c^i\},\{h^i\})=\prod_{i=0}^{d} [ b^i\sqcup \tilde b^{i-1}\sqcup
\tilde h^i, c_i] ^{(-1)^{i}}\in F^*/\{\pm 1\}.
\end{equation}
The convention of powers $(-1)^{i}$ and $(-1)^{i+1}$ has been changed in this
definition, for the purpose of Lemma <ref>.
Let $(C_i)^ \star=\hom_F(C_i,F)$ be the dual cocomplex, with coboundary
C_i^\star\to C^\star_{i-1}$ defined
by $\delta(\theta)=\theta\circ d$. If $(h_i)^\star$ is dual to $h_i$, then
\tor(C_*,\{h_i\}) =\tor ((C_*)^\star, \{ (h_i)^\star\}).
Notice that this lemma uses Definition <ref> of a cochain complex.
instead of the cocomplex one considers the dual complex,
one must re-index the dimension $i$ by $d-i$, then the torsion of the complex
is replaced by its $(-1)^{d+1}$-power!
Then, one may see <cit.> for a proof with this version of
Lemma <ref>.
§.§.§ Twisted chain complexes
Let $K$ be a finite CW-complex. This paper is mostly interested in 3-manifolds
but also in surfaces and in $S^1$, for which there is a canonical
choice of PL-structure.
\rho \colon \pi_1 K \to \mathrm{SL}_n(F)
be a representation of its fundamental group.
Consider the chain complex of vector spaces
C_*(K;\rho):= F^n\otimes_{\rho} C_* (\widetilde K;\ZZ)
where $C_* (\widetilde K;\ZZ )$ denotes the simplicial complex of the universal
covering and
means that one takes the quotient of
F^n\otimes_\ZZ C_* (\widetilde K;\ZZ) $ by the $\ZZ$-module generated by
$$\rho(\gamma)^t v \otimes c- v\otimes \gamma\cdot c ,
where $v\in F$, $\gamma\in\pi_1 K$ and $c\in C_* (\widetilde K;\ZZ)$, and $^t$
stands for transpose.
$$v\otimes \gamma\cdot c= \rho(\gamma)^t v \otimes c \qquad \forall
\gamma\in\pi_1K.$$
Instead of the transpose, one could use the inverse. For some representations
the inverse and transpose are the same or conjugate, but this is not true in
and here this is relevant for duality with cohomology.
The boundary operator is defined by linearity and $\partial (v\otimes
c)=v\otimes\partial c$, for $v\in F$ and $c\in C_* (\widetilde K;\ZZ)$.
The homology of this complex is denoted by
H_*(K; \rho).
Analogously, one considers the cocomplex of cochains
C_*(K;\rho):= \hom_{\pi_1 K} ( C_* (\widetilde K;\ZZ), F^n)
that has a natural coboundary operator to define the cohomology
H^*(K; \rho).
Then $C_*(K;\rho)$ is
a cocomplex of finite dimensional $F$-vector spaces.
Choose $\{v_1,\ldots,v_n\}$ a $F$-basis for $F^n$ and let $\{ e^i_1,\ldots,
denote the set of $i$-cells of $K$. Then
$c_i=\{v_r\otimes\tilde e^i_s\mid r\leq n, s\leq j_i \}$ is a $F$-basis for
Let $h_i$ be a basis for $H_i(K; \rho)$. One can now define the torsion by means
of chain complexes:
The torsion of $(K,\rho, \{h_i\})$ is
\tor(K,\rho, \{h_i\}) = \tor( C_*(K;\rho), \{c_i\}, \{h_i\})\in F^*/\{\pm 1\}.
* This torsion does not depend on the lifts of the cells $\tilde
e_i$ nor the basis $\{v_1,\ldots,v_n\}$ of $F^n$.
* It does not depend either on the conjugacy class
of $\rho$, taking care that the bases for the homology are in correspondence
via the natural isomorphism between
the homology groups of the conjugate representations.
The torsion lies in $F^*/\{\pm 1\}$, but there are ways to avoid the sign
* When both the Euler characteristic $\chi(K)$ and $n=\dim\rho$ are
even, then the sign of this torsion is also well defined,
i.e. it lives
in $F^* $.
* When $\chi(K)$ is even but $n=\dim\rho$ is odd, the ordering of the cells is
relevant: If two cells are permuted in the construction,
then the sign of the torsion is changed.
To overcome this and to get an invariant in $F^*$, Turaev <cit.>
noticed that given an ordering of the basis in homology with
constant real coefficients, $H_*(K;\mathbb R)$, there is a natural way to order
the cells of $K$, (so that the torsion with trivial coefficients is then
So if one has an orientation of the the homology $H_*(K;\mathbb R)$, then
there is a choice for the sign of the torsion.
It is a topological invariant but not in its relative version,
cf <cit.>. Namely:
* It is an invariant of the simple homotopy type of $K$. Hence, by Chapman's
theorem <cit.>,
$\tor(K,\rho, \{h_i\}) $ is invariant by homeomorphisms.
Working with manifolds of dimension $\leq 3$, uniqueness of the triangulation
up to subdivision is an alternative to Chapman's theorem.
* However there is a well defined notion of torsion of a pair of CW-complexes
$(K,L)$, but then the torsion is not a topological
invariant of a pair. In fact Milnor used
Reidemeister torsion of a pair to distinguish two homeomorphic simplicial
complexes that are not combinatorially equivalent <cit.>.
We deal now with the construction using cohomology. Consider elements $(\tilde
e^i_r)^\star\otimes v\colon C_i(\tilde K;\ZZ)\to F ^n$
as the morphism of $\pi_1K$-modules defined by
\left( (\tilde e^i_r)^\star \otimes v \right) (\tilde e^i_s)=\left\{
\begin{array}{ll}
v & \textrm{ if } s=r, \\
0 & \textrm{ if } s \neq r.
\end{array}
\right.
Then $c^i=\{(\tilde e^i_r)^\star \otimes v_j\}_{j\leq n,\ r\leq n_i}$ is a
basis for $C^i(K; \rho)$.
Using this basis, one may define the torsion of the cocomplex $C_*(K;\rho)$,
$\{c^i\}$, and $\{h^i\}$
a basis in cohomology.
Remarks <ref>, <ref>, and <ref>, also
hold true for the torsion defined from cohomology.
By using Lemma <ref>,
we have:
The complexes $C^*(K;\rho)$ and $C_*(K;\rho)$ are dual. In addition
\tor( C^*(K;\rho), \{c^i\}, \{(h_i)^\star\})=
\tor( C_*(K;\rho), \{c_i\}, \{h_i\}).
Hence both homology or cohomology can be used to define the torsion.
Here Poincaré duality and duality between homology and cohomology is not
even if it is used, see <cit.> for instance.
§.§ Geometric properties of combinatorial torsion
This subsection recalls the basic properties of combinatorial torsion, the main
one being Mayer-Vietoris, useful for cut and paste.
Bundles over $S^1$ are also considered, in particular instead of just the
torsion it is more convenient to consider
the twisted Alexander polynomial, which is a torsion by <cit.>. The
subsection finishes with examples.
§.§.§ Mayer-Vietoris
Let $K$ be a CW-complex, with subcomplexes $K_1,\ K_2\subset K$ so that
$K=K_1\cup K_2$. Let $\rho\colon\pi_1K\to \SL_n(F)$ be a representation.
Consider the diagram
of inclusions
\[
\xymatrix{
K_1\cap K_2
\ar[d]^{i_2}
\ar[r]^{i_1} & K_1 \ar[d]^{j_1} \\
K_2 \ar[r]^{j_2} & K.
\]
There is a Mayer-Vietoris exact sequence in homology with twisted coefficients,
if the representations
on $\pi_1K_1$, $\pi_1K_ 2$ and $\pi_1L$ are the restrictions:
\begin{multline}
\label{eqn:MV}
\cdots \to \bigoplus_L H_i(L;\rho) \xrightarrow{i_{1*}\oplus i_{2*}}
H_i(K_1;\rho)\oplus H_i( K_2;\rho)
\xrightarrow{j_{1*}-j_{2*}} H_i(K;\rho) \\
\to \bigoplus_L H_{i-1}(L;\rho)\to\cdots
\end{multline}
where $L$ runs on the connected components of $K_1\cap K_2$. Notice that
different choices of base-points yield canonical isomorphisms
between the homology groups, hence we can consider $K_1\cap K_2$ not
Choose a basis for each of these cohomology groups $h_*$ for $H_*(K;\rho)$,
$h_{1*}$ for $H_*(K_1;\rho)$,
$h_{2*}$ for $H_*(K_2;\rho)$, and $h_{L*}$ for $H_*(L ;\rho)$. The long exact
(<ref>) is viewed as a complex. Its torsion is denoted by $\tor
(\mathcal H, h_{**})$.
Let $K$ be a CW-complex, with subcomplexes $K_1\cap K_2$ so that $K=K_1\cup
K_2$. Let $\rho\colon\pi_1K\to \SL_n(F)$ be a representation and choose basis
\tor (K, h_*;\rho)=\frac{\tor (K_1,h_{1*};\rho)\tor (K_2,h_{2*};\rho)
}{\prod\limits_L\tor (L,h_{L*};\rho) \tor (\mathcal H, h_{**}) }
where $L$ runs on the connected components of $K_1\cap K_2$.
This is used in surgery formulas (e.g. Proposition <ref>) or for the
mapping torus (Proposition <ref>).
The proof can be found in <cit.> or in <cit.>.
§.§.§ Polynomials
Here some properties of twisted Alexander polynomials viewed as torsions are
briefly discussed,
since they are convenient for describing some results of Reidemeister torsion.
See the recent surveys
<cit.> fore more details on twisted Alexander polynomials.
Start with a surjective morphism $\phi\colon\pi_1 K\to \ZZ$. Instead of a
representation $\rho\colon\pi_1K\to \SL_n(F)$, consider the twisted
\rho\otimes \phi\colon\pi_1 K\to \operatorname{GL}_n(F(t))
where $F(t)$ is the field of fractions of the polynomial ring $F[t]$.
If $H_*(K;\rho)=0$, then $H_*(K;\rho\otimes \phi)=0$ <cit.>
and there is a well defined torsion, or twisted polynomial:
\Delta_{K,\rho,\phi}(t)=\tor (K,\rho,\phi)\in F(t)/\pm t^{n\, \ZZ}.
As the determinant is not one, there is an indeterminacy factor $t^{k\, n}$,
for some integer $k\in\ZZ$.
Assume $H_*(K;\rho)=0$. Then
$$ \Delta_{K,\rho,\phi}(1)= \tor (K,\rho).$$
This is proved from the map at the chain level $C_*(K;\rho\otimes\phi)\to
induced by evaluation $t=1$.
If $\Delta_{K,\rho,\phi}=\sum_i a_i t^i$, then its degree can be defined as:
\deg ( \Delta_{K,\rho,\phi} ) =\max \{i-j\mid a_i,a_j\neq 0\}.
If $M^3$ is a three-manifold and $\Sigma\subset M^3$ is a surface dual to
$\phi$ such that
$H_0(\Sigma;\rho)=0$, then
\deg \Delta_{M^3,\rho,\phi} \leq -\chi(\Sigma) n.
This proposition can be proved using Mayer-Vietoris to
a tubular neighborhood of the surface and its exterior.
For a knot ${\mathcal K}\subset S^3$, there is a natural surjection
$\phi\colon\pi_1(S^3-{\mathcal K})\to \ZZ$.
Let $A_{\mathcal K}$ denote its Alexander polynomial.
In <cit.> (this reference is based on a preprint from 1990) Lin defined a
twisted Alexander polynomial $A_{\mathcal K}^\rho$ that lives in $F(t)$,
that was later modified by Wada
Milnor in 1962 for the untwisted Alexander polynomials, and Kitano in 1996 for
the twisted ones, proved:
For a knot exterior $S^3-{\mathcal K}$, we have:
* <cit.> For the trivial representation:
\Delta_{S^3-\mathcal K,1,\phi} =\tor(S^3-{\mathcal K}, \phi)= \frac{A_{\mathcal
* <cit.> For $\rho$ a non-trivial acyclic representation:
\Delta_{S^3-\mathcal K,\rho,\phi}=
\tor(S^3-{\mathcal K},\rho\otimes \phi)= A_{\mathcal K}^\rho(t).
Milnor's formula, for the untwisted polynomial, has been generalized to links
and to other three-manifolds in 1986 by Turaev <cit.>. See also
The following is straightforward from Mayer-Vietoris, Proposition <ref>:
Let $K$ be a CW complex, $f\colon \vert K\vert \to \vert K\vert $ a
with mapping torus $M_f$. Then
$$\Delta _{M_f,\rho,\phi}=\prod_{i=0}^{\dim K} \det (f_i -
t\operatorname{Id})^{(-1)^{i+1}}\, , $$
where $f_i\colon H_i(K;\rho)\to H_i(K;\rho)$ denotes the induced map in
In particular, by Remark <ref>,
\tor(M_f,\rho)=\prod_{i=0}^{\dim K} \det (f_i -
\operatorname{Id})^{(-1)^{i+1}}.
Another relevant issue for this polynomial is Turaev's interpretation of the
twisted Alexander polynomial: namely this polynomial encodes the torsion of
the finite cyclic coverings (<cit.>, see
also <cit.>).
Let $K$, $\rho\colon\pi_1 K\to \SL_n(F)$, and $\phi\colon\pi_1 K\to \ZZ$ be as
Let $K_m\to K$ be the cyclic covering of order $m$ corresponding to the kernel
of $\phi$ composed with the projection
* $H_*(K_m;\rho)=0$ if and only if $\Delta_{M^3,\rho,\phi} (\zeta)\neq 0$
for every $\zeta\in\CC$
satisfying $\zeta^m=1$.
* If $H_*(K_m;\rho)=0$ then
\tor(K_m, \rho)=\prod\limits_{i=0}^{m-1} \Delta_{M^3,\rho,\phi} ( \zeta^i)
for $\zeta\in\CC$ a primitive $m$-root of the unity.
§.§.§ Examples
The first examples below can be found in many references, for instance
[The circle]
Consider the circle $S^1$. A representation $\rho$ of its fundamental group
$\pi_1 S^1\cong\ZZ$ is determined by the image of its generator,
that we denote by $A\in\SL_n(F)$.
Notice that the homology and cohomology of $S^1$ twisted by $\rho$ is
determined by $H^0(S^1;\rho)$,
by duality between homology and cohomology and vanishing of the Euler
Since $H^0(S^1;\rho)$ is isomorphic to the subspace of invariant elements
$(F^n)^{\rho(\ZZ)}=\ker (A-\operatorname{Id} ) $,
\begin{equation*}
H_*(S^1;\rho)=0\textrm{ if and only if }\det (A-\operatorname{Id})\neq 0.
\end{equation*}
On the other hand, $S^1$ is just the mapping torus of the identity map on the
point $*$. The homology of the point is $H_0(*;\rho)\cong F^n$ and the action
return map is multiplication by $A$.
Thus, by Proposition <ref>:
\Delta(S^1,\rho\otimes\phi)=\frac{1}{\det (A-t\operatorname{Id})} \qquad\textrm{
and }\qquad \tor(S^1, \rho)=\frac{1}{\det (A-\operatorname{Id})}.
In fact using Proposition <ref> is a fancy way of computing the
torsion of the circle. It is more natural to view
Proposition <ref> as a generalization of the torsion of the
[The 2-torus]
The homology and the cohomology of the two-torus $T^2$, $H_*(T^2;\rho)$ and
$H^*(T^2;\rho)$, are
determined by $H^0(T^2;\rho)$ which is the subspace of $F^n$ of invariant
elements. This assertion follows from the different dualities (Poincaré, and
and the Euler characteristic. Thus if $F^n$ has no nonzero invariant elements by
$\rho(\pi_1T^2)$, then $H_ *(T^2;\rho)=0$. In this case
\tor(T^2;\rho)=1.
This can be proved viewing $T^2$ as the mapping torus of the identity on $S^1$
and Proposition <ref>,
as the action on $H^1(S^1;\rho)$ is the same as on $H^0(S^1;\rho)$ and the
corresponding terms cancel. See also <cit.>.
[Lens spaces]
Let $p,q$ be integers so that $p\geq 2$, $p\geq q\geq 1$ and $p$ and $q$ are
coprime. View the three-sphere as the unit sphere in $\CC^2$:
$S^3=\{(z_1,z_2)\in \CC^2\mid \vert z_1\vert ^2+\vert z_2\vert^2=1\}$,
and consider the action of
the cyclic group of order $p$ generated by the transformation
\begin{array}{rcl}
S^3 & \to & S^3 \\
(z_1,z_2)& \mapsto & (e^{{2\pi i}/{p}} z_1, e^{ {2\pi i \, q}/{p}} z_2)\, .
\end{array}
The quotient by this action is the Lens space $L(p,q)=S^3/\!\sim\,$.
Consider the Heegaard decomposition into two solid torus $L(p,q)=V_1\cup V_2$,
where $V_1$ and $V_2$ are the torus that lift respectively to
\{(z_1,z_2)\in S^3 \mid \vert z_1\vert \geq \vert z_2\vert\} \qquad\textrm{ and
}\qquad \{(z_1,z_2)\in S^3 \mid \vert z_1\vert \leq \vert z_2\vert\}.
Consider a non-trivial representation $\rho\colon\pi_1 L(p,q)\to \CC^*$. As the
group is cyclic, every irreducible complex representation is one dimensional.
The representation is determined by a non-trivial $p$-th root of unity
$\zeta=e^{\frac{2\pi i\, k}p}$, $k\not\equiv 0\mod p$. If $\rho$ maps the soul
of $V_1$ to $\zeta$, then it maps
the soul of $V_2$ to $\zeta^r$, where $r$ is an integer satisfying $q r\cong
1\mod p$. Applying Mayer-Vietoris (Proposition <ref>) to the pair
$(V_1,V_2)$ and
by the previous examples:
\tor(L(p,q),\rho)=\frac{1}{ (1-\zeta)(1-\zeta^r)}.
Notice that the determinant of $\rho$ is not one, but has norm one. Thus
the topological invariant is obtained once taking the module
$ \vert \tor(L(p,q),\rho)\vert$ and considering all nontrivial $p$-roots
of unity.
This is the original example of Franz <cit.> and Reidemeister
<cit.>. See also <cit.>, for instance.
[$\Sigma\times S^1$]
Consider the product of a compact oriented surface, possibly with boundary, and
the circle, $\Sigma\times S^1$. Let $\rho\colon\pi_1 ( \Sigma\times S^1)\to
\SL_n(\CC)$
be an irreducible representation. In particular, $\rho$ maps $t$ the generator
of the factor $\pi_1 S^1$ to a central matrix, namely
$\rho(t)= \omega \operatorname{Id}$ with $\omega^n=1$.
We view $\Sigma\times S^1$ as the mapping torus of the identity on $\Sigma$
and apply Proposition <ref>.
By irreducibility $H_0(\Sigma;\rho)\cong H_2(\Sigma;\rho)=0$ and $\dim
H_1(\Sigma;\rho)= -n \,\chi(\Sigma)$.
As the action of $t$ on $H_1(\Sigma;\rho)= \CC^{-n \,\chi(\Sigma) }$ is
by $\omega$:
* $\rho$ is acyclic iff $\omega\neq 1$.
* when $\omega\neq 1$, then
\tor( \Sigma\times S^1,\rho)=(\omega-1)^{-n\,\chi(\Sigma)}
[Seifert fibered manifolds]
In <cit.> Kitano computed the Reidemeister torsion of a
Seifert fibered three-manifold with a representation
in $\SL_n(\CC)$. His result is reproduced here. Previously, Freed had computed a
torsion for Brieskorn spheres <cit.>, and of course
Franz <cit.> and Reidemeister <cit.> for lens spaces.
Let $M^3$ be a Seifert fibered manifold, whose base is a compact surface
$\Sigma$, possibly with boundary, with $c$ cone points corresponding to singular
Let $\rho\colon\pi_1 M^3 \to \SL_n(\CC)$ be an irreducible representation. In
particular $\rho$ maps the fibre to $\omega \operatorname{Id}$,
where $\omega^n= 1$. For the $i$-th cone point, let $(\alpha_i,\beta_i)$ denote
the Seifert coefficients, and let $c_i$ denote the loop
such that $c_i^{\alpha_i} f ^{\beta_i}=1$. Let
$\{\lambda_{i,1},\ldots,\lambda_{i,n}\}$ denote the eigenvalues of $\rho(c_i)$.
Choose integers
$(r_i,s_i)$ such that $\alpha_i s_i-\beta_is_i=1$.
Let $\dot\Sigma$ denote the surface $\Sigma$ minus the cone points, i.e. the
space of regular fibres,
so that $\chi(\dot\Sigma)=\chi(\Sigma)-c$.
Assume $M^3$ is not a solid nor a thick torus, then <cit.>:
* $\rho$ is acyclic if and only if $\omega\neq 1$ and
$\lambda_{i,j}^{r_i}\omega^{s_i}\neq 1$, for every $i=1,\ldots,c$,
* If it $\rho$ is acyclic, then
\tor(M^3,\rho)=(\omega-1)^{-n\chi(\dot\Sigma)}\prod_{i=1}^c\prod_{j=1}^n\frac
Not being a solid torus $S^1\times D^2$ nor a thick torus $S^1\times S^1\times
[0,1]$ implies that $\chi(\dot\Sigma)<0$.
When the base $\Sigma$ is orientable, this is a straightforward consequence of
Examples <ref>, <ref>, and <ref>.
When $\Sigma$ is not orientable, choose a reversing orientation curve
$\sigma\subset\dot\Sigma$, so that a tubular neighborhood of $\Sigma$ is
a Möbius band and its complement is orientable, with the same Euler
characteristic. By using the fibration, this decomposes $M^3$ as a Seifert
$N$ with orientable base and $Q$, the orientable circle bundle over the
Möbius strip, with $N\cap Q=\partial Q\cong S^1\times S^1$.
In particular $\pi_1N$ surjects onto $\pi_1M^3$, so the induced representation
on $N$ is irreducible and we use the theorem in the case with orientable base.
The curve $\sigma$ may be chosen so that $\rho\vert_{\pi_1 Q}$ is non-trivial,
hence $Q$ and $\partial Q$ are acyclic, and the assertion about acyclicity
follows from Mayer-Vietoris
(Proposition <ref>) and
the case with orientable base. In addition, in the acyclic case, the torsion
of the 2-torus is trivial (Example <ref>), and so is the torsion of
$Q$ that retracts to a Klein bottle
(hence it is also a mapping torus). Thus $\tor(M^3,\rho)=\tor(N,\rho)$ and the
proof is concluded.
For a Seifert fibered manifold, the torsion is constant on the components of
the variety of representations.
See <cit.> for the asymptotic behavior of these
[Torus knots]
The previous example may be applied to a torus knot, with coefficients $p,q$,
that are relatively prime positive integers. It is Seifert fibered,
with base a disc and two singular fibres, with coefficients $(p,1)$ and
Its components of the variety of representations in $\SL_2(\CC)$ are determined
by the choice of the eigenvalues of the corresponding elements,
$\left\{e^{\pm \pi i\frac{k_1}{p}}\right\}$ and $\left\{e^{\pm \pi
i\frac{k_2}{q} }\right\}$, with $0<k_1 < p $ and $0<k_2< q$ and $k_1\equiv k_2
\mod 2$.
As the fibre is mapped to $(-1)^{k_1}$ times the identity, the representation is
acyclic only for $k_i$ odd. The torsion is
\frac{1}{\left(1-\cos\frac{\pi k_1}{p}\right)\left(1-\cos\frac{\pi
The description of the components of the variety of representations in
$\SL_3(\CC)$ is more involved <cit.>, as there are much more possibilities
for the eigenvalues.
For instance, for the trefoil knot ($p=2$, $q=3$) there are no acyclic
irreducible representations in $\SL_3(\CC)$.
For $p=2$, $q=5$, there are two components of acyclic representations in
$\SL_3(\CC)$ <cit.>.
For one of the components its eigenvalues are $\{e^{\frac{2\pi
i}{3}},e^{\frac{2\pi i}{3}},e^{\frac{4\pi i}{3}}\}$ and
$\{ e^{\frac{2\pi i}{15}}, e^{\frac{8\pi i}{15}}, e^{\frac{20\pi i}{15}}\}$,
for the other, the complex conjugates. For both the torsion is
\frac{4/3}{1+2 \cos\frac{\pi}{5}}= 2-\frac{2}{3}\sqrt{5}.
See <cit.> for a discussion on the torsion of torus
[Volume form on representations of surfaces]
Let $\Sigma$ denote a surface of genus $g> 1$ and $G$ a compact Lie group.
Denote by $X(\Sigma,G)$ the variety of representations of $\pi_1\Sigma$ in $G$
up to
Let $X^*(\Sigma,G)$ denote the subset of representations such that the
commutator of the image is trivial.
This is equivalent to $H^0(\Sigma; \operatorname{Ad}\rho)=0$, and
for $G$ linear, this holds true when $\rho$ is irreducible.
If $\chi_\rho\in X^*(\Sigma,G)$, by definition $H^0(\Sigma ;
\operatorname{Ad}\rho)=0$.
Hence $H^2(\Sigma ; \operatorname{Ad}\rho)=0$
and $H^1(\Sigma ; \operatorname{Ad}\rho)$ is identified to the tangent space
of $X^*(\Sigma ; G)$ at the character of $\rho$
<cit.>, see Theorem <ref>.
Reidemeister torsion on homology defines a volume form, by the formula of change
of basis in homology (<ref>).
Namely, if $2r=\dim X^*(\Sigma,G)$,
\begin{array}{rcl}
\vol_{\tor} \colon \bigwedge^{2r} H^1(\Sigma ; \operatorname{Ad}\rho) & \to
& \RR
\\
h_1\wedge\cdots\wedge h_{2r} & \mapsto & ( {\tor(\Sigma ,\rho,
\{h_1,\ldots,h_{2r}\})} )^{-1}
\end{array}
is a well defined linear isomorphism. The sign can be controlled by means of
The space $X^*(\Sigma ,G)$ has a well defined symplectic structure, due to
and Goldman <cit.>, that we denote by $\omega$.
In particular $\frac{\omega^{r}}{r !}$ is a natural volume form. Witten proved
in <cit.> that they are the same form:
For a compact Lie group $G$
\frac{1}{(2\pi)^{2r}} \vol_{\tor} = \frac{\omega^{r}}{r !}.
The proof uses a symplectic structure on a chain complex, which has been further
developed by Sözen, cf <cit.>.
Johnson used the point of view of volume for constructing the torsion from a
Heegaard splitting,
in a hand written paper
that unfortunately
was never published.
Consider a closed 3-manifold $M^3$ with a Heegaard decomposition: i.e.
$M^3=B_1\cup_\Sigma B_2$,
where $B_1$ and $B_2$ are handlebodies such that
$\Sigma= B_1\cap B_1=\partial B_1=\partial B_2$ is a surface of genus $\geq 2$.
This yields a commutative diagram of fundamental groups and their representation
\[
\xymatrix{
\pi_1 \Sigma
\ar[d]_{i_2*}
\ar[r]^{i_1*} & \pi_1 B_1 \ar[d]^{j_1*} \\
\pi_1 B_2 \ar[r]_{j_2*} & \pi_1 M
\qquad\qquad\qquad
\xymatrix{
& X^*(B_1,G) \ar[l]_{i_1*} \\
X^*(B_2,G) \ar[u]^{i_2*}
& X^*( M^3,G) \ar[l]^{j_2*} \ar[u]_{j_1*}
\]
for $G=\SU$.
Assume we have a representation $\rho\in X^*(M^3,G)$ that is infinitesimally
rigid (i.e. $H^1(M^3 ; Ad\rho)=0$.) Then,
by using a Mayer-Vietoris argument, Johnson shows that $X^*(B_1,G)$ and
$X^*(B_2,G)$ intersect transversally at the character
$\rho_0$ in $X^*(\Sigma,G)$. Johnson uses Reidemeister torsions to define
forms $\operatorname{vol}_{B_i}$ and $X^*(B_i,G)$
and $\operatorname{vol}_{\Sigma}$ on $X^*(\Sigma,G)$ (as the groups involved
are free
or surface groups). He defines
an invariant as the ratio between $\operatorname{vol}_{\Sigma}$ and
$\operatorname{vol}_{B_1}\wedge \operatorname{vol}_{B_2}$.
What he proves is:
\operatorname{tor}(M^3, \rho_0)= \frac{\operatorname{vol}_{B_1}\wedge
\operatorname{vol}_{B_2}}{ \operatorname{vol}_{\Sigma}}\, ,
cf. Proposition <ref>.
Under Johnson hypothesis there are finitely many acyclic conjugacy classes of
representations in $\operatorname{SU}(2)$ and he considers the addition of all
Reidemeister torsions.
Using this Heegaard splitting, this is analogous to Casson's invariant
<cit.>, by taking into account additionally this volume on the
varieties of characters.
The point of view of volume à la Johnson has also been used by Dubois in
<cit.>, we will comment on it in
Example <ref>.
§.§ Analytic torsion
Consider now a smooth compact manifold $M$ and a representation
$$\rho\colon\pi_1M\to \SL_n(\RR).$$
The manifold $M$ has a unique $\mathcal C^1$-triangulation, so one can view it
as a CW-complex and compute its torsion.
Consider the associated flat bundle
\begin{equation}
\label{eqn:flabundle}
E_\rho=\widetilde M\times \RR^n/\pi_1M,
\end{equation}
where $\pi_1M$ acts on the universal covering $\widetilde M$ by deck
transformations and on $\RR^n$ via $\rho$.
The space of $E_\rho$-valuated differential forms is denoted by
$\Omega^p(E_\rho)$, and its de Rham cohomology by
To simplify, we assume that $\rho$ is acyclic,
by de Rham theorem we assume that
\begin{equation}
\label{eqn:deRham}
H^*(M;E_\rho)\cong H^*(M;\rho)=0.
\end{equation}
We choose a Riemannian metric $g$ on $M$ and a metric $\mu$ on the bundle
$E_\rho$ (notice that since we do not assume $\rho$ to be orthogonal,
perhaps the metric $\mu$ cannot be chosen to be flat). This yields a metric on
$\Omega^p(E_\rho)$. Using it,
we may define the adjoint to the differential and the Laplacian $\Delta_p(\rho)$
on $\Omega^p(E_\rho)$.
As it is an elliptic operator, it has a discrete spectrum $0 <
\lambda_0\leq\lambda_1\leq\cdots\to\infty$.
The zeta function is defined on the complex half-plane $\operatorname{Re}(s)\geq
\zeta_p(s)=\sum_{\lambda_i}\lambda_i^{-s}
and it extends to the a meromorphic function on the complex plane, homomorphic
at $s=0$.
The analytic torsion is defined as:
\begin{equation}
\tor_{an}(M,\rho,g,h)=\exp\Big( \frac12 \sum_{p=0}^{\dim M} (-1)^p \,p
\zeta_p'(0) \Big).
\end{equation}
In <cit.> Müller proves that if $\dim M$ is odd and, as we assume
$\rho$ is acyclic, then it is independent of $g$ and $h$.
It can also be defined using the trace of the heat operator:
\begin{multline}
\label{eqn:traceheatkernel}
\tor_{an}(M,\rho) = \\ \exp\left( \frac12 \sum_{p=0}^{\dim M} (-1)^p p
\frac{d\phantom{s}}{d s}
\left.\left(
\frac{1}{\Gamma(s)}
\int_0^\infty
\left( \operatorname{Tr}( e^{-t \Delta_p(\rho)} ) t^{s-1} d\, t \right)
\right) \right|_{s=0} \right)\, .
\end{multline}
In the following theorem notice that we use the convention for Reidemeister
opposite to the usual in analytic torsion.
Let $M$ be a closed hyperbolic manifold of odd dimension and
$\rho\colon\pi_1(M)\to \SL_n(\mathbb{R})$. Then
\tor_{an}(M,\rho)=\frac{1}{\vert \tor(M,\rho)\vert}.
This theorem was first proved by Cheeger <cit.> and Müller
<cit.> independently for orthogonal representations, and later by
Müller <cit.> for unimodular ones. In addition, aciclycity of $\rho$
is not required by choosing orthonormal harmonic basis in cohomology.
This subsection concludes recalling a theorem of Fried.
Let $\rho\colon\pi_1 M\to \operatorname{SO}(n)$ be a representation of a
hyperbolic manifold. For $s\in\CC$
with $\operatorname{Re}(s)$ sufficiently large, consider
R_\rho(s)=\prod_\gamma\det(\operatorname{Id}-\rho(\gamma) e^{-s \, l(\gamma)})
where the product runs over the prime, closed geodesics of $M$ and $l(\gamma)$
denotes the length of $\gamma$.
This is called the Ruelle zeta function.
Let $M$ be a closed hyperbolic manifold of odd dimension and assume that
$\rho\colon\pi_1 M\to \operatorname{SO}(n)$ is acyclic.
Then $R(s)$ extends meromorphically to $\CC$ and
This theorem has been extended by Wotzke <cit.> to other representations
of hyperbolic manifolds, see also <cit.>.
§ TORSION OF HYPERBOLIC THREE-MANIFOLD WITH REPRESENTATIONS IN
This section is devoted to the torsion of orientable hyperbolic 3-manifolds,
using the representations in $\SL_2(\CC)$
obtained as a lift of the holonomy representation (the choice of the lifts
depends on the choice of a spin structure).
It starts with closed manifolds, for which this representation is acyclic.
Then it considers manifolds of finite volume with one cusp, for which it is
also acyclic. Besides the invariant itself,
it analyzes the function on the variety of characters defined by the torsion.
This is applied for instance
to study the behavior of torsion under Dehn filling.
§.§ Torsions from lifts of the holonomy representation
Let $M^3$ be a closed, orientable hyperbolic 3-manifold. Its holonomy
representations is unique up to conjugation:
\operatorname{hol}\colon \pi_1 M^3\to \operatorname{Isom}^+(\mathbb H^3)\cong
\operatorname{PSL}_2(\CC).
To get a natural representation in a linear group one can lift the holonomy to
Such a lift always exists <cit.> and it depends naturally on a choice
of a spin structure, because
the group of isometries is naturally identified with the frame bundle of
$\mathbb H^3$ and
$\SL_2(\CC)$ with the spin bundle, cf. <cit.>.
§.§.§ Lifts of the holonomy
There is natural action of $ H^1(M^3 ; \mathbb{Z}/2\mathbb{Z})$ on the set of
lifts $\varrho$ to $\SL(2,\CC)$ of the holonomy representation:
viewing $ H^1(M^3 ; \mathbb{Z}/2\mathbb{Z})$ as morphisms from $\pi_1M^3$ to
a morphism $\epsilon\colon\pi_1M^3\to \mathbb{Z}/2\mathbb{Z}$ maps a
representation $\varrho$ to $(-1)^\epsilon \varrho$.
* There is a natural bijection between the set of lifts
of the holonomy representation and the set of spin structures.
This is an isomorphism of affine spaces on the vector space $
H^1(M^3 ; \mathbb{Z}/2\mathbb{Z})$.
* If $M^3$ and $\overline M^3$ are the same manifold
with opposite orientations, then there is a natural bijection between spin
on $M^3$ and on $\overline M^3$ so that lifts of the holonomy correspond to
complex conjugates.
Item (<ref>) can be found essentially in <cit.>, see also
A quick way of proving Item (<ref>) is using the bijection of
Item (<ref>), knowing that complex conjugation in
$\PSL_2(\CC)$ is the result of composing with an isometry that changes the
orientation of $\mathbb H^3$. On the other hand,
this bijection can be constructed explicitly
from frame bundles and spin, but details are not given here.
For a spin structure $\sigma$, the corresponding lift of the holonomy, according
Proposition <ref>(<ref>), will be denoted by
\varrho_{\sigma}\colon \pi_1 M^3\to \SL_2(\CC).
The behavior of torsion by mutation is also interesting. Mutation is the
operation that consists in cutting along a genus two surface, applying the
hyperelliptic involution and gluing again.
Notice that one does do not require the surface to be essential, thus the genus
two surface can be replaced by a (properly embedded)
torus with one or two punctures or a sphere with three or four punctures (in
particular a Conway sphere for a knot exterior).
See <cit.>.
Let $(M^3)^\mu$ denote the result of mutation, by <cit.> $(M^3)^\mu$
is hyperbolic with the same volume as $M^3$.
There is a natural correspondence between the spin structures on $M^3$ and the
spin structures on $(M^3)^\mu$.
Here is an explanation of the remark, using the natural bijection between lifts
of the holonomy and spin structures in Proposition <ref>.
Assume that $\Sigma$ separates $M^3$ in two components $M_1$ and $M_2$. Then
\pi_1M^3\cong \pi_1M_1 *_{\pi_1\Sigma}\pi_1 M_2.
If $\varrho_\sigma\colon\pi_1(M^3)\to \SL_2(\CC)$ is a lift of the holonomy,
then $\varrho_{\sigma(\mu)}\colon\pi_1 (M^3)^\mu\to \SL_2(\CC)$ is defined so
$ \varrho_{\sigma(\mu)} \vert_{\pi_1 M_1}=\varrho_\sigma\vert_{\pi_1 M_1}$
and $\varrho_{\sigma(\mu)}\vert_{\pi_1 M_2}$ is conjugate to
$\varrho_\sigma\vert_{\pi_1 M_2}$ by a matrix in $\PSL_2(\CC)$
that realizes the involution $\mu$ on $\Sigma$.
When $\Sigma$ does not separate, then the construction is similar from the
presentation of $\pi_1M^3$ as HNN-extension.
§.§.§ Torsions for closed 3-manifolds
The following theorem is a particular case of Raghunathan's. With other
cohomology results, it is discussed in
Appendix <ref>. In particular the following theorem is stated in
Corollary <ref>.
Let $M^3$ be a closed, orientable, hyperbolic 3-manifold. Then any lift of its holonomy representation is acyclic.
With Mostow rigidity, this yields immediately a topological invariant of the spin manifold.
Let $M^3$ be a compact, oriented, hyperbolic 3-manifold with spin structure $\sigma$. The torsion of $(M^3,\sigma)$ is defined as
\begin{equation}
\label{eqn:tau}
\tau(M^3,\sigma):=\tor(M^3, \varrho_\sigma)\in \mathbb{C}^*
\end{equation}
where $\varrho_\sigma=\widetilde{\mathrm{hol}}$ is the lift of the holonomy $\mathrm{hol}$ corresponding to the spin structure $\sigma$.
There is no sign indeterminacy, i.e. it is a well defined complex number,
because $ \varrho_\sigma $ is a representation in $\mathbb C^2$,
which is even dimensional,
and $\chi(M^3)=0$, see Remark <ref>.
Here are some of its properties.
The torsion $ \tau(M^3,\sigma)$ in Definition <ref> has the following properties:
* It is a topological invariant of the spin manifold $(M^3,\sigma)$.
* There are examples of manifolds $M^3$ with two spin structures $\sigma$ and $\sigma'$ such that
$\tau(M^3,\sigma)\neq \tau(M^3,\sigma')$.
* Let $\overline M^3$ denote the manifold $M^3$ with opposite orientation. If one changes the orientation and the spin structure accordingly
as in Proposition <ref>, then the torsion is the complex conjugate
\tau(\overline M^3,\overline \sigma)= \overline{ \tau(M^3,\sigma)}.
* Let $(M^3)^\mu$ denote the result of mutation. If $\sigma^\mu$ denotes the corresponding spin structure as in Remark <ref>, then
\tau((M^3)^\mu, \sigma^\mu) = \tau(M^3,\sigma).
* Let $M^3$ be an oriented hyperbolic manifold with one cusp and with spin structure $\sigma$.
The set of modules of the torsions obtained by Dehn filling on $M^3$,
$\vert \tau(M^3_{p/q},\sigma)\vert $ so that $\sigma$ extends to $M^3_{p/q}$, is dense in the interval
$$\left[ \frac14\left\vert \tau(M^3,\sigma)\right\vert,+\infty\right).
Item (<ref>) follows from uniqueness of the hyperbolic structure, by Mostow rigidity.
To prove (<ref>) it suffices to compute an example, this is done in Corollary <ref>.
Item (<ref>) is straightforward from Proposition <ref>. Item (<ref>) is proved in <cit.>.
Finally, (<ref>) is proved later when discussing cusped manifolds, as this will follow immediately from a surgery formula.
Item (<ref>) shows that this torsion is not obviously related to the
hyperbolic volume.
The following theorem, a particular case of <cit.>, finds a
relation (we use the convention of torsion opposite to <cit.>):
Let $M^3$ be a compact oriented hyperbolic 3-manifold. Assume that $M_n^3\to M^3$ is a sequence of coverings such that the injectivity radius of $M_n^3$ converges to infinity.
\lim_{n\to\infty}
\frac{\log\vert \tau(M_n^3,\sigma)\vert}{ \operatorname{vol}(M_n^3) } =
\frac{11}{12 \pi} .
\lim_{n\to\infty} \frac{\log\vert \tau(M_n^3,\sigma)\vert} {\deg ( M_n^3\to M^3)
} = \frac{11}{12 \pi} \operatorname{vol}(M^3) .
This theorem relies on analytic torsion and on $L^2$-torsion, as $\frac{-11}{12
\pi}$ is the $L^2$-torsion of $\mathbb H^3$. The proof uses the $L^2$-Laplacian
hyperbolic space, and it is based on approximations of averages of the trace of the difference of heat kernels,
see Equation (<ref>).
They require the notion of strong acyclicity (the property in Theorem <ref>) to avoid eigenvalues of the Laplacian approaching to zero.
This has been generalized in <cit.>, in particular without requiring that the
$M_n^3$ are coverings. See also <cit.>.
§.§.§ Cusped hyperbolic 3-manifolds
In this subsection, $M^3$ denotes a finite volume hyperbolic manifold, i.e. a manifold whose ends are cusps.
Assume that $M^3$ has a single cusp.
This is done not only to simplify notation, but because with more cusps some
further issues need to be discussed <cit.>.
Again one has:
Let $M^3$ be a closed, orientable, hyperbolic 3-manifold with one cusp. Then any lift to $\SL_2(\CC)$ of its holonomy representation is acyclic.
This is proved for instance in <cit.>, it is a particular case of
Theorem <ref> in Appendix <ref>.
With more cusps this is may not hold true for all lifts of the holonomy
representation, i.e. for all spin structures.
It is true provided that for each cusp the trace of the peripheral elements is
not identically $+2$
(for some elements it is $-2$). This is always the case if there is a single
cusp <cit.>.
One may as well define the same torsion as in Definition <ref>:
Let $M^3$ be a compact, oriented, hyperbolic 3-manifold with one cusp, and let $\sigma$ denote a spin structure on
$M^3$. The torsion of $(M^3,\sigma)$ is defined as
\begin{equation}
\label{eqn:taucusp}
\tau(M^3,\sigma):=\tor(M^3, \varrho_\sigma )\in \mathbb{C}^*\, ,
\end{equation}
where $\varrho_\sigma=\widetilde{\mathrm{hol}}$ is the lift of the holonomy corresponding to $\sigma$.
This torsion has the same properties as in the closed case, Proposition <ref>.
§.§.§ The twisted polynomial
It is relevant to mention the twisted polynomial for hyperbolic knot exteriors corresponding to a lift of the holonomy $\varrho$
constructed by Dunfield, Friedl, and Jackson in
Given a hyperbolic knot ${\mathcal K}\subset S^3$, choose the spin structure on
$M^3=S^3-{\mathcal K}$ such that the trace of the meridian is $+2$
(the trace of the longitude is always -2 by <cit.>, see also
<cit.>) and consider the abelianization $\phi\colon\pi_1
M^3\to \ZZ$.
Dunfield, Friedl, and Jackson
study the polynomial
\Delta_{\mathcal K}(t):=\Delta_{M^3, \varrho\otimes\phi}(t)
following the notation of Subsection <ref>.
By Proposition <ref> its degree is $\leq 2 (2 g({\mathcal K})+1)$, where $g({\mathcal K})$ is the genus of the knot, i.e. the minimal genus of a Seifert surface.
Numerical evidence (knots up to 15 crossings) yield them to conjecture:
For a hyperbolic knot $\mathcal K$:
* $\deg \Delta_{\mathcal K}(t) = 2 (2 g({\mathcal K})+1)$.
* $\Delta_{\mathcal K}$ is monic if and only if ${\mathcal K}$ is s fibered knot.
The equality of the degree has been proved by Morifuji and Tan for some
families of two bridge knots, see <cit.> and references therein. Agol
and Dunfield have shown:
For libroid hyperbolic knots,
\deg \Delta_{\mathcal K}(t) = 2 (2 g({\mathcal K})+1).
Being libroid means the existence of a collection of disjointly embedded
minimal genus Seifert surfaces in the exterior of the knot
so that their open complement is a union of books of $I$-bundles, in a way
that respects the structure of sutured manifold.
See <cit.>.
In the remarkable paper <cit.> the authors also conjecture that being
monic determines whether the knot is fibered or not, and rise many interesting
questions about this polynomial and its relationship with other invariants.
§.§ Torsion on the variety of characters
Let $M^3$ be a hyperbolic, oriented manifold with one cusp. A relevant
difference with the closed case is the fact that the holonomy of $M^3$ can be
deformed in the variety of representations
(to holonomies of non-complete structures).
§.§.§ The distinguished curve of characters
The variety of $\SL_2(\CC)$-representations of $M^3$ is the set
\hom(\pi_1M^3,\SL_2(\CC)),
which it is an affine algebraic set: if a generating set of $\pi_1M^3$ has $k$
elements, then $\hom(\pi_1M^3,\SL_2(\CC))$ embeds in $\SL_2(\CC)^k\subset
\CC^{4 k}$, by mapping a representation to the image of its generators. The
algebraic equations are induced by the relations of the group.
The group $\PSL_2(\CC)$ acts on $
\hom(\pi_1M^3,\SL_2(\CC))
$ by conjugation, and the affine algebraic quotient is the variety of characters
X(M^3):= X(M^3,\SL_2(\CC))= \hom(\pi_1M^3,\SL_2(\CC))/\! / \PSL_2(\CC).
This is defined in terms of the invariant functions: $X( M^3,\SL_2(\CC))$ is the algebraic affine set
whose function ring $\CC[ X(M^3,\SL_2(\CC)) ]$ is the ring of invariant functions
\CC[ \hom(\pi_1M^3,\SL_2(\CC))]^{\PSL_2(\CC)}.
By <cit.>, see also <cit.> each
component that contains the lift of the holonomy of $M^3$ is a curve.
An irreducible component of $X(M^3,\SL_2(\CC))$ that contains a lift of the
holonomy is called
a distinguished component and it is denoted by $X_0(M^3)$.
For many manifolds, e.g. for 2-bridge knot exteriors, there is a unique
distinguished component. A priori there could be more components,
but the definition makes sense because they would be isomorphic.
More precisely,
there are two characters of the holonomy representation in $\PSL_2(\CC)$ that
are complex conjugate from each other, that correspond to the different
orientations. When lifting them to $\SL_2(\CC)$, this gives
$2 \vert H^1(M^3;\ZZ/2\ZZ) \vert $
characters, two for each spin structure.
The corresponding components $X_0(M^3)$ are isomorphic by means of
the natural action of $H^1(M^3;\ZZ/2\ZZ)$ and complex
Recently, Casella, Luo, and Tillmann <cit.>
have shown an example of hyperbolic manifold with one cusp $M^3$ such that
the holonomy characters of the different orientations lie
in different components of $X(M^3,\PSL_2(\CC))$. To my knowledge, the following
question is still open:
Once $M^3$ is oriented,
are all the lifts of the oriented holonomy contained in a single irreducible
of $X(M^3,\SL_2(\CC))$?
The distinguished component $X(M^3,\SL_2(\CC))$ is a curve and it was studied
by Thurston in his proof of the hyperbolic Dehn filling theorem
<cit.>. More precisely, in a neighborhood of the
holonomy of the complete structure of $M^3$, the representations are holonomies
of incomplete structures, and in some cases the completion is a Dehn
filling. This is discussed in Paragraph <ref>.
§.§.§ Torsion on the distinguished curve of characters
We say that a character $\chi\in X(M^3)$ is trivial if it takes values in $\{\pm 2\}$, i.e. it is a lift or the trivial character in $\PSL_2(\CC)$.
An irreducible character is the character of a unique conjugacy class of representations.
A reducible character can correspond to more conjugacy classes, but if the character is
non-trivial, then either all representations with this character are acyclic, either
none of them is (Lemma <ref>).
Define the torsion function on $X_0(M^3)$ minus the trivial character:
\begin{equation}
\label{eqn:torsionfunct2}
\mathbb{T}_M(\chi_\rho)=\left\{
\begin{array}{ll}
\tor(M^3,\rho) & \textrm{if }\rho\textrm{ is acyclic;} \\
0 & \textrm{if } \chi_\rho \textrm{ is non-trivial and } \rho \textrm{
\end{array}
\right.
\end{equation}
where $\chi_\rho$ denotes the character of $\rho$.
For a hyperbolic oriented manifold with one cusp, the torsion defines
a rational function on $X_0(M^3)$,
$\mathbb{T}_M\in \CC(X_0(M^3))$,
which is regular away from the trivial character.
In particular, if the trivial character is not contained
in $X_0(M^3)$ (e.g. if $b_1(M^3)=1$), then $\mathbb{T}_M\in \CC[X_0(M^3)]$,
i.e. it is holomorphic (with no poles)
\mathbb{T}_M \colon X_0(M^3)\to \CC.
We prove that the trivial representation cannot be approached by irreducible ones when $b_1(M^3)=1$
in Appendix <ref>.
This is always the case when $M^3$ is a knot exterior.
The dimension of each cohomology group is upper semi-continuous on the representation
(see <cit.>,
this is a particular case of the semi-continuity theorem <cit.>).
With Lemma <ref>, we can conclude that
acyclicity holds true in a dense open Zariski domain $U\subset X_0(M^3)$, after removing
the trivial character if required.
The fact that the function $\mathbb{T}_M$ is algebraic on this domain is clear, as this is defined from polynomials
on the entries of $\rho$.
Invariance by conjugation is one of the properties of the torsion.
Thus it remains to deal with the points where it is not acyclic.
Recall that by Lemma <ref> a representation with nontrivial character is acyclic
if and only all representations with the same character are.
First notice that $H^0(M^3;\rho)$ is trivial when the character
$\chi_\rho$ is non-trivial, because this cohomology group
is naturally isomorphic to the space of invariants $\CC^{\rho(\pi_1 M^3)}$. More precisely,
$\CC^{\rho(\pi_1 M^3)}$
is non-trivial only when all elements in $\rho(\pi_1 M^3)$ have $1$ as eigenvalue, which
means that their trace is $2$, i.e. $\chi_\rho$ is trivial.
Thus by duality $H_0(M^3;\rho)= 0$ when $\chi_\rho$ is non-trivial.
Now fix a representation $\rho_1$ which is not acyclic and non-trivial.
Non-acyclicity implies that $H_1(M^3;\rho_1)\neq 0$ and $H_2(M^3;\rho_1)\neq 0$,
as $H_0(M^3;\rho_1) = 0$, the homotopical
of $M^3$ is $2$, and $\chi(M^3)=0$.
Notice that $M^3$ has the simple homotopy type of a 2-complex (see
<cit.> for instance), that can be used to compute the
Using the notation of Section <ref>,
fix a basis $\{v_1,v_2\}$ for $\CC^2$ and lifts of cells $\tilde e^i_j$ of a triangulation of $M^3$.
Then define a family of basis $c_i(\rho)$ by varying the representation in $v_k\otimes_\rho \tilde e^i_j$.
Now choose $\tilde b_1(\rho)=c_2(\rho)$
and $\tilde b_0(\rho)$, a linear combination of $c_1(\rho)$ with constant coefficients (though $c_1(\rho)$ changes
with $\rho$), so that $\partial(\tilde b_0(\rho_1))=c_0(\rho_1)$.
The function
\begin{equation}
\label{eqn:ratiotau2}
\rho\mapsto [\partial \tilde b_1(\rho)\sqcup \tilde b_0(\rho),c_1(\rho) ]/[\partial \tilde b_0( \rho),c_0(\rho)]
\end{equation}
is well defined in the set where its denominator does not vanish. This is a Zariski open set
that contains $\rho_1$.
On this set $\partial \tilde b_1(\rho)=\partial c_2(\rho)$ has maximal rank iff $\rho$ is acyclic, thus the function
(<ref>) vanishes when $\rho$ is not acyclic, and when $\rho$ is acyclic
(<ref>) is the torsion.
For the figure eight knot, in <cit.> Kitano computes it:
\mathbb{T}_M(\rho)=2-2\tr(\rho(m))=2-2\chi_\rho(m)\, ,
where $m$ denotes the meridian of the knot. Notice that here the function only
depends on $\tr(\rho(m))$,
namely the evaluation of the character at $m$, and one does not
need to describe the variety of characters.
In general, as $ \tr(\rho(m))$
is a non-constant function on the curve $X_0(M^3)$, $\mathbb{T}_M(\rho)$ and $ \chi_\rho(m)= \tr(\rho(m)) $ are related by a polynomial equation;
compare with <cit.>.
§.§.§ Dehn filling space
Again let $M^3$ be an oriented, finite volume hyperbolic manifold with one cusp.
Consider the peripheral torus $T^2$, which is the boundary of a compact core of $M^3$.
Choose a frame on the peripheral torus $T^2$, i.e. two simple closed curves
that generate $\pi_1 T^2$, denote this frame by
$\{m,l\}$. The notation suggests that the canonical choice for a knot exterior is the pair meridian-longitude.
A Dehn filling on $M^3$ is the result of gluing a solid torus $D^2\times S^1$ to a compact core $ M^3$ along the boundary.
Up to homeomorphism, this manifold depends only on the (unoriented) homology class in $T^2$ of the meridian,
i.e. the curve $\partial D^2\times \{*\}$. This curve is written as $\pm (p\, m+q\, l)$, and the Dehn filling is
denoted by $M^3_{p/q}$.
When $\vert p\vert +\vert q\vert$ is sufficiently large, by Thurston's
theorem $M_{p/q}$ is hyperbolic. To prove it, he introduces the Thurston's parameters of the Dehn filling space,
by writing, for representations close to the holonomy of the complete hyperbolic structure,
\begin{equation}
\label{eqn:u}
\rho(m)=\pm \begin{pmatrix}
e^{u/2} & 1 \\ 0 & e^{-u/2}
\end{pmatrix}
\qquad
\rho(l) = \pm \begin{pmatrix}
e^{v/2} & f(u) \\ 0 & e^{-v/2}
\end{pmatrix}
\end{equation}
with $u,v\in\CC$ in a neighborhood of the origin.
For the holonomy of the complete structure, $u=v=0$ and write
\cs= \cs(l,m)=f(0)\in \CC-\RR.
The parameter $u\in U \subset \CC$ as above is called the Thurston parameter and $\cs= \cs(l,m)\in \CC-\RR$ the cusp shape
of the complete structure.
The Thurston parameter $u$ is in fact a parameter of a double branched covering of a neighborhood of the lift of the holonomy $\varrho$,
as the local parameter of $X_0(M^3)$ is
$\tr(\rho(m))= \pm 2 \cosh\frac{u}{2} $.
For the complete structure, the peripheral group acts as a lattice on a horosphere, that we view as $\CC$. Up to affine equivalence, this lattice is given by
$m\mapsto 1$, $l\mapsto \cs$.
There exists a (standard) neighborhood of the origin $U\subset \CC$ such that:
* The map $ U\to X_0(M^3)$ such that $u\mapsto \chi_\rho$, where $\rho$ is as in (<ref>) is a double branched covering
of a neighborhood in $X_0(M^3)$ of the character of the holonomy
of the complete structure.
* $v$ is an analytic odd function on $u$ that satisfies
$$v(u)=\cs u+ O(u^3).
* $f(u)=\sinh(v)/\sinh(u)$.
The generalized Dehn filling coefficients are the
$ such that
\begin{equation}
\label{eqn:pq}
p\,u+q\, v=2\pi i
\end{equation}
when $u\neq 0$ and
$\infty$ when $u=0$.
The generalized Dehn filling coefficients define a homeomorphism between $U$ and a neighborhood of $\infty$ in
If a pair of coprime integers $(p,q)\in\ZZ^2$ lies in the image of this homeomorphism,
then $M^3_{p/q}$ is hyperbolic, with holonomy whose restriction to $M^3$ satisfies (<ref>).
We are interested in properties of $M^3_{p/q}$.
$$\vol (M^3_{p/q})=\vol (M^3)-\pi^2\frac{\operatorname{Im}(\cs)}{|p+\cs q|^2}+ O\left(\frac{1}{|p+\cs q|^4 }\right).$$
We are also interested in the complex length of the soul of the solid torus
added by Dehn filling, which is a short geodesic.
Let $r$ and $s$ be integers satisfying $p\, s-q\, r=1$, this complex length is
$r\, u+s\, v$. A straightforward computation yields
The complex length of the core of the solid torus $M_{p/q}^3$ is
r\, u+s\, v= 2\pi i\frac{r}{p}+ \frac{v}{p}= 2\pi i\frac{s}{q}+ \frac{u}{q}
Given a spin structure $\sigma$ on $M^3$, it may extend or not to $M_{p,q}^3$. It is easy to give a characterization
using the bijection of Proposition <ref>.
A spin structure $\sigma$ on $M$ extends to $M_{p,q}$ iff the corresponding lift of the holonomy $\varrho_{\sigma}$
(for the complete structure on $M^3$) satisfies
\tr(\varrho_{\sigma}( p\, m+ q\, l ))=-2.
§.§.§ Torsion for Dehn fillings
Let $\vert p\vert +\vert q\vert$ be sufficiently large and let $\sigma$ be a spin structure on $M^3$ extensible to $M_{p/q}^3$.
Let $\rho_{p/q,\sigma}$ denote the lift of the holonomy of the restriction to
$M^3$ of the hyperbolic structure on $M_{p/q}^3$, and $\chi_{p/q,\sigma}$ the corresponding character.
Let $\lambda_{p/q}$ denote the complex length of the soul of the filling torus.
There is a surgery formula:
Let $M^3$ be as above. For $\vert p\vert +\vert q\vert$ sufficiently large:
\tau(M_{p/q}^3,\sigma)= \frac{\mathbb{T}_{M}(\chi_{p/q,\sigma} )}{ 2
(1-\cosh(\lambda_{p/q}/2 ) ) }.
This proposition can be proved using Mayer-Vietoris (Proposition <ref>)
to the pair
$(M^3, D^2\times S^1 )$. The proof can be found for instance in
for the figure eight knot,
and in <cit.> for twist knots,
but it applies to every cusped manifold, and it is essentially done too in <cit.>. The only computation required is the torsion of the
solid torus added by Dehn filling, or its core geodesic, which is (by Example <ref>):
\frac{1}{
\det \left( \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\begin{pmatrix}
e^{\lambda_{p/q}/2} & 0 \\
0 & e^{-\lambda_{p/q}/2}
\end{pmatrix}
\right) }= \frac1{2 (1-\cosh(\lambda_{p/q}/2 ))}.
The complex length $\lambda_{p/q}$ is defined up to addition of a term in $2\pi i \ZZ$. The spin structure (or equivalently a choice of lift of the holonomy)
determines $\lambda_{p/q}$ up to $4\pi i\ZZ$, as the holonomy of this curve is conjugate to
\begin{pmatrix}
e^{\lambda_{p/q}/2} & 0 \\ 0 & e^{-\lambda_{p/q}/2}
\end{pmatrix}.
There exist closed hyperbolic manifolds $N^3$ with two spin structures $\sigma$ and $\sigma'$ so that
$\tau(N^3,\sigma)\neq \tau(N^3,\sigma')$.
The manifolds are obtained by Dehn filling on the figure eight knot. Consider
the sequence $p_n/ q_n= 2 n$ so that both spin structures
on the figure eight knot exterior $M^3$ extend to $M^3_{p_n/q_n}$, i.e.
$H_1(M^3_{2 n} ; \ZZ/2\ZZ)\cong \ZZ/2\ZZ$.
The corresponding lifts differ on the sign of the trace of the meridian.
As the core of the solid torus is homologous to the meridian, the
sign of the traces of these cores differ.
Hence for one of the lifts the complex length (modulo $4\pi i\ZZ$) is $\lambda_{2n}$, for the other one it is
$\lambda_{2n}+ 2\pi i$
by Remark <ref>. By Remark <ref>, one may approximate
\begin{equation}
\label{eqn:approx}
\lambda_{2n}=\frac{2\pi i}{2n}+ O\left(\frac{1}{n^2}\right).
\end{equation}
The goal is to show that the limit
\begin{equation}
\label{eqn:limit}
\lim_{n\to+\infty} \frac{\vert \tau(M_{2 n},\sigma )\vert }{ \vert \tau(M_{2 n},\sigma' )\vert}
\end{equation}
is not $1$. By
Proposition <ref> it is the product of the limits:
\begin{equation}
\label{eqn:limitlambdas}
\lim_{n\to\infty}\frac{1-\cosh( \lambda_{2n}/2 ) }{ 1-\cosh( (\lambda_{2n} + 2\pi i)/2) }
=\lim_{n\to\infty}\frac{1-\cos\frac{\pi} {2n}}{1+\cos\frac{\pi} {2n}} = 0
\end{equation}
(using the approximation (<ref>)) and
\begin{equation}
\label{eqn:limitfunctions}
\lim_{n\to\infty}
\frac{\mathbb{T}_M(\chi_{2n,\sigma}
)}{\mathbb{T}_M(\chi_{2n,\sigma'} )}
\frac{\tau(M^3,\sigma)}{\tau(M^3,\sigma')}.
\end{equation}
As for one of the spin structures the trace of the meridian is $2$ and for the other $-2$, by Example <ref>,
the limit in (<ref>) is
Thus the limit (<ref>) vanishes, hence it is not $1$.
The previous argument applies to any knot exterior, the precise value of the torsion is not needed.
Corollary <ref> holds true also for cusped manifolds,
by applying an analogous formula for partial surgery.
The set of modules of the torsions $\left\vert \tau(M^3_{p/q},\sigma)\right\vert $ obtained by Dehn filling on $M^3$,
so that the spin structure $\sigma$ on $M^3$ extends to $M^3_{p/q}$, is dense in
\left[\frac14{\vert \tau(M^3,\sigma)\vert},+\infty\right).
By the surgery formula, Proposition <ref>, it suffices to show that
\left\vert 1- \cosh(
is dense in
the interval $[0,2]$.
Let $r, s\in\ZZ$ be such that $p s-q r = 1$, by Remark <ref>, $\lambda_{p/q}= 2\pi i \frac{r}p+\frac up$ and $u\to 0$.
Then the result follows from the density of $\cos(\pi \frac{r}{p})$ in $[-1,1]$.
Notice that extendibility of $\sigma$ is just determined by a condition
$a\, p + b\, q\equiv 0 \mod 2$,
for $a, b\in \ZZ/2\ZZ$ depending on $\sigma$.
As the surgery formula shows, this density is a consequence of the contribution of the core geodesics.
Notice that in Theorem <ref> the hypothesis require the length of geodesics to be bounded below away from zero.
§.§.§ Branched coverings on the figure eight
We next consider another family of examples. We consider it only for the figure eight knot, but it generalizes to other manifolds.
Consider $M_ n^3$ the n-th cyclic branched covering of $S^3$, branched along the figure eight knot. It is hyperbolic for $n\geq 4$.
Its torsion may be computed by means of the twisted Alexander polynomial, by using the formula à la Fox,
Proposition <ref>,
due to Turaev
see also <cit.>.
More precisely, this polynomial is
p_{\chi}( t )=t^2-2\chi(m) t + 1,
by (<ref>),
cf. <cit.>.
Notice that its value at $1$ agrees with Example <ref>.
The holonomy of $M_ n^3$ extends to a holonomy of the quotient orbifold $M_
(\ZZ/n\ZZ)$ in $\PSL_2(\CC)$.
It induces a representation of the exterior of the
not $S^3-\mathcal K$ that lifts to $\SL_2(\CC)$, that we denote by $\rho_n$.
Since a rotation of angle $2\pi /n$ has trace $\pm 2\cos ( \pi/n)$, we have
\chi_{\rho_n}(m)= \tr(\rho_n(m))= \pm 2 \cos ( \pi/n).
Using Proposition <ref>, if $\Sigma_n\subset M_n$ is the lift of the branching locus, then
\[
\tor(M_n^3-\Sigma_n,\sigma)=\prod_{k=0}^{n-1} p_{\chi_n}( e^{2\pi i\frac{k}{n}}) =
\prod_{k=0}^{n-1} \left( e^{4\pi i\frac{k}{n}}\mp 4\cos({\pi}/{n} )
e^{2\pi i\frac{k}{n}}+1\right).
\]
By the surgery formula (Proposition <ref>) and knowing that the complex length of the core geodesic is
$ \approx \frac{\sqrt{3}\,\pi}n + O\left(\frac{1}{n^3}\right) $ (Remark <ref>),
\begin{equation}
\label{eq:foxsurgery}
\tor(M_n^3,\sigma)= \frac{1/2}{1-\cosh \left( \frac{\sqrt{3}\,\pi}n +
O\left(\frac{1}{n^3}\right) \right)}
\prod_{k=0}^{n-1} \left( e^{4\pi i\frac{k}{n}}\mp 4\cos( {\pi}/{n} )
e^{2\pi i\frac{k}{n}}+1\right).
\end{equation}
In particular,
\begin{equation}
\label{eqn:limlog/n}
\lim_{n\to\infty} \frac{\log\vert \tor(M_n^3,\sigma)\vert}{n}=
\lim_{n\to\infty} \frac{1}{n} \sum_{k=0}^{n-1} \log \left\vert e^{4\pi
i\frac{k}{n}}\pm 4\cos( {\pi}/{n} ) e^{2\pi i\frac{k}{n}}+1 \right\vert.
\end{equation}
Here we used that $\lim_{x\to 0^+} x\log x =0$ to get rid of the first term, outside the product, in (<ref>).
As $t^2 \mp 4 t + 1$ has no roots in the unit circle, the limit
(<ref>) converges to
\frac{1}{2\pi}\int_{\vert z\vert =1} \log\vert z^2\pm 4 z +1 \vert .
Thus by Jensen's formula
\lim_{n\to\infty} \frac{\log\vert \tor(M_n^3,\sigma)\vert}{n}
= \log\vert 2+\sqrt{3}\vert .
This approach uses the ideas on Mahler measure of <cit.>. It applies to a hyperbolic knot provided that the torsion polynomial has no roots in the unit circle.
It is easy to prove that the roots of the polynomial are not roots of the unity, but this does not discard roots in the unit circle. For
the adjoint representation, the nonexistence of roots in the unit circle has been proved by Kapovich <cit.>.
§ TORSIONS FOR HIGHER DIMENSIONAL REPRESENTATIONS OF HYPERBOLIC THREE-MANIFOLDS
This section is devoted to further Reidemeister torsions naturally associated to hyperbolic three-manifolds,
those obtained by using the (finite dimensional and linear) representations of $\SL_2(\CC)$.
§.§ Representations of $\SL_2(\CC)$
Let $V_{k+1}$ be the space of degree $k$ homogeneous polynomials in two variables:
V_{k+1}:=\{ P\in \mathbb{C}[X,Y]\mid P \textrm{ is homogeneous of degree } k\}.
The group
$\SL_2(\CC)$ acts naturally on $V_{k+1}$ by precomposition on polynomials:
\begin{array}{ccl}
\SL_2(\CC)\times V_{k+1} & \to & V_{k+1} \\
(A, P) & \mapsto & P\circ A^t
\end{array}
We consider the transpose matrix $A^t$ so that the action is on the left.
We could also have considered the inverse instead of the transpose, as there exist a matrix $B\in \SL_2(\CC)$
such that $A^{-1}=B A^t B^{-1}$ for every $A\in \SL_2(\CC)$.
This defines a representation
\operatorname{Sym}^{k}\colon \SL_2(\CC) \to \SL_{k+1}(\CC).
Not only $\operatorname{Sym}^{k}$ has determinant $1$, but preserves a non-degenerate bilinear form that is
symmetric for $k$ even, and skew-symmetric for $k$ odd.
\operatorname{Sym}^{k}\colon \SL_2(\CC) \to
\left\{
\begin{array}{ll}
\operatorname{SO} (2l+1,\mathbb{C}) & \textrm{for } k=2l \\
\operatorname{Sp} (2l+2,\mathbb{C}) & \textrm{for } k=2l +1 \\
\end{array}
\right.
For $k=1$, $\operatorname{Sym}$ is the standard representation and the bilinear form is the determinant
of the $2\times 2$ matrix obtained from two vectors. Then
the form invariant for $\operatorname{Sym}^k$ is the $k$-th symmetric product
of this determinant.
We shall also consider the complex conjugate $ \overline{ \operatorname{Sym}^{k}}$, which is antiholomorphic.
Classification of irreducible representations of $\SL_2(\CC)$, cf. <cit.>.
* Any irreducible holomorphic representation of $\SL_2(\CC)$ is equivalent to $\operatorname{Sym}^{k}$ for a unique $k\geq 0$.
* Any irreducible representation of $\SL_2(\CC)$ is equivalent to
$$\operatorname{Sym}^{k_1,k_2}:= \operatorname{Sym}^{k_1}\otimes \overline{ \operatorname{Sym}^{k_2} } $$
for a unique pair of integers $k_1,k_2\geq 0$.
Notice that $\operatorname{Sym}^{k}=\operatorname{Sym}^{k,0}$ and that $\operatorname{Sym}^{0}=\operatorname{Sym}^{0,0}$ is the trivial representation in $\CC$.
The adjoint representation $\operatorname{Ad}\colon\SL_2(\CC)\to
\operatorname{SO}(\mathfrak{sl}_2(\CC))$ is equivalent to
the invariant form being the Killing form.
The representation $\operatorname{Sym}^{k,k}$ takes values in $\SL_{(k+1)^2}(\RR)$. More precisely, the image
of $\operatorname{Sym}^{k,k}$ is contained in $\operatorname{SO}(p,q)$, with
p=\frac{k^2+3k+2}{2}\quad\textrm{ and }\quad q= \frac{k^2+k}{2}.
For instance, $\operatorname{Sym}^{1,1}\colon \PSL_2(\CC) \to
\operatorname{SO}_0(3,1)$ is the canonical isomorphism between
different presentations of the group of orientation preserving isometries of hyperbolic 3-space.
This representation is used to study infinitesimal deformations of conformally flat structures on a hyperbolic 3-manifold.
The representation $\operatorname{Sym}^{2,2}\colon\PSL_2(\CC) \to
\operatorname{SO}_0(6,3)$ is used in the study of
infinitesimal deformations of projective structures on a hyperbolic 3-manifold.
Let $\varrho_\sigma\colon\pi_1 M^3\to \SL_2(\CC)$ denote a lift corresponding to a spin structure $\sigma$, we shall denote its composition with
$ \operatorname{Sym}^{k_1,k_2}$ by
\begin{equation}
\label{eqn:varrho}
\varrho_\sigma^{k_1,k_2}:= \operatorname{Sym}^{k_1,k_2}\circ \varrho_\sigma,
\end{equation}
and $ \varrho_\sigma^{k}= \varrho_\sigma^{k,0}$.
Let $M^3$ be a compact hyperbolic orientable 3-manifold and
$\widetilde{\mathrm{hol}}$ a lift of its holonomy. If $k_1\neq k_2$ then
H^*(M^3 ; \varrho_\sigma^{k_1,k_2} )=0\, .
This theorem, due to Raghunathan, is commented in Appendix <ref>, with other facts about cohomology.
§.§ Higher torsions for closed manifolds
From Theorem <ref>, we can define torsions.
Let $M^3$ be a closed oriented hyperbolic three-manifold, with spin structure $\sigma$,
and let $\varrho_\sigma^{k_1,k_2}= \operatorname{Sym}^{k_1,k_2}\circ \varrho_\sigma$ be as in (<ref>).
For $k_1\neq k_2$, define
\begin{equation}
\label{eqn:taukk}
\tau^{k_1,k_2}(M^3,\sigma):=\tor(M^3, \varrho_\sigma^{k_1,k_2} ) \in
\mathbb{C}^*/{\pm 1}\, ,
\end{equation}
and for $k\geq 1$
\begin{equation}
\label{eqn:tauk}
\tau^{k}(M^3,\sigma):= \tau^{k,0}(M^3,\sigma)= \tor(M^3, \varrho_\sigma^{k}) .
\end{equation}
When $k=1$ one obtains the torsion of Definition <ref>:
\tau^1(M^3,\sigma)=\tau^{1,0}(M^3,\sigma)=\tau(M^3,\sigma).
* For $k_1$ or $k_2$ odd, there is no sign indeterminacy as it is a
representation in $\mathbb{C}^{(k_1+1)(k_2+1)}$, which is even dimensional
(Remark <ref>).
* For $k_1+k_2$ even, $\operatorname{Sym}^{k_1}\otimes \overline{\operatorname{Sym}^{k_2}} $ factors through $\operatorname{PSL}_2(\mathbb{C})$,
hence it does not depend on the spin structure. In addition, the
sign indeterminacy can be avoided by using Turaev's refined torsion and an orientation in homology, provided by Poincaré duality <cit.>.
* By construction,
* When $k_1=k_2$, $\varrho^{k,k}$ may be not acyclic (for instance
when it contains a totally geodesic surface <cit.>), but sometimes it can be acyclic (e.g. almost all
Dehn fillings on two bridge knots for $k=1$ by <cit.> or on the figure eight knot for $k=2$ by
When $\varrho^{k,k}$ is acyclic, then $\tau^{k,k}(M^3,\sigma)$ is also well defined.
* Changing the orientation. Let $M^3$ and $\overline M^3$ denote the same manifold with opposite orientations and let $\sigma$
and $\overline\sigma$ denote the corresponding spin structures as in Proposition <ref>. Then
\tau^{k_1,k_2}(\overline M^3, \overline \sigma)= (-1)^{b_1(M^3) (k_1+k_2)}\, \overline{ \tau^{k_1,k_2}(M^3,\sigma)}.
where $b_1(M^3)$ denotes the first Betti number.
Hence if $M^3$ is amphicherical, then $\tau^{k_1,k_2}(M^3,\sigma)$ is real for
$b_1(M^3) (k_1+k_2)$ even and purely imaginary for $b_1(M^3) (k_1+k_2)$ odd.
In <cit.> W. Müller found a beautiful theorem about the asymptotic behavior when $k\to\infty$:
Let $M$ be a compact, oriented, hyperbolic 3 manifold with a spin structure $\sigma$. Then
\begin{equation}
\label{eqn:muller}
\lim_{k\to\infty} \frac{ \log\vert \tau^{k}(M^3,\sigma)\vert}{k^2} = \frac
1{4\pi}\operatorname{vol} (M^3)\, .
\end{equation}
In particular, the volume can be computed from the sequence of torsions.
Again, with our convention our torsion is the inverse of the usual analytic torsion, so
Equation (<ref>) has opposite sign to <cit.>.
As noticed in <cit.>, this theorem implies that there are finitely many manifolds with the same sequence of torsions.
Is there a finite collection of integers $0<k_1<\cdots<k_n$ such that there are finitely many manifolds
with the same $k_1,\ldots, k_n$-torsions?
The answer is yes under the extra hypothesis that the volume is bounded above.
More precisely, an infinite sequence of manifolds
with bounded volume accumulates (for the geometric topology) to a finite set of cusped manifolds with finite volume,
this is Jørgensen-Thurston properness of the volume function <cit.>.
For such a sequence of manifolds $M_n^3$, $\tau^2(M_n^3,\sigma)\to \infty$, by Corollary <ref>, see also <cit.>.
A priory, for some $N\in \NN$ there could exists an infinite sequence of hyperbolic manifolds whose volume goes to infinity and
with the same $k$-torsions for $k\leq N$. By Theorem <ref> the injectivity radius of this sequence would not converge to infinity.
Müller proves Theorem <ref> using analytic torsion and Ruelle zeta functions. There is first a theorem of Wotzke <cit.> that generalizes Fried's theorem
(Theorem <ref>) and from this he obtains a functional equation that relates the volume, the Ruelle zeta functions and the torsions.
Theorem <ref> has been generalized by Müller and Pfaff <cit.> by working directly on the trace of the heat kernel in (<ref>).
Let $M^3$ be a compact, oriented, hyperbolic 3-manifold with a spin structure $\sigma$. Then
\begin{equation}
\lim_{k_1\to\infty} \frac{ \log\vert \tau^{k_1,k_2}(M^3,\sigma)\vert}{k_1^2} =
\frac 1{4\pi}(k_2+1)\operatorname{vol} (M^3)\, .
\end{equation}
Again, the sign convention here differs from <cit.>.
§.§ Functions on the variety of characters
In this subsection $M^3$ denotes an orientable hyperbolic 3-manifold of finite volume, with a single cusp.
The subsection discusses the torsion as function on the variety of characters.
§.§.§ Generic cohomology on the distinguished component
Some cohomology preliminaries are required before defining the torsion
of $\operatorname{Sym}^{k_1,k_2}$ as function on $X_0(M^3)$.
To simplify, use the notation
\rho^{k_1,k_2}:=\operatorname{Sym}^{k_1,k_2}\circ\rho
for any representation $\rho\colon\pi_1M^3\to \SL_2(\CC)$.
A character $\chi\in X_0(M^3)$ is called $(k_1,k_2)$-exceptional if
there exists a representation $\rho\in\hom(\pi_1M^3,\SL_2(\CC))$ with character
$\chi_\rho=\chi$ such that $H^0(M^3;\rho^{k_1,k_2})\neq 0$.
The set of $(k_1,k_2)$-exceptional characters is denoted by $\EE^{k_1,k_2}$.
If a character is irreducible, then the set of representations with this character is
precisely a conjugation orbit, but for reducible characters there may be representations
with the same characters that are not conjugate.
Notice that in Proposition <ref>
we have shown that $\EE^{1,0}$ consist of the trivial character, when
it belongs to $X_0(M^3)$ (hence $\EE^{1,0}= \emptyset$ when $b_1(M^3)=1$).
If $k_1\neq k_2$, then a $(k_1,k_2)$-exceptional character is reducible. In particular
the $(k_1,k_2)$-exceptional set $\EE^{k_1,k_2}$ is a finite subset of $X_0(M^3)$.
The proof of this lemma is provided in Appendix <ref> (Lemma <ref>).
For further results, the following definition is convenient.
A subset of $X_0(M^3)$ is generic if it is non empty and Zariski open.
Here the ground field is assumed to be $\CC$ when working with an
holomorphic representation $\operatorname{Sym}^{k}$, and $\RR$ for
$\operatorname{Sym}^{k_1,k_2}$ when $k_2\neq 0$.
For the generic behavior of other cohomology groups, one must distinguish the case where $k_1$ or $k_2$ is odd from the case
they are both even. All proofs are postponed to Appendix <ref>.
Let $M^3$ be orientable, hyperbolic and with one cusp. Assume $k_1\neq k_2$.
* If $k_1$ or $k_2$ is odd, then
\left\{ \chi_\rho\in X_0(M^3) \mid \rho^{k_1,k_2}\textrm{ is acyclic}\right\}
is generic.
* If both $k_1$ and $k_2$ are even, then
\begin{equation*}
\left\{ \chi_\rho\in X_0(M^3) \mid \dim H_i(M^3 ; \rho^{k_1,k_2})=1 \textrm{
for } i=1, 2
\textrm{ and 0 otherwise} \right\}
\end{equation*}
is generic.
When $k_1$ and $k_2$ are even, we need to discuss the choice of basis in homology and perhaps we
still need to remove a Zarsiki closed subset.
We shall consider, for $k_1$ and $k_2$ even,
\begin{equation}
\label{eqn:FF}
\FF^{k_1,k_2}= \left\{ \chi_\rho\in X_0(M^3) \mid \dim H^0(T^2 ;
\rho^{k_1,k_2})\neq 1\right\}\cup \EE^{k_1,k_2} \, ,
\end{equation}
which is Zarsiki closed subset of $X_0(M^3)$ (over $\CC$ for $k_2=0$, over $\RR$ otherwise).
In particular it is finite when $k_2=0$ or $k_1=0$.
Let $T^2$ denote the peripheral torus. If both $k_1$ and $k_2$ are even, then
$ \FF^{k_1,k_2}$ is a Zarsiki closed subset of $X_0(M^3)$ (over $\CC$ for $k_2=0$ or $k_1=0$,
over $\RR$ otherwise).
In particular it is finite when $k_2=0$ or $k_1=0$.
Consider the cap product
\cap\colon H^0(T^2; \rho^{k_1,k_2})\times H_i(T^2; \CC)\to H_i(T^2;
\rho^{k_1,k_2}).
It can be described as follows. Choose $a\in H^0(T^2; \rho^{k_1,k_2})$, $a\neq
0$, and view it as an element of $\CC^{(n_1+1)(n_2+1)}$ invariant by
the action of $\rho^{k_1,k_2} (\pi_1 T^2)$. Any element in $H_i(T^2; \CC)$ is represented by a simplicial cycle: $z\in C_*(K; \CC)$ with $\partial z=0$
(here $K$ is a triangulation of $T^2$). Choose a lift of $z$ to the universal
covering, $\tilde z\in C_*(\widetilde K; \CC)$, then
$a\otimes \tilde z\in C_i(K;\rho^{k_1,k_2})$ is a cocycle and
\begin{equation}
\label{eqn:cap}
a\cap [z]= [a\otimes \tilde z] ,
\end{equation}
where the brackets denote the class in homology.
In the next proposition $[T^2]\in H_2(T^2; \ZZ)$ denotes a fundamental class, $\langle \cdot \rangle$ the linear span,
and $i\colon T^2\to M^3$ the inclusion.
Let $k_1\neq k_2$ be even integers. For $0\neq a\in H^0(T^2; \rho^{k_1,k_2})$
and $0\neq \gamma\in H_1(T^2; \CC)$, the set
\begin{multline*}
\left\{ \chi_\rho\in X_0(M^3) \mid \langle i_*(a\cap \gamma ) \rangle =
H_1(M^3; \rho^{k_1,k_2})\textrm{ and } \right.
\\
\left. \langle i_*(a\cap [T^2]) \rangle = H_2(M^3; \rho^{k_1,k_2})
\right\}
\end{multline*}
is generic.
§.§.§ Higher torsions on the variety of characters
Again assume that $M^3$ is an oriented hyperbolic 3-manifold of finite volume with a single cusp.
We define the Reidemeister torsion on the distinguished curve of characters
For $k_1$ or $k_2$ odd, with $k_1\neq k_2$, and for $\chi_\rho\in X_0(M^3)- \EE^{k_1,k_2}$,
where $\EE^{k_1,k_2}$ is as in Definition <ref>,
\TT_M^{k_1,k_2}(\chi_\rho)=
\begin{cases}
\tor(M^3, \rho^{k_1,k_2}) & \textrm{if } \rho^{k_1,k_2} \textrm{ is acyclic;} \\
0 & \textrm{otherwise.}
\end{cases}
For $k_1$ and $ k_2$ even, with $k_1\neq k_2$, $\gamma$ a peripheral curve, and $\chi_\rho\in X_0(M^3)- \FF^{k_1,k_2}$
(where $ \FF^{k_1,k_2}$ is as in (<ref>)), define
\TT_{M,\gamma}^{k_1,k_2}(\chi_\rho)=
\begin{cases} \tor(M^3, \rho^{k_1,k_2}, \{a\cap \gamma, a\cap [T^2]\}) & \textrm{if } \langle a\cap \gamma \rangle= H_1(M^3; \rho^{k_1,k_2}), \\
& \quad \langle a\cap [T^2] \rangle= H_2(M^3;
\rho^{k_1,k_2}); \\
0 & \textrm{otherwise.}
\end{cases}
We also use the notation
\TT_M^{k}= \TT_M^{k,0} \qquad\textrm{ and }\qquad \TT^{k}_{M, \gamma}=
\TT_{M,\gamma}^{k,0}
for $k$ odd or even respectively.
The proof of Proposition <ref> with minor changes yields the following:
For $k_1\neq k_2$, $\TT_M^{k_1,k_2}$ and $ \TT_{M,\gamma}^{k_1,k_2}$ are
nonzero (real) analytic functions on $X_0(M^3)- \EE^{k_1,k_2} $ for $k_1$ or
$k_1$ odd,
or $X_0(M^3)- \FF^{k_1,k_2} $ for $k_1$ and $k_2$ even,
holomorphic when $k_2=0$ and antiholomorphic when $k_1=0$.
When $H^1(M^3;\ZZ)\cong \ZZ$ there is a well defined Alexander polynomial $\Delta(t)$, and we can give conditions for the regularity of
$\TT_M^{k}$ and $\TT_M^{k, \gamma}$ on the whole curve $X_0(M^3)$.
Assume that $H^1(M^3;\ZZ)\cong \ZZ$ and that no root of $ \Delta(t)$ is a root of unity. Then for $k$ odd $\EE^{k,0}=\emptyset$.
Hence for $k$ odd $ \TT_{M}^{k}$ is a holomorphic function
on $X_0(M^3)$.
The proof of Proposition <ref> applies here with a minor modification. Notice that the condition on the homology implies that the trivial character
does not lie in $X_0(M^3)$. On the other hand, the reducible characters in $X_0(M^3)$ are the characters of the composition of a surjection
$$ \pi_1 M^3\to \ZZ $$
with a representation that maps the generator of the cyclic group to
A= \begin{pmatrix}
\lambda & 0 \\
0 & \lambda^{-1}
\end{pmatrix}
with $\Delta(\lambda^2)=0$ <cit.>.
Since $\lambda$ is not a root of unity and $k$ is odd, $\operatorname{Sym}^k(A)$ has no eigenvalue equal to one, hence $H^0(M^3;\rho^k)=0$ for every
$\rho$ with character in $ X_0(M^3)$.
Assume that $M^3=S^3-\mathcal{K}$ is a knot exterior in the sphere. Assume also
that no root of $ \Delta(t)$ is a root of unity and all roots have multiplicity one.
Then for $k$ even, $\TT^k_{M, \gamma}$ can be extended to the whole curve $X_0(M^3)$.
There are two issues to discus: the dimension of $H^0(T^2;\rho)$ and the
vanishing or not of $H^0(M^3;\rho)$.
For the first one, the assumption $M^3=S^3-\mathcal{K}$ implies that $\rho$ is
never trivial on the peripheral torus $T^2$.
It my happen that $\rho(\pi_1T^2)$ is non-trivial but
has an invariant subspace of dimension $\geq 2$. However, this issue only
occurs when $\rho(\pi_1T^2)$ is contained in a 1-parameter subgroup $G$ of
$\SL_2(\CC)$ conjugate to the group of diagonal matrices. Then
$\operatorname{Sym}^{k}(G)$ has only one invariant subspace, that varies
analytically on $\rho$.
Thus the element $a$ in the expression of the cup products can be chosen to
depend analytically on $\rho$.
Next discuss $H^0(M^3;\rho)$.
The argument in Proposition <ref> fails in the case $k$ even because $\operatorname{Sym}^k$
has always an invariant subspace, thus for those reducible representations $H^0(M^3;\rho^k)\neq 0$ and further
discussion on reducible representations is required.
Namely, at a reducible representation,
for the corresponding abelian representation $\rho$ as in the proof of Proposition <ref>, the hypothesis that $\lambda$ is
not a root of unity implies that $\operatorname{Sym}^k(A)$ has a single eigenvalue equal to one, hence
$\dim H^0(M^3;\rho^k)=1$.
The results of <cit.>, and the hypothesis that the root is simple, yield that there are,
up to conjugation, two representations $\rho'$ and $\rho''$ that are non abelian and have the same character as $\rho$.
In particular
$H^0(M^3;(\rho')^{k_1,k_2})\cong H^0(M^3;(\rho'')^{k_1,k_2})=0$. Now, using the arguments of Proposition <ref>, we may define the torsion in a Zariski dense subset
of the component of $\hom(\pi_1M^3, \SL_2(\CC))$ that corresponds to $X_0(M^3)$. It suffices to consider an open neighborhood of $\rho'$ for the usual topology that projects to a neighborhood of
$\chi_\rho$ <cit.>, which proves that the torsion on a punctured neighborhood of $\chi_\rho$ extends to $\chi_\rho$.
The assertion for the case $k=2$ has been proved by Yamaguchi in <cit.>. More precisely he computes the precise value of $\TT^{2}_{M,\gamma}$ at the reducible character.
Some torsion functions are computed here for the figure eight knot exterior.
Let $\theta=\theta_m$ denote the evaluation of a character of a meridian:
$ \theta_m(\chi)=\chi(m)$ (i.e. the trace of a meridian). As mentioned
Example <ref>,
in <cit.> Kitano computes:
\mathbb{T}_M=\mathbb{T}_M^{1} =2-2 {\theta}.
Further computations with the help of symbolic software yield:
\begin{eqnarray*}
\mathbb{T}_M^{3} &= &- \left( {\theta}^{2}-2\,{\theta}-2 \right) ^{2}\\
\mathbb{T}_M^{5} &=&
2\, \left( {\theta}-1 \right) \left( {\theta}^{8}+2\,{\theta}^{7}-13\,{\theta}^{6}-20\,{\theta}^
{5}+49\,{\theta}^{4}+48\,{\theta}^{3}-33\,{\theta}^{2}-18\,{\theta}-18 \right) \\
\mathbb{T}_M^{7} &=&
- \left( {\theta}-1 \right) ^{2} \left( 2\,{\theta}^{7}-4\,{\theta}^{6}-21\,{\theta}^{5}+19
\,{\theta}^{4}+57\,{\theta}^{3}+13\,{\theta}^{2}-18\,{\theta}-6 \right) ^{2}\\
\mathbb{T}_M^{9} &=&
2\, \left( {\theta}-1 \right) \left( {\theta}^{12}+2\,{\theta}^{11}-13\,{\theta}^{10}-13\,{
{\theta} }^{9}+27\,{\theta}^{8}-{\theta}^{7}+95\,{\theta}^{6} +90\,{\theta}^{5}
\right.\\ & & \left.
}^{3}+61\,{\theta}^{2}+12\,{\theta}-6 \right) ^{2}.
\end{eqnarray*}
Remark that these torsions are functions on the variable $\theta=\theta_m$, the trace of the meridian. This is not always the case,
for instance for $\TT_{M,\gamma}^2$, computed in <cit.> (and in Section <ref>) and described below.
We also computed:
\begin{multline*}
\mathbb{T}_M^{2,1}=
13+3\,{\theta}^{4}-3\, \overline{\theta} ^{3}-14\,{\theta}^{2}-3\,
\overline{\theta} ^{2}+13\,\overline{\theta}+{\theta}^{4}\overline{\theta}
+4\,{\theta}^{2} \overline{\theta} ^{3}+2\,{\theta}^{2}
\overline{\theta} ^{2} \\ -16\,{\theta}^{2}\overline{\theta}-{\theta}^{4}
\overline{\theta} ^{3}+{\theta}^{4} \overline{\theta} ^{2}
\eta\overline{\eta} \left( {\theta}^{2}\overline{\theta}-{\theta}^{2}-\overline{\theta}-1
\right)
\end{multline*}
where $\eta$ is a variable that satisfies
$\eta^2= ({\theta}^2-1)({\theta}^2-5)$.
This variable can be written in terms of the traces of other elements, see Equation (<ref>).
In subsection <ref> other torsions are computed.
If $m$ and $l$ denote respectively the meridian and the longitude, by (<ref>), (<ref>), (<ref>):
\TT_{M,l}^2= 5-2\theta^2, \qquad \TT_{M,m}^2= \frac12 \eta, \quad \textrm{ and } \quad \TT_{M,l}^4=8 (2-\theta^2).
§.§ Evaluation at the holonomy
§.§.§ Cohomolgy at the character of the holonomy
In order to evaluate the torsion functions at the lift of the holonomy,
some results on the cohomology for this representation are required.
Again the proofs are postponed to Appendix <ref>.
Let $\varrho\colon\pi_1 M^3\to\SL_2(\CC)$ be a lift of the holonomy.
* If $k_1+k_2$ is odd, then $\varrho^{k_1,k_2}$ is acyclic.
* If $k_1+k_2$ is even, $k_1\neq k_2$, then
\dim H_i(M^3; \varrho^{k_1,k_2})= \begin{cases}
1 & \textrm{if } i=1,2\, ,\\
0 & \textrm{otherwise.}
\end{cases}
Notice that when both $k_1$ and $k_2$ are odd, the homology of $\varrho^{k_1,k_2}$ differs from the generic homology of $ \operatorname{Sym}^{k_1,k_2}$ on $X_0(M^3)$, there is a discontinuity in the dimension
of the cohomology, which is generically trivial.
We also need to discuss the basis. Let $i\colon T^2\to M^3$ denote the
Let $\varrho\colon\pi_1 M^3\to\SL_2(\CC)$ be a lift of the holonomy. Assume that $k_1+k_2$ is even.
* $\dim H^0(T^2;\varrho)=1$.
* For $0\neq a\in H^0(T^2;\varrho)$, $\langle i_*(a\cap [T^2]) \rangle = H_2(M^3; \varrho^{k_1,k_2})$.
* For $0\neq a\in H^0(T^2;\varrho)$ and $0\neq\gamma\in H_1(T_2;\ZZ)$:
* $\langle i_*(a\cap \gamma) \rangle = H_1(M^3; \varrho^{k_1,k_2})$ if $k_1=0$ or $k_2=0$;
in addition
\begin{equation}
\label{eqn:cuspsh}
i_*(a\cap \gamma_1)=\cs(\gamma_1,\gamma_2) \, i_*(a\cap \gamma_2)
\end{equation}
where $\cs(\gamma_1,\gamma_2) $ is the cusp shape (Definition <ref>).
* $ i_*(a\cap \gamma ) = 0$ if $k_1\neq 0$ and $k_2\neq 0$.
§.§.§ Torsion at the characters of the holonomy
Let $\chi_\varrho$ be the character of a lift $\varrho$ of the holonomy representation
* When $k_1+ k_2$ is odd, $\TT_M^{k_1,k_2}(\chi_\varrho)\neq 0$.
In particular, when $k$ is odd, $\TT_M^{k}(\chi_\varrho)\neq 0$.
* When $k_1\neq k_2$ are both odd, $\TT_{M}^{k_1,k_2}(\chi_\varrho) = 0$.
* When $k_1\neq k_2$ are both even and $k_1,k_2\neq 0$, for $0\neq\gamma\in H_1(T_2;\ZZ)$, we have $\TT_{M,\gamma}^{k_1,k_2}(\chi_\varrho) =0$.
* For $k$ even and $0\neq\gamma\in H_1(T_2;\ZZ)$, $\TT_{M,\gamma}^{k}(\chi_\varrho) \neq 0$. In addition
\frac{\TT_{M,\gamma_2}^{k}(\chi_\varrho)}{\TT_{M,\gamma_1}^{k}(\chi_\varrho)}= \cs(\gamma_2,\gamma_1),
where $0\neq\gamma_1,\gamma_2\in H_1(T_2;\ZZ)$.
For item (<ref>), Theorem <ref> guarantees that the representation is acyclic and therefore $\TT_M^{k_1,k_2}(\chi_\varrho)\neq 0$.
For (<ref>), the same theorem tells that the cohomology of $\varrho^{k_1,k_2}$ is nonzero in dimension 1 and 2. Since it is generically zero, what we have is
a discontinuity in the dimension of the cohomology groups. On the other hand, the zeroth homology group of $\varrho^{k_1,k_2}$ vanishes, thus
$\TT_{M}^{k_1,k_2}(\chi_\varrho) = 0$. The proof of (<ref>) is analogous, using the vanishing of the cup product in dimension one (not two)
of the cap product in Proposition <ref>. The non-vanishing of the cup product of Proposition <ref> for $k_1$ even and $k_2=0$
yields the first part of item (<ref>), the second part follows from Equation (<ref>).
For any peripheral curve $\gamma$,
\begin{equation}
\label{eqn:MFP}
\lim_{k\to\infty} \frac{ \log\left\vert \TT_{M,\gamma}^{2k}(\chi_\varrho) \right\vert }{(2 k)^2} =
\lim_{k\to\infty} \frac{ \log\left\vert \TT_{M}^{2k+1}(\chi_\varrho) \right\vert }{(2 k+1)^2} =
\frac 1{4\pi}\operatorname{vol} (M^3)\, .
\end{equation}
Again, the sign convention is the opposite to <cit.>.
This theorem is proved from Müller's Theorem <ref> in the closed case, by using the approximation of the cusped manifolds by closed manifolds obtained by Dehn filling.
Since Müller's proof uses Ruelle zeta functions, the key point is to understand geodesics (of bounded lengths) of the closed manifolds that approximate a cusped one.
§.§.§ Dehn filling
The following is a generalization to other representations of the formula for Dehn filling in Proposition <ref>.
In particular the same context and notation is used.
Let $M^3$ be as above. For $\vert p\vert +\vert q\vert$ sufficiently large:
* For $k_1$ or $k_2$ odd,
\tau^{k_1,k_2}(M_{p/q}^3,\sigma)=
{\mathbb{T}_M^{k_1,k_2}(\chi_{p/q,\sigma} )}{
\prod_{i_1=0}^{k_1}\prod_{i_2=0}^{k_2}
\frac 1{ 1-(\lambda_{p/q})^{2 i_1-k_1} (\overline {\lambda}_{p/q})^{2 i_2-k_2}
} } \, .
* For $k_1$ and $k_2$ even, $k_1\neq k_2$,
\tau^{k_1,k_2}(M_{p/q}^3)=
{\mathbb{T}_{M,pm+ql}^{k_1,k_2}(\chi_{p,q} )}{
\prod_{(i_1,i_2)\in \mathcal I}
\frac 1{ 1-(\lambda_{p/q})^{2 i_1-k_1} (\overline {\lambda}_{p/q})^{2 i_2-k_2}
} } \, ,
where $(i_1,i_2)\in \mathcal I$ if $ 0\leq i_1 \leq {k_1}$, $0\leq i_2\leq {k_2}$, and $i_1\neq \frac{k_1}{2}$ or $i_2\neq \frac{k_2}{2}$.
\lim_{p^2+q^2} \vert \tau^{2k}(M_{p/q}^3,\sigma)\vert =+\infty \, .
Use first Proposition <ref> (<ref>) . Since
the character $\chi_{p,q}$ converges to $\chi_\varrho$,
$\TT_{M,m}^{2k}(\chi_\varrho)\neq 0$, and
$\vert\cs(pm+ql ,m )\vert=\vert p+q\,\cs(l,m)\vert\to\infty$, one has
\vert {\mathbb{T}_{M,pm+ql}^{2 k}(\chi_{p,q} )}\vert \to\infty \, .
As $\vert\lambda_{p/q}\vert\to 1$, the surgery formula in
Proposition <ref> yields the result.
The same proof as Corollary <ref> yields:
Given a spin structure,
The set of modules of the torsions $\vert \tau^{2k_1+1,k_2}(M_{p/q}^3,\sigma)\vert $ obtained by Dehn filling on $M^3$,
so that $\sigma$ can be extended, is dense in
\left[ \frac14{\left\vert \tau^{2k_1+1,k_2} (M^3,\sigma)\right\vert},+\infty\right).
When $k_1+k_2$ is even and $k_1 k_2\neq 0$, then $\TT^{k_1,k_2}_M(\chi_\varrho)=0$ (for $k_i$ odd) and $\TT^{k_1,k_2}_{M,\gamma}(\chi_\varrho)=0$
(for $k_i$ even). In this case, we cannot get conclusions from the surgery
§.§ Quantum invariants
When $(k_1,k_2)=(2,0)$, $\operatorname{Sym}^2$ is the adjoint representation and its torsion
occurs in the volume conjecture.
The role of the torsion in the expansion
of the path integral is already mentioned in
the work of Witten <cit.> and Bar-Natan Witten <cit.>. Of
course the work of Kashaev <cit.>
and Murakami-Murakami <cit.> play a key role in the conjecture.
For a knot $\mathcal K$, let $J_N(\mathcal K;z)$ denote the colored Jones
polynomial of $\mathcal K$. If the knot is hyperbolic,
let $u$ denote Thurston's parameter of the Dehn filling space (Definition <ref>).
Denote the corresponding character by $\chi_u$ and
let $\operatorname{CS}(\mathcal K,\chi_u)$ denote the $\CC$-valued
Chern-Simons invariant of a representation with character $\chi_u$,
namely the real part is minus the Chern-Simons invariant and
the imaginary part is the volume of the representation.
The following version is
taken from <cit.> by H. Murakami, who attributes it to <cit.>.
Let $K$ be a hyperbolic knot. There exists a neighborhood $U\subset\CC$ of the
origin such that for $u\in U- \pi i \QQ$, we have the following asymptotic
J_n(\mathcal K;e^{\frac{2\pi i+u}{N}}) \, \underset{N\to\infty}{\sim}\,
\frac{\sqrt{-\pi}}{2\sinh(\frac{u}{2})} \left( {\TT}^{2}_{\mathcal
K,m}(\chi_{u}) \right)^{-\frac{1}{2}}
\, \left( \frac{N}{2\pi i+u}\right)^{\frac 12} \, e^{\frac{N}{2\pi i+u}
\operatorname{CS}(\mathcal K,\chi_u) }
where $m$ denotes the meridian of the knot.
Again this paper uses a different convention for torsion from
<cit.> and the other references, as it is the opposite to the
convention for analytic torsion.
It has been checked for the figure eight knot and $u$ real, $0< u <
\log((3+\sqrt{5})/2)$ by Murakami in <cit.>.
For torus knots, the volume is zero, but not the Chern Simons invariant nor the torsion, and the asymptotic computations of
Dubois-Kashaev <cit.> and Hikami-Murakami <cit.> support the corresponding conjecture for torus knots.
Related to this conjecture, the torsion is also involved in a potential
function, introduced by Yokota <cit.>. From a diagram of the projection
of a knot $\mathcal K$, in
<cit.> Ohtsuki and Takata define
$\omega_ 2(\mathcal K)^{-1}$ as the modified Hessian of the potential function
of the diagram. They justify (formally) that $\sqrt{\omega_ 2(\mathcal K)}$ is
the term that appears
in the asymptotic development of the Kashaev invariant and therefore they conjecture
\frac{1}{\omega_ 2(\mathcal K)}= \pm { 2\,i \, {\TT}^{2}_{\mathcal K,m}(\chi_{\varrho})}.
In <cit.> they prove that the conjecture holds true for two bridge knots.
There is also a remarkable contribution of Dimofte and Garoufalidis <cit.>,
that define an invariant from an ideal triangulation of a knot exterior and
an enhanced Neumann-Zagier datum.
Enhanced Neumann-Zagier datum means that, besides the complex collection
shape parameters of the ideal hyperbolic tetrahedra, they use matrices with integer coefficients
that describe how to glue the tetrahedra and a collection of
integers that code a combinatorial flattening (introduced in <cit.> by Neumann
to calculate the Chern-Simons invariant combinatorially).
Form these data, Dimofte and Garoufalidis construct an invariant of the hyperbolic manifold and they check numerically that
it equals ${1}/{{\TT}^{2}_{\mathcal K,m}(\chi_{\varrho})}$ up to some
§ REPRESENTATIONS IN $\PSL_{N+1}(\CC)$ AND THE ADJOINT
In <cit.> I considered the torsion corresponding to the adjoint representation as a function on the variety of characters
in $\SL_2(\CC)$, in fact $\PSL_2(\CC)$.
Kitayama and Terashima <cit.> considered the analog for $\PSL_{n+1}(\CC)$, i.e. the torsion for the adjoint representation.
Along the section $M^3$ denotes an oriented finite volume hyperbolic
three-manifold with one cusp, the case with more cusps would require further
but here it is essentially the same. Consider the composition of the holonomy with
$$\varrho^{n}\colon\pi_1M^3\to \PSL_{n+1}(\CC).
As we deal with representations in $\PSL_2(\CC)$, the holonomy does not need to be lifted to $\SL_2(\CC)$.
The character of $\varrho^{n}$ is a smooth point of the variety of characters $X(M^3,\PSL_{n+1}(\CC))$, of local dimension $n$.
It is locally parametrized by the symmetric functions on any peripheral curve.
There exists a unique irreducible component of the character variety $X(M^3,\PSL_{n+1}(\CC))$ that contains the character of
This component is called the distinguished component and it is denoted by
\[
\]
In the proof of Theorem <ref>, one shows that $\mathfrak{sl}_{n+1}(\CC)^{\Ad \varrho^{n}(\pi_1 T^2)}\cong \CC^n$.
In addition, if $\{a_1,\ldots, a_{n}\}$ is a basis for
$\mathfrak{sl}_{n+1}(\CC)^{\Ad \varrho^{n}(\pi_1 T^2)}$, then
\{ a_1\cap [T^2],\ldots, a_{n}\cap [T^2]\}
is a basis for $H_2(M^3; \Ad\varrho^n)$
\{ a_1\cap [\gamma],\ldots, a_{n}\cap [\gamma]\}
a basis for $H_1(M^3; \Ad\varrho^n)$, here $\gamma$ is a peripheral curve, non-trivial in $H_1(T^2;\ZZ)$.
See <cit.> for details.
For a generic character $\chi_\rho$ in $ X_0(M^3,\PSL_{n+1}(\CC))$,
$\mathfrak{sl}(n,\CC)^{\Ad \rho(\pi_1 T^2)}\cong \CC^n$ and a similar
applies to find a basis for $H_i(M^3; Ad\rho)$. Generalizing the construction of <cit.>, Kitayama and Terashima
<cit.> define the torsion. Here we consider the possibility that the function vanishes, but before an exceptional set must be removed.
A representation $\rho\colon\pi_1 M^3\to\PSL_{n+1}(\CC)$ is exceptional if:
* $\mathfrak{sl}_{n+1}(\CC)^{\Ad\rho(\pi_1M^3)}\neq 0$ or
* $\dim \left(\mathfrak{sl}_{n+1}(\CC)^{\Ad\rho(\pi_1T^2)}\right)> n$, for
the peripheral torus $T^2$.
The set of characters of exceptional representations is denoted by
$$\EE_{n+1}\subset X_0(M^3,\PSL_{n+1}(\CC) ).$$
The set $\EE_{n+1}$ is Zariski closed in $X_0(M^3,\PSL_{n+1}(\CC) )$.
The defining properties are Zariski closed in $\hom(\pi_1M^3,\PSL_{n+1}(\CC))$ and invariant by conjugation, so the projection
to $X_0(M^3,\PSL_{n+1}(\CC)) $ is a Zariski closed subset, see <cit.> for instance.
Let $\chi_\rho\in X_0(M^3,\PSL_{n+1}(\CC))-\EE_{n+1}$, choose $\{a_1,\ldots,
a_{n}\}$ a basis for $\mathfrak{sl}_{n+1}(\CC)^{ \Ad \rho(\pi_1 T^2)}$
and let
$\gamma$ be a peripheral curve.
* If $ \{ a_1\cap [\gamma],\ldots, a_{n}\cap [\gamma]\}$ is a basis for $H_1(M^3;\Ad\rho)$, define
\begin{multline}
\T_{(M,\gamma)}(\chi_\rho)= \operatorname{TOR}(M^3, \{ a_1\cap [\gamma],\ldots, a_{n}\cap [\gamma]\}, \\
\{ a_1\cap [T^2],\ldots, a_{n}\cap [T^2]\});
\end{multline}
* otherwise set
\begin{equation}
\T_{(M,\gamma)}(\chi_\rho)=0.
\end{equation}
This yields a function
\begin{equation}
\T_{M,\gamma}\colon X_0(M^3,\PSL_{n+1}(\CC))-E_{n+1} \to \CC\, .
\end{equation}
It can be checked that it is a well defined regular function,
using the ideas of <cit.> and Proposition <ref>. For instance,
when $ \{ a_1\cap [\gamma],\ldots, a_{n}\cap [\gamma]\}$ is a basis for $H_1(M^3;\Ad\rho)$,
then $ \{ a_1\cap [T^2],\ldots, a_{n}\cap [T^2]\}$ is a basis for $H_2(M^3;\Ad\rho)$
(because $H_2(M^3;\Ad\rho)$ is naturally isomorphic to $H_2(T ^2;\Ad\rho)\cong
H^0(T ^2 ; \Ad\rho)$).
When there may be confusion, the index $n+1$ may be included in the notation:
\begin{equation}
\T_{M,\gamma}^{n+1}\colon X_0(M^3,\PSL_{n+1}(\CC))-E_{n+1} \rightarrow \CC.
\end{equation}
The following proposition relates $\T_{M,\gamma}$ with the torsion $\TT^{2n}_{(M,\gamma)}$ when we consider
symmetric powers of representations in $\PSL_2(\CC)$.
For a generic character $\chi_\rho\in X_0(M^3,\PSL_2(\CC))$, including the holonomy $\chi_\varrho$,
\T_{M,\gamma}^{n+1}(\chi_{\rho^n})=\prod_{i=1}^n \TT_{M,\gamma}^{2 i}(\chi_\rho).
For a generic character $\chi_\rho\in X_0(M^3,\PSL_2(\CC))$, including the holonomy $\chi_\varrho$,
\T_{M,\gamma}^{n+1}(\chi_{\rho^n})=\T_{M,\gamma}^{n}(\chi_{\rho^{n-1}})\, \TT_{M,\gamma}^{2 n}(\chi_\rho).
The proof is straightforward from Klebsh-Gordan formula:
\begin{equation}
\label{eqn:adKG}
\Ad\circ\operatorname{Sym}^n=\bigoplus_{i=1}^n \operatorname{Sym}^{2i}
\end{equation}
and multiplicativity of the torsion for sums of representations.
The next theorem due to Weil <cit.>, see also <cit.>, gives a nice interpretation to $\T_{M,\gamma}$:
Let $\Gamma$ be a finitely generated group
and $\rho$
an irreducible representation with character
$\chi_\rho\in X(\Gamma, \PSL_{n+1}(\CC) )-\EE_{n+1}$. Then
there is a natural isomorphism
T^{Zar}_{\chi_\rho} X(\Gamma, \PSL_{n+1}(\CC) )\cong H^1(\Gamma ; \Ad\rho)
where $T^{Zar}$ denotes the Zariski tangent space as a scheme.
In particular, if $\phi\colon\Gamma\to\Gamma'$ is a group morphism, then the
induced map in cohomology corresponds to the tangent map
d\phi^* \colon T^{Zar}_{\chi_\rho} X(\Gamma', \PSL_{n+1}(\CC) )\to
T^{Zar}_{\phi^*\chi_\rho} X(\Gamma, \PSL_{n+1}(\CC) ).
Few comments are in order here. First at all, the condition that $\chi_\rho\not\in\EE_{n+1}$ is used to say
that the infinitesimal commutator of $\Ad\rho$ is trivial, i.e. $H^0(\pi_1 M^3;\Ad\rho)=0$.
the variety of characters is not a variety but a scheme: the defining polynomial ideal may be not reduced, thus we must consider the Zariski tangent space of
the scheme, perhaps not reduced.
Finally, just mention that there are generalizations of this result when $\rho$ is reducible, see <cit.>.
This interpretation has a nice application for surface bundles over the circle, following again <cit.>.
Assume that $M^3$ is a bundle over a circle, with fibre a punctured surface
$\Sigma$ and monodromy $\varphi\colon\Sigma\to \Sigma$, i.e.
M^3=\Sigma\times [0,1]/ (x,1)\sim (\varphi(x),0).
It has a natural epimorphism $\pi_1M^3\to \ZZ$, corresponding to the projection of the fibration $M^3\to S^1$.
The induced map on the monodromy is denoted by
\varphi_*\colon X(\Sigma,\PSL_n(\CC))\to X(\Sigma,\PSL_n(\CC))
and the restriction to $\pi_1\Sigma$ of characters in $X(M^3,PSL_n(\CC))$
restrict to the fixed point set of
$X(\Sigma,\PSL_n(\CC))$. By Weil's theorem, the map induced in
$H^1(\Sigma;\Ad\rho)$ can be interpreted as the differential
Thus, using Proposition <ref>, the twisted polynomial is
\det (d \varphi_*- t \operatorname{Id}).
Its evaluation at $t=1$ vanishes because $H^*(M^3;\Ad\rho)\neq 0$. The polynomial is divisible by $(t-1)^n$,
corresponding to the invariant curve $\gamma=\partial \Sigma$, as $\dim
H^1(\gamma; \Ad\rho)=n $. An argument
using the exact sequences and the basis of homology yields the following result:
If $M^3$ is a punctured surface bundle, and $\gamma$ the boundary of the fibre. Then
\T_{M,\gamma} =\left.\frac{\det (d \varphi_*- t \operatorname{Id})}{(t-1)^{n}}\right\vert_{t=1}.
This result is very useful for computing the torsion, it allows to obtain it from the variety of characters or moduli
spaces without knowing the representation. For instance Kitayama and Terashima use cluster algebras to compute it
<cit.>. In Subsection <ref> we recall the method of <cit.> to compute it.
§.§ Local parameters and change of curve
Consider $\mathfrak{h}\subset \mathfrak{sl}_{n+1}(\CC)$ the Cartan subalgebra of diagonal matrices,
in particular a diagonal matrix in $\PSL_{n+1}(\CC)$ lies in $\exp \mathfrak{h}$.
For a generic character $\chi\in X_0(M^3,\PSL_{n+1}(\CC))$, one would like to consider in a
neighborhood $U$ of $\chi$:
\begin{equation}
\label{eqn:log}
\begin{array}{rcl}
\log_\gamma\colon U\subset X_0(M^3,\PSL_{n+1}(\CC))&\to& \mathfrak{h}\\
\chi_\rho & \mapsto & \log \rho(\gamma)
\end{array}
\end{equation}
but a priori this may not be well defined.
Notice that there are indeterminacies due to the action of the Weyl group (permutation of elements in the diagonal)
and indeterminacies due to the complex logarithm. This motivates the following definition.
A representation $\rho\in \hom(\pi_1M^3,\PSL_{n+1}(\CC))$ is chamber regular is there exists a peripheral element
$\gamma\in\pi_1 T^2$ such that $\rho(\gamma)$ has $n+1$ different eigenvalues.
In terms of Lie groups this is a regularity condition:
$\rho(\pi_1T^2)$ is contained in a Cartan subgroup and in the interior of the Weyl chamber of $\PSL_{n+1}(\CC)$.
By <cit.>,
the symmetric functions on the eigenvalues of $\rho(\gamma)$
define a local biholomorphism in a neighborhood of $\varrho^{n}$, hence all eigenvalues of
$\rho(\gamma)$ are different in a Zariski open set. Thus:
The set of chamber regular
characters is a non-empty Zariski open subset of $X_0(M^3,\PSL_2(\CC))$.
For a chamber regular character and a peripheral element $\gamma\in\pi_1 T^2$,
there exist a neighborhood $U \subset X_0(M^3,\PSL_{n+1}(\CC)) $ such that the logarithm $\log_\gamma$
as in (<ref>) is defined in $U$. Notice that the eigenvalues of the image of $\gamma$
do not need to be different, provided that there is an element in the peripheral group with different eigenvalues.
Next consider a nonzero element $a\in \mathfrak{h}$. Using the Killing form, we
define $a^\star\in\mathfrak{h}$ to be the pairing with $a$.
Let $\rho$ be a chamber regular representation, $\gamma$ a peripheral curve and $a\in\mathfrak{h}$.
Viewing $H_1(M^3;\Ad\rho)$ as cotangent space:
\begin{equation}
\label{eqn:cotangent}
a\cap[\gamma] = d ( a^\star\log_\gamma).
\end{equation}
This must be compared with <cit.>.
Before proving this lemma, let us discuss its consequences.
Let $\chi_\rho\in X_0(M^3,\PSL_{n+1}(\CC))$ be a chamber regular character and $\gamma$ a peripheral curve. Then
$\T_{M,\gamma}(\chi_\rho)\neq 0$ if and only if $\chi_\rho$ is a scheme reduced smooth point of $X_0(M^3,\PSL_{n+1}(\CC))$
and $\log_{\gamma}$ is a local parameter.
The proof follows easily from Lemma <ref>, and we just sketch it.
Namely, the condition of being scheme reduced and smooth is equivalent to saying that $ \dim H^1(M^3;\Ad\rho)=n$.
By the standard arguments of the long exact sequence of the pair $(M^3, T^2)$ this implies that $ \dim H^0(T^2;\Ad\rho)=n$
and that $\{a_1\cap [T^2],\ldots,a_n\cap [T^2]\}$ is a basis for $H^2(M^3;\Ad\rho)$.
Then Lemma <ref> yields
that $\{a_1\cap [\gamma],\ldots,a_n\cap [\gamma]\}$ is a basis for $H^1(M^3;\Ad\rho)$
iff $\log_{\gamma}$ is a local parameter.
For a chamber regular character, the condition of Proposition <ref>
is equivalent to the notion of $\gamma$-regularity of
Notice finally that for any non-trivial peripheral curve $\gamma$, for the lift of the holonomy $\varrho$, even if it is not
chamber regular,
$\T_{M,\gamma}(\chi_{\varrho^n})\neq 0$, by <cit.>.
Let $\gamma_1, \gamma_ 2$ be two peripheral elements. In a Zariski open domain in $X_0(M^3,\PSL_{n+1}(\CC))$
\frac{ \T_{(M,\gamma_1)} }{ \T_{(M,\gamma_2)} }
= \pm \operatorname{J} (\log_{\gamma_1} \log_{\gamma_2}^{-1} ).
Notice that the Jacobian
\operatorname{J} (\log_{\gamma_1} \log_{\gamma_2}^{-1} )
is well defined generically on the distinguished component $X_0(M^3,\PSL_{n+1}(\CC))$, i.e. in a non-empty open Zariski set, as $\log_{\gamma_2}$ is a local parameter in an open set.
In addition, this Jacobian is independent of the parametrization of $\mathfrak{h}$, as any change of parametrization cancels in the quotient.
The proof of Proposition <ref> is straightforward from Lemma <ref> and the formula of change of
basis in homology (<ref>).
Proposition <ref> does not cover $\chi_{\varrho^n}$, the character of $\operatorname{Sym}^n$ of the holonomy of the complete hyperbolic structure, as it is not
chamber regular. However, Corollary <ref> and Proposition <ref>(<ref>) yield:
{\T_{M,\gamma_2}^{n+1}(\chi_{\varrho^n})}= \cs(\gamma_2,\gamma_1)^n \,
where $\cs(\gamma_2,\gamma_1)$ denotes the cusp shape (Definition <ref>).
Assume that $\rho$ is a generic representation so that $\rho(\gamma)$ is diagonal with different
eigenvalues. Let $\mathfrak{h}$ denote the Cartan algebra of diagonal matrices,
and choose a non-zero element $a\in\mathfrak{h}$.
In particular $a\cap\gamma\in H_1(M^3;\Ad\rho)$.
Consider the dual of $a$:
\begin{equation*}
\begin{array}{rcl}
a^\star\colon \mathfrak{h} & \to & \CC \\
h & \mapsto & B(a,h)
\end{array}
\end{equation*}
where $B$ denotes the Killing form.
Consider also a first order deformation $\rho_t$ of $\rho$, i.e.
\begin{equation}
\label{eqn:firstorder}
\rho_t =(1+t\, \dot{\rho}+O(t^2))\rho
\end{equation}
where $\dot\rho\colon\pi_1M^3\to \mathfrak{sl}_{n+1}(\CC)$ is a group cocycle, that we project to
Consider finally the Kronecker pairing
\begin{equation*}
\langle\cdot \rangle\colon H^k(M^3;\Ad\rho)\times H_k(M^3;\Ad\rho)\to \CC
\end{equation*}
defined as follows. Let $z\in C^k(M^3;\Ad\rho) $ be a cocycle, with
$z\colon C_k(\widetilde M^3;\ZZ)\to \mathfrak{sl}_{n+1}(\CC)$,
and let $h\otimes m\in C_k(M^3;\Ad\rho)$, where $h\in \mathfrak{sl}_{n+1}(\CC)$ and $m \in C_k(\widetilde M^3;\ZZ)$. At the (co-)chain level,
the Kronecker pairing is
\begin{equation}
\label{eqn:kronecker}
\langle z, h\otimes m\rangle = B(h,z(m))
\end{equation}
where $B$ denotes again the Killing form. This induces a non-degenerate pairing between homology and cohomology <cit.>.
After all those preliminaries, to establish the lemma one must prove the following equality
\begin{equation}
\label{eqn:pairing}
\langle \dot\rho, a\cap[\gamma]\rangle = \left.\frac{d\phantom{t}}{dt}
a^\star\log\rho_t(\gamma)\right\vert_{t=0} \, .
\end{equation}
To prove (<ref>), start with the group cohomology version of (<ref>) in the current context (see <cit.>):
\begin{equation}
\label{eqn:kroneckergroup}
\langle \dot\rho , a\cap [\gamma]\rangle = B(a, \dot\rho(\gamma)).
\end{equation}
From (<ref>) evaluated at $\gamma$ and taking logarithms:
\begin{equation}
\label{eqn:loga}
\log \rho_t(\gamma) = \log \rho(\gamma) + t\, \dot{\rho}(\gamma)+
O(t^2)) \, .
\end{equation}
Then (<ref>) follows from $a^\star$ applied to (<ref>),
then differentiating and applying
(<ref>) to the result.
Lemma <ref> may be related to the results of Goldman <cit.>.
[Volumes on $X(\mathcal K,SU(2))$]
In <cit.> Dubois makes a relevant contribution of torsions as volume form on
the variety of characters of a knot in $\operatorname{SU}(2)$. Here the Cartan algebra has dimension one
and the logarithm of a matrix in $\operatorname{SU}(2)$ is an angle.
Dubois volume form is, up to sign,
\vol_{\tor}= \pm \frac{d\varphi_\gamma } { \T_{m,\gamma}}
where $\varphi_\gamma$ denotes the angle of the representation of the peripheral curve $\gamma$.
Notice that by Proposition <ref>, this form is independent of the choice of the peripheral curve.
In <cit.>, using Turaev's refinement of torsion and a good choice of $\varphi_\gamma$,
Dubois avoids the sign indeterminacy.
He also shows that it equals to a volume form defined from a Heegaard splitting, à la Johnson (Example <ref>).
This is related to the construction of an orientation on the space of representations of Heusener <cit.>.
Proposition <ref> suggests that a similar volume form can be constructed for representations in
$\operatorname{SU}(n+1)$, taking a convenient parametrization of the Cartan algebra (i.e. infinitesimal angles).
It remains to know also whether the variety of characters of a knot in $\operatorname{SU}(n+1)$ is non-empty and $n$-dimensional.
This holds true for instance for two bridge knots. Another issue is to compute explicitly the variety of $\operatorname{SU}(n+1)$
for a knot.
§.§ An example
Let me use Proposition <ref> to compute $\T_{M,\gamma}$ for the figure eight knot
for $\PSL_2(\CC)$ and $\PSL_3(\CC)$.
For $\PSL_2(\CC)$ this is done in <cit.>, but I recall it here for completeness.
Let $\Gamma=\pi_1 M$ denote the fundamental group of the figure eight knot
exterior. The presentation
\Gamma=\langle a,b, m \mid m\, a\, m^{-1}= a\, b,\, m\, b\, m^{-1}= b\, a\,
\rangle
corresponds to the fact that it is fibered over the circle,
with fibre a punctured torus, whose fundamental group is the free group
$F_2=\langle a, b\mid\rangle$. The element $m$ is also a meridian curve of the knot.
The monodromy $\phi\colon F_2\to F_2$ satisfies
$$\phi(a)=a\, b \qquad \textrm{ and } \qquad
\phi(b)=b\, a\, b.
Since $F_ 2$ is the derived subgroup of $\Gamma$, every representation of $\Gamma$ in $\PSL_{n+1}(\CC)$
restricts to a representation of $F_2$ in $\SL_{n+1}(\CC) $ that is fixed by the monodromy $\phi^*$.
The set of fixed characters is denoted by
The following lemma is proved in <cit.> (using that $F_2$ is the commutator of $\Gamma$):
For $n=1,2$, the restriction map induces an isomorphism
\overline{X_{irr}(\Gamma,\PSL_{n+1}(\CC))} \cong
\overline{X_{irr}(F_2,\SL_{n+1}(\CC))^{\phi^*}},
where the subindex $_{irr}$ stands for irreducible characters.
§.§.§ Computations for $X(M^3,\PSL_{2}(\CC))$
By Fricke-Klein theorem, the variety of characters $X(F^2,\SL_2(\CC))$ is isomorphic to $\CC^3$. More precisely, defining
\begin{array}{ll}
& \alpha_1(\rho)=\tr(\rho(a)) \, ,\\
& \alpha_2(\rho)=\tr(\rho(b)) \, , \\
& \alpha_3(\rho)=\tr(\rho(a\,b)) \, .
\end{array}
Fricke-Klein theorem asserts that $(\alpha_1,\alpha_2,\alpha_3)$ are global coordinates of $X(F^2,\SL_2(\CC))$.
Using the relations of traces, $\forall A,B\in\SL_2(\CC)$:
\[
\begin{array}{ll}
\tr(A^{-1})&= \tr(A), \\
\tr(A B)&= \tr(A B^{-1})- \tr(A)\tr(B),
\end{array}
\]
we may deduce:
\begin{array}{ll}
\phi^*(\alpha_1)&=\alpha_3 \, ,\\
\phi^*(\alpha_2)&=\alpha_2 \alpha_3-\alpha_1 \, ,\\
\phi^*(\alpha_3)&=\alpha_3^2 \alpha_2-\alpha_1 \alpha_3-\alpha_2 \, .
\end{array}
Thus $ \phi^*(\alpha_i)=\alpha_i$ is equivalent to
\alpha_3=\alpha_1,\qquad \alpha_1+\alpha_2=\alpha_1\alpha_2 \, .
Then, the torsion polynomial is
\begin{equation}
\label{eqn:tor82t}
{\det ( d \phi_* - t \operatorname{Id})}= (t-1)(t^2+(1-2 \alpha_1 \alpha_2) t+1).
\end{equation}
Removing the factor $(t-1)$ and evaluation at $t=1$ yields
\begin{equation}
\label{eqn:tor82}
\T_{M,l}= 3-2\alpha_1\alpha_2 = 3-2\alpha_1-2\alpha_2
\end{equation}
Using that the trace of the longitude $l=[a,b]$ satisfies
we get
$\T_{M,l}^2=17+4\theta_l$. If $\theta_m$ denotes the trace of the meridian,
the distinguished component $X_0(\Gamma,\SL_2(\CC))$ is the curve
\begin{equation}
\label{eqn:X0fig8}
\alpha_1^2+\alpha_1-1= \theta_m ^2\,(\alpha_1-1).
\end{equation}
(Let me emphasize that $\theta_l$ and $\theta_m$ denote traces and not
I made this choice because $\tau$ is already used for torsion.)
To get the variety $\PSL_2(\CC)$ characters instead of $\SL_2(\CC)$ just replace $\theta_m^2$ by a new variable.
From (<ref>) and $x_1+x_2=x_1x_2$ we deduce:
\begin{equation}
\label{eqn:x1+x2}
\alpha_1+\alpha_2= \theta_m^2-1 \qquad \textrm{ and }
\qquad \alpha_1-\alpha_2= \pm\sqrt{(\theta_m ^2-5) (\theta_m-1)} \, .
\end{equation}
\begin{equation}
\label{eqn:T2l}
\T_{M,l}= \TT_{M,l}^2= 5-2\theta^2_m.
\end{equation}
Proposition <ref> can be worked out <cit.> to yield:
\begin{equation}
\label{eqn:T2m}
\T_{M,m}= \TT_{M,m}^2= \pm \frac{\alpha_1-\alpha_2}{2}=\pm \left( \alpha_1+\frac{1-\theta_m^2}{2} \right)=\pm \frac{1}{2}
\sqrt{(\theta_m^2-5)(\theta_m^2-1)}.
\end{equation}
§.§.§ Computations for $X(M^3,\PSL_{3}(\CC))$
By a theorem of Lawton <cit.> and Will <cit.>:
is a double branched covering of $\CC^8$ with coordinates
\begin{array}{ll}
& \beta_1(\rho)=\tr(\rho(a)) \\
& \beta_2(\rho)=\tr(\rho(a^{-1})) \\
& \beta_3(\rho)=\tr(\rho(b)) \\
& \beta_4(\rho)=\tr(\rho(b^{-1})) \\
& \beta_5(\rho)=\tr(\rho(a\,b)) \\
& \beta_6(\rho)=\tr(\rho(b^{-1}a^{-1})) \\
& \beta_7(\rho)=\tr(\rho(a^{-1}b)) \\
& \beta_8(\rho)=\tr(\rho(a\,b^{-1}))
\end{array}
and the trace of $l= [a,b]$ and its inverse, $\vartheta_{l^{\pm 1}}$, are a degree
two extension of the coordinates $\beta_1,\ldots,\beta_8$, and
$\vartheta_{l }$ and $\vartheta_{l^{- 1}}$ are Galois conjugate.
As $\phi(l)=l$, we may work in $\CC^8$.
Following <cit.>, $\phi^*$ can be computed as
\begin{array}{ll}
& \phi^*(\beta_1)= \beta_5 \\
& \phi^*(\beta_2)= \beta_{{6}} \\
& \phi^*(\beta_3)= -\beta_{{1}}\beta_{{4}}+\beta_{{3}}\beta_{{5}}+\beta_{{8}} \\
& \phi^*(\beta_4)= -\beta_{{2}}\beta_{{3}}+\beta_{{4}}\beta_{{6}}+\beta_{{7}} \\
& \phi^*(\beta_5)= -\beta_{{1}}\beta_{{4}}\beta_{{5}}+\beta_{{3}}{\beta_{{5}}}^{2}-\beta_{{3}}\beta_{{6}}+\beta_{{5}}\beta_{{8}}+\beta_{{2}} \\
& \phi^*(\beta_6)= -\beta_{{2}}\beta_{{3}}\beta_{{6}}+\beta_{{4}}{\beta_{{6}}}^{2}-\beta_{{4}}\beta_{{5}}+\beta_{{6}}\beta_{{7}}+\beta_{{1}} \\
& \phi^*(\beta_7)= \beta_{{3}} \\
& \phi^*(\beta_8)= \beta_{{4}}.
\end{array}
Now, setting $\beta_i=\phi^*(\beta_i)$, we get rid of four variables (we are left with
$\beta_1$, $\beta_2$, $\beta_3$, and $\beta_4$), and we deduce that
$X(F^2,\SL_3(\CC))^{\phi^*}$ has three components:
* $V_0$, with equations $\beta_1=\beta_2$, $\beta_3=\beta_4$.
* $V_1$, with equations $\beta_1=\beta_2=1$,
* $V_2$, with equations $\beta_3=\beta_4=1$.
The component $V_0$ is the restriction of the distinguished component
$X_0(M^3,\PSL_{3}(\CC) )$. On this component,
the torsion polynomial is
\begin{multline}
{\det ( d \phi_* - t \operatorname{Id})} =
{(t-1)^2} \left( t^2 + (2-\beta_{{1}}\beta_{{3}}) t +1 \right) \\ \times
\left(
t^4 + ( -\beta_{{1}}\beta_{{3}}-2\,\beta_{{1}}-2\,\beta_{{3}} ) t^3 +
( 6\,\beta_{{1}}\beta_{{3}}+2 ) t^2+ ( -\beta_{{1}}\beta_{{3}}-2\,\beta_{{1}}-2\,\beta_{{3}} ) t + 1
\right) \, .
\end{multline}
After getting rid of $(t-1)^2$ and evaluating at $t=1$, we get
\begin{equation}
\label{eqn:TMl}
\T_{M,l}=( 4-\beta_1\beta_3) 4(1-\beta_1)(1-\beta_3) \, .
\end{equation}
Next Corollary <ref> is used to compute $\TT^4_{M,l}$.
The relation between traces in $\SL_2(\CC)$ and their image in $\SL_3(\CC)$
via $\operatorname{Sym}$ yields
\begin{array}{l}
\beta_1= \beta_2 =\alpha_1^2-1\, ,\\
\beta_3=\beta_4 =\alpha_2^2-1 \, .
\end{array}
Using these identities in (<ref>) we get
\T_{M,l}\circ\operatorname{Sym}= (3-2\alpha_1\alpha_2) 4 (2-\alpha_1^2)(2-\alpha_2^2).
Thus by applying Corollary <ref> and (<ref>), one gets
\begin{equation}
\label{eqn:t84}
\TT^4_{M,l}= 4 (2-\alpha_1^2)(2-\alpha_2^2)=8(1-\alpha_1\alpha_2)=8(2-\theta_m^2).
\end{equation}
§ NOT APPROXIMATING THE TRIVIAL REPRESENTATION
For a manifold $M^3$ there are components of $X(M^3,\PSL_2(\CC))$ that consist of characters of abelian representations.
When $b_1(M^3)=1$ those components are curves, and their union is denoted by
The number of
components of $X^{ab}(M^3,\PSL_2(\CC))$
depends on the torsion of $H_1(M^3;\ZZ)$.
Being irreducible is a Zariski open property for a character <cit.>. On the other hand, every reducible character is also the character
of an abelian representation. This yields that $X^{ab}(M^3,\PSL_2(\CC))$ are precisely the components consisting only of reducible characters.
Assume that $b_1(M^3)=1$. Then the trivial character belongs to a single irreducible component of $X(M^3,\PSL_2(\CC))$, which is one of the curves of
If $b_1(M^3)=1$, then the trivial character does
not belong to $X_0(M^3,\PSL_2(\CC))$.
The proof uses the projection
\begin{equation}
\label{eqn:proj}
\begin{array}{rcl}
\pi\colon \hom(\pi_1 M^3,\PSL_2(\CC)) & \to & X(M^3,\PSL_2(\CC)) \\
\rho & \mapsto &\chi_\rho
\end{array}
\end{equation}
and the dimension of its fibre. For the trivial character $\chi_0$, a representation $\rho\in \pi^{-1}(\chi_0)$ is conjugate to
\begin{equation}
\label{eqn:rhoab}
\rho(\gamma)= \pm \begin{pmatrix}
1 & h(\gamma) \\
0 & 1
\end{pmatrix},
\qquad \forall \gamma\in\pi_1M^3,
\end{equation}
where $h\colon \pi_1M^3\to\CC$ is a group homomorphism.
(Not to be confused with the cusp shape of peripheral representations, as $h$
is defined in the whole group $\pi_1M^3$.)
Conjugating the representation (<ref>) by a diagonal matrix means replacing the
morphism $h$ by a multiple. Thus, as $b_1(M^3)=1$, there are only two orbits
by conjugation in $\pi^{-1}(\chi_0)$:
the trivial and the non-trivial morphism $h\colon \pi_1M^3\to\CC$.
By looking at the dimension of the stabilizers,
these orbits have dimension either 0
(for $h$ trivial) or 2 (for $h$ non-trivial). Hence the dimension of
$\pi^{-1}(\chi_0)$ is 2. On the other hand, on components $Y$ of
$X(M^3,\PSL_2(\CC))$ that contain irreducible representations,
the generic dimension of $\pi^{-1}$ is 3, the dimension of $\PSL_2(\CC)$.
Since this dimension is upper semi-continuous, the trivial character cannot
belong to an irreducible component of $X(M^3,\PSL_2(\CC))$ that contains
irreducible characters. Hence it must belong to a component whose characters
are all reducible.
Let $\rho_1,\rho_2\in\hom(\pi_1M^3,\SL_2(\CC))$ have the same character. If $\chi_{\rho_1}=\chi_{\rho_2}$ is nontrivial,
then $\rho_1$ is acyclic if and only if $\rho_2$ is acyclic.
For every character there is a unique closed orbit by conjugation, so that every
other orbit accumulates to it, see <cit.>.
So we may assume that the conjugation orbit of $\rho_2$ accumulates to $\rho_1$.
By semi-continuity, $\rho_1$ acyclic implies
that so is $\rho_2$. Next assume that $\rho_1$ is not acyclic. Up to conjugacy, there exists a
group homomorphism $\phi \colon \pi_1M^3\to\ZZ$ and $\lambda\in\CC-\{0,\pm 1\}$
such that
\rho_1(\gamma)=\begin{pmatrix} \lambda^{\phi(\gamma)} & 0 \\
0 & \lambda^{-\phi(\gamma)}
\end{pmatrix}
\quad\textrm{ and }\quad
\rho_2(\gamma)=\begin{pmatrix} \lambda^{\phi(\gamma)} & f(\gamma) \\
0 & \lambda^{-\phi(\gamma)}
\end{pmatrix}
for every $\gamma\in\pi_1M^3$.
Here $f\colon \pi_1M^3\to\CC$ is a crossed morphism:
$f(\gamma_1\gamma_2)= f(\gamma_1) + \lambda^{2\phi(\gamma)} f(\gamma_2)$,
for all $\gamma_1,\, \gamma_2 \in \pi_1M^3 $.
The homology of $\rho_1$ decomposes as a direct sum of $\pi_1M^3$-modules:
and $\{0\}\oplus\CC$, and both are nonzero (one is dual from the other). The
$\rho_2$-module does not decompose, but there is an
exact sequence of $\pi_1M^3$-modules
0\to \CC\oplus\{0\} \to \CC^2\to \CC\to 0
where $ \pi_1M^3$ acts on $\CC^2$ via $\rho_2$, and the action on the other
modules is the same as for $\rho_1$.
From the corresponding long exact sequence in homology it follows easily that $\rho_2$ is not acyclic.
§ COHOMOLOGY ON THE VARIETY OR CHARACTERS
The aim of this appendix is to provide references and proofs for the result in cohomology of Section <ref>.
§.§ The complete structure
Let $M^3$ be a hyperbolic orientable 3-manifold and
\varrho=\widetilde{\mathrm{hol}}\colon\pi_1Mł\to\SL_2(\CC)
a lift of its holonomy. As before denote by
\begin{equation}
\label{eqn:rhok1k2}
\varrho^{k_1,k_2}:= \operatorname{Sym}^{k_1,k_2}\circ \varrho \colon\pi_1 M^3\to \SL_{(k_1+1)(k_2+1)}(\CC).
\end{equation}
\begin{equation}
\label{eqn:Ek1k2}
E_{k_1,k_2}= \widetilde M \times_{\varrho^{k_1,k_2}} \CC^{(k_1+1)(k_2+1)}
\end{equation}
be the flat bundle twisted by ${\varrho^{k_1,k_2}}$ as in (<ref>).
In particular its de Rham cohomology is isomorphic to the simplicial cohomology of $ \varrho^{k_1,k_2}$
by de Rham theorem (<ref>).
Choosing a Hermitian metric on the fibre of the bundle $E_{k_1,k_2}\to M^3$ and a Riemannian metric on $M^3$, there is a product on
$E_{k_1,k_2}$-valued differential forms
$\Omega^*(M^3; E_{k_1,k_2})$ by integration on $M^3$, that we denote by $\langle\cdot,\cdot\rangle$. In particular it makes sense to talk about
$L^2$-forms, as the forms with finite norm.
For $k_1\neq k_2$, there exists a uniform constant $c_{k_1,k_2} > 0$ with the following property.
For every hyperbolic orientable 3-manifold $M^3$, and every differential
form $\omega\in\Omega^*(M^3; E_{k_1,k_2})$
with compact support,
\langle \Delta \omega, \omega\rangle \geq c_{k_1,k_2} \langle \omega,\omega\rangle.
This property implies strong acyclicity, as it yields that the spectrum
of $\Delta$ is bounded below by
the uniform constant $c_{k_1,k_2} > 0$.
Let $M^3$ be a closed, orientable, hyperbolic 3-manifold, then ${\varrho^{k_1,k_2}}$ is acyclic for $k_1\neq k_2$.
In particular $\varrho^{k}=\operatorname{Sym}^{k}\circ \widetilde{\mathrm{hol}}$ is acyclic for $k\geq 1$.
This corollary does not hold true when $k_1=k_2$. Millson <cit.> showed that it fails when $k_1=k_2>0$ and $M^3$ contains a
totally geodesic embedded surface, by means of bending.
§.§.§ The finite volume case
We next discuss the consequences in the finite volume case of Theorem <ref>.
The following corollary does not assume finite volume.
Let $M^3$ be an orientable hyperbolic 3-manifold and
$\widetilde{\mathrm{hol}}$ a lift of its holonomy.
For $k_1\neq k_2$ every closed $L^2$-form in $\Omega^*(M^3; E_{k_1,k_2})$ is exact. In particular every element
in $H^* (M^3; \varrho^{k_1,k_2}
)$ is represented by a form that is not $L^2$.
In order to apply this corollary, we need first to compute the homology and cohomology of the peripheral torus.
All the information on the dimension is given by $ H^0(T^2;\varrho^{k_1,k_2})$,
which is the set of invariants of the module $\CC^{(k_1+1)(k_2+1)}$ by the action of
$\varrho^{k_1,k_2}(\pi_1 T^2)$.
The following implies Item (a) of Proposition <ref>.
Let $M^3$ be as above and let $T^2\subset M^3$ be the peripheral torus.
The invariant subspace of the peripheral group is
H^0(T^2;\varrho^{k_1,k_2})\cong \left\{
\begin{array}{ll}
0 & \textrm{if } k_1+k_2 \textrm{ is odd,} \\
\CC & \textrm{if } k_1+k_2 \textrm{ is even.}
\end{array}
\right.
The lift of the holonomy restricted to $\pi_1T^2\cong \ZZ^2$ is written as
(n_1,n_2)\mapsto (-1)^{\epsilon(n_1,n_2)} \begin{pmatrix}
1 & n_1 + n_2 \cs \\
0 & 1
\end{pmatrix}
where $\cs\in\CC -\RR $ is the cusp shape in Definition <ref> and
$\epsilon\colon\ZZ^2\to\ZZ/2\ZZ$ is a surjection (here it is relevant that
$\epsilon$ is non-trivial).
From the representation, it is straightforward that if $k_1+k_2$ is odd, then there is an element whose
eigenvalues are all $(-1)$, and for $k_1+k_2$ even, the subspace of invariants is generated by the monomial $X^{k_1+1} \bar X ^{k_2+1}$.
From Poincaré duality we get information on $H^2$, but also on $H^1$ because $\chi(T^2)=0$. We also know the dimension of the homology
groups by duality. Thus we have:
Let $M^3$ and $T^2$ as above.
* If $k_1+k_2$ is odd then
H_*(T^2;\varrho^{k_1,k_2})=0\, .$$
* If $k_1+k_2$ is even, then
\dim H^i(T^2;\varrho^{k_1,k_2})=\dim H_i(T^2;\varrho^{k_1,k_2})=
\left\{
\begin{array}{ll}
1 & \textrm{for } i=0,2, \\
2 & \textrm{for } i=1, \\
0 & \textrm{otherwise. }
\end{array}\right.
Let $M^3$ be a hyperbolic orientable 3-manifold with a single cusp, and let $T^2$ be a peripheral torus.
* When $k_1+k_2$ is odd, then $H_*(M^3;
\varrho^{k_1,k_2})=0$.
* When $k_1+k_2$ is even with $k_1\neq k_2$, then
* $H_i(M^3; \varrho^{k_1,k_2} )=0$ for $i\neq 1,2$,
* $H_2(M^3;
\varrho^{k_1,k_2})\cong H_2(T^2; \varrho^{k_1,k_2})\cong \CC$,
* $H_1(M^3; \varrho^{k_1,k_2})\cong \CC$.
The group $H^0(M^3 ; \varrho^{k_1,k_2})$ vanishes, as this is the subspace of
fixed vectors, and both $\varrho$ and
$\operatorname{Sym}^{k_1}\otimes \overline{\operatorname{Sym}^{k_2}}$ are irreducible.
By Corollary <ref>, the map
H^i(M^3 ; T^2;\varrho^{k_1,k_2})\to H^i(M^3;\varrho^{k_1,k_2})
vanishes. Thus $ H^i(M^3;\varrho^{k_1,k_2})\to H^i(T^2;\varrho^{k_1,k_2})$ is injective,
by the long exact sequence in cohomology of the pair. With Poincaré duality
and duality between homology and cohomology, we get Item (<ref>).
Poincaré duality also yields that $H^1(T^2; \varrho^{k_1,k_2})\to H^2(M^3,T^2; \varrho^{k_1,k_2})$
is surjective. Then the lemma follows from the long exact sequence in cohomology and the duality
between homology and cohomology.
§.§.§ A basis in cohomology
To describe a basis for $H_i(M^3; \varrho^{k_1,k_2})$ when $i=1,2$, recall the
cap product defined in Equation (<ref>).
\cap\colon H^0(T^2; \varrho^{k_1,k_2})\times H_i(T^2; \CC)\to H_i(T^2;
\varrho^{k_1,k_2}).
Let $i\colon T^2\to M^3$ denote the inclusion, and $i_*\colon
H_j(T^2;\varrho^{k_1,k_2} )\to H_j(M^3;\varrho^{k_1,k_2} )$,
the induced map.
Let $M^3$ and $T^2$ be as above, $k_1\neq k_2$, $k_1+k_2$ even. Choose $a\in
H^0(T^2; \varrho^{k_1,k_2})$, $a\neq 0$. Then:
* $i_*(a\cap [T^2])$ is a basis for
$H_2(M^3;\varrho^{k_1,k_2})$, where $[T^2]\in H_2(T^2;\ZZ)$ is a fundamental
* $i_*(a\cap [\gamma])$ is a basis
for $H_1(M^3;\varrho^{k_1,0})$ or $H_1(M^3;\varrho^{0,k_2})$, for any
$[\gamma]\in H_1(T^2;\ZZ)$, $[\gamma]\neq 0$.
* If $\varrho$ denotes a lift of the holonomy of
the complete structure,
for any pair of peripheral curves $0\neq [\gamma_1],[\gamma_2] \in H^1(T^2;\ZZ)
$ and for $a\in H^0(T^2; \varrho^{k,0})$
a\cup [\gamma_2]=\cs(\gamma_2,\gamma_1) a\cap [\gamma_1]
in $H_1(T^2; \rho^{k,0})$.
* When $k_1k_2\neq 0$, the cap product
$$\cap\colon H^0(T^2;\varrho^{k_1,k_2})\times H_1(T^2;\CC)\to H_1
is the trivial map (i.e. identically zero).
For Item (<ref>), $a\cap [T^2]$ is a basis for $H_ 2(T^2; \varrho^{k_1,k_2})$, by Poincaré duality and Lemma <ref>.
Item (<ref>) is proved in <cit.> for $k_2=0$. It holds true for $k_1=0$ by complex conjugation.
To prove the other two items, we choose a cell decomposition of the torus from a
square with opposite edges identified. Namely
there is a $2$-cell $e^2$ represented by a square, whose sides are two copies of
the $1$-cells: $e_1^1$ and $e_2^1$ (with respective homology classes $[e_1^1]=[{m}]$ and $[e_2^1]=[{l}]$ respectively) and the vertices are four copies of the $0$-cell.
Choose lifts to the universal covering so that
\begin{equation}
\label{eqn:partial}
\partial \tilde e^2= (1-l)\tilde e^1_1 + (m-1)\tilde e^2_1,
\end{equation}
where $m$ and $l$ generate
$\pi_1 T^2$. We may assume that
\varrho(m)=
\pm \begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix}
\qquad
\textrm{ and }
\qquad
\varrho(l)=
\pm \begin{pmatrix}
1 & \eta \\
0 & 1
\end{pmatrix}
\eta =\cs(l,m)\in\CC-\RR
We prove now (<ref>). $\operatorname{Sym}^k$ acts on the space of degree $k$
homogeneous polynomials on $X$ and $Y$; the one dimensional space invariant by $ \varrho^{k}=\varrho^{k,0}$
is generated by $a= X^k$ (Lemma <ref>). By (<ref>):
\partial( X^{k-1}Y\otimes \tilde e^2) = -\eta \, a \otimes \tilde e^1_1 + a \otimes \tilde e^2_1 \\
which in cohomology translates to $-\eta\, a\cap [m]+ a\cap [l]=0$. This proves (<ref>) for a system of
generators $\{[m],[l]\}$ of $H_1(T^2;\ZZ)$, and it holds in general by linearity.
We prove finally (<ref>).
$\operatorname{Sym}^{k_1}\otimes\overline{\operatorname{Sym}^{k_2} }$ acts on the space of degree $k_1$ homogeneous polynomials on $X$ and $Y$,
multiplied by degree $k_2$ homogeneous polynomials on $\bar X$ and $\bar Y$;
the one dimensional space invariant by $ \varrho^{k_1,k_2}$ is generated by $a=
X^{k_1} \bar X^{k_2}$.
By (<ref>):
\begin{align*}
\partial( X^{k_1-1}Y\bar X^{k_2}\otimes \tilde e^2) &= -\eta \, a \otimes \tilde e^1_1 + a \otimes \tilde e^2_1 \\
\partial( X^{k_1}\bar X^{k_2-1}Y\otimes \tilde e^2) &= -\bar \eta \, a \otimes \tilde e^1_1 + a \otimes \tilde e^2_1
\end{align*}
Namely, in homology $-\eta \, a \cap [m] + a\cap [l]=-\bar \eta \, a \cap [m] + a\cap [l]=0$.
As $\eta\not\in \RR$, the claim follows.
§.§ Generic representations in the distinguished component.
Next comes the proof of genericity results used in Paragraph <ref>.
Recall the notation
\rho^{k_1,k_2}:=\operatorname{Sym}^{k_1,k_2} \rho
Recall also that a character $\chi\in X_0(M^3)$ is called $(k_1,k_2)$-exceptional if
there exists a representation $\rho\in\hom(\pi_1M^3,\SL_2(\CC))$ with character
$\chi_\rho=\chi$ such that $H^0(M^3;\rho^{k_1,k_2})\neq 0$.
The set of $(k_1,k_2)$-exceptional characters is denoted by $\EE^{k_1,k_2}$.
If $k_1\neq k_2$, then a $(k_1,k_2)$-exceptional character is reducible. In particular
the $(k_1,k_2)$-exceptional set $\EE^{k_1,k_2}$ is a finite subset of $X_0(M^3)$.
In the holomorphic case ($k_2=0$), assume that $H^0(M^3;\rho^{k})\neq 0$.
Then there is a non-trivial subspace of $\CC^{k+1}$ fixed by $\operatorname{Sym}^k(\overline{\rho(\pi_1M^3)})$, where
$ \overline{\rho(\pi_1M^3)}$ denotes the Zariski closure of $\rho(\pi_1M^3)$. Since $\operatorname{Sym}^k$ is irreducible, this means that
$ \overline{\rho(\pi_1M^3)}$
is not the full group $\SL_2(\CC)$, which in the holomorphic setting means that $\rho$ is reducible.
When $k_2\neq 0$, one can only work with the real Zariski closure, and the previous argument yields that either $\rho$ is reducible or it is
contained in a real subgroup conjugate to $\PSL_2(\RR)$ or $\operatorname{SU}(2)$. The restriction of
$\operatorname{Sym}^{k_1}\otimes \overline{\operatorname{Sym}^{k_2} }$ to those real subgroups is equivalent to
$\operatorname{Sym}^{k_1}\otimes {\operatorname{Sym}^{k_2} }$, which by Klebsh-Gordan formula decomposes as
\begin{equation}
\label{eq:KGExc}
\operatorname{Sym}^{k_1}\otimes {\operatorname{Sym}^{k_2} }=
\operatorname{Sym}^{k_1+k_2} \oplus
\operatorname{Sym}^{k_1+k_2-2} \oplus
\cdots \oplus
\operatorname{Sym}^{\vert k_1-k_2\vert }.
\end{equation}
As $k_1\neq k_2$ the powers of $ \operatorname{Sym}$ in (<ref>) are non-trivial, hence the argument in the holomorphic case applies again.
Recall that we say that
a property is generic when it holds true for a non-empty Zariski open subset of $X_0(M^3)$, and that
the ground field is either $\CC$ or $\RR$, depending on whether the discussion is in the holomorphic setting,
for $\operatorname{Sym}^{k}$, or not, for $\operatorname{Sym}^{k_1,k_2}$.
Let $M^3$ be a hyperbolic manifold with one cusp and let $k_1\neq k_2\in \NN$ be such that $k_1$ or $k_2$
is odd.
Then the set
\{ \chi_\rho\in X_0(M^3) \mid \dim H^0(T^2;\rho^{k_1,k_2})=0\}
is a non-empty Zariski open subset of the curve $X_0(M^3)$.
By upper semi-continuity of the dimension of the cohomology (see the proof of Proposition <ref>), it suffices to show that
$\dim H^0(T^2;\rho^{k_1,k_2})=0$ for some representation $\rho$ with character
in $X_0(M^3)$. When $k_1$ is odd and $k_2$ is even, or vice-versa,
then it holds for the lift of the holonomy of the complete structure, by Lemma <ref>.
If both $k_1$ and $k_2$ are odd, consider an orbifold Dehn filling on $M^3$ with filling curve $\gamma$ that consists in adding a solid torus
with singular core curve
with branching order $n$. By the Dehn filling theorem, for $n$ large enough it is hyperbolic and the restriction $\rho$ of its holonomy has character in $X_0(M^3)$.
The holonomy of the curve $\gamma$ is conjugate to
\rho(\gamma)= - \begin{pmatrix} e^{\frac{\pi i}n} & 0 \\ 0 & e^{-\frac{\pi i}n} \end{pmatrix}.
As the core of the geodesic has non-trivial length, the complex length of its holonomy has nonzero real part. Thus we can find a peripheral curve $\gamma'$ with
\rho(\gamma')= \begin{pmatrix} {\lambda} & 0 \\ 0 & \frac1{\lambda} \end{pmatrix}
with $\lambda$ not real nor unitary. This yields that $\rho^{k_1,k_2}(\gamma')$ has no eigenvalue $1$, hence $H^0(T^2;\rho^{k_1,k_2})=0$.
Next lemma implies Lemma <ref>, as $\FF^{k_1,k_2}= (X_0(M^3)-Z^{k_1,k_2} )\cup \EE^{k_1,k_2}$.
Let $M^3$ be a hyperbolic manifold with one cusp and let $k_1\neq k_2\in \NN$ be such that $k_1$ and $k_2$ are even.
* The set of characters
Z^{k_1,k_2}=\{ \chi_\rho\in X_0(M^3) \mid \dim H_0(T^2;\rho^{k_1,k_2})=1\}
is a non-empty Zariski open subset of the curve $X_0(M^3)$.
For any $[\gamma]\in H^1(T^2;\ZZ)$, $[\gamma]\neq 0$, the set
Z^{k_1,k_2}_\gamma=\{\chi_\rho\in Z^{k_1,k_2}\mid i_*(a\cap [\gamma]) \textrm{ is a basis for } H_1(M^3;\rho^{k_1,k_2})\}
is non-empty and Zariski open.
For (<ref>), given any non-trivial peripheral element $\gamma\in\pi_1 T^2$,
for a generic character $\chi_\rho\in X_0(M^3)$, $\rho(\gamma)$ is a diagonal matrix, with eigenvalues
different from $\pm 1$ and so that $\rho^{k_1,k_2}(\gamma)$ has only one eigenvalue equal to one.
This property is in fact Zariski open (over $\CC$ when $k_2=0$, over $\RR$
otherwise), cf. the proof of Lemma <ref>.
Next comes the proof of (<ref>). Any nonzero element in $ H^1(T^2;\ZZ)$ is represented by a simple closed curve
$\gamma$. Consider the orbifold $\mathcal O_n$ obtained by Dehn filling $M^3$ along $\gamma$, so that the
soul of the solid torus is the branching locus, with branching index $n\in\NN$. For $n$ sufficiently large,
$\mathcal O_n$ is hyperbolic, and its holonomy restricts to a representation with character $\chi_\rho\in X_0(M^3)$.
As $\operatorname{Sym}^{k_1}\otimes \overline{\operatorname{Sym}^{k_2}}$ of the holonomy of $\mathcal O_n$ is acyclic,
a Mayer-Vietoris argument yields that $a\cap\gamma$ is a basis for $H^1(M^3;\rho^{k_1,k_2})$. Then the argument
for a generic $\rho$ follows from semi-continuity.
Notice that for all but finitely many curves $ \gamma$ we may take a trivial branching index $n=1$ and work with manifolds instead of orbifolds.
However, in order to consider all filling slopes, one needs to work with orbifolds.
For any character $\chi_\rho$ in $ Z^{k_1,k_2}_\gamma$, by the long exact sequence in cohomology and Poincaré duality,
the inclusion induces an isomorphism, $H_2(M^3;\rho^{k_1,k_2})\cong H_2(T^2;\rho^{k_1,k_2})$. In addition, Poincaré
duality again gives a natural isomorphism $ H_2(T^2;\rho^{k_1,k_2})\cong H^0(T^2;\rho^{k_1,k_2})$ which yields that $a\cap [T^2]$
is a basis for $ H_2(T^2;\rho^{k_1,k_2})$. Thus Lemma <ref>(b) yields Proposition <ref>.
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1511.00069
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CHAPTER: EXACT SOLUTION OF THE PLANAR MOTION OF THREE
ARBITRARY POINT VORTICES
Service de physique du solide et de résonance magnétique,
CEN Saclay, France. DPhG/PSRM/1697/80
We give an exact quantitative solution for the motion of three vortices of any strength,
which Poincaré showed to be integrable. The absolute motion of one vortex is
generally biperiodic: in uniformly rotating axes, the motion is periodic. There are two
kinds of relative equilibrium configuration: two equilateral triangles and one or three
colinear configurations, their stability conditions split the strengths space into three
domains in which the sets of trajectories are topologically distinct.
According to the values of the strengths and the initial positions, all possible
motions are classified. Two sets of strengths lead to generic motions other than
biperiodic. First, when the angular momentum vanishes, besides the biperiodic regime
there exists an expansion spiral motion and even a triple collision in a finite time, but
the latter motion is nongeneric. Second, when two strengths are opposite, the system also
exhibits the elastic diffusion of a vortex doublet by the third vortex.
For given values of the invariants, the volume of the phase space of this
Hamiltonian system is proportional to the period of the reduced motion, a well
known result of the theory of adiabatic invariants. We then formally examine the
behaviour of the quantities that Onsager defined only for a large number of
interacting vortices.
§ INTRODUCTION
Known from a long time, the problem of the motion of a system of point vortices in
interaction presents a particular interest for the study of bidimensional
turbulence. As predicted by Onsager (1949) the study of the thermodynamics of a
large system of vortices shows the possibility of
temperatures and eddy
viscosities (see e.g. Lundgren and Pointin 1977). The case of a small number of
vortices was studied, long ago, by Lord Kelvin and Mayer (1878). They used
experimental methods to obtain some results on the stability of simple geometric
configurations. Recently Novikov (1975) pointed out the interest of the three
vortex system as the first elementary interaction process in isotropic turbulence
kinetics. Using properties of the triangle he was able to solve the problem of
three identical vortices.
While this work was under submission, the referees pointed out to us the existence
of a paper to appear. In this paper, Hassan Aref (March 1979), extending the work
of Novikov with a symmetry-preserving presentation, qualitatively solved the
relative motion of three vortices and undertook a classification of the topology
of the phase space; however, he gave no indication on the nature of the absolute
motion and no quantitative results, except for two special cases of great
importance which he solved completely: the direct or exchange scattering when two
strengths are equal and opposite to the third one, and the self-similar motion of
a triple collision or a triple expansion to infinity when both the inertia
momentum and the angular momentum vanish.
In this paper, we present a method which quantitatively gives the absolute motion
in all cases of strengths or initial conditions. The main idea is to introduce
reduced variables with a very simple geometrical interpretation and whose number
matches the number of degrees of freedom of the system. For three vortices, this
can be done by defining one complex variable $\zeta$ which characterizes the shape
of the triangle. Knowing the motion of $\zeta$, which happens to be periodic, is
then sufficient to derive the behaviour of any physically interesting variable. We
thus have obtained detailed results, which resort mainly to fluid mechanics and
partly to the field of differentiable dynamical systems.
In the first part, we describe the problem, introduce the reduced motion and
discuss its advantages and inconvenients. In the second part, we give the general
solution: generically, i.e. for arbitrary strengths and initial positions, the
absolute motion of any given vortex is the product of a periodic motion of period
$T$ by a uniform rotation or, stated in other words, the motion is periodic when
referred to uniformly rotating axes centered at the center of vorticity; in the
case of a vortex plasma (zero total strength), the uniform rotation is merely
replaced by a uniform translation. Due to its frequent occurrence in nature, we
give special consideration to the case of two or three equal strengths: in a given
domain of initial conditions, vortices with equal strengths have the same motion,
up to a shift of $\frac{T}{2}$ or $\frac{T}{3}$ in time and a rotation in space.
The third part, following the Smale's method of study of a differentiable
dynamical system, is devoted to the finding of the bifurcation set, i.e. the set
of values of the invariants for which the nature of the motion changes. This
reduces to the finding of the relative equilibria, which are shown to be of two
types, the same than those of the three body problem of celestial mechanics: two
equilateral triangles and one or three colinear configurations. The number and
stability of these relative equilibria are discussed, thus leading to a separation
of the strengths space in three principal domains where the sets of orbits are
topologically different; an important result is that, whatever be the strengths
domain, the phase space is always multiconnected; for strengths on the boundaries
of these domains, the motion must be studied separately for it presents special
In the fourth part, we study the absolute motion whenever it differs from the general
biperiodic case, i.e. when strengths are on the boundaries of their limiting domains or
when initial positions are those of a relative equilibrium. At a relative equilibrium the
motion is a rigid body rotation. Otherwise two new generic motions are found. First, for
strengths such that the angular momentum vanishes, the vortices can go to infinity in a
spiral motion with a fixed asymptotic shape of the triangle; there even exists a spiral
motion ending in a triple collision in a finite time, but it is nongeneric since it
happens only for a zero value of the inertia momentum. Second, when two strengths are
opposite, besides the doubly periodic motion an elastic scattering can occur in which a
vortex doublet is diffused by the third vortex; if moreover the third vortex has the
strength of one of the two others, a third generic motion exists which is an exchange
The fifth part consists in formally examining the behaviour of the thermodynamical
quantities, of course meaningless for this integrable system, that Onsager defined
for a large number of vortices; the volume of the phase space is found to be equal
to the period of the relative motion; among the features which could be indicative
for a large number of vortices is a possible lack of ergodicity due to a multiple
connexity of the phase space.
§ DESCRIPTION OF THE PROBLEM
We consider three point vortices $M_{j}$ $(j=1,2,3)$ in a plane with strengths
$\kappa_{j}$ and given initial positions. Their motion is ruled by the first order
differential system:
\begin{equation}
\forall\,j=1,2,3\enskip {:}\enskip 2\pi i\frac{\D\bar{z}_{j}}{\D t}=\sum_{\substack{\ell=1\\
\ell \ne j}}^{3}\frac{\kappa_{\ell}}{z_{j}-z_{\ell}},
\end{equation}
where $z=x\,{+}\,iy$ and the bar denotes the complex conjugation. This system is
equivalent to the set of Hamilton equations for the Hamiltonian:
\begin{equation}
j}}\vphantom{\sum}_{\substack{\\ \\ \\ \\ <}}\sum_{\substack{\ell\\[2pt]
\ell}}\kappa_{j}\kappa_{\ell}\enskip{\rm Log}|z_{j}-z_{\ell}|,
\end{equation}
in the conjugate variables $\sqrt{|\kappa_{j}|}x_{j}$ and $\sqrt{|\kappa_{j}|}$
sign $(\kappa_{j})y_{j}$, $j=1,2,3$; therefore $H$ is an invariant and remains
equal to the energy $E$. The invariance of $H$ under translation and rotation
yields two other invariants:
\begin{eqnarray*}
&&\hbox{the impulse } B=\sum_{j}\kappa_{j}z_{j}=X+iY,\\[4pt]
&& I=\sum_{j}\kappa_{j}|z_{j}|^{2},
\end{eqnarray*}
which Poincaré called inertia momentum.
In fact we shall use, instead of $I$, the invariant
\begin{equation}
j}}\vphantom{\sum}_{\substack{\\ \\ \\ \\ <}}\sum_{\substack{\ell\\[2pt]
\ell}}\kappa_{j}\kappa_{\ell}|z_{j}-z_{\ell}|^{2}=(\kappa_{1}+\kappa_{2}+\kappa_{3})I-|B|^{2},
\end{equation}
which depends only on the relative positions of the vortices. There are $6-4=2$ remaining
degrees of freedom and we suppose the problem to be nondegenerate:
$\kappa_{1}\kappa_{2}\kappa_{3}(z_{2}-z_{3})(z_{3}-z_{1})(z_{1}-z_{2})\ne 0$.
Among the six Poisson brackets built from the four known integrals of motion $H,J,X,Y$,
only one is nonzero:
\[
\{X,Y\}=\sum_{j}\kappa_{j}=K.
\]
Since a Hamiltonian system with $2N$ variables is integrable in the sense of Liouville
when it has $N$ independent invariants in involution, the three vortex system $(N=3)$ is
therefore integrable (Poincaré, 1893). The purpose of this paper is to integrate it.
Let us remark that this problem is not affected by the adjunction of an external
velocity field made of a uniform translation and a uniform rotation, since the new
equations of motion
\[
i\frac{\D\bar{z}_{j}}{\D t}=\sum_{\ell}'\frac{\kappa_{\ell}}{z_{j}-z_{\ell}}+2\pi
i\bar{v}+2\pi \omega\bar{z}_{j},\quad \omega \in {\mathbb R},\quad v \in {\mathbb
\]
reduce to the original ones by the change of variables
\[
Z_{j}=\frac{iv}{\omega}+\left(z_{j}-\frac{iv}{\omega}\right)e^{-i\omega t}.
\]
§.§ The reduced motion
The main idea is to match the number of variables and the number of degrees of
freedom of this system, so as to keep the minimum number of independent variables.
For this purpose, let us define a time dependent complex plane $\zeta$ in which
two of the three point vortices remain fixed. This can be achieved by the
following transformation
\begin{equation}
\zeta=\frac{(\kappa_{2}+\kappa_{3})z-(\kappa_{2}z_{2}+\kappa_{3}z_{3})}{z_{2}-z_{3}},
\end{equation}
where 2 and 3 number two vortices whose sum of strengths is nonzero; the affixes of these
two vortices become $\kappa_{3}$ and $-\kappa_{2}$ under the transformation.
It is clearly seen from the definition how a reduced point is geometrically
deduced from an absolute point. We shall simply note $\zeta$ the transformed
of $M_{1}$:
\[
\zeta=\xi+i\eta=\frac{Kz_{1}-B}{z_{2}-z_{3}},
\]
$\zeta$ therefore represents the shape of the triangle, and the inverse
transformation is represented by
\begin{gather}
z_{1}=z_{1},\quad z_{2}=\frac{(\kappa_{3}K-\kappa_{1}\zeta)z_{1}+(\zeta-\kappa_{3})B}{s\zeta},\nn\\[4pt]
\end{gather}
where $s=\kappa_{2}+\kappa_{3}\ne 0$.
The reader will have noticed the main disadvantage of the above definition of two reduced
coordinates $\xi,\eta$: it does not reflect the invariance of the problem under the
permutations of the three elements $(\kappa_{j},z_{j})$ and therefore every result we can
get may be uneasy to interpret. Nevertheless, the advantages are numerous. First, every
physical quantity can systematically be expressed, as we shall soon see, as a function of
$\zeta$ and $Kz_{1}-B$ only; moreover, since $\zeta$ is invariant under a change of
length and $Kz_{1}-B$ extensive in the lengths, such a physical function will quite
generally be the product of a function of $\zeta$ by a function of $Kz_{1}-B$ and we are
going to see that this uncoupling between intensive and extensive variables will enable
us to solve the motion not only qualitatively but also quantitatively. Secondly, unlike
Novikov and Aref, we do not have to eliminate some unphysical portions of our $\zeta$
plane (which will be seen to be the initial conditions plane) since every $\zeta$ point
describes a physical situation. Thirdly, this reduction leaves only the required number
of degrees of freedom.
§ THE SOLUTION FOR THE GENERAL CASE
To obtain the absolute motion we need only determine the evolution of $\zeta$ and
$z_{1}$, since we have the parametric representation (5). Provided $K$ and $B$ do
not simultaneously vanish the motions of $z_{1}$ and $\zeta$ are ruled by:
\begin{eqnarray}
2\pi i\frac{\D\bar{z}_{1}}{\D t}&=&\frac{1}{Kz_{1}-B}\frac{s^{2}\zeta(\zeta+d)}{(\zeta+\kappa_{2})(\zeta-\kappa_{3})},\\[3pt]
2\pi i\frac{\D\bar{\zeta}}{\D t}&=&\frac{-1}{|Kz_{1}-B|^{2}}
\frac{s|\zeta|^{2}[\bar{\zeta}(\zeta^{2}+d\zeta-Q)-sK(\zeta+d)]}
\end{eqnarray}
where $d=\kappa_{2}-\kappa_{3}$ and
We can express the invariants $E$ and $J$ as functions of $z_{1}$ and $\zeta$:
\begin{gather}
\left|\frac{\zeta+\kappa_{2}}{s}\right|^{2\kappa_{1}\kappa_{3}},
\end{gather}
expressions where we notice the factorized dependency on $\zeta$ and $z_{1}$.
Unless $Q$ and $J$ simultaneously vanish, at least one of the two above equations
expresses $|Kz_{1}-B|$ as a function of $\zeta$ and from (7) we obtain a first order
differential system for $\zeta$. For example if $J$ is nonzero the elimination of $z_{1}$
between (7) and (8) gives:
\begin{equation}
i\frac{\D\bar{\zeta}}{\D t}=\frac{-(\kappa_{1}|\zeta|^{2}+\kappa_{2}\kappa_{3}K)[\bar{\zeta}(\zeta^{2}+d\zeta-Q)-sK(\zeta+d)]}
\end{equation}
There is no need to solve this system since the equation of the reduced
trajectory is given by the straightforward elimination of $z_{1}$ between (8)
and (9):
\begin{equation}
\frac{\kappa_{1}|\zeta|^{2}+\kappa_{2}\kappa_{3}K}{sQ}\left|\frac{\zeta-\kappa_{3}}{s}\right|^{-2\frac{\kappa_{1}\kappa_{2}}{Q}}
\left|\frac{\zeta+\kappa_{2}}{s}\right|^{-2\frac{\kappa_{1}\kappa_{3}}{Q}}=\frac{J}{Q}e^{4\pi\frac{E}{Q}}.
\end{equation}
This represents in the $\zeta$ plane a set of closed orbits, indexed by the
non-dimensional variable $c=\frac{J}{Q}e^{\frac{4\pi E}{Q}}$ which is invariant
under a change of length or a change of unit of vorticity.
The strengths space. The numbering of regions reflects the ternary
The set of orbits is symmetrical relative to the $\xi$ axis and also, when
$\kappa_{2}$ equals $\kappa_{3}$, to the $\eta$ axis. When two vortices are close
to each other, they remain as such and therefore the $\zeta$ curves are near to
circles in the vicinity of $-\kappa_{2}, \kappa_{3}$ and $\infty$.
Figures <ref> to <ref> show examples of the $\zeta$ plane.
$\zeta$ plane in the domain 351 ($Q<0$, $\Delta > 0$)domain 351$\kappa_{1}=-2,\kappa_{2}=1,\kappa_{3}=4$.
$\zeta$ plane in the domain 1 ($Q>0$, $\Delta < 0$) domain 1$\kappa_{1}=-2,\kappa_{2}=5,\kappa_{3}=9$.
$\zeta$ plane in the domain 12 ($Q<0$, $\Delta < 0$) domain 12$\kappa_{1}=-6,\kappa_{2}=7,\kappa_{3}=11$.
$\zeta$ plane on the line $Q=0$. The nonperiodic domain is hatched $Q=0$ domain
(12 and 1) $\kappa_{1}=-2,\kappa_{2}=3,\kappa_{3}=6$.
$\zeta$ plane on the line (11 and 12). The diffusion domain is hatched
domain (11 and 12) $\kappa_{1}=-2,\kappa_{2}=2,\kappa_{3}=3$.
Since there is in general no stationary point on the orbit, the reduced motion is
periodic and the period is expressed by
\begin{equation}
T=-\frac{\pi J}{s}\oint\frac{|(\zeta+\kappa_{2})(\zeta-\kappa_{3})|^{2}}
{(\kappa_{1}|\zeta|^{2}+\kappa_{2}\kappa_{3}K)^{2}}\frac{\D |\zeta|^{2}}{{\rm
\end{equation}
When $J$ is zero we use (9) instead of (8) to obtain a similar expression, a
result which shows that the condition $J=0$ alone represents nothing special as
one would believe in the Aref classification (for more details see the third part
of this paper). To have a dimensionless result we can take as unit of time
$T_{u}=\frac{4\pi^{2}J}{QK}$ which is, as we shall see later, the period of the
absolute motion when the vortices are in a configuration of relative equilibrium;
therefore, in every domain of the multiply-connected $\zeta$ plane,
$\frac{T}{T_{u}}$ depends only on $c$.
§.§ The absolute motion
Using the parametric representation of the $z_{j}$'s we easily obtain the
following relations:
\begin{equation}
\frac{Kz_{1}-B}{s\zeta}=\frac{Kz_{2}-B}{\kappa_{3}K-\kappa_{1}\zeta}=
\frac{Kz_{3}-B}{-\kappa_{2}K-\kappa_{1}\zeta}.
\end{equation}
Then, using (8) or (9), we conclude that, when the reduced motion is periodic, the
modulus of $z_{j}-\frac{B}{K}$ has the period $T$ of the reduced motion. As to the
arguments of these affixes, after one period they have all been increased by the
same value, modulo $2\pi$:
\begin{equation}
\left\{\begin{array}{@{}l@{}}
\ds \Delta\varphi_{1}=\left[\arg\left(z_{1}-\frac{B}{K}\right)\right]_{o}^{T}\\[14pt]
\ds \qquad\;=\oint \frac{-K {\rm Re}\{(\zeta^{2}+d\zeta)(\bar{\zeta}^{2}+d\bar{\zeta}-\kappa_{2}\kappa_{3})\}}{2|\zeta|^{2}(\kappa_{1}|\zeta|^{2}+\kappa_{2}\kappa_{3}K){\rm Im}(\zeta^{2}+d\zeta)}\D|\zeta|^{2}=\Delta\varphi,\\[14pt]
\ds \Delta\varphi_{2}=\Delta\varphi_{1}+\left[\arg\left(\frac{\kappa_{3}K-\kappa_{1}\zeta}{s\zeta}\right)\right]_{o}^{T}=\Delta \varphi\ {\rm modulo}\ 2\pi,\\[14pt]
\ds \Delta\varphi_{3}=\Delta\varphi_{1}+\left[\arg\left(\frac{-\kappa_{2}K-\kappa_{1}\zeta}{s\zeta}\right)\right]_{o}^{T}=\Delta \varphi\ {\rm modulo}\ 2\pi.\\
\end{array}\right.
\end{equation}
For the motion it means that, after a time interval of $T$, the shape and size of the
triangle are again the same, i.e. the new positions are deduced from their initial values
by a rotation of $\Delta \varphi$ around the center of vorticity. Depending on the domain
of initial conditions, the number of turns around the barycentrum may vary from one
vortex to another by an integer value. Therefore, when $K$ is nonzero, the absolute
motion of any vortex is the product of a uniform rotation about the barycentrum and of a
periodic motion, the two periods being the same for the three vortices:
\[
\forall\,j=1,2,3\enskip{:}\enskip
\varphi\frac{t}{T}}f_{j}\left(\frac{t}{T}\right),
\]
where $f_{j}$ is periodic with period 1. In other words, in uniformly rotating
axes centered at the barycentrum, the absolute motion is periodic.
Figure <ref> shows the absolute trajectory of $M_{1}$ both in fixed axes and
in rotating axes, for $\vec{\kappa}=(-2,1,4)$ and $\zeta_{o}=-\frac{3K}{2}$ (the
$\zeta$ orbit is the small curve surrounding $P$ in Figure <ref>).
Volume $\Omega$ and temperature $\tau$ versus energy in the domain $0(k_{j}K>0)$.
Data for these qualitative figures are taken from
Absolute motion $z_{1}(t)$ (left) and rotating motion $z_{1}(t)$
$e^{-i\Delta\varphi\frac{t}{T}}$ (right), $-T < t < T$, for arbitrary strengths and
initial conditions: $\vec{\kappa}=(-2,1,4)$, $\zeta_{o}=-\frac{3K}{2}$.Data are: $c=1.726$, $\frac{T}{T_{u}}=0.262$,
Absolute motion of the three vortices for an elastic diffusion.
$\vec{\kappa}=(-2,2,3)$, $\zeta_{o}=-\frac{20}{21}K$ $(c=1.170)$.
Some symmetries may exist in the absolute motion: for instance if the $\zeta$ axis is an
axis of symmetry of the $\zeta$ orbit, then at intervals distant of $\frac{T}{2}$ the
vortices are colinear; if we choose for origin of time an instant of colinearity, the
absolute trajectory of every vortex, when run between 0 and $T$, possesses an axis of
symmetry (see Figure <ref>).
-1Let us now mention an important result concerning this dynamical
system: since integral (14) is a continuous function of both the strengths
$\frac{\vec{\kappa}}{K}$ and the initial conditions, the angle $\Delta \varphi$ is
generically incommensurable with $2\pi$, i.e. commensurability occurs only for a
set of strengths and initial conditions of zero measure. We conclude that
generically, due to the incommensurability of $\Delta \varphi$ with $2\pi$, the
trajectory of a given vortex completely fills an annulus centered at the
barycentrum (see Landau and Lifchitz Figure <ref> Chap. III).
In the case of a vortex plasma $(K=0)$ and when the barycentrum is at infinity
$(B\ne 0)$, the variation of $z_{j}$ over one period does not depend on $j$ and is
given by
\begin{equation}
\frac{i}{B}[\bar{z}_{j}]_{o}^{T}=\oint\frac{(\zeta^{2}+d\zeta)(\bar{\zeta}^{2}+d\bar{\zeta}-\kappa_{2}\kappa_{3})}
{2\kappa_{1}|\zeta|^{4}\;{\rm Im}(\zeta^{2}+d\zeta)}\D|\zeta|^{2},
\end{equation}
which evaluates to a real quantity. The rotation has become a uniform translation
in the direction normal to the direction of the impulse. The common mean velocity
of the vortices is $[z_{1}]/T$ and, for instance in the case of an initial
equilateral triangle where the absolute motion is a uniform translation, this
velocity evaluates to $\frac{iQ}{2\pi B}$.
Let us conclude all that with a geometrical remark: the relations
\[
\kappa_{1}|Kz_{1}-B|^{2}+ \kappa_{2} \kappa_{3}K|z_{2}-z_{3}|^{2}=sJ,
\]
\[
2\pi\frac{\D|z_{2}-z_{3}|^{2}}{\D t}=\frac{2\kappa_{1}s^{2}\;{\rm Im}(\zeta^{2}+d\zeta)}
\]
show that the extrema of $|z_{j}-\frac{B}{K}|$ occur simultaneously with those of
$|z_{\ell}-z_{m}|$ $(j,\ell,m$ permutation of $1,2,3$) and that they are reached
when the triangle is either flat or isosceles with $M_{j}$ as a summit.
§.§ The absolute motion for two or three equal strengths
An important practical case is that of the invariance of a $\zeta$ orbit under one
of the six permutations of the three elements $(\kappa_{j},z_{j})$. For instance
the permutation (132) acts by: $\kappa_{2}\to \kappa_{3}$, $\kappa_{3}\to
\kappa_{2}$ and is seen to leave the set of $\zeta$ orbits invariant provided
$\kappa_{2}=\kappa_{3}$. Since most of the vortices encountered in nature have
equal, or opposite, strengths, we shall consider this particular case in some
-1Let us therefore assume $\kappa_{2}=\kappa_{3}$ and consider a $\zeta$
orbit having the origin as a center of symmetry (e.g. the circle $J=0$ on the
Figure <ref> assumed continuously deformed so as to admit the origin as a
center of symmetry; note that, on the same figure, the orbit surrounding $T_{1}$
has not the required property). Having chosen such an initial condition $t=0$,
$\zeta=\zeta_{o}$, during the evolution there will happen a time when $\zeta$
evaluates to $-\zeta_{o}$; this time is necessarily equal to the half-period $T/2$
and, since the states ($t=0$, $\zeta=\zeta_{o}$) and $(t=\frac{T}{2}$,
$\zeta=-\zeta_{o}$) are identical from a point of view of initial conditions, we
conclude to the identity of the motions for $0< t < \frac{T}{2}$ and for
$\frac{T}{2}<t<T$; the triangle at $t=\frac{T}{2}$ is deduced from the triangle at
$t=0$ by some fixed rotation around the barycentrum (a change of size and a
translation are excluded because of the invariants) and of course by the exchange
of 2 and 3 $(B=0)$:
\begin{gather*}
\forall\,t\enskip{:}\enskip z_{1}(t)=z_{1}\left(t-\frac{T}{2}\right)e^{i\alpha},\quad z_{2}(t)=z_{3}\left(t-\frac{T}{2}\right)e^{i\alpha},\\
\end{gather*}
By iterating, we see that $2\alpha$ is equal to $\Delta \varphi$ modulo $2\pi$.
Therefore the absolute motions of 2 and 3 are identical, up to a rotation in space
and a translation in time. Every absolute trajectory has two independent axes of
symmetry and therefore, an infinity: they are defined by the barycentrum and the
absolute positions of the vortex when the triangle is either isosceles or flat
(i.e. when $\zeta$ crosses one of its two symmetry axes).
If $K$ is zero, i.e. $\vec{\kappa}=(-2,1,1)$, the conclusions are similar, with a
translation of $L/2$ instead of a rotation, $L$ being the translation after one
period; the initial conditions which have the required symmetry are defined by
Let us now study the absolute motion of three identical vortices. The set of
orbits is invariant under the six permutations, but a given orbit is only
invariant under two or three of them. Two regimes are found:
(1) $\sqrt[3]{2}<c$. The shortest of the three mutual distances is always the same,
say $M_{2}M_{3}$, the $\zeta$ orbit is invariant by a permutation of 2 and 3 and
the above conclusions apply. The three vortices are colinear at intervals of
$T/2$. Choosing for $t=0$ a colinear configuration, the triangle is isosceles at
$t=\frac{T}{4}+n\frac{T}{2}$, $n\in Z$ and the summit of the triangle is then
always the same vortex $M_{1}$.
(2) $1<c<\sqrt[3]{2}$, i.e. a vicinity of the equilateral configurations. The $\zeta$
orbit is invariant under any circular permutation and, using quite similar
arguments, we deduce for instance $(B=0)$:
\begin{gather*}
\forall\,t\enskip{:}\enskip
\end{gather*}
with $3\alpha=\Delta \varphi$ modulo $2\pi$. This time, the three motions are
identical. The vortices are never colinear. Choosing for $t=0$ an isosceles
configuration, we see that the triangle regains the same shape at
$t=n\frac{T}{3}$, successively with the summits $1,2,3$ (or $1,3,2$ depending on
the initial conditions), and a second isosceles shape at
The period $T$, that Novikov gave as a hyperelliptic integral, is reducible to an
ordinary elliptic integral (see Appendix I).
§ THE RELATIVE EQUILIBRIA AND THE BIFURCATION SET
In two fundamental papers linking topology to mechanics, Smale (1970) explains how
to split the study of any dynamical system into two simpler problems. He first
defines the integral manifolds as the set of points in the phase space with given
values of the invariants, or better as the quotient of that set by the symmetry
group of the system. Then the first problem is to find the topology of the
integral manifolds of the phase space and more precisely to find the bifurcation
set, i.e. the set of values of the invariants $(E, J)$ for which this topology,
and therefore the nature of the motion, changes. The second problem, which has
been solved above at least in the general case, is the study of the dynamical
systems on the integral manifolds.
Since we do not want to insist on the mathematics, we shall only give the
bifurcation set, i.e., for every value of the strengths, we shall determine the
values of $E$ and $J$ which cause a qualitative change in the absolute motion;
such a research will introduce separating lines in the strengths space.
Our discussion will therefore take place in two different spaces: the space of
strengths (parameter space) and the space of invariants, spaces which we are now
going to describe in more detail.
Due to the homogeneity of the equations of motion, the strengths space may be
represented by its section by a plane $K$ $=$ constant and, in this plane, by a
figure invariant under a rotation of $\frac{2\pi}{3}$ around the point
$\kappa_{1}=\kappa_{2}=\kappa_{3}$. Its dimension is therefore 2 and we shall
represent a point by its polar coordinates:
\begin{equation}
\rho\,\cos\theta=\frac{\sqrt{3}(\kappa_{2}-\kappa_{3})}{2K},\quad
\rho\,\sin\theta=\frac{-2\kappa_{1}+\kappa_{2}+\kappa_{3}}{2K},
\end{equation}
$\theta$ describing any interval of amplitude $\frac{2\pi}{6}$ (see
Figure <ref>).
Summary of results.
12pt 3cdegeneracy
strengths domain sign (Q) sign ($\Delta$) $c=-\infty$ $c=0$ $c=1$ $c=+\infty$ topology
0 $+$ $-$ 0 0 $2,1,2,3$ H M H M H
1 $+$ $-$ 1 $3,2,1,2$ 0 H M H M H
1 0 $-$ $-$ $0,1,2,1$ 2 2 M E H E M
1 1 $-$ $-$ $1,2,1$ 2 2, 1 E M H E M
1 2 $-$ $-$ $2,1$ 2 2, 1, 0 E M H M E
1 4 0 $-$ $+$ $0,1$ 2 2 M E M
1 5 0 $-$ $+$ 0 1, 2 2 M E M
1 5 1 $-$ $+$ 1 2 2, 1 M E M
$K=0$, $\kappa_{1}<0<\kappa_{2}<\kappa_{3}$ $-$ $+$ 0 2 2 M E M
$\Delta=0$ (11 and 351) $-$ 0 1, 1 2 2, 1 E M P M ȷH
$\Delta=0$ (10 and 140) $-$ 0 0, 1, 1 2 2 M E P M ȷH
$\prod (\kappa_{j}+\kappa_{l})=0$ (11 and 12) $-$ $+$ 2, 1 2 2, 1 E M H X ȷH
id. (10 and 11) $-$ $+$ 1, 2, 1 2 2 X H E M ȷH
id. (140 and 351) $-$ $-$ 1 2 2 X M
$Q=0$ (1 and 12) 0 $-$ 0 2, 1, 2, 4 1 M H M
point B $(-5,4,4)$ $-$ 0 0, 1 2 2 M Q M
point A $(-1,1,1)$ $-$ $-$ 2, 2 3 2 X H X
point C $(-2,\sqrt{3},2)$ $-$ 0 1, 1 2 2 X P M
The strengths domains are defined in Figure <ref>. Under the heading
“degeneracy” are listed by interval of $c$ the number of different states, i.e.
of $\zeta$ orbits, having the same value of $c$; for instance, in the domain 11,
the limiting values of $c$ are $-\infty,0,+\infty$ (at the points $M_{j}$), 1 (at
the triangles), $c_{1}, c_{2},c_{3}$ (at the colinear relative equilibria), they
define 6 intervals, hence a sequence of 6 degeneracy numbers; when $Q$ is zero,
$e^{4\pi E/K^{2}}$ is used in place of $c$ for the classification.
The column “topology” lists the sequence of the nature of the real
remarkable $\zeta$ points along the real axis: $M$ stands for a point $M_{j}$ (we
omit $M_{1}$ at infinity), $E, H, P$ for elliptic, hyperbolic, parabolic, $X$ for
an $M$ coinciding with $E$ or $H$ and $Q$ for the only higher-order point we
found. The information on the nature of the triangles is contained in the column
sign $(Q)$. All this is sufficient to draw every $\zeta$ plane.
For given values of the strengths, the space of invariants is a priori
bidimensional, since the center of vorticity is not a relevant invariant except
when $K$ is zero. Let us compare this space to the space of initial conditions. As
we have seen, an initial condition is an orbit in the $\zeta$ plane, since two
sets of absolute positions $z_{i}$ whose $\zeta$ values belong to the same $\zeta$
orbit evolve in the same absolute motion, up to a translation in time and a
translation, rotation and scale change in space. On a given orbit, $c$ is
constant, but, inversely, the equation $c=\hbox{cst}$ represents a finite number (between
0 and 4, see Table <ref>) of orbits. From this fact, we draw two conclusions:
first, the space of invariants $(E, J)$ is in fact of dimension one, two points
being identified if they lead to the same value of
$c=\frac{J}{Q}e^{4\pi\frac{E}{Q}}$; second, an initial condition is characterized
by, and therefore equivalent to, a value of the invariant $c$ plus an index of
region in the $\zeta$ plane (or of sheet in the phase space). We can therefore
identify the space of initial conditions to the product of the one-dimensional
space of invariants by the finite set of the indices of region. Accordingly, the
most precise graphical representation will be the orbits of the $\zeta$ plane, but
we shall also use for simplicity a plane $(E, J)$ or an axis $c$ with some
handwritten information on it to take into account the integer index.
Let us now proceed to the determination of the bifurcation set. We exclude the set
of collisions, represented in the $\zeta$ plane by three points of affixes
$\infty$, $\kappa_{3},-\kappa_{2}$ where $c$ evaluates to $+0$, $-0$, $+\infty$ or
$-\infty$ depending on the signs of $J, Q$ and the strengths; these limiting
values of $c$ will therefore belong to the bifurcation set.
The nature of the motion (at least its topological nature since it is always
generically biperiodic) changes when there is a stationary point on the $\zeta$
orbit; such points, where the velocity of $\zeta$ vanishes, are given by the
solutions of the complex equation
\begin{equation}
\bar{\zeta}(\zeta^{2}+d\zeta-Q)-sK(\zeta+d)=0,
\end{equation}
equivalent to
\[
\frac{K\bar{z}_{1}-\bar{B}}{z_{2}-z_{3}}+
\frac{K\bar{z}_{2}-\bar{B}}{z_{3}-z_{1}}+
\frac{K\bar{z}_{3}-\bar{B}}{z_{1}-z_{2}}=0.
\]
These points are also the critical points of the function $(\xi,\eta)\to c$ and
they give all the relative equilibrium configurations of the three vortices, i.e.
the states for which the system moves generally like a rigid body. The solutions
of (17) are:
(a) one or three colinear configurations, defined by the real zeros
of $P(\zeta)\equiv \zeta^{3}+d\zeta^{2}-(sK+Q)\zeta-sKd$ (points
$P_{1},P_{2},P_{3}$ of Figures <ref> to <ref>),
(b) two equilateral triangles, defined by the zeros of $\zeta^{2}+d\zeta+sK-Q$, i.e.
$\zeta=\frac{-d\pm is\sqrt{3}}{2}$ (points $T_{1}$ and $T_{2}$ of
Figures <ref> to <ref>); these stationary triangles, already known to
Kelvin for identical vortices, therefore exist for vortices of any strengths.
(c) When $Q=0$, the isolated zero $\zeta=-d$ of $P$ and every point of the
circle $|\zeta|^{2}=sK$ on which lie the summits of the equilateral triangles and
the two other real zeros of $P$. On this circle $J$ is equal to zero.
The corresponding absolute motions will be described later. It is interesting to
remark that, except for some values of the strengths like for instance $Q=0$, the
relative equilibria of the three vortex system are qualitatively the same than
those of the three body problem of celestial mechanics, where there are two
equilateral triangles and three colinear configurations; in fact, a simple
geometric reasoning shows the same qualitative composition of the set of relative
equilibria for every planar three body motion ruled by a two body central
The study of the stability of the relative equilibria is given in Appendix II and
only three different behaviours are found (in celestial mechanics, only two cases
arise: the three colinear configurations are always unstable, the two triangles
are stable for $(m_{1}+m_{2}+m_{3})^{2}-27(m_{1}m_{2}+m_{2}m_{3}+m_{3}m_{1})>0$,
unstable otherwise), depending on the signs of two polynoms of the strengths of
even degree:
$\Delta > 0$, $Q < 0$: unstable triangles, one stable aligned configuration,
$\Delta < 0$, $Q < 0$: unstable triangles, two stable aligned configurations, one unstable,
$\Delta < 0$, $Q > 0$: stable triangles, three unstable aligned configurations,
\[
\Delta=-32\sum_{j\ne l}\kappa_{j}^{3}\kappa_{l}-61 \sum_{j<
\]
The important role played by $Q$ can easily be understood if we notice that the
angular momentum of the system
$\sum\kappa_{j}(x_{j}\frac{\D y_{j}}{\D t}-y_{j}\frac{\D x_{j}}{\D t})$ is precisely
$\frac{Q}{2\pi}$. Like for the planar three body problem (see Smale II), we are
going to see that its sign is a basic element of classification of the topology of
the phase space and that very special behaviours of the motion occur when it
>From (7) we see that there is no other $\zeta$ orbit where the nature of the
motion changes. For given values of the strengths, the bifurcation set is
therefore the union, in the $\zeta$ plane, of the three points $\infty$,
$\kappa_{3},-\kappa_{2}$ and the three or five orbits going through the stationary
points (see Figures <ref> to <ref>, where some other ordinary orbits
have been added). Note that the circle
$|\zeta|^{2}=-\frac{\kappa_{2}\kappa_{3}K}{\kappa_{1}}$ on which $J$ is zero does
not belong to the bifurcation set, except when $Q$ is zero as we shall see.
However, when we represent the bifurcation set in the $(E,J)$ plane, the line
$J=0$ seems to belong to it for it is for some values of the strengths the
limiting curve $c\to 0$ associated with two vortices at the same location;
Figure <ref> shows the same bifurcation set as Figure <ref>, but
represented in the plane $(E,J)$.
Bifurcation set in the domain 1 $(Q>0$, $\Delta<0$), not to scale. The
numbers are the number of periodic solutions in each region.
-1Special behaviour for the motion will be obtained when two or more
elements of the bifurcation set come into coincidence. This will give us
separating lines in the strengths plane. The only conditions of coincidence
between the solutions of (17) and the set (d) of collisions $(\zeta=\infty$,
$\kappa_{3},-\kappa_{2})$ are:
(i) coincidence of (a) and (d): $\prod_{j\ne l}(\kappa_{j}+\kappa_{l})=0$,
(ii) coincidence of two zeros of $P\,{:}\,\Delta=0$.
We must add to the above mentioned lines the three lines given by
$\kappa_{1}\kappa_{2}\kappa_{3}=0$ which are forbidden.
$\prod(\kappa_{j}+\kappa_{l})$ is represented by three lines, $Q=0$ by the circle
$\rho=1$ and $\Delta=0$ by a quartic of equation:
\[
\rho^{4}+8\rho^{3}\sin 3\theta+18\rho^{2}-27=0,
\]
the form of which reflects the ternary symmetry.
The intersections of these lines define three distinct remarkable points that we
shall study explicitly:
\begin{eqnarray*}
&&\hbox{point A: $\rho=2$},\quad \hbox{$\ds 3\theta=\frac{3\pi}{2}$ (strengths $-1,1,1$)},\\[4pt]
&&\hbox{point B: $\rho=3$},\quad \hbox{$\ds 3\theta=\frac{3\pi}{2}$ (strengths $-5,4,4$)},\\[4pt]
&&\hbox{point C: $\rho=\sqrt{5}$},\quad \hbox{$\ds
\sin\;3\theta=\frac{-11\sqrt{5}}{25}$ (strengths $-2,\sqrt{3},2$).}
\end{eqnarray*}
The points A, B and some numerical values of the strengths correspond to cases of
integrability for the period (see Appendix I).
These lines define different domains in the strengths space, which has been
represented on Figure <ref>. When we stay in one of these domains, the
topology of the bifurcation set does not change, and Table <ref> summarizes
the results concerning the bifurcation set for all domains of the strengths space.
The classification for the bifurcation set is mainly based on the signs of $Q$ and
$\Delta$, i.e. on the number and stability of the relative equilibria; among the
three quantities, i.e. arithmetic, geometric and harmonic means of the strengths,
whose signs were proposed by Aref as a basis for a classification, only the third
one $\frac{Q}{3\kappa_{1}\kappa_{2}\kappa_{3}}$ is relevant, although the factor
$\kappa_{1}\kappa_{2}\kappa_{3}$ prevents it from being invariant under a
strengths reversal, an operation equivalent to a time reversal which must leave
invariant any candidate to a classification; moreover, for any $N$, the formula
$\sum'\kappa_{j}\kappa_{l}=0$, not $\sum\frac{1}{\kappa_{j}}=0$, expresses the
associated physical property of invariance of the energy under a change of length.
In the next part, we describe the behaviour of the three vortex system when it is
nongeneric, i.e. when either the strengths are on the boundaries of the domains
represented on Figure <ref> or when the initial conditions are those of a relative
equilibrium. We shall proceed by studying first the relative equilibria associated with
nonsingular strengths, then the strengths on the separating lines of the strengths space
and finally the three particular points A, B, C of the strengths space.
§ ABSOLUTE MOTIONS FOR NONGENERIC STRENGTHS
OR INITIAL CONDITIONS
§.§ Absolute motions for the ordinary relative equilibria
Ordinary means that we exclude $\zeta$ points which are the coincidence of two elements
of the bifurcation set: this is equivalent to $J$ nonzero and $\zeta$ not a multiple zero
of $P$.
For an initial condition in the vicinity of such a $\zeta$ point, every point obeys the
general motion, except those lying on the two curves intersecting at an unstable $\zeta$
point, in which case the motion is asymptotic to the motion at the stationary point,
motion which we are going to determine:
(a) $K\ne 0$. From the time variation law
\[
2\pi\frac{\D |Kz_{1}-B|^{2}}{\D t}=\frac{-2s^{2}{\rm
\]
it follows that the distance from each point to the barycentrum remains constant.
The absolute motion is therefore a solid rotation around the barycentrum;
moreover, using
\[
\sum_{j=1}^{N}\kappa_{j}z_{j}\frac{\D\bar{z}_{j}}{\D t}=\frac{Q}{2\pi i},
\]
whose imaginary part yields
\[
\sum_{j=1}^{N}\kappa_{j}|z_{j}|^{2}\frac{\D\arg z_{j}}{\D t}=\frac{Q}{2\pi},
\]
we see that the common angular velocity of every vortex is independent of time and
remains equal to $\frac{Q}{2\pi I_{o}}$, where $I_{o}$ is the inertia momentum
relative to the barycentrum
\[
I_{o}=\sum \kappa_{j}\left|z_{j}-\frac{B}{K}\right|^{2}=\frac{J}{K},
\]
and $\frac{Q}{2\pi}$ is the angular momentum of the system.
The period of the uniform rotation is therefore
\[
\]
a formula valid for any number of vortices when there exists a solid rotation.
A particular case is $Q=0$, for which the only isolated relative equilibrium is
$\zeta=-d$, i.e.
$\kappa_{2}\kappa_{3}z_{1}+\kappa_{3}\kappa_{1}z_{2}+\kappa_{1}\kappa_{2}z_{3}=0$; all
the vortices remain at rest. This situation is of course unstable (free vortices cannot
have a stable rest position since the complex velocity is a nonconstant meromorphic
function of the affixes), and the small motions have for pulsation
$\omega=\pm\frac{9i\kappa_{1}\kappa_{2}\kappa_{3}}{2\pi J}$
(b) $K=0$. The three stationary points are ordinary.
$\zeta=0$, stable: the impulse $B$ is zero and the affixes verify:
\[
\frac{z_{2}-z_{3}}{\kappa_{1}}=\frac{z_{3}-z_{1}}{\kappa_{2}}=
\frac{z_{1}-z_{2}}{\kappa_{3}}.
\]
The inertia momentum is nonzero and is the same at every point, and the absolute motion
is a uniform solid rotation of period
\[
\]
equal to that of the small motions.
At the two unstable equilateral triangles, $B$ is nonzero and the absolute motion is a
uniform translation of velocity
\[
\forall\,j\enskip{:}\enskip\frac{\D z_{j}}{\D t}=\frac{iQ}{2\pi \bar{B}},
\]
orthogonal to the impulse.
§.§ Absolute motions on the boundaries of the strengths domains
These lines are:
§.§ $Q=0$. Triple collision in a finite time, expanding motion
In addition to the already studied isolated stationary point, there exists a
circle of stationary $\zeta$ points (see Figure <ref>), on which $J$ is zero,
On this circle lie the two summits of the equilateral triangles and the two other
zeros of $P$. These four points are the points of contact of the circle $J=0$ with
the set of $\zeta$ trajectories whose equation is now:
\[
\left|\frac{\zeta-\kappa_{3}}{s}\right|^{-2\frac{\kappa_{1}\kappa_{2}}{K_{2}}}
\left|\frac{\zeta+\kappa_{2}}{s}\right|^{-2\frac{\kappa_{1}\kappa_{3}}{K_{2}}}=e^{\frac{4\pi
\]
We shall first study the absolute motion when the $\zeta$ point lies on the circle
$J=0$, then examine its neighborhood and finally deduce the bifurcation set.
Let us assume $\zeta$ on the circle $J=0$. This circle is no longer a trajectory and the
$\zeta$ point stays at rest. The two conditions $Q\,{=}\,0$, $J\,{=}\,0$ express that
both the energy and the inertia momentum relative to the barycentrum are invariant under
a change of length, and therefore nothing prevents the vortices from going to infinity or
to zero; we see below that both cases are possible. The absolute motion is ruled by
Equation (6) which implies that $\frac{d\rho_{1}^{2}}{dt}$ and
$\rho_{1}^{2}\frac{d\varphi_{1}}{dt}$ are constant in time ($\rho_{1},\varphi_{1}$ are
polar coordinates of $M_{1}$, $B$ is chosen zero). Since the shape of the triangle is
conserved, Equation (6) integrates as in
\begin{eqnarray*}
\forall\,j&=&1,2,3\enskip{:}\enskip z_{j}=z_{j,o}\left(1-\frac{t}{t_{c}}\right)^{1/2-i\omega t_{c}},\\
\hbox{i.e.}\quad
\sqrt{1-\frac{t}{t_{c}}}&=&\frac{\rho_{j}}{\rho_{j,o}}=\exp\{(\varphi_{j}-\varphi_{j,o})/(-2\omega
\end{eqnarray*}
where the characteristic time $t_{c}$ and the initial angular velocity $\omega$
are defined by
\[
\frac{\kappa_{\ell}}{(\bar{z}_{j}(z_{\ell}-z_{j}))_{o}}.
\]
-1$\omega$ never vanishes and has the sign of $K$. $\frac{1}{t_{c}}$
vanishes and changes sign when $\zeta$ is one of the four points already mentioned
where the circle is tangent to the set of $\zeta$ curves. We conclude that, for a
$\zeta$ point of the circle $J=0$ distinct of these four points, every vortex runs
a logarithmic spiral whose pole is the barycentrum, the shape of the triangle
remains constant and, depending on the sign of $t_{c}$, the triangle either
expands to infinity in an infinite time or collapses on the barycentrum in a
finite time $t_{c}$. At the time of this triple collision all the denominators of
the equations of motion (1) simultaneously vanish like $(t_{c}-t)^{1/2}$. After
the collision, the system is made of a single motionless vortex of strength $K$
located at what was the center of vorticity.
When $\frac{1}{t_{c}}$ vanishes, the spiral motion degenerates into a uniform
circular motion whose period $T=\frac{2\pi}{\omega}$ can also be written as
\[
\]
for the two aligned configurations and twice the same expression for the
equilateral triangles.
Let us now examine the motion elsewhere in the $\zeta$ plane $(J\ne 0)$.
Figure <ref> shows that two generic situations exist depending on whether the
$\zeta$ orbit intersects or not the circle $J=0$ of singular points; the limiting
$\zeta$ orbits, which belong to the bifurcation set, are $E=0$ (tangent at
$T_{1},T_{2}$) and $E=E(P_{3})$. For an energy $E$ outside the interval
$[E(P_{3}),0]$, the motion is the usual biperiodic one.
Inside the energy interval $]E(P_{3}),0[$, striped on Figure <ref>, every $\zeta$
trajectory stops on the circle (note that it cannot cross it) and, since $J$ is the
product of $|z_{2}-z_{3}|^{2}$ by a function of $\zeta$ vanishing on the circle, every
$\zeta$ point having a nonzero $J$ and the energy of a curve intersecting the circle
yields an expanding motion in which the trajectory of every vortex is asymptotic to a
logarithmic spiral going to infinity. No motion exists which is asymptotic to the triple
collision in a finite time: therefore the points of the half circonference (from $P_{1}$
to $T_{1}$ and from $P_{3}$ to $T_{2}$) where such a collision exists are repulsive
points, while the other half is made of attractive points.
Bifurcation set for $Q=0$, not to scale.
Finally, the bifurcation set, which we have also represented in the $(E,J)$ plane
on Figure <ref>, is the union of the following lines of the $\zeta$ plane:
the set of collisions $(\zeta=\infty$, $-\kappa_{2},\kappa_{3}$), the circle $J=0$
(not an orbit), the orbit going through $P_{2}$, the orbit $E=0$ tangent to the
circle at $T_{1},T_{2}$, the orbit $E=E(P_{3})$ tangent to the circle at $P_{3}$
and that portion of the orbit $E=E(P_{1})$ which is interior to the circle $J=0$.
§.§ $\Delta=0$
Two of the three colinear relative equilibria coincide except at the three points
of retrogression $\rho=3$ of the quartic $\Delta=0$ (strengths $-5,4,4$) where the
three of them coincide and at the six points $\rho=\sqrt{5}$ (strengths
$-2,\sqrt{3},2$) where two of the strengths are opposite (see next case).
For the absolute motion, there is nothing qualitatively changed: in the vicinity
of the double point $\zeta_{o}$, the orbits are equivalent to the cubics of
\[
\]
and the period, which behaves like $J\int\frac{\D\xi}{\eta}$, diverges like
$|c-c_{o}|^{-1/16}$; the absolute motion is still biperiodic, although the
triangle spends quite a long time in nearly aligned states. On the orbit $c=c_{o}$
passing through the unstable $\zeta_{o}$ point, the absolute motion is asymptotic
to the usual circular motion, the time law being different: $\xi-\xi_{o}\sim
t^{-2}$, $\eta\sim t^{-3}$.
§.§ $\Pi(\kappa_{j}+\kappa_{\ell})=0$. Elastic diffusion
At least two vortices have opposite strengths. To fix the ideas, let us assume
$\kappa_{1}+\kappa_{2}=0$. The case $\vec{\kappa}=(-1,1,1)$ will be described in
next section. The point $\zeta=\kappa_{3}$, which lies on the circle $J=0$, is
singular in the sense that for every $c$ value there exists a $\zeta$ orbit
passing through $\kappa_{3}$, which forbids a reduced periodic motion.
Therefore two generic situations exist, as shown on Figure <ref>: outside of
the striped domain which contains $J=0$, the absolute motion is biperiodic. Inside
this domain, for every $c$ value there is one and only one $\zeta$ orbit and the
$\zeta$ point is attracted by $\kappa_{3}$ in an infinite time: it tends to
$\kappa_{3}$ normally to the real axis following a law
\[
i(\zeta-\kappa_{3})t\to \frac{2\pi s}{\kappa_{1}}e^{-4\pi E/Q}.
\]
In the absolute space, the point $M_{3}$ stops and the two others go to infinity
together with a motion asymptotic to a uniform translation in the direction normal
to the line joining the final position of $M_{3}$ to the barycentrum, exactly as
if they were alone and obeyed the motion of two vortices of opposite strengths: at
the limit, the translation velocity is $\frac{\sqrt{-Q}}{2\pi}e^{2\pi E/Q}$ and
the mutual distance is $e^{-2\pi E/Q}$.
This creation of a doublet can also be interpreted in terms of the elastic diffusion of
the doublet by the third vortex: if we take for initial condition $\zeta$ near to $K_{3}$
with $\eta/K$ positive, the vortices 1 and 2 are initially a doublet moving towards the
third vortex; then the three mutual distances become the same order of magnitude, i.e.
there is interaction; finally, the doublet emerges and goes to infinity away from the
third vortex with unchanged mutual distance and velocity, but in a different direction.
This is exactly a process of elastic diffusion. Figure <ref> shows typical absolute
motions. We define the scattering angle $\Delta(\phi)$ by the total variation along the
trajectory of $\arg(z_{1})$ ($B$ is chosen zero) and a dimensionless impact parameter by
$\frac{\overline{OH}}{\overline{OM_{3}}}$ (Figure <ref>) where $H$ is the projection
of the middle of $M_{1}M_{2}$ on the line defined by the barycentrum and the final
position of $M_{3}$:
\begin{eqnarray*}
\frac{\overline{OH}}{\overline{OM_{3}}}&=&\frac{{\rm Re}(\bar{z}_{3}\frac{z_{1}+z_{2}}{2})}{\bar{z}_{3}z_{3}}\\[4pt]
\frac{s\kappa_{3}}{-2r\kappa_{1}}+\frac{s-\kappa_{1}}{-2\kappa_{1}}=\frac{c+1}{2},
\end{eqnarray*}
where $r$ is the radius of curvature in $\kappa_{3}$:
\[
\]
The impact parameter is then our dimensionless invariant $c$, up to a linear
transformation. As to the scattering angle, it can be computed from an integral taken
along the $\zeta$ orbit:
\begin{eqnarray*}
\Delta \varphi_{1}=[\arg\, z_{1}]_{t=-\infty}^{+\infty}=\oint -sK\frac{{\rm %RC
\end{eqnarray*}
This integral can be carried out exactly for $c=0$, when the $\zeta$ orbit is a
\begin{eqnarray*}
\Delta \varphi_{1}&=&\int_{o}^{2\pi}\frac{2K+d}{2(K+d)}\left(1-\frac{Kd}{2K^{2}+2Kd+d^{2}+2K(K+d)\cos\theta}\right)\D\theta\\[4pt]
&=&\left\{\begin{array}{@{}l@{\quad}l@{}} %RC
\pi&{\rm if}\ |\kappa_{1}|>|K|\\[7pt]
\ds \left(1-2\frac{K}{\kappa_{1}}\right)\pi&{\rm if}\ |\kappa_{1}|<|K|.\\
\end{array}\right.
\end{eqnarray*}
The bifurcation set for this case is made, in the $\zeta$ plane, of the following
lines: the set of collisions $(\infty,\kappa_{3},-\kappa_{2})$, the boundary
between the two regimes, the point $P_{1}$ and the orbit $c=c(P_{2})$.
§.§ The three particular $\vec{\kappa}$ points
(a) Point A (strengths $-1,1,1$). Direct and exchange diffusion.
The $\zeta$ plane looks like the one Figure <ref> assumed continuously
deformed so as to admit the origin as a center of symmetry: $M_{3}=P_{1}$,
symmetric of $M_{2}$, and $P_{2}=0$.
Aref (1979) has extensively studied the motion and we shall only briefly summarize it.
The results obtained for $\kappa_{1}+\kappa_{2}=0$ still apply, but a new physical
situation arises from the existence of $\zeta$ orbits which go from $\zeta=\pm K$ to
$\zeta=\mp K$ (these are all the orbits which cross the segment $T_{1}T_{2}$): the motion
is then an exchange scattering in which the incident pair (1 2) is different from the
outgoing doublet (1 3).
To sum up, three generic situations exist, depending on the initial conditions:
* $|\frac{\zeta}{K}\pm 1|>2$: in uniformly rotating axes, the absolute motion is
* $|\frac{\zeta}{K}\pm 1|<2$ and $-8<c<1$: exchange scattering;
the integral giving the scattering angle is to be taken from 0 to $\pi$ only.
* $|\frac{\zeta}{K}\pm 1|<2$ and $c$ outside $[-8,1]$: direct scattering as previously described.
The reduced forms of all the elliptic integrals giving the period and the
scattering angle are gathered in Appendix I.
A particular case of exchange scattering is easy to solve and will give an idea of
the motion:
For $c=0$, the $\zeta$ orbit is a half-circle and, computing
$\frac{\D^{2}z_{1}}{\D t^{2}}$ by deriving Equation (6), we find zero, which means a
uniform linear motion $M_{1}$, hence a scattering angle of $\pi$. The motion of
$\zeta=Ke^{i\theta}$ is ruled by $\frac{\D\theta}{\D t}=\frac{v}{a}\sin^{2}\theta$
which integrates as $\zeta=K\frac{ia-vt}{\sqrt{a^{2}+v^{2}t^{2}}}$ where we have
noted $a=2e^{2\pi E/K^{2}}$, $v=\frac{K}{\pi a}$.
The origin of time being chosen when $M_{1}$ is the summit of an isosceles
rectangle triangle, the absolute motions take place on three parallel
straight lines:
\[
\left\{\begin{array}{@{}l@{}}
\ds z_{1}=a+i\;v\;t\\[4pt]
\ds z_{2}=\frac{a+i\;v\;t}{2}+\frac{i}{2}\sqrt{a^{2}+v^{2}\;t^{2}}\\[10pt]
\ds z_{3}=\frac{a+i\;v\;t}{2}-\frac{i}{2}\sqrt{a^{2}+v^{2}\;t^{2}}.\\
\end{array}\right.
\]
The exchange scattering process is clearly seen on the above equations, and every
other exchange scattering motion can be thought of as a continuous deformation of
this one.
The bifurcation set is made of the set of collisions and of the boundaries of the
domains limiting the three generic situations; note that the circle
$J=0(|\zeta|=|K|)$ does not belong to it. This set is simple enough to be
represented without any ambiguity by a c axis with the number and type of
solutions in each interval:
(b) Point B summit of the quartic (strengths $-5,4,4$).
Nothing special happens to the motion. At the triple point $\zeta=0$, the vortex of
strength $\frac{-5K}{3}$ is motionless as it coincides with the barycentrum, and the two
other vortices, which are symmetric with regard to the barycentrum, obey the circular
motion. The $\zeta$ orbits around this stable point are equivalent to the quartics of
\[
\left(\frac{\xi}{K}\right)^{4}=0,
\]
and they correspond to ordinary motions; however the period, which still behaves
like $J\int \frac{\D\xi}{\eta}$, diverges as $(c+2^{4/3})^{-1/4}$.
(c) Point C (strengths $-2, \sqrt{3}, 2$) conjunction of $\Delta=0$ and
$\Pi(\kappa_{j}+\kappa_{\ell})=0$. This point has no new properties: the
previously studied singularities only add without interfering.
§ VOLUME OF THE PHASE SPACE
The Hamiltonian system has an invariant element of volume of the phase space equal
\[
\D v=\kappa_{1}\kappa_{2}\kappa_{3}\D x_{1}\wedge \D y_{1}\wedge \D x_{2} \wedge \D y_{2}
\wedge \D x_{3}\wedge \D y_{3}.
\]
Since there exist four real invariants $E, J, X, Y$ $(X+iY=B)$, we want the
density of states $\frac{\D v}{\D E\;\D J\;\D X\;\D Y}$ after integration over two
independent variables of the phase space.
By using the two successive changes of variables $(z_{1},z_{2},z_{3})\to
(z_{2}-z_{3},\zeta,B)$ and
\[
\]
whose jacobians are respectively
\[
\frac{D(z_{2}-z_{3},\zeta,B)}{D(z_{1},z_{2},z_{3})}=\frac{-sK}{z_{2}-z_{3}},
\]
\[
\frac{D(E,J)}{D(\zeta,\bar{\zeta})}=\frac{i\kappa_{1}^{2}\kappa_{2}\kappa_{3}}{2\pi}|z_{2}-z_{3}|^{2}\frac{{\rm
Im} (\zeta^{2}+d\zeta)}{|(\zeta-\kappa_{3})(\zeta+\kappa_{2})|^{2}},
\]
we obtain
\begin{eqnarray*}
\D v&=&\frac{\pi}{\kappa_{1}s^{2}K^{2}}\frac{|(\zeta-\kappa_{3})^{2}(\zeta+\kappa_{2})^{2}|}{{\rm
&& \ \D(x_{2}-x_{3}) \wedge \D(y_{2}-y_{3})\wedge \D E\wedge \D J \wedge \D X\wedge \D Y,
\end{eqnarray*}
and we still have to integrate over $x_{2}-x_{3}$ and $y_{2}-y_{3}$. Since
$|z_{2}-z_{3}|$ moves in time according to
\[
2\pi\frac{\D |z_{2}-z_{3}|^{2}}{\D t}=2\kappa_{1}s^{2}\frac{{\rm
\]
the integration is quite easy to perform and we finally get
\[
\D v=\frac{\pi}{K^{2}}T(E,J) \D E\wedge \D J\wedge \D X\wedge \D Y,
\]
which shows that the density of states is the period of the reduced motion. This
is a well known result of the theory of adiabatic invariants (see e.g. Landau and
Lifchitz), for energy and time are conjugate variables: when $J$ is constant and
$E$ slowly varying, then the product $ET$ is constant since the volume of the
phase space if conserved.
To take into account the fact that the phase space is the union of disconnected
parts, we must sum the above expression over the number (between 0 and 4, see
Table <ref>) of different domains associated to given values of $E$ and $J$.
The resulting volume $\Omega(J,E)$ obeys the scaling law:
\[
\Omega(E,J)=\frac{\pi}{K^{2}}\sum_{{\rm
\]
Onsager (1949) defined the entropy $S$ and the temperature $\tau$ of an assembly
of a large number of interacting vortices:
\[
S=k_{B}\;{\rm Log}\,\Omega,\quad \frac{1}{\tau}=\frac{\D S}{\D E}.
\]
Although it makes no sense to speak of thermodynamics about an integrable system,
there may be some interest for the understanding of the behaviour of a large
number of vortices to examine what the Onsager's theory gives when formally
applied to the three-vortex system.
The main hypothesis made by Onsager is that the total amount of volume
$\int_{-\infty}^{+\infty}\Omega \D E$ available to the system is finite, an hypothesis
equivalent to assume the system confined in a box since the phase space and the
configuration space are the same. Due to the scaling law for $\Omega$, the integral
$\int\Omega \D E$ will be either finite and proportional to $J$ or infinite, depending on
the strengths of the vortices. The first arising question is therefore: when $J$ is kept
constant, is the integral
$\int\Omega(E,J)\D E=\frac{Q}{4\pi}\int_{cQJ>0}\Omega\frac{\D c}{c}$ finite or not, i.e. are
all the singularities of $T$ integrable or not?
The singularities of $T$ are: the set of collisions, the unstable relative
equilibria and, since $J$ is kept nonzero, the limit $e^{4\pi E/Q}\to 0$ in the
domains $\kappa_{1}\kappa_{2}\kappa_{3}K<0$ only.
(a) Two vortices close to each other (vortex $j$ alone): then
\[
\frac{4\pi^{2}J}{(K-\kappa_{j})^{2}\kappa_{j}}\left(\frac{Qc}{\kappa_{j}(K-\kappa_{j})}\right)^{-\frac{Q\kappa_{j}}{\kappa_{1}\kappa_{2}\kappa_{3}}}\to
\]
and the singularity $\int\;T$ $\D E=\frac{QJ}{4\pi}\int\frac{T}{J}\frac{\D c}{c}$ is
(b) $\zeta$ tends to an unstable $\zeta_{0}$ value, whose $c$ value is $c_{o}$.
The equivalent hyperbola having for equation:
\begin{eqnarray*}
\frac{c-c_{o}}{c_{o}}+\frac{J\,\kappa_{1}\kappa_{2}\kappa_{3}}{2Q(\kappa_{1}|\zeta_{0}|^{2}+\kappa_{2}\kappa_{3}K)^{2}}\\[3pt]
\end{eqnarray*}
the period is equivalent to
\[
T\sim {\rm cst}\,J\int \frac{\D \bar{\zeta}}{G'_{\zeta}(\zeta,\bar{\zeta})}\sim {\rm cst}\
J\ {\rm Log}|c-c_{o}|,
\]
and the singularity is therefore integrable.
(c) $|\zeta|^{2}\to - \frac{\kappa_{2}\kappa_{3}K}{\kappa_{1}}$ (possible in every
domain, except 0 and 150).
Since $c$ tends to zero with $J$ being kept constant and nonzero, then $e^{4\pi E/Q}$
tends to zero, therefore the period $T=\hbox{cst}$ $e^{-4\pi E/Q}$ has a nonintegrable
In conclusion, the volume of the phase space, with $J$ being constant, is finite
and proportional to $J$ in the domains 0, 150 and $K=0$ and infinite elsewhere.
The unit of time we chose, i.e. $4\pi^{2}\frac{J}{QK}$, is a posteriori convenient
for it is proportional to the volume.
Let us now examine the behaviour of the thermodynamical quantities, keeping in mind that
any conclusion is meaningless for three vortices and can only be indicative for a larger
system. For instance in the domain 0 (i.e. $\kappa_{j}K>0$), the function $E_{o}\to
\int_{E<E_{o}}$ $\Omega \D E$ is of course increasing, but the integrand $\Omega$, which is
zero at the edges $E\to \pm \infty$, positive and integrable, has three infinite maxima
at finite values of the energy (Figure <ref>). Each of these maxima corresponds to
an unstable relative equilibrium configuration or, in other words, to a point of the
phase space which links two disconnected domains. This feature somehow complicates the
correspondence between energy and temperature and there exists some numerical evidence
(Lundgren and Pointin, 1977, and references herein) of a possible lack of ergodicity
which could come out of a multiple connexity of the phase space. Another interesting
observation is that, when the energy tends to $+\infty$, the “temperature” of the three
vortex system tends to a constant, negative value (Fig. <ref>), a fact already
noticed for a large number of vortices by Lundgren and Pointin (page 334), C.E. Seyler
(1974), Edwards and Taylor (1974, page 262).
§ CONCLUSION
In addition to the fact of being an exactly soluble three body problem, the three
vortex system is very interesting in connection with the theory of turbulence.
Unfortunately its number of degrees of freedom is too small to yield a chaotic
behaviour (the threshold for such a behaviour is 3) and this was confirmed by the
results: nonperiodic behaviours are obtained only for very particular values of
the parameters. A four vortex system (see some preliminary results in Conte,
1979), with its 3 independent degrees of freedom and because we do not know about
its integrability, is the really interesting dynamical system to study in order to
have some hints about the integrability of the N vortex system.
§ ACKNOWLEDGEMENTS
We want to thank Y. Pomeau for many fruitful discussions which led to the discovery of
the appropriate plane. We also greatly appreciated the formal Reduce-like computer
language AMP (Drouffe, 1976) which helped us to establish the numerous necessary
The work is the first chapter of the unpublished Thèse d'État of the first
author. It is an honor and a pleasure for us to dedicate it to Professor Hao
Bailin and to wish him a long life.
H. Aref (1979), Motion of three vortices, Phys. Fluids 22, 393–400.
Bateman manuscript project, (1953), Higher transcendental functions, vol. II,
chapter XIII; A. Erdélyi Editor, Mc Graw Hill.
R. Conte (1979), Thèse d'Etat, Université de Paris VI.
J.M. Drouffe, AMP language, (1976), same address as authors.
S.F. Edwards and J.B. Taylor (1974), Negative temperature states of
two-dimensional plasmas and vortex fluids, Proc. Roy. Soc. London, A 336,
W. Gröbner and Hofreiter (1965), Integraltafel, vol. 4, Springer-Verlag.
L. Landau and E. Lifchitz (1960), Mechanics, Pergamon Press.
T.S. Lundgren and Y.B. Pointin (1977), Statistical mechanics of two-dimensional
vortices, Journal of statistical physics, 17, 323–325.
A.M. Mayer (1878), Floating magnets, Nature, 18, 258.
E.A. Novikov (1975), Dynamics and statistics of a system of vortices, JETP 41, 937–943.
L. Onsager (1949), Statistical hydrodynamics, Nuovo Cimento, 6 suppl.,
H. Poincaré (1893), Théorie des tourbillons, pages 77–84, Deslis
frères, Paris.
C.E. Seyler, Jr. (1974), Partition function for a two-dimensional plasma in the
random phase approximation, Phys. Rev. Letters 32, 515–517.
S. Smale (1970), Topology and mechanics, Inventiones math., 10, 305–331 and
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§ APPENDIX I
§ CASES OF INTEGRABILITY
For practical applications, it may be of interest to find which values of the
$\kappa_{j}$'s lead to integrable expressions for the period (12). A first case is
when two strengths are equal: $\kappa_{2}=\kappa_{3}$; then $\xi\eta$ can be
expressed only with $|\zeta|^{2}$, using (11), and the period is a simple integral
in the variable $|\zeta|^{2}$, which can be easily integrated numerically.
Another case is when, the strengths being rational, the algebraic curve (11) is of
genus one or zero (the genus of an algebraic curve of degree $n$ is equal to
$\frac{(n-1)(n-2)}{2}$ minus the number of double points). The only curve of genus
zero is the circle $J=0$ but then the period is given by another non-integrable
expression. If the trajectory is of genus one, the abelian integral expressing the
period can always be reduced to an elliptic integral by a birational
transformation of the coordinates (see e.g. Bateman 1953). For small integer
values of the strengths, there is some chance of finding curves of genus one.
Let us just mention three particular cases.
$\underline{\vec{\kappa}=(1,1,1) K/3}$. This belongs to the first but not to the
second case (degree 6, genus 4 in general). The period is expressed by the
hyperelliptic integral in $u=\frac{|\zeta|^{2}}{\kappa_{1}^{2}}$:
\[
\frac{T}{T_{u}}=\frac{1}{2\pi}\oint \frac{-9(u+3)\;{\rm
sign}(\xi\eta)}{\sqrt{27c^{3}(u+1)^{2}-2(u+3)^{3}}\sqrt{2(u+3)^{3}-27c^{3}(u-1)^{2}}}\D u.
\]
Novikov gave this expression in the variable
$b=\frac{6}{u+3}=\frac{3\kappa_{1}^{2}|z_{2}-z_{3}|^{2}}{J}$ which always remains
between 0 and 2 but he did not integrate it:
\[
\frac{T}{T_{u}}=\frac{3}{4\pi c^{3}}\oint \frac{{\rm
sign}(\xi\eta)}{\sqrt{f(b)}\sqrt{-g(b)}}\D b
\]
with $f(b)\equiv b(b-3)^{2}-\frac{4}{c^{3}}$, $g(b)\equiv
b(b-\frac{3}{2})^{2}-\frac{1}{c^{3}}$. This hyperelliptic integral happens to be
reducible to an elliptic integral (Bolza, 1898, mentioned in the tables of
Gröbner and Hofreiter, 1965) of the variable $z=\frac{g(b)}{3b}$, due to the
\begin{gather*}
\varphi(z)\equiv
\frac{1}{4c^{6}}=\frac{f(b)[h(b)]^{2}}{27b^{3}},\\[4pt]
\frac{\D z}{\D b}=\frac{6h(b)}{9b^{2}}
\end{gather*}
with $h(b)\equiv b^{3}-\frac{3}{2}b^{2}+\frac{1}{2c^{3}}$; this gives for the
\[
\frac{T}{T_{u}}=\frac{1}{8\pi c^{3}}\oint \frac{{\rm sign}(\xi\eta){\rm
sign}(h(b))}{\sqrt{-z\varphi(z)}}\D z
\]
Let us call $b_{1}<b_{2}<b_{3}$ the zeros of $f$, $b_{4}<b_{5}<b_{6}$ those of $g$
($b_{4}$ and $b_{5}$ are not real for $1<c^{3}<2$), $b_{7}<b_{8}<b_{9}$ those of
$h$ and $z_{1}<z_{2}<z_{3}$ those of $\varphi$. The correspondence is
$(b_{1},b_{9})\to z_{1}$, $(b_{2},b_{8})\to z_{2}$, $(b_{3},b_{7})\to z_{3}$,
which gives the following values of the period for the two domains:
\begin{gather*}
1<c^{3}<2\enskip{:}\enskip {\rm sign}(K)\oint =2\int_{b_{1}}^{b_{2}}\D b=6\int_{z_{1}}^{z_{2}}\D z,\\[3pt]
\frac{T}{|T_{u}|}=\frac{3}{2\pi c^{3}}\frac{K(k)}{\sqrt{(z_{3}-z_{2})(-z_{1})}},\quad k^{2}=\frac{z_{3}(z_{2}-z_{1})}{(z_{3}-z_{2})(-z_{1})},\quad z_{1}<z_{2}<0<z_{3}\\[3pt]
b_{7}<b_{1}<b_{4}<b_{8}<b_{5}<b_{9}<b_{6}<b_{2}<b_{3},\quad z_{1}<0<z_{2}<z_{3}.
\end{gather*}
Two equivalent expressions lead to the period, according to whether $\zeta$ turns
around $M_{1}$ or another vortex:
\begin{gather*}
\hspace*{-.05pc}\hbox{($\zeta$ around $M_{1}$)}\hspace*{2.5pc}:\quad {\rm sign}(K)\oint=4\int_{b_{1}}^{b_{4}}\D b=4\int_{z_{1}}^{o}\D z\\[4pt]
\hbox{($\zeta$ around $M_{2}$ or $M_{3}$)}:\quad {\rm sign}(K)\oint=2\int_{b_{5}}^{b_{6}}\D b=4\int_{z_{1}}^{o}\D z\\[4pt]
\frac{T}{|T_{u}|}=\frac{K(k)}{\pi c^{3}\sqrt{z_{3}(z_{2}-z_{1})}},\quad
\end{gather*}
$\underline{\vec{\kappa}=(-1,1,1,)K}$ where three generic situations exist. We
assume $K>0$.
The $\zeta$ curves are bicircular quartics of genus one and we derive below the
normal forms of the scattering angle and the period of the reduced motion:
\begin{eqnarray*}
\Delta \varphi_{1}&=&\oint\frac{(u+u_{o})\;{\rm sign}(\xi\eta)}{2u\sqrt{(u-u_{-})(u-u_{+})}\sqrt{(u-1)(u_{o}-u)}}\D u\\[4pt]
\frac{T}{T_{u}}&=&\frac{1}{2\pi}\oint\frac{4\;{\rm
sign}(\xi\eta)}{c(u-1)\sqrt{(u-u_{-})(u-u_{+})}\sqrt{(u-1)(u_{o}-u)}}\D u
\end{eqnarray*}
with the notations $u=|\frac{\zeta}{K}|^{2}$, $u_{o}=1+\frac{8}{c}$,
$u_{\pm}=\frac{4}{c}-1\pm \frac{4}{c}$ $\sqrt{1-c}$. The variable $b$ in Aref is
related to $u$ by $b=\frac{6}{1-u}$. $K, E$ and $\Pi$ are the complete elliptic
integrals of the first, second and third kind, the last one being defined
as[In the tables of Gradshteyn and Ryzhik $4^{{\rm th}}$ edition, the
definition 8.111.4 is not consistent with the rest of the book; many formulae
concerning elliptic integrals are wrong, among them 3.132.5, 3.132.6, 3.138.8,
\[
\Pi(n,k)=\int_{0}^{1}\frac{\D x}{(1-nx^{2})\sqrt{(1-x^{2})(1-k^{2}x^{2})}}
\]
First regime (exchange scattering). $-8<c<1$. $\oint {\rm
sign}(\xi\eta)\D u=2\int_{1}^{u_{-}}\D u$.
There is no discontinuity for $c=0$ where $\Delta \varphi_{1}$ evaluates to $\pi$.
\begin{eqnarray*}
&&-8<c<0\enskip{:}\enskip \Delta \varphi_{1}=\frac{-2}{\sqrt{(1-u_{o})(u_{-}-u_{+})}}\\[5pt]
&&\qquad {\rm with}\ k^{2}=\frac{(1-u_{-})(u_{o}-u_{+})}{(1-u_{o})(u_{-}-u_{+})},\quad n=\frac{u_{+}(u_{-}-1)}{u_{-}-u_{+}},\\[5pt]
&&0<c<1\enskip{:}\enskip \Delta \varphi_{1}=\frac{2}{\sqrt{(u_{o}-u_{-})(u_{+}-1)}}\left[2K(k)+\frac{8}{c}\Pi(n,k)\right]\\[5pt]
&&\qquad {\rm with}\ k^{2}=\frac{(u_{o}-u_{+})(u_{-}-1)}{(u_{o}-u_{-})(u_{+}-1)},
\quad n=\frac{u_{o}(1-u_{-})}{u_{o}-u_{-}}
\end{eqnarray*}
Second regime (direct scattering). $c<-8$ or $1<c$. $\oint {\rm
sign}(\xi\eta)\D u=2\int_{1}^{u_{o}}\D u$
\begin{eqnarray*}
&&c<-8\enskip{:}\enskip \Delta \varphi_{1}=\frac{-2}{\sqrt{(1-u_{-})(u_{o}-u_{+})}}\\[4pt]
&&\qquad {\rm with}\ k^{2}=\frac{(1-u_{o})(u_{-}-u_{+})}{(1-u_{-})(u_{o}-u_{+})},\quad n=\frac{u_{+}(u_{o}-1)}{u_{o}-u_{+}},\\[4pt]
&&1<c\enskip{:}\enskip \Delta \varphi_{1}=(u_{o}^{-\frac{1}{4}}-u_{o}^{\frac{1}{4}})K(k)+\frac{(1+\sqrt{u_{o}})^{2}}{2}\Pi(n,k)\\[4pt]
&&\qquad {\rm with}\
k^{2}=\frac{(u_{o}-1)^{2}-4(\sqrt{u_{o}}-1)^{2}}{16\sqrt{u_{o}}}, \quad
\end{eqnarray*}
Third regime (biperiodic). $0<c<1$ $\oint {\rm
sign}(\xi\eta)\D u=4\int_{u_{+}}^{u_{o}}\D u$
\begin{eqnarray*}
&&\Delta \varphi_{1}=\frac{4}{\sqrt{(u_{-}-1)(u_{o}-u_{-})}}\left[2K(k)-\frac{8}{c+8}\Pi(n,k)\right]\\
&&\qquad {\rm with}\ k^{2}=\frac{(u_{-}-1)(u_{o}-u_{+})}
{(u_{+}-1)(u_{o}-u_{-})},\quad n=\frac{u_{+}-u_{o}}{u_{o}(u_{+}-1)}
\end{eqnarray*}
$\underline{\vec{\kappa}=(-1,2,2)\frac{K}{3}}$, a case with two possible regimes
(biperiodic, expanding).
The $\zeta$ trajectories are the Cassini ovals, whose genus is one. The period for
instance is given by
\[
sign}(\xi\eta)}{4(u-3)^{2}\sqrt{(u+1)^{2}-\alpha}\sqrt{\alpha-(u-1)^{2}}}\D u
\]
with $u=|\frac{\zeta}{\kappa_{2}}|^{2}$,
$\alpha=|(\frac{\zeta}{\kappa_{2}})^{2}-1|^{2}=16$ $e^{18\pi E/K^{2}}$. Its
reduced form is not very compact and we shall not give it here.
§ APPENDIX II
§ STABILITY OF THE RELATIVE EQUILIBRIA
In order to obtain the shape of the $\zeta$ trajectories in the vicinity of the relative
equilibria we must determine whether they are of elliptic or hyperbolic nature.
The points $\infty,-\kappa_{2},\kappa_{3}$ are elliptic, neighbouring orbits are
circles described with a uniform circular motion of period:
\[
\]
We now assume that $\zeta_{o}$ is the affix of an ordinary relative equilibrium
(the case of two coincident r.e. is studied elsewhere in the paper), which implies
$J \ne 0$ and we study the vicinity of the equilateral triangles and of the
aligned configurations.
By writing $f(\zeta,\bar{\zeta})$ for the right-hand side of Equation (10), the
small motions of a $\zeta$ point in the vicinity of a stationary point $\zeta_{o}$
are ruled by:
\[
2\pi i\frac{\D \bar{\zeta}}{\D t}=(\zeta-\zeta_{o})\frac{\partial f}{\partial \zeta}
(\zeta_{o},\bar{\zeta}_{o})+(\overline{\zeta-\zeta_{o}})\frac{\partial f}{\partial
\bar{\zeta}}(\zeta_{o},\bar{\zeta}_{o}),
\]
or, in real matricial notation:
\[
2\pi\frac{\D}{\D t}\binom{\xi}{\eta}=M\binom{\xi-\xi_{o}}{\eta-\eta_{o}}=
\left(\begin{array}{@{}l@{\quad}l@{}}
\alpha'+\beta'&\alpha-\beta\\
\alpha+\beta&-\alpha'+\beta'\\
\end{array}\right)\binom{\xi-\xi_{o}}{\eta-\eta_{o}}
\]
with $\alpha+i\alpha'=\frac{\partial f}{\partial \zeta}(\zeta_{o},\bar{\zeta}_{o}) =\mu$,
$\beta+i\beta'=\frac{\partial f}{\partial \bar{\zeta}}(\zeta_{o},\bar{\zeta}_{o})=\nu$.
The stability condition is: ${\rm tr}(M)=0$, ${\rm det}(M)>0$. We find:
\[
\frac{-(\kappa_{1}|\zeta|^{2}+\kappa_{2}\kappa_{3}K)}{J(\zeta+\kappa_{2})(\zeta-\kappa_{3})}=
\frac{\mu}{2|\zeta|^{2}+d\bar{\zeta}-sK}=\frac{\nu}{\zeta^{2}+d\zeta-Q}
\]
with the condition (17). The trace of $M$ is therefore zero. If $\zeta_{o}$ is
elliptic, then the small motions have the period $4\pi^{2}/\sqrt{{\rm det}(M)}$.
We now divide the study according to the two types of stationary points.
(a) The equilateral triangles
${\rm det}(M)=\frac{3Q^{3}}{J^{2}}$. The stability condition is $Q>0$ and, when
this is fulfilled, the period of the small motions is
$\frac{4\pi^{2}J}{\sqrt{3}Q^{3/2}}$; comparing with the period of the absolute
motion which will be derived later, we find:
\[
\left(\frac{T_{r}}{T_{a}}\right)^{2}=1-\frac{1}{2K^{2}}\sum_{j}\sum_{\ell >
\]
The overall rotation is therefore quicker than the small motions with equality
only for identical strengths.
(b) The aligned configurations
\[
{\rm det}\
\]
with the notation
\begin{eqnarray*}
\end{eqnarray*}
and the condition: $\zeta$ is a real zero of $P$.
The determinant of $M$ changes sign when the resultant of $P$ and $F_{1}F_{2}$
vanishes. We find:
\[
{\rm res}(P,F_{1})=-3s^{2}Q^{2},\quad {\rm res}(P,F_{2})=s^{2}\Delta.
\]
Then we obtain the nature of the aligned configurations in the parameter space:
\begin{eqnarray*}
\hbox{$\Delta > 0$ $(Q < 0)$}&:& \hbox{one stable configuration}\\
\hbox{$\Delta < 0$ and $Q<0$}&:& \hbox{two stable configurations, one unstable}\\
\hbox{$\Delta < 0$ and $Q>0$}&:& \hbox{three unstable configurations}.
\end{eqnarray*}
It is worth observing that in the present problem the nonlinear stability is the same as
the linear one.
\end
|
1511.00358
|
Department of Physics,
Graduate School of Science,
University of Tokyo, Tokyo, 113-0033, Japan
Research center for the early universe,
Graduate School of Science,
University of Tokyo, Tokyo, 113-0033, Japan
Institute for cosmic ray research, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan
A rapidly spinning neutron star (NS) would emit a continuous gravitational wave (GW)
detectable by the advanced LIGO, advanced Virgo, KAGRA and proposed third generation detectors
such as the Einstein Telescope (ET).
Such a GW does not propagate freely,
but is affected by the Coulomb-type gravitational field of the NS itself.
This effect appears as a phase shift in the GW depending on the NS mass.
We have shown that mass of an isolated NS can, in principle, be determined
if we could detect the continuous GW with two or more frequency modes.
Indeed, our Monte Carlo simulations have demonstrated that mass of a NS with
its ellipticity $10^{-6}$ at 1 kpc is typically measurable
with precision of $20\%$ using the ET, if the NS is
precessing or has a pinned superfluid core and emits GWs with once and twice the spin frequencies.
After briefly explaining our idea and results, this paper concerns with the effect of misalignment angle
(“wobble angle” in the case of a precessing NS) on the mass measurement precision.
§ INTRODUCTION
The distribution of masses of neutron stars (NSs) gives insights to understand
their birth mechanisms and evolution histories.
For instance, the mass measurements of the massive pulsar PSR J1614–2230 <cit.>
and PSR J0348+0432 <cit.>
have great impacts on exotic matter physics and studies of NS interiors.
Currently, mass measurements of NSs are mostly limited to those in
binaries <cit.>. Since possible mutual mass
transfers in binaries may change the mass distributions of the
component NSs, it is desirable to measure masses of as many isolated NSs as possible.
We have proposed a new method to measure mass of an isolated NS by detecting Coulomb-type phase shifts
imprinted in gravitational waves (GWs) by the NS gravitational field <cit.>.
In that paper, we have shown that the mass of a NS at 1 kpc distance
and with its ellipticity $10^{-6}$ is typically measurable
with precision of $20\%$ using the third generation GW telescope such
as the Einstein Telescope (ET) <cit.>,
if the NS is precessing or has a pinned superfluid core and emits GWs with once and twice the spin frequencies.
In the following sections, we will review the idea of our method, and
then comment on our assumptions on misalignment angles in
our original paper <cit.>, namely the misalignment angle of the freely-precessing
NSs and their effect on mass measurement precision.
§ METHOD
Let us consider an isolated NS with mass $M$ located at the coordinate origin
$r=0$. We assume that the NS emits GW with multiple frequencies
$\omega_{\alpha} = 2\pi f_{\alpha}$($\alpha = 1,2,\cdots$).
The two polarization modes of GWs from the NS can be written as
\begin{align}
\bar h_{+} &= \sum_{\alpha}A_{+,\alpha}\cos\Psi_{\alpha},\quad
\bar h_{\times} = \sum_{\alpha}A_{\times,\alpha}\sin\Psi_{\alpha},\\
\Psi_{\alpha} &\equiv - \omega_{\alpha} t + \omega_{\alpha} r + \Phi_{\alpha},\\
\Phi_{\alpha} &\equiv 2\omega_{\alpha}M \ln[2\omega_{\alpha}r] + \phi_{R\alpha},
\end{align}
where $\phi_{R\alpha}$ denotes the constant reference GW phases.
The logarithmic term in $\Phi_{\alpha}$ is the Coulomb phase shift due
to the static part of the NS gravitational field <cit.>.
All the searches for GWs from rapidly rotating NSs conducted so far
have not taken into account these Coulomb phase shifts.
Our method to measure the mass $M$ of a NS and results are as follows <cit.>.
Suppose that we determine GW phases $\Phi_{\alpha}$ for two or more frequency modes (say, $\omega_{1}$ and
$\omega_{2}$ with the ratio $K \equiv \omega_{2}/\omega_{1}$).
If these two modes have such a property that
\begin{align}
\phi_{R2}-K\phi_{R1} = 0,
\label{eq:condition_on_phase}
\end{align}
we can determine the mass from
the following combination,
\begin{align}
%\zeta \equiv
\Phi_{2} - K \Phi_{1} = 2\omega_{2}M\ln K
\simeq 0.06
\left(\frac{M}{1.4M_\odot}\right)
\left(\frac{f_2}{1{\rm kHz}}\right)
\left(\frac{\ln K}{\ln2}\right).
\label{eq:mass_estimator}
\end{align}
The condition (<ref>) is satisfied for
a freely-precessing isolated bi-axial NS, which may emit
GWs (approximately) at once and twice the spin frequencies ($K=2$).
Another possibility can be found in the model proposed by Jones <cit.>.
He showed that if a tri-axial NS contains a pinned superfluid core and none of the axes of the principal
moments of inertia of the solid crust aligns with the spin angular
velocity vector of the core, then such a NS emits GWs at once and twice the
spin frequencies. Notably, the condition (<ref>) is hold
when such a NS is bi-axial (but none of the axes of the principal moments of
inertia is aligned with the spin axis).
The GWs from these two models of NSs have (See
A_+,1 = 14 h_0sin2θsinιcosι,
A_×,1 = 14 h_0 sin2θsinι,
A_+,2 = 12 h_0sin^2θ( 1+cos^2ι),
A_×,2 = h_0sin^2θcosι,
where the inclination angle $\iota$ is defined as the angle between the rotational axis and
the line-of-sight, and
the misalignment angle $\theta$ is defined as the angle between the principal axis and the angular momentum axis.
The overall amplitude is $h_0 = 4\varepsilon I \omega_1^2/r$ where
$I$ denotes the star's average moment of inertia and the ellipticity $\varepsilon$ quantifies
the degree of the non-axisymmetry.
Assuming these models, we have performed Monte-Carlo simulations
to study the measurement precision of NS mass using Eq. (<ref>) for three-year ET observation <cit.>.
For those simulations, we have randomly selected sets of waveform parameters $\left(\theta, \iota, \phi_R, \psi, \alpha, \delta \right)$
where $\phi_R$ is the reference phase, $\psi$ is the GW polarization phase, and $\alpha$ and $\delta$ are the sky position of the NS.
Uniform distributions are assumed for those parameters.
The NS moment of inertia $I$ and its ellipticity $\varepsilon$ are set to be
$10^{45}$ g$\cdot$cm$^2$ and $10^{-6}$
<cit.>, respectively.
We adopted several representative values of $r$ (1 kpc, 10 kpc, and 50 kpc) and $f_1$ (300 Hz and 500 Hz).
Then, We estimate the measurement error in the NS mass for each simulation.
In the ideal case where all the waveform parameters except for $\phi_{R}$ and $M$ are known, the measurement error in the GW
phase $\Phi_{\alpha}$ is $1/\rho$ where $\rho$ is the signal-to-noise ratio (SNR).
Hence, GW detections with $\rho \gtrsim 100$ for both modes may suffice to
estimate the NS mass as indicated in Eq. (<ref>).
When all the waveform parameters are unknown in advance, the
correlations among the parameters degrade the mass measurement precision.
To improve the mass measurement precision, we have assumed that
the spin frequency, the spin-down rate, the sky position of the NS
are known in advance from electromagnetic observations or GW
observations by the second generation GW detectors such as the advanced LIGO <cit.>,
advanced Virgo <cit.>, and KAGRA <cit.>.
The resulting cumulative distributions of mass measurement precision are plotted in
the Fig. 1 of <cit.>. For example, we
found that the mass of the NS with its spin frequency 500 Hz and its ellipticity
is typically measurable with an accuracy of 20% using the ET.
§ MISALIGNMENT ANGLE
We have assumed a uniform distribution for the misalignment angle
$\theta$ in our Monte Carlo simulations in the previous paper <cit.>.
In the case of a NS containing a pinned superfluid core, there seems to be neither observational nor theoretical constraint on $\theta$.
In fact, a uniform distribution for the misalignment angle
$\theta$ is assumed in this model <cit.>.
A possible range of the misalignment angle $\theta$ in the case of
a freely precessing NS is not well-known.
Two known examples indicate smaller misalignment angles:
$\theta\simeq 3$ degrees (PSR B1828-11) <cit.>
and $\theta\simeq 0.8$
degrees (PSR B1642-03) <cit.>. On the other hand,
the theoretical maximum possible value of $\theta$ is estimated by Eq. (24) of
<cit.> as $\theta_{\rm max} \simeq 10$ degrees $(500
{\rm Hz}/f_1)(u_{\rm break}/{10^{-2}})$ where $u_{\rm break}$ is the
breaking strain of the NS crust. For the
freely precessing magnetars recently discovered in
parameter degeneracies prevent from inferring misalignment angles.
In any case, one may find it questionable to
use a uniform distributions for $\theta$ in the case of a precessing
To see how the misalignment angle affects the measurement
precision of NS mass in our method, we have conducted 10,000 Monte Carlo simulations for three-year ET observations
similar to the previous paper, but for several fixed misalignment angles.
In these simulations,
the sky position, polarization angle, reference phase, and inclination angles
are randomly chosen from uniform distributions.
The distance of $r = 1$ kpc and spin frequency of $f = 500$ Hz are assumed.
Fisher matrix method is used to estimate measurement errors of the waveform parameters.
As in the previous simulations, the frequency, spin-down rate, sky position of each NS are assumed to be known.
The results are shown in Fig. <ref>.
This figure indicates that
masses of more than 70% of NSs are measurable with a precision of $\Delta M/M \simeq 0.2$
if $10^{\circ} \leq \theta \leq 80^{\circ}$.
On the other hand, if the misalignment angles are smaller than $10$
degrees, as suggested by PSR B1828-11 and PSR B1642-03, even the third
generation GW telescope cannot determine the NS mass with sufficient
precision (e.g. $\Delta M/M \simeq 0.8$ for 20% of NSs in the case of $\theta = 3^{\circ}$).
This is because the amplitudes for both modes become smaller for the smaller misalignment angle
as indicated by Eqs. (<ref>) and (<ref>).
The cumulative distribution functions for relative
measurement errors in NS mass $\Delta M_{\text{NS}}/M_{\text{NS}}$.
This figure is obtained by Monte Carlo simulations where the NS sky position,
polarization angle, reference GW phase, and inclination angles are
randomly chosen from uniform distributions.
The distance of $r = 1$ kpc, spin frequency of $f = 500$Hz, and three year
observation by the ET are assumed.
The solid lines correspond to NSs with the misalignment angles of
$\theta = 45^{\circ}$ (black line), $80^{\circ}$ (red line), $87^{\circ}$ (blue line), $89^{\circ}$ (green line),
while the dashed lines correspond to $\theta = 10^{\circ}$ (cyan line) and $3^{\circ}$ (magenta line), respectively.
While a NS with a pinned superfluid core may have any
values of $\theta$, a freely precessing NS may be limited by $\theta < 10$ degrees.
Mass measurement precision for a freely precessing NS
may be worse than that for a NS with a pinned superfluid core.
§ ACKNOWLEDGMENTS
We thank Toshio Nakano for informing us that the misalignment angles
are not determined for the freely precessing magnetars
due to parameter degeneracy.
We also thank Bruce Allen for pointing out
that mass measurement precision would be degraded for freely
precessing stars with $\theta$ close to zero.
This work is supported by JSPS Fellows Grant No. 26.8636 (K. E.),
JSPS Grant-in-Aid for Young Scientists Grant No. 25800126, and the MEXT Grant-in-Aid for
Scientific Research on Innovative Areas (Grant Number 24103005) (Y. I.).
§ REFERENCES
|
1511.00375
|
By combining a parameterized Hermitian matrix, the realignment matrix of the bipartite density matrix $\rho$ and the
vectorization of its reduced density matrices, we present a family of separability criteria, which are stronger than
the computable cross norm or realignment (CCNR) criterion. With linear contraction methods, the proposed criteria can
be used to detect the multipartite entangled states that are biseparable under any bipartite partitions. Moreover, we show by examples that the
presented multipartite separability criteria can be more efficient than the corresponding multipartite
realignment criterion based on CCNR, multipartite correlation tensor criterion and multipartite covariance matrix criterion.
§ INTRODUCTION
Quantum entanglement, as an intrinsical feature of quantum mechanics, provides the basic physical resource in quantum
information and computation <cit.>. It leads to a fundamental problem of how to distinguish
between entangled states and separable states. Nevertheless this problem is extremely difficult to solve and
has been proven to be NP-hard <cit.>. In the last decades, a variety of operational methods have been
proposed to detect entanglement such as the positive partial transpose (PPT) criterion or Peres-Horodecki criterion
<cit.>, realignment criteria <cit.>, correlation matrix or tensor
criteria <cit.>, covariance matrix criteria
<cit.>, entanglement witnesses <cit.>, separability criteria via
measurements <cit.> and so on; see, e.g., <cit.> for comprehensive surveys.
Among the criteria mentioned above, the most popular one is the PPT
criterion <cit.>, which bases on the fact that the partial
transpose of a separable state is positive semidefinite. Moreover,
this criterion is sufficient and necessary for the separability of
$m\times n$ quantum sates with $mn\le 6$ <cit.>.
However, it only provides necessary conditions for separability of
states with higher dimensions, since there exist entangled $2\times
4$ and $3\times 3$ states with positive partial transposes
<cit.>. Thus, it is crucial to check whether a given
PPT state is entangled or not with $mn>6$.
Another well-known one is the computable cross norm or realignment (CCNR) criterion <cit.>, which is
very easy to apply and shows a dramatic ability to detect many PPT entangled states. The multipartite case of this criterion
was considered in <cit.>. The authors showed that the partial realignment can detect the tripartite entangled
states with biseparability under any bipartite partitions, and that the PPT criterion and the CCNR criterion are equivalent to the
permutations of the density matrix's indices. After that, the generalizations of CCNR criterion were investigated in
<cit.>. In <cit.>, the authors made use of the symmetric function of Schmidt coefficients to improve the CCNR criterion further.
Recently, the CCNR criterion was used to study the entanglement conditions for any two-mode continuous-variable state
with permutational symmetry <cit.>.
In <cit.>, Zhang et al. presented a separability criterion (for simplicity, we call it the Z-R criterion) based
on the entry realignment of $\rho-\rho^A\otimes \rho^B$, which was shown to be strictly stronger than the CCNR criterion
and the correlation matrix criterion <cit.>. A generalization of Z-R criterion was studied in <cit.>.
However, the Z-R criterion is still a strong one.
In this paper, based on a parameterized Hermitian matrix, the realignment matrix of the bipartite density matrix $\rho$ and
the vectorization of reduced density matrices $\rho^A$ and $\rho^B$, we construct realignment matrices with larger
scales. Then, separability criteria for bipartite quantum systems that are stronger than the CCNR criterion are presented.
Meanwhile, the new criteria exhibit
comparative detection abilities of entanglement compared with the Z-R criterion. Finally, by linear contraction methods introduced
in <cit.>, the proposed criteria are valid to detect the multipartite entanglement states that
are biseparable under any bipartite partitions, while the Z-R criterion fails to be applied in a similar way. Moreover, two examples show that
the obtained multipartite separability criteria can outperform the corresponding multipartite realignment criterion based on CCNR <cit.>,
multipartite correlation tensor criterion <cit.>, and multipartite covariance matrix criterion <cit.>.
The remainder of the paper is arranged as follows. In Section 2, we
first give the realignment methods, and then introduce the new
separability criteria. Theoretical analysis and an example are employed to show the
efficiency of the presented criteria. In Section 3, the proposed
criteria in Section 2 are extended to the multipartite case.
Meanwhile, two examples are supplemented to show the performance of
the multipartite separability criteria. Finally, some concluding
remarks are made in Section 4.
§ SEPARABILITY CRITERIA FOR BIPARTITE STATES
For a matrix $X=(x_{ij})\in \mathbb{C}^{m\times n}$, the vector $vec(X)$ is defined as
\[
\]
where $T$ stands for the transpose.
Let $Y$ be an $m\times m$ block matrix with $n\times n$ subblocks $Y_{i,j}$, $i,j=1,\cdots,m$. Then the realignment matrix of $Y$ <cit.> is defined as
\[
\mathcal{R}(Y)=\left(vec(Y_{1,1}) \cdots, vec(Y_{m,1}),\cdots, vec(Y_{1,m}), \cdots, vec(Y_{m,m}) \right)^T.
\]
For any quantum state $\rho$ in $\mathbb{C}^{d_A}\otimes \mathbb{C}^{d_B}$, we define
\begin{equation*}
\mathcal{N}_{\alpha,\ell}^{G}(\rho)=\left( {{\begin{array}{*{20}c}
G& {\alpha \omega_{\ell}(\rho^B)^T} \\
{\alpha \omega_{\ell} (\rho^A)} &\mathcal{R}(\rho) \\
\end{array} }} \right),
\end{equation*}
where $G$ is a given Hermitian matrix, $\alpha$ is an arbitrary real
number, $\ell$ is an arbitrary natural number, $\rho^A$ ($\rho^B$)
is the reduced density matrix of the $A$ ($B$) subsystem, and for
any complex matrix $X$,
\[
\omega_{\ell} (X) = \underbrace {\left( {\begin{array}{*{20}{c}}
{vec(X)} & \cdots & {vec(X)} \\
\end{array}} \right)}_{\ell\;\; columns}.
\]
We denote by $||\cdot ||_{\text{tr}}$, $||\cdot||_2$,
$\text{Tr}(\cdot)$ and $E_{m\times n}$ the trace norm (i.e. the sum
of singular values), the spectral norm (i.e. the maximum singular
value), the trace and the $m\times n$ matrix with all entries being
$1$, respectively. The following theorem establishes the new separability criterion
based on $\mathcal{N}_{\alpha,\ell}^{G}(\rho)$ for bipartite states.
Theorem 2.1. Let $G-\alpha^2 E_{\ell\times \ell}$ be positive semidefinite. If the state $\rho$
in $\mathbb{C}^{d_A}\otimes \mathbb{C}^{d_B}$ is separable, then
\[
{\left\| {\mathcal{N}_{\alpha ,\ell }^{G}(\rho )} \right\|_{\text{tr}}} \le 1+\text{Tr}(G).
\]
Proof. Since $\rho$ is separable, it can be written as a convex combination of pure product states, i.e.,
\[
\rho=\sum\limits_i p_i\rho_i^A\otimes \rho_i^B,
\]
where $0\le p_i\le 1,\sum\nolimits_i p_i=1,$ $\rho_i^A$ and $\rho_i^B$
are pure states of the $A$ and $B$ subsystems, respectively. One gets
\begin{align}
\label{eq:th211}\left\|\mathcal{N}_{\alpha ,\ell }^{G}(\rho )\right\|_{\text{tr}}=\left\|\sum\limits_i p_i
\mathcal{N}_{\alpha,\ell }^{G}(\rho_i^A\otimes \rho_i^B )\right\|_{\text{tr}}\le \sum\limits_i p_i \left\|
\mathcal{N}_{\alpha ,\ell }^{G}(\rho_i^A\otimes \rho_i^B )\right\|_{\text{tr}}.
\end{align}
Thus, we only need to give the upper bound of
\[
\left\|\mathcal{N}_{\alpha,\ell}^{G}(\rho_i^A\otimes \rho_i^B )\right\|_{\text{tr}}={\left\| {\left( {\begin{array}{*{20}{c}}
G & \alpha \omega_{\ell}(\rho_i^B)^T \\
{\alpha \omega_{\ell}(\rho_i^A)} & {vec(\rho_i^A)vec(\rho_i^B)^T} \\
\end{array}} \right)} \right\|_{\text{tr}}},
\]
where we have used the equality <cit.>
\[
\mathcal{R}(\rho_i^A\otimes \rho_i^B)=vec(\rho_i^A)vec(\rho_i^B)^T.
\]
Since the equality $||vec(|x\rangle\langle x|)||_2=|||x\rangle||^2_2$
holds for any vector $|x\rangle$, there exist unitary matrices $U$ and $V$ such that
\begin{align*}
U\;vec(\rho_i^A)&=\left( {\begin{array}{*{20}{c}}
{\|vec(\rho_i^A)\|_2} &0& \cdots & {0} \\
\end{array}} \right)^T=\left( {\begin{array}{*{20}{c}}
{1} &0& \cdots & {0} \\
\end{array}} \right)^T,\\
V\;vec(\rho_i^B)&=\left( {\begin{array}{*{20}{c}}
{\|vec(\rho_i^B)\|_2} &0& \cdots & {0} \\
\end{array}} \right)^T=\left( {\begin{array}{*{20}{c}}
{1} &0& \cdots & {0} \\
\end{array}} \right)^T.
\end{align*}
Furthermore, we have
\begin{align}
\label{eq:thm212}\left( {\begin{array}{*{20}{c}}
{{I_{\ell}}} & {} \\
{} & U \\
\end{array}} \right)\mathcal{N}_{\alpha,\ell }^{G}(\rho_i^A\otimes\rho_i^B )\left( {\begin{array}{*{20}{c}}
{{I_{\ell}}} & {} \\
{} & {{V^\dag}} \\
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{G} & {\alpha N} \\
{\alpha M} & P \\
\end{array}} \right): = S^G_{\alpha,\ell},
\end{align}
where $I_{\ell}$ is the ${\ell}\times{\ell}$ identity matrix, and
\[M = {\left( {\begin{array}{*{20}{c}}
1 & 1 & \cdots & 1 \\
0 & 0 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 0 \\
\end{array}} \right)_{d_A^2 \times \ell}},~
N = {\left( {\begin{array}{*{20}{c}}
1 & 0 & \cdots & 0 \\
1 & 0 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
1 & 0 & \cdots & 0 \\
\end{array}} \right)_{\ell \times d_B^2}},~
P = {\left( {\begin{array}{*{20}{c}}
1 & 0 & \cdots & 0 \\
0 & 0 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 0 \\
\end{array}} \right)_{d_A^2 \times d_B^2}}.\]
The matrix $S^G_{\alpha,\ell}$ can be repartitioned as
\begin{align}
\label{eq:thm213} S^G_{\alpha,\ell}=\left( {\begin{array}{*{20}{c}}
{\mathcal{W}_{\alpha,\ell}^G} & {0} \\
{0 } & 0 \\
\end{array}} \right),
\end{align}
\[
\mathcal{W}_{\alpha,\ell}^G=\left( {\begin{array}{*{20}{c}}
{G} & {\alpha E_{\ell\times 1}} \\
{\alpha E_{1\times \ell} } & 1 \\
\end{array}} \right).
\]
Since the matrix $G-\alpha^2
E_{\ell\times \ell}$ is positive semidefinite, from <cit.>, $\mathcal{W}_{\alpha,\ell}^G$ is also positive semidefinite. Due to the fact that
the trace norm of a Hermitian positive semidefinite matrix equals to
its trace, we get, by (<ref>) and (<ref>),
\begin{align*}
\label{eq:purestate}
\left\|\mathcal{N}_{\alpha,\ell}^{G}(\rho_i^A\otimes \rho_i^B
\end{align*}
and then, by (<ref>),
\[
{\left\| {\mathcal{N}_{\alpha ,\ell }^{G}(\rho )} \right\|_{\text{tr}}} \le 1+\text{Tr}(G).
\]
$\hfill \Box$
The CCNR criterion <cit.> claims that any separable state $\rho$ in
$\mathbb{C}^{d_A}\otimes \mathbb{C}^{d_B}$ satisfies the inequality
\[
||\mathcal{R}(\rho)||_{\text{tr}}\le 1.
\]
It is obvious that Theorem 2.1 reduces to the CCNR criterion when $\alpha$ is chosen to be $0$.
For the case $\alpha\neq 0$, the following proposition shows that Theorem 2.1 is stronger than the CCNR criterion.
Proposition 2.1.Theorem 2.1 is stronger than the CCNR criterion when $\alpha \neq 0$.
Proof. For any state $\rho$ in $\mathbb{C}^{d_A}\otimes \mathbb{C}^{d_B}$, we have, by <cit.>,
\[
||\mathcal{N}_{\alpha,\ell}^G(\rho)||_{\text{tr}}\ge ||G||_{\text{tr}}+||\mathcal{R}(\rho)||_{\text{tr}}=\text{Tr}(G)+||\mathcal{R}(\rho)||_{\text{tr}}.
\]
Thus, if $||\mathcal{N}_{\alpha,\ell}^G(\rho)||_{\text{tr}}\le 1+\text{Tr}(G)$, one has $||\mathcal{R}(\rho)||_{\text{tr}}\le 1$.
$\hfill \Box$
By choosing some special parameterized matrices $G$, we
obtain the following two corollaries for detecting entanglement of
bipartite states.
Corollary 2.1. If the state $\rho$ in $\mathbb{C}^{d_A}\otimes \mathbb{C}^{d_B}$ is separable, then
\[
{\left\| {\mathcal{N}_{\alpha ,\ell }^{\ell\alpha^2 I_{\ell}}(\rho )} \right\|_{\text{tr}}} \le 1+\ell^2\alpha^2 .
\]
Proof. We take $G=\ell \alpha^2 I_{\ell}$. Obviously $G-\alpha^2 E_{\ell\times \ell}$ is positive semidefinite.
Then, from Theorem 2.1, the conclusion holds.
$\hfill \Box$
Corollary 2.2. If the state $\rho$ in $\mathbb{C}^{d_A}\otimes \mathbb{C}^{d_B}$ is separable, then
\[
{\left\| {\mathcal{N}_{\alpha ,\ell }^{\alpha^2 E_{\ell\times \ell}}(\rho )} \right\|_{\text{tr}}} \le 1+\ell \alpha^2.
\]
Proof. From $G=\alpha^2 E_{\ell\times \ell}$, we get $G-\alpha^2 E_{\ell\times \ell}=0$. Hence the conclusion holds by Theorem 2.1. $\hfill \Box$
In the following, we consider the problem of the selection of
$\ell$. By adding a row and a column to $G$, we get an
$(\ell+1)\times(\ell+1)$ Hermitian matrix
\begin{align*}
\bar G=\left( {\begin{array}{*{20}{c}}
{\tau} & {\eta^\dag} \\
{\eta} & G \\
\end{array}} \right),
\end{align*}
where $\eta\in\mathbb{C}^\ell, \tau\in \mathbb{R}$. By an analogous proof of Proposition 2.1, we can derive a more general result immediately.
Proposition 2.2. If $\bar G-\alpha^2 E_{(\ell+1)\times (\ell+1)}$ is positive semidefinite,
then the separability criterion ${\left\| {\mathcal{N}_{\alpha ,\ell+1 }^{\bar G}(\rho )} \right\|_{\text{tr}}} \le 1+\text{Tr}(\bar G)$
is stronger than the separability criterion
${\left\| {\mathcal{N}_{\alpha ,\ell }^{G}(\rho )} \right\|_{\text{tr}}} \le 1+\text{Tr}(G).$
From Proposition 2.2, it is obvious that Corollaries 2.1-2.2 can detect more entanglement when $\ell$ gets larger.
The Z-R criterion given in <cit.> is based on the realignment of $\rho-\rho^A\otimes \rho^B$. It states that, for any separable
state $\rho$ in $\mathbb{C}^{d_A}\otimes \mathbb{C}^{d_B}$, the inequality
\[
||\mathcal{R}(\rho-\rho^A\otimes \rho^B)||_{\text{tr}}\le \sqrt{(1-\text{Tr}((\rho^A)^2))(1-\text{Tr}((\rho^B)^2))}
\]
must hold. This criterion is also stronger than the CCNR criterion,
but the exact relation between this criterion and Theorem 2.1 needs
to be established.
The following well-known example shows the power of Corollaries 2.1 and 2.2. Nevertheless, we only report the results from Corollary 2.1,
since Corollary 2.1 has a close performance to Corollary 2.2 by numerical calculations.
Example 2.1. The following $3\times 3$ PPT entangled state was introduced in <cit.>:
\[
\rho=\frac{1}{4}\left(I_9-\sum\limits_{k=0}^4 |\xi_k\rangle\langle\xi_k|\right),
\]
\begin{align*}
(|1\rangle-|2\rangle)|0\rangle,~ |\xi_4\rangle=\frac{1}{3}(|0\rangle+|1\rangle+|2\rangle)(|0\rangle+|1\rangle+|2\rangle).
\end{align*}
We consider the mixture of $\rho$ with white noise:
\[
\rho_p=\frac{1-p}{9}I_9+p\rho.
\]
The CCNR criterion and the Z-R criterion can detect entanglement of $\rho_p$ for $0.8897\le p\le 1$ and $0.8822\le p\le 1$, respectively.
The latter entanglement condition $0.8822\le p\le 1$ can also be obtained by Corollary 2.1 when one of the conditions $\ell\ge 12, \alpha\ge 3.4640$
and $\ell\ge 1, \alpha\ge 11.6590$ holds.
Although Example 2.1 shows that Corollary 2.1 is not better than
the Z-R criterion, their detection abilities for entanglement are
comparative. More importantly, Theorem 2.1 can be extended to
multipartite states by linear contraction methods
<cit.> directly, see Section 3.
§ SEPARABILITY CRITERIA FOR MULTIPARTITE STATES
In this section, Theorem 2.1 and its corollaries are applied for multipartite systems. An $n$ partite sate $\rho$ in
$ \mathbb{C}^{d_1}\otimes \cdots\otimes \mathbb{C}^{d_n}$ is separable (or fully separable) <cit.> if and only if it can be represented as
\[
\rho=\sum\limits_i p_i\rho_i^1\otimes \cdots\otimes \rho_i^n,
\]
where $p_i\ge 0, \sum\nolimits_i p_i=1$, and $\rho_i^1,\cdots,\rho_i^n$ are pure states of subsystems.
Under the condition that the linear map $\mathcal{L}_{(k)}$ acting on the $k$ chosen subsystems is contractive on product
states $\sigma_{j_1}\otimes\sigma_{j_2}\otimes\cdots\otimes\sigma_{j_k}$, where $1\le j_1<j_2<\cdots<j_k\le n$, Horodecki et al.
<cit.> presented the following separability criterion: if a state $\rho$ in
$ \mathbb{C}^{d_1}\otimes \cdots\otimes \mathbb{C}^{d_n}$ is separable, then
\begin{align}
\label{eq:multi} ||\mathcal{L}_{(k)}\otimes \mathcal{I}_{(n-k)}(\rho)||_{\text{tr}}\le 1,
\end{align}
where the map $\mathcal{I}_{(n-k)}$ means that the remaining $n-k$ subsystems are left untouched.
To extend Theorem 2.1 to the multipartite case, we define the following map:
\begin{equation}
\label{contractionmap}\mathcal{M}_{\alpha,\ell}^{G}(\rho)=\frac{1}{1+\text{Tr}(G)}\left( {{\begin{array}{*{20}c}
\text{Tr}(\rho)G& {\alpha \omega_{\ell}(\rho^B)^T} \\
{\alpha \omega_{\ell} (\rho^A)} &\mathcal{R}(\rho) \\
\end{array} }} \right), \;\;\forall\;\; \rho \;\;\text{in}\;\; \mathbb{C}^{d_A}\otimes \mathbb{C}^{d_B},
\end{equation}
where $G$, $\ell$ and $\alpha$ are defined as in Theorem 2.1.
$\text{Tr}(\rho)$ is only to guarantee the linearity of the map in
the extension. From Theorem 2.1, the map (<ref>) is
contractive on any product state $\sigma^A\otimes \sigma ^B$. Thus,
due to Horodeckis' separability criterion (<ref>) for
multipartite systems, we get the following separability criterion based
on $\mathcal{M}_{\alpha,\ell}^{G}$.
Theorem 3.1. If the state $\rho$ in $ \mathbb{C}^{d_1}\otimes \cdots\otimes \mathbb{C}^{d_n}$ is separable, then
\[
\left\|\mathcal{M}_{\alpha,\ell}^{G,(2)}\otimes \mathcal{I}_{(n-2)}(\rho)\right\|_{\text{tr}}\le 1,
\]
where $\mathcal{M}_{\alpha,\ell}^{G,(2)}$ denotes the map $\mathcal{M}_{\alpha,\ell}^{G}$ acting on any chosen 2 subsystems.
Under the combination of the CCNR criterion and (<ref>), Horodecki et al. <cit.> showed that, if
the state $\rho$ in $ \mathbb{C}^{d_1}\otimes \cdots\otimes \mathbb{C}^{d_n}$ is separable, then
\[
\left\|\mathcal{R}^{(2)}\otimes \mathcal{I}_{(n-2)}(\rho)\right\|_{\text{tr}}\le 1,
\]
where $\mathcal{R}^{(2)}$ denotes the realignment map $\mathcal{R}$ acting on any chosen 2 subsystems. For simplicity, we
call it the H-R criterion. Surprisingly, this criterion can detect the tripartite entangled state which is biseparable under
any bipartite partitions <cit.>. From Proposition 2.1, it can be found that the H-R criterion should be weaker than
Theorem 3.1.
The Z-R criterion is based on the realignment of $\rho-\rho^A\otimes
\rho^B$, but this realignment is not linear on quantum states.
Hence, the Z-R criterion cannot be extended to the multipartite case
by contraction methods. Nevertheless, the Z-R criterion can be
generalized to an analog of permutation separability criterion for
multipartite systems <cit.>. However, the obtained
multipartite separability criterion is only valid for systems of even number of subsystems.
The following examples illustrate the efficiency of Theorem 3.1 compared with the H-R criterion <cit.>, the
multipartite correlation tensor criteria <cit.> and <cit.>, and
the multipartite covariance matrix criterion <cit.>.
Example 3.1. Consider the tripartite state <cit.>:
\[
\rho_{ABC}=\frac{1}{8}\left(I_8-\sum\limits_{k=1}^4 |\phi_k\rangle\langle\phi_k|\right),
\]
\[
|\phi_1\rangle=|0,1,+\rangle,~|\phi_2\rangle=|1,+,0\rangle,~|\phi_3\rangle=|+,0,1\rangle,~|\phi_4\rangle=|-,-,-\rangle,~ \pm=\frac{1}{\sqrt{2}}(|0\rangle\pm |1\rangle).
\]
It was shown in <cit.> that this state is biseparable under any bipartite partitions $A|BC$, $B|CA$ and $C|AB$, but it is still entangled.
To verify the efficiency of Theorem 3.1, we consider the mixture of
$\rho_{ABC}$ with white noise:
\[
\rho^p_{ABC}=\frac{1-p}{8}I_8 +p\rho_{ABC}.
\]
Clearly, $\rho^p_{ABC}$ is also biseparable under any bipartite partitions $A|BC$, $B|CA$ and $C|AB$.
Let $\mathcal{R}_{(BC)}$ and $\mathcal{M}_{\alpha,\ell}^{G,(BC)} $
be the maps $\mathcal{R}$ and $\mathcal{M}_{\alpha,\ell}^{G}$ acting
on $B$ and $C$ subsystems, respectively. By the H-R criterion, the
entanglement of $\rho^p_{ABC}$ for $0.873529 \le p\le 1 $ can be
detected. Meanwhile, from Theorem 3.1 and Corollary 2.1, we choose
$G=\ell\alpha^2 I_{\ell}$. Table 1 displays the entangled conditions
for different values of $\alpha$ and $\ell$. It is easy to see
that Theorem 3.1 is more efficient than the H-R criterion. Moreover,
the detection ability of Theorem 3.1 becomes slightly stronger when
$\alpha$ and $\ell$ get larger.
However, the corresponding multipartite correlation tensor criteria <cit.> and the multipartite
covariance matrix criterion <cit.> cannot detect any entanglement in $\rho^p_{ABC}$.
$\alpha$ $\ell=1$ $\ell=10$ $\ell=100$ $\ell=500$
$1$ $0.845476\le p\le 1$ $0.831017\le p\le 1$ $0.828701\le p\le 1$ $0.828483\le p\le 1$
$10$ $0.828701\le p\le 1$ $0.828455\le p\le 1$ $0.828430\le p\le 1$ $0.828428\le p\le 1$
$100$ $0.828430\le p\le 1$ $0.828428\le p\le 1$ $0.828428\le p\le 1$ $0.828427\le p\le 1$
Entanglement conditions of $\rho^p_{ABC}$ from Theorem 3.1 with $G=\ell\alpha^2 I_{\ell}$.
Example 3.2. A perturbation of the GHZ state leads to the
following tripartite qubit state used in <cit.>:
\[
|\psi'_{GHZ}\rangle=\frac{1}{\chi}(|000\rangle+\epsilon |110\rangle+|111\rangle),
\]
where $\epsilon$ is a given real parameter, and $\chi$ denotes the
normalization. We consider the mixture of this state with white
\[
\rho_{GHZ'}^p=\frac{1-p}{8}I_8+ p|\psi'_{GHZ}\rangle \langle \psi'_{GHZ}|.
\]
In the numerical demonstration, the maps $\mathcal{R}_{(BC)}$ and $\mathcal{M}_{\alpha,\ell}^{G,(BC)}$ are used for the H-R criterion
and Theorem 3.1, respectively. From Corollary 2.1, the parameterized matrix $G$ is simply chosen to be $G=\ell\alpha^2I_\ell$ with
$\alpha=\ell=10$. It should be noted that, for tripartite systems, the multipartite correlation tensor criterion <cit.>
is equivalent to the criterion <cit.>.
Table 2 gives the results for different values of $\epsilon$.
It follows that Theorem 3.1 outperforms the H-R criterion, the multipartite correlation tensor criterion, and the multipartite
covariance matrix criterion for different values of $\epsilon$.
$\epsilon$ H-R M-T M-C Theorem 3.1
$0$ $0.3344\le p\le 1$ $0.4118\le p\le 1$ $--$ $0.3334\le p\le 1$
$10^{-5}$ $0.3344\le p\le 1$ $0.4118\le p\le 1$ $p=1$ $0.3334\le p\le 1$
$10^{-3}$ $0.3344\le p\le 1$ $0.4118\le p\le 1$ $0.9981\le p\le 1$ $0.3334\le p\le 1$
$10^{-1}$ $0.3340\le p\le 1$ $0.4118\le p\le 1$ $0.8341\le p\le 1$ $0.3339\le p\le 1$
$1$ $0.3899\le p\le 1$ $0.4256\le p\le 1$ $0.4286\le p\le 1$ $0.3849\le p\le 1$
Entanglement conditions of $\rho_{GHZ'}^p$ with different values of $\epsilon$ from the
H-R criterion $($H-R$)$, the multipartite correlation tensor criterion $($M-T$)$, the multipartite covariance matrix
criterion $($M-C$)$, and Theorem 3.1. The symbol “$--$" denotes that no entanglement can be detected.
§ CONCLUSION
In this paper, by introducing a Hermitian matrix and a real
parameter, we first realigned the density matrix and its reduced
density matrices, and then proposed a family of separability criteria,
which, by a strict proof, are stronger than the well-known CCNR
criterion. In general, the new criteria become more efficient when
the involved parameter $\ell$ gets larger. Second, due to the
special choices of $G$, we gave two simple separability criteria, i.e.,
Corollaries 2.1 and 2.2, which are easy to apply and exhibit, by
examples, comparative abilities of entanglement detection compared
with the Z-R criterion. Finally, by linear contraction methods, the
presented criteria were extended to the multipartite case. Two
examples showed that the presented multipartite separability criteria
can be more efficient than the H-R criterion, the multipartite
correlation tensor criterion, and the multipartite covariance matrix
There are still many problems that need to be further addressed. For example, the exact relations between Theorem 2.1 and the Z-R
criterion should be clarified further. How to choose the parameterized matrix and the parameters $\alpha, \ell$ such that the
proposed criteria can detect more entanglement should be further investigated. Whether some other criteria can be generalized
and improved by the methods used in this paper is also an interesting problem.
§ ACKNOWLEDGMENTS
This work is supported by the NSFC (11105226, 11275131), the
Fundamental Research Funds for the Central Universities (15CX08011A, 24720122013), and the Project-sponsored by SRF for
ROCS, SEM.
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|
1511.00051
|
School of Electrical & Computer Engineering, Purdue University, West Lafayette, IN 47907, USA
Synaptic memory is considered to be the main element responsible for learning and cognition in humans. Although traditionally non-volatile long-term plasticity changes have been implemented in nanoelectronic synapses for neuromorphic applications, recent studies in neuroscience have revealed that biological synapses undergo meta-stable volatile strengthening followed by a long-term strengthening provided that the frequency of the input stimulus is sufficiently high. Such “memory strengthening” and “memory decay” functionalities can potentially lead to adaptive neuromorphic architectures. In this paper, we demonstrate the close resemblance of the magnetization dynamics of a Magnetic Tunnel Junction (MTJ) to short-term plasticity and long-term potentiation observed in biological synapses. We illustrate that, in addition to the magnitude and duration of the input stimulus, frequency of the stimulus plays a critical role in determining long-term potentiation of the MTJ. Such MTJ synaptic memory arrays can be utilized to create compact, ultra-fast and low power intelligent neural systems.
§ INTRODUCTION
With significant research efforts being directed to the development of neurocomputers based on the functionalities of the brain, a seismic shift is expected in the domain of computing based on the traditional von-Neumann model. The $BrainScaleS$ <cit.>, $SpiNNaker$ <cit.> and the IBM $TrueNorth$ <cit.> are instances of recent flagship neuromorphic projects that aim to develop brain-inspired computing platforms suitable for recognition (image, video, speech), classification and mining problems. While Boolean computation is based on the sequential fetch, decode and execute cycles, such neuromorphic computing architectures are massively parallel and event-driven and are potentially appealing for pattern recognition tasks and cortical brain simulations. To that end, researchers have proposed various nanoelectronic devices where the underlying device physics offer a mapping to the neuronal and synaptic operations performed in the brain. The main motivation behind the usage of such non-von Neumann post-CMOS technologies as neural and synaptic devices stems from the fact that the significant mismatch between the CMOS transistors and the underlying neuroscience mechanisms result in significant area and energy overhead for a corresponding hardware implementation. A very popular instance is the simulation of a cat's brain on IBM's Blue Gene supercomputer where the power consumption was reported to be of the order of a few $\sim MW$ <cit.>. While the power required to simulate the human brain will rise significantly as we proceed along the hierarchy in the animal kingdom, actual power consumption in the mammalian brain is just a few tens of watts.
In a neuromorphic computing platform, synapses form the pathways between neurons and their strength modulate the magnitude of the signal transmitted between the neurons. The exact mechanisms that underlie the “learning” or “plasticity” of such synaptic connections are still under debate. Meanwhile, researchers have attempted to mimic several plasticity measurements observed in biological synapses in nanoelectronic devices like phase change memories <cit.>, $Ag-Si$ memristors <cit.> and spintronic devices <cit.>, etc. However, majority of the research have focused on non-volatile plasticity changes of the synapse in response to the spiking patterns of the neurons it connects corresponding to long-term plasticity <cit.> and the volatility of human memory has been largely ignored. As a matter of fact, neuroscience studies performed in <cit.> have demonstrated that synapses exhibit an inherent learning ability where they undergo volatile plasticity changes and ultimately undergo long-term plasticity conditionally based on the frequency of the incoming action potentials. Such volatile or meta-stable synaptic plasticity mechanisms can lead to neuromorphic architectures where the synaptic memory can adapt itself to a changing environment since sections of the memory that have been not receiving frequent stimulus can be now erased and utilized to memorize more frequent information. Hence, it is necessary to include such volatile memory transition functionalities in a neuromorphic chip in order to leverage from the computational power that such meta-stable synaptic plasticity mechanisms has to offer.
Fig. <ref> (a) demonstrates the biological process involved in such volatile synaptic plasticity changes. During the transmission of each action potential from the pre-neuron to the post-neuron through the synapse, an influx of ionic species like $Ca^{2+}, Na^{+}$ and $K^{+}$ causes the release of neurotransmitters from the pre- to the post-neuron. This results in temporary strengthening of the synaptic strength. However, in absence of the action potential, the ionic species concentration settles down to its equilibrium value and the synapse strength diminishes. This phenomenon is termed as short-term plasticity (STP) <cit.>. However, if the action potentials occur frequently, the concentration of the ions do not get enough time to settle down to the equilibrium concentration and this buildup of concentration eventually results in long-term strengthening of the synaptic junction. This phenomenon is termed as long-term potentiation (LTP). While STP is a meta-stable state and lasts for a very small time duration, LTP is a stable synaptic state which can last for hours, days or even years <cit.>. A similar discussion is valid for the case where there is a long-term reduction in synaptic strength with frequent stimulus and then the phenomenon is referred to as long-term depression (LTD).
Such STP and LTP mechanisms have been often correlated to the Short-Term Memory (STM) and Long-Term Memory (LTM) models proposed by Atkinson and Shiffrin <cit.> (Fig. <ref>(b)). This psychological model partitions the human memory into an STM and an LTM. On the arrival of an input stimulus, information is first stored in the STM. However, upon frequent rehearsal, information gets transferred to the LTM. While the “forgetting” phenomena occurs at a fast rate in the STM, information can be stored for a much longer duration in the LTM.
In order to mimic such volatile synaptic plasticity mechanisms, a nanoelectronic device is required that is able to undergo meta-stable resistance transitions depending on the frequency of the input and also transition to a long-term stable resistance state on frequent stimulations. Hence a competition between synaptic memory reinforcement or strengthening and memory loss is a crucial requirement for such nanoelectronic synapses. In the next section, we will describe the mapping of the magnetization dynamics of a nanomagnet to such volatile synaptic plasticity mechanisms observed in the brain.
(a) A synapse is a junction joining the pre-neuron to the post-neuron. Incoming action potential from the pre-neuron results in the influx of ionic elements like $Ca^{2+}$ which, in turn, results in the release of neurotransmitters at the synaptic junction. This causes short-term synaptic plasticity (STP) while frequent action potentials result in long-term potentiation (LTP). (b) Such STP and LTP mechanisms can be related to the psychological model of human memory where memory transitions from a temporary short-term memory (STM) to a long-term memory (LTM) based on the frequency of rehearsal of the input stimulus.
§ FORMALISM
(a) An MTJ structure consists of a FL separated from a PL by a TB. Initially the MTJ synapse is in the low conductive AP state. On receiving an input stimulus it transitions to the high conductive P state conditionally depending on the time interval between the inputs. The STP-LTP behavior can be explained from the energy landscape of the FL. (b) STP behavior exhibited in the MTJ synapse. The MTJ structure was an elliptic disk of volume $\frac{\pi}{4} \times 40 \times 40 \times 1.5 nm ^3$ with saturation magnetization of $M_s = 1000 KA/m$ and damping factor, $\alpha = 0.0122$. The AP and P conductances of the MTJ were taken to be 0.5mS and 1mS respectively. The input stimulus was taken to be $100\mu A$ in magnitude (assuming $\eta=50\%$) and $1ns$ in duration. The time interval between the pulses was taken to be $6ns$. (c) The MTJ synapse undergoes LTP transition incrementally when the interval between the pulses is reduced to $3ns$.
Let us first describe the device structure and principle of operation of an MTJ <cit.> as shown in Fig. <ref>(a). The device consists of two ferromagnetic layers separated by a tunneling oxide barrier (TB). The magnetization of one of the layers is magnetically “pinned” and hence it is termed as the “pinned” layer (PL). The magnetization of the other layer, denoted as the “free layer” (FL), can be manipulated by an incoming spin current $\textbf{I}_s$. The MTJ structure exhibits two extreme stable conductive states – the low conductive “anti-parallel” orientation (AP), where PL and FL magnetizations are oppositely directed and the high conductive “parallel” orientation (P), where the magnetization of the two layers are in the same direction.
Let us consider that the initial state of the MTJ synapse is in the low conductive AP state. Considering the input stimulus (current) to flow from terminal T2 to terminal T1, electrons will flow from terminal T1 to T2 and get spin-polarized by the PL of the MTJ. Subsequently, these spin-polarized electrons will try to orient the FL of the MTJ “parallel” to the PL. It is worth noting here that the spin-polarization of incoming electrons in the MTJ is analogous to the release of neurotransmitters in a biological synapse.
The STP and LTP mechanisms exhibited in the MTJ due to the spin-polarization of the incoming electrons can be explained by the energy profile of the FL of the MTJ. Let the angle between the FL magnetization, $\widehat {\textbf {m}} $, and the PL magnetization, $\widehat {\textbf {m}}_P$, be denoted by $\theta$. The FL energy as a function of $\theta$ has been shown in Fig. <ref>(a) where the two energy minima points ($\theta=0^{0}$ and $\theta=180^{0}$) are separated by the energy barrier, $E_{B}$. During the transition from the AP state to the P state, the FL has to transition from $\theta=180^{0}$ to $\theta=0^{0}$. Upon the receipt of an input stimulus, the FL magnetization proceeds “uphill” along the energy profile (from initial point 1 to point 2 in Fig. <ref>(a)). However, since point 2 is a meta-stable state, it starts going “downhill” to point 1, once the stimulus is removed. If the input stimulus is not frequent enough, the FL will try to stabilize back to the AP state after each stimulus. However, if the stimulus is frequent, the FL will not get sufficient time to reach point 1 and ultimately will be able to overcome the energy barrier (point 3 in Fig. <ref>(a)). It is worth noting here, that on crossing the energy barrier at $\theta=90^{0}$, it becomes progressively difficult for the MTJ to exhibit STP and switch back to the initial AP state. This is in agreement with the psychological model of human memory where it becomes progressively difficult for the memory to “forget” information during transition from STM to LTM. Hence, once it has crossed the energy barrier, it starts transitioning from the STP to the LTP state (point 4 in Fig. <ref>(a)). The stability of the MTJ in the LTP state is dictated by the magnitude of the energy barrier. The lifetime of the LTP state is exponentially related to the energy barrier <cit.>. For instance, for an energy barrier of $31.44KT$ used in this work, the LTP lifetime is $\sim 12.4$ hours while the lifetime can be extended to around $\sim 7$ years by engineering a barrier height of $40KT$. The lifetime can be varied by varying the energy barrier, or equivalently, volume of the MTJ.
The STP-LTP behavior of the MTJ can be also explained from the magnetization dynamics of the FL described by Landau-Lifshitz-Gilbert (LLG) equation with additional term to account for the spin momentum torque according to Slonczewski <cit.>,
\begin{equation}
\label{llg}
\frac {d\widehat {\textbf {m}}} {dt} = -\gamma(\widehat {\textbf {m}} \times \textbf {H}_{eff})+ \alpha (\widehat {\textbf {m}} \times \frac {d\widehat {\textbf {m}}} {dt})+\frac{1}{qN_{s}} (\widehat {\textbf {m}} \times \textbf {I}_s \times \widehat {\textbf {m}})
\end{equation}
where, $\widehat {\textbf {m}}$ is the unit vector of FL magnetization, $\gamma= \frac {2 \mu _B \mu_0} {\hbar}$ is the gyromagnetic ratio for electron, $\alpha$ is Gilbert’s damping ratio, $\textbf{H}_{eff}$ is the effective magnetic field including the shape anisotropy field for elliptic disks calculated using <cit.>, $N_s=\frac{M_{s}V}{\mu_B}$ is the number of spins in free layer of volume $V$ ($M_{s}$ is saturation magnetization and $\mu_{B}$ is Bohr magneton), and $\textbf{I}_{s}=\eta \textbf{I}_Q$ is the spin current generated by the input stimulus $\textbf{I}_Q$ ($\eta$ is the spin-polarization efficiency of the PL). Thermal noise is included by an additional thermal field <cit.>, $\textbf{H}_{thermal}=\sqrt{\frac{\alpha}{1+\alpha^{2}}\frac{2K_{B}T_{K}}{\gamma\mu_{0}M_{s}V\delta_{t}}}G_{0,1}$, where $G_{0,1}$ is a Gaussian distribution with zero mean and unit standard deviation, $K_{B}$ is Boltzmann constant, $T_{K}$ is the temperature and $\delta_{t}$ is the simulation time step. Equation <ref> can be reformulated by simple algebraic manipulations as,
\begin{equation}
\begin{aligned}
\frac{1+\alpha^2}{\gamma} \frac {d\widehat {\textbf {m}}} {dt} =-(\widehat {\textbf {m}} \times \textbf {H}_{eff})- & \alpha (\widehat {\textbf {m}} \times \widehat {\textbf {m}} \times \textbf {H}_{eff})\\
&+\frac{1}{qN_{s}} (\widehat {\textbf {m}} \times \textbf {I}_s \times \widehat {\textbf {m}})
\end{aligned}
\end{equation}
Hence, in the presence of an input stimulus the magnetization of the FL starts changing due to integration of the input. However, in the absence of the input, it starts leaking back due to the first two terms in the RHS of the above equation.
It is worth noting here that, like traditional semiconductor memories, magnitude and duration of the input stimulus will definitely have an impact on the STP-LTP transition of the synapse. However, frequency of the input is a critical factor in this scenario. Even though the total flux through the device is same, the synapse will conditionally change its state if the frequency of the input is high. We verified that this functionality is exhibited in MTJs by performing LLG simulations (including thermal noise). The conductance of the MTJ as a function of $\theta$ can be described by,
\begin{equation}
\begin{aligned}
G = G_{P}.\cos^{2}\left( \frac{\theta}{2}\right)+G_{AP}.\sin^{2}\left(\frac{\theta}{2}\right)
\end{aligned}
\end{equation}
where, $G_{P}$ ($G_{AP}$) is the MTJ conductance in the P (AP) orientation respectively. As shown in Fig. <ref>(b), the MTJ conductance undergoes meta-stable transitions (STP) and is not able to undergo LTP when the time interval of the input pulses is large ($6 ns$). However, on frequent stimulations with time interval as $3 ns$, the device undergoes LTP transition incrementally. Fig. <ref>(b) and (c) illustrates the competition between memory reinforcement and memory decay in an MTJ structure that is crucial to implement STP and LTP in the synapse.
§ RESULTS AND DISCUSSIONS
We demonstrate simulation results to verify the STP and LTP mechanisms in an MTJ synapse depending on the time interval between stimulations. The device simulation parameters were obtained from experimental measurements <cit.> and have been shown in Table I.
TABLE I. Device Simulation Parameters
Parameters Value
Free layer area $\frac{\pi}{4} \times 40 \times 40 nm^2$
Free layer thickness $ 1.5 nm$
Saturation Magnetization, $M_{S}$ 1000 $KA/m$ <cit.>
Gilbert Damping Factor, $\alpha$ 0.0122 <cit.>
Energy Barrier, $E_{B}$ 31.44 $K_{B}T$
Spin polarization strength of PL, $\eta$ 0.5
MTJ conductance 0.5-1$mS$
Pulse magnitude $100\mu A$
Pulse width, $t_{PW}$ $1ns$
Temperature, $T_K$ $300K$
(a) Stochastic LLG simulations with thermal noise performed to illustrate the dependence of stimulation interval on the probability of LTP transition for the MTJ. The MTJ was subjected to 10 stimulations, each stimulation being a current pulse of magnitude $100 \mu A$ and $1 ns$ in duration. However, the time interval between the stimulations was varied from $2 ns$ to $8 ns$. While the probability of LTP is 1 for a time interval of $2ns$, it is very low for a time interval of $8 ns$, at the end of the 10 stimulations. (b) Average MTJ conductance plotted at the end of each stimulation. As expected, the average conductance increases faster with decrease in the stimulation interval. The results have been averaged over 100 LLG simulations.
PPF (average MTJ conductance after 2nd stimulus) and PTP (average MTJ conductance after 10th stimulus) measurements in an MTJ synapse with variation in the stimulation interval. The results are in qualitative agreement to PPF and PTP measurements performed in frog neuromuscular junctions <cit.>.
STM and LTM transition exhibited in a $34 \times 43$ MTJ memory array. The input stimulus was a binary image of the Purdue University logo where a set of 5 pulses (each of magnitude $100\mu A$ and $1ns$ in duration) was applied for each ON pixel. While the array transitioned to LTM progressively for frequent stimulations at an interval of $T=2.5ns$, it “forgot” the input pattern for stimulation for a time interval of $T=7.5ns$.
The MTJ was subjected to 10 stimulations, each stimulation being a current pulse of magnitude $100 \mu A$ and $1 ns$ in duration. As shown in Fig. <ref>, the probability of LTP transition and average device conductance at the end of each stimulation increases with decrease in the time interval between the stimulations. The dependence on stimulation time interval can be further characterized by measurements corresponding to paired-pulse facilitation (PPF: synaptic plasticity increase when a second stimulus follows a previous similar stimulus) and post-tetanic potentiation (PTP: progressive synaptic plasticity increment when a large number of such stimuli are received successively) <cit.>. Fig. <ref> depicts such PPF (after 2nd stimulus) and PTP (after 10th stimulus) measurements for the MTJ synapse with variation in the stimulation interval. The measurements closely resemble measurements performed in frog neuromuscular junctions <cit.> where PPF measurements revealed that there was a small synaptic conductivity increase when the stimulation rate was frequent enough while PTP measurements indicated LTP transition on frequent stimulations with a fast decay in synaptic conductivity on decrement in the stimulation rate. Hence, stimulation rate indeed plays a critical role in the MTJ synapse to determine the probability of LTP transition.
The psychological model of STM and LTM utilizing such MTJ synapses was further explored in a $34\times 43$ memory array. The array was stimulated by a binary image of the Purdue University logo where a set of 5 pulses (each of magnitude $100\mu A$ and $1ns$ in duration) was applied for each ON pixel. The snapshots of the conductance values of the memory array after each stimulus have been shown for two different stimulation intervals of $2.5ns$ and $7.5ns$ respectively. While the memory array attempts to remember the displayed image right after stimulation, it fails to transition to LTM for the case $T=7.5ns$ and the information is eventually lost $5ns$ after stimulation. However, information gets transferred to LTM progressively for $T=2.5ns$. It is worth noting here, that the same amount of flux is transmitted through the MTJ in both cases. The simulation not only provides a visual depiction of the temporal evolution of a large array of MTJ conductances as a function of stimulus but also provides inspiration for the realization of adaptive neuromorphic systems exploiting the concepts of STM and
LTM. Readers interested in the practical implementation of such arrays of spintronic devices are referred to Ref. <cit.>.
§ CONCLUSIONS
The contributions of this work over state-of-the-art approaches may be summarized as follows. This is the first theoretical demonstration of STP and LTP mechanisms in an MTJ synapse. We demonstrated the mapping of neurotransmitter release in a biological synapse to the spin polarization of electrons in an MTJ and performed extensive simulations to illustrate the impact of stimulus frequency on the LTP probability in such an MTJ structure. There have been recent proposals of other emerging devices that can exhibit such STP-LTP mechanisms like $Ag_{2}S$ synapses <cit.> and $WO_{X}$ memristors <cit.>. However, it is worth noting here, that input stimulus magnitudes are usually in the range of volts (1.3V in <cit.> and 80mV in <cit.>) and stimulus durations are of the order of a few msecs (1ms in <cit.> and 0.5s in <cit.>). In contrast, similar mechanisms can be exhibited in MTJ synapses at much lower energy consumption (by stimulus magnitudes of a few hundred $\mu A$ and duration of a few $ns$). We believe that this work will stimulate proof-of-concept experiments to realize such MTJ synapses that can potentially pave the way for future ultra-low power intelligent
neuromorphic systems capable of adaptive learning.
§ ACKNOWLEDGEMENTS
The work was supported in part by, Center for Spintronic Materials, Interfaces, and Novel Architectures (C-SPIN), a MARCO and DARPA sponsored StarNet center, by the Semiconductor Research Corporation, the National Science Foundation, Intel Corporation and by the National Security Science and Engineering Faculty Fellowship.
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1511.00606
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$^1$A. V. Rzhanov Institute of Semiconductor Physics,
Novosibirsk 630090, Russia
$^2$Novosibirsk State University, Novosibirsk,
630090, Russia
$^3$Experimental and Applied Physics, University of
Regensburg, D-93040 Regensburg, Germany
We measure the quantum capacitance and probe thus directly the electronic density of states of the high mobility, Dirac type of two-dimensional electron system, which forms on the surface of strained HgTe. Here we show that observed magneto-capacitance oscillations probe – in contrast to magnetotransport - primarily the top surface. Capacitance measurements constitute thus a powerful tool to probe only one topological surface and to reconstruct its Landau level spectrum for different positions of the Fermi energy.
PACS numbers
1 73.25.+i, 2 73.20.At, 3 73.43.-f
Three dimensional topological insulators (3D TI) represent a new class of materials with insulating bulk and conducting two-dimensional surface states <cit.>. The properties of these surface states are of particular interest as they have a spin degenerate, linear Dirac like dispersion with spins locked to their electron’s $k$-vector <cit.>. Strained epilayers of HgTe, examined here, constitute a 3D TI with high electron mobilities allowing the observation of Landau quantization and quantum Hall steps down to low magnetic fields <cit.>. While unstrained HgTe is a zero gap semiconductor with inverted band structure <cit.>, the degenerate $\Gamma 8$ states split and a gap opens at the Fermi energy $E_F$ if strained. This system is a strong topological insulator <cit.>, explored so far by transport <cit.>, ARPES <cit.>, photoconductivity and magneto-optical experiments <cit.>; also the proximity effect at superconductor/HgTe has been investigated <cit.>. Since these two-dimensional electron states (2DES) have high electron mobilities of several $10^5$ cm$^2$/V$\cdot$s, pronounced Shubnikov- de Haas (SdH) oscillations of the resistivity and quantized Hall plateaus commence in quantizing perpendicular magnetic fields <cit.>, stemming from both, top and bottom 2DES. The origin of the oscillatory resistivity is Landau quantization which strongly modifies the density of states (DoS). Capacitance spectroscopy allows to directly probe the thermodynamic DoS $dn/d\mu$ ($n$ = carrier density, $\mu$ = electrochemical potential) denoted as $D$, of a 3D TI. The total capacitance measured between a metallic top gate and a two-dimensional electron system (2DES) depends, besides the geometric capacitance, also on the quantum capacitance $e^2 D$ , connected in series and reflecting the finite density of states $D$ of the surface electrons <cit.>; $e$ is the elementary charge. Below, the quantum capacitance of the top surface layer is denoted as $e^2 D_{\rm t}$, the one of the bottom layer by $e^2 D_{\rm b}$.
(a) Schematic cross section of the heterostructures studied. The Dirac surface states, shown in red, enclose the strained HgTe layer. (b) Top view of the device showing the Hall geometry with 8 potential probes at the side, covered by the top gate (yellow). (c) Schematic illustration of the “three plate capacitor” formed by the metallic top gate electrode and top and bottom layer (red) of the two-dimensional Dirac surface electrons with density of states $D_{\rm t}$ and $D_{\rm b}$ of top and bottom layer, respectively. The geometric capacitance $C_{\rm tb} = \varepsilon_{\rm HgTe} \varepsilon_0 A / d_{\rm HgTe}$ is determined by the gate area $A$, dielectric constant and thickness of CdTe, SiO$_2$ and Si$_3$N$_4$ layers. The electric field (black arrows) is partially screened by the top surface layer. (d) Equivalent circuit of the three plate capacitor with the quantum capacitances $A e^2 D_{\rm t}$ and $A e^2 D_{\rm b}$ in series with the respective geometrical capacitances. The equivalent circuit is the one introduced in <cit.>, but extended by the quantum capacitance of the bottom surface, $A e^2 D_{\rm b}$. A similar equivalent circuit was recently introduced in <cit.>.
The experiments are carried out on strained 80 nm thick HgTe films, grown by molecular beam epitaxy on CdTe (013) <cit.>. The Dirac surface electrons have high electron mobilities of order $4\times10^5$ cm$^2$/V$\cdot$s. The cross-section of the structure is sketched in Fig. 1a. For transport and capacitance measurements, carried out on one and the same device, the films were patterned into Hall bars with metallic top gates (Fig. 1b). For gating, two types of dielectric layers were used, giving similar results: 100 nm SiO$_2$ and 200 nm of Si$_3$N$_4$ grown by chemical vapor deposition or 80 nm Al$_2$O$_3$ grown by atomic layer deposition. In both cases, TiAu was deposited as a metallic gate. The measurements were performed at temperature $T=1.5$ K and in magnetic fields $B$ up to 13 T. Several devices from the same wafer have been studied. For magnetotransport measurements standard lock-in technique has been applied with the excitation AC current of 10-100 nA and frequencies from 0.5 to 12 Hz. For the capacitance measurements we superimpose the dc bias voltage $V_g$ and a small ac bias voltage (see Fig. 1c) and measure the ac current flowing across our device phase sensitive using lock-in technique. The typical ac voltage was 50 mV at a frequency of 213 Hz. The absence of both leakage currents and resistive effects in the capacitance were controlled by the real part of the measured ac current. In order to avoid resistive effects in high magnetic fields additional measurements at lower frequencies (up to several Hz) were performed.
When the Fermi level (electrochemical potential) is located in the bulk gap the system can be viewed as a “three-plate” capacitor where the top and bottom surface electrons form the two lower plates (see Fig. 1c and the corresponding equivalent circuit in Fig. 1d). From this equivalent circuit follows that, as long as $D_b$ does not vanish, the measured total capacitance is more sensitive to changes of $D_{\rm t}$ than of $D_{\rm b}$; the explicit connection between $D_t$, $D_b$ and the total capacitance $C$ is given in the Supplemental Material. The ratio of $(dC/dD_{\rm t}) / (dC/dD_{\rm b})=\Bigl( A D_{\rm b} e^2 + C_{\rm tb} \Bigr)^2 / C_{\rm tb}^2$ is significantly larger than unity since $A D_{\rm b} e^2 + C_{\rm tb}$ is (at $B = 0$) at least a factor of two larger than $C_{\rm tb}$ . Here, $C_{\rm tb}$ is the geometric capacitance $C_{\rm tb} = \varepsilon_{\rm HgTe} \varepsilon_0 A / d_{\rm HgTe}$ between top and bottom layer, with $\varepsilon_{\rm HgTe} \approx 21$ <cit.>, the dielectric constant of HgTe, $d_{\rm HgTe}$ is the thickness of the HgTe film, $A$ is the gated TI area and $\varepsilon_0$ the dielectric constant of vacuum. Therefore, the measured capacitance reflects primarily the top surface’s DoS, $D_{\rm t}$. In the limit $C_{\rm tb} \rightarrow 0$ (infinite distance to bottom surface) or ($e^2 D_{\rm b}\rightarrow 0$) (no charge on bottom surface) the total capacitance $C$ is given by the expression usually used to extract the DoS $D$ of a two-dimensional electron system: $1/C = 1/C_{\rm gt} + 1/A e^2 D$ with the geometric capacitance $C_{\rm gt}=\varepsilon_{\rm gt} \varepsilon_0 A / d_{\rm gt}$, where $\varepsilon_{\rm gt}$ is the dielectric constant of the layers between gate and top 2DES, and $d_{\rm gt}$, is the corresponding thickness <cit.>. Note that $C_{\rm gt} \ll e^2 D_{\rm t}$ therefore the variations of the DoS cause only small changes of the measured value of $C$.
(a) Typical $\rho_{xx}(V_g)$ and $\rho_{xy}(V_g)$ traces measured at $B = 0 $ and $B = 4$ T, respectively. $\rho_{xx}$ displays a maximum and $\rho_{xy}$ changes sign in the vicinity of the charge neutrality point (CNP). (b) Capacitance measured at $B= 0 $ and $B = 2$ T. The pronounced minimum of the zero field capacitance corresponds to the reduced DoS when $E_F$ is in the gap. From that we can estimate the band edges to be at $E_v = 2.2$ V and $E_c = 4.4$ V. The quantum oscillations of the capacitance stem from the oscillations of the DoS. (c) Hall conductivity $\sigma_{xy}(V_g)$ measured for $B = 4$ T (black), 7 T (red), and 10 T (green). Quantized steps occur both on the hole side and on the electron side. (d) 2D color map of $\sigma_{xx}(V_g, B)$ . For the plot each $\sigma_{xx}(V_g)$ trace has been normalized with respect to the average value, set to 1. The red color stands for $\sigma_{xx}$ maxima while blue color displays minima. From the distance between blue $\sigma_{xx}$ minima we extract a filling rate of $\alpha_{\rm total} = 7.6\times 10^{10}$ cm$^{-2}$/V (see text). This allows constructing a LL fan chart (dashed yellow lines) which describes the low filling factors $\nu$ well. (e) 2D color map of $\delta C(B) = C(B) - C(0)$ shown as function of $B$ and $V_g$. As in (d), the red color stands for LL maxima (DoS maxima) while blue color display the gaps between LLs. The LL fan chart is the same as in d. CNP, $E_c$, and $E_v$ are marked on the x-axis.
Typical $\rho_{xx}$ and $\rho_{xy}$ traces as function of the gate voltage $V_g$ are shown in Fig. 2a. $\rho_{xx}$ displays a maximum near $V_g = 1.5$ V, whereas $\rho_{xy}$ changes sign; this occurs in the immediate vicinity of the charge neutrality point (CNP)<cit.>. The corresponding capacitance $C(V_g)$ at $B = 0$ in Fig. 2b exhibits a broad minimum between 2.2V̇ and 4.5 V and echoes the reduced density of states $D_{\rm t}$ and $D_{\rm b}$ of the Dirac 2DES when the Fermi energy $E_F$ is in the gap of HgTe. For $V_g>4.5$ V, $E_F$ moves into the conduction band where surface electrons coexist with the bulk ones. There, the capacitance (and thus the DoS) is increased and grows only weakly with increasing $V_g$. Reducing $V_g$ below 2.2 V shifts $E_F$ below the valence band edge so that surface electrons and bulk holes coexist. A strong positive magnetoresistance, a non-linear Hall voltage and a strong temperature dependence of $\rho_{xx}$ provide independent confirmation that $E_F$ is in the valence band<cit.>. Due to the valley degeneracy of holes in HgTe and the higher effective mass, the DoS, and therefore the measured capacitance $C$ is highest in the valence band.
For $B$ well below 1 T both $C(V_g)$ and $\rho_{xx}(V_g)$ start to oscillate and herald the formation of Landau levels (LLs). The $C(V_g)$ trace oscillates around the zero field capacitance, shown for $B=2$ T in Fig. 2b. These oscillations, reflecting oscillations in the DoS, are more pronounced on the electron side (to the right of the CNP). This electron-hole asymmetry stems mainly from the larger hole mass, leading to reduced LL separation on the hole side. At higher fields Hall conductivity $\sigma_{xy}$ and resistivity $\rho_{xy}$ (not shown) become fully quantized, for both electron and hole side. $\sigma_{xy}(V_g)$, shown for $B = 4$ T, 7 T, and 10 T in Fig. 2c, shows quantized steps of height $e^2/h$ , ( $h$ = Planck’s constant) as expected for spin-polarized 2DES.
An overview of transport and capacitance data in the whole $V_g$ and $B$ space is presented in Figs. 3d and 3e as 2D color maps (see the Supplementary Material for additional data). We start with discussing the $\sigma_{xx}$ data, calculated from $\sigma_{xx} = \rho_{xx}/(\rho_{xx}^2 + \rho_{xy}^2)$, in Fig. 2d first. The sequence of $\sigma_{xx}$ maxima and minima is almost symmetrical to the CNP where the electron and hole densities are equal. At fixed magnetic field the distance $\Delta V_g$ between neighboring $\sigma_{xx}$ minima corresponds to a change in a carrier density $\Delta n$ from which we can calculate, with the LL degeneracy $eB/h$, the filling rate $dn/dV_g = \alpha_{\rm total} = 7.6\times10^{10}$ cm$^{-2}$/V at 10 T. Comparison of electron densities extracted in the classical Drude regime with densities taken from the periodicity of SdH oscillations have shown that $\sigma_{xx}$ oscillations at high $B$ reflect the total carrier density in the TI, i.e. charge carrier densities in the bulk plus in top and bottom surfaces <cit.>. We therefore conclude that the filling rate $\alpha_{\rm total}$ describes the change of the total carrier density $n$ with $V_g$ and is directly proportional to the capacitance per area $C/A = edn/dV_g$. $\alpha_{\rm total}= 7.6\times10^{10}$ cm$^{-2}$/V corresponds to a capacitance of $C/A = e \alpha_{\rm total} = 1.22\times10^{-4}$ F/m$^{2}$, a value close to the calculated capacitance $C_{\rm gt}^{\rm calc}/A = 1.45\times 10^{-4}$ F/m$^{2}$ using thickness and dielectric constant of the layers (see Figs. 1a and c and the Supplemental Material). Using this $\alpha_{\rm total}$ extracted at 10 T, the Landau level fan chart, i.e. the calculated positions of the $\sigma_{xx}$ minima as function of $V_g$ and $B$, fits the data for low filling factors $\nu$ quite well. For filling factors larger than $\nu = 2$ on the electron side the fan chart significantly deviates from experiment and is discussed using higher resolution data below. On the hole side where the SdH oscillations stem from bulk holes, fan chart and experimental data match almost over the whole $(V_g, B)$ range. Remarkably, minima corresponding to odd filling factors are more pronounced than even ones, reflecting the large hole $g$-factor. We now turn to the magneto-capacitance data $\delta C = C(V_g, B) - C(V_g, B = 0)$ shown in Fig. 2e. The data are compared with the same fan chart derived from transport. On the electron side, the experimental $\delta C$ minima display a reduced slope compared to the calculated fan chart, pointing to a reduced filling rate. This is a first indication that the capacitance does not reflect the total carrier density in the system but predominantly the fraction of the top 2DES only. On the hole side the LL fan chart fits the data quite well but in contrast to transport, LL features are less well resolved there. This asymmetry is connected to the different effective masses; the enhanced visibility in transport is due to that fact that SdH oscillations depend on $D^2$ while the capacitance depends on $D$ only.
(a) 2D color map of $d^2 \sigma_{xx}/dV_g^2$ at low fields between 0 and 1.5 T. The second derivative brings out quantum oscillation more clearly. In each row the root mean square value was normalized to 1. Red color again indicates large $\sigma_{xx}$ values, blue color low $\sigma_{xx}$, corresponding to LL gaps. The experimental data charts can be fitted by two fan charts originating at $V_g^0$ and $V_g^1$. The different slope of the two fan charts resembles the different filling rates $\alpha_{\rm top}^{\rm gap}$ and $\alpha_{\rm top}^{\rm bulk}$ (see text). The red arrows and the two dashed lines starting at $V_g \approx 4.4$ V mark extra features discussed in the text; the gap region and CNP are marked by dashed vertical lines. (b) Corresponding $\delta C(V_g, B)$ data. The two fan charts are the same as the ones in (a). (c) As (a), but for fields between 0 and 4 T. For $V_g > 4.5$ V the data can no longer be described by simple fan charts; the pattern is entangled in a complicated way, suggesting that electrons of top and bottom surface and the bulk contribute to the conductivity oscillations. (d) Capacitance data corresponding to (c) show a quite regular Landau fan chart. The yellow and white lines in the gap and for $E_F < E_C$, respectively, correspond to the same filling rates $\alpha_{\rm top}^{\rm gap}$ and $\alpha_{\rm top}^{\rm bulk}$ as in (a) and (b). This indicates that the capacitance oscillations stem from the top surface only.
Previous transport experiments have shown that the periodicity of the SdH oscillations is changed at low magnetic fields, reflecting a reduced carrier density. Tentatively, this was ascribed to SdH oscillations stemming from the top surface only <cit.>. When top and bottom surface electrons have different electron densities and mobilities it is expected that Landau level splitting - in sufficiently low $B$ - is in transport first resolved for the layer with the higher density and mobility, i.e. higher partial conductivity and lower LL level broadening. This expectation is consistent with previous experimental observation <cit.>. If, indeed, the low field SdH oscillations resemble the carrier density of a single Dirac surface, capacitance oscillations which probe preferentially the top surface and SdH oscillations should have the same period. Thus we compare $\sigma_{xx}$ and capacitance $\delta C$ in Fig. 3a and 3b at low $B$ up to 1.5 T. In this low field region the capacitance (Fig. 3b) shows overall uniform oscillations of $\delta C$. The position of the maxima, corresponding to different LLs, is perfectly fitted by two fan charts featuring a distinct crossover at about $V_g = 4.4$ V. The crossover is due to $E_F$ entering the conduction band which causes a reduced filling rate of the electrons responsible for the quantum oscillations at $V_g>4.4$ V. Apparently, the quantum oscillations are caused by the top surface electrons only while the bulk electrons and electrons on the bottom surface merely act as a reservoir. Note, that the filling rate into the top surface state, probed in experiment, gets reduced when the filling rate into bulk states $\neq 0$, since the total filling rate must be constant <cit.>. From the distance of $\delta C$ maxima at constant $B$ we can extract the filling rate $\alpha_{\rm top}^{\rm gap}$ in the gap (2.2 V $<V_g <$ 4.4 V) and when $E_F$ is in the conduction band, $\alpha_{\rm top}^{\rm bulk}$. From Fig. 3b we obtain $\alpha_{\rm top}^{\rm gap}=5.25\times 10^{10}$ cm$^{-2}$/V and $\alpha_{\rm top}^{\rm bulk}=3.3\times 10^{10}$ cm$^{-2}$/V. This means that in the gap $\alpha_{\rm top}^{\rm gap}/\alpha_{\rm total}=70$ %
of the total filling rate apply to the top surface while the remaining 30 %
can be ascribed to the bottom surface. The reduced filling rate $\alpha_{\rm top}^{\rm bulk}$ for $E_F$ in the conduction band is $0.44\cdot \alpha_{\rm total}$ and hence the remaining filling rate of 56 %
is shared between bulk and back surface filling. We note that we obtain reasonable values for the filling rates only when we assume spin-resolved LL degeneracy. Since there is no signature of spin splitting down to 0.6 T, where the oscillations get no longer resolved one can conclude that the quantum oscillations always stem from non-degenerate LLs, proving the topological nature of the charge carriers. The extrapolation of the two fan charts towards $B \rightarrow 0$ defines two specific points on the $V_g$-axis, denoted as $V_g^{0}=1.25$ V and $V_g^{1}=-0.5$ V. These points correspond to vanishing electron density on the top surface $n_{\rm top}$ in case the respective filling rates $\alpha_{\rm top}^{\rm gap}$ and $\alpha_{\rm top}^{\rm bulk}$ would stay constant over the entire $V_g$ range. This is not the case as $\alpha_{\rm top}^{\rm bulk} = {\rm constant}$ only applies for $E_F$ in the conduction band and $\alpha_{\rm top}^{\rm gap} = {\rm constant}$ for $E_F$ in the gap. Moving $E_F$ into the valence band greatly reduces this filling rate. Therefore $V_g^{0}$ and $V_g^{1}$ correspond only to virtual zeroes of the electron density while the real one is much deeper in the valence band.
To get a better resolution of the low field SdH oscillation we plot $d^2 \sigma_{xx} / dV_g^2$ in Fig. 3a; as before red regions indicate $\sigma_{xx}$ maxima. The same LL fan chart in Figs. 3b and 3a, fits both, transport and capacitance, quite well. Hence, at sufficiently low $B$, both $\sigma_{xx}$ and $\delta C$ oscillations resemble the carrier density of the top surface only. However, even at low $B$ striking deviations in $d^2 \sigma_{xx} / dV_g^2$ pop up which get more pronounced at higher $B$ (see below). Two faint lines at $B = 1 \ldots 1.5$ T and $V_g = 2.5 \ldots 3.5$ V which appear between the LLs of the top surface and are marked by red arrows in Fig. 3a, are ascribed to SdH oscillations stemming from the bottom surface. More differences occur once $E_F$ enters the conduction band ($V_g > 4.4$ V). New maxima appear in the fan chart and form rhomb like structures. These anomalous structures, completely absent in capacitance, get even more intriguing at higher $B$, displayed in Fig. 3c. Data taken at $B$ up to 4 T, displayed in Figs. 3c and 3d, show marked differences between transport and capacitance. $\delta C$ maxima, given by the red regions in Fig. 3d are, as before, well described by the same two LL fan chart with the same filling rates as in Figs. 3a and b. The filling factors given on top of Fig. 3d are the ones of the electrons in the top surface only, while
the filling factors given in Fig. 3c are the ones of the total carrier density, determined by the total filling rate and the position of the $\sigma_{xy}$ plateaus in Fig. 2c. The transport data for $E_F$ in the gap show splitting of the Landau levels and for $V_g > 4.4$ V, i.e. for $E_F$ in the conduction band, a very complex structure with crossing LLs evolves, which is strikingly different from the one observed in the capacitance data. We thus conclude that in transport experiments the three available transport channels (top, bottom surface electrons, bulk electrons) contribute to the signal and lead to a complicated pattern of the quantum oscillations as a function of $B$ and $V_g$. The oscillations of the quantum capacitance, in contrast, stem preferentially from the top surface and allow probing the LL spectrum of a single Dirac surface in a wide range of $B$ and $V_g$. Corrections to that likely occur at high $B$ and $V_g$: a level splitting at ($V_g \approx 7$ V, $B\approx 3$ T) in Fig. 3d suggest that signals from bulk or back surface can affect also $\delta C$ at higher $B$, although to a far lesser degree when compared to transport.
In summary, we present first measurements of the quantum capacitance of a TI which directly reflect the DoS of Dirac surface states. The oscillations of the quantum capacitance in quantizing magnetic fields allow tracing the LL structure of a single Dirac surface. The complimentary information provided by transport and capacitance experiments is promising in getting a better understanding of the electronic structure of TIs, the latter being particularly important for potential applications of this new class of materials.
We acknowledge funding by the Elite Network of Bavaria and by the German Science Foundation via SPP 1666. This work was partially supported by RFBR grants No 14-02-31631, 15-32-20828 and 15-52-16008.
Supplementary Information: “Probing quantum
capacitance in a 3D topological insulator"
D. A. Kozlov, D. Bauer, J. Ziegler, R. Fischer, M. L. Savchenko,
Z. D. Kvon, N. N. Mikhailov, S. A. Dvoretsky and D. Weiss
§ SAMPLES DETAILS
The experiments were carried out on strained 80 nm thick HgTe films,
grown by molecular beam epitaxy (MBE) on CdTe (013)
<cit.>. The Dirac surface electrons have high
electron mobilities of order $\mu\approx
4\times10^5$ ńm$^2$/V$\cdot$s <cit.>. The cross
section of a structure is sketched in
Fig. <ref>a. For transport and capacitance
measurements, carried out on one and the same device, the films
were patterned into Hall bars with metallic top gates
(Fig. <ref>b).
For gating, two types of dielectric layers were used, giving
similar results: 100 nm SiO$_2$ and 200 nm of Si$_3$N$_4$ grown
by chemical vapor deposition or 80 nm Al$_2$O$_3$ grown by atomic
layer deposition (not shown in Fig. <ref>a). In
both cases, TiAu was deposited as a metallic gate. The
measurements of the total capacitance $C$ between the gate and the
HgTe layer were performed at temperature $T=1.5$ K and in
magnetic fields up to 13 T. Several devices from the same wafer
have been studied. For magnetotransport measurements standard
lock-in technique has been applied with excitation AC currents of
10-100 nA at frequencies $\omega/2\pi$ ranging from 0.5 to
12 Hz. For the capacitance measurements we superimpose the DC
bias voltage with a small AC bias voltage (see Fig. 2c) and
measure with lock-in technique the AC current flowing across our
device phase sensitively. The typical AC voltage was 50 mV at a
frequency of 213 Hz. The absence of both leakage currents and
resistive effects in the capacitance were controlled by the real
part of the measured AC current. These resistive effects occur
when the condition $RC \omega <<1$ becomes invalid, for example
when the series resistance $R$, i.e. the resistance of the two-dimensional electron gas
increases at integer filling factors and high magnetic fields. In
order to suppress resistive effects in high magnetic fields
additional measurements at lower frequencies (down to several Hz)
were performed (see below).
a, Cross section of the heterostructure with SiO$_2$/Si$_3$N$_4$ insulator. The
Dirac surface states, shown in red, enclose the strained HgTe
layer. b, Top view of the device showing the Hall geometry with 10
potential probes, covered by the top gate (yellow); the gate is shown in more detail in the optical micrograph on the bottom.
§ SAMPLES BAND DIAGRAM AND EQUIVALENT CIRCUIT
§.§ Electrostatic and band diagram
Following the papers of Stern <cit.>, Smith <cit.>
and Luryi <cit.> we introduce an equivalent circuit for the
sample's capacitance based on simple one-dimensional electrostatics. The
discussion is limited to the case where the Fermi level is
located in the gap, so that contributions of bulk holes and
electrons are absent. The top and bottom surface layers are
treated as infinitely thin, negatively charged surfaces with
finite density of states (DoS), i.e. we ignored the spatial
distribution of the surface electrons. Finally, we treated the
insulating layer as uniform with an average dielectric constant
and with the total thickness of the layers given in
Fig. <ref>.
A cartoon of the resulting simplified band diagram is presented
for zero gate voltage $V_g$ in Fig. <ref>a. Without
applied $V_g$ the Fermi level is constant across the structure.
The density of electrons $n_{\rm t}$ and $n_{\rm b}$ on top and
bottom surface, respectively, depends on the position of the Fermi
level with respect to the Dirac point (which is, other than
sketched in Fig. <ref>a below the valence band edge).
For the sake of simplicity we assume that the densities are the
same at $V_g = 0$. This simplification is justified since we are
only interested in changes of the carrier density with gate
voltage and not in absolute values of the carrier density.
a and b, Simplified band diagrams of the structure
studied for $E_{\rm F}$ is in the bulk gap at $V_g = 0$
(panel a) and $V_g>0$ (panel b). c, d and e, Equivalent
circuit for a simple capacitor where one plate consists of a two-dimensional electron gas (c), for our three plate capacitor with $E_{\rm F}$ in the bulk gap (d) and for a three plate capacitor with $E_{\rm F}$ in the conduction band or valence band, respectively (e).
For an applied gate voltage $V_g \neq 0 $ we assume that the
voltage drop occurs exclusively across the SiO$_2$/Si$_3$N$_4$
insulator leading to an energy difference $e V_g$ between the
Fermi level in the gate metal and the HgTe layer. Here, $e$ is the
elementary charge. Applying a positive gate voltage $V_g >0$
increases the carrier density in top and bottom layer by $\Delta
n_{\rm top}$ and $\Delta n_{\rm bott}$, respectively. Due to
screening of the electric field by the top surface layer the
induced electron density is higher in the top layer than in the
bottom one; while the Fermi level throughout the HgTe layer is
constant the different electron density in top and bottom
surface layer causes an electrical potential drop between bottom
and top surfaces of $e\phi_{\rm HgTe}$. The change of carrier
density in top and bottom layer can be written as $\Delta \mu_{\rm
top (b) }= \Delta n_{\rm top (bott)} / D_{\rm t (b)}$, where
$D_{\rm t(b)}$ is the density of states (DoS) of the electrons on
the top (bottom) surface. Thus the following relation holds:
\begin{equation} \label{form1}
\Delta \mu_{\rm top} = \Delta \mu_{\rm bot} + e \phi_{\rm
\end{equation}
Using charge neutrality and Gauss's law of electrostatics the potential drop in the HgTe layer can be calculated from the electric field $E_{\rm HgTe} = e \Delta n_{\rm b} /
\varepsilon_{\rm HgTe} \varepsilon_0$, induced by the additional
electron density $\Delta n_{\rm b}$ on the bottom layer:
\begin{equation} \label{form2}
\phi_{\rm HgTe} = E_{\rm HgTe} d_{\rm HgTe} =
e \Delta n_{\rm bot} d_{\rm
HgTe} / \varepsilon_{\rm HgTe} \varepsilon_0 .
\end{equation}
$\varepsilon_{\rm HgTe}$ is the dielectric constant of HgTe and
$d_{\rm HgTe}$ is the distance between the electron wave functions of top and bottom surfaces layers, approximated by the thickness of the HgTe layer, $d_{\rm HgTe}$. Combining formulas <ref> and
<ref> gives a relation between $\Delta n_{\rm
top}$ and $\Delta n_{\rm bot}$:
\begin{equation}
\Delta n_{\rm top} / D _{\rm t}= \Delta n_{\rm bot} / D _{\rm b} + e^2\Delta n_{\rm bot} d_{\rm
HgTe} / \varepsilon_{\rm HgTe} \varepsilon_0 .
\end{equation}
This is the expression given by Luryi <cit.> generalized
for non-parabolic dispersion of the surface electrons.
§.§ The equivalent circuit
In the main text we discuss the equivalent circuit of a three plate
capacitor as a model for the capacitance measured in a topological
insulator with metallic gate and top and bottom surface layers.
The equivalent circuit is applicable when the Fermi level is in
the bulk gap of HgTe and is shown in Fig. 1d and the Supplementary
Figure <ref>d. A similar equivalent circuit has been
recently suggested for the same system but a slightly different
configuration <cit.>. The four capacitors drawn in
Supplementary Figure <ref>d represent either
geometrical ($C_{gt}$ and $C_{tb}$) or quantum ($A e^2 D_{\rm t}$
and $A e^2 D_{\rm b}$) capacitances. Here, $C_{gt} = A
\varepsilon_0 \varepsilon_{\rm ins} / d_{\rm ins}$ is the geometrical
capacitance associated with the insulating layer between the gate
and the top surface of the HgTe layer; $C_{tb} = A \varepsilon_0
\varepsilon_{\rm HgTe} / d_{\rm HgTe}$ is responsible for the
potential drop across the HgTe layer between electrons on the top
and bottom surface; $A e^2 D_{\rm t}$ and $A e^2 D_{\rm b}$ are
the corresponding quantum capacitances representing the finite DoS
of the corresponding HgTe surface states; in all cases $A$ is the
gate area. The equivalent circuit in Fig. <ref>d
describes the electrostatics of the system: Indeed, the charge on
the quantum capacitors is the one induced by the gate voltage,
$\Delta n_{\rm top}$ and $\Delta n_{\rm bot}$ and the voltage
across the geometrical ones is, as it should be, proportional to
the corresponding induced charge and the thickness of the
respective dielectric layer.
The circuit becomes simpler if one removes the influence of the
bottom surface: this can be done by increasing the distance
between top and bottom layer ($C_{\rm tb} \rightarrow 0$) or by
removing the charge carriers ($e^2 D_{\rm b}\rightarrow 0$). Then only two capacitors are present in the reduced circuit
(Fig. <ref>c); this is the situation relevant for a
gated two-dimensional electron system. In contrast, if the Fermi
level is lying in the bulk electron or hole band (see
Fig. <ref>e) the situation becomes more complex. In
this case bulk carriers are characterized by a particular wave
function across the HgTe layer. A simple approach to model this is
to replace the capacitor $C_{\rm tb}$ by two capacitors $C_{\rm
tc}$ and $C_{\rm cb}$ representing the potential drop between the
maximum of the bulk carriers' wave function (denoted by subscript
"c" at a point around the center of the HgTe layer) and the top and bottom of the HgTe layer, respectively. For
this the relation $C_{\rm tb}^{-1}=C_{\rm tc}^{-1} + C_{\rm
cb}^{-1}$ holds.
The total sample's capacitance $C$ for the circuit shown in
Fig. <ref>c can be easily derived using Ohm's law and is given by:
\begin{equation}\label{CSample}
\frac{A}{C}
= %\frac{A}{C_{\rm gt}} + \frac{A}{C_1 + C_2} =
\frac{A}{C_{\rm gt}} + \frac{1}{e^2 D_{\rm t}
+ \frac{1}{ \frac{A}{C_{\rm tb}} + \frac{1}{e^2 D_{\rm b}}}} =\\
A C_{\rm gt}^{-1} +
\Bigl[
e^2 D_{\rm t} + \Bigl( A C_{\rm tb}^{-1} + (e^2 D_{\rm b})^{-1} \Bigr)^{-1}
\Bigr]^{-1}
\end{equation}
In the actual device the relation $C_{\rm gt}/A \ll e^2 D_{\rm t},
e^2 D_{\rm b}$ holds, therefore $C \approx C_{\rm gt}$
and any variation of the DoS, e.g. in a magnetic field, leads only to a small change of the total capacitance $C$. An important result which can be derived from Eq. <ref>
is the sensitivity of the total capacitance $C$ to a change
of the DoS of top and bottom layer, $D_{\rm t}$ and $D_{\rm b}$, respectively:
\begin{equation}
dC/dD_{\rm t}=
\frac{
A e^2 C_{\rm gt}^2 C_{\rm tb}^2
( C_{\rm tb} + A e^2 D_{\rm b} )^2
\Bigl[
C_{\rm gt}(C_{\rm tb}+A e^2 D_{\rm b} )+
A e^2 \bigl( C_{\rm tb} ( D_{\rm b} + D_{\rm t} ) + A e^2 D_{\rm b} D_{\rm t} \bigr)
\Bigr]^{2}
\end{equation}
\begin{equation}
dC/dD_{\rm b}=
\frac{
A e^2 C_{\rm gt}^2 C_{\rm tb}^2
\Bigl[
C_{\rm gt}(C_{\rm tb}+A e^2 D_{\rm b} )+
A e^2 \bigl( C_{\rm tb} ( D_{\rm b} + D_{\rm t} ) + A e^2 D_{\rm b} D_{\rm t} \bigr)
\Bigr]^{2}
\end{equation}
The ratio of both quantities
\begin{equation}
\frac{
dC/dD_{\rm t}
dC/dD_{\rm b}
\frac{
(C_{\rm tb} + A D_{\rm b} e^2)^2
C_{\rm tb}^2
\Bigl(1 +
\frac{
A D_{\rm b} e^2 }
{ C_{\rm tb}}
\Bigr)^2
\end{equation}
is always larger than 1 as long as $D_{\rm b}\neq 0$ which means that the total capacitance $C$ is more sensitive to oscillations of the top layer, e.g., the same oscillation amplitude of $D_{\rm
t}$ and $D_{\rm b}$ will lead to significantly different
oscillation amplitudes in the measured total capacitance $C$.
§.§ Numeric evaluations
Here, we provide the values of the capacitances of our devices.
The insulating layer underneath the metallic gate consists of
Si$_3$N$_4$ (200 nm layer with $\varepsilon = 7.5$), SiO$_2$
(100 nm with $\varepsilon = 3.5$), CdTe (40 nm with $\varepsilon
= 10.2$), CdHgTe (20 nm with $\varepsilon \approx 13$) and HgTe
(5-7 nm with $\varepsilon \approx 21$) and $C_{\rm gt}^{\rm calc}
/ A = 1.45 \times 10^{-10}$ F/mm$^2$. The calculated value of the
specific geometrical capacitance $C_{\rm gt}^{\rm calc}/A$ is very
close to the experimental one, $C_{\rm gt}/A = e \alpha_{\rm total}
= 1.22\times 10^{-10}$ F/mm$^2$ obtained from the filling rate
$\alpha_{\rm total} = 7.6\times10^{10}$ cm$^{-2}$/V$\cdot$s (see
main text).
Using the gate area $A = 0.05$ mm$^2$ (see
Fig. <ref>b) the sample's geometric capacitance is expected to
be $C_{\rm gt} \approx 6$ pF which is in line
with the measured value ($\approx 23$ pF) if one subtracts the
parasitic capacitance of our set up, being around
17 pF.
With the known values of the surface electrons' effective mass $m^* =
0.030 m_0$ <cit.> one can estimate the specific
quantum capacitances: $e^2 D_{\rm t(b)}^{\rm eval} = e^2 m^* /2
\pi \hbar^2 \approx 10^{-8}$ F/mm$^2$ which is almost two orders of magnitude
higher than the specific geometrical capacitance $C_{\rm gt}^{\rm
Finally, the evaluated specific capacitance of the $80$ nm thick HgTe layer with
$\varepsilon = 21$ is $C_{\rm tb}^{\rm calc}/A =
2.4\times 10^{-9}$ F/mm$^2$. From this value and with Eq. 3 one
can derive the relation between filling rates of electrons on top
and bottom surfaces: $\alpha_{\rm top}/\alpha_{\rm
bott}\equiv(dn_{\rm top}/dV_g) / (dn_{\rm bot}/dV_g) = \Bigl(
\frac{D_{\rm t}}{ D_{\rm b}} + \frac{Ae^2 D_{\rm t} }{C_{\rm
tb}^{\rm eval}} \Bigr) \approx 5.1$.
However, in our experiment, described in the main text, we
obtain $\alpha_{\rm top} = 0.7 \times \alpha_{\rm total}$,
$\alpha_{\rm bot} = 0.3 \times \alpha_{\rm total}$ and
$\alpha_{\rm top} / \alpha_{\rm bot} \approx 2.3$. This
discrepancy suggests that the equivalent circuit in
Fig. <ref>d does not fully describe the experimental
situation. Most likely this deviation stems from the fact that we
neglected the dependence of the surface states' dispersion on the transversal electric field described
in <cit.>. Phenomenologically, this shortcoming can be
corrected by introducing a larger effective capacitance of $C_{\rm
tb}^{\rm calc}/A = 7.5\times 10^{-9}$ F/mm$^2$, indicating an
enhanced screening in HgTe. This can be achieved by a higher value
of the dielectric constant of HgTe or a reduced effective layer
We note that for both values of $C_{\rm tb}$ given above, the
ratio $(dC/dD_{\rm t}) / (dC/dD_{\rm b})$ is larger than 5.5,
meaning that the sample capacitance is predominantly sensitive to
changes of the DoS of the top surface.
§ DENSITIES AND PARTIAL FILLING RATES
a, Gate voltage dependence of the different charge carrier species. $n_{\rm top}$ is the electron density of the top surface; $n_{\rm bot}$ the one of the bottom surface; $N_{\rm s}$ is the total density of both surfaces plus the bulk electron density; $P_{\rm s}$ is the bulk hole density. The dashed lines extrapolate the corresponding
charge carrier species in regions where data can not be directly extracted. b, The absolute value of the partial filling rates between surface and bulk carriers. The bluish area on both panels corresponds the bulk gap region. "CNP", "$E_v$" and "$E_c$" mark the charge neutrality point, top of the valence band and bottom of the conductive band, respectively. Values of the partial filling rates given by dashed lines can not be directly extracted from experiment but are taken from the corresponding equivalent circuit.
The gate voltage dependence of the different carrier species is shown in
Fig. <ref>a. The total electron density $N_{\rm s}$ and
the bulk hole density $P_{\rm s}$ were obtained from the low field transport data using the classical
(two-carrier) Drude model as described in <cit.>. The
top surface electron density $n_{\rm top}$ was obtained from the periodicity $\Delta (1/B)$ of the capacitance oscillations, $n_{\rm top}=e(h\Delta (1/B))^{-1}$, with the period $\Delta (1/B)$ on the $1/B$ scale and Planck's constant $h$.
For $V_g < 1.5$ this technique
is not applicable since the oscillations are dominated by bulk holes. The
density of the bottom surface electrons was obtained from
$n_{\rm bot } = N_{\rm s} - n_{\rm top}$ which holds as long as $E_{\rm
F}$ is located in the bulk gap, i.e. $V_{\rm g} =
2.2\ldots4.5$ V. In our experiment it is impossible to directly
measure the density of electrons on the bottom surface since their
response is masked by electrons on the top surface. Consequently
the value of $n^{\rm bot }$ can not be derived when the Fermi
level is in the valence (because of unknown value of
$n^{\rm top}$) or the conductance band (because of the unknown value of
$n^{\rm bulk}$). However, based on the equivalent circuit shown in
Fig. <ref>d and e one can make an educated guess on the partial filing rates. The result of such analysis is shown
in Fig. <ref>b.
§ RESISTIVE EFFECTS IN THE CAPACITANCE
As mentioned in the main text, resistive effects occur when the
condition $RC \omega <<1$ becomes invalid, for example when the
resistance $R$ of a two-dimensional electron gas connected in series to the capacitance increases
at integer filling factors and high magnetic fields. The absence
of both leakage currents and resistive effects in the capacitance
were controlled by the real part of the measured AC current. A direct manifestation of resistive effects is the
frequency-dependence of the measured $C(V_{\rm g})$ traces. A corresponding example, taken from a sample with larger gate area and therefore larger capacitance is shown in Fig. <ref>. The capacitance minimum appearing at the charge neutrality point (CNP) and connected to a gap opening at higher magnetic fields shows a pronounced frequency dependence. The resistive effects are weak up to 34 Hz. With increasing frequency, however, the minimum becomes deeper and the capacitance signal no longer reflects the density of states. Remarkably, in all our samples
the resistive effects first appear close the CNP and are much less
pronounced at integer filling factors at the same values
of magnetic field. Thus the remaining capacitance oscillations reflect the density of states.
Frequency dependence of the magnetocapacitance data:
$C(V_{\rm g})$ measured at a magnetic field of 3.9 T and at
frequencies $f = 18, 24, 34, 54$ and $103$ Hz. The dashed line
corresponds to the zero field $C(V_{\rm g})$ trace.
§ EXPERIMENTAL DATA
Magnetocapacitance data: a, $C(V_g)$ dependencies at
magnetic fields from 0 (on top) to 13 T (bottom) using
steps of 1 T. Starting at $B=1$ T each subsequent trace is shifted by -0.05 pF on
the Y-axis. The filling factors $v = -1$ and 1 are indicated. b,
Differential magnetocapacitance $\delta C(V_g) =
C(V_g)|_{B}-C(V_g)|_{B=0}$ dependencies at magnetic fields from 0
(top) to 2 T (bottom) using steps of 0.25 T between the traces. Starting at $B=0.25$ T each subsequent trace is shifted by
-0.025 pF on the Y-axis. The thin lines
correspond to the raw data and the thicker ones are the data after
Fourier filtering.
Strong magnetic field magnetotransport data: a,
$\rho_{xx}(V_g)$ dependencies at magnetic fields from 0 (top) to 10 T (bottom). Starting at
$B=9$ T each subsequent trace is shifted by 1 T on the Y-axis. b, $\sigma_{xx}(V_g)$ dependencies from 0 (on
the top) to 10 T (bottom). Each trace was
normalized by the average value $<\sigma_{xx}(V_g)> =
1$ (see text) and shifted by 1 T on the Y-axis. The filling factors $\nu =$ -2, -1, 1, and 2 are marked.
Small magnetic field magnetocapacitance data: a,
$\sigma_{xx}(V_{\rm g})$ dependencies at magnetic fields from 0
(on the top) to 1.5 T (bottom). Starting at $B=1.4$ T each trace was shifted by 0.1 T on the Y-axis.
b - The second derivative
$d^2\sigma_{xx}(V_{\rm g})/dV_{\rm g}^2$ of the traces of
panel (a). The SdH oscillations come out more clearly in the second derivative.
Each trace was normalized by its RMS value and shifted on the Y-axis.
In this section we show some of the raw data and outline how they were prepared for the color maps in the main article.
Fig. <ref>a displays the capacitance as a function of gate voltage $V_{\rm g}$ and for magnetic fields between 0 and 13 T.
In the color maps we plot the differential magnetocapacitance $\delta C(V_g) =C(V_g)|_{B}-C(V_g)|_{B=0}$, shown for magnetic fields up to 2 T in Fig. <ref>b. In order to suppress the noise we used standard band-pass Fourier filtering with the cutoff frequencies $f_1 = 0.1-0.3$ T$^{-1}$ and $f_2 = 1-5$ T$^{-1}$ depending on the magnetic field range.
The corresponding transport data measured on the same device are plotted in Fig. <ref>a. Together with the corresponding $\rho_{xy}(V_g)$ traces (not shown) the resistivity data were converted into $\sigma_{xx}(V_g)$ traces shown in Fig. <ref>b. Since the value of $\sigma_{xx}$ varies over several orders of magnitude and in order to improve the clarity of the color maps (Fig. 3d) each $\sigma_{xx}(V_g)$ trace was normalized with respect to its average value using $\sigma_{xx}^{\rm normalized}(V_g)= \sigma_{xx}(V_g) / <\sigma_{xx}>$, where $<...>$ means averaging over the whole $V_g$ range. After the normalization procedure the average value of each trace is equal to 1.
The $\sigma_{xx}(V_g)$ data taken at lower magnetic fields are shown in the Fig. <ref>a. To work out the oscillations more clearly we plot the second derivative $d^2\sigma_{xx}(V_{\rm g})/dV_{\rm g}^2$ of the traces, shown in panel a of Fig. <ref>b. In order to visualize the low-field oscillations clearly on the color maps each $d^2\sigma_{xx}(V_{\rm g})/dV_{\rm g}^2$ trace was normalized by its root mean square (RMS) value taken over the whole $V_g$ scale. Finally we applied standard low-pass Fourier filtering to remove random noise which is quite pronounced for the lowest field ($B \leq0.3$ T) traces.
|
1511.00193
|
We propose to study a new type of Backward stochastic differential equations, driven by a family of Itô's processes. We prove existence and uniqueness of the solution, and investigate stability and comparison theorem.
Keys Words: Backward stochastic differential
equation; Family of Itô's processes; Dynamic sublinear expectation operator; $m$-stability; Hedging claims under model uncertainty.
AMS Classification(1991): 60H10,
§ INTRODUCTION.
Originally motivated by questions arising in
stochastic control theory, the theory of backward stochastic differential
equations (BSDEs for short) has found important applications in fields as stochastic control, mathematical finance, Dynkin games and the second order PDE theory (see, for example, <cit.> and the references therein).
BSDEs have been introduced long time ago by J. B. Bismut <cit.> both
as the equations for the adjoint process in the stochastic version of Pontryagin maximum principle as well as the model behind the
Black and Scholes formula for the pricing and hedging of options in mathematical finance. However the first published paper on nonlinear
BSDEs appeared only in 1990, by Pardoux and Peng <cit.>. The classical BSDE consists of an equation of the form:
\begin{equation}\label{equa0}
Y_t = \xi + \dint_t^T f(s,Y_s,Z_s)ds - \dint_t^T Z_s .dW_s,
\quad\quad 0\leq t\leq T.
\end{equation}
driven by a $d$-dimensional Brownian motion $W$, with a deterministic terminal time $T>0$, a generator $f:[0,T]\times \Omega\times\R\times\R^d\rightarrow\R$ and an $\F_T$-measurable terminal value $\xi$, where $({\F}_t)_{t\leq T}$ is the natural
filtration of $(W_t)_{t\leq T}$ augmented by the null sets. The solution of this equation, denoted by $eq(f,\xi)$, is a pair of adapted processes $(Y,Z)$ with values in $\R\times\R^d$ and $Y_T=\xi$. The existence and uniqueness result of Pardoux and Peng assumes the uniform Lipschitz assumption on the generator $f$ in $y$ and $z$. So it is supposed that there exists a positive constant $K$ such that:
|f(s,\omega,y,z)-f(s,\omega,y',z')|\leq K(|y-y'|+|z-z'|).
The proof of this Theorem is done in two steps. The first step consider the particular case where the generator $f$ does not depend on the variables $y$ and $z$. The process $M$ defined for $t\in[0,T]$ by
M_t=\E\left(\xi+\int_0^T f(s)\,ds|\F_t\right),
is a martingale, so by using the martingale representation theorem there exists an $\R^d$-valued integrable process $Z$ such that $M_t=M_0+\int_0^t Z_s.dW_s$. We define then the process $Y$ by
Y_t=\E\left(\xi+\int_t^T f(s)\,ds|\F_t\right)=\xi+\int_t^T f(s)\,ds-\int_t^T\,Z_s.dW_s .
The second step is based on a fixed point theorem: by introducing the Banach space $\H_{T,\beta}(\R^k)$ associated to the norm
\|X\|_{T,\beta}=\left(\E\int_0^T\,e^{\beta s}|X_s|^2\,ds\right)^{1/2} ,
we define the mapping $\Phi:\H_{T,\beta}(\R)\times\H_{T,\beta}(\R^d)\rightarrow \H_{T,\beta}(\R)\times\H_{T,\beta}(\R^d)$ by $\Phi(y,z)=(Y,Z)$ where $(Y,Z)$ is the solution of the BSDE with generator $f(s,y_s,z_s)$. Such solution exists from the first step. It is proved that $\Phi$ is a contraction for a specific value of $\beta$ and then admits a unique fixed point.
Several papers extended these results by taking a more general driving martingale or by assuming weak assumptions on the generator $f$ (Among others, see Antonelli <cit.>, El Karoui and Huang <cit.>, Ma, Protter and Yong <cit.>, Pardoux and Peng <cit.>, Peng <cit.> and Essaky and Hassani <cit.>). For example the Pardoux-Peng result can be extended easily to the new equation $eq(f,\xi,S)$:
\begin{eqnarray}
\label{eq2}
\end{eqnarray}
driven by an Itô process $S$ of the form $dS_t=\mu^S_t\,dt+\sig^S_t.dW_t$, where $\mu^S$ and $\sigma^S$ are respectively $\R^d$-valued and $\R^d\otimes\R^d$-valued predictable processes. By supposing that the matrix-valued process $\sig^S$ is invertible and that the process $(\sig^S)^{-1}\mu^S$ is uniformly bounded, we assure that $S$ has a unique martingale measure denoted by $\bQ^S$, equivalent to the physical probability $\P$. We remark that if $(Y^S,Z^S)$ is the solution of the equation $eq(f,\xi,S)$, then $(Y^S,Z^S\sig^S)$ is the solution of the equation $eq(f^S,\xi)$ where the generator $f^S$ is defined by $f^S(t,y,z)=f(t,y,z(\sig^S_t)^{-1})-z(\sig^S_t)^{-1}\mu^S_t$.
In this paper we consider a family $D$ of Itô's processes instead of a single process $S$, such that each element of $D$ admits a unique equivalent martingale measure. We propose to solve the equation $eq(f,\xi,D)$:
\begin{eqnarray}
\label{eqq3}
\hY_t=\xi+\int_t^T\,f(s,\hY_s,\hZ_s) ds-\int_t^T\,\hZ_s.d\hS_s,
\end{eqnarray}
for which the solution is a triplet $(\hY,\hZ,\hS)$ satisfying the equation (<ref>) and such that $\hS\in D$ and the process $\int_0^.\,\hZ_s.d\hS_s$ is a $\cE$-martingale with $\cE$ the dynamic sublinear expectation operator associated to the set of probability measures $\Q:=\{\bQ^S:\;S \in D\}$. We study existence and uniqueness of the solution, stability and comparison theorem.
This work is mainly motivated by pricing and hedging problems in Finance under model misspecification setting or the presence of some type of ambiguity.
§ MAIN RESULT
Let $(\Omega, {\F}, ({\F}_t)_{t\leq T}, P)$ be a stochastic
basis on which is defined a $d$-dimensional Brownian motion
$(W_t)_{t\leq T}$ such that $({\F}_t)_{t\leq T}$ is the natural
filtration of $(W_t)_{t\leq T}$ and ${\F}_0$ contains all
$P$-null sets of $\F$. Note that $({\F}_t)_{t\leq T}$
satisfies the usual conditions, i.e. it is right continuous and
Let us now introduce the following notations :
$\bullet$ ${\L}^{2}_T(\R)$ denotes the space of $\F_T-$measurable random variables $\xi$ satisfying $\E|\xi|^2<\infty$.
$\bullet$ ${\S}^{2}_T(\R)$ is the space of predictable processes $Y$ such that
\|Y \|^2 = \E \sup_{s\leq T} |Y_s|^2 <\infty.
$\bullet$ ${\H}^{2}_T(\R^d)$ is the space of predictable processes $Z$ such that
\|Z \|^2 = \E \dint_0^T |Z_s|^2 ds <\infty.
$\bullet$ $\mathcal{P}$ denotes a set of predictable processes.
$\bullet$ $\S$ is the set of $\R^d$-valued Itô's processes $S$ of the form $dS=\mu^S\,dt+\sigma^S\,dW$.
$\bullet$ $D$ is the set of process $S\in\S$ such that the process $\theta^S:=(\sig^S)^{-1}\mu^S$ belongs to $\mathcal{P}$ and bounded.
We assume the following assumption (H) :
(1) $\mathcal{P}$ is predictably convex: for all $X^1,X^2\in\mathcal{P}$ and a $\{0,1\}$-valued predictable process $h$, we have $X\in\mathcal{P}$ where $X_t=h_tX^1_t+(1-h_t)X^2_t$ for $t\in [0,T]$.
(2) For any predictable process $Z$ with values in $\R^d$, there exists some $S\in D$ such that
ess\inf_{S'\in D}(\theta^{S'}_t. Z_t)=\theta^{S}_t. Z_t,
\,\,\mbox{for all}\,\, t\in [0,T].
(3) The terminal value $\xi\in \L_{T}^2(\R)$, the generator $f$ is uniformly $K-$Lipschitz with respect to $y$ and $z$ and $f(.,0,0)\in\H_{T}^2(\R)$.
Let us now introduce the definition of our BSDE driven by a family of Itô's processes.
A solution of the following BSDE $eq(f,\xi,D)$
\begin{eqnarray}
\label{eq3}
\hY_t=\xi+\int_t^T\,f(s,\hY_s,\hZ_s) ds-\int_t^T\,\hZ_s.d\hS_s,
\end{eqnarray}
is a triplet $(\hY,\hZ,\hS)$ satisfying equation (<ref>) such that $\hS\in D$ and the process $\int_0^.\,\hZ_s.d\hS_s$ is a $\cE$-martingale with $\cE$ the dynamic sublinear expectation operator associated to the set of probability measures $\Q:=\{\bQ^S:\;S \in D\}$.
The main result of this paper is as follows.
Under the assumption (H), the equation $eq(f,\xi,D)$ has a unique solution.
Before proving Theorem <ref>, we recall the definition of the $m$-stability property and state some intermediate results.
A family $\Q$ of probability measures, all elements of which are equivalent to $\P$, is called multiplicativity stable ($m$-stable) if for all elements $\bQ^1, \bQ^2\in \Q$ with density processes $\Lam^1, \Lam^2$ and for all stopping time $\tau \leq T$, it holds that $\Lam_T := \Lam^1_{\tau}\Lam^2_{T}/\Lam^2_{\tau}$ is the density of some $\bQ\in\Q$.
The set $\Q:=\{\bQ^S:\;S\in D\}$ is $m$-stable.
. First each $\bQ^S$ is defined via its Radon-Nikodym density $\Lam^S_T$, given by
\Lam^S_T=\exp\left\{\int_0^T\,\theta^S_s.dW_s-\frac{1}{2}\int_0^T\,|\theta^S_s|^2\,ds\right\}.
For $i=1,2$, let $S^i\in D$ with $\Lam^i_T$ the density of $\bQ^{S^i}$. Let a stopping time $\tau$ and define the probability measure $\bQ$ by its density $\Lam_T=\Lam^1_\tau\Lam^2_T/\Lam^2_\tau$ where $\Lam^i_\tau=\E(\Lam^i_T|\F_\tau)$. We shall prove that there exists some $S\in D$ such that $\bQ=\bQ^S$. For this we define the two processes $\mu$ and $\sig$ by $\mu_t=\1_{t<\tau}\mu^1_t+\1_{t\geq \tau}\mu^2_t$ and $\sig_t=\1_{t<\tau}\sig^1_t+\1_{t\geq\tau}\sig^2_t$ for $t\in [0,T]$ where $\mu^i$ and $\sig^i$ are associated to $S^i$ and define the process $S$ by $dS=\mu\,dt+\sig.dW$. We verify easily from assumption (H)(1) by taking $h_t=\1_{\{t>\tau\}}$ that $\theta:=\sig^{-1}\mu\in\mathcal{P}$. So $S\in D$ and then $\bQ=\bQ^S$.
For the sake of clarity, we recall the following result which is a simplified
version of Proposition 3.1 in <cit.>.
For a family of standard parameters $(f,\xi)$ and $(f^{\alpha},\xi)$, with $\alpha$ from an arbitrary index set, let $(Y,Z)$ and $(Y^{\alpha}, Z^{\alpha})$ denote respectively the solution to the corresponding BSDEs $eq(f,\xi)$ and $eq(f^{\alpha},\xi)$. If there exists a parameter $\overline{\alpha}$ such that
f(t, Y_t, Z_t)=ess\inf_{\alpha} f^{\al}(t, Y_t, Z_t)= f^{\overline{\alpha}}(t, Y_t, Z_t)\,\ dP \times dt- a.e.,
then $Y_t = ess\inf_{\alpha} Y_t^{\alpha} =Y_t^{\overline{\alpha}}$ holds for all $t \leq T$, $P$-a.s..
We suppose for the next two propositions that $\sig^S$ is the identity matrix for all $S\in D$. We define the dynamic sublinear expectation operator $\cE$, associated to the set of probability measures $\Q=\{\bQ^S:\;S\in D\}$, by $\cE_t(X)=ess\sup_{S\in D}\E^{\bQ^S}(X|\F_t)$.
Let $g$ be a square integrable adapted process and let $(Y,Z)$ be the solution of the equation $eq(\bg,\xi)$ where $\bg(t,z)=ess\sup_{S\in D}(g_t-\mu^S_t z)$. Then for $t\in\T$
. For $S\in D$, let $(Y^S,Z^S)$ be the solution of the equation $eq(g,\xi,S)$. Then
We remark also that $(Y^S,Z^S)$ is the solution of the equation $eq(g^S,\xi)$ where $g^S(t,z)=g_t-\mu^S_t z$ and then from Proposition <ref> we deduce that
Y_t=ess\sup_{S\in D}Y^S_t=ess\sup_{S\in D}\E^{\bQ^S}\left(\xi+\int_t^T\,g_s\,ds|\F_t\right).
Proposition <ref> is then proved.
Under assumption H(2)-(3),
there exists a unique solution $(Y,Z)$ to the equation $eq(\hf,\xi)$ where
$$\hf(t,y,z)=ess\sup_{S\in D}(f(t,y,z)-\mu^S_t z).$$
Moreover, we have
. Along the proof, $C$ will denote a generic constant which may vary from line to line. The Proof is based on the Picard's approximation scheme. Let $(Y^0,Z^0) =(0,0)$ and define $(Y^{n+1},Z^{n+1})$ be the solution of the following BSDE:
\begin{equation}\label{nonlinear}
Y^{n+1}_t=\xi+\int_t^T ess\sup_{S\in D}\left[f(s,Y^{n}_s,Z^{n}_s)-\mu^S_s Z^{n}_s\right]\,ds-\int_t^T\,Z^{n+1}_s.dW_s.
\end{equation}
Let $n, m\in \N$ and $\beta >0$. Applying Itô's formula to
$(Y_{t}^{n+1} -Y_{t}^{m+1})^2 e^{\beta t}$ we get
\begin{equation}\label{yass}
\begin{array}{ll}
&(Y_{t}^{n+1} -Y_{t}^{m+1})^2 e^{\beta t} +
\integ{t}{T}e^{s\beta}|Z_s^{n+1} - Z_s^{m+1}|^2ds
+\integ{t}{T}\beta e^{\beta s}(Y_{s}^{n+1} -Y_{s}^{m+1})^2ds
\\ & =
2\integ{t}{T}e^{s\beta}(Y_{s}^{n+1} -Y_{s}^{m+1})
(\hf(s,Y_{s}^{n} ,Z_{s}^{n})-\hf(s,Y_{s}^{m},Z_{s}^{m}))ds
\\ &
\qquad-2\integ{t}{T}e^{s\beta}(Y_{s}^{n+1}
-Y_{s}^{m+1})(Z_{s}^{n+1}- Z_{s}^{m+1}).dW_{s},
\end{array}
\end{equation}
where $\hf(t,Y,Z)=f(t,Y,Z)-\mu^{\hS}_t Z$, for some ${\hS}\in D$.
From assumptions (H)(3), it follows that for every $\varepsilon >0$,
\begin{array}{ll}
& \quad 2(Y_{s}^{n+1} -Y_{s}^{m+1}) (\hf(s,Y_{s}^{n}
,Z_{s}^{n})-\hf(s,Y_{s}^{m},Z_{s}^{m}))\\ & \quad\qquad\leq
2\varepsilon\mid Y_{s}^{n+1} -Y_{s}^{m+1}\mid^2
+\frac{K^2}{\varepsilon}\mid Y_{s}^{n}-Y_{s}^{m}\mid^2+\frac{(K+C)^2}{\varepsilon}\mid Z_{s}^{n}-Z_{s}^{m}\mid^2,
\end{array}
where we have used the fact that $\mu^{\hS}$ is bounded by a positive constant $C$. By taking $\varepsilon = \frac{\beta}{2}$, it follows then that
\begin{equation}\label{eqmea}
\begin{array}{ll}
&(Y_{t}^{n+1} -Y_{t}^{m+1})^2 e^{\beta t} +
\integ{t}{T}e^{s\beta}|Z_s^{n+1} - Z_s^{m+1}|^2ds
\\ & \leq
\frac{2K^2}{\beta}\integ{t}{T}e^{s\beta}\mid Y_{s}^{n}
-Y_{s}^{m}\mid^2 ds +\frac{2(K+C)^2}{\beta}\integ{t}{T}e^{s\beta}\mid Z_{s}^{n}
-Z_{s}^{m}\mid^2 ds
\\ & \qquad -2\integ{t}{T}e^{s\beta}(Y_{s}^{n+1}
-Y_{s}^{m+1})(Z_{s}^{n+1}- Z_{s}^{m+1}).dW_{s}.
\end{array}
\end{equation}
Using a localization procedure, we have
\begin{equation}\label{equaa}
\begin{array}{ll}
& \E\integ{0}{T}e^{s\beta}|Z_s^{n+1} - Z_s^{m+1}|^2ds
\\ & \leq \frac{2(K+C)^2(T+1)}{\beta}\E\bigg[\displaystyle\sup_{s\leq T}e^{s\beta}\mid Y_{s}^{n}
-Y_{s}^{m}\mid^2 + \integ{0}{T}e^{s\beta}\mid Z_{s}^{n}
-Z_{s}^{m}\mid^2 ds\bigg].
\end{array}
\end{equation}
It follows from (<ref>) and Davis-Burkholder-Gundy
inequality that there exists a constant $c>0$, such that
\begin{array}{ll}
&\E\displaystyle\sup_{t\leq T}(Y_{t}^{n+1} -Y_{t}^{m+1})^2 e^{\beta t}
\\ & \leq
\frac{2(K+C)^2(T+1)}{\beta}\E\bigg[\displaystyle\sup_{s\leq T}e^{s\beta}\mid Y_{s}^{n}
-Y_{s}^{m}\mid^2 + \integ{0}{T}e^{s\beta}\mid Z_{s}^{n}
-Z_{s}^{m}\mid^2 ds\bigg]\\ & \qquad+c\E\bigg(\integ{0}{T}(e^{s\beta})^2\mid
Y_{s}^{n+1} -Y_{s}^{m+1}\mid^2\mid Z_{s}^{n+1}- Z_{s}^{m+1}\mid^2
\\ &\leq \frac{2(K+C)^2(T+1)}{\beta}\E\bigg[\displaystyle\sup_{s\leq T}e^{s\beta}\mid Y_{s}^{n}
-Y_{s}^{m}\mid^2 + \integ{0}{T}e^{s\beta}\mid Z_{s}^{n}
-Z_{s}^{m}\mid^2 ds\bigg]\\ & \qquad+\frac12\E\displaystyle\sup_{t\leq T}(Y_{t}^{n+1} -Y_{t}^{m+1})^2
+\frac{c^2}{2}\E\integ{0}{T}e^{s\beta}\mid Z_{s}^{n+1}-
Z_{s}^{m+1}\mid^2 ds.
\end{array}
By using inequality (<ref>), we get
\begin{array}{ll}
&\E\sup_{t\leq T}(Y_{t}^{n+1} -Y_{t}^{m+1})^2 e^{\beta t}
\\ &\leq\frac{2(K+C)^2(T+1)}{\beta}(\frac{c^2}{2} +1)\E\bigg[\displaystyle\sup_{s\leq T}e^{s\beta}\mid Y_{s}^{n}
-Y_{s}^{m}\mid^2 + \integ{0}{T}e^{s\beta}\mid Z_{s}^{n}
-Z_{s}^{m}\mid^2 ds\bigg]\\ & \qquad+\frac12\E\sup_{t\leq T}(Y_{t}^{n+1} -Y_{t}^{m+1})^2
\end{array}
and then
\begin{array}{ll}
&\E\sup_{t\leq T}(Y_{t}^{n+1} -Y_{t}^{m+1})^2 e^{\beta t}
\\ &\leq \frac{4(K+C)^2(T+1)}{\beta}(\frac{c^2}{2} +1)\E\bigg[\displaystyle\sup_{s\leq T}e^{s\beta}\mid Y_{s}^{n}
-Y_{s}^{m}\mid^2 + \integ{0}{T}e^{s\beta}\mid Z_{s}^{n}
-Z_{s}^{m}\mid^2 ds\bigg].
\end{array}
Coming back to equation (<ref>), we have
\begin{equation}\label{equa2}
\begin{array}{ll}
&\E\displaystyle\sup_{t\leq T}(Y_{t}^{n+1} -Y_{t}^{m+1})^2 e^{\beta t} +
\E\integ{0}{T}e^{s\beta}|Z_s^{n+1} - Z_s^{m+1}|^2ds
\\ & \leq
\frac{4(K+C)^2}{\beta} (c^2 +2)(T+1)\bigg(\E\displaystyle{\sup_{t\leq T}}(Y_{t}^{n} -Y_{t}^{m})^2
e^{t\beta}+\E\integ{0}{T}e^{s\beta}\mid Z_{s}^{n}
-Z_{s}^{m}\mid^2 ds\bigg).
\end{array}
\end{equation}
$\beta \geq 16(K+C)^2(c^2+2)(T+1)$ and
$$\Gamma^{n+1, m+1} = \bigg(\E\sup_{t\leq T}(Y_{t}^{n+1} -Y_{t}^{m+1})^2
\E\integ{0}{T}e^{s\beta}|Z_s^{n+1} - Z_s^{m+1}|^2ds\bigg)^{\frac12},$$
it follows that, for every $n\geq m$
\Gamma^{n+1, m+1}\leq \dfrac{1}{2} \Gamma^{n, m}\leq \bigg(\dfrac12\bigg)^m
\Gamma^{n-m+1, 1}.
Using similar arguments as above and the fact that $f$ is $K-$Lipschitz and $f(.,0,0)\in\H^2_T(\R)$, it is not difficult to prove that there exists a positive constant C such that
\Gamma^{n-m+1, 1}\leq C.
\Gamma^{n+1, m+1}\leq C\bigg(\dfrac12\bigg)^{m},
and then
\lim_{n, m\rightarrow +\infty}\E\sup_{t\leq T}(Y_{t}^{n}
-Y_{t}^m)^2 =0,\qquad \lim_{n, m\rightarrow +\infty}\E\integ{0}{T}\mid Z_{s}^{n}
-Z_{s}^{m}\mid^2 ds =0.
Consequently, the sequence $ (Y^n,Z^n)$ converge to $(Y,Z)$ in ${\S}^{2}_T(\R)\times {\H}^{2}_T(\R^d)$.
Let $(\overline{Y}, \overline{Z})$ be the solution, which
exists according to the previous proposition, of the following BSDE
\overline{Y}_t=\xi+\int_t^T ess\sup_{S\in D}\left[f(s,Y_s,Z_s)-\mu^S_s Z_s\right]\,ds-\int_t^T\,\overline{Z}_s.dW_s.
It is not difficult to prove that there exists a constant $C>0$ such
\begin{array}{ll}
\E\sup_{t\leq T}(Y_{t}^{n+1} -\overline{Y}_{t})^2
+\E\integ{0}{T}|Z_s^{n+1} - \overline{Z}_s|^2ds \\ & \quad \leq C \bigg[\E\sup_{t\leq
T}(Y_{t}^{n} -Y_t)^2+\E\integ{0}{T}|Z_s^{n} - Z_s|^2ds\bigg].
\end{array}
\lim_{n\rightarrow +\infty}\Big[\E\displaystyle{\sup_{t\leq T}}(Y_{t}^{n+1}
-\overline{Y}_{t})^2 + \E\integ{0}{T}|Z_s^{n+1} -
\overline{Z}_s|^2ds\Big]=0.
It follows that
\E\sup_{s\leq T}\mid Y_s -\overline{Y}_s \mid^2 =
0,\,\,\mbox{and}\,\, \E\integ{0}{T}|Z_s - \overline{Z}_s|^2ds =0.
Therefore $Y = \overline{Y}$ and $Z=\overline{Z}$, and then $(Y,Z)$ satisfies $eq(\xi,\hf)$.
Thanks to Proposition <ref>, it follows from Equation (<ref>) that
\begin{eqnarray}
\label{**}}{Y^{n+1}_t=\cE_t\left(\xi+\int_t^T\,f(s,Y^{n}_s,Z^{n}_s)\,ds\right).
\end{eqnarray}
By taking the limit in (\ref{**}) and using the Fatou property of $\cE$ we obtain that
Since there exists some $\hS\in D$ such that $\hf(t,Y_t,Z_t)=f(t,Y_t,Z_t)-\mu^{\hS}_t.Z_t$. So $(Y,Z,\hS)$ is the solution of $eq(f,\xi,S)$, we take the expectation with respect to $\bQ^S$ in both parts of this equation and obtain that
Therefore the second assertion is obtained. Proposition \ref{p2} is then proved.
\eop
Now we prove Theorem \ref{t1}.
\bop\,\textit{of Theorem} \ref{t1}.
{\bf Existence of a solution:} Let $(Y,Z)$ be the solution of the equation $eq(\hf,\xi)$, where
\hf(t,y,z)=ess\sup_{S\in D}\left(f(t,y,z)-\theta^S_t.z\right),
and let $\hS\in D$ such that $\hf(t,Y,Z)=f(t,Y,Z)-\theta^{\hS}_t Z$. By applying Proposition \ref{p2} to the family $\{\theta^S:\;S\in D\}$ instead of the family $\{\mu^S:\;S\in D\}$ we get that
But for $\hZ=Z(\sig^{\hS})^{-1}$ we have that
Then for all $t\in [0,T]$,
\cE_t\left(\int_t^T\,\hZ_s.d\hS_s\right)=0.
Therefore $(Y,\hZ,\hS)$ is a solution of the equation $eq(f,\xi,D)$.
{\bf Uniqueness of the solution:} Let $(\hY^1,\hZ^1,\hS^1)$ and $(\hY^2,\hZ^2,\hS^2)$ two solutions of the equation $eq(f,\xi,D)$. By applying It\^o formula to the semi-martingale $e^{\beta s}(\hY^1_s-\hY^2_s)^2$ from $s=t$ to $s=T$ we obtain that
\begin{eqnarray}
\label{***}
e^{\beta t}(\delta\hY_t)^2+\beta\int_t^T\,e^{\beta s}(\delta\hY_s)^2\,ds+\int_t^T\,e^{\beta s}|\delta(\hZ_s\sig_s)|^2\,ds
\end{eqnarray}
$$=e^{\beta T}(\delta\hY_T)^2+2\int_t^T\,e^{\beta s}\delta\hY_s\, \delta f(s)\,ds-\int_t^T\,e^{\beta s}\delta\hY_s\, d(\delta M)_s,
with $\delta\hY=\hY^1-\hY^2$, $\delta(\hZ\sig)=\hZ^1\sig^{\hS^1}-\hZ^2\sig^{\hS^2}$, $\delta f(s)=f(s,\hY^1,\hZ^1\sig^{\hS^1})-f(s,\hY^2,\hZ^2\sig^{\hS^2})$, $\delta M=M^1-M^2$ and $dM^i=\hZ^i.d\hS^i$ for $i=1,2$.
We need to show that the random variable $K:=\int_t^T\,e^{\beta s}\delta\hY_s\, d\delta M_s$ has a positive expected value under a certain probability $\bQ\in\Q$. Let define $\mu_t=\1_{(\delta\hY_t\geq 0)}\mu^1_t+\1_{(\delta\hY_t< 0)}\mu^2_t$, $\sig_t=\1_{(\delta\hY_t\geq 0)}\sig^1_t+\1_{(\delta\hY_t< 0)}\sig^2_t$ and the process $S$ by $dS=\mu dt+\sig.dW$. From assumption {\bf (H)}(1) we have $S\in D$ and since $M^1$ and $M^2$ are $\bQ^S$-super martingales, then
\eta_1:=\E^{\bQ^S}\left(\int_t^T\,e^{\beta s}(\delta\hY_s)_- d M^1_s\right)\leq 0,
\eta_2:=\E^{\bQ^S}\left(\int_t^T\,e^{\beta s}(\delta\hY_s)_+ dM^2_s\right)\leq 0,
where $(\delta\hY_s)_+$ and $(\delta\hY_s)_-$ are respectively the positive and the negative parts of $\delta\hY_s$. We have also that
\eta:=\int_t^T\,e^{\beta s}(\delta\hY_s)_+ d\delta M^1_s+\int_t^T\,e^{\beta s}(\delta\hY_s)_- d\delta M^2_s=\int_t^T\,e^{\beta s}(\delta\hY_s)Z_s dS_s,
with $Z_s=\1_{\delta\hY_s\geq 0}\hZ^1_s+\1_{\delta\hY_s< 0}\hZ^2_s$. Therefore $\eta_3:=\E^{\bQ^S}(\eta)=0$ and $\E^{\bQ^S}(K)=-\eta_1-\eta_2+\eta_3\geq 0$.
Now by taking the expectation in (\ref{***}) with respect to $\bQ=\bQ^S$ and by following the same techniques as in Proposition 2.1 in \cite{EPQ} we obtain that $\delta\hY\equiv 0$ and $\delta(\hZ \sig)\equiv 0$.
\eop
\begin{rem}
It should be pointed out that our existence result hold true if we suppose that $det(\sigma^S(\sigma^S)^{tr})\neq 0, dP \times dt-$a.e. for all $S\in\S$ and the set $D$ is taken as follows $D=\{S\in\S:\;\theta^S:=\sigma^S(\sigma^S(\sigma^S)^{tr})^{-1}\mu^S\in\mathcal{P},\,\,\mbox{and bounded}\}$, where $\mathcal{P}$ satisfying assumption {\bf (H)}(1).
\end{rem}
An immediate consequence of Theorem \ref{t1} concerns the generalization of the martingale representation of a square integrable random variable.
\begin{cor}
\label{c1}
For any $\xi\in \L^2_T(\R)$, there exists a real number $x_0$, a driver $\hS\in D$ and a square integrable predictable $\R^d$-valued process $\hZ$ such that
\xi=x_0+\int_0^T\,\hZ_s.d\hS_s,
and the process $\left(\int_0^t\,\hZ_s.d\hS_s\right)_{t\in\T}$ is a $\cE$-martingale, i.e for all $t<u$ we have
\cE_t\left(\int_0^u\,\hZ_s.d\hS_s\right)=\int_0^t\,\hZ_s.d\hS_s.
\end{cor}
\begin{rem}
\label{r1}
The triplet $(\hY,\hZ,\hS)$ is the unique solution of the equation $eq(f,\xi, D)$ if and only if $\hY=Y$, $\hZ=Z(\sigma^{\hS})^{-1}$ and $ess\inf_{S\in D}(\theta^S.Z)=\theta^{\hS}.Z$ where the pair $(Y,Z)$ is the unique solution of the equation $eq(\hf,\xi)$ with $$\hf(t,y,z)=ess\sup_{S\in D}(f(t,y,z)-\theta^S_t z).$$
\end{rem}
Thanks to the previous remark we obtain comparison theorem of solutions as a direct consequence of Theorem 2.2 in \cite{EPQ}.
\begin{thm}
\label{t2}
Let $(\hY^i,\hZ^i,\hS^i)$ be the solution of the equation $eq(f^i,\xi^i, D)$ for $i=1,2$. We suppose that $\xi^1\geq \xi^2$ a.s and that $\delta_2 f_t:=f^1(t,\hY^2_t,\hZ^2{\hat \sig}^2_t)-f^2(t,\hY^2_t,\hZ^2{\hat \sig}^2_t)\geq 0$ a.s. Then for a.s any time $t$ we have $\hY^1_t\geq\hY^2_t$. Moreover if $\hY^1_t=\hY^2_t$ on a $\F_t$-measurable set $A$, then $\hY^1_s=\hY^2_s$ on $[t,T]\times A$.
\end{thm}
\bop. We have $f^1(t,\hY^2_t,\hZ^2{\hat \sig}^2_t)\geq f^2(t,\hY^2_t,\hZ^2{\hat \sig}^2_t)$, then $\hf^1(t,\hY^2_t,\hZ^2{\hat \sig}^2_t)\geq \hf^2(t,\hY^2_t,\hZ^2{\hat \sig}^2_t)$. Thanks to Theorem 2.2 in \cite{EPQ} and Remark \ref{r1} we obtain the result.
\eop
Another consequence of remark \ref{r1} concerns the explicit solution of linear BSDE.
\begin{cor}
\label{c2}
Let $(\alpha,\gamma)$ be a bounded $\R\times\R^d$-valued predictable process, $\varphi\in\H_{T}^2(\R)$ and $\xi\in\L^2_T(\R)$. Then the linear BSDE $eq(\xi,f,D)$ with $f(s,y,z)=\varphi_s+y\alpha_s+z.\gamma_s$,
Y_t=\xi+\int_t^T \left(\varphi_s+Y_s\alpha_s+Z_s.\gamma_s\right)\,ds-\int_t^T\,Z_s.dS_s ,
has a unique solution $(\hY,\hZ,\hS)$ such that $\hS$ is solution of the minimization problem
ess\inf_{S\in D}(\theta^S.Z)=\theta^{\hS}.Z,
$$ for all predictable processes $Z$ and
$\hY$ is given by
\hY_t=\Gamma_{t}^{-1}\E\left(\xi\,\Gamma_{T}+\int_t^T \varphi_s\Gamma_{s}\,ds|\F_t\right),
where $\Gamma_{s}$ is the adjoint process defined for $s\geq 0$ by the forward linear SDE:
d\Gamma_{s}=\Gamma_{s}\left(\alpha_s\,ds+(\sigma^{\hS}_s)^{-1}(\gamma_s-\mu^{\hS}_s).dW_s\right) ,
and $\Gamma_{0}=1$.
\end{cor}
Next we illustrate previous results by an example of a geometric Brownian motion with volatility uncertainty.
\begin{ex} We consider the case $d=1$ and define the geometric Brownian motion $S$, solution of the equation
dS_t=\mu S_t dt+ \sigma^S S_t dW_t,
and define the family $D$ as the set of processes $S$ that satisfy $\sigma^S\in [\sigma^1,\sigma^2]$ where $\sigma^1$ and $\sigma^2$ are two positive real constants. So the equation $eq(f,\xi,D)$ has the unique solution $(\hY,\hZ,\hS)$ given by
$$\hY=Y, \;\hZ=\dfrac{Z}{\sigma(Z)},\; d\hS=\mu\hS dt+ \sigma(Z)\hS dW,$$
with $\sigma(Z)=\sigma^1\1_{(Z\geq 0)}+\sigma^2\1_{(Z< 0)}$ and the pair $(Y,Z)$ is the unique solution of the equation $eq(\hf,\xi)$ where $\hf(t,y,z)=f(t,y,z)-\dfrac{\mu}{\sigma(z)} z$.
\end{ex}
\section {Application to hedging claims under model uncertainty.}
We consider a financial market, which is composed of a riskless asset and $d$ risky assets. We suppose that the price of these $d+1$ assets is modelled as follows: $S^0\equiv 1$ and $S=(S^1,\ldots,S^d)$ is solution of the stochastic differential equation:
dS_t=S_t\left(\mu_t dt+ \sigma_t.dW_t\right),
where $\mu$ and $\sigma$ are respectively $\R^d$-valued and $\R^d\otimes\R^d$-valued predictable processes. By supposing that the matrix-valued process $\sig$ is invertible and that the process $\sig^{-1}\mu$ is uniformly bounded, we assure that $S$ has a unique martingale measure denoted by $\bQ^S$, equivalent to the physical probability $\P$, and therefore every contingent claim with payoff value $H$ at maturity time $T$ can be fully hedged, which means that there exists an $\R^d$-valued strategy $\phi$ and a price $\E^{\bQ^S}(H)$ such that $H=\E^{\bQ^S}(H)+\int_0^T\,\phi_s.dS_s$. In the Markovian case and for an European type option contract $H=f(S_T)$ we can express the strategy $\phi$ as follows: we define the function $u(t,x)=\E(f(S_{T-t})|S_0=x)$ and found out that $H=u(0,S_0)$ and $u$ is the solution of the partial differential equation:
\partial_t u+\mu\partial_x u+\frac{1}{2}\sigma\sigma^*\partial^2_x u=0,
$$ and $u(T,x)=f(x)$. In a general setting we express the strategy $\phi$ in terms of the solution of a backward stochastic differential equation: $H=Y_T$ where $(Y,Z)$ is the solution of BSDE:
We may ask if the full hedge is still possible and what is the price if we suppose some uncertainty on the parameters $\mu$ and/or $\sigma$. More precisely we shall consider the situation where the vector valued parameter $\theta_t:=\sig_t^{-1}\mu_t$ varies in a random interval $[h_t,g_t]$ for $t\in[0,T]$. We denote $D=\{S\in\S:\;\theta\in[h,g]\,\,\mbox{a.s.}\}$.
\begin{thm}
\label{t3}
Let a contingent claim $H$ with $\cE(H)<\infty$. Then $H=\hY_T$ where $(\hY,\hZ,\hS)$ is the solution of the BSDE $eq(0,H,D)$:
\hY_t=H-\int_t^T\,\hZ_s.d\hS_s,
and the process $\hS$ is given by $d\hS=\sigma(\theta^0 dt + dW)$ with $\theta^0=(h^i\1_{(Z^i>0)}+g^i\1_{(Z^i\leq 0)})_{i=1\ldots d}$ and $(Y,Z)$ is the solution of the BSDE:
Y_t=H-\int_t^T\,(h_s.Z^-_s+g_s.Z^+_s)dt -\int_t^T\,Z_s.dW,
with $Z^{+}=((Z^{1})^{+},\ldots,(Z^{d})^{+})$ and $Z^{-}=((Z^{1})^{-},\ldots,(Z^{d})^{-})$, where $z^{+}$ and $z^{-}$ denote respectively the positive and the negative parts of $z$.
\end{thm}
\begin{rem}
\label{r2}
We refer to \cite{EPQ} among other references for more details and motivations on BSDE's and their applications in numerous domains.
\end{rem}
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\end{document}
|
1511.00302
|
Error bounds for multivariate Laplace approximation]Simple error bounds for the multivariate Laplace approximation under weak local assumptions
Piotr Majerski, Faculty of Applied Mathematics,
AGH University of Science and Technology,
Al. Mickiewicza 30, 30-059 Kraków, Poland
[2000]41A60, 41A63, 41A80, 41A25
The paper provides new upper and lower bounds for the multivariate Laplace approximation under weak local assumptions. Their range of validity is also given. An application to an integral arising in the extension of the Dixon's identity is presented. The paper both generalizes and complements recent results by Inglot and Majerski and removes their superfluous assumption on vanishing of the third order partial derivatives of the exponent function.
§ NOTATION AND PRELIMINARIES
For a given function $f:\mathbb{R}^d\to\mathbb{R}$, with $d\geqslant2$, $\dot{f}(\mathbf{0})$ and
$\ddot{f}(\mathbf{0})$ is used to denote, respectively, the gradient vector and the Hessian matrix of the function $f$ evaluated
at $\mathbf{0}=(0,\ldots,0)^\prime\in\mathbb{R}^d$. $d^2f(\boldsymbol0,\boldsymbol t)$, $d^3f(\boldsymbol0,\boldsymbol t)$ and $d^4f(\boldsymbol0,\boldsymbol t)$ will respectively denote the second, the third and the fourth order total derivative of $f$ at $\mathbf{0}$ with respect to $\mathbf{t}=(t_1,\ldots,t_d)^\prime\in\mathbb{R}^d$, so in particular $d^2f(\boldsymbol0,\boldsymbol t)=\boldsymbol t'\ddot{f}(\mathbf{0})\boldsymbol t$, and
\[
d^3f(\mathbf{0},\mathbf{t}):=\sum_{i=1}^d\sum_{j=1}^d\sum_{k=1}^d \frac{\partial^3f(\mathbf{0})}{\partial t_i\partial t_j\partial t_k}t_it_jt_k.
\]
For $r>0$ and a positive definite $d\times d$ matrix $A$, we define $B_r:=\{\mathbf{t}\in\mathbb{R}^d:~\mathbf{t}^\prime\mathbf{t}\leqslant r^2\}$ and $E_r(A):=\{\mathbf{t}\in\mathbb{R}^d:~\mathbf{t}^\prime A\mathbf{t}\leqslant r^2\}$.
Finally, concerning the smoothness of functions, for a nonnegative integer $k$ by $f\in {\rm C}^k(\Omega)$ or $f\in {\rm C}^k(\boldsymbol t_0)$ we mean that $f$ is of class ${\rm C}^k$ on $\Omega$ or some neighborhood of the point $\boldsymbol t_0$, respectively. We shall use an analogous notation for the Hölder continuous class of functions ${\rm C}^{k,\alpha}$, with $\alpha\in(0,1]$.
For $n>0$ and an integer $d\ge2$ consider the Laplace integral
\[
J(n):=\int_{\Omega}e^{-nf(\boldsymbol t)}g(\boldsymbol t)d\boldsymbol t,
\]
where $\Omega\subset\mathbb{R}^d$ is a measurable set and $f,g$ are two real valued measurable functions defined on $\Omega$. Under classical assumptions, i.e.
(L1) $J(n)$ exists and is finite for every large $n$,
(L2) $f$ attains a separate absolute minimum value at an interior point $\boldsymbol t_0$ of $\Omega$,
(L3) $f\in {\rm C}^2(\boldsymbol t_0)$, $\dot{f}(\boldsymbol t_0)=\boldsymbol 0$, with $\boldsymbol0=(0,\ldots,0)'\in\mathbb{R}^d$; $\ddot{f}(\boldsymbol t_0)>0$,
(L4) $g$ is continuous at $t_0$ and $g(t_0)\neq0$,
it is well known (<cit.>) that as $n\to\infty$, $J(n)$ is asymptotic to $\tilde{J}(n)$, where
\[
\tilde{J}(n):=\frac{e^{-n f(\boldsymbol t_0)}g(\boldsymbol t_0)}{\sqrt{\det(\ddot{f}(\boldsymbol t_0))}}\left(\frac{2\pi}{n}\right)^\frac{d}{2}.
\]
There are two broad issues concerning the quality of this approximation, namely the rate at which its relative error
converges to 0 (inseparably with the associated regularity of $f$ and $g$) and the expression for the coefficient which appears in the error term. Let us first turn to the second question and, for the moment, notice only that typically this rate is $n^{-1}$. Several authors devoted their efforts to evaluating this constant (as well as these appearing in a further asymptotic expansion), always encountering an inevitable complexity in the obtained formulae, no matter if they considered a general setup as Kirwin (<cit.>) or a special form of the exponent function $f$ as Kaminski and Paris (<cit.>). This is actually unavoidable even in the one dimensional case (see, e.g., <cit.>), which is not considered in this paper.
From a practical point of view, rather than an asymptotic constant, one needs the upper and lower bounds for $E(n)$ instead, valid for $n\ge n_0$, say, with a benefit of a knowledge of $n_0$. The upper bounds for $|E(n)|$ was first derived in the paper <cit.>, where the two dimensional case ($d=2$) was considered in detail. Under global regularity conditions ($f\in {\rm C}^4(\Omega)$, $g\in{\rm C}^2(\Omega))$) and assuming some restrictions on the rate of growth of $f$ over $\Omega$ McClure and Wong provided such bounds valid for every $n$ for which $J(n)$ exists, and proved the typical rate $n^{-1}$ of the relative error of Laplace approximation. The main advantage of this result is its range of validity. On the other hand, even for $d=2$, it gives complicated coefficients of the main error term.
Recently, in <cit.> Inglot and Majerski introduced a new proof the Laplace approximation and obtained for any $d\ge2$ simple upper and lower bounds for $E(n)$, which essentially depend the smallest eigenvalue of the Hessian matrix $\ddot{f}(\boldsymbol t_0)$ only.
Despite this result requires a weak local regularity, $f\in{\rm C}^{3,1}(\boldsymbol t_0)$, it holds only for a constant $g$ and
under the hypotheses that the third order partial derivatives of $f$ all vanish at $\boldsymbol t_0$. Especially the last condition is constricting and was conjectured superfluous.
In the presented paper we shall prove that similar results actually hold without these constraints, i.e. when just $f\in{\rm C}^{3,1}(\boldsymbol t_0)$ and $g\in{\rm C}^{1,1}(\boldsymbol t_0)$. We also pay attention to make our results easier to use and remove two technical parameters used in <cit.>. Thus, up to these minor convenience aimed changes the present results do generalize their counterparts from <cit.>.
To complete the discussion on the rate of convergence of $E(n)$ it is worth mentioning here, that the typical rate $n^{-1}$ is not the only possible and is slower for less regular functions $f$ and $g$, but improvement of their regularity is insufficient alone to speed it up. In fact it follows from <cit.> that:
* if $f\in {\rm C}^{2,\alpha}(\boldsymbol t_0)$, $g\in{\rm C}^{0,\beta}(\boldsymbol t_0)$ for some $\alpha,\beta\in(0,1]$ then $E(n)$ of the order $n^{-\alpha\wedge \beta}$;
* if $f\in{\rm C}^{k,\alpha}(\boldsymbol t_0)$ with $k+\alpha>4$, $g$ is a constant function and when for $i=3,\ldots,k-1$ all $i$th order partial derivatives vanish at $\boldsymbol t_0$, then $E(n)$ is of the order $n^{-1-(k+\alpha-4)/2}$.
An application of our results to previously inaccessible examples considered earlier in <cit.> and <cit.> is presented. In particular we find that for $d=2$ our estimations give a better main error term than those obtained by the McClure and Wong technique and it turns out that a range of a factual validity of our bounds is far better than that indicated by our theoretical result. The latter is inevitable in the Inglot and Majerski approach and fortiori in the present one. We, however discuss a remedy to this flaw by sacrificing the quality of the constant in the main error term.
Finally, in addition to the presented application we point out that a motivation to study the foundations of the Laplace expansion comes from different fields of mathematical statistics (for some recent papers see <cit.>) but is also useful in such branches of science as natural language processing (see, e.g., <cit.> and the references therein) and statistical mechanics and quantum field theory (cf, <cit.>).
§ MAIN RESULT
We first deal with integrals of a fully exponential form, i.e.
$$I(n):=\int_{\Omega}e^{-n f(\mathbf{t})}d\mathbf{t}.$$
For simplicity we shall consider the normalized situation, which means that $f$ attains its global minimum $\mathbf{t_0}$ at $\mathbf{0}\in\mathbb{R}^d$ and $f(\mathbf{0})=0$. This is, of course, no restriction since putting $f_0(\mathbf{t}):=f(\mathbf{t}+\mathbf{t_0})-f(\mathbf{t_0})$ and $\Omega_0:=\{\mathbf{t}-\mathbf{t_0},~~{\mathbf t}\in \Omega\}$ one can write
\[
\int_{\Omega}e^{-n f(\mathbf{t})}d\mathbf{t}=e^{-n f(\mathbf{t_0})}\int_{\Omega_0}e^{-n f_0(\mathbf{t})}d\mathbf{t}.
\]
In what follows $||\cdot||$ will denote the Euclidean norm.
Let $\Omega\subset \mathbb{R}^d,\;d\geqslant 2$ be a measurable set, let $f:\Omega\to\mathbb{R}$ be a measurable function, and suppose that for some $n_1>0$ the integral
$I(n)=\int_{\Omega}e^{-n f(\mathbf{t})}d\mathbf{t}$ exists and is finite for all $n\geqslant n_1$. Let $\dot f$ vanish at $\mathbf{0}$, $\ddot f(\mathbf{0})$ exist and be positive definite with $\lambda_{\rm{min}}$ being its smallest eigenvalue, all third order partial derivatives of $f$ exist at $\boldsymbol0$. Denote
D:=\frac{d^{3/2}}6\max_{i,j,k}\left|\frac{\partial ^3f(\mathbf{\mathbf{0}})}{\partial t_i\partial t_j\partial t_k}\right|,\qquad i,j,k\in\{1,\ldots,d\}.
Assume that
(A1) there exist positive real numbers $r$, $C$ and $\alpha>1$ such that $B_r\subset\Omega$ and for every $\mathbf{t}\in B_r$
$$\left|f(\mathbf{t})-\frac12d^2f(\mathbf{0},\mathbf{t})-\frac16d^3f(\mathbf{0},\mathbf{t})\right|\leqslant C ||\mathbf{t}||^{2+\alpha};
(A2) there exist $\delta>0$ and $\Delta>0$ such that $f$ is convex on $B_{\delta}\subset\Omega$ and, moreover, for every $\mathbf{t}\in \Omega\setminus B_{\delta\wedge r}$
$$f(\mathbf{t})\geqslant \Delta.
Then for every $n\geqslant n_0$ with $n_0$ given explicitly below, the following bounds hold true
\begin{equation}\label{Lap_appr_statement_Lower}
I( n)\geqslant\frac{1}{\sqrt{\mbox{\rm det} \ddot{f}(\mathbf{0})}}
\left(\frac{2\pi}{ n}\right)^{d/2}\left[1-\frac{K_{\alpha,1}}{ n^{\alpha/2}}-\frac{K_l}{ n^{1+\alpha}}\right],
\end{equation}
\begin{equation}\label{Lap_appr_statement_Upper}
I( n)\leqslant\frac{1}{\sqrt{\mbox{\rm det} \ddot{f}(\mathbf{0})}}
\left(\frac{2\pi}{ n}\right)^{d/2}\left[1+\frac{K_{\alpha,1}}{ n^{\alpha/2}}+\frac{K_1}{n}+\frac{K_{\alpha,2}+K_u}{ n^{\alpha}}\right],
\end{equation}
\begin{equation}\label{constant_K}
\end{equation}
\begin{equation}\label{constant_K_1}
% K_1=\frac{D^2\Gamma(3+d/2)}{(\lambda_{\rm{min}}/2)^3\Gamma(d/2)},
\end{equation}
\begin{equation}\label{constant_K_l}
\end{equation}
\begin{equation}\label{constant_K_u}
K_u=\frac{7\sqrt{\mbox{\rm det}\ddot{f}(\mathbf{0})}}{4(2\pi)^{d/2}}I( n_1)e^{\xi n_1/2},
\end{equation}
$\xi=(r^2\lambda_{\rm{min}})\wedge (2\Delta)$ and $(x)_a=\Gamma(x+a)/\Gamma(x)$ is the Pochhammer symbol ($a\ge0$).
Moreover, $n_0:=\inf (N_1\cap N_2)$, with
\[
n\ge1:~{d}\leqslant(d+2\alpha){\log n}\leqslant \xi n
\right\},
\]
\[
n\ge1:~D\left(\frac{d+2\alpha}{\lambda_{\rm{min}}}\right)^{3/2}\cdot\frac{\log^{3/2}n}{n^{1/2}}+ C\left(\frac{d+2\alpha}{\lambda_{\rm{min}}}\right)^{1+\alpha/2}\cdot\frac{\log^{1+\alpha/2}n}{n^{\alpha/2}}\leqslant7/4
\right\}.
\]
Without loss of generality one can set $\Omega:=\mathbb{R}^d$. In fact, the suitable estimations in the proof below are eventually done by integrating over $\mathbb{R}^d$.
Let $U$ be the upper triangular matrix with positive diagonal entries defined as $\ddot{f}(\mathbf{0})=U^\prime U$. For $ n\geqslant 1$ put $\varepsilon=\sqrt{\frac{(d+2\alpha)\log n}{n}}$ and note that $\varepsilon=\varepsilon( n)\to0$ as $ n\to\infty$. Then $I( n)$ can be written as
\[
I( n)=\int_{E_\varepsilon(\ddot{f}(\mathbf{0}))}e^{- n f(\mathbf{t})}d\mathbf{t}+\int_{(E_\varepsilon(\ddot{f}(\mathbf{0})))^c}e^{- n f(\mathbf{t})}d\mathbf{t}=I_1( n)+I_2( n),
\]
where ${(E_\varepsilon(\ddot{f}(\mathbf{0})))^c}:=\mathbb{R}^d\setminus {E_\varepsilon(\ddot{f}(\mathbf{0}))}$.
Let $ n_2\geqslant 1\vee n_1$ be the smallest number such that
\begin{equation}\label{eq:lambda2}
\frac{d}{n}\leqslant(d+2\alpha)\frac{\log n}{ n}\leqslant \xi
\end{equation}
for all $ n\geqslant n_2$. It then follows from Rayleigh-Ritz theorem (<cit.>) that for
$ n\geqslant n_2$ it holds $\varepsilon^2\leqslant \lambda_{\rm{min}} r^2$ and $E_\varepsilon(\ddot f(\mathbf{0}))\subset B_r$. Using (A1) we have
\begin{align}
I_1( n)&=\int_{E_\varepsilon(\ddot{f}(\mathbf{0}))}e^{- n f(\mathbf{t})}d\mathbf{t}\notag\\
&\leqslant \int_{E_\varepsilon(\ddot{f}(\mathbf{0}))}e^{- n\mathbf{t}^\prime\ddot{f}(\mathbf{0})\mathbf{t}/2 -nd^3f(\mathbf{0},\mathbf{t})/6 + C n ||\mathbf{t}||^{2+\alpha}}d\mathbf{t}\notag\\
&=\frac{n^{-d/2}}{\det U} \int_{B_{\sqrt{ n}\varepsilon}}e^{-||\mathbf{u}||^2/2} \exp\left\{-\frac{d^3f(\mathbf{0},U^{-1}\mathbf{u})}{6n^{1/2}}+ \frac{C||U^{-1}\mathbf{u}||^{2+\alpha}}{ n^{\alpha/2}}\right\}d\mathbf{u}\label{ineq:I1U1},
\end{align}
where in (<ref>) we have applied the substitution $\mathbf{u}=\sqrt{ n}U\mathbf{t}$.
The condition
\begin{equation}\label{eq:lambda0}
D\left(\frac{d+2\alpha}{\lambda_{\rm{min}}}\right)^{3/2}\cdot\frac{\log^{3/2}n}{n^{1/2}}+ C\left(\frac{d+2\alpha}{\lambda_{\rm{min}}}\right)^{1+\alpha/2}\cdot\frac{\log^{1+\alpha/2}n}{n^{\alpha/2}}\leqslant7/4
\end{equation}
insures that the quantity in curly brackets in (<ref>) is less or equal $7/4$ for all $\mathbf{u}\in B_{\sqrt{n}\varepsilon}$. Let thus $n_0\geqslant n_2$ be the smallest number such that the inequality in (<ref>) holds for all $ n\geqslant n_0$. Using the the relation $\|U^{-1}\mathbf{u}\|\leqslant \lambda_{\rm{min}}^{-1/2} \|\mathbf{u}\|$, the definition of $\varepsilon$ and the inequality $e^x\leqslant1+x+x^2$ ($x\in(-\infty,7/4]$), we get from (<ref>) for $ n\geqslant n_0$
\begin{align}
I_1(n)&\leqslant\frac{n^{-d/2}}{{\rm det}\,U}\int_{B_{\sqrt{n}\varepsilon}}e^{-\|\mathbf{u}\|^2/2}
\Bigg\{1+\frac{C\|\mathbf{u}\|^{2+\alpha}}{\lambda_{\rm{min}}^{1+\alpha/2}n^{\alpha/2}}-
\frac{d^3f(\mathbf{0},U^{-1}\mathbf{u})}{6n^{1/2}}\notag\\
\frac{C\|\mathbf{u}\|^{2+\alpha}d^3f(\mathbf{0},U^{-1}\mathbf{u})}{3n^{(1+\alpha)/2}\lambda_{\rm{min}}^{1+\alpha/2}}+
\frac{\left(d^3f(\mathbf{0},U^{-1}\mathbf{u})\right)^2}{36n}\Bigg\}d\mathbf{u}.\notag
\end{align}
Observe that for every $R>0$, $\beta\geqslant0$ and for every $d\times d$ matrix $A$ we have
\begin{equation}\label{eq:int_d3}
\int_{B_R}e^{-\|\mathbf{u}\|^2/2}\|\mathbf{u}\|^\beta d^3f(\mathbf{0},A\mathbf{u})d\mathbf{u}=0.
\end{equation}
Hence, using the inequality
\[
\int_{B_{\sqrt{n}\varepsilon}}e^{-\|\mathbf{u}\|^2/2}d\mathbf{u}\leqslant (2\pi)^{d/2}
\]
we get for $n\geqslant n_0$
\begin{align}
\det U n^{d/2} I_1( n)&\leqslant (2\pi)^{d/2}+
\frac{C}{\lambda_{\rm{min}}^{1+\alpha/2} n^{\alpha/2}}\int_{\mathbb{R}^d} ||\mathbf{u}||^{2+\alpha}e^{-||\mathbf{u}||^2/2}d\mathbf{u}\notag\\
& +\frac{C^2}{\lambda_{\rm{min}}^{2+\alpha} n^{\alpha}}\int_{\mathbb{R}^d} ||\mathbf{u}||^{4+2\alpha}e^{-||\mathbf{u}||^2/2}d\mathbf{u}+\frac{D^2}{\lambda_{\rm{min}}^3n}\int_{\mathbb{R}^d} ||\mathbf{u}||^6e^{-||\mathbf{u}||^2/2}d\mathbf{u}\notag\\
\frac{K_{\alpha,1}}{ n^{\alpha/2}}+\frac{K_{\alpha,2}}{ n^{\alpha}}+\frac{K_1}{n}\right].\label{ineq:I1U}
\end{align}
Arguing in the same way, using again (A1), the inequality $e^x\geqslant 1+x$ and (<ref>) we obtain for $ n\geqslant n_0$
\begin{equation}\label{ineq:I1L}
{\det U} n^{d/2}I_1( n)\geqslant \int_{B_{\sqrt{ n}\varepsilon}} e^{-||\mathbf{u}||^2/2}d\mathbf{u}-\left(2\pi\right)^{d/2}\frac{K_{\alpha,1}}{ n^{\alpha/2}}.
\end{equation}
Set $R^2=n\varepsilon^2$. Using Lemma 2.1 from <cit.> we have for $n\ge n_2$
\begin{align}
\int_{B_{\sqrt{ n}\varepsilon}} e^{-||\mathbf{u}||^2/2}d\mathbf{u}&\ge (2\pi)^{d/2}\left(1-\frac{1}{\sqrt{\pi d}}\frac{R^2}{R^2-d+2}\exp\left\{-\frac12\left(R^2-d-(d-2)\log\frac{R^2}{d}\right)\right\}\right)\notag\\
&=(2\pi)^{d/2}\left(1-n^{-(\alpha+1)}\frac{e^{{d}/2}}{\sqrt{\pi d}}\frac{R^2}{R^2-d+2}\exp\left\{-\frac{d-2}{2(d+2\alpha )}R^2+\left(\frac{d}2-1\right)\log\frac{R^2}{d}\right\}\right),\notag
\end{align}
where the equality $R^2=2(\alpha+1)\log n+(d-2)/(d+2\alpha)R^2$ has been used. Observing that $R^2/(R^2-d+2)$ decreases with $R^2$ while the exponent attains its maximum value at $R^2=d+2\alpha$ we get for $n\ge n_2$ the following lower bound for the integral $\int_{B_{\sqrt{ n}\varepsilon}} e^{-||\mathbf{u}||^2/2}d\mathbf{u}$,
\[
(2\pi)^{d/2}\left(1-n^{-(\alpha+1 )}(e/2)\sqrt{d/\pi}(1+2\alpha/d)^{d/2-1}\right).
\]
This together with (<ref>) and the fact that $I_2( n)\geqslant 0$ proves the lower bound in (<ref>) for $ n\geqslant n_0$.
In order to prove the upper bound in (<ref>) one has to upper bound $I_2( n)$.
To this end observe first that from $({\rm A2})$ and (<ref>) we have on the set $(E_{\varepsilon}(\ddot{f}(\mathbf{0})))^c\cap B_{\delta\wedge r}^c$, for every $n\ge n_2$
\begin{equation}\label{eq:EcBc}
e^{-(n-n_1)f(\mathbf{t})}\le e^{-n\Delta}e^{n_1\Delta}
\le n^{-\frac{d}2-\alpha}e^{n_1\xi/2}.
\end{equation}
Now let us turn to the set $(E_{\varepsilon}(\ddot{f}(\mathbf{0})))^c\cap B_{\delta\wedge r}$. For every $\mathbf{t}$ from this intersection define $\mathbf{t}^*=(\varepsilon/(\mathbf{t}'\ddot{f}(\mathbf{0})\mathbf{t})^{1/2})\cdot \mathbf{t}$ and note that
$\mathbf{t}^*$ belongs to the line segment joining $\mathbf{0}$ with $\mathbf{t}$. Observe also that $(\mathbf{t}^*)'\ddot{f}(\mathbf{0})\mathbf{t}^*=\epsilon^2$.
Convexity of $f$ together with the fact that $\mathbf{0}$ is its minimizer imply that $f(\mathbf{t})\ge f(\mathbf{t}^*)$,
hence from the assumption (${\rm A1}$) for every $n\ge n_2$
\[
f(\mathbf{t})\ge f(\mathbf{t}^*)\ge\frac12(\mathbf{t}^*)'\ddot{f}(\mathbf{0})\mathbf{t}^*+\frac16d^3f(\mathbf{0},\mathbf{t}^*)-C\|\mathbf{t}^*\|^{2+\alpha}
\]
on the set $(E_{\varepsilon}(\ddot{f}(\mathbf{0})))^c\cap B_{\delta\wedge r}$.
\[
\|\mathbf{t}^*\|\le\lambda_{{\rm min}}^{-1}(\mathbf{t}^*)'\ddot{f}(\mathbf{0})\mathbf{t}^*
\]
\[
\left|\frac16d^3f(\mathbf{0},\mathbf{t}^*)\right|\le D\|\mathbf{t}^*\|^3\le
\frac{D}{\lambda_{{\rm min}}^{3/2}}((\mathbf{t}^*)'\ddot{f}(\mathbf{0})\mathbf{t}^*)^{3/2}
\]
we get for every $n\ge n_2$
\begin{equation}\label{eq:fLowerEcB}
f(t)\ge \frac12\varepsilon^2-\frac{D}{\lambda_{{\rm min}}^{3/2}}\varepsilon^3-\frac{C}{\lambda_{{\rm min}}^{1+\alpha/2}}\varepsilon^{2+\alpha}
\end{equation}
on the set $(E_{\varepsilon}(\ddot{f}(\mathbf{0})))^c\cap B_{\delta\wedge r}$. We also have for $n\ge n_0\ge n_2$
\begin{align}
&\exp\left\{-(n-n_1)\left(\frac12\varepsilon^2-\frac{D}{\lambda_{{\mathrm min}}^{3/2}}\varepsilon^3 -\frac{C}{\lambda_{{\mathrm min}}^{1+\alpha/2}}\varepsilon^{2+\alpha}\right)\right\}\notag\\
&=\exp\left\{\frac{n}2\varepsilon^2\right\}\exp\left\{n\left(\frac{D}{\lambda_{{\mathrm min}}^{3/2}}\varepsilon^3 +\frac{C}{\lambda_{{\mathrm min}}^{1+\alpha/2}}\varepsilon^{2+\alpha}\right)\right\}\exp\left\{n_1\left(\frac{\varepsilon^2}2-\frac{D}{\lambda_{{\mathrm min}}^{3/2}}\varepsilon^3 -\frac{C}{\lambda_{{\mathrm min}}^{1+\alpha/2}}\varepsilon^{2+\alpha}\right)\right\}\notag\\
&\le \frac74n^{-({d}/2+\alpha)}\exp\left\{n_1\left(\frac{\varepsilon^2}2-\frac{D}{\lambda_{{\mathrm min}}^{3/2}}\varepsilon^3 -\frac{C}{\lambda_{{\mathrm min}}^{1+\alpha/2}}\varepsilon^{2+\alpha}\right)\right\}\notag\\
&\le \frac74n^{-({d}/2+\alpha)}\exp\left\{n_1\frac{\varepsilon^2}2\right\}\le\notag\frac74e^{\xi n_1/2}n^{-({d}/2+\alpha)},
\end{align}
where we used (<ref>) and (<ref>). Merging the preceding inequality with (<ref>) and (<ref>) we get for $n\ge n_0$
\begin{align}
\int_{(E_{\varepsilon}(\ddot{f}(\mathbf{0})))^c}e^{-nf(\mathbf{t})}d\mathbf{t}&=
\int_{(E_{\varepsilon}(\ddot{f}(\mathbf{0})))^c}e^{-(n-n_1)f(\mathbf{t})}e^{-n_1f(\mathbf{t})}d\mathbf{t}\notag\\
&\le\notag\frac74e^{\xi n_1/2}n^{-({d}/2+\alpha)}\int_{(E_{\varepsilon}(\ddot{f}(\mathbf{0})))^c}e^{-n_1f(\mathbf{t})}d\mathbf{t}\notag\\
&\le\notag\frac74e^{\xi n_1/2}n^{-({d}/2+\alpha)}I(n_1),\notag
\end{align}
which completes the proof.
Clearly, the threshold $7/4$ in the condition (<ref>) ensues from the range of validity of the inequality $e^x\le 1+x+x^2$ and this arbitrary value may impose a strong restriction on the value $n_0$ and thus may limit the range of applicability of the result. When the condition (<ref>) is more restrictive than (<ref>) one may apply instead an inequality of a form $e^x\le1+x+(1+a)x^2$ valid for $x\le x_a$ with $a>0$ (and hence $x_a>7/4$) chosen such that (<ref>) holds for all $n\ge n_2$. This will, however, be done at a cost of enlarging the constants $K_1$, $K_{\alpha,2}$ and $K_u$ which should then be respectively rewritten as $aK_1$, $aK_{\alpha,2}$ and $4/7x_a$. Such a trade-off between the quality of the upper bound in (<ref>) and its range of applicability is illustrated in example <ref>.
On the other hand a completely opposite situation may arise, where the above-mentioned constants in (<ref>) are unsatisfactory but the left hand side of (<ref>) leaves plenty of space for adjustment. In such a situation one may take a suitable $a\in(-1/2,0)$ and improve $K_1$, $K_{\alpha,2}$ and $K_u$ at a cost (if any) of enlarging $n_0$.
The form of the upper bound in theorem <ref> involves two possible leading terms of the main error. Depending on $\alpha$ the rate of the main error term is either $n^{-\alpha/2}$ or $n^{-1}$. Inglot and Majerski proved that the rate $n^{-\alpha/2}$ is not underestimated when third order partial derivatives all vanish at $\mathbf{0}$, and hence the $n^{-1}$ term would be absent in our theorem <ref>. The following example shows that it is also the case without this condition for $\alpha<2$.
Let $f(\boldsymbol t)=\sum_{i=1}^dt_i^2+\sum_{i=1}^dt_i^3+\sum_{i=1}^d|t_i|^{3+\gamma}$, where $\gamma\in(0,1)$. The assumtions of theorem <ref> all hold. In particular for every $\mathbf{t}\in\mathbb{R}^d$
\[
\left|f(\mathbf{t})-\frac{1}{2}d^2f(\boldsymbol0,\boldsymbol t)-\frac16d^3f(\mathbf{0},\mathbf{t})\right|\le ||\mathbf{t}||^{3+\gamma},
\]
which means that (A1) holds for each $r>0$ with $\alpha=\gamma+1$ and $C=1$. The condition (A2) holds with $\delta=r$. Moreover, $\ddot{f}(\mathbf{0})={\rm diag}(2,\ldots,2)'$, so $\lambda_{\rm min}=2$. All the unmixed third order partial derivatives at $\mathbf{0}$ are equal 6, mixed being equal $0$, which yields $d^3f(\boldsymbol0,\boldsymbol t)/6=\sum_{i=1}^dt_i^3$.
We have
\begin{equation}\label{eq:ex1}
\int_{\mathbb{R}^d}e^{-n f(\mathbf{t})}d\mathbf{t}=\left(\int_{-\infty}^{\infty}e^{-n(x^2+x^3+|x|^{3+\gamma})}dx\right)^d=(i(n))^d.
\end{equation}
Using the inequalities $e^{-x}\ge1-x$ $(x\in\mathbb{R})$ and $e^{-x}\le1-x+x^2/2$ ($x>0$) we get respectively
\[
i(n)\ge\int_{-\infty}^{\infty}e^{-nx^2}\left(1-nx^3-n|x|^{3+\gamma})\right)dx=\sqrt{\frac\pi n}\left(1-\frac{\Gamma(2+\frac{\gamma}{2})}{\sqrt{\pi}}\cdot n^{-(1+\gamma)/2}\right),
\]
\begin{align}
\int_0^\infty e^{-n(x^2+x^3+x^{3+\gamma})}dx &\le
\frac12\sqrt{\frac{\pi}{n}}\left(1-\frac{\Gamma(2+\frac{\gamma}{2})}{\sqrt{\pi}}\cdot n^{-(1+\gamma)/2}-\frac{1}{\sqrt{\pi n}}\right.\notag\\
\frac{\Gamma(7/2+\gamma)}{\sqrt{\pi}}n^{-(1+\gamma)}\right)\notag.
\end{align}
Put $a_n:=\sqrt{2\log n/n}$. We have
\begin{align}
\int_{-\infty}^{-a_n}e^{-n(x^2+x^3+(-x)^{3+\gamma})}dx&=\int_{a_n}^{\infty}e^{-(n-1)(x^2-x^3+x^{3+\gamma})}e^{-(x^2-x^3+x^{3+\gamma})}dx\notag\\
& \le e^{-na_n^2}e^{na_n^3-na_n^{3+\gamma}}\int_0^{\infty}e^{-(x^2-x^3+x^{3+\gamma})}dx\notag\\
& \le C_1n^{-2},\notag
%\le i(1)e^{-(n-1)}\le2.35 e^{-n},
\end{align}
with $C_1=\exp(0.3)\int_0^{\infty}e^{-(x^2-x^3+x^{3+\gamma})}dx$. Moreover, since $x^3-x^{3+\gamma}$ is increasing for sufficiently small $x>0$, it holds that for $n$ large enough and $x\le a_n$
\[
n(x^3-x^{3+\gamma})\le\frac{(2\log n)^{3/2}}{n^{1/2}}\left(1-\left(\frac{2\log n}{n}\right)^{\gamma/2}\right)<7/4.
\]
Thus, using the inequality $e^{x}\le1+x+x^2$, $(x\le7/4)$,
\begin{align}
\int_{0}^{a_n}e^{-nx^2}e^{nx^3-nx^{3+\gamma}}dx&\le \int_{0}^{a_n}e^{-nx^2}\left(1+nx^3-nx^{3+\gamma}+n^2(x^3-x^{3+\gamma})^2\right)dx\notag\\
&\le \int_0^{\infty}e^{-nx^2}(1+nx^3+n^2x^6)dx-n\int_{0}^{a_n}e^{-nx^2}x^{3+\gamma}dx\notag\\
&=\notag \frac12\sqrt{\frac{\pi}{n}}\left(1+\frac{1}{\sqrt{\pi n}}+\frac{15}8n^{-1}-\frac{\Gamma(2+\frac{\gamma}{2})-\Gamma(2+\frac{\gamma}{2},2\log n)}{\sqrt{\pi}}\cdot n^{-(1+\gamma)/2}\right),\notag
\end{align}
where $\Gamma(\cdot,\cdot)$ denotes the incomplete gamma function. In view of a fact $\Gamma(k,2\log n)\sim (2\log n)^{k-1}n^{-2}$ as $n\to \infty$ (<cit.>), we infer that with $C_2<15/8$ for every large $n$ it holds
\[
i(n)\le\sqrt{\frac{\pi}{n}}\left(1-\frac{\Gamma(2+\frac{\gamma}{2})}{\sqrt{\pi}}\cdot n^{-(1+\gamma)/2}+C_2n^{-1}\right).
\]
Hence from (<ref>) and the multinomial theorem we conclude that
\begin{align}
I(n)= \left(\frac{\pi}{n}\right)^{d/2}\left(1-\frac{d\Gamma(2+\frac{\gamma}{2})}{\sqrt{\pi}}\cdot n^{-(1+\gamma)/2}+O(n^{-1})\right).
\end{align}
Let $\Omega$ and $f$ satisfy assumptions of theorem <ref>. Let $g:\Omega\to\mathbb{R}$ be a measurable function and $g(\boldsymbol0)\neq0$. Suppose $g\in {\rm C}^{1,1}(B_r)$, so that
(A3) there exists $M>0$ such that for every $\boldsymbol t\in B_r$
$$\left|g(\boldsymbol t)-g(\boldsymbol 0)-\dot{g}(\boldsymbol 0)'\boldsymbol t\right|\le M\left\|t\right\|^2.$$
Let $J(n)=\int_{\Omega}e^{-n f(\boldsymbol t)}g(\boldsymbol t)dt$ and suppose there exists $n_3>0$ such that $J(n)$ is finite for all $n\geqslant n_3$. Then for every $n\geqslant n_4$, where $n_4$ is given by (<ref>), it holds
\[
J(n)=\frac{g(\boldsymbol0)}{\sqrt{\mbox{\rm det} \ddot{f}(\boldsymbol0)}}\left(\frac{2\pi}{n}\right)^{d/2}\left(1+E(n)\right),
\]
\begin{align}
\notag\\
E(n)&\leqslant \frac{K_{\alpha,1}}{n^{\alpha/2}}+\frac{K_1+K_2+K_3}{n}+\frac{K_{\alpha,2}+K_{ul}}{n^{\alpha}}+
\frac{K_{\alpha,3}}{n^{1+\alpha/2}}+\frac{K_4}{n^2}+\frac{K_{\alpha,5}}{n^{1+\alpha}}+\frac{K_{\alpha,6}}{n^{(3+\alpha)/2}},\notag
\end{align}
with $K_{\alpha,1},\;K_{\alpha,2},\;K_1$ and $K_l$ are given by (<ref>), (<ref>) and (<ref>), respectively, and
\[
% K_{\beta}=\frac{M\,\Gamma((d+\beta)/2)}{\left|g(\boldsymbol0)\right|(\lambda_{\rm{min}}/2)^{\beta/2}\Gamma(d/2)},\;
% K_{3,\beta}=\frac{MD^2\Gamma(3+(\beta+d)/2)}{|g(\boldsymbol0)|(\lambda_{\rm{min}}/2)^{3+\beta/2}\Gamma(d/2)},
\]
\[
\;K_4=\frac{MD^2(2/\lambda_{\rm{min}})^4(d/2)_4}{|g(\boldsymbol0)|},
\]
\[
% K_{\alpha,1,\beta}=\frac{C\,M\,\Gamma(1+(\alpha+\beta+d)/2)}{\left|g(\boldsymbol0)\right|(\lambda_{\rm{min}}/2)^{1+(\alpha+\beta)/2}\Gamma(d/2)},
% \;K_{\alpha,2,\beta}=\frac{C^2M\,\Gamma(2+\alpha+(d+\beta)/2)}{\left|g(\boldsymbol0)\right|(\lambda_{\rm{min}}/2)^{2+\alpha+\beta/2}\Gamma(d/2)},
\;K_{\alpha,6}=\frac{2CD\|\dot{g}(\boldsymbol0)\|\,(2/\lambda_{\rm{min}})^{(3+\alpha)/2}(d/2)_{(3+\alpha)/2}}{\left|g(\boldsymbol0)\right|},
\]
\[
K_{ul}=\frac{7\sqrt{\mbox{\rm det}\,\ddot{f}(\boldsymbol0)}}{4\left|g(\boldsymbol0)\right|(2\pi)^{d/2}}e^{n_3\xi/2}\int_{\mathbb{R}^d}e^{-n_3 f(\boldsymbol t)}|g(\boldsymbol t)|d\boldsymbol t.
\]
Moreover, $n_4\ge n_0$ is the smallest number such that the following inequality holds true
\begin{equation}\label{eq:lambda4}
\frac{M(d+2\alpha)}{\lambda_{\rm min}}\frac{\log n}{n}+\|\dot{g}(\boldsymbol0)\|\sqrt{\frac{d(d+2\alpha)}{\lambda_{\rm min}}}\cdot\sqrt{\frac{\log n}{n}}\le g(\boldsymbol0).
\end{equation}
Proof. The proof is a slight extension that of theorem 2.3 in <cit.> and an extension of our equality (<ref>). The condition (<ref>) insures that for every $t\in B_r$ the both bounds for $g$ in the inequalities
\[
g(\boldsymbol 0)+\dot{g}(\boldsymbol 0)'\boldsymbol t-M\|t\|^2 \le g(\boldsymbol t)\le g(\boldsymbol 0)+\dot{g}(\boldsymbol 0)'\boldsymbol t+M\|t\|^2
\]
are nonnegative.
A consequence of the above theorem is the following corollary, which provides the rate $n^{-1}$ under the weakest regularity assumption on $f$ and $g$.
Let the conditions (L1)-(L4) all hold. Let moreover $f\in{\rm C}^{2,1}(\boldsymbol t_0)$ and $g\in{\rm C}^{1,1}(\boldsymbol t_0)$. Then it holds
\[
J(n)=\frac{e^{-n f(\boldsymbol t_0)}g(\boldsymbol t_0)}{\sqrt{\mbox{\rm det} \ddot{f}(\boldsymbol t_0)}}\left(\frac{2\pi}{n}\right)^{d/2}\left(1+O(n^{-1})\right).
\]
§ APPLICATION
Following <cit.> consider the sum
\[
S(s,n)=\sum_{k=0}^{2n}(-1)^{k+n}{2n \choose k}^s,
\]
where $n$ and $s$ are two positive integers. This formula is known in an explicit form for at most $s=3$ (called Dixon's identity) and no such extension exists for $s\ge4$. Here we consider an approximation of $S(s,n)$ as $n\to\infty$ for $s$ fixed. To this end we shall use its following integral representation <cit.>
\[
S(d+1,n)=\frac{2^{2n(d+1)}}{\pi^d}\cdot\int_{(-\pi/2,\pi/2)^d}\left\{\sin\!\left(\sum_{i=1}^d \varphi_i\right)\prod_{i=1}^d\cos\varphi_i\right\}^{2n} d\boldsymbol\varphi,
\]
with $\boldsymbol\varphi=(\varphi_1,\ldots,\varphi_d)'$. As the integrand is symmetrical over values of $\boldsymbol\varphi$ corresponding to $\sum_{i=1}^d \varphi_i>0$ or $<0$, in order to investigate the asymptotic behavior of $S(d+1,n)$ it suffices to consider the integral
\[
T(d,n):=\int_{\Omega'} e^{-2n h(\boldsymbol\varphi)}d\boldsymbol\varphi,
\]
where $\Omega'=\{\boldsymbol\varphi\in\mathbb{R}^d:~\varphi_i<\pi/2,~\varphi_1+\ldots\varphi_d>0\}$, and $h(\boldsymbol\varphi)=-\log\sin(\sum_{i=1}^d \varphi_i)-\sum_{i=1}^d\log\cos\varphi_i$. The unique minimizer of $h$ in $\Omega'$ is $\hat{\boldsymbol\varphi}=(\hat\alpha,\ldots,\hat\alpha)'$ with $\hat\alpha=\pi/2(d+1)$ and $h(\hat{\boldsymbol\varphi})=-(d+1)\log\cos(\hat\alpha)$, for $\sin(d\hat\alpha)=\cos(\hat\alpha)$. In order to apply theorem <ref> write
\[
T(d,n)=\cos^{2n(d+1)}(\hat{\alpha})\int_\Omega e^{-2n f(\boldsymbol\varphi)}d\boldsymbol\varphi,
\]
\[
\Omega=\{\boldsymbol\varphi\in\mathbb{R}^d:~\varphi_i<d\hat{\alpha},~\varphi_1+\ldots\varphi_d>-d\hat{\alpha}\}
\]
and $f(\boldsymbol\varphi)=h(\boldsymbol\varphi+\hat{\boldsymbol\varphi})-h(\hat{\boldsymbol\varphi})=
\log\cos(\sum_{i=1}^d\varphi_i-\hat\alpha)-
\sum_{i=1}^d\log\cos(\varphi_i+\hat\alpha)-
h(\hat{\boldsymbol\varphi})$. Now $\mathbf{0}$ minimizes $f$ over $\Omega$ and
$\dot{f}(\mathbf{0})=0$. We also have $\ddot{f}(\mathbf{0})=[(\delta_{ij}+1)\cos^{-2}(\hat{\alpha})]_{i,j=1,\ldots,d}$, where $\delta_{ij}$ denotes the Kronecker delta, hence $\det(\ddot{f}(\mathbf{0}))=(d+1)\cos^{-2d}(\hat{\alpha})$ and $\lambda_{\rm{min}}=\cos^{-2}(\hat{\alpha})$. Moreover,
\[
\frac{\partial ^3f(\boldsymbol0)}{\partial t_i\partial t_j\partial t_k}=-2(1-\delta_{ij}\delta_{jk})(\cos^{-3}\cdot\sin)(\hat{\alpha}).
\]
For fixed $\eta\in(0,1)$ let us define the ball $B_{\eta\sqrt{d}\hat{\alpha}}$. It is easy to verify that $B_{\eta\sqrt{d}\hat{\alpha}}\subset \Omega_{\eta}\subset \Omega$, where $\Omega_\eta=\{\boldsymbol\varphi\in\mathbb{R}^d:~|\varphi_i|\le\eta\sqrt{d}\hat{\alpha},~|\varphi_1+\ldots\varphi_d|\le\eta d\hat{\alpha}\}$. An application of Taylor's theorem yields for $\boldsymbol\varphi\in B_{\eta\sqrt{d}\hat{\alpha}}$
\[
\left|f(\boldsymbol\varphi)-\frac12d^2f(\boldsymbol0,\boldsymbol \varphi)-\frac16d^3f(\boldsymbol0,\boldsymbol\varphi)\right|=
\frac1{24}\left|d^4(\boldsymbol\xi,\boldsymbol\varphi)\right|,
\]
where $\boldsymbol\xi\in B_{\eta\sqrt{d}\hat{\alpha}}\subset \Omega_\eta$. We have
\[
\frac{\partial ^4f(\boldsymbol\xi)}{\partial t_i\partial t_j\partial t_k\partial t_l}=2\left(u\left(\sum_{i=1}^d\xi_i-\hat{\alpha}\right)
+\delta_{ij}\delta_{jk}\delta_{kl}\cdot u\left(\xi_l+\hat{\alpha}\right)\right),
\]
where $u:=(\cos^{-4}\cdot(1+2\sin^2))$. It is seen that $u(\sum_{i=1}^d\varphi_i-\hat\alpha)\le u((\eta d+1)\hat\alpha)$ and $u(\varphi_i+\hat\alpha)\le u((\eta\sqrt{d}+1)\hat\alpha)$ on $\Omega_\eta$. One thus obtains for every $\boldsymbol\varphi\in B_{\eta\sqrt{d}\hat{\alpha}}$
\begin{equation}\label{eq:Ex3C}
\left|f(\boldsymbol\varphi)-\frac12d^2f(\boldsymbol0,\boldsymbol \varphi)-\frac16d^3f(\boldsymbol0,\boldsymbol\varphi)\right|\le
\end{equation}
\[
\frac{1+2\sin^2}{\cos^4}\left(\frac{(\eta d+1)\pi}{2(d+1)}\right)+\frac{1+2\sin^2}{\cos^4}\left(\frac{(\eta\sqrt{d}+1)\pi}{2(d+1)}\right)
\right\},
\]
thus, the condition (A1) holds. To verify (A2) note that $f$ is convex on $\Omega$ since $\ddot{f}(\boldsymbol\varphi)=(\cos^{-2}(\sum_{i=1}^d\varphi_i-\hat{\alpha})+\delta_{ij}\cos^{-2}(\varphi_j+\hat{\alpha}))_{i,j=1,\ldots,d}$ is postive definite as a matrix of the quadratic form
\[
\cos^{-2}(\varphi_1+\hat{\alpha})x_1^2+\ldots+\cos^{-2}(\varphi_d+\hat{\alpha})x_d^2+\cos^{-2}(\sum_{i=1}^d\varphi_i-\hat{\alpha})(x_1+\ldots +x_d)^2.
\]
We thus may take $\delta=r=\eta\sqrt{d}\hat{\alpha}$. Finding a reasonable estimation for $\Delta$ in such a general setting seems a difficult task. One way to do this is to use the (A1) condition, from which $f(\boldsymbol\varphi)\ge\boldsymbol\varphi\ddot{f}(\boldsymbol 0)\boldsymbol\varphi'/2+d^3f(\boldsymbol 0,\boldsymbol\varphi)/6-C\|\boldsymbol\varphi\|^4$ on $B_r$ and because $|d^3f(\boldsymbol 0,\boldsymbol\varphi)/6|\le D\|\boldsymbol\varphi\|^3$ and $\boldsymbol\varphi\ddot{f}(\boldsymbol 0)\boldsymbol\varphi'\ge\lambda_{\rm min}\|\boldsymbol\varphi\|^2$ we infer that for every $t\in B_r$
\[
f(\boldsymbol\varphi)\ge\frac{\lambda_{\rm min}}2r^2-Dr^3-Cr^4.
\]
Hence, in view of convexity of $f$ one may take $\Delta:=\lambda_{\rm min}r^2/2-Dr^3-Cr^4$, provided $\Delta>0$, i.e. $\eta$ is small enough.
Denoting $I(n):=\int_\Omega e^{-2nf(\boldsymbol\varphi)}d\boldsymbol\varphi$, we finally conclude that for every $\eta\in(0,1)$ such small that $\Delta$ defined above is positive, with $r:=\eta\sqrt{d}\hat\alpha$ and for every large enough $n\ge1$ there hold
\begin{align}
\left\{1-\frac{K_{\alpha,1}}{2n}-\frac{K_l}{8n^3}\right\},\notag\\
\left\{1+\frac{K_{\alpha,1}+K_1}{2n}+\frac{K_{\alpha,2}+K_u}{4n^2}\right\},\notag
\end{align}
with $K_{\alpha,1}=C\cos^4(\hat\alpha)d(d+2)$, $K_1=\sin^2(\hat\alpha)d^4(d+2)(d+4)$,
$K_{\alpha,2}=C^2\cos^8(\hat\alpha)d(d+2)(d+4)(d+6)$, $K_l$ given by (<ref>) and
$K_u=(7/4)\sqrt{d+1}\cos^{-d}(\hat\alpha)(2\pi)^{-d/2}I(1)e^{\xi/2}$, where $I(1)=(2^d-1)(\pi/4)^d$.
Recalling that $\hat\alpha=\pi/2(d+1)$ one sees consistency in the behavior of our main error term of the upper and lower bound as $d\to\infty$, for $K_{\alpha,1}\asymp d^4$ and $K_1\asymp d^4$.
We shall consider the case $s=3$ and compare our estimations with those obtained previously in <cit.>. For $d=2$ $\Delta>0$ whenever $\eta\le0.36$. Taking example value $\eta=1/3$ we find $C=7.7$, $K_{\alpha,1}/2=17.4$ and $K_1/2=64$. Hence for every large $n$ we have
\[
S(3,n)=\frac{3^{3n+\frac12}}{2\pi n}\left(1+\frac{A}{n}+O\!\left(\frac1{n^2}\right)\right),
\]
where $A\le 81.4$. Note that taking $\eta$ small enough one can near the constant $C$ arbitrarily close to $C^{(best)}=16/9$, and consequently reach $A^{(best)}=68$. As we shall see this is a very poor estimate for $A$, which is a consequence of a poor estimation of $C$.
We shall now adopt the method used by McClure and Wong (<cit.>), who proposed prior elimination of cross product term $xy$ in the Taylor expansion of $f(\varphi_1,\varphi_2)$ via a linear transformation. This can be achieved using the Cholesky decomposition of $\ddot{f}(0,0)$. Thus, following <cit.> and applying to our $I(n)$ the change of variables
\[
\varphi_1=(\sqrt{3}/2)x-(1/2)y,~~~~\varphi_2=y
\]
one obtains
\[
\]
\begin{align}
f(x,y)= &-\log\cos \left(\frac{\sqrt3}{2}x-\frac12y+\frac{\pi}6\right)-\log\cos \left( -\frac{\sqrt3}2x-\frac12y+\frac{\pi}6 \right)\notag\\
&-\log\cos \left( y+\frac\pi6 \right) +3\log\frac{\sqrt3}2,\notag
\end{align}
and $\Omega=\left\{(x,y)\in\mathbb{R}^2:\frac{\sqrt3}2x-\frac12y<\frac\pi3,~ y<\frac\pi3,~ \frac{\sqrt3}2x+\frac12y>-\frac\pi3\right\}$.
Take $r\in(0,\pi/3)$ and note that $B_r\subset\mathbb{R}^2$ is enclosed by the hexagonal set $\Omega_r:=\{(x,y)\in\mathbb{R}^2:~|\frac{\sqrt3}2x-\frac12y|\le r,~ |y|\le r,~ |\frac{\sqrt3}2x+\frac12y|\le r\}$. Considering the univariate function $-\log\cos(t+\pi/6)$ whose Maclaurin expansion is of a form
\[
\]
we can apply the Taylor theorem with remainder to each of the three first terms defining $f(x,y)$ and obtain on $B_r$
\begin{align}
&=\frac1{12}\left\{ \left(\frac{\sqrt3}{2}x-\frac12y\right)^4 u(\xi_1)+\left(-\frac{\sqrt3}{2}x-\frac12y\right)^4 u(\xi_2)+y^4 u(\xi_3)\right\},\notag
\end{align}
where $u(t):=(\cos^{-4}(1+2\sin^2)) (t+\pi/6)$ and $\xi_i\in(-r,r)$, for $i=1,2,3$. Thus, using the relation
\[
\]
we infer
\begin{equation}\label{ex:Cbound}
\left|f(x,y)-x^2-y^2-\frac{\sqrt{3}}9(y^3-3x^2y)\right|\le C(x^2+y^2)^2,
\end{equation}
with $C=(3/32)(\cos^{-4}(1+2\sin^2))(r+\pi/6)$. Following <cit.> put $r^2=\pi^2/108$ and note that the condition (A1) holds with $C=0.9238$, $\alpha=2$ and $\lambda_{\rm{min}}=2$. Moreover, $\delta=r$, $f$ is convex and reasoning as in the previous example we find $\Delta=0.06863$. Hence we have
\begin{align}
S(3,n)&\ge\frac{{3}^{3n+1/2}}{2\pi n}
\left\{1-\frac{K_{\alpha,1}/2}{n}-\frac{K_l/8}{n^3}\right\},\label{eq:S3nApprLo}\\
S(3,n)&\le\frac{{3}^{3n+1/2}}{2\pi n}
\left\{1+\frac{(K_{\alpha,1}+K_1)/2}{n}+\frac{(K_{\alpha,2}+K_u)/4}{n^2}\right\},\label{eq:S3nApprUp}
\end{align}
where $K_{\alpha,1}/2=0.9238$, $(K_{\alpha,1}+K_1)/2=1.072$, $K_l/8=0.1355$, $K_{\alpha,2}=20.48$ and $(K_{\alpha,2}+K_u)/4=5.439$.
Using their device McClure and Wong <cit.> proved that (in our notation and emending an obvious slip of the pen in the leading term in (5.37); cf. <cit.>)
\[
S(3,n)=\frac{{3}^{3n+1/2}}{2\pi n}
\left\{1+\tilde{E}_1(n)+\tilde{E}_2(n)\right\},
\]
where $|\tilde{E}_1(n)|\le 1.8245/n$ and $|\tilde{E}_2(n)|\le (7/3)e^{-n\pi^2/72}$. We conclude that our main error term $1.072$ significantly improves the known estimation from <cit.>. We also note that for every $n\ge2$ the relative error given in our bounds is better estimated than the compared one. For instance, when $n=1$ their result implies that the error is within $\mp 3.9$ whereas our gives the interval $(-1.1,6.5)$. For $n=2$ these are $\mp2.7$ and $(-0.48,1.9)$; for $n=5$: $\mp1.5$ and $(-0.18,0.43)$; for $n=10$: $\mp0.77$ and $(-0.093,0.16)$ while for $n=100$: $\mp0.018$ and $(-0.0092,0.011)$.
It is however well known that the approach used in this paper fails to indicate a satisfactory value for $n_0$, i.e. the minimum value of $n$ insuring the validity of inequalities in (<ref>) and (<ref>) (cf. <cit.>). While the right hand side of (<ref>) already indicates $n_0\ge240$, the requirement (<ref>) yields $n_0=1479$. Meanwhile, numerical simulations not presented here showed that the inequalities work for every positive integer $n$. In order to make the condition (<ref>) less restrictive than (<ref>) one could use a weaker inequality $e^x\le1+x+2.2x^2$ $(x<3.39)$ in the proof of the theorem <ref> yielding $n_0=240$ at a cost of enlarging the constants $(K_{\alpha,1}+K_1)/2$ and $(K_{\alpha,2}+K_u)/4$ to $1.2497$ and $11.5833$, respectively. On the other hand, even if one could find a better estimation for $\Delta$, for a given $r^2=\pi^2/108$ the restriction imposed by (<ref>) gives at least $n_0\ge169$, for $\xi\ge r^2\lambda_{\rm min}$.
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|
1511.00460
|
Version v8
Physik und Zentrum für Materialwissenschaften,
Philipps-Universität, 35032 Marburg, Germany
73.20.-r, 78.47.J-, 79.60.-i, 79.60.Bm
We combine tunable mid-infrared (MIR) pump pulses with time- and
angle-resolved two-photon photoemission to study ultrafast
photoexcitation of the topological surface state (TSS) of
It is revealed that MIR pulses permit a direct excitation of
the unoccupied TSS owing to an optical coupling across the Dirac
The novel optical coupling provokes asymmetric transient populations
of the TSS at ${\pm}k_{||}$, which mirrors a macroscopic photoexcited
electric surface current.
By observing the decay of the asymmetric population, we
directly investigate the dynamics of the long-lived
photocurrent in the time domain.
Our discovery promises important advantages of photoexcitation by MIR pulses for
spintronic applications.
Three-dimensional (3D) topological insulators (TIs) belong to a new class of
materials which are characterized by an insulating bulk and a metallic
topological surface state (TSS) <cit.>.
The most remarkable properties of the TSS are its Dirac-cone-like energy
dispersion and its chiral spin texture in $k$-space <cit.>.
The latter incorporates a protection against backscattering and is
very promising for spintronic applications.
Optical coupling to the chiral spin texture of TSSs offers
many interesting phenomena, such as optical control of
the spin <cit.>, surface
transport <cit.>,
and topological phases <cit.>.
To exploit these exotic properties, a detailed understanding
of the optical excitation and the subsequent electron dynamics are essential keys.
A number of studies have investigated the ultrafast electron dynamics
by optical methods like reflectivity <cit.>, second harmonic
generation <cit.>, or by optically triggered detection of
photocurrents <cit.>.
These experiments, however, hardly show a pure photoexcitation of
TSSs, since the bulk response typically governs the total signal.
Time- and angle-resolved two-photon photoemission (2PPE) is a
particularly suited technique for this purpose <cit.>,
because it can directly image the optically excited electron
population by energy-momentum mapping and makes it possible to
follow the ultrafast carrier dynamics in the time domain by pump-probe
This technique has been successfully applied for the study of the
electron dynamics in 3D TIs <cit.> using pump laser pulses at 800-nm or
in the visible range.
In contrast to higher-lying states, such as image-potential states,
however, the relevant TSS close to the Fermi energy ($E_{\rm F}$)
is in these studies only indirectly excited by a delayed filling
from states far above $E_{\rm F}$.
Under such conditions, a coherent optical control of the TSS population
is difficult and its decay dynamics is masked.
In this Letter, we demonstrate that tunable low-energy pump
pulses in the mid-infrared (MIR) regime are capable to induce a
direct optical transition between the occupied and unoccupied
part of the TSS across the Dirac point (DP). We show this for
the example of $p$-doped Sb$_{2}$Te$_{3}$, in which the most
part of its Dirac cone is initially
unoccupied <cit.>.
In contrast to a bulk mediated indirect population, this
resonant TSS-TSS transition makes it possible to generate an
asymmetry between the transient population of opposite parallel
momenta $k_{||}$, which directly mirrors a macroscopic spin
polarized photocurrent within the TSS.
By monitoring the decay of this asymmetry, we identify different scattering
mechanisms of the electrons that carry the photocurrent.
[](Color online)
(a) Excitation scheme for the population of
the TSS in Sb$_2$Te$_3$ with MIR pump pulses and subsequent photoemission with
ultraviolet probe pulses.
(b) and (c) Angle-resolved 2PPE spectra for
0.37 eV pump and 5.16 eV probe pulses 50 fs after MIR excitation at 300 K and
80 K, respectively. Red arrows indicate the energies
of maximum population enhancement in the TSS.
(d) and (e) show the results acquired at 80 K using
0.33 eV and 0.31 eV pump pulses, respectively. (f) shows a spectrum for
2.58 eV pump pulses for comparison.
Details of our optical setup are described in the Supplemental
Material <cit.>.
Electrons were excited into initially unoccupied states above
$E_{\rm{F}}$ with 100-fs MIR laser pulses of tunable
photon energy ($h\nu_1$=0.25-0.37 eV).
The transient population was subsequently probed by photoemission of
these electrons using ultraviolet (UV) laser pulses ($h\nu_2$=5.16 eV,
80 fs) [Fig. <ref> (a)].
Both $p$-polarized beams were focused on the sample into a spot
with a diameter of $\sim$100 $\mu$m using a non-collinear geometry.
The experiments were carried out in a $\mu$-metal shielded UHV chamber
at a base pressure of $4\times 10^{-11}$ mbar.
Photoelectrons were collected along the high symmetry
line $\bar{\rm{\Gamma}}$-$\bar{{\rm{K}}}$ by a hemispherical analyzer
(Specs Phoibos 150) with a display-type detector.
A single crystal of $p$-doped Sb$_{2}$Te$_{3}$ was cleaved in
situ by the Scotch tape method at a pressure of $3\times
10^{-10}$ mbar followed by a rapid recovery back to the base pressure
within a minute.
The Dirac point (DP) of the sample was located $\sim$150 meV
above $E_{\rm{F}}$.
During the measurements the sample temperature was maintained at 300 K
or 80 K.
We start by discussing the optical excitation process into the unoccupied TSS
by means of Figures <ref> (b) and (c), which show angle-resolved 2PPE spectra
of Sb$_{2}$Te$_{3}$ using a pump photon energy $h\nu_1$ of
0.37 eV at 300 K and 80 K, respectively.
To avoid interfering 2PPE signals from the image-potential states,
which are excited by the UV pulses and probed by the MIR pulses,
a 50 fs delay of the UV probe pulses with respect to the MIR pump
pulses has been used.
Even at this small delay, a considerable population of the TSS can
already be observed.
It extends only up to 350 meV above $E_{\rm{F}}$ due to the low excitation energy.
Curiously, the population of the TSS is pronounced at a specific energy
of $\sim$300 meV (red arrows).
It becomes even more pronounced at 80 K [Fig. <ref> (c)], where also
a considerable population near the lower edge of bulk conduction band
(BCB) at 270 meV can be observed.
The specific energy of the population enhancement in the TSS significantly
depends on $h\nu_1$: it clearly shifts to lower energy with
decreasing $h\nu_1$ even for small changes of $h\nu_1$ [Fig. <ref> (c)-(e)].
The data also clearly shows that the population at $-k_{||}$ is
much more prominent compared to that at $+k_{||}$, which represents a strong
asymmetric population in $k$-space.
The population enhancement and its strong asymmetry is not observed
for visible pump pulses ($h\nu_1$=2.58 eV) and the same UV probe pulses
[Fig. <ref> (f)].
Thus, it turns out that theses findings are unique for the optical excitation
with MIR pulses.
[](Color online)
Energy position of the pronounced 2PPE signal created
in the TSS as a function of the MIR photon energy acquired at 300 K
(red circle) and 80 K (blue circle), respectively.
The solid and the dashed line show the simulated energy dependence
for TSS-TSS transitions and TSS-BCB transitions (see text for details).
In order to reveal the origin of the pronounced population at a specific energy,
we have investigated its dependence on $h\nu_1$ by systematic tuning of the MIR pulses.
Figure <ref> summarizes this dependence for 300 K (red symbols) and 80 K (blue symbols).
Obviously, the energy position of the population enhancement is proportional
to the excitation energy for both temperatures.
This is a clear indication that it is induced by a specific direct optical
coupling to the TSS.
In principle, three different optical excitation processes can result
in a population of the TSS within this energy range:
Firstly, transitions from the bulk valence band (BVB) to the BCB.
This process, however, can neither raise the population at a specific
energy in the TSS, nor should it show a clear $h\nu_1$ dependence, because both
bands can be coupled over a wide energy range.
Secondly, TSS-BCB (BVB-TSS) transitions for which the initial (intermediate) state
is the TSS and the intermediate (initial) state is a bulk state.
Thirdly, resonant transitions between the occupied and unoccupied
part of the TSS across the DP (TSS-TSS transition).
[](Color online)
(a) Time-evolutions of the 2PPE signals $I_{+k}$ (red), $I_{-k}$ (blue)
and their difference ${\Delta}I$=$I_{-k}-I_{+k}$ (green) for different
energies at 80 K.
$I_{\pm{k}}$ is respectively obtained by integrating the 2PPE signal within
the windows shown in Fig. <ref> (c). The inset depicts the
maximum of $A$=${\Delta}I/(I_{+k}+I_{-k})$ as a function of energy.
(b) Representative line profiles from (a) at the direct excitation energy at
300 K (top) and 80 K (bottom).
Solid lines indicate the fitting results of the rate equation model [see the main text].
The fitting parameters are indicated in the figure.
The inset shows logarithmic plots of the normalized $I_{\pm{k}}$.
(c) Experimental geometry where a mirror plane of the
surface coincides with the plane of incidence (red square) perpendicular to
the detection plane of photoelectrons (blue square).
The triangle depicts the $C_{3v}$ surface symmetry.
(d) Energy and momentum scheme for the decay of the asymmetric population
due to elastic scattering within the TSS ($\Gamma^e_{k}$), inelastic scattering
($\Gamma^i$) and transport ($\Gamma^t$).
It can be easily understood that TSS-BCB and BVB-TSS transitions, which are transitions
into (from) a continuum of states, should show a linear dependence on $h\nu_1$
with a slope of unity as depicted by the dashed line in Fig. <ref>.
Obviously, this dependence cannot describe our experimental data.
The $h\nu_1$ dependence of TSS-TSS transitions, on the other hand,
is given by the dispersion of the TSS.
The population enhancement for this process appears at those
intermediate state energies for which the energy difference between
the initial state in the occupied part of the TSS and the intermediate
state in the unoccupied part of the TSS just matches $h\nu_1$.
For the simulation of this process, we have used data
on the dispersion of the occupied part of the TSS in Sb$_{2}$Te$_{3}$
from Ref. <cit.> and for the unoccupied part from Ref. <cit.>.
The almost linear dispersion of the TSS results in a simple linear
dependence on $h\nu_1$ as depicted by the solid line in Fig. <ref>
which can excellently reproduce our experimental results.
The simulation furthermore predicts that TSS-TSS
transitions are not possible for $h\nu_1$ below 0.27 eV at which
the initial state becomes unoccupied.
This is also in very good agreement with our finding that
the population in the TSS strongly drops for $h\nu_1$ below 0.25 eV.
It is therefore concluded that MIR pulses are able to drive a
direct optical transition from the lower into the upper part of the TSS.
One may wonder why such a transition between states of opposite
chiral spin textures across the DP are allowed,
since spin-flip excitations are forbidden in the dipole approximation.
However, if spin-orbit coupling plays an important role,
the TSS cannot be fully spin polarized and this selection rule is softened.
In fact, a spin polarization of below 80% has been discussed for the TSS in different
3D TIs <cit.>.
In addition, a hybridization with bulk states <cit.>
possibly results in a further reduction of the spin polarization <cit.>.
We now turn to the asymmetry of the excited population which is only observed
for direct excitation with MIR pulses.
Its time and energy evolution is shown in Fig. <ref> (a), where we have plotted the
transient 2PPE intensity at ${\pm}k$ as well as its difference for
different energies as depicted by the five integration windows in Fig. <ref> (c).
For all energies, the 2PPE intensity shows a fast rise within the time resolution of $\approx 200$ fs
and a subsequent decay within a few picoseconds.
This is in strong contrast to the delayed dynamics observed for 800-nm, or visible excitation
and corroborates the direct excitation process of the TSS.
The asymmetry $A=\Delta I/(I_{-k}+I_{+k})$ reaches up to 60% at the direct excitation energy
and strongly drops for energies below.
We observe such large asymmetry along $\bar{\Gamma}$-$\bar{\rm K}$
for $p$-polarized MIR pulses but almost no contrast between
opposite helicities of circular polarized light <cit.>.
This is surprising for our geometry where a mirror plane of the
surface coincides with the plane of incidence
[Fig. <ref> (c)].
It indicates that the three-fold symmetry of the surface might
be broken.
Possible reasons for such symmetry break can be oriented
step edges on the cleaved surface which superimpose a one-fold
symmetry, a distortion of the first quintuple layer with
respect to the underlying bulk, or a non-perfect azimuthal
orientation of the sample, which was, however, oriented within
better than 5$^\circ$ <cit.>.
Independent of its actual origin, a k-space asymmetry of the
population in the intermediate state directly reflects a
photocurrent parallel to the surface <cit.>.
This is verified by the observation of a distinct dynamics of
the 2PPE signal for opposite momenta at the direct excitation
energy [Fig. <ref> (b)], which is most clearly seen in the
logarithmic plot of the normalized intensities
[inset of Fig. <ref> (b)].
decay at $-k$ is
initially faster as compared to the
decay at $+k$ due to momentum scattering which progressively equalizes the asymmetry.
On a longer timescale, both signals decay with a common time constant due to inelastic decay.
In accordance with the intensity contrast, the difference of the decay dynamics is more pronounced at 80 K
as compared to 300 K [Fig. <ref> (b)].
These findings unambiguously reveal that the novel direct optical excitation
generates a transient photocurrent in the TSS.
To analyze the dynamics of the photocurrent in more detail, we
use a rate-equation model which is depicted in
Fig. <ref> (d). It describes the populations $n_{+k}$ and
$n_{-k}$ at the direct excitation energy where no indirect
filling from higher-lying states can occur:
\begin{eqnarray*}
\frac{dn_{+}}{dt}&=&a{\delta}(t)-{\Gamma^{e}_{k}}{n_{+}}+{\Gamma^{e}_{k}}{n_{-}}-{\Gamma^{d}}{n_{+}}\\
\frac{dn_{-}}{dt}&=&b{\delta}(t)-{\Gamma^{e}_{k}}{n_{-}}+{\Gamma^{e}_{k}}{n_{+}}-{\Gamma^{d}}{n_{-}}
\label{eq:rate_carrier}
\end{eqnarray*}
Here, ${\delta}(t)$ is the temporal intensity profile of the
Gaussian shaped MIR laser pulse, and $a$ and $b$ indicate the
different excitation probabilities of electrons at $+k_{||}$
and $-k_{||}$, respectively. In our model, $n_{+k}$ and
$n_{-k}$ exchange electrons with an elastic scattering rate
($\Gamma^{e}_{k}$) which includes momentum and spin scattering.
Both mechanisms are closely related to each other due to the
spin structure of the Dirac cone but might only be disentangled
by a direct observation of the spin dynamics.
Both populations mutually decay with an effective total population
decay rate ($\Gamma^{d}$).
Beside the inelastic decay into lower-lying states ($\Gamma^{i}$),
$\Gamma^{d}$ includes interband scattering into the BCB with subsequent bulk
transport ($\Gamma^{t}$), because the direct excitation energy is close to the
BCB bottom <cit.>.
$\Gamma^{d}$ is thus defined as
By assuming that $I_{\pm k}{\propto}n_{\pm k}$, the difference ${\Delta}{I}=I_{+k}-I_{-k}$
directly reflects the photocurrent in the TSS.
With the two rate equations above, it is described by:
\begin{eqnarray*}
\frac{d{\Delta}I}{dt}&=&(b-a){\delta}(t)-({2\Gamma^{e}_{k}+\Gamma^{d}}){\Delta}I
\label{eq:rate_current}
\end{eqnarray*}
The photocurrent thus decays exponentially with a time constant $\tau^{c}=1/({2\Gamma^{e}_{k}+\Gamma^{d}}$)
which is governed by both elastic momentum scattering and total population decay.
Best fits for this model of the experimental data $I_{\pm{k}}$ and ${\Delta}{I}$
are shown as solid lines in Fig. <ref> (b).
Clearly, even this simple model can reproduce the experimental data
for both temperatures very well.
$\Gamma^{d}$ is for both temperatures much larger compared to $\Gamma^{e}_{k}$ through $\Gamma^{i}$,
because electron-hole pair creation in the incompletely filled VB
is an important inelastic decay channel for $p$-doped samples.
$\tau^{c}$=0.42 (0.52) ps at 80 (300) K is therefore governed by the overall population decay of the TSS.
The increase of $\Gamma^{d}$ at 80 K as compared to 300 K can be explained
by an enhancement of $\Gamma^{t}$,
which shows quantitatively good agreement with our recent work on
Sb$_{2}$Te$_{2}$Se <cit.>.
In contrast, $\Gamma^{e}_{k}$ shows no significant change with temperature
although the Debye temperature of Sb$_2$Te$_3$ ($\theta_D=162$ K <cit.>) is
well between the two investigated temperatures.
We thus conclude that phonon scattering plays only a minor role
for $\Gamma^{e}_{k}$ and surface imperfections like steps or
defects are most likely the main factors for elastic
Beyond defect scattering, interband scattering into and from
the BCB, where backscattering is allowed, might also be
possible due to the large wavefunction overlap of the TSS with
the BCB.
Both scattering processes effectively increase $\Gamma^{e}_{k}$
which ultimately limits the lifetime of the photocurrent even if $\Gamma^{d}$ can be suppressed.
Indeed, $\Gamma^{e}_{k}$ becomes the main factor for the transport
properties under static electric fields if the sample is close
to charge neutrality.
In any case, the decay time $\tau^{e}_{k}=1/\Gamma^{e}_{k}=2.5$ ps is quite long,
if for example, compared to dephasing times of quantum beats
between image potential states on well prepared noble metal
surfaces <cit.>.
Such a slow randomization of momentum holds a general advantage
also for application under static conditions.
If the main process of $\Gamma^{e}_{k}$ is in fact defect scattering,
high-quality thin film samples might further increase the mobility of
electrons in the TSS.
In conclusion, we have shown that MIR pulses generate a novel optical
coupling between the occupied and the unoccupied part of the TSS
across the DP, which permits an ultrafast direct excitation of the TSS.
Our data unambiguously reveal that this novel direct excitation
generates an unbalanced transient electron population in
$k$-space, which directly mirrors a photoexcited electric
current in the TSS.
Even if the polarization dependence of the current generation
is not fully understood yet, our discovery opens a pathway for
a coherent optical control of the TSS via an ultrafast optical
excitation with MIR pulses.
We thank H. Bentmann and F. Reinert for providing us Sb$_{2}$Te$_{3}$
samples and gratefully acknowledge funding by the Deutsche
Forschungsgemeinschaft through SPP1666 and GU495/2.
K. K. acknowledges support from JSPS Postdoctral Fellowship for Research Abroad.
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1511.00537
|
Abstract: Dvořák et al.
introduced a variant of the Randić index of a graph $G$, denoted
by $R'(G)$, where $R'(G)=\sum_{uv\in E(G)}\frac 1 {\max\{d(u),
d(v)\}}$, and $d(u)$ denotes the degree of a vertex $u$ in $G$.
The coloring number $col(G)$ of a graph $G$ is the smallest number
$k$ for which there exists a linear ordering of the vertices of $G$
such that each vertex is preceded by fewer than $k$ of its
neighbors. It is well-known that $\chi(G)\leq col(G)$ for any graph
$G$, where $\chi(G)$ denotes the chromatic number of $G$. In this
note, we show that for any graph $G$ without isolated vertices,
$col(G)\leq 2R'(G)$, with equality if and only if $G$ is obtained
from identifying the center of a star with a vertex of a complete
graph. This extends some known results. In addition, we present some
new spectral bounds for
the coloring and achromatic numbers of a graph.
Keywords: Chromatic number; Coloring number; Achromatic number; Randić index
§ INTRODUCTION
The Randić index $R(G)$ of a (molecular) graph $G$ was
introduced by Milan Randić <cit.> in 1975 as the sum of
$1/\sqrt{d(u)d(v)}$ over all edges $uv$ of $G$, where $d(u)$ denotes
the degree of a vertex $u$ in $G$. Formally,
$$R(G)=\sum\limits_{uv\in E(G)}\frac{1}{\sqrt{d(u)d(v)}}.$$
This index
is useful in mathematical chemistry and has been extensively
studied, see <cit.>. For some recent results on the Randić index,
we refer to <cit.>.
A variation of the Randić index of a graph $G$ is called the
Harmonic index, denoted by $H(G)$, which was defined in
<cit.> as follows:
$$H(G)=\sum_{uv\in E(G)} \frac 2
In 2011 Dvořák et al. introduced another
variant of the Randić index of a graph $G$, denoted by $R'(G)$, which has been further studied by Knor et al <cit.>.
$$R'(G)=\sum_{uv\in E(G)}\frac{ 1 }{\max\{d(u), d(v)\}}.$$
It is clear from the definitions that for a graph $G$,
\begin{equation} R'(G)\leq H(G)\leq R(G). \end{equation}
The chromatic number of $G$, denoted by $\chi(G)$, is
the smallest number of colors needed to color all vertices of $G$
such that no pair of adjacent vertices is colored the same. As
usual, $\delta(G)$ and $\Delta(G)$ denote the minimum degree and the
maximum degree of $G$, respectively. The coloring number $col(G)$ of
a graph $G$ is the least integer $k$ such that $G$ has a vertex
ordering in which each vertex is preceded by fewer than $k$ of its
neighbors. The $degeneracy$ of $G$, denoted by $deg(G)$, is defined
as $deg(G)=max\{\delta(H): H\subseteq G\}$. It is well-known (see
Page 8 in <cit.>) that for any graph $G$,
\begin{equation}col(G)=deg(G)+1. \end{equation}
List coloring is an extension of coloring of graphs, introduced by
Vizing <cit.> and independently, by Erdős et al. <cit.>.
For each vertex $v$ of a graph $G$, let $L(v)$ denote a list of
colors assigned to $v$. A list coloring is a coloring $l$ of
vertices of $G$ such that $l(v)\in L(v)$ and $l(x)\neq l(y)$ for any
$xy\in E(G)$, where $v, x, y\in V(G)$. A graph $G$ is $k$-choosable
if for any list assignment $L$ to each vertex $v\in V(G)$ with
$|L(v)|\geq k$, there always exists a list coloring $l$ of $G$. The
list chromatic number $\chi_l(G)$ (or choice number) of $G$ is the
minimum $k$ for which $G$ is $k$-choosable.
It is well-known that for any graph $G$,
\begin{equation}\chi(G)\leq \chi_l(G)\leq col(G)\leq \Delta(G)+1.
\end{equation}
The detail of the inequalities in (3) can be found in a
survey paper by Tuza <cit.> on list coloring.
In 2009, Hansen and Vukicević <cit.> established the
following relation between the Randić index and the chromatic
number of a graph.
(Hansen and Vukicević <cit.>) Let $G$ be a
simple graph with chromatic number $\chi(G)$ and Randić index
$R(G)$. Then $\chi(G)\leq2R(G)$ and equality holds if $G$ is a
complete graph,
possibly with some additional isolated vertices.
Some interesting extensions of Theorem 1.1 were recently obtained.
(Deng et al <cit.> ) For a graph $G$,
$\chi(G)\leq 2H(G)$ with equality if and only if $G$ is a complete
graph possibly with some additional isolated vertices.
(Wu, Yan and Yang <cit.> ) If $G$ is a graph of order $n$ without isolated vertices, then
with equality if and only if $G\cong K_n$.
Let $n$ and $k$ be two integers such that $n\geq k\geq 1$. We denote the graph obtained from identifying the center of the star $K_{1, n-k}$ with a vertex of the complete graph $K_k$ by $K_k\bullet K_{1,n-k}$. In particular, if $k\in\{1, 2\}$, $K_k\bullet K_{1,n-k}\cong K_{1,n-1}$; if $k=n$, $K_k\bullet K_{1,n-k}\cong K_n$.
The primary aim of this note is to prove stronger versions of Theorems
1.1-1.3, noting the inequalities in (1).
For a graph $G$ of order $n$ without isolated vertices, $col(G)\leq
2R'(G)$, with equality if and only if $G\cong K_k\bullet K_{1,n-k}$ for some $k\in\{1, \ldots, n\}$.
For a graph $G$ of order $n$ without isolated vertices, $\chi(G)\leq
2R'(G)$, with equality if and only if $G\cong K_k\bullet K_{1,n-k}$ for some $k\in\{1, \ldots, n\}$.
For a graph $G$ of order $n$ without isolated vertices, $\chi_l(G)\leq
2R'(G)$, with equality if and only if $G\cong K_k\bullet K_{1,n-k}$ for some $k\in\{1, \ldots, n\}$.
For a graph $G$ of order $n$ without isolated vertices, $col(G)\leq 2H(G)$, with equality if and
only if $G\cong K_n$.
The proofs of these results will be given in the next section.
A complete $k$-coloring of a graph $G$ is a
$k$-coloring of the graph such that for each pair of different
colors there are adjacent vertices with these colors. The achromatic number of $G$, denoted by $\psi(G)$, is the maximum
number $k$ for which the graph has a complete $k$-coloring. Clearly,
$\chi(G)\leq \psi(G)$ for a graph $G$. In general,
$col(G)$ and $\psi(G)$ are incomparable. Tang et al. <cit.> proved that for a graph $G$,
$\psi(G)\leq 2R(G)$.
In Section 3, we prove new bounds for the coloring
and achromatic numbers of a graph in terms of its spectrum, which strengthen $col(G) \le 2R(G)$ and $\psi(G) \le 2R(G)$. In
Section 4, we provide an example and propose two related conjectures.
§ THE PROOFS
For convenience, an edge $e$ of a graph $G$ may be viewed as a
2-element subset of $V(G)$ and if a vertex $v$ is an end vertex of
$e$, we denote the other end of $e$ by $e\setminus v$. Moreover,
$\partial_G(v)$ denotes the set of edges which are incident with $v$
in $G$.
First we need the following theorem, which will play a key role in the
proof of Theorem 1.4.
If $v$ is a vertex of
$G$ with $d(v)=\delta(G)$, then
$$R'(G)-R'(G-v)\geq 0,$$
with equality if and only if
$N_G(v)$ is an independent set of $G$ and
$d(w)<d(v_i)$ for all $w\in N(v_i)\setminus \{v\}$.
Let $k=d_G(v)$. The result is trivial for $k=0$. So, let $k>0$ and
let $N_G(v)=\{v_1, \cdots, v_k\}$ and $d_i=d_G(v_i)$ for each $i$.
Without loss of generality, we may assume that $d_1\geq \cdots \geq
d_k$. Let
$$E_1=\{e: e\in\partial_G(v_1)\setminus \{vv_1\}\ \text{such that}\
d_G(e\setminus v_1)<d_G(v_1)\ \text {and}\ e\not\subseteq N_G(v)
\},$$
$$E_1'=\{e: e\in\partial_G(v_1)\setminus \{vv_1\}\ \text{such that}\ \
d_G(e\setminus v_1)=d_G(v_1)\ \text {and}\ e\subseteq N_G(v)\},$$
and for an integer $i\geq 2$,
$$E_i=\{e: e\in\partial_G(v_i)\setminus \{vv_i\}\ \text{such that}\
d_G(e\setminus v_i)<d_G(v_i)\ \text {and}\ e\not\subseteq
$$E_i'=\{e: e\in\partial_G(v_i)\setminus \{vv_i\}\ \text{such that}\ \
d_G(e\setminus v_i)=d_G(v_i)\ \text {and}\ e\subseteq N_G(v)\}
\setminus (\cup_{j=1}^{j=i-1} E_j').$$
Let $a_i=|E_i|$ and
$b_i=|E_i'|$ for any $i\geq 1 $. Since $a_i+b_i\leq d_i-1$,
$$R'(G)-R'(G-v)=\sum_{i=1}^k \frac 1 {d_i}-\sum_{i=1}^k \frac {a_i+b_i}
{d_i(d_i-1)}\geq 0,$$
with equality if and only if $a_i+b_i=d_i-1$
for all $i$, i.e., $N_G(v)$ is an independent set of $G$ and
$d(w)<d(v_i)$ for all $w\in N(v_i)\setminus \{v\}$.
If $T$ is a tree of order $n\geq 2$, then $R'(T)\geq 1$, with equality if and only if
By induction on $n$. It can be easily checked that $R'(K_{1,n-1})=1$. So, the assertion of the lemma is true for $n\in\{2, 3\}$. So, we assume that $n\geq 4$.
Let $v$ be a leaf of $T$. Then $T-v$ is a tree of order $n-1$. By Theorem 2.1, $R'(T)\geq R'(T-v)\geq 1$. If $R'(T)=1$, then $R'(T-v)=1$, and by the induction hypothesis, $T-v\cong K_{1, n-2}$. Let $u$ be the center of $T-v$.
We claim that $vu\in E(T)$. If this is not, then $v$ is adjacent to a leaf of $T-v$, say $x$, in $T$. Thus $d_T(x)=2$. However, since $R'(T)=R'(T-v)$ and $u\in N_T(x)$, by Theorem 2.1 that $d_T(x)>d_T(u)=n-2\geq 2$, a contradiction. This shows that $vu\in E(T)$ and $T\cong K_{1,n-1}$.
The proof of Theorem 1.4:
Let $v_1, v_2, \ldots, v_n$ be an ordering of all vertices of $G$
such that $v_i$ is a minimum degree vertex
of $G_i=G-\{v_{i+1}, \ldots, v_n\}$ for each $i\in \{1, \ldots,
n\}$, where $G_n=G$ and $d_G(v_n)=\delta(G)$. It is well known that $deg(G)=\max\{d_{G_i}(v_i):\ 1\leq
i\leq n\}$ (see Theorem 12 in <cit.>). Let $k$ be the maximum number such that
$deg(G)=d_{G_k}(v_k)$, and $n_k$ the order of $G_k$. By Theorem 2.1,
\begin{equation} 2R'(G)\geq 2R'(G_{n-1})\geq \cdots\geq
\begin{eqnarray*}
2R'(G_k)&=& \sum\limits_{uv\in
&\geq& \sum\limits_{uv\in
&\geq& \frac{\Delta(G_k)+(n_k-1)\delta(G_k)}{\Delta(G_k)} \\
&\geq& \delta(G_k)+1\\
&=& col(G).
\end{eqnarray*}
It follows that
\begin{equation} \text {if}\ R'(G_k)=\delta(G_k)+1,
\text { then}\ G_k\cong K_k.\end{equation}
Now assume that $col(G)=2R'(G)$. Observe
that $col(G)=\max\{col(G_i):$ $G_i$ is a component of $G\}$ and $R'(G)=\sum_{i} R'(G_i)$. Thus, by the assumption that $col(G)=2R'(G)$ and $G$ has no isolated vertices, $G$ is connected and $col(G)\geq 2$. If $col(G)=2$, then $G$ is a tree. By Corollary 2.1, $G\cong K_{1,n-1}$.
Next we assume that $col(G)\geq 3$. By (4) and (5) we have $R'(G)=\cdots=R'(G_k)$,
$G_k\cong K_k$ and thus $col(G)=k$. We show $G\cong K_k\bullet K_{1,n-k}$ by induction on $n-k$.
It is easy to check that $col(K_n)=n+1=2R'(K_n)$ and thus the result is for $n-k=0$.
If $n-k=1$, then
$G_k=G-v_n$ and $R'(G_k)=R'(G)$, by Theorem 2.1,
$N_G(v_n)$ is an independent set of $G$. Combining with $G_k\cong K_k\ (k\geq 3)$,
$d_{G_n}(v_n)=1$. Thus $G=K_k\bullet K_{1,1}$.
Next assume that $n-k\geq 2$.
We consider $G_{n-1}$. Since $col(G_{n-1})=2R'(G_{n-1})$, by the induction hypothesis
$G_{n-1}\cong K_k\bullet K_{1,n-1-k}$. Without loss of generality, let $N_{G_{n-1}}(v_{k+1})=\cdots= N_{G_{n-1}}(v_{n-1})=\{v_k\}$. So, it remains to show that $N_G(v_n)=\{v_k\}$.
Claim 1. $N_G(v_n)\cap \{v_{k+1}, \ldots, v_{n-1}\}=\emptyset$.
If it is not, then $\{v_{k+1}, \ldots, v_{n-1}\}\subseteq N_G(v_n)$, because $d_G(v_n)=\delta(G)$.
By Theorem 2.1, $d_G(v_{n-1})>d_G(v_k)$. But, in this case, $d_G(v_{n-1})=2<d_G(v_k)$, a contradiction.
By Claim 1, $N_G(v_n)\subseteq \{v_1, \ldots, v_k\}$.
Since $N_G(v_n)$ is an independent set and $\{v_1, \ldots, v_k\}$ is a clique of $G$, $|N_G(v_n)\cap \{v_1, \ldots, v_k\}|=1$.
If $N_G(v_n)=\{v_i\}$, where $i\neq k$, then $d_G(v_i)=k$. By Theorem 2.1, $d_G(v_i)>d_G(v_k)$.
However, $d_G(v_k)\geq n-2\geq k$, a contradiction. This shows $N_G(v_n)=\{v_k\}$ and hence $G\cong K_k\bullet K_{1,n-k}$.
It is straightforward to check that
$$2R'(K_k\bullet K_{1,n-k})=2\times(\frac {k-2} 2+1)=k
The proofs of Corollaries 1.5 and 1.6:
By (3) and Theorem 1.4, we have $\chi(G)\leq \chi_l(G)\leq 2R'(G)$. If $\chi(G)=2R'(G)$
(or $\chi_l(G)=2R'(G)$), then $col(G)=2R'(G)$. By Theorem 1.4, $G\cong K_k\bullet K_{1,n-k}$. On the other hand, it is easy to check that
$$\chi(K_k\bullet K_{1,n-k})=\chi_l(K_k\bullet K_{1,n-k})=k=2R'(K_k\bullet K_{1,n-k}).$$
The proof of Corollary 1.7:
By (1) and Theorem 1.4, we have $col(G)\leq 2H(G)$. If $col(G)=2H(G)$, then $col(G)=2R'(G)$.
By Theorem 1.4, $G\cong K_k\bullet K_{1,n-k}$. It can be checked that $k=2H(K_k\bullet K_{1,n-k})$ if and only if $k=n$, i.e., $G\cong K_n$.
§ SPECTRAL BOUNDS
§.§ Definitions
Let $\mu = \mu_1 \ge ... \ge \mu_n$ denote the eigenvalues of the adjacency matrix of $G$ and let $\pi, \nu$ and $\gamma$ denote the numbers
(counting multiplicities) of positive, negative and zero eigenvalues respectively. Then let
\[
s^+ = \sum_{i=1}^\pi \mu_i^2 \mbox{ and } s^- = \sum_{i=n-\nu+1}^n \mu_i^2.
\]
Note that $\sum_{i=1}^n \mu_i^2 = s^+ + s^- = tr(A^2) = 2m$.
§.§ Bounds for $\psi(G)$ and $col(G)$
For a graph G, $\psi(G) \le 2m/\sqrt{s^+} \le 2m/\mu \le 2R(G)$.
Ando and Lin <cit.> proved a conjecture due to Wocjan and Elphick <cit.> that $1 + s^+/s^- \le \chi$ and consequently $s^+ \le 2m(\chi -
1)/\chi$. It is clear that $\psi(\psi - 1) \le 2m$. Therefore:
\[
s^+ \le \frac{2m(\chi - 1)}{\chi} \le \frac{2m(\psi -
1)}{\psi} \le \frac{4m^2}{\psi^2}.
\]
Taking square roots and re-arranging completes the first half of the
Favaron et al <cit.> proved that $R(G) \ge m/\mu$.
Therefore $\psi(G) \le 2m/\mu \le 2R(G)$.
Note that for regular graphs, $2m/\mu = 2R' = 2H = 2R = n$ whereas for almost all regular graphs $2m/\sqrt{s^+} < n$.
For all graphs, $col(col - 1) \le 2m$.
As noted above, $s^+ \le 2m(\chi - 1)/\chi$ and $\chi(\chi - 1) \le 2m$.
\[
\sqrt{s^+}(\sqrt{s^+} + 1) = s^+ + \sqrt{s^+} \le \frac{2m(\chi - 1)}{\chi} +
\frac{2m}{\chi} = 2m.
\]
We can show that $deg(G) \le \mu(G)$ as follows.
\[
deg(G) = \max(\delta(H) : H \subseteq G) \le \max(\mu(H) : H
\subseteq G) \le \mu(G).
\]
Therefore $col(G) \le \mu + 1$, so:
\[
col(col - 1) \le \mu(\mu + 1) \le \sqrt{s^+}(\sqrt{s^+} + 1) \le 2m.
\]
We can now prove the following theorem, using the same proof as for
Theorem 3.1.
For a graph G, $col(G) \le 2m/\sqrt{s^+} \le 2m/\mu \le 2R(G)$.
§.§ Bounds for $s^+$
The proof of Lemma 3.2 uses that $s^+ + \sqrt{s^+} \le 2m$, from which it follows that:
\[
\sqrt{s^+} \le \frac{1}{2}(\sqrt{8m + 1} - 1).
\]
This strengthens Stanley's inequality <cit.> that:
\[
\mu \le \frac{1}{2}(\sqrt{8m + 1} - 1).
\]
However, $\sqrt{2m - n + 1} \le (\sqrt{8m + 1} - 1)/2$, and Hong <cit.> proved for graphs with no isolated vertices that $\mu \le \sqrt{2m - n + 1}.$ Elphick et al <cit.> recently conjectured that for connected graphs $s^+ \le 2m - n + 1$, or equivalently that $s^- \ge n - 1$.
§ EXAMPLE AND CONJECTURES
A Grundy $k$-coloring of $G$ is a
$k$-coloring of $G$ such that each vertex is colored by the smallest
integer which has not appeared as a color of any of its neighbors.
The Grundy number $\Gamma(G)$ is the largest integer $k$, for
which there exists a Grundy $k$-coloring for $G$. It is clear that
for any graph $G$,
\begin{equation} \Gamma(G)\leq \psi(G) \ \text{and}\ \chi(G)\leq \Gamma(G)\leq
\Delta(G)+1. \end{equation}
Note that each pair of $col(G)$ and $\psi(G)$, or $col(G)$ and $\Gamma(G)$, or $\psi(G)$ and $\Delta(G)$ is incomparable in general.
As an example of the bounds discussed in this paper, if $G=P_4$, then
$$2H(G) = 3.67 \ \text{and}\ 2R(G) = 3.83$$
$$\mu_1(G) = 1.618 \ \text{and}\ \mu_2 = 0.618$$
$$2m/\mu_1(G) = 3.71 \ \text{and}\ 2m/\sqrt{s^+} = 3.46.$$
We believe the following conjectures to be true.
For any graph $G$, $\psi(G)\leq 2R'(G)$.
In view of (6), a more tractable conjecture than the above is as follows.
For any graph $G$, $\Gamma(G)\leq 2R'(G)$.
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|
1511.00283
|
Department of Mathematics. Harvey Mudd College, Claremont, CA 91711
rjeffs@g.hmc.edu, omar@g.hmc.edu, nsuaysom@g.hmc.edu, awachtel@g.hmc.edu, nyoungs@g.hmc.edu
Determining how the brain stores information is one of the most pressing problems in neuroscience. In many instances, the collection of stimuli for a given neuron can be modeled by a convex set in $\mathbb{R}^d$. Combinatorial objects known as neural codes can then be used to extract features of the space covered by these convex regions. We apply results from convex geometry to determine which neural codes can be realized by arrangements of open convex sets. We restrict our attention primarily to sparse codes in low dimensions. We find that intersection-completeness characterizes realizable 2-sparse codes, and show that any realizable 2-sparse code has embedding dimension at most $3$. Furthermore, we prove that in $\R^2$ and $\R^3$, realizations of 2-sparse codes using closed sets are equivalent to those with open sets, and this allows us to provide some preliminary results on distinguishing which 2-sparse codes have embedding dimension at most $2$.
§ INTRODUCTION
A fundamental problem in convex geometry is to understand the intersection behavior of convex sets. Classical theorems in this area include Helly's theorem and its many variations, which show that the presence of lower order intersections of convex sets in $\R^d$ can enforce intersections of higher order <cit.>. Recent work of Tancer <cit.> on the representability of simplicial complexes provides a sharp bound on the dimension in which intersection patterns of convex sets can be realized. We consider the problem of simultaneously realizing intersection patterns along with other relationships between convex sets, such as containment. This problem is motivated by one of the challenges of mathematical neuroscience: determining how the structure of a stimulus space is represented in the brain.
Neurons respond to stimuli in an environment; the set of all such stimuli is called the stimulus space $X$. Usually, we consider $X\subset \R^d$. If we are considering data from $n$ neurons $\{1,...,n\}$ which respond to stimuli in $X$, the receptive field for neuron $i$ is the subset $U_i$ of the stimulus space $X$ for which neuron $i$ is highly responsive. Throughout this article, we assume the sets $U_i$ are convex. Indeed, experimental data on many types of neurons, such as place cells <cit.> or orientation-tuned neurons <cit.>, make it evident that receptive fields are often very well approximated by convex sets. Hence, for such neurons, the regions of stimulus space in which multiple neurons fire can be modeled by intersections of convex sets. Because of this, the mathematical theory developed by Helly, Tancer, and others can inform us about the possible arrangements of receptive fields in a given dimension.
Helly's theorem, however, cannot inform us about all types of receptive field arrangements. For example, if $U_i$, $U_j$ are receptive fields which properly intersect, the neural data will differentiate between $U_i\subset U_j$ and $U_i\not\subset U_j$, but Helly's theorem merely notes that $U_i$ and $U_j$ intersect. We thus go beyond the usual scope of convex geometry to consider the problem of finding arrangements of convex sets which fully realize the information present in the neural data, including containments. This problem was posed originally in <cit.>. In order to address this issue, we must first describe how neural data is represented mathematically.
A neural code on $n$ neurons is a set of binary firing patterns $\C\subset\{0,1\}^n$, representing neural activity. Elements of $\C$ are referred to as codewords.
The firing of a neuron is an all-or-nothing event, and so a codeword $c \in \C$ represents a data point in which a specific set of neurons are simultaneously firing; with neuron $i$ active if $c_i=1$ and inactive if $c_i=0$. For instance, the codeword $0011$ represents a data point at which neurons 3 and 4 were active, while neurons 1 and 2 were not. In the receptive field context, the presence of this codeword in $\C$ indicates that $(U_3 \cap U_4 ) \backslash (U_1\cup U_2) \neq \emptyset$.
Let $\U = \{U_1,\ldots,U_n\}$ be a collection of sets in $\R^d$.
The associated neural code $\C(\U) \subset \{0,1\}^n$ is the set of firing patterns representing the regions in the arrangement:
$$\C(\U) \od \left \{ c \in \{0,1\}^n \ \bigg | \ \left(\bigcap_{c_i = 1} U_i \right) \setminus \left(\bigcup_{c_j = 0} U_j \right) \neq \emptyset \right \}.$$
Any collection of sets $\U$ in $\R^d$ can give rise to an associated neural code. However, as we have mentioned, the receptive fields $U_i$ are generally presumed to be convex. One of our main motivating example is that of place cells, whose receptive fields are generally seen to be convex as explained in <cit.>.
We additionally assume the receptive fields $U_i$ are open, since by restricting to open sets, we force all sets in our realization to be full-dimensional; furthermore, their intersections, if nonempty, must also be full-dimensional, which corresponds to non-degenerate cases in reality. These assumptions are consistent with the literature <cit.>. However, many of our proofs will require that we shift between closed and open convex sets which are associated to the same code. We therefore make the following definition:
If $\U=\{U_1,...,U_n\}$ is a collection of open (respectively, closed) convex sets in $\R^d$ for which $\C = \C(\U)$, then we say that $\C$ is open (closed) convex realizable in $\R^d$, and that $\U$ is an open (closed) convex realization of $\C$.
Then, for any code $\C$, we define $d(\C)$ to be the minimum dimension $d$ such that $\C$ has an open convex realization in $\R^d$, if such a dimension $d$ exists. If $\C$ is not realizable with open convex sets in any dimension, we say $d(\C) = \infty$. Such codes do exist; see Figure <ref>.
An open convex realization of the code $\C = \{000, 100, 010, 110, 011\}$ in $\R^2$, with each region labelled with its corresponding codeword. This shows $d(\C)\leq 2$. It can be shown that, in fact, $d(\C)=1$.
The code $\C = \{000, 010, 001, 110, 101\}$ is not open convex realizable in $\R^d$ for any $d<\infty$. If it were, we could pick points $p\in (U_1\cap U_2)\backslash U_3$ and $q\in (U_1\cap U_3)\backslash U_2$. The line segment $\overline{pq}$ is contained in $U_1$ by convexity; to move from $p$ to $q$ along $\overline{pq}$, we must leave $U_2$ and enter $U_3$. If we leave $U_2$ before entering $U_3$ that would indicate the presence of codeword 100, which is not in the code; if we enter $U_3$ before leaving $U_2$ that would indicate the codeword 111, which is not in the code. Since all sets are open, these are the only possibilities.
The support of a vector $c\in\{0,1\}^n$, denoted $\supp(c)$, is the set of indices of value 1, or the set of all firing neurons:
$$\supp (c) \od \{i \mid c_i =1 \}$$
The support of a code $\C\subset \{0,1\}^n$ is the set of the supports of its codewords:
$$\supp (\C) \od \{\supp (c) \mid c \in \C \}.$$
We assume that there are instances when none of the neurons of interest are firing; hence, we will always assume that the codeword $00\cdots 0$ is present in any code.
Let $\C = \{000, 101, 110, 111\}$. Then $\supp(101)=\{1,3\}$, $\supp(111)=\{1,2,3\}$, and $\supp(\C)=\{\emptyset,\{1,3\},\{1,2\},\{1,2,3\}\}$.
Recent work, for example that of Miesenböck et al. <cit.> shows the utility and importance of sparsity in neural codes. For practical reasons, our definition of `sparse' differs slightly from the usual low average weight definition often used in coding literature (see for example <cit.>) and we use instead a low maximum weight definition, as defined here.
A code $\C$ is $k$-sparse if $|\supp(c)| \le k$ for all $c\in \C$.
We begin the program of studying $k$-sparse code by focusing on $2$-sparse codes, where there is already rich mathematics to be found. Our fundamental motivating questions are the following:
Which 2-sparse codes are open convex realizable?
If $\C$ is an open convex realizable 2-sparse code, what is its minimum embedding dimension $d(\C)$?
In Section 2, we will answer the first question entirely. In Theorem <ref> we show that 2-sparse codes are open convex realizable exactly when they are closed under intersection. In the process, we show in Lemma <ref> that for such codes it is equivalent to find a closed convex realization, as it may be transformed to an open convex realization in $\R^2$ or $\R^3$. It immediately follows from this and work of Tancer <cit.> that any 2-sparse code has a convex open realization in $\R^3$. In Section 3, we exhibit a class of 2-sparse codes with $d\leq 2$, and as well as a class with $d=3$.
§ EQUIVALENCE WITH GRAPHS
This section is dedicated to proving Theorem <ref>, which establishes that a $2$-sparse code is realizable exactly when its support is closed under intersection, and for such codes $\C$, $d(\C) \leq 3$. An important construct in proving this theorem is that of simplicial complexes.
In order to prove this theorem, we make use of the simplicial complex of a code, which is introduced below.
A simplicial complex on a finite set $S$ is a family $\Delta$ of subsets of $S$ such that if $X\in \Delta$ and $Y \subseteq X$, then $Y \in \Delta$.
Often, the set $S$ under consideration will be $[n] = \{1,...,n\}$. In a situation where $S=\{v_1,...,v_n\}$, we will typically refer to any set in a simplicial complex on $S$ by its set of indices.
The simplicial complex of a code $\C$ is the smallest simplicial complex containing $\supp(\C)$; this is denoted $\Delta(\C)$.
The $k$-skeleton of a simplicial complex $\Delta$ is the simplicial complex $\Delta_k$ given by the collection of sets in $\Delta$ of size at most $k + 1$.
At left, a geometric representation of simplicial complex on $S=\{v_1,v_2,v_3,v_4\}$ with $\Delta = \{ \emptyset, \{1\}, \{2\}, \{3\}, \{4\}, \{1,2\}, \{1,3\}, \{2,3\}, \{2,4\}, \{1,2,3\} \}$. At right, a geometric representation of the 1-skeleton of $\Delta$
If $\C$ is 2-sparse, then $\Delta(\C)$ consists only of $0, 1$, and $2$ element sets. We can therefore think of $\Delta(\C)$ as a graph, with 1-element sets corresponding to vertices and $2$-element sets as edges between them. Note that since $\Delta(\C)$ is a simplicial complex, if $\{i,j\}\in \Delta(\C)$, then both $\{i\}$ and $\{j\}$ must be in $\Delta(\C)$ as well; hence this association is well-defined. The formal relationship between $2$-sparse codes and graphs is captured by the following definition.
Let $\C\subset\{0,1\}^n$ be a neural code. The graph of $\C$, denoted $G_\C$, is the graph whose vertex set is $[n]$, with $i$ adjacent to $j$ if $\{i,j\} \in \supp(\C)$.
Note that $G_\C$ is essentially the 1-skeleton of $\Delta(\C)$. In particular, for a 2-sparse code, $\Delta(\C)$ and $G_\C$ contain exactly the same information because $\Delta(\C)$ is equal to its 1-skeleton.
The graph $G_{\C}$ for $\C = \{000, 100, 010, 110, 011\}$; see Figure <ref> for a realization of $\C$.
As we saw in Figure <ref>, there exist 2-sparse codes which are not convex in any dimension. The following lemma generalizes the obstruction presented in that figure.
Let $\C$ be a 2-sparse code. If $\C$ has a convex open realization in any dimension, then $\supp(\C)$ is closed under intersection.
Suppose $\C$ is a 2-sparse code with open convex realization $\U = \{U_1,...,U_n\}$. Since $\C$ is 2-sparse, $|\supp(c)|\in \{ 0,1,2\}$ for every $c\in \C$.
If $|\supp(c)|=1$ is at most $1$, then $\supp(c)\cap \supp(c') \in \supp(\C)$ because it is either $\emptyset$ or $\supp(c)$. It then remains to show that $\supp(c) \cap \supp(c') = \{i\} \in \supp(\C)$ when $\supp(c) = \{i,j\}$ and $\supp(c')=\{i,k\}$ with $j \neq k$. In this case $U_i \cap U_j$ and $U_i \cap U_k$ are nonempty so there exists points $p\in U_i\cap U_j$, $q\in U_i \cap U_k$. Consider the line segment $\overline{pq}$ connecting $p$ and $q$. Since $U_i$ is convex, $\overline{pq}$ is contained in $U_i$. For each $m\in [n]\backslash\{i\}$, consider the set $L_m = \overline{pq}\cap U_i\cap U_m$; note that any two such sets are disjoint, and that $L_j$ and $L_k$ are nonempty. If the sets $\{L_m\}$ partition the line $\overline{pq}$, then this would disconnect $\overline{pq}$ in the subspace topology, but as $\overline{pq}$ is connected, this is impossible. Thus, there must be some point on $\overline{pq}$ which is contained in $U_i$ only. The existence of this point implies $\{i\} \in \supp(\C)$ as desired.
The conclusion of the previous lemma is in fact a characterization of 2-sparse codes with an open convex realization. This is the content of the next theorem, one of our main results:
A 2-sparse code $\C$ has an open convex realization if and only if $\supp(\C)$ is closed under intersection. Furthermore, if $\C$ is realizable then $d(\C) \leq 3$.
In order to prove Theorem <ref>, we will repeatedly make geometric augmentations to existing realizations. In order to make such augmentations without changing the underlying code, we must ensure that subset containment relations between sets are maintained. In the 2-sparse case, the following definition encapsulates the key relationships that must be maintained:
Let $\U = \{U_1,\ldots, U_n\}$ be a collection of sets in $\R^d$. For any ordered pair $(U_i,U_j)$ we distinguish three possible relations between $U_i$ and $U_j$:
* (Type A Disjointness: $U_i\cap U_j = \emptyset$; i.e., $\{i,j\} \not\in \supp(\C)$.
* (Type B) Containment: $U_j\subset U_i$; i.e., there exists a codeword $c\in \C(\U)$ so that $\{i,j\}\subseteq \supp(c)$ and any codeword whose support contains $i$ must also have $j$ in its support.
* (Type C) Proper Intersection: $U_i\cap U_j$ is nonempty and $U_j\setminus U_i$ is nonempty; i.e., there exist codewords $c_1,c_2 \in \C(\U)$ so that $\{i,j\}\subset \supp(c_1)$, $j\in \supp(c_2)$ and $i\notin \supp(c_2)$.
The Type A, Type B and Type C set relationships effectively characterize the structure of a 2-sparse code; indeed 2-sparse codes are completely determined by the pairwise relationships of the sets in a given realization of them. We explicitly state this in the following proposition:
Let $\U$ and $\U'$ be collections of sets in $\R^d$ so that $\C(\U)$ and $\C(\U')$ are both 2-sparse. Then $\C(\U) = \C(\U')$ if and only if for every ordered pair $(i,j)$ the relation between $U_i$ and $U_j$ is the same as the relation between $U_i'$ and $U_j'$.
We now introduce the geometric underpinnings of the augmentations we will apply to realizations of codes.
Given $\epsilon>0$ and $A \subset \R^d$, the trim of $A$ by $\epsilon$ is the set
\[\trim(A,\epsilon) \od\interior \{ p\in \R^d \mid B_\epsilon(p)\subseteq A\}.\]
The inflation of $A$ by $\epsilon$ is the set
\[\inflate(A,\epsilon) \od \{a + x \ | \ a\in A, \ x\in \R^d \text{ with } ||x||<\epsilon\}\]
If $\mathcal A = \{A_1,\ldots, A_n\}$ is a collection of sets, then $\trim(\mathcal A,\epsilon) = \{\trim(A_1,\epsilon),\ldots,\trim(A_n,\epsilon)\}$, and $\inflate(\mathcal A, \epsilon) = \{\inflate(A_1,\epsilon),\ldots,\inflate(A_n,\epsilon)\}$.
For any convex set $A\subset \R^d$ and $\epsilon>0$, the following statements hold:
* $\trim(A,\epsilon)$ is an open convex set
* $\closure{\trim(A,\epsilon)}$ is contained in the interior of $A$
* $\inflate(A,\epsilon)$ is an open convex set
For (1), we need only prove convexity, and we may assume $\trim(A,\epsilon)$ is nonempty. Let $p$ and $q$ be points in $\trim(A,\epsilon)$; then $B_\epsilon(p)$ and $B_\epsilon(q)$ are contained in $A$, and hence so is the convex hull of their union. This convex hull contains the line segment $\overline{pq}$. For (2), note that $\closure{\trim(A,\epsilon)}\subseteq \trim(A, \epsilon/2) \subseteq \operatorname{int}(A)$. Finally, (3) follows from the fact that $A$ is convex and $\{x \in \R^d \ | \ ||x|| < \epsilon\}$ is open and convex.
Given a 2-sparse code $\C$ with an open convex realization $\U = \{U_1,\ldots,U_n\}$, there exists some $\epsilon > 0$ so that $\trim(\U,\epsilon)$ is also a realization of $\C$.
Our plan is as follows: for each set $U_i$, we find an $\epsilon_i$ such that $\trim(U_i, \epsilon_i)\neq \emptyset$, and for each pair $\{i,j\}$ we find an $\epsilon_{ij}$ such that $\trim(\{U_i, U_j\}, \epsilon_{ij})$ preserves their relationship type (Type A, Type B or Type C). We then let $\epsilon$ be the minimum of all $\epsilon_i$ and $\epsilon_{ij}$, and show that $\trim(\U,\epsilon)$ is a realization of the original code $\C$.
To start, for each $i$ with $U_i$ nonempty, there must be some point $p$ and $\delta_i>0$ with $B_{\delta_i}(p)\subset U_i$. Let $\epsilon_i = \delta_i/2$. Let $\varepsilon_1 = \min_{i\in [n]} \epsilon_i$. Now, for each pair $\{i,j\}$, we choose $\epsilon_{ij}$ depending on the relationship type between $U_i$ and $U_j$.
* Type A: If $U_i\cap U_j = \emptyset$, set $\epsilon_{ij} = \min\{\epsilon_i, \epsilon_j\}$
* Type B: If $U_i = U_j$, set $\epsilon_{ij} = \min\{\epsilon_i, \epsilon_j\}$. If $U_i\subsetneq U_j$, note that $U_j\backslash U_i$ has nonempty interior. Thus there exists some point $p$ and some $\delta_{ij}>0$ with $B_{\delta_{ij}}(p)\subset U_j\backslash U_i$. Let $\epsilon_{ij} = \min\{\delta_{ij}/2, \epsilon_i\}$.
* Type C: If $U_i\cap U_j\neq \emptyset$, but neither $U_i\subset U_j$ nor $U_j\subset U_i$ is true, note that $U_i\cap U_j$ is open and therefore there exists a point $p$ and $\epsilon'>0$ with $B_{\epsilon'}(p)\subset U_i\cap U_j$. There exists also points $p_i$, $p_j$ in $U_i\backslash U_j$, $U_j\backslash U_i$ respectively, with corresponding $\hat\epsilon$ and $\tilde \epsilon$ such that $B_{\hat\epsilon}(p_i) \subset U_i\backslash U_j$ and $B_{\tilde\epsilon}(p_j)\subset U_j\backslash U_i$. Pick $\epsilon_{ij} = \min\{\epsilon_i, \epsilon_j, \hat\epsilon/2, \tilde\epsilon/2, \epsilon'/2\}$.
Let $\varepsilon_2 = \min_{i,j} \epsilon_{ij}$, and finally, let $\epsilon = \min\{\varepsilon_1, \varepsilon_2\}$. Since $\trim(\epsilon, U) \subset U$, and originally there were no triple intersections, by construction it is impossible for $\trim(\epsilon, \U)$ to have triple intersections. Thus, $\C(\trim(\epsilon, \U))$ is still 2-sparse. We now show that $\C(\trim(\epsilon, \U)) = \C$.
If the codeword with support $\{i,j\}$ is in $\C(\trim(\epsilon,\U))$, then $\trim(\epsilon,U_i)\cap \trim(\epsilon, U_j)\neq \emptyset$. As $\trim(\epsilon, U)\subset U$, this implies that $U_i\cap U_j\neq \emptyset$. Since $\C$ is 2-sparse, the codeword with support $\{i,j\}$ is in $\C$. On the other hand, if the codeword with support $\{i,j\}$ is in $\C$, then $U_i\cap U_j\neq \emptyset$, and so we are in case $(1), (2)$, or $(3)$ above. By our choice of $\epsilon$, we ensure that in each case $\trim(U_i,\epsilon)\cap \trim(U_j,\epsilon) \neq \emptyset$, and hence (as the code is 2-sparse) the codeword with support $\{i,j\}$ is in $\C(\trim(\U,\epsilon))$.
If a codeword with support $\{i\}$ is in $\C(\trim(\U, \epsilon))$, then $\trim(U_i, \epsilon) \backslash \bigcup_{j\in [n], j\neq i} \trim(U_j,\epsilon) \neq \emptyset.$ We then know that $U_i\backslash \bigcup_{j\in [n], j\neq i}U_j \neq \emptyset$. If it were not, then we would have $U_i\subset \bigcup_{j\in I}U_j$ for some index set $I$. However, this is impossible: if $|I| = 1$, then $U_i\subset U_j$, but then $\trim(U_i, \epsilon) \subset \trim(U_j, \epsilon)$. If $|I|>1$, then $U_i\subset \bigcup_{j\in I} U_j$. But then the 2-sparsity of $\C$ means we would see the codewords $\{i,j\}$ and $\{i,k\}$ in $\C$ for $j,k\in I$ but not their intersection $\{i\}$, contradicting Lemma <ref>. Hence, the codeword with support $\{i\}$ is in $\C$.
Now, suppose a codeword with support $\{i\}$ is in $\C$, and let $J = \{j\ | \ U_i\cap U_j \neq \emptyset\}$. If $|J|\leq 1$ then we are in case $(1), (2)$, or $(3)$ above, and by our choice of $\epsilon$ we know there is a codeword with support $\{i\}$ in $\C(\trim(\U, \epsilon))$. If $|J|\geq 2$, let $j,k\in J$. Then by our choice of $\epsilon$, we know $\trim(U_i,\epsilon)\cap \trim(U_j, \epsilon)\neq \emptyset$ and $\trim(U_i,\epsilon)\cap \trim(U_k,\epsilon)\neq \emptyset$, and hence the codewords with supports $\{i,j\}$ and $\{i,k\}$ are in $\trim(\U, \epsilon)$. By Lemma <ref>, we know the codeword with support $\{i\}$ is also in $\C(\trim(\U, \epsilon))$.
Let $\C$ be a 2-sparse code with a closed convex realization $\V = \{V_1,\ldots, V_n\}$ in which every set is bounded. Then there exists some $\epsilon>0$ such that $\inflate(\V,\epsilon)$ is an open convex realization of $\C$.
Consider the partial ordering on $\V$ given by set inclusion. We will use this ordering to inflate the sets in $\V$ iteratively (possibly by different $\epsilon$ factors) and then argue that we can obtain a uniform $\epsilon$ for which $\inflate(\V,\epsilon)$ is an open convex realization of $\C$. In this iterative process, if $V_i=V_j$ for any $i \neq j$, we apply the process simultaneously to $V_i$ and $V_j$. As such, it is sufficient for our proof to assume $V_i \neq V_j$ for any $i \neq j$.
To start, begin with a fixed index $i$ for which $V_i$ is maximal in $\V$ with respect to inclusion. All sets in $\V$ are closed and bounded so for any $j$ with $V_j \cap V_j = \emptyset$, $V_i$ has positive distance $d_{i,j}$ to $V_j$. Let $\displaystyle \delta_i=\min_{V_i \cap V_j = \emptyset} d_{i,j}$. Now if there are $j,k\neq i$ with $V_j\cap V_k\neq\emptyset$, then $V_i$ has positive distance $d_{i,j,k}$ to $V_j\cap V_k$; take $\delta'_i$ to be the minimum of all such $d_{i,j,k}$. Furthermore, let $\delta''_i>0$ be such that for all $j$ with $V_j\not\subset V_i$, we have $V_j\not\subset\inflate(V_i,\delta''_i)$. Finally, choose $\epsilon_i < \min\left\{\frac{\delta_i}{2}, \frac{\delta'_i}{2},\frac{\delta''_i}{2}\right\}.$
If we replace $V_i$ by $\closure{\inflate(V_i,\epsilon_i)}$, then the three subset relationship types for the ordered pairs $(V_i,V_j)$ where $j \neq i$ are maintained:
* Type 1: Disjointness is preserved since $\epsilon_i$ is at most half the distance from $V_i$ to any set disjoint from it.
* Type 2: Containment is preserved since we are only making $V_i$ bigger.
* Type 3: Proper intersection is preserved by our choice of $\epsilon_i$.
By a similar argument, the subset relationship of the order pair $(V_j,V_i)$ for any $j \neq i$ is also preserved after replacing $V_i$ by $\closure{\inflate(V_i,\epsilon_i)}$. Thus replacing $V_i$ by $\closure{\inflate(V_i,\epsilon_i)}$ yields a new realization of $\C$.
For any subsequent step in our iterative process, choose a set $V_i \in \V$ for which every member of the set $\{V_j \in \V \ | \ V_j \supset V_i\}$ has already been inflated. Choose $\epsilon_i$ in the same way as previously described with the additional caveat that if $V_i \subset V_j$ $\epsilon_i < \epsilon_j$. A similar argument shows that replacing $V_i$ by $\closure{\inflate(V_i,\epsilon_i)}$ yields a new realization of $\C$. Once we have inflated every set in the realization we can let $\epsilon = \min_{i\in [n]} \epsilon_i$ and observe that $\inflate(\U, \frac{\epsilon}{2})$ is an open convex realization of $\C$.
This result allows us to prove the following useful fact.
Let $\C$ be a 2-sparse code. Then $\C$ has an open convex realization in $\R^d$ if and only if $\C$ has a closed convex realization in $\R^d$.
$(\Rightarrow)$ Let $\U$ be an open convex realization of $\C$. Applying Lemma <ref>, there is an $\epsilon>0$ such that $\U' = \trim(\U, \epsilon)$ is an open realization of $\C$. Since the closure of each $U_i'$ is contained in $U_i$ (by Proposition <ref>), $\U'$ is an open convex realization of $\C$ in which two sets intersect if and only if their closures do. Let $\V = \{\closure{U_1'},\ldots,\closure{U_n'}\}$. No triple intersections are present in $\V$ since these would correspond to triple intersections in $\U$. Thus by Proposition <ref> it suffices to show that $\V$ preserves the relations between sets in $\U'$. Disjointness is preserved since sets in $\U$ intersect if and only if their closures do. Containment is preserved under taking closures. Lastly, proper intersection is preserved since if $U_i\setminus U_j$ is nonempty then there exist limit points of $U_i$ that are not limit points of $U_j$.
$(\Leftarrow)$ Let $\V$ be a closed convex realization of $\C$. For every nonempty intersection $V_i\cap V_j$, let $p_{i,j}$ be a point in this intersection. Furthermore, if some set $V_i$ is not contained in any other $V_j$, let $p_i\in V_i\setminus \bigcup_{j\neq i}V_j$. Then set $V$ to be the convex hull of all these $p_i's$ and $p_{i,j}'s$. Replacing each $V_i$ by $V_i\cap V$ yields a realization of $\C$ in which every set is closed, convex, and bounded. Applying Lemma <ref>, we obtain an open convex realization of $\C$ in $\R^d$.
Although it may not be immediately clear from the proof, the condition that $\C$ is 2-sparse is necessary for Lemma <ref>. The 2-sparse condition is in fact best possible, since there exist 3-sparse codes which have closed convex realizations in $\R^2$, but for which open convex realizations exist only in $\R^3$ or higher. One such example is the code
$$\C = \{0000,1000,0100,0010,0001, 1110, 1001, 0101, 0011\}$$
(see <cit.> for more details). Even more strikingly, there exist codes with a closed convex realization in $\R^2$ that have no open convex realization in any dimension. This emphasizes how special realizations of 2-sparse codes are.
We can now use the previous lemmas to relate the convexity of a 2-sparse code $\C$ to the convexity of its associated graph $G_\C$. We first need a technical lemma.
Let $\U$ be an open convex realization of a 2-sparse code $\C$. Then if $U_j\not\subset U_k$ for any $k\neq j$, there is a point $p\in \partial U_j \backslash \bigcup_{k\neq j} U_k$.
Recall that for any set $U \subset \R^d$, $\partial U$ is the boundary of $U$. Consider the sets $\{\partial U_j\cap U_k\}_{k\neq j}$. These sets are disjoint: if not, then there exists $p \in (\partial U_j \cap U_k)\cap (\partial U_j \cap U_\ell)$. As $p\in U_k\cap U_\ell$, there exists $\epsilon>0$ with $B_\epsilon(p)\subseteq U_k\cap U_\ell$. But then $B_\epsilon(p)\cap U_j\neq \emptyset$, as $p \in \partial U_j$, so $U_j\cap U_k\cap U_\ell\neq \emptyset$ contradicting that $\C$ is 2-sparse.
Now, note that the disjoint sets $\{\partial U_j\cap U_k\}_{k\neq j}$ are open in the subspace topology with respect to $\partial U_j$, and hence they cannot partition $\partial U_j$ since $\partial U_j$ is connected. Thus, there exists $p\in \partial U_j\backslash \bigcup_{k\neq j} U_k$.
Let $\C$ be a 2-sparse code and let $d\ge 2$. Then $\C$ has an open convex realization in $\R^d$ if and only if $\supp(\C)$ is closed under intersection and $G_\C$ has an open convex realization in $\R^d$.
$(\Rightarrow)$ We know from Lemma <ref> that if $\C$ has a realization then its support is closed under intersection. We will show that given a realization $\U$ of $\C$, we can construct a realization of $G_\C$.
Since $\C$ is 2-sparse, we know $\C$ and $\Delta(\C)$ have the same $2$-element sets in them. We will show that we can adjust the realization of $\C$ to obtain any singletons $\{i\}$ which appear in $\Delta(\C)$ but not in $\C$.
Let $\{i\}\in \Delta(\C)\backslash \supp(\C)$. If there exist $j,k$ such that $\{i,j\}$ and $\{i,k\}$ are both in $\supp(\C)$, then as $\supp(\C)$ is closed under intersection, we know $\{i\}\in \supp(\C)$. Thus, there is exactly one $j$ such that $\{i,j\}\in \supp(\C)$.
Note right away that in the realization $\U$ we have $U_i \subset U_j$ since $\{i,j\}$ is the only set in the support where $i$ appears. It suffices to transform $\U$ so that $U_i$ and $U_j$ intersect, but $U_i$ also contains points not in any other set in the realization.
If $U_j\subset U_i$, then $U_i=U_j$, and we can replace $U_j$ with an open ball properly contained in $U_i$ to obtain the desired result. Otherwise, $U_j$ may intersect many other sets in the realization, but cannot be contained in them, since this would imply a triple intersection between the containing set, $U_j$, and $U_i$. Apply Lemma <ref> to obtain $\epsilon>0$ for which $\U' = \trim(\U,\epsilon)$ is an open realization of $\C$. Define the sets $V_k = \partial U'_j \cap \overline{U'_k}$; note that each $V_k$ is closed. Furthermore, these sets are disjoint, since if $p\in V_k\cap V_\ell$, then $p\in U_j\cap U_k \cap U_\ell$ in the original realization which is impossible for a 2-sparse code. Since $\partial U_j'$ is connected and the $V_k$ are disjoint closed sets, $\bigcup_{k\neq j} V_k\subsetneq \partial U'_j$; let $p\in \partial U'_j\backslash \bigcup_{k\neq j} V_k$. Then $p$ has positive distance to all sets $U'_k$ with $k\neq j$ so there is some $\epsilon'>0$ with $B_{\epsilon'}(p)\cap U'_k = \emptyset$ for all $k\neq j$. Replacing $U_i'$ with $B_{\epsilon'}(p)$ will create a realization of a code $\C'$ with $\supp(\C') = \supp C \cup \{i\}$. Repeating this step as many times as necessary, we obtain a realization of $\Delta(\C)$.
$(\Leftarrow)$ Suppose $\U$ is an open convex realization of $\Delta(\C)$. Note that if $\{i,j\} \in \supp(\Delta(\C))$, it is also in $\supp(\C)$ since $\C$ is 2-sparse. Now, suppose $\{i\} \in \supp(\Delta(\C))\backslash \supp(\C)$. Then there is at most one $j\neq i$ such that $\{i,j\}\in \supp(\C)$ as $\C$ is closed under intersection. If there is such a $j$, replace $U_i$ with $U_i\cap U_j$ which is an open convex set; if there is no such $j$, then remove $U_j$ entirely. This gives a convex realization of $\Delta(\C)\backslash\{i\}$, and we can repeat this operation as many times as necessary to obtain a realization of $\C$.
The above lemma can be summarized as follows: realizing a 2-sparse code and realizing its simplicial complex are equivalent, as long as $\supp(\C)$ is closed under intersection. This equivalence is our main tool in proving Theorem <ref> and obtaining a complete classification of which 2-sparse codes are convex in $\R^3$.
$(\Rightarrow)$ The fact that $\supp(\C)$ is closed under intersection follows directly from Lemma <ref>.
$(\Leftarrow)$ Since $\supp(\C)$ is closed under intersection, we know by Lemma <ref> that it is equivalent to find a realization for $G_\C$. But $\C$ is 2-sparse, and so Lemma <ref> tells us that it suffices to find a closed convex realization for $G_\C$. The graph $G_\C$ is a 1-dimensional simplicial complex. An augmentation of construction of Tancer <cit.> (see the proof of Theorem 3.1 therein) leads to a realization of a $1$-dimensional simplicial complex in $\R^{3}$.
This proves the desired result.
Theorem <ref> makes it very straightforward to check whether a 2-sparse code has an open convex realization in $\R^3$. The challenge that lies ahead is determining the minimal embedding dimension for a given 2-sparse code. We begin investigating this problem in the next section.
§ REALIZATION RESULTS
Throughout this discussion we will often refer to “realizations” of a graph. It is important to note that while a graph is the intersection graph of its realization, finding convex sets whose intersection graph is the graph of concern is not always sufficient. In particular, if a collection of convex sets has a triple with nonempty intersection then it is not a realization of a graph, since graphs only encode intersections of order two.
In this section, we begin the program of classifying 2-sparse codes based on minimal embedding dimension. Recall from Lemma <ref> that realizing a 2-sparse code $\C$ is equivalent to realizing its graph $G_\C$, so throughout this section we refer to realizing $G_\C$ rather than $\C$ itself. Our main contribution is that it is difficult to determine graph theoretic criteria on $G_\C$ that predict $d(\C)$. Topological properties inherent to $G_\C$ seem to potentially influence $d(\C)$: for instance in Proposition <ref>, we observe $d(\C)=2$ if $G_\C$ is planar and in Proposition <ref> if $G_\C$ is not planar one can construct a related graph whose code has minimal embedding dimension $3$. However, planarity does not strictly govern minimal embedding dimension, as all complete and complete bipartite graphs are realizable in $\R^2$. To start, our first proposition establishes that specific ubiquitous graphs have open convex realizations in $\R^2$.
The following graphs have an open convex realization in $\R^2$:
* planar graphs,
* the complete $k$-partite graph $K_{n_1,n_2,\ldots,n_k}$ with part sizes $n_1,n_2,\ldots,n_k$
* any graph $G$ with vertex set $\{v_1,v_2,\ldots,v_n,u_1,\ldots,u_k\}$ where the induced subgraph on the vertices $v_1,v_2,\ldots,v_n$ is complete and $\{v_1,v_2,\ldots,v_n\} \supseteq N_G(u_k) \supseteq N_G(u_{k-1}) \supseteq \cdots \supseteq N_G(u_1)$.
In all cases, we find a closed convex realization of the given graph $G$, which by Lemma <ref> implies the existence of an open convex realization. For (1), we first recall the Circle Packing Theorem
which says that for any planar graph $G$ with vertex set $\{v_1,\ldots,v_n\}$, there exist disjoint disks $C_1,C_2,\ldots,C_n$ in $\R^2$ such that $C_i$ is tangent to $C_j$ if and only if $v_i$ is adjacent to $v_j$, and $C_i \cap C_j = \emptyset$ otherwise. See Figure <ref> for an illustration of how these disks are constructed.
A planar graph $G$ and the corresponding closed realization using the Circle Packing Theorem.
For (2), we first find a realization for the complete graph $K_n=K_{1,1,\ldots,1}$ ($n$ copies of $1$ here). Consider the line segments $\ell_1,\ell_2,\ldots,\ell_n$ where $\ell_i$ has endpoints $(i,0),(0,n+1-i)$, and observe that $\ell_i \cap \ell_j \neq \emptyset$ for any $i \neq j$. Moreover, no three of these lines are concurrent. This gives a closed convex realization of $K_n$. Now to realize $K_{n_1,n_2,\ldots,n_k}$, start with a closed convex realization of $K_k$ as constructed in the realization of (2). Replace each line segment $\ell_i$ with $n_i$ disjoint parallel translates of $\ell_i$ that are arbitrarily close in distance to $\ell_i$, and call these segments $s_{i1},s_{i2},\ldots,s_{in_i}$. Observe that by construction, $s_{ij} \cap s_{ij'} = \emptyset$ for any $j \neq j'$. Moreover, $s_{ij} \cap s_{i'j'} \neq \emptyset$ for $i \neq i'$ because $l_i \cap l_{i'} \neq \emptyset$ and $s_{ij}$ and $s_{i'j'}$ are arbitrarily close and parallel to $\ell_i$ and $\ell_{i'}$ respectively. Moreover, if any three line segments $s_{ij},s_{i'j'},s_{i''j''}$ had a point in common, then $l_i,l_{i'},l_{i''}$ would, which they don't. Hence the union of the sets $\{s_{i1},s_{i2},\ldots,s_{in_i}\}_{i=1}^k$ gives a closed convex realization of $K_{n_1,n_2,\ldots,n_k}$. See Figure <ref> for examples of the constructions in the proof of (2).
A closed convex realization of $K_5$ (left) and a closed convex realization of $K_{2,4,3}$ (right), as constructed in the proof of Proposition <ref>
It remains to prove (3). Without loss of generality, we assume $N_G(u_k)=\{v_1,v_2,\ldots,v_r\}$, indexed in such a way that each set $N_G(u_j)$ is $\{v_1,v_2,\ldots,v_s\}$ for some $s$. To realize $G$, first start with a realization of $K_n$ as in the proof of (2), where $v_j$ is represented by $\ell_j$ for each $j$. Now, extend each line segment $\ell_j$ for $1 \leq j \leq r$ so that $(0,j)$ remains as an endpoint, the slope remains the same, but the lower endpoint has $y$-coordinate $-k$. Then, for each $s$ with $1 \leq s \leq k$, introduce a line segment $\ell'_s$ that lies on the line in the $xy$-plane given by $y=s$, and only intersects the line segments in the set $\{\ell'_j \ | \ j \in N_G(u_s)\}$. The line segments $\ell_1,\ldots,\ell_n,\ell'_1,\ldots,\ell'_k$ give a closed realization of $G$. See Figure <ref> for an example of this construction.
A closed convex realization of the graph $G$ with vertices $v_1,v_2, v_3, v_4,v_5, u_1,u_2,u_3$ where the induced graph on $v_1,...,v_5$ is complete, and $N(u_3) = \{v_1,v_2,v_3\}$, $N(u_2) = \{v_1,v_2\}$ and $N(u_1) = \{v_1\}$.
Thus far, we have exhibited classes of graphs that can be realized in $\R^2$, which included graphs $G_\C$ that are planar. We know show how to augment any nonplanar graph to create a graph that can not be realized in $\R^2$.
Let $G$ be a nonplanar graph. Let $G'$ be the graph obtained from $G$ by replacing each edge $v_i v_j$ by a length 2 path $v_i, v_{ij}, v_j$ (we refer to this as the edge subdivision of $G$ throughout). Then $G'$ does not have an open convex realization in $\R^2$, and hence its minimal embedding dimension is $3$.
Suppose by contradiction that $G'$ has an open convex realization in $\R^2$. Let the graph $G$ have vertex set $\{v_1,v_2,..,v_n\}$, so $G'$ has as its vertices $\{v_i \ | \ i=1,...,n\}$ together with vertices $\{v_{ij} \ | \ v_iv_j\in E(G)\}$ where for any $i,j$, $v_{ij}$ is adjacent only to $v_i$ and $v_j$. Suppose the open convex realization $\mathcal{U}$ of $G'$ consists of the sets $\{U_i\}$ and $\{U_{ij}\}$ where for any $i$, $U_i$ is the open convex set corresponding to $v_i$, and for any $i \neq j$ with $v_iv_j\in E(G)$, $U_{ij}$ is the open convex set corresponding to $v_{ij}$.
Fix $i,j$ such that $v_i$ and $v_j$ are adjacent in $G$. Let $p_i$ and $p_j$ be points in $U_i$ and $U_j$ respectively, that do not lie in any other sets in $\U$. Since $U_i \cap U_{ij}$ and $U_j \cap U_{ji}$ are nonempty, we can find points $x_{ij}$ and $x_{ji}$ in $U_i \cap U_{ij}$ and $U_j \cap U_{ij}$, respectively. Let $x_{ij}x_{ji}$ intersect $\partial U_i$ and $\partial U_j$ at point $p_{ij}$ and $p_{ji}$, respectively. Define the path $P_{ij}$ from $p_i$ to $p_j$ by concatenating the line segments $p_ip_{ij}$, $p_{ij}p_{ji}$, and $p_{ji}p_j$ in that order.
Now additionally fix indices $k,l$. We claim that the two different paths $P_{ij}$, $P_{kl}$ can only intersect at the points $p_i$,$p_j$,$p_k$ or $p_l$, if anywhere. To do so, it is enough to show that among any pair of line segments, one chosen from $\{p_ip_{ij},p_{ij}p_{ji},p_{ji}p_j\}$ and one from $\{p_kp_{kl},p_{kl}p_{lk},p_{lk}p_l\}$, their intersection (if it exists), must be one of the points $p_i$,$p_j$,$p_k$ or $p_l$. We split this into three cases:
First, consider the intersection of $p_ip_{ij}$ and $p_kp_{kl}$. If $i=k$ then the two segments can only intersect at $p_i$, unless $j=l$, in which case the segments were the same segments to begin with. If $i \neq k$, then observe that $p_ip_{ij} \in U_i$, $p_kp_{kl} \in U_k$ and $U_i \cap U_k$ is empty because $v_i$ and $v_k$ are not adjacent in $G'$. A similar argument establishes our desired result when the pair of segments in question are $\{p_ip_{ij} , p_{kl}p_k\}$, $\{p_{ij}p_i, p_{kl}p_k\}$ and $\{p_{ij}p_i, p_{k}p_{kl}\}$.
Second, consider the intersection of $p_{ij}p_{ji}$ and $p_{kl}p_{lk}$. Notice that $p_{ij}p_{ji} \subseteq U_{ij}$ and $p_{kl}p_{lk} \subseteq U_{kl}$. Since $v_{ij}$ and $v_{kl}$ are not adjacent in $G'$, $U_{ij} \cap U_{kl}$ is empty, so the two paths in question can not intersect.
Finally, consider the intersection of $p_{i}p_{ij}$ and $p_{kl}p_{lk}$. Suppose that $i=k$. When $j=l$, the segments in question are $p_ip_{ij}$, $p_{ij}p_{ji}$ but these are from the same path $P_{ij}$ so we need not consider this situation. When $j \neq l$, $p_{i}p_{ij} \subseteq U_i \cup U_{ij}$, and $p_{il}p_{li} \subseteq U_{il}\backslash U_i$. Since $j \neq l$, $U_{ij} \cap U_{il} = \emptyset$, and hence $(U_i \cup U_{ij}) \cap (U_{il}\backslash U_i) = \emptyset$, so the two segments in question do not intersect. A similar argument establishes the result when $j = l$. It remains to establish the desired result when $i \neq l,k$. Suppose for a contradiction that $p_ip_{ij}$ intersects $p_{kl}p_{kl}$. Since $p_ip_{ij} \subseteq U_i \cup \partial U_i$, and $p_{kl}p_{lk} \subseteq U_{lk}$, this implies $(U_i \cup \partial U_i) \cap U_{lk}$ is nonempty. However, this is impossible because $U_i \cap U_{lk} = \emptyset$ (because $v_i$ and $v_{lk}$ are not adjacent in $G'$) and $\partial U_i \cap U_{lk} = \emptyset$.
The above argument establishes that two distinct paths $P_{ij},P_{kl}$ can only intersect at their endpoints. Construct a graph $G''$ on the same vertex set as $G$ with two vertices adjacent precisely when they are adjacent in $G$, but with each edge $v_iv_j$ drawn precisely along the path $P_{ij}$. The graph $G''$ is a planar embedding of $G$, contradicting $G$ is planar.
§ FUTURE DIRECTIONS
This paper initiated the program of studying $k$-sparse codes with a full characterization of the structure of $2$-sparse codes. Section 2 was dedicated to a topological and analytic investigation of such codes in order to achieve a full characterization through Theorem <ref>, which additionally told us that any realizable $2$-sparse code has minimal embedding dimension at most $3$. Section 3 then began the study of differentiating $2$-sparse codes by embedding dimension through Proposition <ref> and Proposition <ref>. The most pressing questions are how these investigations generalize when $k>2$.
For a particular $k$, how can we characterize which k-sparse codes are realizable? More specifically, given a positive integer $\ell$, for which $k$-sparse codes is $d(\C)=\ell$?
In investigating the minimum embedding dimension of a $k$-sparse code, certain dimension bounds can be used. For example, suppose $\C$ is a $k$-sparse code with $\Delta = \Delta(\C)$, and let $f_d(\Delta)$ be the number of codewords in $\Delta$ with support size $d+1$. Then, by applying the Fractional Helly theorem, we find $k>f_d(\Delta)/\binom{n-1}{d}$; this was noted in <cit.>. Similar to this, many known bounds rely solely on the combinatorial information in the code and in particular the simplicial complex $\Delta(\C)$. However, in addressing the question of whether a $k$-sparse code is realizable at all, an investigation into the topology of the matter can provide insight beyond what is apparent from the combinatorics. This is especially evident from the developments in Section <ref>. The key idea there was shifting from one realization of a code to another by shrinking or expanding sets. The question then for $k$-sparse codes for $k>2$ is what analogous topological operations to realizations preserve the underlying code.
Given a convex realization $\U=\{U_1,...,U_n\}$ of a code $\C$ in $\R^d$, what topological maps can be applied to the sets $U_i$ so that the resulting sets still form a convex realization of $\C$?
§ ACKNOWLEDGMENTS
The authors thank Dean's Office and the Department of Mathematics at Harvey Mudd College for their summer research support, and thank Carina Curto, Chad Giusti, Elizabeth Gross, Vladimir Itskov, Bill Kronholm and Anne Shiu for fruitful conversations.
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Spectro-photometry for Transit Spectroscopy
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Faculty of Science, Guilin University of Aerospace Technology, Guilin, Guangxi, P.R. China.
Faculty of Science, Guilin University of Aerospace Technology, Guilin, Guangxi, P.R. China.
Department of Modern Physics , University of Science and Technology of China, Hefei, Anhui, China.
We investigate decoherence of quantum superpositions induced by gravitational time dilation and spontaneous emission between two atomic levels. It has been shown that gravitational time dilation can be an universal decoherence source.
Here, we consider decoherence induced by gravitational time dilation only in the situation of spontaneous emission. Then, we obtain that the coherence of particle's position state depends on reference frame due to the time dilation changing the distinguishability of emission photon from two positions of particle. Changing the direction of light field can also result in the difference about the coherence of quantum superpositions. For observing the decoherence effect mainly due to gravitational time dilation, time-delayed feedback can be utilized to increase the decoherence of particle's superpositions.
03.65.Yz, 04.62.+v, 42.50.-p
§ INTRODUCTION
Quantum phenomenon has been observed by numerous experiments on microscopic scales. However, on macroscopic scales, it is difficult to find quantum effects, such as quantum superpositions. A lot of physicists have been looking up the root of quantum-to-classical transition for decades. The reason can be divided into two categories: coarsened measurement and decoherence<cit.>.
Commonly viewpoint is that decoherence plays a prominent role in quantum-to-classical transition. There are two routes to explain decoherence: one route is that system interacts with external environments, the other is taken in wave function collapse<cit.>, which need not external environments.
The latter one is often inspired by general relativity and makes a fundamental modification on quantum theory. Recently, Igor at al.<cit.> demonstrated the existence of decoherence induced by gravitational time dilation without any modification of quantum mechanics. This work motivates further study on decoherence due to time dilation.
Spontaneous emission between two atomic levels inevitably occurs. We research decoherence due to time dilation during spontaneous emission. Without spontaneous emission, decoherence will not occur in our model only by time dilation. As we all know, spontaneous emission can induce decoherence. We find that gravitational time dilation can reduce or increase the decoherence due to spontaneous emission in different reference frames (different zero potential energy point). It is attributed to the fact that in different reference frame, the distinguishability of emission photon from different positions is different. The direction of emission light also influences the coherence of quantum superpositions in fixed direction of gravitational field.
In order to make the decoherence due to time dilation stronger than due to spontaneous emission, time-delayed feedback control<cit.> is used.
The rest of paper is arranged as follows. In section II, we present the model about the decoherence of quantum superpositions due to time dilation during spontaneous emission. Coherence of particle's position in different reference frame is explored in section III. In section IV, we discuss the influence of different directions of emission light. In section V, a
time-delayed feedback scheme is utilized to increase decoherence induced by gravitational time dilation. We deliver a conclusion and outlook in section VI.
§ MODEL
Firstly, let us simply review the gravitational time dilation which causes clocks to run slower near a massive object. Given a particle of rest mass $m$ with an arbitrary internal Hamiltonian $H_0$, which interacts with the gravitational potential $\Phi(x)$. The total Hamiltonian $H$ is described by<cit.>
\begin{eqnarray}
\end{eqnarray}
where $H_{ext}$ is external Hamiltonian. For a free particle, $H_{ext}=mc^2+p^2/2m+m\Phi(x)$. In Eq.(1), the last term, $-H_0p^2/(2m^2c^2)$, is simply the velocity-dependent special relativistic time dilation. The coupling with position, $H_0\Phi(x)/c^2$, represents the gravitational time dilation. When we consider slowly moving particles, $p\approx0$, the gravitational time dilation will be the main source of time dilation. It will not be canceled by the velocity-dependent special relativistic time dilation.
We consider that an atom with two levels is in superposition of two vertically distinct positions $x_1$ and $x_2$.
The atom is coupled to a single unidirectional light field, as depicted in Fig. 1.
Decoherence of an atom is induced by gravitational time dilation under the situation of spontaneous emission. The dotted line represents a homogeneous gravitational field $\Phi(x)\approx g x$, where $g =9.81m/s^2 $ is the gravitational acceleration on Earth. Initial atom is in the superposition state: $1/\sqrt{2}(|x_1\rangle+|x_2\rangle)$. The direction of emitting photon is along the solid line, x-direction, which is contrary with the direction of gravitational field.
The whole system interacts with a homogeneous gravitational field $\Phi(x)\approx g x$ which generates the gravitational time dilation. The total system-field Hamiltonian is described by ($\hbar=1$)
\begin{eqnarray}
H=[mc^2+m g x_1+w_1(1+g x_1/c^2)|1\rangle\langle1|+w_2(1+g x_1/c^2)|2\rangle\langle2|]|x_1\rangle\langle x_1|\nonumber
\\+\sqrt{\kappa_1/2\pi}(1+g x_1/c^2)|x_1\rangle\langle x_1|\int dw [a b^\dagger(w)\exp(-i w x_1/c)+H.c]\nonumber\\
+[mc^2+m g x_2+ w_1(1+g x_2/c^2)|1\rangle\langle1|+ w_2(1+g x_2/c^2)|2\rangle\langle2|]|x_2\rangle\langle x_2|\nonumber\\
+\sqrt{\kappa_2/2\pi}(1+g x_2/c^2)|x_2\rangle\langle x_2|\int dw [a b^\dagger(w)\exp(-i w x_2/c)+H.c]+\int dw w b^\dagger(w)b(w),
\end{eqnarray}
where $w_1$ and $w_2$ ($w_1>w_2$) are eigenvalues for the atomic level 1 and 2, respectively, and operator $a=|2\rangle\langle1|$.
$\kappa_1$ and $\kappa_2$ denote the coupling constants in position $x_1$ and $x_2$, respectively. Without extra control, the two coupling constants should be same: $\kappa_1=\kappa_2=\kappa$. The last term in Eq.(2) represents the free field Hamiltonian, and the filed modes, $b(w)$, satisfy $[b(w),b^\dagger(w')]=\delta(w-w')$.
Using Pauli operator, $\sigma_z=|1\rangle\langle1|-|2\rangle\langle2|$, to simplify the Eq.(2), we can obtain the new form of system-field Hamiltonian
\begin{eqnarray}
H=[E_1+w_0/2(1+g x_1/c^2)\sigma_z]|x_1\rangle\langle x_1|+\sqrt{\kappa/2\pi}(1+g x_1/c^2)|x_1\rangle\langle x_1|\int dw [a b^\dagger(w)\exp(-i w x_1/c)+H.c]\nonumber\\
+[E_2+w_0/2(1+g x_2/c^2)\sigma_z]|x_2\rangle\langle x_2|+\sqrt{\kappa/2\pi}(1+g x_2/c^2)|x_2\rangle\langle x_2|\int dw [a b^\dagger(w)\exp(-i w x_2/c)+H.c]\nonumber\\+\int dw w b^\dagger(w)b(w),
\end{eqnarray}
where $E_i=mc^2+\frac{(w_1+w_2)}{2}(1+g x_i/c^2)$ for $i=1,2$ and $w_0=w_1-w_2$.
We consider the initial field in the vacuum state and the atom in the state $|1\rangle\frac{|x_1\rangle+|x_2\rangle}{\sqrt{2}}$.
Then, the atom will spontaneously emit photon. According to there being only a single excitation conservation between system and field<cit.>, the system state in any time $t$ can be solved analytically, see Appendix.
§ COHERENCE OF PARTICLE'S POSITION
The quantum coherence of particle's position state can be quantified by the interferometric visibility $V(t)$, as shown in Eq.(27) in Appendix. When the time $t$ satisfy $\lambda_1\kappa t\gg1$ and $\lambda_2\kappa t\gg1$, the amplitude of excitation state $C_1\approx0$ and $C_2\approx0$. Then, we arrive at
\begin{eqnarray}
\exp[-\lambda_1^2\kappa\tau],
\end{eqnarray}
where $\lambda_i=1+g x_i/c^2$ for $i=1,2$.
From the above equation, we can see that the decoherence comes from the spontaneous emission (when $\lambda_1=\lambda_2=1$) and the gravitational time dilation. Spontaneous emission can generate the decoherence due to the fact that photon is emitted from different positions, which leads to having a phase difference $w\tau$, where $w$ denotes the frequency of photon.
And we achieve that coherence depends on the reference frame. Different zero potential energy point (different value of $\lambda_1$ ) will give different coherence strength. The counterintuitive result occurs because in different frame the phase difference will become different so that the distinguishability of emitting photon from two positions is different. Reducing the zero potential point (increasing the value of $\lambda_1$), the phase difference will increase because of time dilation. For a fixed position difference $\Delta=g(x_2-x_1)/c^2$, the quantum coherence can be rewritten
\begin{eqnarray}
\exp[-\lambda_1^2\kappa\tau].
\end{eqnarray}
There is an optimal value of $\lambda_1$, which can give the maximal quantum coherence, as shown in Fig. 2.
Diagram of quantum coherence $V$ changing with $\lambda_1$. The quantum coherence depends on reference frame. The parameters are given by: $w_0/\kappa=10^6$, $\Delta=10^{-6}$, $\kappa \tau=10^{-2}$.
In an optimal reference frame, one can obseve the maximal coherence: $V$ is close to 1.
In order to observe the decoherence induced by gravitational time dilation, it need to satisfy that the decoherence effect from time dilation is stronger than only from spontaneous emission ($\lambda_1=\lambda_2=1$):
\begin{eqnarray}
\frac{2\kappa\lambda_1(\lambda_1+\Delta)}{\sqrt{[\kappa(\lambda_1^2+(\lambda_1+\Delta)^2)]^2+(w_0\Delta)^2}}\exp[-\lambda_1^2\kappa\tau]
\ll\exp[-\kappa\tau].
Noting that the value of $\lambda_2-\lambda_1$ is generally small in experiment, the condition $\exp[(\lambda_1^2-1)\kappa\tau]\gg1$ is necessary for observing decoherence mainly induced by gravitational time dilation.
When one changes the direction of emitting photon, the quantum coherence will change accordingly,
\begin{eqnarray}
\exp[-\lambda_2^2\kappa\tau].
\end{eqnarray}
It is due to that the phase difference changes with the direction of emitting photon, becoming $-w\tau$. Different directions of emitting photon in the fixed gravitational field will generate different quantum coherence $V$.
Then, we consider general three-dimensional space: the emitting photon can be along any direction, as shown in Fig. 3.
Diagram shows that the photon can spontaneously emit in any direction. Here $\theta$ denotes the angle between direction of photon and x-direction, which can change from 0 to $\pi$.
We obtain the quantum coherence of particle's position state as following:
\begin{eqnarray}
\int_0^{\pi/2} d\theta\sin\theta\cos^2\theta
\exp[(iw_0\lambda_1-\lambda_1^2\kappa)\tau\cos\theta]+\exp[-(iw_0\lambda_2+\lambda_2^2\kappa)\tau\cos\theta]|,\\
&=\frac{3\kappa\lambda_1\lambda_2}{\sqrt{[\kappa(\lambda_1^2+\lambda_2^2)]^2+[w_0(\lambda_2-\lambda_1)]^2}}|[-2 + \exp(k_1) (2 - 2 k_1 + k_1^2)]/k_1^3+[-2 + \exp(k_2) (2 - 2 k_2 + k_2^2)]/k_2^3|,\\
&\textmd{in which},\nonumber\\
&k_j=[(-1)^jiw_0\lambda_j-\lambda_j^2\kappa]\tau, \ \textmd{for} \ j=1,2,
\end{eqnarray}
where the coupling strength between atom and light field changes with the direction of emitting photon, becoming $\sqrt{\kappa/2}\cos\theta $<cit.>.
For $w_0\lambda_j\tau\ll1$ and $\lambda_j^2\kappa\tau\ll1$, $V_3\approx\frac{2\kappa\lambda_1\lambda_2}{\sqrt{[\kappa(\lambda_1^2+\lambda_2^2)]^2+[w_0(\lambda_2-\lambda_1)]^2}}(1-\lambda_1^2\kappa\tau-\lambda_2^2\kappa\tau)<V'<V.$ It means that the quantum coherence in general three-dimensional space is smaller than in one-dimensional space of fixed direction.
For $w_0\lambda_j\tau\gg1$ and $\lambda_j^2\kappa\tau\gg1$, $V_3\approx\frac{2\kappa\lambda_1\lambda_2}{\sqrt{[\kappa(\lambda_1^2+\lambda_2^2)]^2+[w_0(\lambda_2-\lambda_1)]^2}}|\cos3\varphi|(3/[(w_0^2\lambda_1^2+(\lambda_1^2\kappa\tau)^2]^{3/2}+3/[(w_0^2\lambda_1^2+(\lambda_1^2\kappa\tau)^2]^{3/2})\geq V,$ with $\cos\varphi=\lambda_1^2\kappa\tau/\sqrt{w_0^2\lambda_1^2+(\lambda_1^2\kappa\tau)^2}$. It means that in new condition the quantum coherence in general three-dimensional space is larger than in one-dimensional space of fixed direction.
The root of generating $V_3\neq V$ is the phase difference changing from $w\tau$ to $w\tau\cos\theta$.
§ TIME DELAY FEEDBACK
When one chooses the center of two positions as the zero potential point, the interferometric visibility reads
\begin{eqnarray}
\exp[-(1-\Delta/2)^2\kappa\tau].
\end{eqnarray}
In order to observe the decoherence from gravitational time dilation, not from the spontaneous emission, it is necessary to satisfy condition $V_c\ll\exp[-\kappa\tau]$. However, the value of $\Delta$ is very small in experiment. So, the condition is hard to meet. We can utilize the time delay feedback <cit.> to increase the decoherence from gravitational time dilation.
Diagram of time delay feedback. The center of two positions is chosen as zero potential point. So, in the new reference frame, the position $x_1$ ($x_2$) is transformed to be $\Delta c^2/2g$ ($-\Delta c^2/2g$). Here, we just consider that the photon emits along the fixed x-direction, which can be easily generalized to the case of three-dimensional space. At $r+\Delta c^2/2g$, a mirror is put to reflect the light field, leading to that a time-delay light field is fed back to system-field interaction.
As shown in Fig. 4, the light field is reflected by a mirror. The whole system-field Hamiltonian can be described by
\begin{eqnarray}
&H=\sqrt{\kappa/2\pi}(1+g x_1/c^2)|x_1\rangle\langle x_1|\int dw \{a b^\dagger(w)2\exp[-i w(r+\Delta c^2/2g)]\cos(2w r)+H.c\}\nonumber\\
&+\sqrt{\kappa/2\pi}(1+g x_2/c^2)|x_2\rangle\langle x_2|\int dw\{a b^\dagger(w)2\exp[-i w(r+\Delta c^2/2g)]\cos[ w (2r+2\Delta c^2/g)/c]+H.c\}\nonumber\\
&+\int dw w b^\dagger(w)b(w)+[E_1+w_0/2(1+g x_1/c^2)\sigma_z]|x_1\rangle\langle x_1|+[E_2+w_0/2(1+g x_2/c^2)\sigma_z]|x_2\rangle\langle x_2|,
\end{eqnarray}
Using the way in Appendix, we can obtain the quantum coherence at time $t\gg1$. With the feedback, the spontaneous emission is suppressed due to superposition effect. The total system-field wave function can also be described by Eq.(13) in Appendix.
When the conditions $w_0(1+\Delta/2)2r/c=n\pi$ and $w_0(1-\Delta/2)(2r+2\Delta c^2/g)/c\neq m\pi$ hold, for $t\gg1$ the amplitudes $|C_2|^2=\exp[-n\pi\kappa(1+\Delta/2)/w_0]$ and $|C_1|\simeq0$, where $n,m=1,2,3\cdot\cdot\cdot$. When $w_0\gg\kappa$ and $n=1$, $|C_2|^2\simeq1$. So, we achieve that the quantum coherence $V_c\simeq0$. Without gravitational time dilation, the quantum coherence is much larger than 0. So, utilizing time-delayed feedback scheme can satisfy that the decoherence induced by the gravitational time dilation is far less than by spontaneous emission.
§ CONCLUSION AND OUTLOOK
We explore the decoherence of an atom's positions induced by the gravitational time dilation only in the situation of spontaneous emission. As the phase difference of photon emitted from two positions are different in different reference frames, the quantum coherence of superposition state of positions depends on the reference frame. So one can choose proper reference frame to observe the decoherence from the gravitational time dilation. It is worth mentioning that the direction of emitting photon will influence the quantum coherence. So comparing the case of fixed emitting direction with the case of any direction, there are some differences about quantum coherence. When one chooses the center of two positions as the zero potential point, the decoherence induced by the gravitational time dilation is difficult to be far larger than by spontaneous emission. The time delay feedback can be used to increase the decoherence from the time dilation with proper conditions.
In this article, we only discuss the decoherence of an atom with two energy levels induced by gravitational time dilation. It is interesting to research the decoherence of many particles with many energy levels induced by time dilation with spontaneous emission. In this case we believe that it will increase the decoherence effect from the gravitational time dilation. And considering extra drive is the further research direction. In this situation, due to the fact that a single excitation conservation between system and field do not hold, the question will become complex and rich.
§ ACKNOWLEDGEMENT
This work was supported by the National Natural Science Foundation of China under Grant No. 11375168.
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$ \mathbf{ APPENDIX}$
In the single photon limit, the total system-field wave function is described by
\begin{eqnarray}
|\Psi(t)\rangle=C_1|x_1\rangle|1\rangle|0\rangle+\int dw C_{1w}b^\dagger(w)|x_1\rangle|2\rangle|0\rangle+C_2|x_2\rangle|1\rangle|0\rangle+\int dw C_{2w}b^\dagger(w)|x_2\rangle|2\rangle|0\rangle.
\end{eqnarray}
The variables $C_1$, $C_{1w}$, $C_2$ and $C_{2w}$ denote the corresponding amplitudes of the four states at time $t$.
Applying the Schr$\ddot{o}$dinger equation in the rotating frame, we arrive at the following set of partial differential equations:
\begin{eqnarray}
i\partial_tC_1=[E_1+w_0/2(1+g x_1/c^2)]C_1+(1+g x_1/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(wx_1/c-w)t]C_{1k},\\
i\partial_tC_{1w}=[E_1-w_0/2(1+g x_1/c^2)]C_{1k}+(1+g x_1/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(-wx_1/c+w)t]C_{1},\\
i\partial_tC_2=[E_2+w_0/2(1+g x_2/c^2)]C_1+(1+g x_2/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(wx_2/c-w
i\partial_tC_{2w}=[E_2-w_0/2(1+g x_2/c^2)]C_{2k}+(1+g x_2/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(-wx_2/c+w)t]C_{2}.
\end{eqnarray}
Substituting $C'_1=\exp[-i(E_1+w_0/2(1+g x_1/c^2))t]C_1, C'_{1k}=\exp[-i(E_1-w_0/2(1+g x_1/c^2))t]C_{1k},$
$C'_2=\exp[-i(E_2+w_0/2(1+g x_2/c^2))t]C_2, C'_{2k}=\exp[-i(E_2-w_0/2(1+g x_2/c^2))t]C_{2k}$ into above equations, we arrive at the following simplified equations:
\begin{eqnarray}
i\partial_tC'_1=(1+g x_1/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(wx_1/c+(w_0(1+g x_1/c^2)-w)t]C'_{1w},\\
i\partial_tC'_{1w}=(1+g x_1/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(-wx_1/c+(w-w_0(1+g x_1/c^2))t]C'_{1},\\
i\partial_tC'_2=(1+g x_2/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(wx_2/c+(w_0(1+g x_2/c^2)-w)t]C'_{2w},\\
i\partial_tC'_{2w}=(1+g x_2/c^2)\sqrt{\kappa/2\pi}\int dw \exp[i(-wx_2/c+(w-w_0(1+g x_2/c^2))t]C'_{2}.
\end{eqnarray}
Eq.(15) and Eq.(17) are integrated formally and inserted into Eq.(14) and Eq.(16),respectively. Utilizing the integral
\begin{eqnarray}
\int dw\exp[i((w_0(1+g x_1/c^2)-w)t]=2\pi\delta(t),
\end{eqnarray}
we can analytically solve the set of partial differential equations.
At time $t$, using the initial values $C_1(0)=1/\sqrt{2}$ and $C_2(0)=1/\sqrt{2}$, we obtain
\begin{eqnarray}
C'_1(t)=1/\sqrt{2}\exp[-1/2\lambda_1^2\kappa t],\\
C'_{1w}=\frac{1-\exp[-1/2\lambda_1^2\kappa t-i(\lambda_1w_0-w)t]}{\lambda_1^2\kappa+i(\lambda_1w_0-w)}\sqrt{\kappa/2\pi}\lambda_1\exp[iwx_1/c],\\
C'_2(t)=1/\sqrt{2}\exp[-1/2\lambda_2^2\kappa t],\\
C'_{2w}=\frac{1-\exp[-1/2\lambda_2^2\kappa t-i(\lambda_2w_0-w)t]}{\lambda_2^2\kappa+i(\lambda_2w_0-w)}\sqrt{\kappa/2\pi}\lambda_2\exp[iwx_2/c],
\end{eqnarray}
where $\lambda_i=1+g x_i/c^2$ for $i=1,2$.
The quantum coherence of position state can be quantified by the interferometric visibility
\begin{eqnarray}
V(t)&=&2|C_1^*C_2+\int dwC^*_{1w}\int dw'C_{2w'}|\nonumber\\
&=&2|\exp[i(x_2-x_1)w_1g/c^2t]{C'_1}^*C'_2+\exp[i(x_2-x_1)w_2g/c^2t]\int dwC'^*_{1w}\int dw'C'_{2w'}|,
\end{eqnarray}
where the term $\int dwC'^*_{1w}\int dw'C'_{2w'}$ can be integrated by residue theorem. We can arrive at
\begin{eqnarray}
&\int dwC'^*_{1w}\int dw'C'_{2w'}=\frac{\kappa\lambda_1\lambda_2}{\kappa(\lambda_1^2+\lambda_2^2)+iw_0(\lambda_2-\lambda_1)}
\{\exp[-1/2\lambda_1^2\kappa\tau+ iw_0\lambda_1\tau]-\exp[-1/2\lambda_1^2\kappa t+i\lambda_1w_0t+i\xi(\tau-t)]-\nonumber\\
&\exp[-1/2\lambda_2^2\kappa t-i\lambda_2w_0t+i(w_0\lambda_1+i\lambda_1^2\kappa)(\tau+t)]+\exp[-1/2\lambda_1^2\kappa (t+\tau)-1/2\lambda_1^2\kappa t+i(\lambda_2-\lambda_1)w_0t+i\lambda_1w_0\tau]\},\\
&\textmd{in which},\nonumber\\
&\textmd{for}\ t<\tau, \ \xi=w_0\lambda_1+i\lambda_1^2\kappa,\ \ \textmd{for}\ t\geq\tau,\ \xi=w_0\lambda_2-i\lambda_2^2\kappa.
\end{eqnarray}
|
1511.00159
|
Department of Mathematics, Yazd University, 89195-741, Yazd, Iran
Let $G$ be a simple graph of order $n$.
The domination polynomial of $G$ is the polynomial
$D(G, x)=\sum_{i=1}^n d(G,i) x^i$,
where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. The $n$-barbell graph $Bar_n$ with $2n$ vertices, is formed by joining two copies of a complete graph $K_n$ by a single edge. We prove that for every $n\geq 2$, $Bar_n$ is not $\mathcal{D}$-unique, that is, there is another non-isomorphic graph with the same domination polynomial. More precisely, we show that for every $n$, the $\mathcal{D}$-equivalence class of barbell graph, $[Bar_n]$, contains many graphs, which one of them is the complement of book graph of order $n-1$, $B_{n-1}^c$. Also we present many families of graphs in $\mathcal{D}$-equivalence class of $K_{n_1}\cup K_{n_2}\cup \cdots\cup K_{n_k}$.
Keywords: Domination polynomial; $\mathcal{D}$-unique; Equivalence; Generalize barbell graphs.
AMS Subj. Class.: 05C60, 05C69
§ INTRODUCTION
All graphs in this paper are simple of finite orders, i.e., graphs are undirected with no loops or
parallel edges and with finite number of vertices. The complement $G^c$ of a graph $G$, is a graph with the same vertex set as $G$ and with the property that two vertices are adjacent in $G^c$ if and only if they are not adjacent in $G$.
For any vertex $v\in V(G)$, the open neighborhood of $v$ is the
set $N(v)=\{u \in V (G) | uv\in E(G)\}$ and the closed neighborhood of $v$
is the set $N[v]=N(v)\cup \{v\}$. For a set $S\subseteq V(G)$, the open
neighborhood of $S$ is $N(S)=\bigcup_{v\in S} N(v)$ and the closed neighborhood of $S$
is $N[S]=N(S)\cup S$. A set $S\subseteq V(G)$ is a dominating set if $N[S]=V$, or equivalently,
every vertex in $V(G)\backslash S$ is adjacent to at least one vertex in $S$.
The domination number $\gamma(G)$, is the minimum cardinality of a dominating set in $G$.
For a detailed treatment of domination theory, the reader is referred to <cit.>.
Let ${\cal D}(G,i)$ be the family of dominating sets of a graph $G$ with cardinality $i$ and
let $d(G,i)=|{\cal D}(G,i)|$.
The domination polynomial $D(G,x)$ of $G$ is defined as
${D(G,x)=\sum_{ i=\gamma(G)}^{|V(G)|} d(G,i) x^{i}}$ (see <cit.>). This polynomial is the generating polynomial for the number of dominating sets of each cardinality.
Calculating the domination polynomial of a graph $G$ is difficult in general, as the smallest power of a non-zero term is the domination number $\gamma (G)$ of the graph, and determining whether $\gamma (G) \leq k$ is known to be NP-complete <cit.>. But for certain classes of graphs, we can find a closed form expression for the domination polynomial. Two graphs $G$ and $H$ are said to be dominating equivalent,
or simply ${\cal D}$-equivalent, written $G\sim H$, if
$D(G,x)=D(H,x)$. It is evident that the relation $\sim$ of being
${\cal D}$-equivalence
is an equivalence relation on the family ${\cal G}$ of graphs, and thus ${\cal G}$ is partitioned into equivalence classes,
called the ${\cal D}$-equivalence classes. Given $G\in {\cal G}$, let
\[
[G]=\{H\in {\cal G}:H\sim G\}.
\]
We call $[G]$ the equivalence class determined by $G$.
A graph $G$ is said to be dominating unique, or simply
$\mathcal{D}$-unique, if $[G] = \{G\}$ <cit.>. Determining $\mathcal{D}$-equivalence class of graphs is
one of the interesting problems on equivalence classes.
A question of recent interest concerning this equivalence relation $[\cdot]$ asks
which graphs are determined by their domination polynomial. It is known
that cycles <cit.> and cubic graphs of order $10$ <cit.> (particularly, the Petersen
graph) are, while if $n\equiv 0 (mod\, 3)$, the paths of order $n$ are not <cit.>. In <cit.>, authors
completely described the complete $r$-partite graphs which are $\mathcal{D}$-unique. Their results in the
bipartite case, settles in the affirmative a conjecture in <cit.>.
Let $n$ be any positive integer and $Bar_n$ be $n$-barbell graph with $2n$ vertices which is formed by joining
two copies of a complete graph $K_n$ by a single edge.
In this paper, we consider $n$-barbell graphs and study their domination polynomials.
We prove that for every $n\geq 2$, $Bar_n$ is not $\mathcal{D}$-unique. More precisely, in Section 2, we show that for every $n$, $[Bar_n]$ contains many graphs, which one of them is $2K_n$ and another one is the complement of book graph of order $n-1$, $B_{n-1}^c$. In Section 3, we present many graphs in $[K_{n_1}\cup K_{n_2}\cup\cdots\cup K_{n_k}]$.
§ $\MATHCAL{D}$-EQUIVALENCE CLASSES OF SOME GRAPHS
In this section, we study the $\mathcal{D}$-equivalence classes of some graphs. First we consider the domination polynomial of the complement
of book graph.
The $n$-book graph $B_n$ can be constructed by bonding $n$ copies of the cycle graph $C_4$ along a common edge $\{u, v\}$, see Figure <ref>.
The book graph $B_n$.
The following theorem gives a formula for the domination polynomial of $B_n$.
For every $n \in \mathbb{N}$,
\[ D(B_n,x)=(x^2+2 x)^n(2x+1) + x^2(x+1)^{2n}- 2x^n.\]
Domination polynomials, exploring the nature and location of roots of domination polynomials of book graphs has studied in <cit.>. Here, we consider the domination polynomial of the complement of the book graphs. We shall prove that the $n$-barbell graph $Bar_n$ and $B_{n-1}^c$ have the same domination polynomial.
The Turán graph $T(n,r)$ is a complete multipartite graph formed by partitioning a set of $n$ vertices into $r$ subsets, with sizes as equal as possible, and connecting two vertices by an edge whenever they belong to different subsets. The graph will have $(n~ mod ~ r)$ subsets of size $\lceil\frac{n}{r}\rceil$, and $r - (n~ mod ~ r)$ subsets of size $\lfloor\frac{n}{r}\rfloor$. That is, a complete $r$-partite graph
\[ K_{\lceil\frac{n}{r}\rceil, \lceil\frac{n}{r}\rceil,\ldots,\lfloor\frac{n}{r}\rfloor,\lfloor\frac{n}{r}\rfloor}.\]
The Turán graph $T(2n,n)$ can be formed by removing a perfect matching, $n$ edges no two of which are adjacent, from a complete graph $K_{2n}$. As Roberts (1969) showed, this graph has boxicity exactly $n$; it is sometimes known as the Robert's graph <cit.>. If $n$ couples go to a party, and each person shakes hands with every person except his or her partner, then this graph describes the set of handshakes that take place; for this reason it is also called the cocktail party graph. So, the cocktail party graph $CP(t)$ of order $2t$ is the graph with vertices $b_1, b_2, \cdots, b_{2t}$
in which each pair of distinct vertices form an edge with the exception of the pairs
$\{b_1 , b_2 \}, \{b_3 , b_4\}, \ldots, \{b_{2t- 1}, b_{2t}\}$.
The following result is easy to obtain.
For every $n\in \mathbb{N}$, $D(CP(n),x) = (1+x)^{2n}-2nx-1$.
Figure <ref> shows the complement of the book graph $B_n^c$.
Complement of the book graph $B_n^c$.
The vertex contraction $G/u$ of a graph $G$ by a vertex $u$ is the operation under
which all vertices in $N(u)$ are joined to each other and then $u$ is deleted (see<cit.>).
The following theorem is useful for finding the recurrence relations for the domination polynomials of arbitrary graphs.
Let $G$ be a graph. For any vertex $u$ in $G$ we have
\[
D(G, x) = xD(G/u, x) + D(G - u, x) + xD(G - N[u], x) - (1 + x)p_u(G, x),
\]
where $p_u(G, x)$ is the polynomial counting the dominating sets of $G - u$ which do not contain any
vertex of $N(u)$ in $G$.
The following theorem gives a formula for the domination polynomial of the complement of the book graph.
For every $n\in \mathbb{N}$,
$$D(B_n^c,x) = ((1+x)^{n+1}-1)^2.$$
Consider graph $B_{n}^c$ and vertex $v$ in the Figure <ref>. By Theorem <ref>, we have:
\begin{eqnarray*}
D(B_n^c, x)&=& x D(B_n^c/v, x) + D(B_n^c - v, x) + x D(B_n^c - N[v], x) - (1 + x)p_v(B_n^c, x)\\
&=& (x +1) D(B_n^c-v, x) + xD(K_{n+1},x) - (1 + x)(D(K_{n+1},x)-(n+1)x-nx^2)\\
&=&(x +1) D(B_n^c-v, x) - D(K_{n+1},x)+x(1+x)(1+n(1+x))\\
&=&(x +1) D(B_n^c-v, x) - ((1+x)^{n+1}-1)+x(1+x)(1+n(1+x)),
\end{eqnarray*}
where $(B_n^c/v) \simeq B_n^c-v$.
Now, we use Theorem <ref> to obtain the domination polynomial of the graph $B_n^c-v$. We have
\begin{eqnarray*}
D(B_n^c-v, x)&=& x D(B_n^c-v/u, x) + D((B_n^c - v)-u, x) \\
&&+ x D((B_n^c-v) - N[u], x) - (1 + x)p_u(B_n^c-v, x).
\end{eqnarray*}
Since $(B_n^c-v/u) \simeq (B_n^c-v)-u\simeq CP(n)$ and using Lemma <ref>, we have
\begin{eqnarray*}
D(B_n^c-v, x)&=& (x +1) D(CP(n), x) + x(D(K_{n},x)) - (1 + x)(D(K_{n},x)-nx)\\
&=&(x +1) D(CP(n), x) - D(K_{n},x)+nx(1+x)\\
&=&(x +1)((1+ x)^{2n} -(1 + 2nx)) - ((1+x)^n-1)+nx(1+x)\\
\end{eqnarray*}
\begin{eqnarray*}
D(B_n^c, x)&=&(x +1) ((1+x)^n((1+x)^{n+1}-1)-nx(1+x)-x)\\
&& - ((1+x)^{n+1}-1)+x(1+x)(1+n(1+x))\\
\end{eqnarray*}
The $n$-barbell graph is the graph on $2n$ vertices which is formed by joining
two copies of a complete graph $K_n$ by a single edge, known as a bridge, shown in Figure <ref>. We denote this graph by $Bar_n$. For this graph, we shall calculate this domination polynomial. We need the following definition and theorems.
The barbell graph of order 16, $Bar_8$.
An irrelevant edge is an edge $e\in E(G)$, such that $D(G, x) = D(G-e, x)$, and a vertex $v \in V(G)$ is domination-covered,
if every dominating set of $G-v$ includes at least one vertex adjacent to $v$ in $G$ <cit.>.
We need the following theorems to obtain the domination polynomial of barbell graph $Bar_n$.
Let $G = (V,E)$ be a graph. An vertex $ v\in V$ of $G$ is domination-covered if and only if there is a vertex $u\in N[v]$ such that $N[u] \subseteq N[v]$.
Let $G = (V,E)$ be a graph. An edge $e = \{u, v\} \in E$ is an irrelevant edge
in $G$, if and only if $u$ and $v$ are domination-covered in $G-e$.
For every $n\geq 2$ and $n\in \mathbb{N}$,
$$D(Bar_n,x) = ((1+x)^{n}-1)^2.$$
Let $e$ be an edge joining two $K_n$ in barbell graph.
By Theorem <ref> two end vertices of edge $e$ are domination-covered in $Bar_n-e$. So, by Theorem <ref> the edge $e$ is an irrelevant edge of $Bar_n$. Therefore
$$D(Bar_n,x)=D(Bar_n-e,x)=D(K_n\cup K_n,x)=((1+x)^{n}-1)^2.$$
The following corollary is an immediate consequence of Theorems <ref> and <ref>.
For each natural number $n$, $Bar_n$ and $B_{n-1}^c$ have the same domination polynomial. More precisely, for every $n$, $[Bar_n]\supseteq \{Bar_n,B_{n-1}^c,K_n\cup K_n\}$, and $[B_{n-1}^c]\supseteq \{Bar_n,B_{n-1}^c,K_n\cup K_n\}$.
Here, we present some other families of graphs whose are in the $[Bar_n]$. Let to define the generalized barbell graphs. As we know, the $Bar_n$ is formed by joining two copies of a complete graph $K_n$ by a single edge. We like to join two copies with more edges as follows:
Suppose that $\{u_1,...,u_n\}$ and $\{v_1,...,v_n\}$ are the vertices of two copies of complete graph of order $n$, $K_n$ and $\mathcal{K}_n$.
The generalized barbell graph is denoted by $Bar_{n,t}$ and is a graph with $V(Bar_{n,t})=\{u_1,...,u_n\}\cup \{v_1,...,v_n\}$ and
$$E(Bar_{n,t})=E(K_n)\cup E(\mathcal{K}_n)\cup\big\{u_iv_j|1\leq i\leq n-1, 1\leq j\leq n-1\big\},$$
where $\Big|\big\{u_iv_j|1\leq i\leq n-1, 1\leq j\leq n-1\big\}\Big|=t$.
As examples see two non-isomorphic graphs $Bar_{3,2}$ in Figure <ref>. Notice that $B_{n-1}^c$ is one of the specific case of $B_{n,(n-1)(n-2)}$. The left graph in Figure <ref>, is $B_2^c$.
Two generalized barbell graphs $Bar_{3,2}$.
We have the following theorem.
For every $n\geq 3$ and $n\in \mathbb{N}$,
$$D(Bar_{n,t},x) = ((1+x)^{n}-1)^2.$$
We prove this Theorem by induction on $t$. Suppose that $t = 1$, Then by Theorem <ref>, the result holds. Assume that the result holds for $t= (n-1)^2-1$. Let $t=(n-1)^2$ and $e$ be the additional edge of $Bar_{n,t}$ to the $Bar_{n,t-1}$. By Theorem <ref> two end vertices of edge $e$ are domination-covered in $Bar_n-e$. So, by Theorem <ref> the edge $e$ is an irrelevant edge of $Bar_{n,t-1}$. Therefore by the induction hypothesis we have the result.
The following corollary is an immediate consequence of Theorems <ref> and <ref>.
For each natural number $n$ and $t\leq (n-1)^2$, $Bar_n$ and $Bar_{n,t}$ have the same domination polynomial.
The following example shows that, except for the generalized barbell graphs, there are other graphs in $\mathcal{D}$-equivalence classes of $Bar_n$.
All connected graphs in $[Bar_3]$ are the graphs $Bar_3,~Bar_{3,2},~Bar_{3,3},~Bar_{3,4}$ and two graphs in Figure <ref>.
Two graphs in $[Bar_3]$.
§ SOME GRAPHS IN $[K_{N_1}\CUP K_{N_2}\CUP\CDOTS\CUP K_{N_K}]$
We observed that, for each natural number $n$ and $t\leq (n-1)^2$, the domination polynomials of $Bar_n$ and $Bar_{n,t}$ is $((1+x)^n-1)^2$. In this section, we present graphs whose domination polynomials are $\prod_{i=1}^k((1+x)^{n_i}-1)$. For this purpose, we construct families of graphs from a path $P_k$ which we denote them by $S(G_1,G_2,...,G_k)$ in the following definition.
The graph $S(G_1,G_2,...,G_k)$ is a graph which obtain from a path $P_k$ with the vertices $\{v_1, v_2, \ldots, v_k\}$, by
substituting a graph $G_i$ of order $n_i\geq 3$, for every vertex $v_i$ of $P_k$, such that
* for $i=1,k$, the graphs $G_i$ have at least one vertex of degree $n_i-1$ and other $G_i$'s have at least two vertices of degree $n_i-1$, and
* in the graph $S(G_1,G_2,...,G_k)$, the end vertices of each edge $e_i$ in the path graph, $P_k$ are one vertex of degree $n_i-1$ in graphs $G_{i-1}$ and $G_i$.
We have the following result for graph $S(G_1,G_2,...,G_k)$.
For every natural number $k\geq 2$,
$$D(S(G_1,G_2,...,G_k),x) = D(G_1,x)D(G_2,x)\ldots D(G_k,x).$$
In particular if $G_i=K_{n_i}$ and $n_i\geq 3$, then
$$D(S(K_{n_1},K_{n_2},...,K_{n_k}),x) = \prod_{i=1}^k D(K_{n_i},x)=\prod_{i=1}^k((1+x)^{n_i}-1).$$
Let $e_i$ $(1\leq i\leq k)$ be the edge joining $G_{i-1}$ and $G_i$ in $S(G_1,G_2,...,G_k)$.
By Theorem <ref> two end vertices of edge $e_i$ are domination-covered in $S(G_1,G_2,...,G_k)-e_i$. So, by Theorem <ref> every edge $e_i$ is an irrelevant edge of $S(G_1,G_2,...,G_k)$. Therefore we have the result.
We shall generalize the graphs $S(G_1,G_2,...,G_k)$ in Definition <ref> such that this generalized graphs and $S(G_1,G_2,...,G_k)$ have the same
domination polynomial. Suppose that $GS_t(K_{n_1},K_{n_2},...,K_{n_k})$
be a family of graphs in the form of $S(K_{n_1},K_{n_2},...,K_{n_k})$ such that the complete graphs $K_{n_i}$ with $V(K_{n_i})=\{u_1,...,u_{n_i}\}$ and $K_{n_{i+1}}$ with $V(K_{n_{i+1}})= \{v_1,...,v_{n_{i+1}}\}$ are joined with $t$ following edges
\[\big\{u_iv_j|1\leq i\leq n_{i}-1, 1\leq j\leq n_{i+1}-1\big\}.\]
Similar to the proof of the Theorem <ref>, we have the following theorem:
For each natural number $t$, all graphs in the family of $GS_t(K_{n_1},K_{n_2},...,K_{n_k})$ have the same domination polynomial. More precisely, the domination polynomial of each $H$ in $GS_t(K_{n_1},K_{n_2},...,K_{n_k})$ is equal to $\prod_{i=1}^k((1+x)^{n_i}-1)$.
In this paper, we studied the $\mathcal{D}$-equivalence classes of barbell graphs $Bar_n$. We showed that, for each natural number $n$, $2K_n$, $Bar_n$, $Bar_{n,t}$ and the complement of the book graph of order $n-1$, $B_{n-1}^c$ have the same domination polynomial, i.e., $[Bar_n]=[Bar_{n,t}]=[B_{n-1}^c]=[K_n\cup K_n]$.
Example <ref>, implies that except for these kind of graphs, there are another graphs in this class. Therefore, exact characterization of graphs in $[Bar_n]$ remains as an open problem. Also we presented many families of graphs whose are in $[K_{n_1}\cup K_{n_2}\cup...\cup K_{n_k}]$, but similar to Example <ref>, there are another graphs in this class. So, exact characterization of graphs in $[K_{n_1}\cup K_{n_2}\cup...\cup K_{n_k}]$ remains as another open problem.
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|
1511.00494
|
LU TP 15-50
November 2015
$^{1}$ Department of Astronomy and Theoretical Physics, Lund University, 223-62 Lund, Sweden.
$^{2}$ Instituto de Física e Matemática, Universidade
Federal de Pelotas,
Caixa Postal 354, CEP 96010-900, Pelotas, RS, Brazil
The current uncertainty on the gluon density extracted from the global parton analysis is large in the kinematical range of small values of the Bjorken - $x$ variable and low values of the hard scale $Q^2$. An alternative to reduces this uncertainty is the analysis of the exclusive vector meson photoproduction in photon - hadron and hadron - hadron collisions. This process offers a unique opportunity to constrain the gluon density of the proton, since its cross section is proportional to the gluon density squared. In this paper we consider current parametrizations for the gluon distribution and estimate the exclusive vector meson photoproduction cross section at HERA and LHC using the leading logarithmic formalism. We perform a fit of the normalization of the $\gamma h$ cross section and the value of the hard scale for the process and demonstrate that the current LHCb experimental data are better described by models that assume a slow increasing of the gluon distribution at small - $x$ and low $Q^2$.
12.38.Bx, 13.60.Le
One of the basic ingredients to estimate the hadronic cross sections are the parton distributions functions (PDFs). Theoretically, at large energies the hadrons are dominated by gluons, with its behaviour at small-$x$ being determined by the QCD dynamics at high parton densities. Consequently, a precise determination of the gluon distribution is fundamental to probe a possible transition between the linear and non linear regimes of the QCD dynamics <cit.>. Experimentally, our understanding about the partonic structure of the proton has been significantly improved by the results obtained in $ep$ collisions at HERA, which have obtained very precise data in a broad range in photon virtualities $Q^2$ and Bjorken - $x$ values, imposing the tightest constraints on the existing PDFs (For recent reviews see, e.g. Refs. <cit.>). However, the behaviour of the gluon distribution at small- $x$ is still poorly known as can be observed in Fig. <ref>, where we compare the predictions for the gluon distribution obtained by different groups
<cit.> that perform the global analysis of the existing experimental data using the DGLAP evolution equations <cit.> in order to determine the parton distributions. Consequently, additional measurements are necessary to pin down the gluon distribution. An alternative is the analysis of the experimental results for the heavy quark production at forward rapidities in $pp$ collisions at the LHC energies. Recent results <cit.> have analysed the impact of the LHCb data in the determination of the gluon distribution. Another promising observable is the cross section for the diffractive production of vector mesons, which in the leading logarithmic approximation is proportional to the gluon density squared <cit.>. This process was analysed at HERA and is currently been studied in photon - induced interactions at hadronic colliders. In recent years a series of experimental results at RHIC <cit.>, Tevatron <cit.> and LHC <cit.> demonstrated that the study of photon - induced interactions in hadronic colliders is feasible and can be used to probe e.g. the nuclear gluon distribution <cit.>, the dynamics of the strong interactions <cit.>, the Odderon <cit.>, the mechanism of quarkonium production <cit.> and the photon flux of the proton <cit.>. It has stimulated the improvement of the theoretical description of these processes as well as the proposal of new forward detectors to be installed in the LHC <cit.>.
[width=0.45]gluon_pdf.eps [width=0.45]gluon_pdf_upsilon.eps
Comparison between the gluon distributions obtained by different groups <cit.> in the global analysis of the experimental data. We present the results for two different values of the hard scale: $Q = M_{J/\psi}/2$ (left panel) and $M_{\Upsilon}/2$ (right panel).
The basic idea in the photon-induced processes is that
a ultra relativistic charged hadron (proton or nuclei)
give rise to strong electromagnetic fields, such that the photon stemming from the electromagnetic field
of one of the two colliding hadrons can interact with one photon of
the other hadron (photon - photon process) or can interact directly with the other hadron (photon - hadron
process) <cit.>. In photon-hadron processes the total cross section can be factorized in
terms of the equivalent flux of photons into the hadron projectile and the photon-hadron production cross section, with the corresponding rapidity distribution being a direct probe of the photon - hadron cross section for a given energy. In the particular case of vector meson photoproduction in $pp$ collisions, the experimental data for a given rapidity $y$ gives access to the behaviour of the gluon distribution of the proton for $x = M_V/\sqrt{s} \exp(-y)$, where $M_V$ is the mass of the vector meson and $\sqrt{s}$ is the center - of - mass energy of the hadron - hadron collision.
This property was the main motivation for the proposition presented in Ref. <cit.>, which was improved by several authors in the last fourteen years <cit.>. Our goal in this paper is to complement these previous studies by performing a phenomenological analysis of the exclusive vector meson photoproduction which can illuminate several aspects of the formalism and about the current parametrizations for the gluon distribution obtained in the global parton analysis. Basically, we consider different models for $xg$ and estimate the cross section for the photoproduction of vector mesons in $\gamma p$ interactions using the leading logarithmic formalism <cit.>. We demonstrate that assuming the usual value of the factorization scale $\bar{Q} = M_V/2$, the HERA and LHCb data for the $J/\Psi$ photoproduction are not described by these models. Taking into account that this formalism have been derived at leading order, which implies a uncertainty in the choice of $\bar{Q}$, we determine its value by fitting the $\gamma h$ data. Using these best fit parameters we estimate the rapidity distributions for the $J/\Psi$, $\Psi(2S)$ and $\Upsilon$ production in $pp$ collisions at the LHC and compare with the recent LHCb data <cit.>. Our results indicate that the current experimental data for the exclusive vector meson production in $pp$ collisions can only be described by models which predict a slow increasing of the gluon distribution at small - $x$.
Exclusive vector meson photoproduction in $pp$ collisions.
Lets initially present a brief review of the formalism for the calculation of the vector meson production in $pp$ collisions. The process is represented in Fig. <ref>, with its rapidity distribution being given by
\begin{equation} \label{eq:dsdy}
\frac{d\sigma}{dy}(h_1 h_2 \rightarrow h_1 \otimes V \otimes h_2) = \mathcal{S}^2(W_+)\left[\omega_+\frac{dN_{\gamma/h_1}(\omega_+)}{d\omega_+}\right]\sigma_{\gamma h_2\rightarrow V h_2}(y) + \mathcal{S}^2(W_-)\left[\omega_-\frac{dN_{\gamma/h_2}(\omega_-)}{d\omega_-}\right]\sigma_{\gamma h_1 \rightarrow V h_1}(-y)
\end{equation}
where we have taken into account that the two incident protons can be the source of the photon and $\otimes$ characterizes the presence of a rapidity gap in the final state. The photon - hadron center of mass energies squared $W^2_\pm$ and the photon energies $\omega_\pm$ are given by
\begin{equation}
W^2_\pm = e^{\pm |y|}\sqrt{s}M_V,\quad \omega_\pm = \frac{M_V}{2}e^{\pm |y|},
\end{equation}
where $\sqrt{s_{h_1h_2}}$ is the hadron - hadron center - of - mass energy, $M_V$ is the mass of the vector mass and $y$ its rapidity.
Moreover, the factors $\mathcal{S}^2(W_\pm)$ characterizes absorptive corrections which can destroy the rapidity gaps generated in exclusive processes <cit.>. The elastic photon flux for the proton can be expressed by <cit.>
\begin{eqnarray}
\frac{dN_{\gamma/p}(\omega)}{d\omega} = \frac{\alpha_{\mathrm{em}}}{2 \pi\, \omega} \left[ 1 + \left(1 -
\frac{2\,\omega}{\sqrt{s_{NN}}}\right)^2 \right]
\left( \ln{\Omega} - \frac{11}{6} + \frac{3}{\Omega} - \frac{3}{2 \,\Omega^2} + \frac{1}{3 \,\Omega^3} \right) \,,
\label{eq:photon_spectrum}
\end{eqnarray}
with the notation $\Omega = 1 + [\,(0.71 \,\mathrm{GeV}^2)/Q_{\mathrm{min}}^2\,]$ and
$Q_{\mathrm{min}}^2= \omega^2/[\,\gamma_L^2 \,(1-2\,\omega /\sqrt{s_{NN}})\,] \approx
\gamma_L)^2$. The main input for the calculation of the rapidity distribution [ Eq. (<ref>)] is the photon - hadron cross section for the vector meson production $\sigma_{\gamma h \rightarrow V h}$.
In the leading logarithmic approximation, the cross section for the vector meson production off any hadronic target, including a nucleus, at small-$x$ and for a sufficiently hard scale, is proportional to the square of the gluon parton density of the target.
To lowest order, the $\gamma h \rightarrow V h$ $(h = p, \,A)$
amplitude can be factorized into the product of the $\gamma
\rightarrow q \overline{q}$ transition $(q = c, \,b)$, the scattering of the
$q\overline{q}$ system on the target via (colorless) two-gluon
exchange, and finally the formation of the quarkonium from the
outgoing $q\overline{q}$ pair. The heavy meson mass $M_{V}$
ensures that perturbative QCD can be applied to photoproduction.
The calculation was
performed some years ago to leading logarithmic ($\log(\overline{Q}^2)$) approximation,
assuming the produced vector meson quarkonium system to be
nonrelativistic <cit.> and improved in distinct aspects
Assuming a non-relativistic wave function for the vector meson one have that the $t=0$
differential cross section of photoproduction of heavy vector
mesons in leading order approximation is given by <cit.>
\begin{eqnarray}
\left.\frac{d\sigma^{\gamma h \rightarrow Vh}}{dt}\right|_{t=0} = {\cal{N}} \frac{\pi^3 \Gamma_{e^+ e^-}M_V^3}{48 \alpha_\mathrm{em}}\left[ \frac{\alpha_s(\bar{Q}^2)}{\bar{Q}^4} xg_h(x, \bar{Q}^2) \right]^2
\,\,, \label{sigela}
\end{eqnarray}
where $xg_h$ is the target gluon distribution and $x =
4\overline{Q}^2/W^2$, with $W$ the $\gamma h$ center of mass energy,
$\overline{Q}^2$ the characteristic hard scale of the processes and ${\cal{N}} = 1$ at leading order (see below). Moreover, $\Gamma_{ee}$ is the
leptonic decay width of the vector meson. In Refs. <cit.> the authors have estimated the
relativistic corrections [${\cal{O}}(4\%)$] , the real part contribution of the production amplitude [${\cal{O}}(15\%)$], the skewness effect of off-diagonal partons [${\cal{O}}(20\%)$] and next-to-leading order corrections [${\cal{O}}(40\%)$]
to the LO exclusive heavy vector meson production, given by Eq. (<ref>).
It is important to emphasize that magnitude of these corrections is still a subject of discussions <cit.>. In particular, the value of the hard scale $\bar{Q}$ is not fixed reliably at leading order. Such corrections have direct impact in the normalization of the cross section and in its energy dependence. In what follows we take into account of the contributions associated to real part of the scattering amplitude and to the skewness effect, which is equivalent to multiply the Eq. (<ref>) by a factor $(1 + \beta) R_g^2$, where
\begin{equation}
\beta = \tan \frac{\pi \lambda}{2},\quad
R_g = \frac{2^{2\lambda + 3}}{\sqrt{\pi}} \frac{\Gamma(\lambda + 5/2)}{\Gamma(\lambda + 4)}
\end{equation}
\begin{equation}
\lambda = \frac{\partial \ln[xg(x,\bar{Q}^2)]}{\partial \ln 1/x}\,\,.
\end{equation}
In order to estimate the total cross section we assume an exponential parametrization for the small $|t|$ behaviour of the amplitude, which implies
\begin{equation}
\sigma_{\gamma h \rightarrow Vh} = \frac{1}{b_V} \left.\frac{d\sigma^{\gamma h \rightarrow Vh}}{dt}\right|_{t=0}
\end{equation}
with $b_V$ being given by the following parametrization <cit.>
\[
b_V(W) = 4.9 + 0.24 \ln (W/\unit[90]{GeV}),
\]
which is compatible with the HERA data. Moreover, we consider the values of mass and electronic decay widths of the vector mesons as given in Ref. <cit.>.
Finally, in our calculations of the rapidity distribution we will assume that the absorptive corrections $\mathcal{S}^2(W_\pm)$ are given by the gap survival probability computed in <cit.> using the model proposed in Ref. <cit.>.
Energy dependence of the $J/\Psi$ photoproduction cross section obtained assuming that $\bar{Q} = M_{J/\Psi}/2$. Data from Refs.
In Fig. <ref> we present the predictions for the gluon distribution obtained at leading order by the different groups <cit.> that perform the global parton analysis of the experimental data to extract the parton distributions. We show the results for two different values for the hard scale: $Q = M_{J/\Psi}/2$ (left panel) and $Q = M_{\Upsilon}/2$ (right panel). These values are usually assumed as being $\bar{Q}$ in the calculations of the exclusive $J/\Psi$ and $\Upsilon$ photoproduction cross sections, respectively. We have that the distinct predictions differ significantly, mainly at small - $x$ and low $Q$, which demonstrate that the global analysis do not reliably determine the gluon in this kinematical range. Basically, these distinct behaviours of the gluon distribution for low values of the hard scale are directly connected with the different assumptions for the initial conditions of the DGLAP equations considered by the distinct groups. With the increasing of the hard scale, the predictions becomes less dependent of these assumptions, and the distinct predictions becomes similar at very large $Q$. These results demonstrate the importance of probes of the gluon distribution at small - $x$ and low $Q$.
In Fig. <ref> we present the resulting predictions for the energy dependence of the exclusive $J/\Psi$ photoproduction cross section obtained from Eq. (<ref>) assuming $\bar{Q} = M_{J/\Psi}/2$. The experimental data from Refs.
<cit.> are presented for comparison. We obtain that the different models for the gluon distribution are not able to describe the normalization and/or the energy dependence of the data. The differences present in Fig. <ref> are amplified in the
exclusive $J/\Psi$ photoproduction cross section due to its quadratic dependence on $xg$. In particular, the distinct $x$ - behaviour predicted by the NNPDF and CT10 parametrizations for
$Q = M_{J/\Psi}/2$, with $xg$ decreasing in the range $10^{-4} \le x \le 10^{-2}$ before to increase for $x < 10^{-4}$, implies the anomalous behaviour in $\sigma_{\gamma p \rightarrow J/\Psi p}$ observed in Fig. <ref>.
A possible interpretation of the results presented in Fig. <ref> is that the different models fail to describe the data since our predictions for
$\sigma_{\gamma p \rightarrow J/\Psi p}$ have been obtained at leading order, which can be strongly affected by theoretical uncertainties associated to higher order corrections <cit.>. It is generally believed that these higher order corrections should be important, but a full calculation still remains a challenge. In order to analyse the possible impacts of the higher order corrections in our phenomenological analysis, in what follows we will assume that the general behaviour of the cross section, Eq. (<ref>), will remain unchanged after the inclusion of the higher order corrections and that they can be effectively incorporated by taking the values of the normalization and the hard scale from a fit to the $\gamma h$ data. In other words, we will assume that the main effect of the higher order corrections will be the modification of the normalization of the cross section and that they can be taken into account by an appropriate choice of the hard scale (For a similar approach see <cit.>). Basically we will assume $\bar{Q} = \xi M_V/2$ and will take $\xi$ and ${\cal{N}}$ in Eq. (<ref>) as free parameters to be determined by fitting the experimental data for the $\sigma_{\gamma p \rightarrow V p}$ cross section. We consider
the experimental data from Fermilab, HERA and LHC for the $J/\Psi$, $\Psi(2S)$ and $\Upsilon$ production and performed the minimization of $\chi^2/\mathrm{d.o.f.}$ in order to determine $\xi$ and ${\cal{N}}$. These new parameters are then used to calculate the rapidity distribution for the vector meson photoproduction in $pp$ collisions.
In Table <ref> we present the fitted parameters for the different processes and a comparison between the predictions and the HERA data are shown in the left panel of the Figs. <ref>, <ref> and <ref>. In general, the results gives reasonable values for the $\chi^2/\mathrm{d.o.f.}$, inside the 98% confidence level. We obtain smaller values of $\chi^2/\mathrm{d.o.f.}$ for heavier states, mainly due the small number of experimental data. In contrast, for the $J/\psi$ data, a small $\chi^2/\mathrm{d.o.f.}$ is only obtained using CT10 and NNPDF parametrizations, which describe the data in the full energy range, with the other parametrizations being able to describe the data in a restrict range. Moreover, as can be verified from the analysis of the Figs. <ref>, <ref> and <ref>, the predictions of the distinct parametrizations differ significantly in the kinematical range beyond the HERA data. It is important to emphasize that the results for $\xi$ and ${\cal{N}}$ obtained using the CT10 and NNPDF parametrizations indicate that
in order to describe the data for $J/\Psi$ and $\Psi(2S)$ production a large amount of higher order corrections are necessary. In contrast, these corrections are small for the $\Upsilon$ case.
In the right panel of the Figs. <ref>, <ref> and <ref> we present the corresponding predictions for the rapidity distributions for the vector meson photoproduction in $pp$ collisions at $\sqrt{s} = 7$ TeV. We compare our predictions with the recent data from the LHCb Collaboration <cit.>. As the rapidity distributions probe a large range of $\gamma p$ center of mass energies, the differences present in the predictions for
$\sigma_{\gamma p \rightarrow V p}$ are amplified in $d\sigma/dy$, particularly for large rapidities in the case of the lighter mesons. For $J/\Psi$ and $\Psi(2S)$ production the models characterized by a strong increasing of the gluon distribution at small - $x$ and small hard scales (Alekhin, MMHT and GJR) predict a double peak structure in the distributions. In contrast, the CT10 and NNPDF parametrizations predict a flat $y$ distribution in a large rapidity range. These parametrizations describe quite well the LHCb data for the $J/\Psi$ production and overestimate the $\Psi(2S)$ data.
It is important to emphasize that we have obtained these predictions using the model for
$\mathcal{S}^2(W_\pm)$ proposed in <cit.>. Another aspect is the description of the
$\Psi(2S)$ wave function used in our calculations, which is currently a subject of intense debate.
In the case of the $\Upsilon$ production, we obtain that the different models are not able to describe the current LHCb data, with the predictions at central rapidities being largely distinct. These results can indicate e.g. smaller values for $\mathcal{S}^2(W_\pm)$ than those proposed in <cit.>. Certainly, the future CMS data for $y \approx 0$, which are currently under analysis, will be important to get more definitive conclusions.
3c|$J/\psi$ 3c|$\Psi(2S)$ 3c$\Upsilon$
Parametrization $\xi$ $\mathcal{N}$ $\chi^2/\mathrm{d.o.f.}$ $\xi$ $\mathcal{N}$ $\chi^2/\mathrm{d.o.f.}$ $\xi$ $\mathcal{N}$ $\chi^2/\mathrm{d.o.f.}$
Alekhin02 0.879 0.180 3.818 0.816 0.133 1.520 8.488$\cdot 10^{-2}$ 1.087$\cdot 10^{-4}$ 0.624
CT10 3.412 45.041 1.179 3.783 59.331 1.682 0.614 0.188 0.312
NNPDF 4.373 139.670 1.215 5.064 208.837 1.871 0.939 1.063 0.312
MMHT14 1.035 0.297 7.226 0.641 0.101 1.220 0.135 3.690 1.820
GJR08 2.202 0.755 11.740 3.436 7.799 8.824 14.257 1.428$\cdot 10^{3}$ 3.707
Values of the free parameters $\xi$ and $\mathcal{N}$ for the different gluon parametrizations obtained by the minimization of $\chi^2/\mathrm{d.o.f.}$ for the distinct processes. The $\chi^2/\mathrm{d.o.f.}$ values are show for comparison.
Left panel: Energy dependence of the exclusive $J/\psi$ photoproduction cross section obtained using the parameters described in the Table <ref>. Right panel: Predictions for the rapidity distribution for the exclusive $J/\psi$ photoproduction in $pp$ collisions at $\sqrt{s} = 7$ TeV. Data from Refs.
Left panel: Energy dependence of the exclusive $\Psi(2S)$ photoproduction cross section obtained using the parameters described in the Table <ref>. Right panel: Predictions for the rapidity distribution for the exclusive $\Psi(2S)$ photoproduction in $pp$ collisions at $\sqrt{s} = 7$ TeV. Data from Refs.
Left panel: Energy dependence of the exclusive $\Upsilon$ photoproduction cross section obtained using the parameters described in the Table <ref>. Right panel: Predictions for the rapidity distribution for the exclusive $\Upsilon$ photoproduction in $pp$ collisions at $\sqrt{s} = 7$ TeV. Data from Refs.
A comment is in order. In our study we have considered leading order gluon distributions in order to be theoretically consistent with the fact that our expression for the total cross section have been derived at the leading logarithmic approximation. However, we have verified that our main conclusions are not modified if next - to - leading - order (NLO) gluon distributions are used as input in the calculations. Basically, the NLO corrections in the global analysis implies that the associated gluon distributions are not so steep at small - $x$ and low values of the hard scale. As a consequence, a better description of the HERA data for the vector meson production is feasible. For example, in the case of the GJR parametrization, the value of $xg$ for $x = 10^{-4}$ and $\mu = M_{J/\Psi}/2$ decreases by a factor 2. The resulting values for the $\chi^2/\mathrm{d.o.f.}$ for the description of the HERA data becomes
1.5 / 2.2 / 1.9 for the $J/\Psi / \Psi(2S) / \Upsilon$ production, respectively. However, these values still are larger than those obtained using the LO and NLO CT10 parametrizations. Moreover, as the difference between the GJR and CT10 gluon distributions increases at smaller values of $x$, probed at larger values of $|y|$, the resulting GJR (NLO) predictions for the rapidity distributions are not able to describe the LHC data. Similar results are obtained using the NLO gluon distributions derived in Refs. <cit.>. In the case of the CT10 and NNPDF parametrizations, we have that the impact of the NLO corrections is small, with the resulting gluon distributions being similar to those obtained at leading order. As a consequence, the associated predictions for the rapidity distributions are similar to those presented in the Figs. <ref>, <ref> and <ref>. These results indicate that the HERA and LHC data are better described when the gluon distribution present a slow increasing at small - $x$ and low $Q$, in agreement with conclusion obtained at leading order.
Finally, lets summarize our main conclusions. Exclusive vector meson photoproduction offers a unique opportunity to constrain the gluon density of the proton in the kinematical range of small - $x$ and low $Q^2$, which is the range in which the global analysis do not reliable determine the gluon distribution. In this paper we have performed a phenomenological study of this process using the leading logarithmic formalism and different models for $xg$ predicted by distinct groups that perform the global analysis of the experimental data in order to obtain the PDFs. We demonstrated that the combination of anyone of these models and the leading logarithmic formalism is not able to describe the experimental data for the $J/\Psi$ photoproduction. Assuming that the main modifications in the formalism due to higher order corrections are the change in the normalization of the cross section and in the value of the hard scale probed in the process, we have fitted the current experimental data for the $J/\Psi$, $\Psi(2S)$ and $\Upsilon$ photoproduction and used this new set parameters as input to calculate the rapidity distributions for the vector meson photoproduction in $pp$ collisions. We have demonstrated that a better description of the $\gamma h$ data is obtained when the gluon distribution presents a slow increasing at small - $x$ and low $Q$. Moreover, our results indicated that the LHCb data also are better described by these models. However, results for $\Psi(2S)$ and $\Upsilon$ shown that the normalizations are not perfectly described, which can indicated the a more detailed analysis about the magnitude of the factor $\mathcal{S}^2(W_\pm)$ for the different final states is important in future. Future experimental data, particularly at central rapidities, will be important to constrain the different models.
A final comment is in order. In this paper, in order to estimate the exclusive photoproduction of vector mesons, we have assumed the leading logarithmic formalism and different solutions of the linear DGLAP equation. Our goal was to try to describe the current experimental data for the exclusive vector photoproduction by improving this formalism and assuming that non linear effects in the QCD dynamics can be disregarded in this process. Our results indicate that it is not an easy task. Certainly, more definitive conclusions could be obtained using the full NLO expression for the cross section, which still is in progress. However, it is important to emphasize that the vector meson photoproduction at HERA and LHC is quite well described using the color dipole formalism, taking into account the non linear effects in the QCD dynamics and assuming $\mathcal{S}^2(W_\pm) = 1$ (See e.g. Refs <cit.>). Such aspects demonstrated that future experimental results for the exclusive vector meson photoproduction in hadronic colliders are fundamental to understanding the QCD dynamics at high energies.
This work was partially financed by the Brazilian funding
agencies CNPq, CAPES and FAPERGS.
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|
1511.00240
|
[footnoteinfo]The authors gratefully acknowledge the financial support form the Fonds
National de la Recherche, Luxembourg (8864515)
Johan]Johan Thunbergjohan.thunberg@uni.lu,
Johan]Jorge Goncalvesjorge.goncalves@uni.lu,
Hu]Xiaoming Huhu@math.kth.se
[Johan]Universite du Luxembourg
Campus Belval
7, avenue des Hauts-Fourneaux
L-4362 Esch-sur-Alzette
[Hu]KTH Royal Institute of Technology, S-100 44 Stockholm, Sweden
Attitude synchronization, formation control, multi-agent systems, networked robotics.
This paper addresses the consensus problem and the formation problem on $SE(3)$ in multi-agent systems
with directed and switching interconnection topologies.
Several control laws are introduced for the consensus problem. By a simple transformation, it is shown that the proposed control laws can be used
for the formation problem. The design is first conducted on the kinematic level, where the velocities are the control laws. Then, for rigid bodies in space,
the design is conducted on the dynamic level, where the torques and the forces are the control laws. On the kinematic level, first two
control laws are introduced that explicitly use
Euclidean transformations, then
separate control laws are defined for the rotations and the translations.
In the special case of purely rotational motion, the consensus problem
is referred to as consensus on $SO(3)$ or attitude synchronization. In this problem,
for a broad class of
local representations or parameterizations of $SO(3)$, including the Axis-Angle Representation, the
Rodrigues Parameters and the Modified Rodrigues Parameters, two types of control laws
are presented that look structurally the same for any choice of local representation. For these two control laws we provide conditions on the initial rotations and the connectivity of the graph such that the system reaches consensus on $SO(3)$. Among the contributions of this
paper, there are conditions for when exponential rate of convergence occur. A theorem is provided showing that for any choice of local representation for the rotations, there is a change of coordinates such that the transformed system has a well known structure.
§ INTRODUCTION
This work addresses the problem of continuous time consensus and formation control on $SE(3)$ for switching interconnection topologies.
We start by designing control laws
on a kinematic level, where the velocities are
control signals and then continue to design
control laws on the dynamic level for rigid bodies in space,
where the forces and
the torques are the control laws.
The main
focus in this paper is on the kinematic control laws.
This approach is not justified from a physical perspective. Nevertheless, there are reasons why this
path is still reasonable to take.
Firstly, dynamics are often platform dependent, and
especially in the robotics community it is desired
to specify control laws on the kinematic level. Secondly,
a deeper understanding of how the geometry of $SE(3)$ (and in particular $SO(3)$) affects the
control design can be acquired by designing the control
laws on a kinematic level, since we are then
working directly in the tangent space of $SE(3)$ or
On $SE(3)$, consensus control is a special case of formation control, but actually formation control
can be seen as a special case of consensus control,
a fact that will be used in this paper. The approach
is to develop consensus control laws,
which after a simple transformation can be used as
formation control laws.
By taking this approach we can use existing theory
for consensus in order to provide convergence results
for the formation problem.
The consensus control problem on $SO(3)$ comprises a subset
of the consensus control problem on $SE(3)$, but it
is, from many perspectives, the most challenging part of the control design. Hence, most emphasis will be taken towards this problem.
Whereas the translations are elements of $\mathbb{R}^3$, the rotations are elements of the compact manifold $SO(3)$, the group of orthogonal matrices in $\mathbb{R}^{3\times3}$ with determinant equal to $1$.
There is a wide range of applications for the proposed
control laws, e.g., satellites or spacecraft that shall reach a certain formation, multiple robotic arms that shall hold a rigid object or
cameras that shall look in some desired
directions (or the same direction in case of consensus).
For rigid bodies in space, i.e., spacecraft or satellites,
there has recently been an extensive research on the consensus on $SO(3)$ problem <cit.>. In that problem, the goal is
to design a control torque such that the rotations
of the rigid bodies become synchronized or reach consensus.
There are also adjacent problems, such as the problem where a group of spacecraft shall follow a leader while synchronizing the rotations between each other <cit.>. In the recent work by
D. Lee et al. <cit.>, a dynamic level control
scheme is presented for spacecraft formation flying with collision
In this work we propose six kinematic control laws.
The first two are constructed as elements of $se(3)$;
they are linear functions of
the transformations or the relative transformations
of neighboring agents. The third and fourth are
defined for the rotations only. They are constructed for
the tangent space of $SO(3)$ using the angular velocity. Finally, the fifth and the sixth control laws are
defined for the translations only. They are constructed in the
tangent space of $\mathbb{R}^3$.
All the control laws lead to
consensus (or equivalently formation) under different assumptions on the graphs,
the initial conditions and measurable entities.
The results for consensus on $SO(3)$ expands
on the publications <cit.>, by considering
a larger class of local representations.
Moreover, Proposition <ref>
provides the result that for certain topologies
and all the considered local representations, the rate of convergence is exponential.
An interesting geometric insight is provided in
Theorem <ref> where it is shown that for any
of the local representations considered, if the
second rotation control law is used and the rotations
initially are contained inside the injectivity region, there is a change of coordinates so that
the system has a well known-structure.
Towards the end of this paper we also consider the second order dynamics and
torque control laws for rigid bodies in space. We use methods similar to backstepping in order to generalize the kinematic control laws to this scenario. This generalization is only performed
for the case of time-invariant
The paper proceeds as follows. In Section <ref>,
preliminary concepts are defined such as Euclidean transformations,
rotations, translations, network topologies and switching signal functions. The concept of local representations for the rotations is also
are introduced. In Section <ref>, the problem formulation
is given. Section <ref>
introduces the six kinematic control laws, which are categorized into
two groups. Convergence results for the first group of control laws
are provided
in Section <ref>, whereas the second group is treated in Section <ref>. In Section <ref> – for the
application of rigid bodies in space – we
provide results for control laws on the dynamic level.
§ PRELIMINARIES
§.§ Euclidean transformations, rotations, and translations
We consider a system of $n$ agents with states in $SE(3)$, the group of Euclidean
transformations. This means that each agent $i$ has a
$$G_i(t) = \begin{bmatrix}
R_i(t) & T_i(t) \\
0 & 1
\end{bmatrix} \in SE(3)$$
at each time $t \geq t_0$.
The matrix $R_i(t)$ is an element of $SO(3)$, the matrix group which is
defined by
$$SO(3) = \{R \in \mathbb{R}^{3 \times 3}: R^TR = I, \text{det}(R) = 1\}.$$
The vector $T_i(t)$ is an element in $\mathbb{R}^3$.
Each agent has a corresponding rigid body.
We denote the world coordinate frame by $\mathcal{F}_W$ and the instantaneous body coordinate
frame of the rigid body of each agent $i$ by $\mathcal{F}_i$. Let $R_i(t) \in SO(3)$
be the
rotation of $\mathcal{F}_i$ in the world frame $\mathcal{F}_W$ at
time $t$ and let $R_{ij}(t) \in SO(3)$ be the rotation of
$\mathcal{F}_j$ in the frame $\mathcal{F}_i$, i.e.,
$$R_{ij}(t) = R_i^T(t)R_{j}(t).$$
We refer to $R_i(t)$ as
absolute rotation and $R_{ij}(t)$ as relative rotation.
The vector $T_i(t)$ is the position of agent $i$ in $\mathcal{F}_W$
at time $t$. The relative positions between agent $i$ and agent $j$
in the frame $\mathcal{F}_i$ at time $t$ is
$$T_{ij}(t) = R_i^T(t)(T_j(t) - T_i(t)),$$
which in general is different from $T_j(t) - T_i(t)$, the relative
positions between agent $i$ and agent $j$ in the world frame.
In the same way as for the rotations, we refer to $T_i(t)$ as
absolute translation and $T_{ij}(t)$ as relative translation.
The relative Euclidean transformation
\begin{align*}
& G_{ij}(t) = G_i^{-1}(t)G_j(t) \\
& = \begin{bmatrix}
R_i^T(t)R_j(t) & R_i^T(t)(T_j(t) - T_i(t)) \\
0 & 1
\end{bmatrix}
\end{align*}
contains both the relative rotation and the
relative translation.
From now on, in general we suppress the explicit time-dependence
for the variables, i.e., $G_i$ should be
interpreted as $G_i(t)$.
§.§ Local representations for the rotations
For a vector $p = [p_1, p_2, p_3]^T$ in $\mathbb{R}^3$ we define
$\widehat{{p}} = p^{\wedge}$ by
\begin{equation}\label{eq:hatmap}
\widehat{{p}} = p^{\wedge} =
\begin{bmatrix}
0 & -p_3 & p_2\\
p_3 & 0 &-p_1\\
-p_2 &p_1 & 0
\end{bmatrix}.
\end{equation}
We also define $(\cdot)^{\vee}$ as the inverse of $(\cdot)^{\wedge}$, i.e.,
$(p^{\wedge})^{\vee} = p$.
We consider local representations or parameterizations of $SO(3)$.
Often we simply refer to them as representations or parameterizations.
In this context, what is meant by a local
representation is a diffeomorphism $f: B_r(I) \rightarrow B_{r', 3}(0) \subset \mathbb{R}^3$,
where $B_r(I)$ is an open geodesic ball around the identity matrix in $SO(3)$ of
radius $r$ less than or equal to $\pi$, and $B_{r', 3}(0)$ is an open ball around the point $0$ in $\mathbb{R}^3$ with
radius $r'$. $\bar{B}_{r}(I)$ and $\bar{B}_{r',3}(0)$ are the closures
of said balls. If we write $B_{r,3}$ or $B_r$, this is short
hand notation for $B_{r,3}(0)$ or $B_r(I)$ respectively. The same goes for the closed balls. The local representations can be seen as coordinates
in a chart covering an open ball around the identity matrix in $SO(3)$.
A set in $SO(3)$ is convex if any geodesic shortest path segment between any two points in the set
is contained in the set. The set is strongly convex if there is a unique geodesic shortest path segment contained in the set <cit.>.
If $r = \pi$, $B_r(I)$
comprises almost all of $SO(3)$ (in terms of measure), and $B_r(I)$ is convex if and only if
$r \leq \pi/2$. The radius $r$ is referred to as the radius of injectivity. The parameterizations that we use have the following
special structure
\begin{equation}
f(R) = g(\theta)u,
\end{equation}
where $\theta$ is the geodesic distance between $I$ and
$R$ on $SO(3)$, also referred to as the Riemannian distance,
written as $d(I,R)$. The variable $u \in \mathbb{S}^2$ is the rotational axis
of $R$, and $g: (-r, r) \rightarrow \mathbb{R}$ is an odd, analytic and
strictly increasing function such that $f$ is a diffeomorphism.
On $B_{\pi}(I)$ the vector $u$ and the positive variable $\theta$ are obtained
as functions of $R$ in the following way
$$\theta = \cos^{-1}\bigg ( \frac{\text{trace}(R) -1}{2}\bigg ), \quad u = \frac{1}{2\sin(\theta)}
\begin{bmatrix}
r_{32} - r_{23} \\
r_{13} - r_{31} \\
r_{21} - r_{12}
\end{bmatrix},$$
where $R = [r_{ij}]$.
Let us denote $y_i = f(R_i)$ and $y_{ij} = f(R_{ij}).$
It holds that
{y}_{ij} = -{y}_{ji}, \text{ but in general } {y}_{j} - {y}_{i} \neq {y}_{ij}.$
For each representation, i.e., choice of $g$, $r \leq \pi$
is the largest radius such that $f$ is a diffeomorphism.
The radius $r$ is the radius of injectivity and depends on the representation, but we suppress this explicit dependence and
throughout this paper,
$r$ corresponds to the representation at hand, i.e., the one we have chosen to consider at the moment. For the representation at hand we also define
$$r' = \sup_{s \uparrow r}g(s).$$
Some common representations are:
* The Axis-Angle Representation, in which case $g(\theta) = \theta$ and
$r = r' =\pi$. This representation is almost global. The set $SO(3)\backslash B_{\pi}(I)$ has measure zero in $SO(3)$.
The Axis-Angle Representation is obtained from the
logarithmic map by
\begin{align*}
{{x}}_i& = (\text{Log}(R_i))^{\vee}, \\
{{x}}_{ij}& = (\text{Log}(R_i^TR_j))^{\vee}.
\end{align*}
In the other direction, a rotation matrix $R_i$ is obtained via the exponential
map by
$$R_i(x_i) = \text{exp}(\widehat{x}_i).$$
The matrix $R_{ij}$ is obtained by
$$R_{ij}(x_i, x_j) = \text{exp}(\widehat{x}_i)^T\text{exp}(\widehat{x}_j).$$
The function $ \text{exp}_{R_i}$ is the exponential map at $R_i$. Using this notation, the function $\text{exp}$ is short hand notation for $\text{exp}_{I}$.
* The Rodrigues Parameters, in which case $g(\theta) = \tan(\theta/2)$. The corresponding
$r$ and $r'$ are equal to $\pi$ and $\infty$ respectively.
* The Modified Rodrigues Parameters, in which case $g(\theta) = \tan(\theta/4)$, $r = \pi$ and $r' = 1$.
This representation is obtained from the rotation matrices by a second order Cayley transform <cit.>.
* The representation $(R -R^T)^{\vee}$, in which case $g(\theta) = \sin(\theta)$,
and the corresponding $r$ and $r'$ are $\pi/2$ and $1$ respectively. This representation is popular because it is easy to express
in terms of the rotation matrices. Unfortunately, since $r = \pi/2$, only $B_{\pi/2}(I)$ is covered.
* The Unit Quaternions, or rather parts of it. The unit quaternion $q_i$, expressed as a function
of the Axis-Angle Representation $x_i = \theta_iu_i$ of $R_i \in B_{\pi}(I)$, is given by
$$q(x_i) = (\cos(\theta_i/2), \sin(\theta_i/2)u_i)^T \in \mathbb{S}^3.$$
This means that we can choose the last three elements of the unit
quaternion vector as our representation, i.e., $\sin(\theta_i/2)u_i$,
in which case $r = \pi$.
The unit quaternion representation is popular since the mapping from $SO(3)$
to the quaternion sphere is a Lie group homomorphism.
Let ${x}_i(t)$ and ${x}_{ij}(t)$ denote the axis-angle
representations of the rotations $R_i(t)$ and $R_{ij}(t)$,
respectively. In the following, since we are only addressing
representations of (subsets of) $B_{\pi}(I)$, we choose ${x}(t) = [{x}_1^T(t), {x}_2^T(t), \ldots, {x}_n^T(t)]^T \in (B_{\pi,3}(0))^n$
as the state of the system instead of $(R_1(t), \ldots, R_n(t)) \in (B_{\pi}(I))^n.$
Note that since $\theta_i = \|x_i\|$, it holds that $g(\theta_i) = g(\|x_i\|)$. The variables $y_i$ and $y_{ij}$ can be seen as functions of $x_i$ and $x_i, x_j$
\begin{align*}
y_i(x_i) & = (f \circ \text{exp})(\widehat{x}_{i}), \\
y_{ij}(x_i, x_j) & = (f \circ \text{exp})(\text{Log}(R_i^T(x_i)R_j(x_j))).
\end{align*}
Since $x_i$ and $x_j$ are elements of the vector $x$, we can write $y_i(x)$ and $y_{ij}(x)$.
When we write $y_i(t)$ and $y_{ij}(t)$, this
is equivalent to $y_{i}(x(t))$ and $y_{ij}(x(t))$ respectively.
If we want to emphasize the dependence of the initial condition,
instead of writing $x(t)$ (or $y(t)$) we write $x(t,t_0,x_0)$
(or $y(t,t_0,y_0))$ where $x_0$ is the initial state and
$t_0$ is the initial time.
§.§ Kinematics
We denote the instantaneous angular velocity of $\mathcal{F}_i$
by ${\omega}_i$. From now on, until Section <ref>, we assume that $\omega_i$ is the
control variable for the rotation of agent $i$.
The kinematics for $R_i$ is given by
$$\dot{R}_i = R_i\widehat{\omega}_i,$$
where $R_i\widehat{\omega}_i$ is an element of the tangent space $T_{R_i}SO(3)$.
The kinematics
is given by
\begin{align}\label{dynamics}
\dot{{x}}_{i} & = L_{{x}_{i}}{\omega}_{i},
\end{align}
where the Jacobian (or transition) matrix $L_{{x}_i}$ is given by
\begin{equation}\label{jacobian}
L_{{x}_i} = L_{\theta_i {u}_i} = I_3 +
\frac{\theta}{2}\widehat{{u}}_i+ \bigg (1 -
\frac{\text{sinc}(\theta_i)}{\text{sinc}^2(\frac{\theta_i}{2})}\bigg
\end{equation}
The proof is found in <cit.>.
The function $\mathrm{sinc}(\beta)$ is
defined so that $\beta\mspace{3mu}\mathrm{sinc}(\beta) = \sin(\beta)$ and $\mathrm{sinc}(0) =
1$. It was shown in <cit.> that
$L_{\theta {u}}$ is invertible for $\theta \in
(-2\pi,2\pi)$. Note however that $\theta \in [0, \pi)$ here.
The linear velocity of agent $i$, expressed in $\mathcal{F}_i$, is
denoted by $v_i$. Up until Section <ref>,
we assume that $v_i$ is the control variable
for the translation of agent $i$. The time derivative of $T_i(t)$ is given by
$$\dot{T}_i(t) = R_i(t)v_i.$$
$$\xi_i = \begin{bmatrix}
\widehat{\omega}_i & v_i \\
0 & 0
\end{bmatrix}.$$
It holds that
$$\dot{G}_i(t) = G_i(t)
\xi_i.$$
§.§ Dynamics
The dynamics for agent $i$ is given by
\begin{align*}
\dot{G}_i & = G_i\xi_i \\
\dot{\xi}_i & =
\begin{bmatrix}
(J_i^{-1}(-\widehat{\omega}_iJ_i\omega_i + \boldsymbol{\tau}_i))^{\wedge} & (\frac{\boldsymbol{f}_i}{m_i} - \widehat{\omega}_iv_i) \\
0 & 0
\end{bmatrix},
\end{align*}
where $J_i$ is the inertia matrix, $m_i$ is the mass, $\boldsymbol{\tau_i}$ is the control torque, and $\boldsymbol{f}_i$ is control force – the latter two are given as a bold symbols
since we do not want to mix them up with other defined entities.
§.§ Connectivity
A directed graph (or digraph) $\mathcal{G} = (\mathcal{V},
\mathcal{E})$ consists of a set of nodes, $\mathcal{V} = \{1, ..., n\}$
and a set of edges $\mathcal{E} \subset \mathcal{V} \times \mathcal{V}$.
Each node in the graph corresponds to a unique agent.
We also define neighbor sets or neighborhoods.
Let $\mathcal{N}_i \in
\mathcal{V}$ comprise the neighbor set (sometimes referred to simply as neighbors)
of agent $i$, where $j \in \mathcal{N}_i$
if and only if $(i,j) \in \mathcal{E}$. We assume that $i \in
\mathcal{N}_i$ i.e., we restrict the collection of graphs
to those for which $(i,i) \in \mathcal{E}$ for all $i \in \mathcal{V}$.
A directed path of $\mathcal{G}$ is an ordered sequence of distinct nodes in $\mathcal{V}$
such that any consecutive pair of nodes in the sequence corresponds to an edge in the graph.
An agent $i$ is connected to an agent $j$ if there is a directed path starting in $i$ and ending in $j$.
A digraph is strongly connected if each node $i$
is connected to all other nodes.
A digraph is quasi-strongly connected if there exists a rooted
spanning tree or a center, i.e., at least one node such
that all other nodes are connected to it.
An adjacency matrix $\mathcal{A} = [a_{ij}]$ for a graph
$\mathcal{G} = (\mathcal{V},
\mathcal{E})$ is a matrix where $a_{ij} \geq 0$ for all $i,j$, and furthermore $a_{ij} > 0$
if and only if $(i,j) \in \mathcal{E}$ for all $i,j$. Given our definition of graph, i.e., Definition <ref>,
there are infinitely many adjacency matrices for a graph.
From Definition <ref> we see that there are $2^{n^2}$
directed graphs with $n$ nodes, i.e., the power set of the
edge set. Since we assume that
$(i,i)$ is an edge in the graph for all $i$, there are $2^{n^2 - n}$ graphs we consider.
For $k \in \{1, \ldots, 2^{n^2 - n}\}$ we associate a corresponding unique graph $\mathcal{G}_k = (\mathcal{V}, \mathcal{E}_{k})$ and a
unique adjacency matrix ${A}_k$.
The ${A}_k$ matrices are constructed in the following way.
We construct a positive adjacency matrix ${A}' = [a_{ij}]$ for the complete (fully connected) graph. For the matrix ${A}_k = [a_{ij}^k]$,
it holds that $a_{ij}^k = a_{ij}$ if $(i,j) \in \mathcal{E}_k$,
otherwise $a_{ij}^k = 0$. Thus, if $(i,j) \in \mathcal{E}_k$,
we can write $a_{ij}$ instead of $a_{ij}^k$.
Now, for each agent $i$ there are $2^{n-1}$ unique neighborhoods
$\mathcal{N}_i^l$, where $l \in \{1, \ldots, 2^{n-1}\}$.
Given $k \in \{1, \ldots, 2^{n^2 - n}\}$, for agent $i$
there is a unique $l \in \{1, \ldots, 2^{n - 1}\}$ such that
$\mathcal{N}_i^l$ is the neighborhood of agent $i$ in the graph $\mathcal{G}_k$.
Also, if each agent $i$ has chosen an $l \in \{1, \ldots, 2^{n - 1}\}$
such that $\mathcal{N}_i^l$ is the neighborhood of agent $i$,
then there is a unique $k \in \{1, \ldots, 2^{n^2 - n}\}$ such that
$\mathcal{G}_k$ is the graph for the system.
We are now ready to address time-varying graphs.
In order to do so, for each agent $i$, we introduce
a switching signal function
$$\sigma^i: \mathbb{R} \rightarrow \{1, \ldots, 2^{n-1}\},$$
which is piece-wise constant and right-continuous.
Let $\{\tau^i_k\}$ be the monotonically strictly increasing
sequence of times for which $\sigma^i$
is discontinuous.
We assume that there is a positive lower bound $\tau_D$ between two consecutive switches, i.e.,
$$\sup_{k}(\tau^i_{k+1} - \tau^i_k) > \tau_D \quad \text{ for all } i.$$
The time-varying neighborhood of agent $i$ is
Given the set of switching signal functions $\sigma^i$ we
can construct a piece-wise constant and right-continuous
switching signal function for the graph of the
multi-agent system. This switching signal function $\sigma$
has range $\{1, \ldots, 2^{n^2 - n}\}$ and
switching times
$$\{\tau_k\} = \bigcup_{i}\{\tau_{l}^i\},$$
where $\{\tau_k\}$ is monotonically strictly increasing in $k$. Note that
for $\sigma$ it is not
necessarily true that there is a positive lower bound on the dwell time
between two consecutive switches as is the case for $\sigma^i$.
between any two switching times,
$\sigma(t)$ is equal to the $k$ for which the graph $\mathcal{G}_k$ it holds that
the neighborhood of each agent $i$ is equal to $\mathcal{N}_i^{\sigma^i(t)}$.
The union graph of $\mathcal{G}_{\sigma(t)}$ during the time
interval $[t_1,t_2)$ is defined by
\begin{equation*}
\mathcal{G}([t_1, t_2))
= \textstyle\bigcup_{t\in[t_1, t_2)} \mathcal{G}_{\sigma(t)}
= (\mathcal{V},\textstyle\bigcup\nolimits_{t\in[t_1, t_2)}\mathcal{E}_{\sigma(t)}),
\end{equation*}
where $t_1 < t_2 \leq +\infty$.
The graph $\mathcal{G}_{\sigma(t)}$ is uniformly
(quasi-) strongly connected if there is $T^{\sigma}>0$ such that the
union graph $\mathcal{G}([t, t + T^{\sigma}))$ is (quasi-) strongly connected for all $t$.
The idea of using an individual switching signal $\sigma^i$ for each agent, is that each agent shall be able to choose independently
which neighbors it decides to receive information from.
Instead of using the term communication graph for
$\mathcal{G}_{\sigma(t)}$, we deliberately use the terms
neighborhood graph, connectivity graph or interaction graph. Direct communication does not necessarily take place
between the agents in practice. Instead, they can choose to just observe each
other via cameras or other sensors, i.e., indirect communication.
§ CONSENSUS AND FORMATION CONTROL
§.§ Consensus
We start this section by introducing the consensus
problem on $SE(3)$. Consensus on $SE(3)$
means that, as time tends to infinity,
the set of transformations $(G_1(t), G_2(t), \ldots, G_n(t)) \in (SE(3))^n$ approaches
the consensus set where all the transformations are
equal. The problem is to construct a distributed control law
for each agent $i$, where only information from the neighbors $\mathcal{N}_i$ is used in the control law, such that the system reaches consensus. This information could be the relative transformations to the neighbors or the absolute transformations of the neighbors.
An other desired property is that
the velocities $\xi_i$ tend to zero sufficiently fast so that the
transformations converge to a static transformation.
When we say that the Euclidean transformations of the
agents approaches the consensus set, we mean that
the rotations
$$(R_1(t), R_2(t), \ldots, R_n(t)) \in SO(3)^n$$
approach $\{(R_1, \ldots, R_n) \in (\bar{B}_{q}(I))^n: R_1 = \hdots = R_n\}$ and the translations
$$T^{\text{tot}} = [T_1^T(t), T^T_2(t), \ldots, T^T_n(t)]^T \in \mathbb{R}^{3n}$$
approach the set
where all the translations are equal. For the translations
the convergence is defined in terms of the Euclidean metric.
For the rotations, the convergence is defined in terms of the
Riemannian metric on $SO(3)$. If the rotations are contained within the region of injectivity of a local parameterization, asymptotic stability in terms of the Riemannian metric on $SO(3)$ is equivalent to asymptotic stability using the Euclidean metric in the parameterization domain for $x$.
consensus problem on $SE(3)$ might seem uninteresting in practice,
since for rigid bodies in space it is not physically possible
to reach consensus in the positions. There are two reasons for
considering this problem anyway. Firstly, if we look at the consensus
problem as two subproblems, consensus in the rotations and consensus
in the positions, the former is still interesting in practice and
has received a great deal of attention lately. Secondly and more importantly, the consensus control problem is equivalent to the formation
control problem after a change of coordinates. Thus, all the control laws we develop for the consensus
control problems can also be used for the formation control problem after a simple
transformation. This will be elaborated more in Section <ref>.
The subproblem of reaching consensus in the rotations is referred to as the attitude synchronization problem or consensus on $SO(3)$. Then we shall find a feedback control law ${\omega}_i$ for each agent $i$ using the local representations of either absolute rotations or relative rotations so that the absolute rotations of all agents converge to the set where all the rotations are equal as time goes to infinity, i.e.,
\begin{equation}\label{condition1}
\|R_i(t) - R_j(t)\| \rightarrow 0, \text{ for all } i,j, \text{ as } t \rightarrow \infty,
\end{equation}
or equivalently,
\begin{equation*}
\hspace{9mm} \|R_{ij}(t) - I\| \rightarrow 0, \text{ for all } i,j, \:\: \text{as } t \rightarrow \infty.
\end{equation*}
If $y \in (B_{r',3}(0))^n$ it is true that
\begin{align}\label{eq:z1}
R_i =R_j & \Longleftrightarrow x_i = x_j \Longleftrightarrow x_{ij} = 0 \\
\nonumber
& \Longleftrightarrow y_i = y_j \hspace{0.51mm} \Longleftrightarrow y_{ij} = 0 \quad \text{ for all } i,j.
\end{align}
We define the consensus set $\mathcal{A}$ in $\mathbb{R}^{3n}$ as follows:
$$\mathcal{A} = \{z = [z_1^T, z_2^T, \ldots, z_n^T]^T \in \mathbb{R}^{3n}: z_i = z_j \in \mathbb{R}^3, \forall i,j\}.$$
According to (<ref>) and the fact that the
$$R_i \mapsto x_i$$
is a diffeomorphism on $B_{\pi}(I)$, (<ref>) can equivalently be
written as $x(t) \rightarrow \mathcal{A}$ as $t\rightarrow \infty$.
This means that the solution approaches $\mathcal{A}$.
Thus, provided we can guarantee that $y(t) \in (B_{r',3}(0))^n$
for all $t \geq t_0$, where $t_0$ is the initial time,
consensus on $SE(3)$ for the multi-agent system is the following
$$(x(t),T^{\text{tot}}(t)) \rightarrow \mathcal{A} \times \mathcal{A}, \quad \text{ as } t \rightarrow \infty. $$
A stronger assumption on the convergence to $\mathcal{A} \times \mathcal{A}$ is global uniform asymptotic stability of $\mathcal{A} \times \mathcal{A}$
relative to a strongly forward-invariant set, see Definition <ref> and Definition <ref> below. The distance from
a point $z$ in $\mathbb{R}^p$ to a set $\mathcal{D}$ in $\mathbb{R}^p$ is defined by
$$\|z\|_{\mathcal{D}} = \inf_{w \in \mathcal{D}}\|z-w\|.$$
For a time-invariant system, forward invariance or positive invariance of a set means that every solution to the system with initial condition
in the set is forward complete and the solution at any time greater than the initial time is contained in the set. For switched systems we have the following type of invariance.
The $f$-vectors used in the following two definitions are locally
defined in that context.
Consider dynamical systems of the following class. The
dynamical equation is given by
$$\dot{z} = f_{\sigma(t)}(z),$$
where $z(t) \in \mathbb{R}^p$ for some positive integer $p$.
The right-hand side is switching between a finite set $\mathcal{F} = \{f_k\}$
of time-invariant functions according
to a switching signal function $\sigma$. The switching signal function $\sigma$ is well-behaved in the sense that there are only
finitely many switches on any compact time interval.
A set $\mathcal{D} \subset \mathbb{R}^p$ is strongly forward-invariant if
for any time $t_0$, any $z_0 \in \mathcal{D}$ and any such well behaved switching signal function $\sigma$ switching between functions in $\mathcal{F}$, the solution $z(t,t_0,z_0)$
exists, is unique, forward complete and contained in $\mathcal{D}$ for all $t \geq t_0$.
Consider the dynamical system
$$\dot{z} = f(t,z),$$
where $y(t) \in \mathbb{R}^p$ for some positive integer $p$.
A set $\mathcal{D}_1 \subset \mathbb{R}^p$ is globally
uniformly asymptotically stable relative to the compact
strongly forward-invariant set $\mathcal{D}_2$, if
* $\mathcal{D}_1$ is uniformly stable relative to $\mathcal{D}_2$,
i.e., for every $\epsilon > 0$, there is a $\delta(\epsilon) > 0$ such that
\begin{align*}
& (\|z_0\|_{\mathcal{D}_1} \leq \delta, \: z_0 \in \mathcal{D}_2) \Longrightarrow \\
& (\|z(t_2,t_1,z_0)\|_{\mathcal{D}_1} \leq \epsilon \: \textnormal{ for all } t_1, t_2 \textnormal{ where } t_2 \geq t_1),
\end{align*}
* $\mathcal{D}_1$ is globally uniformly attractive relative to
$\mathcal{D}_2$, i.e.,
for every $\epsilon > 0$, there is a $\tau(\epsilon) > 0$ such that
\begin{align*}
& z_0 \in \mathcal{D}_2 \Longrightarrow \\
& (\|z(t_2,t_1,z_0)\|_{\mathcal{D}_1} \leq \epsilon \: \textnormal{ for all } t_1,t_2, \\
& \textnormal{ such that } t_2 \geq t_1 + \tau(\epsilon)).
\end{align*}
One can show that if $\mathcal{A}$ is globally uniformly
asymptotically stable relative to the strongly forward invariant set $(\bar{B}_{q,3}(0))^n$ for $x(t)$ where $q < \pi$, then the set $\{(R_1, \ldots, R_n) \in (\bar{B}_{q}(I))^n: R_1 = \hdots = R_n\}$ is globally
uniformly asymptotically stable relative to $(\bar{B}_{q}(I))^n$ (when the Riemannian metric is used). The notation of strong forward invariance is adopted from <cit.>,
where it is defined for hybrid systems.
§.§ Formation
The consensus problem has many applications in the cases where the motion
is purely rotational, e.g., attitude synchronization for spacecraft or
orientation alignment for cameras. However, as already mentioned, reaching consensus in the
positions is obviously not physically possible for rigid bodies, but reaching
a formation is.
The objective is to make the $G_i^{-1}(t)G_j(t)$ matrices
converge to some desired $G^*_{ij}$ matrices. The $G^*_{ij}$
matrices are assumed to be transitively consistent in that
$$G^*_{ij}G^*_{jk} = G^*_{ik} \text{ for all } i,j,k.$$
A necessary and sufficient condition for transitive consistency <cit.>
of the $G^*_{ij}$ is that there are $G^*_{i}$ such that
$$G^*_{ij} = G^{*-1}_{i}G^*_{j} \text{ for all }i,j.$$
In this light, we formulate the objective in the formation problem as
follows. Given some desired constant Euclidean transformation matrices $G_1^*, \ldots, G_n^*$,
construct a control law for each agent $i$ such that
\begin{align*}
~\|G_1\left(t\right) - Q^{-1}(t)G_1^*\| & \rightarrow 0, \\
~\|G_2\left(t\right) - Q^{-1}(t)G_2^*\| & \rightarrow 0, \\
~& \: \vdots \\
~\|G_n\left(t\right) - Q^{-1}(t)G_n^*\| & \rightarrow 0,
\end{align*}
as $t \rightarrow \infty$, where $Q(t)$ is a Euclidean transformation. This implies that
\begin{align*}
\|G_i^{-1}\left(t\right)G_j\left(t\right) - G_i^{*-1}G_j^*\| \rightarrow 0 \quad \text{ as } \quad t \rightarrow \infty.
\end{align*}
Thus, in some (possibly time-varying) coordinate frame,
the Euclidean transformation of agent $i$ converges to $G_i^*$
as time tends to infinity.
Each matrix $G_i^*$ contains the rotation matrix $R_i^*$ and the
translation $T_i^*$.
On a kinematic level the formation control problem is
equivalent to the consensus problem. Let us define
$$\tilde{G}_i = G_iG_i^{*-1} \: \text{ and } \: \tilde{G}_{ij} = G_i^{*}G_{ij}G_j^{*-1}, \quad \text{for all } i.$$
The kinematics for $\tilde{G}_i$ is given by
$$\dot{\tilde{G}}_i = G_i\xi_iG_i^{*-1} = \tilde{G}_iG_i^{*}\xi_iG_i^{*-1}
= \tilde{G}_i\tilde{\xi}_i,$$
\begin{align*}
\tilde{\xi}_i & = G_i^{*}\xi_iG_i^{*-1} = \begin{bmatrix}
\widehat{\tilde{\omega}}_i & \tilde{v}_i\\
0 & 0
\end{bmatrix} \\
& =
\begin{bmatrix}
R_i^*\widehat{\omega}_iR_i^{*T} & - R_i^*\widehat{\omega}_iR_i^{*T}T_i^* + R_i^*v_i\\
0 & 0
\end{bmatrix}.
\end{align*}
\begin{align*}
{\xi}_i & = G_i^{*-1}\tilde{\xi}_iG_i^* = \begin{bmatrix}
\widehat{{\omega}}_i & {v}_i\\
0 & 0
\end{bmatrix} \\
& =
\begin{bmatrix}
R_i^{*T}\widehat{\tilde{\omega}}_iR_i^{*} & R_i^{*T}(\widehat{\tilde{\omega}}_iT_i^* + \tilde{v}_i)\\
0 & 0
\end{bmatrix}.
\end{align*}
It easy to see that if the system reaches consensus in the $\tilde{{G}}_i$, it also reaches the desired formation.
Thus, a consensus control law $\tilde{\xi}_i$ can be constructed
for each agent and provided that each agent $i$ knows $G_i^*$,
$\xi_i$ is obtained by $\xi_i = G_i^{*-1}\tilde{\xi}_iG_i^{*}$.
In general, unless the design is
limited to the $\omega_i$, the proposed control laws in the next section
should be used for formation control of the $\tilde{\xi}_i$.
On the dynamic level we have that
\begin{align}
\label{eq:formation_dyn_1}
\dot{\tilde{\omega}}_i =~ & R_i^{*}J_i^{-1}\left(-\left(R_i^{*T}\widehat{\tilde{\omega}}_i R_i^{*}\right)J_iR_i^{*T}\tilde{\omega}_i + \boldsymbol{\tau}_i\right), \\
\label{eq:formation_dyn_2}
\dot{\tilde{v}}_i =~ & R_i^{*}\left (\frac{\boldsymbol{f}_i}{m_i} - R_i^{*T} \widehat{\tilde{\omega}}_i\left (\widehat{\tilde{\omega}}_iT_i^* + \tilde{v}_i\right ) \right ) - \left (\dot{\tilde{\omega}}_i\right)^{\wedge} T_i^*.
\end{align}
For the control design on the dynamic level, the approach is to
design a consensus control law for the $\tilde{\omega}_i$ and the
$\tilde{v}_i$ and then track
this desired kinematic control law using methods similar to backstepping.
For control laws designed on the kinematic level, since the
problems of consensus and formation are equivalent, we will only
focus on the consensus problem. The consensus problem more
tractable, since one can use existing theory for that problem. On the dynamic level we will also only
consider the consensus problem – the formation control
laws have a similar structure as the consensus control laws
in this case.
§ KINEMATIC CONTROL LAWS
We use two approaches for the design of the $\xi_i$.
The first approach is to treat $\xi_i$ as one control variable
and design a feedback control law as an expression of the $G_i$,
the second approach is to design $\omega_i$ and $v_i$ separately.
Most emphasis will be on the second approach. The control laws in the first
approach are referred to as the first control laws,
whereas the control laws in the second approach are referred to as the second control laws.
§.§ The first control laws
We propose the following control laws based on absolute and relative transformations respectively.
\begin{align}
\label{eq:approach1:controller:1}
{\xi}_i & = \sum_{j\in\mathcal{N}^{\sigma^i(t)}_i} a_{ij}\left( \left(G_j - G_i) + (G_i^{-1} - G_j^{-1}\right) \right), \\
\label{eq:approach1:controller:2}
{\xi}_i & = \sum_{j\in\mathcal{N}^{\sigma^i(t)}_i} a_{ij}\left( G_{ij} - G_{ij}^{-1} \right).
\end{align}
§.§.§ The second control laws
In the first two control laws below,
$y_i$ and $y_{ij}$ could be any of the local representations considered in Section <ref>.
\begin{align}
\label{chapter2:controller:1}
{\omega}_i & = \sum_{j\in\mathcal{N}^{\sigma^i(t)}_i} a_{ij}({y}_{j} - {y}_{i}), \hspace{29mm}\\
\label{chapter2:controller:2}
{\omega}_i &= \sum_{j\in\mathcal{N}^{\sigma^i(t)}_i} a_{ij}{y}_{ij}, \\
\label{chapter2:controller:1:v}
{v}_i & = \sum_{j\in\mathcal{N}^{\sigma^i(t)}_i} a_{ij}({T}_{j} - {T}_{i}), \hspace{29mm}\\
\label{chapter2:controller:2:v}
{v}_i &= \sum_{j\in\mathcal{N}^{\sigma^i(t)}_i} a_{ij}{T}_{ij}.
\end{align}
The structure of these second control laws and especially
(<ref>) and (<ref>)
are well known from the
literature <cit.>. In Section <ref> we provide
new results on the rate of convergence and regions of attractions for these
control laws in this context. When the control laws are used for
formation instead of consensus, the $\tilde{\xi}_i$ are designed instead of
the $\xi_i$; the controllers are obtained through the relation
$$\tilde{\xi}_i = G_i^{*}\xi_iG_i^{*-1},$$
as given in Section <ref>. As an example, suppose all the $R_i$ rotations and all
the desired $R_i^*$ rotations in the formation are equal to the identity matrix. Then the
agents shall reach a desired formation in the positions only. All the agents
construct $\tilde{v}_i$ according to (<ref>) or (<ref>) and solve for $v_i$ through the following
I & T_i^* \\
0 & 1
\end{bmatrix}^{-1}
\begin{bmatrix}
0 & \tilde{v}_i \\
0 & 0
\end{bmatrix}
\begin{bmatrix}
I & T_i^* \\
0 & 1
\end{bmatrix}
\begin{bmatrix}
0 & \tilde{v}_i \\
0 & 0
\end{bmatrix}
\begin{bmatrix}
0 & v_i \\
0 & 0
\end{bmatrix}.
In this simple case $v_ i = \tilde{v}_i$ and $\xi_i = \tilde{\xi}_i$. However, in general $\xi_i \neq \tilde{\xi}_i$.
The following two sections are devoted to the study of the control laws (<ref>-<ref>).
§ RESULTS FOR THE FIRST CONTROL LAWS
Suppose the graph $\mathcal{G}_{\sigma(t)}$ is time-invariant and strongly connected.
Suppose that each rotation is contained in $B_{\pi/2}(I)$, then if control law (<ref>) is used, the set $(B_{\pi/2,3}(0))^n$ is strongly forward invariant for the dynamics of $x$ and
$$(x(t), T^{\text{tot}}(t)) \rightarrow \mathcal{A} \times \mathcal{A} \quad \text{ as } t \rightarrow \infty.$$
Suppose the graph $\mathcal{G}_{\sigma(t)}$ is time-invariant and
quasi-strongly connected. Suppose $q < \pi/4$, if all the rotations are contained in $\bar{B}_{q}(I)$, then if control law (<ref>) is used, the set $(\bar{B}_{q,3}(0))^n$ is strongly forward invariant for the dynamics of $x$ and
$\mathcal{A} \times \mathcal{A}$
is globally asymptotically stable relative to
$(\bar{B}_{q,3}(0))^n \times \mathbb{R}^{3n}$.
It can be shown that the results in propositions <ref> and <ref> are slightly more general. It is true
that $(x(t), T^{\text{tot}}(t))$ converges to a fixed point in $\mathcal{A} \times \mathcal{A}$, i.e., not a limit cycle. This result is however not shown here.
In proposition <ref>, we only guarantee stability of a set instead of uniform stability of the set.
In the following two proofs, since the graph is time-invariant, we write
$\mathcal{G}$ and $\mathcal{N}_i$ instead of
$\mathcal{G}_{\sigma(t)}$ and $\mathcal{N}^{\sigma^i(t)}_i$ respectively.
The graph Laplacian
matrix $L(\mathcal{G},A)$ for the graph $\mathcal{G}$ with the adjacency matrix $A$, is
$$L(\mathcal{G},A) = D - A,$$
$$D = \text{diag}(d_1, \ldots, d_n) = \text{diag}\left(\sum_{j=1}^n a_{1j}, \ldots, \sum_{j=1}^n a_{nj}\right).$$
Proof of Proposition <ref>:
When the control law (<ref>)
is used, $\omega_i$ is given by the following expression
$${\omega}_i = \sum_{j\in\mathcal{N}_i} a_{ij}({y}_{j} - {y}_{i}),$$
where $y_i = \sin(\theta_i)u_i$ for all $i$. This control law for $\omega_i$ is on the form (<ref>) and we
will later show that, provided the rotations are contained within
the region of injectivity, which in this case is the ball around
the identity with radius $\pi/2$, $x(t)$ approaches $\mathcal{A}$ asymptotically. Also,
$(B_{\pi/2,3}(0))^n$ is forward invariant.
Given the initial states $x_i(t_0)$, since there are finitely many agents, there is a positive $q < \pi/2$ such that $x(t_0) \in (\bar{B}_{q,3}(0))^n$.
$\mathcal{X} = (\bar{B}_{q,3}(0))^n \times \mathbb{R}^{3n}$ and define the
two closed sets
\begin{align*}
\Gamma_2 & = \mathcal{A} \cap (\bar{B}_{q,3}(0))^n \times \mathbb{R}^{3n} \\
\Gamma_1 & = \mathcal{A} \cap (\bar{B}_{q,3}(0))^n \times \mathcal{A}.
\end{align*}
We can choose the state space as $\mathcal{X}$ for
$(x, T^{\text{tot}})$ since this set is
forward invariant, see Proposition <ref> in Section <ref>.
We observe that $\Gamma_1 \subset \Gamma_2 \subset \mathcal{X}.$
On $\mathcal{X}$, the dynamics for $T_i$ is given by
$$\dot{T}_i = \sum_{j\in\mathcal{N}_i}a_{ij}\left(\left(R_i + R_iR_j^T)\right)T_j - \left(R_i + I\right)T_i\right).$$
But on the set $\Gamma_2$ the dynamics for $T_i$ is given by
$$\dot{T}_i = \sum_{j\in\mathcal{N}_i}a_{ij}\left(\left(I + Q^*\right)\left(T_j - T_i\right)\right), \hspace{20mm}$$
where $Q^* \in \bar{B}_{q,3}(0)$ is some constant rotation matrix. On $\Gamma_2$,
the dynamics for $T^{\text{tot}}$ is given by
$$\dot{T}^{\text{tot}} = -(L(\mathcal{G},A) \otimes (I + Q^*)) T^{\text{tot}}.$$
By using the fact that the eigenvalues of $(I + Q^*)$ have
real parts strictly greater than zero, the fact that ($(B_{\pi/2,3}(0))^n$ is forward invariant),
and the fact that $L(\mathcal{G},A)$ is the graph Laplacian matrix for
a strongly connected graph, one can show that
$\Gamma_1$ is exponentially stable relative to $\Gamma_2$.
Now one can use Theorem 8 in <cit.> in order to show that
$\Gamma_1$ is globally attractive relative to $\mathcal{X}.$
Proof of Proposition <ref>:
When the control law (<ref>)
is used, $\omega_i$ is given by the following expression
$${\omega}_i = \sum_{j\in\mathcal{N}_i} a_{ij}y_{ij},$$
where $y_{ij} = \sin(\theta_{ij})u_{ij}$ for all $i$. This control law for $\omega_i$ is on the form (<ref>).
$\mathcal{X} = (\bar{B}_{q, 3}(0))^n \times \mathbb{R}^{3n}$ and define the
two closed sets
\begin{align*}
\Gamma_2 & = \mathcal{A} \cap (\bar{B}_{q, 3}(0))^n \times \mathbb{R}^{3n} \\
\Gamma_1 & = \mathcal{A} \cap (\bar{B}_{q, 3}(0))^n \times \mathcal{A}.
\end{align*}
We observe that $\Gamma_1 \subset \Gamma_2 \subset \mathcal{X}.$
Proposition <ref> in Section <ref> in combination with the fact that the right-hand sides of the $\dot{T}_i$ are well-defined, guarantees that $\mathcal{X}$
is forward invariant and can serve as the state space for
$(x, T^{\text{tot}})$. Also the set $\Gamma_2$ is globally uniformly asymptotically stable relative to $\mathcal{X}$.
On $\mathcal{X}$, the dynamics for $T_i$ is given by
$$\dot{T}_i = \sum_{j\in\mathcal{N}_i}a_{ij}\left(\left(T_j - T_i\right) - R_iR_j^T\left(T_i -T_j\right)\right),$$
but on the set $\Gamma_2$ the dynamics for $T_i$ is given by
$$\dot{T}_i = \sum_{j\in\mathcal{N}_i}a_{ij}\left(T_j - T_i\right).$$
The dynamics for $T^{\text{tot}}$ is given by
$$\dot{T}^{\text{tot}} = -(L(\mathcal{G},A) \otimes I) T^{\text{tot}},$$
where $L(\mathcal{G},A)$ is the graph Laplacian matrix for
a quasi-strongly connected graph. It is well known that the consensus set is
exponentially stable for this dynamics. Thus, the set
$\Gamma_1$ is globally asymptotically stable relative to $\Gamma_2$.
Now one can use Theorem 10 in <cit.> in order to show that
$\Gamma_1$ is globally asymptotically stable relative to $\mathcal{X}.$
§.§ Numerical experiments
In order to illustrate the relation between consensus
and formation the following example is considered. For a system of five agents,
in Figure <ref> the convergence of the $\tilde{G}_i$
variables to consensus and the
convergence of the $G_i$ variables to a desired formation
is shown. The adjacency matrix was chosen to that of a quasi-strongly connected graph with
entries equal to $0$, $1$ or $2$.
The initial rotations are drawn from the uniform distribution
over $B_{\pi/2}(I)$.
Each initial translation vector is drawn from the
uniform distribution over the unit box in $\mathbb{R}^3$. The initial $R_i(0)$ rotations and initial $T_i(0)$
translations are the building blocks of the $G_i(0)$ transformations. The desired $G^*_i$ are constructed in
the same manner as the $G_i(0)$, after which the $\tilde{G}_i(0)$ transformations
are constructed by $\tilde{G}_i(0) = {G}_i(0)G_i^{*-1}$.
For the same initial conditions,
the four upper plots in Figure <ref>
show the convergence when
controller (<ref>) is used, whereas
the four lower plots in Figure <ref> show the convergence when
controller (<ref>) is used. In each
of these four subplots, the first plot is showing the
difference $\|\tilde{G}_i(t) - \tilde{G}_1(t)\|_{\text{F}}$
for all $i$; the second plot is
showing one of the elements of $\tilde{G}_i(t)$ for all $i$ as function of time, this element is the upper left one in the $\tilde{G}_i(t)$, i.e., it is an element of the rotation matrix;
the third plot is showing the difference $\|{G}_i(t) - {G}_1(t)\|_{\text{F}}$
for all $i$ as function of time; the fourth plot is
showing one of the elements of ${G}_i(t)$ as function of time for all $i$. This element is chosen as the upper left element in the ${G}_i(t)$.
The construction of the initial rotations in this example
does not guarantee
that initial rotations are contained in the regions
specified in Proposition <ref>
and Proposition <ref>, yet the convergence is obtained for both control laws. For 1000 simulations with five agents and random quasi-strongly connected
topologies, where the initial rotations are drawn from the uniform distribution over
$SO(3)$ and the translations are drawn from the uniform distribution over the unit cube
in $\mathbb{R}^3$, the $\tilde{G}_i(t)$ transformations converged to consensus 909 respective 910 times for the two different control laws, i.e., a success rate of over 90 $\%$. If the initial
rotations in $\tilde{G}_i$ were drawn from the uniform distribution over $B_{\pi/2}(I)$ the transformations converged to
consensus 1000 respective 1000 times for the two different control laws, i.e., a success rate of 100 $\%$.
These plots illustrate the difference between reaching
consensus in the $\tilde{G}_i$ variables and reaching
a formation in the $G_i$ variables.
These plots show the convergence to
consensus under discrete sampling, additive absolute noise, and
switching between quasi-strongly connected graphs.
Furthermore, numerical experiments were conducted
when there was additive absolute, noise and when the transformations were measured discretely the
topologies were switching.
The noise were random skew symmetric matrices, whose magnitudes
were equal to $0.1$.
Under these conditions the controllers (<ref>) and (<ref>)
were tested
in 100 simulations where the graphs
were switching between quasi-strongly connected topologies and the initial
rotations in $\tilde{G}_i$ were drawn from the uniform distribution over $B_{\pi/2}(I)$. The number of agents was 5. The matrices converged to
consensus in every simulation for both controller (<ref>) and controller (<ref>). In the simulations the
graphs switched with a frequency of $10$, which was the same as the
sampling frequency; the consensus is shown for one simulation in Figure <ref>.
The simulations show stronger results than those presented in
Proposition <ref> and
Proposition <ref>.
The input is constant between sample points. Thus, we can solve the system exactly between those points (it becomes a linear time-invariant system).
The solutions between the sampling points are not shown
in the figure, instead there are straight lines
connecting the solutions at the sample points.
In all simulations the random graphs were created by constructing adjacency matrices in the following way: First an adjacency matrix for a tree graph
was created and then a binary matrix was created where each element in the matrix was drawn from the uniform distribution over $\{0,1\}$. The final adjacency was then chosen as the sum of the adjacency matrix for the tree graph and the binary matrix.
§ RESULTS FOR THE SECOND CONTROL LAWS
§.§ Rotations
Here we address the controllers (<ref>) and (<ref>). We start with
The structure of controller (<ref>) is well known from
the consensus problem in a system of agents with single integrator dynamics and states in $\mathbb{R}^m$ <cit.>.
The question is if this simple control law also works for rotations expressed in any of the local representations that we consider. The answer is yes. For all the convergence results provided in this section it is true that the state $x(t)$ converges to a fixed point, i.e., not a limit cycle.
Suppose $q < r$ and the graph $\mathcal{G}_{\sigma(t)}$ is uniformly strongly connected, then if controller (<ref>) is used,
$(\bar{B}_{q,3}(0))^n$ is strongly forward invariant, $0 \in \mathbb{R}^{mn}$ is uniformly stable
and $(\bar{B}_{q,3}(0))^n \cap \mathcal{A}$ is globally attractive relative to $(\bar{B}_{q,3}(0))^n$.
In order to prove Proposition <ref>,
we use the following proposition.
Suppose controller (<ref>) is used, $\mathcal{G}_{\sigma(t)}$ is uniformly strongly connected,
and $q < r$.
Now, suppose there is a continuously differentiable function
$$V: \mathbb{R}^3 \rightarrow \mathbb{R}$$
such that for any given $k \in \{1, \ldots 2^{n-1}\}$ and
$\bar{x} = [\bar{x}_1^T, \bar{x}_2^T, \ldots, \bar{x}_n^T]^T \in (\bar{B}_{q,3}(0))^n$
* if
$$i \in \textnormal{arg}\max_{j \in \mathcal{N}^{k}_i}(V(\bar{x}_j))$$
it holds that
\begin{equation}\label{eq:t1}
\langle \nabla V(\bar{x}_i) , \sum_{j\in\mathcal{N}^{k}_i} a_{ij}({y}_{j}(\bar{x}) - {y}_{i}(\bar{x})) \rangle \leq 0,
\end{equation}
where $y_i$ is the local representation.
* and equality holds for (<ref>) if and only if $\bar{x}_i = \bar{x}_j$ for all $j \in \mathcal{N}^{k}_i$,
then $(\bar{B}_{q,3}(0))^n$ is strongly forward invariant for the dynamics of $x$ and $(\bar{B}_{q,3}(0))^n \cap \mathcal{A}$ is globally attractive
relative to $(\bar{B}_{q,3}(0))^n$.
The proof of Proposition <ref> is omitted here but follows,
up to small modifications, the
procedure in the proof of Theorem 2.21. in <cit.>. The essential difference between the two
is that besides the fact that in Theorem 2.21. in <cit.> more general
right-hand sides of the system dynamics are considered,
only one switching signal function is used for the system in that theorem,
whereas in in this work we assume individual switching signal functions
for the agents.
Proof of Proposition <ref>:
We verify that (1) and (2) are satisfied in
Proposition <ref> by choosing $V(\bar{x}_i) = \bar{x}_i^T\bar{x}_i$.
Let $i \in \textnormal{arg}\max_{j \in \mathcal{N}^{k}_i}(V(\bar{x}_j))$. Then
\begin{align*}
\langle \nabla V(\bar{x}_i), \sum_{j\in\mathcal{N}^{k}_i} a_{ij}({y}_{j}(\bar{x}) - {y}_{i}(\bar{x})) \rangle & \leq \\
\sum_{j\in\mathcal{N}^{k}_i}(\|\bar{x}_j\|g(\|\bar{x}_j\|) - \|\bar{x}_i\|g(\|\bar{x}_i\|)) & \leq 0,
\end{align*}
where we have used the fact that
$g$ is strictly increasing. The last inequality is strict
if and only if $\bar{x}_i = \bar{x}_j$ for all $j \in \mathcal{N}^{k}_i$.
Instead of using (<ref>), one could use feedback linearization
and construct the following control law for agent $i$,
$$\omega_i = L_{y_i}^{-1}\sum_{j\in\mathcal{N}^{\sigma^i(t)}_i} a_{ij}({y}_{j} - {y}_{i}),$$
where $L_{y_i}$ is the Jacobian matrix for the representation $y_i$. If this feedback linearization
control law is used and the graph $\mathcal{G}_{\sigma(t)}$ is
quasi-strongly connected,
the consensus set, restricted to any closed ball $(\bar{B}_{q,3}(0))^n$
where $q < r$,
is globally uniformly asymptotically stable relative to $(\bar{B}_{q,3})^n$. However, for many representations such as the Rodrigues Parameters, the Jacobian matrix $L_{y_i}$
is close to singular as $y_i$ is close to the boundary of $\bar{B}_{q,3}(0)$. Furthermore, the expression is nonlinear in the $y_i$. This might make this type of
control law more sensitive to measurement errors than (<ref>).
Now we continue with the study of (<ref>) where only local representations of the relative
rotations are available. Under stronger assumptions on the
initial rotations of the agents at time $t_0$ and weaker assumptions on
the graph $\mathcal{G}_{\sigma(t)}$, the following proposition ensures
uniform asymptotic convergence to the consensus set.
$q < r/2$ and the controller (<ref>) is used, then
$(\bar{B}_{q,3}(0))^n$ is strongly forward invariant and $(\bar{B}_{q,3}(0))^n \cap \mathcal{A}$ is globally uniformly asymptotically stable relative to
$(\bar{B}_{q,3}(0))^n$ if and only if $\mathcal{G}_{\sigma(t)}$ is uniformly quasi-strongly connected.
In Proposition <ref>, since only information that is independent of $\mathcal{F}_W$ is used in
(<ref>), the assumption that the rotations initially are contained in $\bar{B}_q(I)$ can be relaxed.
As long as there is a $Q \in SO(3)$ such that all the rotations
are contained in $(\bar{B}_{q}(Q))^n$ initially,
the rotations will reach consensus asymptotically and uniformly with respect to time.
In order to prove Proposition <ref>, we first provide
a theorem, which gives some geometric insight. Then we provide a
Proposition, which guarantees asymptotic stability of the
consensus set.
Suppose that the control law (<ref>) is used and $x \in (B_{q, 3}(0))^n$, where
$q < r/2$. Let
$z_i = \tan(\theta_i/2)u_i$ and $z = [z_1^T, \ldots, z_n^T]^T$.
\begin{align*}
\dot{z}_1 = & \sum_{j \in \mathcal{N}^{\sigma^1(t)}_1}a_{1j}h_{1j}(z_1,z_j)(z_{j} - z_1), \\
\vdots & \\
\dot{z}_n = & \sum_{j \in \mathcal{N}^{\sigma^n(t)}_n}a_{nj}h_{nj}(z_n,z_j)(z_{j} - z_n),
\end{align*}
where $h_{ij}(z_i,z_j) \geq 0$ and $h_{ij}(z_i,z_j) > 0$ if $z_{j} \neq z_i$.
The $h_{ij}$ functions in Theorem <ref> depends on the parameterization $y$.
A proof of Theorem <ref> (up to small modifications
due to the assumptions on the switching signal functions)
can be found in <cit.>. It is based
on the results in <cit.>.
Theorem <ref> states that, after
a change of coordinates to the Rodrigues Parameters,
the system satisfies the well known convexity
assumption that the right-hand side of
each agent's dynamics is inward-pointing <cit.> relative
to the convex hull of its neighbors' positions. There
are many publications addressing this type of dynamics, e.g., <cit.>.
Suppose control law (<ref>) is used, $q < r/2$, and $(\bar{B}_{q,3}(0))^n$ is strongly forward invariant for the dynamics of $x$.
Suppose there is a continuously differentiable function
$$W: \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}^+,$$
such that for any given $k,l \in \{1, \ldots 2^{n-1}\}$ and
$\bar{x} = [\bar{x}_1^T, \bar{x}_2^T, \ldots, \bar{x}_n^T]^T \in (\bar{B}_{q,3}(0))^n$,
* if
$$(i,j) \in \textnormal{arg}\max_{k' \in \mathcal{N}^{k}_i, l' \in \mathcal{N}^{l}_j}(W(\bar{x}_i,\bar{x}_j))$$
it holds that
\begin{align*}
\langle \nabla W(\bar{x}_i,\bar{x}_j), [\left(\sum_{k'\in\mathcal{N}^{k}_i} a_{ij}({y}_{k'}(\bar{x}) - {y}_{i}(\bar{x}))\right)^T, & \\
\left(\sum_{l'\in\mathcal{N}^{l}_j} a_{ij}({y}_{l'}(\bar{x}) - {y}_{j}(\bar{x}))\right)^T] \rangle & \leq 0
\end{align*}
* and equality holds if and only if $\bar{x}_i = \bar{x}_{k'}$ for all $k' \in \mathcal{N}^{k}_i$
and $\bar{x}_j = \bar{x}_{l'}$ for all $l' \in \mathcal{N}^{l}_j$,
then $(\bar{B}_{q,3}(0))^n \cap \mathcal{A}$ is globally uniformly asymptotically stable
relative to $(\bar{B}_{q,3}(0))^n$ if and only if
$\mathcal{G}_{\sigma(t)}$ is uniformly quasi-strongly connected.
The proof of Proposition <ref>, is omitted here, but follows,
up to small modifications, the
procedure in the proof of Theorem 2.22. in <cit.>.
Proof of Proposition <ref>:
Let us define the functions
\begin{align*}
V(x_i) = & \: \: \: \: x_i ^Tx_i \text{ and } \\
W(x_i,x_j) = & (z_j(x_j) - z_i(x_i))^T(z_j(x_j) - z_i(x_i)),
\end{align*}
Using Theorem <ref> and the function $V$ together with Proposition <ref>, along the lines of the
proof of Proposition <ref>, one can show that
$(\bar{B}_{q,3}(0))^n$ is strongly forward invariant
and if $\mathcal{G}_{\sigma(t)}$ is uniformly strongly connected, $(\bar{B}_{q,3}(0))^n \cap \mathcal{A}$ is
globally attractive relative to $(\bar{B}_{q,3}(0))^n$.
Now, since $(\bar{B}_{q,3}(0))^n$ is strongly forward invariant, one can use Theorem <ref> in order to show that $W$ satisfies the criteria in Proposition <ref>. The mapping
$$x_i \mapsto z_i$$
is a diffeomorphism on $(\bar{B}_{r,3}(0))^n$.
The set $(\bar{B}_{q,3}(0))^n \cap \mathcal{A}$ is globally
uniformly asymptotically stable relative to $(\bar{B}_{q,3}(0))^n$.
We can generalize the results in
Proposition <ref> and Proposition <ref>.
Up until now we have assumed that we first
fix a representation $y_i$, $y_{ij}$ and then we use the control laws
(<ref>) and (<ref>) for this representation.
Instead, at each switching time $\tau_k^i$ we can allow
the representation to switch also.
The following proposition addresses a special case when the
the rate of convergence is exponential.
Suppose $\mathcal{G}_{\sigma(t)}$ fulfills the following. At each
time $t$ and for each pair $(i,j)$, the edge $(i,j) \in \mathcal{E}_{\sigma(t)}$ or the edge $(j,i) \in \mathcal{E}_{\sigma(t)}$.
Suppose controller (<ref>) is used
and $g(\theta_i) \geq k\theta_i$ for some $k > 0$.
For $q < r/2$, the set
$\{(R_1, \ldots, R_n) \in (\bar{B}_{q}(I))^n: R_1 = \hdots = R_n\}$ is globally exponentially stable
relative to $(\bar{B}_{q}(I))^n$ for the closed loop dynamics of $(R_1, R_2, \ldots, R_n)$ with respect to the Riemannian metric on $SO(3)$.
Using the results in <cit.> it can be obtained that for the
Axis-Angle Representation the convergence is exponential also for general
uniformly quasi-strongly connected graphs, i.e., not only the
restricted class of graphs considered Proposition <ref>.
In Proposition <ref> (1), since we have assumed that $g$ is analytic, the condition $g(\theta_i) \geq k\theta_i$ can equivalently be formulated as
$g(\theta_i) = \mathcal{O}(\theta_i)$ as $\theta_i \rightarrow 0$. All the local representations
previously addressed fulfill this assumption, e.g., the
Axis-Angle Representation, the Rodrigues Parameters and the Unit Quaternions.
Before we prove Proposition <ref> we
formulate the following lemma.
Suppose $x \in (\bar{B}_{q,3}(0))^n$ where $q < r/2$. If
$$(i,j) \in \textnormal{arg}\max_{(k,l) \in \mathcal{V} \times \mathcal{V}}\|x_{kl}\|,$$
$$x_{ij}^Ty_{ik} \geq 0 \quad \text{ for all } \quad k.$$
The proof of Lemma <ref> follows more or less as a consequence of Theorem <ref> and is omitted here.
Proof of Proposition <ref>:
We already know from Proposition <ref> that the set
$(\bar{B}_{q,3}(0))^n$ is strongly forward invariant and
$(\bar{B}_{q,3}(0))^n \cap \mathcal{A}$ is globally uniformly
asymptotically stable relative to $(\bar{B}_{q,3}(0))^n$. What is
left to prove is that for the special structure of the graph considered,
the rate of convergence is exponential relative to $(B_{q}(I))^n$ when the Riemannian metric is used.
Let us define
$$\alpha = \min_{(k,l) \in \mathcal{V} \times \mathcal{V}}a_{kl},$$
$$V(x) = \max_{(k,l) \in \mathcal{V} \times \mathcal{V}}x_{kl}^Tx_{kl}.$$
At time $t$ let $(i,j)$ be such that $V(x(t)) = x^T_{ij}(t)x_{ij}(t)$.
$$x_{ij}^T\left(\omega_j - \omega_i\right) \leq -\alpha k V(x(t)),$$
where the last inequality is due to Lemma <ref> and the assumption on the
graph $\mathcal{G}_{\sigma(t)}$.
Now one can show that
$$D^+V(x(t)) \leq -\alpha k V(x(t)).$$
By using the Comparison Lemma, one can show that $V$ converges to
zero with exponential rate of convergence.
§.§ Illustrative example
In order to illustrate the convergence of the
rotations to the consensus set, an
illustrative example is constructed where the representation $(R - R^T)^{\vee}$ is chosen
both for control law (<ref>)
and (<ref>). The number
of agents is $5$ and the graph the
graphs were constructed in the same manner as
in Section <ref>. The initial
rotations are drawn from the uniform distribution over $B_{\pi/2}(I)$. The convergence to consensus is shown in Figure <ref>.
Convergence to consensus. For the same initial
rotations, controller (<ref>) and
controller (<ref>) are used. The left
plot shows the errors (in terms of Frobenius norm) between the
first rotation and the other rotations as a function of time
when controller (<ref>) is used;
in the right plot the same type of errors is shown when controller (<ref>) is used.
§.§ Translations
Here we address the controllers (<ref>) and (<ref>)
Controller (<ref>), despite
its appealing structure does in general not guarantee
consensus in the translations. In order to see this,
we consider the following example.
$$\dot{T}_i = \sum_{j \in \mathcal{N}^{\sigma^i(t)}_i}R_i^T(T_j -T_i).$$
\begin{align*}
& \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{bmatrix}T_i(0) = T_i(0) \text{ and } \\
R_i(t) = &
\begin{bmatrix}
-1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 1
\end{bmatrix} \quad \text{ for all } i,t.
\end{align*}
Then, for this particular choice of rotations and initial conditions
for the $T_i$,
$$\dot{T}_i = \sum_{j \in \mathcal{N}^{\sigma^i(t)}_i}(T_i -T_j).$$
$$\dot{T}^{\text{tot}} = (L(\mathcal{G}_{\sigma(t)}, \mathcal{A}) \otimes I)T^{\text{tot}},$$
which is unstable.
One partial result for controller (<ref>) is the following one.
By a change of coordinates one can prove that,
if all the rotations of the agents are the same
and constant and the translations are contained in
the linear subspace spanned by the rotational axis,
the translations converge asymptotically to consensus.
Controller (<ref>) delivers
a much stronger result.
Suppose controller (<ref>) is used
and the graph $\mathcal{G}_{\sigma(t)}$
is uniformly quasi-strongly connected. The set $\mathcal{A}$ is globally asymptotically stable stable.
The proof of the proposition is based on the fact that the closed loop dynamics
given by
$\dot{T}^{\text{tot}} = -L(\mathcal{G}_{\sigma(t)},A)T^{\text{tot}}.$
§ CONTROL ON THE DYNAMIC LEVEL FOR RIGID BODIES IN SPACE
In this section we construct control laws on the dynamic level
for the case of rigid bodies in space.
The dynamical equations for agent $i$ are given by
\begin{equation}\label{eq:s1}
\begin{cases}
\dot{R}_i = R_i\widehat{\omega}_i, \\
\dot{\omega}_i = J_i^{-1}\left(-\widehat{\omega}_iJ_i\omega_i + \boldsymbol{\tau}_i\right), \\
\dot{T}_i = R_iv_i, \\
\dot{v}_i = \frac{\boldsymbol{f}_i}{m_i} - \widehat{\omega}_iv_i,
\end{cases}
\end{equation}
where $J_i$ is the inertia matrix and $\boldsymbol{\tau}_i$ is the
control torque; $\omega_i$ is a state variable.
In the formation problem the goal is to reach consensus in the
$\tilde{(\cdot)}$-variables, and the dynamical equations
for those variables are
\begin{equation}\label{eq:s2}
\begin{cases}
\dot{\tilde{R}}_i & = \tilde{R}_i\widehat{\tilde{\omega}}_i, \\
\dot{\tilde{\omega}}_i & = R_i^{*}J_i^{-1}\left(-\left(R_i^{*T}\widehat{\tilde{\omega}}_i R_i^{*}\right)J_iR_i^{*T}\tilde{\omega}_i + \boldsymbol{\tau}_i\right), \\
\dot{\tilde{T}}_i & = \tilde{R}_i\tilde{v}_i \\
\dot{\tilde{v}}_i & = R_i^{*}\left (\frac{\boldsymbol{f}_i}{m_i} - R_i^{*T} \widehat{\tilde{\omega}}_i\left (\widehat{\tilde{\omega}}_iT_i^* + \tilde{v}_i\right )\right) - \left (\dot{\tilde{\omega}}_i\right )^{\wedge} T_i^*.
\end{cases}
\end{equation}
In this section, we strengthen the assumptions on
$\mathcal{G}_{\sigma(t)}$ by assuming it is time-invariant. Thus, we
denote the time-invariant (also referred to as constant or fixed) graph by $\mathcal{G}$.
The reason for choosing time-invariant graphs
is that we are now considering a second order system, and the methods we use here
are based on backstepping. In order to show stability, we introduce auxiliary error variables,
and in the case of a switching graph, these variables suffer from discontinuities.
One way to avoid this problem is to replace the
discontinuities with continuous in time transitions. This is however
not something we do here.
§.§ Rotations
Only the consensus problem and the first set
of equations, (<ref>), will be considered here. When performing
formation control, the presented control laws below, (<ref>) and (<ref>), are modified slightly.
In both control laws, all the variables should be replaced
by $\tilde{(\cdot)}$-variables, i.e., $x_i$ should be $\tilde{x}_i$ instead, $\tilde{\omega}_i$ should be ${\omega}_i$ instead and so on. The expression $J_i$( is replaced by $J_iR_i^T$(,
and the expression $\widehat{\omega}_iJ_i\omega_i$
is replaced by $\left(R_i^{*T}\widehat{\tilde{\omega}}_i R_i^{*}\right)J_iR_i^{*T}\tilde{\omega}_i$.
Based on the two kinematic control laws (<ref>) and (<ref>), we now propose two torque control laws
for each agent $i$,
where the first one is based on absolute rotations and the second one is based
on relative rotations. The control laws are
\begin{align}
\label{chapter2:torque:controller:3}
\boldsymbol{\tau}_i & = J_i(-x_i + \sum_{j \in \mathcal{N}_i}a_{ij}(L_{x_j}\omega_j - L_{x_i}\omega_i - \bar{\omega}_i)) + \widehat{\omega}_iJ_i\omega_i, \\
\label{chapter2:torque:controller:4}
\boldsymbol{\tau}_i & = J_i( -k_i\bar{\omega}_i' + \sum_{j \in \mathcal{N}_i}a_{ij}L_{-y_{ij}}\omega_{ij}) + \widehat{\omega}_iJ_i\omega_i.
\end{align}
The parameter $k_i$ is a positive gain. The error variables $\bar{\omega}_i$ and $\bar{\omega}_i'$ are by
\begin{align*}
\bar{\omega}_i & = \omega_i - \sum_{j \in \mathcal{N}_i}a_{ij}(x_j - x_i), \\
\bar{\omega}_i' & = \omega_i - \sum_{j \in \mathcal{N}_i}a_{ij}y_{ij}.
\end{align*}
The matrix $L_{y_{ij}}$ is the Jacobian matrix for
$y_{ij}$, i.e.,
\begin{align*}
\dot{y}_{ij} & = L_{-y_{ij}}\omega_{ij},
\end{align*}
$$\omega_{ij} = R_{ij}\omega_j - \omega_i$$
is the relative
angular velocity between agent $i$ and agent $j$.
In the following, the notation $(x_i,\bar{\omega}_i') = [x_i^T,\bar{\omega}_i'^T].$
We collect all the $x_i$ and $\bar{\omega}_i$ into $(x, \bar{\omega}) \in (B_{\pi,3})^n \times (\mathbb{R}^3)^n$
and all the $x_i$ and $\bar{\omega}_i'$ into $(x, \bar{\omega}') \in (B_{\pi,3})^n \times (\mathbb{R}^3)^n$.
Now, given $i \in \mathcal{V}$, the
right-hand side for $(x_i, \bar{\omega}_i)^T$
when the torque control law (<ref>) is used is
\begin{align*}
\dot{x}_i & = L_{x_i}\sum_{j \in \mathcal{N}_i}a_{ij}(x_j - x_i) + L_{x_i}\bar{\omega}_i, \\
\dot{\bar{\omega}}_i & = -x_i - \sum_{j \in \mathcal{N}_i}a_{ij}\bar{\omega}_i,
\end{align*}
whereas the closed loop system for $(x_i, \bar{\omega}'_i)^T$ when the torque
control law (<ref>) is used, is
\begin{align*}
\dot{x}_i & = L_{x_i}\sum_{j \in \mathcal{N}_i}a_{ij}y_{ij} + L_{x_i}\bar{\omega}_i', \quad \quad \: \: \: \\
\dot{\bar{\omega}}_i' & = -k_i\bar{\omega}_i'.
\end{align*}
We note that in (<ref>), each agent $i$ needs to know, not only the absolute
rotations of its neighbors, but also the angular velocities of its neighbors. This requirement
is fair, in the sense that in order to obtain the absolute rotations of the neighbors,
communication is in general necessary. In this case the angular velocities can also
be transmitted. In (<ref>), we see that each agent $i$ needs to know the relative
rotations, relative velocities to its neighbors and the angular velocity of itself. The assumption that agent $i$
knows its own angular velocity is quite strong the sense that this velocity is not to be regarded as relative information.
However, in practice the angular velocity is possible to measure without the knowledge of the
global frame $\mathcal{F}_W$. Thus, the angular velocity is local information.
Suppose $\mathcal{G}$ is strongly connected.
$$\max_{i \in \mathcal{V}} \: x_i^T(t_0)x_i(t_0) + \bar{\omega}_i(t_0)^T\bar{\omega}_i(t_0) \leq q < \pi,$$
i.e., $(x_i(t_0), \bar{\omega}_i(t_0))^T \in \bar{B}_{q,6}$ for all $i$ and some $q < \pi$, then if controller (<ref>) is used,
$\bar{B}_{q,6}$ is invariant for $(x(t), \bar{\omega}(t))$ and $x(t) \rightarrow \mathcal{A}$ and $\omega_i(t) \rightarrow 0$ for all
$i$ as $t \rightarrow \infty$.
In the multi-agent system at hand we have $n$ agents, where each agent $i$ has the state
$(x_i, \bar{\omega}_i)^T$. We first show the invariance of the ball $\bar{B}_{q, 6}$.
$$V((x_i, \bar{\omega}_i)^T) = \frac{1}{2}\left (x_i^Tx_i + \bar{\omega}_i^T\bar{\omega}_i \right ).$$
We see that
\begin{align*}
& \frac{d}{dt}V((x_i, \bar{\omega}_i)^T) \\
= & \sum_{j \in \mathcal{N}_i}a_{ij}(x_i, \bar{\omega}_i)(x_j - x_i, -\bar{\omega}_i)^T \\
= & \sum_{j \in \mathcal{N}_i}a_{ij}((x_i, \bar{\omega}_i)(x_j, 0)^T - (x_i, \bar{\omega}_i)(x_i, \bar{\omega}_i)^T) \\
\leq & \sum_{j \in \mathcal{N}_i}a_{ij}x_i^T(x_j - x_i).
\end{align*}
$$D^+f_{V,6}((x(t), \bar{\omega}(t))^T) \leq 0.$$
Now, by using the Comparison Lemma one can show the invariance.
In order to show the convergence, we define the following function
$$\bar{\gamma}(x, \bar{\omega}) = \sum_{i= 1}^n\xi_i (x_i^Tx_i + \bar{\omega}_i^T\bar{\omega}_i),$$
where $\xi = (\xi_1, \ldots, \xi_n)^T$ is the positive vector chosen such that (the symmetrical part of) $\text{diag}(\xi)L(\mathcal{G},A)$ is positive semi-definite.
We have that
$$\dot{\bar{\gamma}} = -{x}^T({L}' \otimes {I}_3){x} - \sum_{i= 1}^n\xi_i \sum_{j \in \mathcal{N}_i}a_{ij}\bar{\omega}_i^T\bar{\omega}_i.$$
By LaSalle's theorem, $(x(t), \bar{\omega}(t))^T$ will converge to the largest invariant set contained
$$\{(x, \bar{\omega})^T: \dot{\bar{\gamma}}((x, \bar{\omega})^T)~= 0\}$$
as the time goes to infinity. This largest invariant set is contained in the set $\{(x, \bar{\omega})^T: x \in \mathcal{A}, \bar{\omega} = 0\}$.
In the proof of Proposition <ref>, if we look at the dynamics of $(x, \bar{\omega})$, we see that the largest invariant set contained in $\{(x, \bar{\omega})^T: \dot{\bar{\gamma}}((x, \bar{\omega})^T)~= 0\}$ is actually the point $0$. Hence, the system will reach consensus in the
point $x = 0$.
Now let us turn to control law (<ref>).
Suppose $\mathcal{G}$ is quasi-strongly connected.
For any positive $r_1$ and $r_2$ such that $r_1 < r_2 < r/2$ and $q >0$,
there is a $k >0$ such that if $k_i \geq k$ and
$(x_i(t_0), \bar{\omega}_i'(t_0))^T \in \bar{B}_{r_1,3} \times \bar{B}_{q,3}$ for all $i$, then if controller (<ref>) is used
it holds that
$(x_i(t), \bar{\omega}_i'(t))^T \in \bar{B}_{r_2,3} \times \bar{B}_{q,3}$ for all $i, t \geq t_0$ and
$$(x(t), \bar{\omega}'(t))^T \rightarrow (\bar{B}_{r_2,3})^n \cap \mathcal{A} \times \{0\} \quad \text{ as } \quad t \rightarrow \infty.$$
Furthermore, $(\bar{B}_{r_2,3})^n \cap \mathcal{A} \times \{0\}$ is globally asymptotically stable relative to the largest
invariant set contained in $(\bar{B}_{r_2,3})^n \times (\bar{B}_{q,3})^n$ for the dynamics of $(x(t), \bar{\omega}'(t))^T$.
Proof of Proposition <ref>:
Let us define
$$\mathcal{D}^* \subset \mathcal{D} = (\bar{B}_{r_2,3})^n \times (\bar{B}_{q,3})^n,$$
as the largest invariant set
contained in $\mathcal{D}$. The set $\mathcal{D}^*$ is compact
and implicitly a function of $k$ (or the $k_i$).
Now we show that for a proper choice of the constant $k$, it holds that
$$(\bar{B}_{r_1,3})^n \times (\bar{B}_{q,3})^n \subset \mathcal{D}^*.$$
We assume without loss of generality that $t_0 = 0$, and note that
$$\|\bar{\omega}_i'(t)\| =\|\bar{\omega}_i'(0)\|\exp(-k_it) \leq q\exp(-k_it) \leq q\exp(-kt).$$
We choose
$$V(x_i(t)) = x_i^T(t)x_i(t).$$
By using Lemma <ref>, it is possible to show that there exists an interval $[0, t_1)$ on which it holds that
$$D^+(\max_iV(x(t))) \leq qr_2\exp(-kt).$$
By using the Comparison Principle, it follows that
$$\max_iV(x_i(t)) \leq \max_iV(x_i(0)) + qr_2\frac{(1- \exp(-kt))}{k}$$
on $[0, t_1)$.
Now if we choose $k \geq qr_2/(r_2 - r_1)$ we see that $\max_iV(x_i(t)) \leq r_2$
for $t \geq 0$, and we can choose $t_1 = \infty$.
In order to show the desired convergence we use
Theorem 10 in <cit.>, where $\mathcal{X} = \mathcal{D}^*$,
$\Gamma_2 = \mathcal{D}^* \cap ((\bar{B}_{r_2,3})^n \times \{0\})$ and $\Gamma_1 = \mathcal{D}^* \cap (\mathcal{A} \times \{0\})$.
§.§ Translations
Also in this section only the consensus problem and the first set
of equations, (<ref>), are considered here.
We introducing a generalized version of the control law
When the formation control
problem is considered the all variables are
replaced by $\tilde{(\cdot)}$-variables. Furthermore
the expression $m_i($ is replaced by
$m_iR_i^{*T}($, and the expression $\widehat{\omega}_iv_i$
is replaced by $\widehat{\tilde{\omega}}_i(\widehat{\tilde{\omega}}_iT_i^* + \tilde{v}_i)) + (\dot{\tilde{\omega}}_i)^{\wedge} T_i^*$.
The proposed consensus controller
\begin{align*}
\boldsymbol{f}_i = & m_i(- k_i\bar{v}_i + \sum_{j\in\mathcal{N}_i} a_{ij}R_i^T(R_jv_j - R_iv_i) \\
& - \sum_{j\in\mathcal{N}_i} a_{ij}\widehat{\omega}_i R_i^T(T_j - T_i) + \widehat{\omega}_iv_i),
\end{align*}
$$\bar{v}_i = v_i - \sum_{j\in\mathcal{N}_i} a_{ij}{T}_{ij}. \hspace{5mm}$$
The closed loop dynamics is
\begin{align*}
\hspace{10mm} \dot{T}_i & = \sum_{j\in\mathcal{N}_i} a_{ij}(T_j - T_i) + R_i(t)\bar{v}_i \\
\dot{\bar{v}}_i & -k_i\bar{v}_i.
\end{align*}
By treating the time as a variable $z$, we get the following system
\begin{align*}
\dot{z} & = 1 \\
\hspace{10mm} \dot{T}_i & = \sum_{j\in\mathcal{N}_i} a_{ij}(T_j - T_i) + R_i(z)\bar{v}_i \\
\dot{\bar{v}}_i & -k_i\bar{v}_i.
\end{align*}
Let the state of the entire system be $(z,T^{\text{tot}},\bar{v}^{\text{tot}})$,
where $v^{\text{tot}} = [\bar{v}_1^T(t), \bar{v}_2^T(t), \ldots, \bar{v}_n^T(t)]^T \in \mathbb{R}^{3n}$.
Suppose that $R_i(z)$ is well behaved, in the sense that the right-hand side of the dynamics for $(z,T^{\text{tot}},{v}^{\text{tot}})$
is locally Lipschitz, then
the set $\mathbb{R} \times \mathcal{A} \times 0$ is globally asymptotically stable for the system.
Let the state space be $\mathcal{X} = \mathbb{R} \times \mathbb{R}^{3n} \times \mathbb{R}^{3n}$. We define the two closed subsets $\Gamma_1 \subset \Gamma_2$ of $\mathcal{X}$ as follows
\begin{align*}
\Gamma_1 & = \mathbb{R} \times \mathcal{A} \times 0 \\
\Gamma_2 & = \mathbb{R} \times \mathbb{R}^{3n} \times 0 \\
\end{align*}
It is easy to show that $\Gamma_2$ is globally asymptotically stable
relative to $\mathcal{X}$ and $\Gamma_1$ is globally asymptotically
stable relative to $\Gamma_2$.
Now the desired result follows from Theorem 10 in <cit.>.
§.§ Illustrative examples
In Figure <ref> the
convergence to consensus is shown when controllers
(18), (19) and (20) are used. In the simulations, five agents were considered and a random quasi-strongly connected graph was used.
The convergence
to consensus is shown for the rotations, left plots,
and the translation, right plots, when controller
(18) was used together with controller (20) and controller
(19) was used together with controller (20).
The left plots shows the Euclidean distance between
$x_i(t)$ and $x_1(t)$
for $i = 2, \ldots, 5$, and the right plots
show the Euclidean distance between
$T_i$ and $T_1$ as a function of time
for $i = 2, \ldots, 5$.
In controller (19) as well as controller (20)
the $k_i$ were chosen to $3$ for all $i$.
The adjacency matrix was chosen to that of a quasi-strongly connected graph with
entries equal to $0$, $1$ or $2$.
In controller (19) the representation
$(R - R^T)^{\vee}$ was used as the local representation for the $y_{ij}$.
§ CONCLUSIONS
This work has considered the consenus and formation problems
on $SE(3)$ for multi-agent systems with switching
interaction topologies. By a change of coordinates
it was shown that the consensus problem
can be seen as equivalent to the formation problem.
Any control law designed for the consensus problem can, after
change of coordinates, be used for the formation problem.
New kinematic control laws
have been presented as well as new convergence results. It has been shown that the same type of
control laws can be used for many popular local representations of $SO(3)$ such as the Modified Rodrigues Parameters and the Axis-Angle Representation. It has been shown that some
of the control laws guarantee almost global convergence. For non-switching topologies, the
kinematic control laws have been extended to torque
and force control laws for rigid bodies in space.
The proposed control approaches have been justified
by numerical simulations.
Convergence to consensus.
|
1511.00447
|
$^1$ School of Physics, University of New South Wales, Sydney 2052, Australia
$^2$ Mainz Institute for Theoretical Physics, Johannes Gutenberg University Mainz, D 55122 Mainz, Germany
We outline new laser interferometer measurements to search for variation of the electromagnetic fine-structure constant $\alpha$ and particle masses (including a non-zero photon mass).
We propose a strontium optical lattice clock – silicon single-crystal cavity interferometer as a novel small-scale platform for these new measurements.
Our proposed laser interferometer measurements, which may also be performed with large-scale gravitational-wave detectors, such as LIGO, Virgo, GEO600 or TAMA300, may be implemented as an extremely precise tool in the direct detection of scalar dark matter that forms an oscillating classical field or topological defects.
§ INTRODUCTION
Dark matter remains one of the most important unsolved problems in contemporary physics. Astronomical observations indicate that the energy density of dark matter exceeds that of ordinary matter by a factor of five <cit.>. Extensive laboratory searches for weakly interacting massive particle (WIMP) dark matter through scattering-off-nuclei experiments have failed to produce a strong positive result to date, see, e.g., Refs. <cit.>, which has spurred significant interest of late in searching for alternate well-motivated forms of dark matter, such as ultralight (sub-eV mass) spin-0 particles that form either an oscillating classical field or topological defects, see, e.g., Refs. <cit.>.
The idea that the fundamental constants of Nature might vary with time can be traced as far back as the large numbers hypothesis of Dirac, who hypothesised that the gravitational constant $G$ might be proportional to the reciprocal of the age of the Universe <cit.>. More contemporary theories, which predict a variation of the fundamental constants on cosmological timescales, typically invoke a (nearly) massless underlying dark energy-type field, see, e.g., the review <cit.> and the references therein.
Most recently, a new model for the cosmological evolution of the fundamental constants of Nature has been proposed in Ref. <cit.>, in which the interaction of an oscillating classical scalar dark matter field with ordinary matter via quadratic interactions produces both `slow' linear-in-time drifts and oscillating-in-time variations of the fundamental constants <cit.>.
Topological defects, which are stable, extended-in-space forms of dark matter that consist of light scalar dark matter fields stabilised by a self-interaction potential <cit.> and which interact with ordinary matter, produce transient-in-time
variations of the fundamental constants <cit.>.
The oscillating-in-time and transient-in-time variations of the fundamental constants produced by scalar dark matter can be sought for in the laboratory using high-precision measurements, which include atomic clocks <cit.>, highly-charged ions <cit.>, molecules <cit.> and nuclear clocks <cit.>, in which two transition frequencies are compared over time.
Instead of comparing two transition frequencies over time, we may instead compare a photon wavelength with an interferometer arm length, in order to search for variations of the fundamental constants <cit.> (see also <cit.> for some other applications).
In the present work, we outline new laser interferometer measurements to search for variation of the electromagnetic fine-structure constant and particle masses (including a non-zero photon mass).
We propose a strontium optical lattice clock – silicon single-crystal cavity interferometer as a novel small-scale platform for these new measurements.
The small-scale hydrogen maser – cryogenic sapphire oscillator system <cit.> and large-scale gravitational-wave detectors, such as LIGO-Virgo <cit.>, GEO600 <cit.>, TAMA300 <cit.>, eLISA <cit.> or the Fermilab Holometer <cit.>, can also be used as platforms for some of our newly proposed measurements.
§ GENERAL THEORY
Unless explicitly indicated otherwise, we employ the natural units $\hbar = c = 1$ in the present work.
Alterations in the electromagnetic fine-structure constant $\alpha = e^2/\hbar c$, where $-e$ is the electron charge, $\hbar = h/2\pi$ is the reduced Planck constant and $c$ is the photon speed, or particle masses (including a non-zero photon mass $m_\gamma$) produce alterations in the accumulated phase of the light beam inside an interferometer $\Phi = \omega L / c$, since an atomic transition frequency $\omega$ and length of a solid $L \sim N a_{\textrm{B}}$, where $N$ is the number of atoms and $a_{\textrm{B}} = \hbar^2/m_e e^2$ is the Bohr radius ($m_e$ is the electron mass), both depend on the fundamental constants and particle masses. Alterations in the accumulated phase can be expressed in terms of the sensitivity coefficients $K_X$, which are defined by:
\begin{equation}
\label{Sens_coeffns_defn}
\frac{\delta \Phi }{\Phi} = \sum_{X_i = \alpha, m_e, ...} K_{X_i} \frac{\delta X_i}{X_i} + K_{m_\gamma} \left( \frac{m_\gamma}{m_e} \right)^2 ,
\end{equation}
where the sum runs over all relevant fundamental constants $X_i = \alpha, m_e, ...$ (except photon mass).
The sensitivity coefficients depend on the specific measurement that is performed.
In order to define the variation of dimensionful parameters, such as $m_e$, we assume that such variations are due to the interactions of dark matter with ordinary matter, see, e.g., Ref. <cit.>.
The sensitivity coefficients, which we derive below in Sections <ref> and <ref>, are for single-arm interferometers, but are readily carried over to the case of two-arm Michelson-type interferometers, for which the observable quantity is the phase difference $\Delta \Phi = \Phi_1 - \Phi_2$ between the two arms, as we illustrate with a couple of examples in Section <ref>.
One intuitively expects that multiple reflections should enhance observable effects due to variation of the fundamental constants by the effective mean number of passages $N_{\textrm{eff}}$.
This can be readily verified by the following simple derivation.
For multiple reflections of a continuous light source that forms a standing wave (in the absence of variation of the fundamental constants), we sum over all possible number of reflections $n$:
\begin{equation}
\label{App-A_derivation1}
\sum_{n=1}^{\infty} \exp \left[ - n \left( \kappa - i \Phi \right) \right] = \frac{1}{\exp \left( \kappa - i \Phi \right) - 1} ,
\end{equation}
where $\kappa \equiv 1/N_{\textrm{eff}}$ is the attenuation factor that accounts for the loss of light amplitude after a single to-and-back passage along the length of the arm, and $\Phi = 2 \pi m + \delta \Phi$ ($m$ is an integer) is the phase accumulated by the light beam in a single to-and-back passage along the length of the arm. For a large effective mean number of passages, $N_{\textrm{eff}} \gg 1$, and for sufficiently small deviations in the accumulated phase, $N_{\textrm{eff}} \cdot \delta \Phi \ll 1$, the sum in Eq. (<ref>) can be written as:
\begin{align}
\label{App-A_derivation2}
\sum_{n=1}^{\infty} \exp \left[ - n \left( \kappa - i \Phi \right) \right] &\simeq N_{\textrm{eff}} \exp \left(i N_{\textrm{eff}} \cdot \delta \Phi \right) ,
\end{align}
from which it is evident
that the effects of small variations in the accumulated phase are enhanced by the factor $N_{\textrm{eff}}$.
§ VARIATION OF THE ELECTROMAGNETIC FINE-STRUCTURE CONSTANT AND PARTICLE MASSES
Variation of $\alpha$ and particle masses alters the accumulated phase through alteration of $\omega$ and $L \sim N a_{\textrm{B}}$.
There are four main classes of experimental configurations to consider, depending on whether the frequency of light inside an interferometer is determined by a specific atomic transition (i.e., when the high-finesse cavity length is stabilised to an atomic transition) or by the length of a resonator (i.e., when the laser is stabilised to a high-finesse cavity), and whether the interferometer arm length is allowed to vary freely (i.e., allowed to depend on the length of the solid spacer between the mirrors) or its fluctuations are deliberately shielded (i.e., the arm length is made independent of the length of the solid spacer between the mirrors, e.g., through the use of a multiple-pendulum mirror system). We consider each of these configurations in turn.
§.§ Configuration A (atomic transition frequency, free arm length)
The simplest case is when the frequency of light inside an interferometer is determined by an optical atomic transition frequency and the interferometer arm length is allowed to vary freely (i.e., allowed to depend on the length of the solid spacer between the mirrors).
A strontium clock – silicon cavity interferometer in its standard mode of operation falls into this category.
In this case, the atomic transition wavelength and arm length are compared directly:
\begin{align}
\label{Phase_accumulated_1}
\Phi = \frac{\omega L}{c} \propto \left(\frac{e^2}{a_{\textrm{B}} \hbar} \right) \left(\frac{N a_{\textrm{B}}}{c} \right) = N \alpha ,
\end{align}
where the optical atomic transition frequency $\omega$ is proportional to the atomic unit of frequency $e^2/a_{\textrm{B}} \hbar$.
Variation of $\alpha$ thus gives rise to the following phase shift:
\begin{align}
\label{Sensitivity_coefficient_1}
\delta\Phi \simeq \Phi \frac{\delta \alpha}{\alpha} .
\end{align}
We note that the effect of variation of $\alpha$ already appears at the non-relativistic level in Eq. (<ref>), with the corresponding sensitivity coefficient
$K_\alpha = 1$.
For systems consisting of light elements, the relativistic corrections to this sensitivity coefficient are small and can be neglected.
This is in stark contrast to optical clock comparison experiments, for which $K_\alpha = 0$ in the non-relativistic approximation and the contributions to $K_\alpha$ arise solely from relativistic corrections <cit.>.
For a strontium clock – silicon cavity interferometer, which operates on the $^{87}$Sr $^{1}S_0$ $-$ $^{3}P_0$ transition ($\lambda = 698$ nm) and for which the cavity length is $L = 0.21$ m <cit.>, the phase shift in Eq. (<ref>) for a single to-and-back passage of the light beam is:
\begin{align}
\label{Sensitivity_coefficient_1a}
\delta \Phi \simeq 3.8 \times 10^6 \frac{\delta \alpha}{\alpha} .
\end{align}
For comparison, in a large-scale gravitational-wave detector of length $L = 4$ km and operating on a typical atomic optical transition frequency, the phase shift for a single to-and-back passage of the light beam is:
\begin{align}
\label{Sensitivity_coefficient_1b}
\delta \Phi \sim 10^{11} \frac{\delta \alpha}{\alpha} .
\end{align}
As noted in Section <ref>, multiple reflections enhance the coefficients in Eqs. (<ref>) and (<ref>) by the effective mean number of passages $N_{\textrm{eff}}$, which depends on the reflectivity properties of the mirrors used.
For large-scale interferometers, this enhancement factor is $N_{\textrm{eff}} \sim 10^2$.
For small-scale interferometers with highly-reflective mirrors, this enhancement factor can be considerably larger: $N_{\textrm{eff}} \sim 10^5$.
Another possible system in this category is the hydrogen maser – cryogenic sapphire oscillator system, which operates on the $^{1}$H ground state hyperfine transition:
\begin{equation}
\label{omega_H_hf}
\omega \propto \left(\frac{e^2}{a_{\textrm{B}} \hbar}\right) \left[\alpha^2 F_{\textrm{rel}}(Z\alpha)\right] \left[ \mu_p \frac{m_e}{m_p} \right] ,
\end{equation}
where $F_{\textrm{rel}}(Z\alpha) \simeq 1$ is the relativistic Casimir factor and $\mu_p$ is the dimensionless magnetic dipole moment of the proton in units of the nuclear magneton.
In this case, changes in the measured phase have the following dependence on changes in the fundamental constants:
\begin{equation}
\label{Sensitivity_coefficient_1CSO}
\frac{\delta \Phi}{\Phi} \simeq 3 \frac{\delta \alpha}{\alpha} + \frac{\delta m_e}{m_e} - 0.14 \frac{\delta m_q}{m_q} ,
\end{equation}
where $m_q = (m_u + m_d) / 2$ is the averaged light quark mass, and where we have used the calculated values $\delta \mu_p / \mu_p = -0.09 \delta m_q / m_q$ <cit.> and $\delta m_p / m_p = +0.05 \delta m_q / m_q$ <cit.>.
If one performs two simultaneous interferometry experiments with two different transition lines, using the same set of mirrors, then one may search for variations of the fundamental constants associated with changes in the atomic transition frequencies:
\begin{equation}
\label{seismic_shield}
\delta X = \frac{c (\omega_A \delta \Phi_B - \omega_B \delta \Phi_A)}{L (\omega_A \frac{\partial \omega_B}{\partial X} - \omega_B \frac{\partial \omega_A}{\partial X})} .
\end{equation}
In particular, note that shifts in the arm lengths (due to variation of the fundamental constants or undesired effects, such as seismic noise or tidal effects) cancel in Eq. (<ref>).
We also note that atomic clock transition frequencies may also be compared by locking lasers to the atomic transitions and using phase coherent optical mixing and frequency comb techniques to measure the laser frequency difference/ratio.
§.§ Configuration B (atomic transition frequency, fixed arm length)
If fluctuations in the arm length are deliberately shielded (i.e., the arm length is made independent of the length of the solid spacer between the mirrors, e.g., through the use of a multiple-pendulum mirror system), but $\omega$ is still determined by an atomic transition frequency, then changes in the measured phase $\Phi \propto \omega/c \propto m_e e^4 / \hbar^3 c = (m_e c / \hbar) \cdot (e^2 / \hbar c)^2$ have the following dependence on changes in the fundamental constants:
\begin{equation}
\label{Sensitivity_coefficient_4}
\frac{\delta \Phi}{\Phi} \simeq \frac{\delta m_e}{m_e} + 2 \frac{\delta \alpha}{\alpha} .
\end{equation}
§.§ Configuration C (resonator-determined wavelength, free arm length)
When a laser is locked to a resonator mode determined by the length of the resonator, $\omega$ is determined by the length of the resonator, which changes if the fundamental constants change. In the non-relativistic limit, the wavelength and arm length (as well as the size of Earth) have the same dependence on the Bohr radius, and so there are no observable effects if changes of the fundamental constants are slow (adiabatic) and if the interferometer arm length is allowed to vary freely (i.e., allowed to depend on the length of the solid spacer between the mirrors).
Indeed, this may be viewed as a simple change in the measurement units. Transient effects due to the passage of topological defects may still produce effects, since changes in $\omega$ and $L$ may occur at different times.
The sensitivity of laser interferometry to non-transient effects is determined by relativistic corrections, which we estimate as follows. The size of an atom $R$ is determined by the classical turning point of an external atomic electron. Assuming that the centrifugal term $\sim 1/R^2$ is small at large distances, we obtain $(Z_i+1)e^2/R = -E$, where $E$ is the energy of the external electron and $Z_i $ is the net charge of the atomic species (for a neutral atom, $Z_i=0$). This gives the relation: $\delta R/ R = \delta (E/e^2) / |E/e^2|$.
The single-particle relativistic correction to the energy in a many-electron atomic species is given by <cit.>:
\begin{equation}
\label{rel-correxn_FS}
\Delta_n \simeq E_n \frac{(Z\alpha)^2}{\nu (j+1/2)} ,
\end{equation}
where $E_n = - m_e e^4 (Z_i+1)^2 / 2 \hbar^2 \nu^2$ is the energy of the external atomic electron,
$j$ is its angular momentum, $Z$ is the nuclear charge, and $\nu \sim 1$ is the effective principal quantum number. Variation of $\alpha$ thus gives rise to the following phase shift:
\begin{equation}
\label{Sensitivity_coefficient_2}
\frac{\delta \Phi}{\Phi} \simeq 2\alpha^2 \left[\frac{Z_{\textrm{res}}^2}{\nu_{\textrm{res}} (j_{\textrm{res}} + 1/2)} - \frac{Z_{\textrm{arm}}^2}{\nu_{\textrm{arm}} (j_{\textrm{arm}} + 1/2)} \right] \frac{\delta \alpha}{\alpha} .
\end{equation}
Here $Z_{\textrm{res}}$ is the atomic number of the atoms that make up the solid spacer between the mirrors of the resonator, while $Z_{\textrm{arm}}$ is the atomic number of the atoms that make up the arm.
Note that the sensitivity coefficient depends particularly strongly on the factor $Z^2$. $\left|K_\alpha\right| \ll 1$ for light atoms and may be of the order of unity in heavy atoms.
§.§ Configuration D (resonator-determined wavelength, fixed arm length)
If fluctuations in the arm length are deliberately shielded (i.e., the arm length is made independent of the length of the solid spacer between the mirrors) and $\omega$ is determined by the length of the resonator, then changes in the measured phase $\Phi \propto 1/\lambda \propto 1/a_{\textrm{B}}$ have the following dependence on changes in the fundamental constants:
\begin{equation}
\label{Sensitivity_coefficient_3}
\frac{\delta \Phi}{\Phi} \simeq \frac{\delta m_e}{m_e} + \frac{\delta \alpha}{\alpha} .
\end{equation}
A large-scale gravitational-wave detector (such as LIGO, Virgo, GEO600 or TAMA300) in its standard mode of operation falls into this category.
§ NON-ZERO PHOTON MASS
A non-zero photon mass alters the accumulated phase through alteration of $\omega$, $L = N R$ (where $R$ is the atomic radius) and $c$. In particular, if a non-zero photon mass is generated due to the interaction of photons with slowly moving dark matter ($v_{\textrm{DM}} \ll 1$), then the energy and momentum of the photons are approximately conserved and the photon speed changes according to:
\begin{equation}
\label{delta_c/c}
\delta c \simeq - \frac{m_\gamma^2 }{2 \omega^2 } .
\end{equation}
The effects of a non-zero photon mass in atoms are more subtle. The potential of an atomic electron changes from Coulomb to Yukawa-type:
\begin{align}
\label{Coulomb_to_Yukawa}
&V_{\textrm{Coulomb}} (r) = \sum_{i} \frac{e^2}{|\v{r} - \v{r}_i|} - \frac{Ze^2}{r} , \\
=> ~ &V_{\textrm{Yukawa}} (r) = \sum_{i} \frac{ e^{-m_\gamma |\v{r} - \v{r}_i| } e^2 }{|\v{r} - \v{r}_i|} - \frac{ e^{-m_\gamma r} Ze^2 }{r} ,
\end{align}
where the sum runs over all remaining atomic electrons.
For $m_\gamma r \ll 1$, the leading term of the corresponding perturbation reads (we omit the constant terms, which do not alter the atomic transition frequencies and wavefunctions):
\begin{equation}
\label{Yukawa_perturbation}
\delta V (r) = \frac{e^2 m_\gamma^2 }{2 } \left[ \sum_{i}{|\v{r} - \v{r}_i|} - Zr \right] ,
\end{equation}
which for a neutral atom takes the asymptotic forms:
\begin{eqnarray}
\label{Yukawa_perturbation_asymptotic}
\delta V (r) \simeq \left\{ \begin{array}{ll}
- Z e^2 m_\gamma^2 r /2 & \textrm{when $r \ll a_{\textrm{B}}/Z^{1/3}$,}\\
- e^2 m_\gamma^2 r /2 & \textrm{when $r \gg a_{\textrm{B}}/Z^{1/3}$.}
\end{array} \right.
\end{eqnarray}
In the semiclassical approximation, it is straightforward to confirm that the dominant contribution to the expectation value of the operator (<ref>) comes from large distances, $r \gg a_{\textrm{B}}/Z^{1/3}$, where the external electron sees an effective charge of $Z_{\textrm{eff}} = 1$. Therefore, the shift in an atomic energy level $A$ is simply:
\begin{equation}
\label{Yukawa_energy_shift}
\delta E_A \simeq - \frac{e^2 m_\gamma^2 R_A}{2} ,
\end{equation}
where $R_A = \left< A \left| r \right|A \right> $ is the expectation value of the radius operator for state $A$. Typically, $R_A \sim \textrm{several} ~ a_{\textrm{B}}$.
Assuming that the perturbation (<ref>) is adiabatic and that the the dominant contribution to the matrix elements $\left< n \left| \delta V \right|A \right>$ comes from large distances, application of time-independent perturbation theory gives the following shift in the size of the atomic orbit for state $A$:
\begin{align}
\label{delta_R_Yukawa}
\delta R_A \simeq -m_\gamma^2 \sum_{n \ne A} \frac{\left< A \left| er \right|n \right> \left< n \left| er \right|A \right>}{E_{A}^{(0)} - E_{n}^{(0)}} \sim m_\gamma^2 \alpha_A ,
\end{align}
where $\alpha_A$ is the static dipole polarisability of state $A$. Static dipole polarisabilities for the electronic ground states of neutral atoms range from $4.5$ $a_{\textrm{B}}^3$ in hydrogen to $400$ $a_{\textrm{B}}^3$ in caesium <cit.>.
§.§ Configuration A (atomic transition frequency, free arm length)
If $\omega$ is determined by an atomic transition frequency and the interferometer arm length is allowed to vary freely (i.e., allowed to depend on the length of the solid spacer between the mirrors), then a non-zero photon mass produces the following changes in the measured phase $\Phi = \omega L/c$:
\begin{align}
\label{Sensitivity_coefficient_1B}
\frac{\delta \Phi}{\Phi} &\simeq \frac{e^2 m_\gamma^2 (R_{f} - R_{i} ) }{2 \omega} + \frac{m_\gamma^2 \alpha_{\textrm{arm}} }{R_{\textrm{arm}} } + \frac{m_\gamma^2}{2 \omega^2} \simeq \frac{m_\gamma^2}{2 \omega^2} ,
\end{align}
where $R_{f} - R_{i} = \left< f \left| r \right|f \right> - \left< i \left| r \right|i \right>$ is the difference in the orbital size between the final and initial states involved in the radiative atomic transition, and $\alpha_{\textrm{arm}}$ is the static dipole polarisability of the atoms that make up the arm.
The three separate contributions in Eq. (<ref>) scale roughly in the ratio ${\alpha^2:\alpha^2:1}$, respectively, meaning that the contribution from the change in the photon speed dominates.
§.§ Configuration B (atomic transition frequency, fixed arm length)
If fluctuations in the arm length are deliberately shielded (i.e., the arm length is made independent of the length of the solid spacer between the mirrors), but $\omega$ is still determined by an atomic transition frequency, then a non-zero photon mass produces the following changes in the measured phase $\Phi \propto \omega/c$:
\begin{align}
\label{Sensitivity_coefficient_4B}
\frac{\delta \Phi}{\Phi} &\simeq \frac{e^2 m_\gamma^2 (R_{f} - R_{i} ) }{2 \omega} + \frac{m_\gamma^2}{2 \omega^2} \simeq \frac{m_\gamma^2}{2 \omega^2} ,
\end{align}
where we again note that the contribution from the change in the photon speed dominates.
§.§ Configuration C (resonator-determined wavelength, free arm length)
If $\omega$ is determined by the length of the resonator and the interferometer arm length is allowed to vary freely (i.e., allowed to depend on the length of the solid spacer between the mirrors), then a non-zero photon mass produces the following changes in the measured phase $\Phi = 2 \pi L / \lambda$:
\begin{equation}
\label{Sensitivity_coefficient_2B}
\frac{\delta \Phi}{\Phi} \sim m_\gamma^2 \left( \frac{\alpha_{\textrm{arm}} }{R_{\textrm{arm}}} - \frac{\alpha_{\textrm{res}} }{R_{\textrm{res}}} \right) .
\end{equation}
Here $\alpha_{\textrm{res}}$ is the static dipole polarisability of the atoms that make up the solid spacer between the mirrors of the resonator.
The phase shift in Eq. (<ref>) is suppressed by the factor $\sim \alpha^2$ in the static limit (compare with Eqs. (<ref>) and (<ref>) above). However, for time-dependent effects, the phase shift can be significantly larger (see the examples in Section <ref>).
§.§ Configuration D (resonator-determined wavelength, fixed arm length)
If fluctuations in the arm length are deliberately shielded (i.e., the arm length is made independent of the length of the solid spacer between the mirrors) and $\omega$ is determined by the length of the resonator, then a non-zero photon mass produces the following changes in the measured phase $\Phi \propto 1/\lambda$:
\begin{equation}
\label{Sensitivity_coefficient_3B}
\frac{\delta \Phi}{\Phi} \sim - m_\gamma^2 \frac{\alpha_{\textrm{res}} }{R_{\textrm{res}}} .
\end{equation}
Similarly to Eq. (<ref>), the phase shift in Eq. (<ref>) is also suppressed by the factor $\sim \alpha^2$ in the static limit. However, we again note that the phase shift can be significantly larger for time-dependent effects (see Section <ref>).
§ SPECIFIC EXAMPLES
§.§ Oscillating classical dark matter (effects of spatial coherence)
Oscillating classical dark matter exhibits not only temporal coherence <cit.>, but also spatial coherence, with a coherence length given by: $l_{\textrm{coh}} \sim 2\pi / m_\phi v_{\textrm{vir}} \sim 10^3 \cdot 2\pi / m_\phi$, where $m_\phi$ is the dark matter particle mass, and a virial (root-mean-square) speed of $v_{\textrm{vir}} \sim 10^{-3} $ is typical in our local galactic neighbourhood. Our Solar System travels through the Milky Way (and hence relative to galactic dark matter) at a comparable speed $\left< v \right> \sim v_{\textrm{vir}} \sim 10^{-3} $. An oscillating scalar dark matter field takes the form:
\begin{equation}
\label{Coh_spat_DM_field}
\phi \left(\v{r}, t\right) \simeq \phi_0 \cos \left( m_\phi t - m_\phi \left<\v{v}\right> \cdot \v{r} \right) ,
\end{equation}
meaning that measurements performed on length scales $l \lesssim 2\pi / m_\phi v_{\textrm{vir}}$ are sensitive to dark matter-induced effects that arise from differences in the spatial phase term $m_\phi \left<\v{v}\right> \cdot \v{r}$ at two or more points.
(Color online) Passage of dark matter directly onto an arm of a gravitational-wave detector ($L_1 = L_2 = L$).
As a specific example, we consider measurements performed using a large-scale gravitational-wave detector with equal arm lengths that are deliberately shielded from fluctuations, $L_1 = L_2 = L = \textrm{constant}$, and with the emitted photon wavelength determined by the length of the resonator.
Since we are considering slowly moving dark matter ($v_{\textrm{DM}} \ll 1$), changes in the wavelength of the travelling photon are related to changes in $c$ by: $\delta \lambda / \lambda \simeq \delta c \simeq - [m_\gamma(\v{r},t)]^2 / 2 \omega^2$, where the interaction between the photon field and $\phi^2$ may be interpreted as the varying photon mass: $[m_\gamma \left(\v{r}, t\right)]^2 = (m_\gamma)_{\textrm{max}}^2 {\cos^2 \left( m_\phi t - m_\phi \left<\v{v}\right> \cdot \v{r} \right)} $.
For the simplest case when the dark matter is incident directly onto one of the detector arms as shown in Fig. <ref>, the shift in the accumulated phase difference between the two arms is given by:
\begin{equation}
\label{LIGO_example_A}
\delta \left( \Phi_1 - \Phi_2 \right) = \frac{2\pi}{\lambda} \int_{z_0}^{z_0 + L} \left[ \frac{\delta \lambda (z)}{\lambda} - \frac{\delta \lambda (z_0)}{\lambda} \right] dz ,
\end{equation}
and to leading order we find:
\begin{equation}
\label{LIGO_example_B}
\frac{\delta \left( \Phi_1 - \Phi_2 \right)}{\Phi} \simeq \frac{(m_\gamma)_{\textrm{max}}^2 m_\phi \left< v \right> L}{4 \omega^2} \sin \left( 2 m_\phi t + 2 m_\phi \left< v \right> z_0 \right) .
\end{equation}
The shift in the accumulated phase difference between the two arms in Eq. (<ref>) is suppressed by the factor $m_\phi \left< v \right> L < 1$.
§.§ Topological defect dark matter
Topological defect dark matter is intrinsically coherent, both temporally and spatially. As a specific example, we again consider measurements performed using a large-scale gravitational-wave detector with equal arm lengths that are deliberately shielded from fluctuations and with the emitted photon wavelength determined by the length of the resonator. For the case of a 2D domain wall with a Gaussian cross-sectional profile of root-mean-square width $d \sim 1/m_\phi$ and which travels slowly ($v_{\textrm{TD}} \ll 1$) in the geometry shown in Fig. <ref>, the interaction between the photon field and $\phi^2$ may be interpreted as the varying photon mass: $[m_\gamma \left(z, t\right)]^2 = (m_\gamma)_{\textrm{max}}^2 {\exp[-(z + vt)^2 / d^2]} $.
Calculating the shift in the accumulated phase difference between the two arms, Eq. (<ref>), we find to leading order:
\begin{align}
\label{LIGO_example_C}
&\frac{\delta \left( \Phi_1 - \Phi_2 \right)}{\Phi} \simeq \frac{(m_\gamma)_{\textrm{max}}^2 }{2 \omega^2} \left\{ \exp\left[- \frac{\left(z_0 + tv \right)^2}{d^2}\right] \right. \notag \\
&- \left. \frac{\sqrt{\pi}d}{2L} \left[\textrm{erf} \left( \frac{L + tv +z_0}{ d} \right) - \textrm{erf} \left( \frac{tv +z_0}{ d} \right) \right] \right\}
\end{align}
where erf is the standard error function, defined as $\textrm{erf}(x) = \left( 2/\sqrt{\pi} \right) \int_0^x e^{-u^2} du$.
The shift in the accumulated phase difference between the two arms in Eq. (<ref>) is largest for $d \sim L$. For $d \gg L$, the phase shift in (<ref>) is suppressed by the factor $L/d \ll 1$. In the case when $d \ll L$, the phase shift in (<ref>) is suppressed by the factor $d/L \ll 1$ when the topological defect envelops arm 2 but remains far away from arm 1; however, at the times when the topological defect envelops arm 1, there is no such suppression.
§ CONCLUSIONS
We have outlined new laser interferometer measurements to search for variation of the electromagnetic fine-structure constant $\alpha$ and particle masses (including a non-zero photon mass).
We have proposed a strontium optical lattice clock – silicon single-crystal cavity interferometer as a novel small-scale platform for these new measurements.
Our proposed laser interferometer measurements, which may also be performed with large-scale gravitational-wave detectors, such as LIGO, Virgo, GEO600 or TAMA300, may be implemented as an extremely precise tool in the direct detection of scalar dark matter.
For oscillating classical scalar dark matter, a single interferometer is sufficient in principle, while for topological defects, a global network of interferometers is required.
The possible range of frequencies for oscillating classical dark matter is $10^{-8}~\textrm{Hz} \lesssim f \lesssim 10^{13}~\textrm{Hz}$ (corresponding to the dark matter particle mass range $10^{-22}~\textrm{eV} \lesssim m_\phi \lesssim 0.1~\textrm{eV}$), while the timescale of passage of topological defects through a global network of detectors is $T \sim R_{\textrm{Earth}} / v_{\textrm{TD}} \sim 20$ s for a typical defect speed of $v_{\textrm{TD}} \sim 300$ km/s.
The current best sensitivities to length fluctuations are at the fractional level $\sim 10^{-22} - 10^{-23}$ in the frequency range $\sim 20 - 2000$ Hz for a large-scale gravitational-wave detector <cit.> and at the fractional level $\sim 10^{-15} - 10^{-16}$ in the frequency range $\sim 0.01 - 10$ Hz for a silicon-based cavity <cit.>.
§ ACKNOWLEDGEMENTS
We are very grateful to Jun Ye and Fritz Riehle for suggesting the strontium clock – silicon cavity interferometer as a suitable small-scale platform for our newly proposed measurements, and for important discussions.
We are also grateful to an anonymous referee for suggesting the use of phase coherent optical mixing and frequency comb techniques, further to our proposal centred around Eq. (<ref>).
We would like to thank Bruce Allen, Dmitry Budker, Federico Ferrini, Hartmut Grote, Sergey Klimenko, Giovanni Losurdo, Guenakh Mitselmakher and Surjeet Rajendran for helpful discussions.
This work was supported by the Australian Research Council. V. V. F. is grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and support.
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firstpage–lastpage 2015
We check the performance of the PARSEC tracks in reproducing the blue loops of
intermediate age and young stellar populations at very low metallicity.
We compute new evolutionary PARSEC tracks of intermediate- and high-mass stars from 2to 350with enhanced envelope overshooting (EO), EO=2and 4, for very low metallicity, Z=0.0005. The input physics, including the mass-loss rate, has been described in PARSEC V1.2 version.
By comparing the synthetic color-magnitude diagrams (CMDs) obtained from the different sets of models with envelope overshooting EO=0.7(the standard PARSEC tracks), 2and 4, with deep observations of the Sagittarius dwarf irregular galaxy (SagDIG), we find
an overshooting scale EO=2to best reproduce the observed loops. This result is consistent with that obtained by <cit.> for Z in the range 0.001-0.004.
We also discuss the dependence of the blue loop extension
on the adopted instability criterion and find that, contrary to what stated in literature,
the Schwarzschild criterion, instead of the Ledoux criterion, favours the development of blue loops.
Other factors that could affect the CMD comparisons such as
differential internal extinction or the presence of binary systems
are found to have negligible effects on the results.
We thus confirm that, in presence of core overshooting during the H-burning phase,
a large envelope overshooting is needed to reproduce the main features of the central
He-burning phase of intermediate- and high-mass stars.
stars: evolution – stars: interiors – Hertzsprung–Russel (HR)
diagram – stars: massive
§ INTRODUCTION
We have updated the new evolutionary tracks of massive stars from 14to 350in <cit.> and thus built a complete library of evolutionary tracks from very low (M=0.1) to very massive (M=350) stars, from the pre-main sequence to the beginning of central carbon burning. These tracks are computed with PARSEC: Padova TRieste Stellar
Evolution Code. All the input physics, including the mass-loss rate, has been detailed in <cit.>, <cit.>, and <cit.>. In this paper, we present the evolutionary tracks of intermediate- and high-mass stars at very low metallicity, Z=0.0005, and test them against the observations of the Sagittarius dwarf irregular galaxy (SagDIG).
Observed color-magnitude diagrams of SagDIG and evolutionary tracks computed by PARSEC V1.2 with EO=0.7. Cyan lines are tracks of M=0.9, 1.0and 1.1with Z=0.0002, marking the RGB stars. Green lines mark HB evolutionary phases of stars with M=0.85and 0.95and with Z=0.0002. Red lines are tracks of M=1.5, 1.7, 2.3, 3.0, 4.0, 5.0, 8.0, 12.0, 16.0and 20.0with Z=0.0005.
Convection plays a crucial role in stellar structure and evolution, but it has not yet been fully understood. Convection is the macroscopic motions with energy and chemical element transport in the stellar interior. The most widely used theory for convection is the mixing length theory (MLT),
proposed by <cit.>, and the mixing length parameter is calibrated to be $\alpha_{MLT}$=1.74 based on the solar model <cit.>. In the MLT, the convective boundary is the location where the acceleration of the fluid elements is zero ($a$ = 0). However, at this border the elements still have a non-negligible velocity. They are able to cross the boundary and enter the stable radiative region until the velocity is zero ($v$ = 0). The region lies between $a$ = 0 and $v$ = 0 is called the overshooting. <cit.> provided a simple way to calculate the overshooting.
As we know, convection occurs not only in the central core but also in the external envelope. Correspondingly, the extension of convective boundary is described by core overshooting above the convective core and envelope overshooting at the base of the convective envelope, respectively. Much theoretical and observational work supports the existence of core overshooting <cit.>. The main parameter describing core overshooting is the mean free path of convective elements across the border of the unstable region, $l_{c}=\Lambda_{c}$, where is the local pressure scale height. In this work $\Lambda_{c}$=0.5 is adopted <cit.>, which corresponds to 0.25above the unstable region. The consideration of envelope overshooting in stellar models was first suggested by <cit.>. There are two relevant observational constraints: the location of RGB bump in globular clusters and old open cluster, and the extension of blue loops in intermediate- and high-mass stars. Both features can be reproduced better by considering a moderate amount of envelope overshooting EO=0.7<cit.>.
However, <cit.> found that a larger mixing scale, EO=2or 4, is preferred to reproduce the extension of the observed blue loops in nearby star forming dwarf galaxies. This result implies a strong dependence of the mixing scale below the formal Schwarzschild border of the envelope, on the stellar mass or luminosity. In this paper, we calibrate the envelope overshooting parameter by comparing the synthetic and observed color-magnitude diagrams (CMDs) of SagDIG, and further confirm this result.
The presence of blue loops during central He-burning phase was first thoroughly investigated by <cit.> who introduced the core potential $\phi=M_{core}/R_{core}$ to explain the occurrence. On the other hand, many studies show that the proximity of the H-burning shell to the H-He discontinuity marked by the depth of first dredge up triggers the blue loop <cit.>. Thus any factor that moves the discontinuity deeper into the star causes a more extended loop. This conclusion is supported by <cit.> who realized it by enhancing the envelope overshooting. Further <cit.> demonstrated that this phenomenon is essentially due to the removal of the excess helium above the burning shell which causes changes to the mean molecular weight as well as the local opacity and the shell fuel supply. They believed that if this happens faster than the core evolution, the blue loop is triggered.
SagDIG is an ideal candidate to test our models at extremely low metallicity, because it is a very metal-poor star forming galaxy in the Local Group, and also it is nearby ($\approx$1.1 Mpc) which enables the Hubble Space Telescope to well resolve its star content. In section <ref> we briefly describe the photometric data and the observed CMD from which the foreground contamination has been deducted. In section <ref> we discuss the dependence of the blue loop extension on the adopted convection criterion, and present the new evolutionary tracks of intermediate- and high-mass stars computed by PARSEC with different envelope overshooting. In section <ref> we review the technique used to construct the synthetic CMDs, and in section <ref> we show the comparison of different models with the observed CMD of SagDIG. Finally, we discuss and conclude in section <ref>.
§ DATA
§.§ Color-magnitude diagram
To test our models, we use deep observations of SagDIG with the Advanced Camera for Surveys (ACS) on board the Hubble Space Telescope (HST). The observations contain two-epoch data-sets (GO-9820 and GO-10472) that are separated by $\sim$ 2 years. The main body of the galaxy ($\emph{l}=21.06^{\circ}$, $\emph{b}=-16.28^{\circ}$) was imaged in three filters: F475W, F606W and F814W. More details on the observations and data reduction can be found in <cit.>.
Due to its low latitude, SagDIG suffers a heavy Galactic contamination but <cit.> used two epochs data to analyse the relative proper motions of all detected stars and correct for foreground contamination. The CMDs cleaned for such a contamination of SagDIG are shown in Figure <ref>,
for filters F475W and F606W and for filters F606W and F814W, in the left and right panels,
The high resolution HST/ACS observations enable us to distinguish various stellar populations in SagDIG,
as marked also by selected PARSEC evolutionary tracks overplotted in the figures.
For the superposition of the evolutionary tracks we adopt a distance modulus (m-M)$_0$=25.06 and an extinction of
A(F475W)=0.657 mag, A(F606W)=0.520 mag and A(F814W)=0.286 mag, respectively. These values will be discussed
more extensively in a subsequent section.
SagDIG is characterized by the presence of very old populations. Red giant branch (RGB)
stars are indicatively marked by PARSEC evolutionary tracks
of masses M=0.9, 1.0and 1.1(cyan)
while, for horizontal branch (HB) stars we show two helium burning tracks of M=0.85and 0.95(pink).
Since these are meant to be the most metal poor stars in SagDIG, we have adopted a very low metallicity Z=0.0002.
The turnoff stars of these old populations are not visible in the CMDs of Figure <ref>.
Indeed, the deeper region of the observed main sequence,
around a magnitude of $m_{F606W}\sim$27, corresponds to the end of the H-burning phase of the track with
M$\sim$1.5, which is the faintest track plotted in red. This track is just near the
separation mass between low- and intermediate-mass stars, i.e. between those that undergo
or escape the helium flash. The track of M$\sim$1.7already belongs to the
intermediate-mass progeny and shows a well developed red clump.
For all the tracks with mass M$\geq$1.5, we use a larger metallicity, Z=0.0005, as will be discussed below.
We also note the presence of AGB stars in the continuation of the RGB tracks
above the corresponding RGB tips, and few of them likely corresponding to intermediate-mass stars. The AGB population is not modelled in our simulations because the majority of it comes from older stellar populations.
Starting from the red clump we may see the locus of the bluest He-burning (BHeB) intermediate- and high-mass stars, which is clearly separated from the H-burning main sequence (MS).
This locus is marked by the blue loops of the other tracks plotted in the figure, which have
initial masses of M=2.3, 3.0, 4.0, 5.0, 8.0and 12.0.
We note that the upper main sequences in the SagDIG CMDs seem to extend beyond that
of the model of M=12and, for this reason, we also plot
the tracks with M=16.0and 20.0. However these tracks ignite He in the blue side of the CMDs
and do not perform blue loops as the less massive ones.
The clear separation of MS and BHeB stars makes the CMD of this galaxy,
especially the $m_{F606W}$ vs ($m_{F606W}-m_{F814W}$) CMD where the separation is striking, a powerful workbench
for stellar evolution models at low metallicity.
§.§ Metallicity of SagDIG
From the color of the RGB, <cit.> estimated the metallicity of SagDIG [Fe/H]=-2.45 $\pm$ 0.25, while <cit.> derived [Fe/H] in the range from -2.8 to -2.4. By comparing the color differences between the RGB stars in the SagDIG and Galactic globular clusters (GGC) fiducial lines, <cit.> yielded a mean metallicity [Fe/H]=-2.1 $\pm$ 0.2 for the red giants, and <cit.> gave the range [Fe/H]=-2.2 to -1.9 depending on different assumed reddening.
On the other hand, by analysing optical spectrophotometry of $H_{\uppercase\expandafter{\romannumeral2}}$ regions in SagDIG, <cit.> derived an oxygen abundance of 12+log(O/H)=7.42, which is in accordance with the measurement of <cit.> who estimated the O abundance in the range 12+log(O/H)=7.26 to 7.50.
From the latter values we obtain, using for the Sun 12+log10(O/H)=8.83 and Z=0.017 <cit.>,
Z between 4.5E-4 and 7.9E-4. If instead we use the solar values 12+log10(O/H)=8.69 and Z=0.0134 <cit.>,
we obtain Z between 5.0E-4 and 8.7E-4. We thus adopt for the young population of SagDIG a metallixity Z=5E-4 which is the lower value compatible with
spectroscopic observations.
We note that for the most recent populations <cit.> infer a metallicity between Z=0.0001 and Z=0.0004, by fitting
the extension of the blue loops of intermediate- and high-mass stars using the previous Padova isochrones.
Kippenhahn diagrams of central He-burning stars
of $M$=12M$_\odot$ and $Z$=0.004 computed with different convection criteria,
the Schwarzschild criterion (left panel) and the Ledoux criterion (right panel). The core overshooting parameter $\Lambda_{c}$=0.5 is adopted. The brown line marks the location of the H-He discontinuity. The insets show the corresponding evolutionary tracks in the HR diagram.
§ MODELS
Not only is there a clear separation between the H-burning main sequence (MS) and the blue He-burning stars (BHeB) in the CMDs of Figure <ref>, but also red He-burning giants/supergiants (RSG) can be fairly well separated from the older red giant stars. This allows us to test different prescriptions used in building models of intermediate- and high-mass stars, in particular those that are known to affect the extension and duration of the blue loops. We have already performed a similar analysis on three well studied dwarf irregular galaxies (DIGs), Sextans A, WLM and NGC 6822, which are used to constrain
the models at metallicities somewhat higher (Z=0.001-0.004) than the one suitable for SagDIG (Z=0.0005) <cit.>. The result of this analysis indicated that, if one keeps the extent of the core overshooting parameter
fixed on the value resulting from comparison of low- and intermediate-mass stars (i.e. an overshooting length of about 0.25above the unstable core), then the blue loops of intermediate- and high-mass stars are significantly reduced with respect to the models computed without core overshooting. To restore the extent of the blue loops, significant overshooting at the base of the convective envelope is required, with a typical extent of a few .
It is worth noting that any mechanism that increases the size of the fully He exhausted region
tend to shorten or even suppress the blue loops.
This is also the case of models with rotational enhanced central mixing.
In some cases even models computed with the usual instability criterion and without any
extended mixing face the problem of lacking extended blue loops, especially at high metallicities.
The goal of this paper is to test if the conclusion obtained by <cit.> remains valid also
at very low metallicities. For this purpose we model the brightest area of the CMD, which can be reasonably well represented by the last episode of star formation in SagDIG. We thus
focus on stars brighter than $m_{F606W}$=24, where the completeness is 100%, 99.7% and 98.8% in F475W, F606W and F814W, respectively. This corresponds to a mass limit of $\sim$5.0and $\sim$3.0on the MS and on the He-burning phase respectively, as indicated in Figures <ref> and <ref>.
§.§ Comparisons between the Schwarzschild criterion and the Ledoux criterion
Before embarking in this comparison, we stress that PARSEC models make use of the Schwarzschild criterion
for convective instability. The boundary of the convective region can be determined by either the Schwarzschild criterion or the Ledoux criterion.
According to Ledoux, the condition for the onset of convection is more restrictive than the Schwarzschild one
\begin{equation}
\nabla_{rad} > \nabla_{ad} + \nabla_{\mu},
\end{equation}
where $\nabla_{rad}$, $\nabla_{ad}$ and $\nabla_{\mu}$ represent the radiative temperature gradient, adiabatic temperature gradient and molecular weight gradient, respectively. $\nabla_{\mu}$ can be expressed as
\begin{equation}
\nabla_{\mu} = -(\frac{\partial ln{\rho}}{\partial lnT})_{P, \mu}^{-1}(\frac{\partial ln{\rho}}{\partial ln{\mu}})_{P, T}(\frac{dln{\mu}}{dlnP}).
\end{equation}
In a region with homogeneous chemical composition, $\nabla_{\mu} = 0$, as you'd expect, it becomes the Schwarzschild criterion,
\begin{equation}
\nabla_{rad} > \nabla_{ad}.
\end{equation}
There have been claims in literature that the use of the Ledoux criterion could
eventually be more suitable in regions with variable mean molecular weight and it could give rise to more extended blue loops.
This has been shown in the pioneering paper by <cit.> who investigated the effect of the above two instability criteria on the morphology of the tracks and, specifically, on the extension of the blue loops. They found that the adoption of different criteria produces different size of semi-convective regions above the unstable core, and that only the model computed with the Ledoux criterion develops a blue loop.
Notably, the model computed with the Schwarzschild criterion ignites central He at high effective temperature, before reaching the red supergiant phase, and only at the end of central He burning
it slowly moves toward the latter phase.
The different behaviour resulting from the adoption of different instability criteria may be understood by considering the competition between two important structural properties of the model,
the relative size of the H-exhausted core and the mass pocket between the H-burning shell and the H-He discontinuity at the bottom of the convective envelope.
As already shown in <cit.>,
since core overshooting during the H-burning phase
has the effect of increasing the relative mass size of the H-exhausted core, it favours He ignition and burning
in the red phase. On the other hand, it has been shown that the hydrostatic equilibrium location of He-burning models in the HRD is very sensitive to the mean molecular weight in the mass pocket between the H-exhausted core and the discontinuity in the H profile,
left either by the development of intermediate convective regions and/or by the penetration
of the convective envelope. <cit.> have clearly shown that if the chemical
composition of this mass pocket,
which usually has an outward increasing H content, is artificially changed into a helium
rich mixture, the hydrostatic location of the model shifts to the blue region of the HRD.
This explains why the blue loop starts when the H-burning shell reaches the discontinuity of mean molecular weight
at the base of the H-rich envelope at early phase during the central He burning <cit.>.
Three possibilities may arise, depending on when, eventually, the H-shell reaches the H-discontinuity after central H-exhaustion. If it happens very soon during its expansion phase before the star becomes a red giant, the path in the HRD is inverted and the star ignites and burns He as a blue supergiant. Only at central He exhaustion the star will move toward the red giant phase. This behaviour may be typical of most massive stars, especially at low metallicity. It corresponds to the case B (model computed with the Schwarzschild criterion) in <cit.>: that model does not perform a loop because it ignites He already in the blue loop hydrostatic configuration.
The second case happens when the H-shell reaches the discontinuity after He ignition in the red (super-)giant
phase, but early enough during central He burning. In this case the star performs a blue loop in the HRD as in case A (model computed with the Leodux criterion) of
<cit.>. Finally if the H-shell reaches the discontinuity at late time during central He burning, the star does not perform a blue loop and burns central He entirely as a red (super-)giant.
However, this behaviour is not a general property of the
adopted instability criterion. To demonstrate this we show here
two models with M=12computed with the two different instability criteria, and convective core overshooting is also considered in models. As shown in the Kippenhahn diagrams of Figure <ref>, the internal structure of the models is identical at the end of central H burning
and both models are characterized by a rapid red-ward evolution followed by He ignition in the RSG phase,
independently of the adopted instability criterion. The larger size of the H-exhausted core originated
from the overshooting mixing forces both stars to ignite He in the supergiant phase.
However, in the case of the Schwarzschild criterion (upper panel), a large intermediate convective region develops, shifting the location of the H-He discontinuity slightly deeper, as indicated by the brown horizontal line, and effectively decreasing the size of the mass pocket between the H-exhausted core and the
H-He discontinuity.
Instead using the Ledoux criterion (lower panel), which is more restrictive, the intermediate convective regions are smaller or even suppressed, and the resultant mass pocket is relatively larger.
The difference with respect to the previous case is small, but in the former case
the H-burning shell is able to reach the discontinuity and a loop occurs, while in the latter case
this happens too late during He burning and the model does not perform a loop.
Thus the computations shown in Figure <ref> indicate that, in presence of sizable
convective overshooting during central H-burning, the model computed with the Schwarzschild instability criterion
performs the blue loop while the one computed with the Ledoux criterion spends all the He-burning lifetime in the RSG phase.
We thus illustrate that the effect found by <cit.> is not a general property of the
instability criterion applied to massive stars. We instead confirm that the loop is activated if the H-shell
reaches the H-He discontinuity during the early He-burning phase <cit.>. A possible explanation
of why this effect triggers a blue loop has been advanced by <cit.>, who indicated that
this is the hydrostatic equilibrium location of a class of central He-burning models with a fully discontinuous H profile.
§.§ Models with extended envelope overshooting
To conclude this section we add to the CMDs of SagDIG in Figure <ref> the evolutionary tracks computed by assuming a large value of envelope overshoot, EO=4, which is the largest value adopted in <cit.>. These computations have been performed only for initial masses M$\geq$2.1and the
evolutionary tracks are shown in blue color.
The models
run superimposed to the standard PARSEC models computed with EO=0.7up to central He ignition.
Then, models with larger envelope overshooting ignite He at slightly lower luminosities
(the red giant tips are fainter) and they burn central He at a significantly lower luminosity,
both in the early red giant stage and in the blue loop phase.
An interesting effect of a large envelope overshooting is that,
in the low mass range, the new models cross the Cepheid instability strip, being 0.5 magnitude
fainter than the standard ones. We also note that the blue loop is significantly more extended than that in standard models.
These differences become smaller at increasing mass and practically disappear at masses
As expected, the new models reproduce fairly well the region of the observed blue He-burning stars of SagDIG.
We have also computed models with EO=2, which lie between the two extreme cases already discussed.
In the next sections we compare the simulated CMDs obtained by these models with the observed one.
Observed color-magnitude diagrams of SagDIG and evolutionary tracks computed by PARSEC V1.2 with EO=0.7(red) and 4(blue). Blue lines are tracks of M=2.1, 2.3, 3.0, 4.0, 5.0, 8.0, 12.0, 16.0and 20.0with Z=0.0005. Pink lines mark the beginning and end of central H burning of stars with different initial masses, respectively.
§ SIMULATED COLOR MAGNITUDE DIAGRAMS
To construct a synthetic CMD of SagDIG, we follow the same procedure described in <cit.>
as briefly summarized below.
For each of the three values of the envelope overshooting parameter, EO=0.7, 2.0and 4.0, we generate a large mock catalogue in the theoretical plane with the following recipes.
* The star formation rate (SFR) $\psi_m$ is specified as an exponential function of time, $\psi_m \propto exp(t/\tau)$, where $\tau$ is an adjustable parameter and $t$ is the stellar age, spanning from 1 Myr to 1 Gyr.
* The age-metallicity relation (AMR) is replaced by a fixed value $Z$=0.0005 as we have discussed in section <ref>, since our analysis is limited to the youngest and brightest stars.
* The initial mass function (IMF) $\phi_m$ is a broken power law in the form of $\phi_m \propto m^{-\alpha}$,
where $\alpha = 0.4$ for $0.1 \Msun \leq m < 1 \Msun$, while the exponent is parametrized for $1 \Msun \leq m \leq M_{UP}$. The total mass is normalized to 1 in order to express SFR in units of /yr.
* The library of theoretical isochrones is computed by PARSEC, giving the luminosity $L$, the effective temperature $T_{eff}$, the surface gravity $g$ and other physical properties of stars as a function of stellar age $t$, initial mass $M_{ini}$, and metallicity $Z$.
We transform the catalogue into the observational plane by means of
bolometric corrections, as a function of $T_{eff}$, $g$ and $Z$, adopted from <cit.>
and accounting for the distance. We finally add the effect of extinction and photometric errors. We may also include
the effect of binarity as discussed later.
With the simulated catalogue that contains much more stars than the observed data, we perform
hundred realizations of the CMD models to obtain, on one hand, the average and variance of the star luminosity function and, on the other hand, the best-fit model to the observed luminosity function.
We do not try more sophisticated statistical methods to reproduce the observed CMD, because
our aim here is to obtain the best value of the envelope overshooting parameter to be used
with the fixed metallicity Z=0.0005.
Since we care about the extension of the blue loops in intermediate- and high-mass stars, we just select stars with mass M$\geq$1.9, and also set the apparent magnitude limit to $m_{F606W}$=24 in the simulation.
§.§ Simulated photometric errors
<cit.> have estimated the photometric errors for observed stars in SagDIG from artificial star experiments.
To account for photometric errors, we first bin their results in 0.1 mag steps
as a function of the apparent magnitude in each filter, and calculate the median error for each bin.
Then we assign to each star in the mock catalogue an error that is randomly drawn from a Gaussian distribution with the standard deviation derived from the median corresponding to its magnitude <cit.>.
We note that since the observations are very deep, the simulated errors for stars brighter than 24 mag are small ($\leq$0.03 mag), as can be seen in Figure <ref>.
Simulated photometric errors as a function of the apparent magnitude in the F606W filter.
§.§ Foreground and internal extinction
<cit.> estimated a low foreground reddening E(B-V)=0.06 based on the (B-V) vs (V-I) diagram, consistent with the value E(B-V)=0.07 derived by <cit.> from the (V-I) color distribution of foreground stars toward SagDIG, as the blue cut-off of the (V-I) location is a function of reddening along the line of sight. <cit.> inferred E(B-V)$\approx$0.05 in the same way, but using (R-I) color distribution. The infrared dust maps of <cit.> indicate a slightly higher reddening E(B-V)=0.12.
On the other hand, spectroscopic studies of $H_{\uppercase\expandafter{\romannumeral2}}$ regions in SagDIG suggest a higher reddening. <cit.> calculated $c(H_\beta)$=0.33 and E(B-V)=0.22 is derived according to the relation $c(H_\beta)=1.47E(B-V)$ <cit.>. <cit.> obtained a similar value E(B-V)=0.19 from the measurement of the Balmer decrement. As this method is based on the H line ratio, the estimated value include both the foreground and the internal reddening.
As young stars are physically associated with the warm interstellar medium (ISM), it is reasonable to believe they suffer higher reddening compared to old stars. This trend was also found in other dwarf irregular galaxies <cit.>.
<cit.> derived individual star by star extinction from their multi-band data
of Sextans A, WLM and NGC6822 dwarf irregular galaxies and this information was implemented in the
synthetic CMD analysis by <cit.>.
Since for SagDIG we lack multi-band data and cannot repeat the same procedure with the same accuracy, we deal with the extinction in the following ways. The simplest way is to use a single value of the attenuation in each photometric band, as derived from the simultaneous alignment of the observed and modelled MS stars in both the $m_{F606W}$ vs ($m_{F475W}-m_{F606W}$) and $m_{F606W}$ vs ($m_{F606W}-m_{F814W}$) CMDs.
For the superposition of the evolutionary tracks, we adopt a distance modulus (m-M)$_0$=25.06 and an extinction of
A(F475W)=0.657 mag, A(F606W)=0.520 mag and A(F814W)=0.286 mag, respectively. These values will be discussed
more extensively in a subsequent section.
§ RESULTS
Left panels: comparisons of observed and modelled CMDs, EO=0.7, EO=2and EO=4, from top to bottom respectively. The observation and simulation are represented by black and green points, and the corresponding medians and fiducial lines are marked in black and red, respectively. The error bars are calculated as the median absolute deviation. Right panels: the observed (black solid) and average simulated (red dashed) luminosity functions (LFs) with vertical standard deviation bars
obtained from one hundred simulations with fixed parameters. Blue dotted lines indicate the LFs of the best-fit models.
The synthetic $m_{F606W}$ vs ($m_{F606W}-m_{F814W}$) CMDs of SagDIG obtained with the three different values of envelope overshooting are shown in Figure <ref>.
The observed CMD is represented by black points, while the best simulation is shown in green color. The simulation refers to the stars brighter than $m_{F606W}$=24. For ease of comparison, we draw the observed fiducial main sequence,
which is indicated by the almost vertical black line at $(m_{F606W}-m_{F814W}) \sim 0$. It has been
obtained by considering the median of the colors of main sequence stars,
defined as the stars bluer than ($m_{F606W}$-$m_{F814W}$)=($m_{F606W}$-18)/65, in magnitude bins of $\Delta{m_{F606W}}$=0.5 in the range $21 \leq m_{F606W} \leq 24$.
The horizontal bar represents the standard deviation for the corresponding magnitude bin.
It is calculated as the median absolute deviation, which is defined as the median of an array of differences between the colors of stars and the median color. The fiducial main sequence locus derived from the best fit CMD is shown in red color.
Since the evolution up to central H exhaustion is not affected by envelope overshooting, the main sequence loci
are the same in the three simulated CMDs.
The superposition of the observed and simulated main sequence loci has been obtained by assuming a single value of the extinction
for all stars in each of the three different photometric bands, A(F475W)=0.657 mag, A(F606W)=0.520 mag and A(F814W)=0.286 mag respectively. The adopted distance modulus is (m-M)$_0$=25.06.
These values of extinction and distance modulus have also been used to draw the evolutionary tracks on the CMDs in Figures <ref> and <ref>.
Adopting A$_{F606W}$/A$_V$=0.91 (as in the Galactic extinction law with $R_{v}$=3.1), we get A$_V$=0.57.
We compare the quantities A$_{F475W}$/A$_V$ and A$_{F814W}$/A$_V$ with
those corresponding to other typical extinction laws in Figure <ref>.
We see that the value of A$_{F475W}$/A$_V$ lies on the Calzetti extinction curve <cit.>, while the one of A$_{F814W}$/A$_V$ falls slightly below the Galactic one <cit.>.
The reddening curve A$_\lambda$/A$_V$ is shown as a function of $\lambda$. Our values A$_\lambda$/A$_V$ for different photometric bands are shown by green solid dots.
The black and blue line represent the Galactic extinction law<cit.>
and the Calzetti Law <cit.>, respectively.
The models shown in Figure <ref> correspond to the best fit selected from one hundred stochastic realizations made with the same parameters, based on the merit function that measures the agreement between the observed and modelled luminosity functions of stars both in the main sequence and in the blue-loops evolutionary phase. We notice that very bright stars ($m_{F606W} \leq 20$) in models are always much fewer than the observations. The parameters of the star formation rate and initial mass function adopted in the models are listed in Table <ref>.
The synthetic CMDs reproduce fairly well the main features of the observed one, except for the RGB. This is because we focus on newly-formed stars, and exclude stars redder than the line ($m_{F606W}$-$m_{F814W}$)=(29.95-$m_{F606W}$)/7 which, as shown in
Figures <ref> and <ref>, correspond to the RGB of the old populations. We note however that the simulated green-dot sequence in the case with EO=0.7, representing
red giant stars of the intermediate age populations, seems to be more populous than
the corresponding data, while the discrepancy gets smaller
at increasing value of EO.
As discussed in <cit.>, models with enhanced envelope overshooting
produce more extended blue loops, which is
also shown in Figure <ref>. But the simulations show two other interesting properties. One is that, at increasing envelope overshooting, the relative fraction of blue He-burning stars increases while that of red He-burning stars decreases, explaining the better agreement
between the simulated and observed red giants obtained with EO=4.
The other is that the population of yellow He-burning stars, say those with $0.4 \leq (m_{F606W}-m_{F814W}) \leq 0.7$, decreases at increasing envelope overshooting. This is particularly evident in the magnitude range $23 \leq m_{F606W} \leq 24$.
The effect of envelope overshooting on the extension of the blue loops can be appreciated already by eye from Figure <ref>, but in order to render it more clear,
we derive the BHeB stars main locus by using the same method adopted for the main sequence locus.
The bins are the same used for the main sequence stars, though we recall that, at fixed initial mass,
He-burning stars are brighter than H-burning ones. In order to derive the median color we have considered
all stars redder than ($m_{F606W}$-$m_{F814W}$)=($m_{F606W}$-18)/65 and bluer than ($m_{F606W}$-$m_{F814W}$)=0.4.
The meaning of the error bar is the same as the one obtained for MS stars.
The median locus of the observed BHeB stars is drawn in black and it runs almost parallel to the
main sequence locus, but about 0.25 mag redder. The locus of the synthetic BHeB stars is drawn in red.
For EO=0.7, it is 0.1 mag redder than the observed one at magnitudes $23 \leq m_{F606W} \leq 24$. At brighter magnitudes
the difference disappears. The locus of the models with envelope overshoot EO=2runs superimposed to that of the observed data, in the magnitude range $23 \leq m_{F606W} \leq 24$, while
using the models with envelope overshoot EO=4, it is slightly bluer than the observed one,
in the same magnitude range $23 \leq m_{F606W} \leq 24$.
We note that the difference between the observed and modelled BHeB loci is not large, even in the case of EO=0.7. Actually
in the latter case the largest difference is comparable to the standard deviations of the loci themselves. However if we base
our judgment more on the systematics of the effect than on its entity, it is clear that only models with larger envelope overshooting are able to reproduce the extension of the blue loops in the magnitude range $23 \leq m_{F606W} \leq 24$.
§.§ The star formation rate
As in <cit.>, the star formation rate is represented by an exponential parametrization
\begin{equation}
SFR(t) = SFR_0\times exp(\frac{t}{\tau})
\end{equation}
where $t$ represents the stellar age, SFR$_0$ the current value of the SFR and $\tau$ the characteristic $e$-folding time.
The results are shown in Table <ref>. We find that the SFR in SagDIG increases toward recent times ($\tau < 0$). Considering the case of EO=2, the average SFR in the last 100 Myr ($\sim$9E-4 M$_\odot$/yr) is significantly lower than that we obtained for Sextans A ($<$SFR$>$=2.9E-3 M$_\odot$/yr), WLM (($<$SFR$>$=2.7E-3 M$_\odot$/yr)) and NGC 6822 (($<$SFR$>$=3.7E-3 M$_\odot$/yr)). It is worth noting that this average SFR is for the whole galaxy, while in those three galaxies the derived SFR refer to selected star-forming regions.
We also notice that the SFR does not change much for models with different envelope overshooting, but in the case with high envelope overshooting, EO=4,
the SFR turns out to be $\sim$ 30% larger than that in the case of EO=0.7. Correspondingly, the mass formed in the recent burst
amount to M*=9.26E5with EO=4, while it is relatively smaller in the other two cases, M*$\sim$7E5.
Our value is close to that of <cit.> who found
$<$SFR$>$=6.6$\pm$0.8E-4/yr in the age range 0.05-0.2 Gyr. Indeed, in the same period, we find $<$SFR$>$=8.2E-4/yr using models with EO=0.7.
In Table <ref> we also show the masses of the most massive and of the brightest stars found in the simulations.
§.§ Effects of differential extinction and binary stars
A remarkable property of the CMDs shown in Figure <ref>
is that the standard deviation of the observed fiducial sequences are larger than
the modelled ones. This indicates that both the observed main sequence and the
observed BHeB sequence are more dispersed than that predicted by the models,
at least for stars brighter than $m_{F606W}$=24. This cannot be ascribed
to photometric errors because, besides being explicitly included in the simulation,
they are by far too small to explain the effect. Thus other explanations have to be found.
In the case of the main sequence, the discrepancy could be due to the well known main sequence widening effect, i.e. that the termination point of the observed main sequence is cooler than
that predicted by the models. This could be appreciated in the comparison between the observed
CMD and the evolutionary tracks in Figure <ref> where the width of the main sequence
is marked by the pink lines. This discrepancy is one of the motivations that
inspire the presence of extended mixing effects during the central H-burning phase
of intermediate- and high-mass stars <cit.>.
In the case of the BHeB stars, the problem does not have a similar explanation.
The star number counts in the evolved phases are proportional to the evolutionary lifetimes
of the corresponding phases, and to explain a large dispersion one should invoke a mechanism that
is able to slow down the transition from the RSG to the BSG phases which is at present not known.
There are two other effects that could explain the widening of the sequences.
One is differential extinction. We have already mentioned that in previous analysis of DIGs, <cit.> and <cit.> have directly measured and then modelled differential extinction of individual stars.
This effect certainly contributes to the widening of both the main sequence and the blue He-burning sequence and thus helps filling the gap between them.
But unfortunately, at variance with the quoted DIGs, in the case of SagDIG we lack the broad multiband
photometry which allows <cit.> to obtain estimates of attenuation for individual stars.
Nevertheless we may try to estimate the size of this effect, by assuming that the models
are correct. Applying the same method used in <cit.>, we derive a more realistic estimate of the extinction of individual stars,
according to the trend of increasing attenuation at increasing luminosity, as shown in the diagram
A(F606W) vs F606W of Figure <ref>.
Differential extinction is assumed as a certain dispersion around a mean value A(F606W)=-0.02 $\times$ $m_{F606W}$+0.98 (black line).
Adopting the differential extinction we obtain the simulated CMDs shown in Figure <ref>.
While the gap can be partly filled by introducing the differential reddening, it remains evident in the case of EO=0.7.
The gap decreases in the case of EO=2and almost disappears when a high envelope overshooting is adopted, EO=4.
The fiducial lines of the main sequence and of the BHeB stars, drawn in the same way as in Figure <ref>,
show that the fact that the models with the standard value of envelope overshooting, EO=0.7, are not able to reproduce the observed extended loops during central He-burning phase, is not due to neglecting differential extinction.
Even with differential extinction a large value of envelope overshooting (EO$\sim$2.0) is favoured.
Parameters of the CMD simulations of SagDIG
EO $\tau$ $\alpha$ SFR$_0$ $<$SFR$>$* M$_{max}$ M$_{bright}$
yr M$_\odot$/yr M$_\odot$/yr M$_\odot$ M$_\odot$
0.7 -2E9 2.05 8.79E-4 8.57E-4 39 11
2 -2E9 2.25 9.65E-4 9.41E-4 62 13
4 -2E9 2.15 1.18E-3 1.15E-3 30 30
4l*the average SFR in the last 100 Myr
The best model obtained by PARSEC V1.2 with EO=0.7, 2and 4from top to bottom with differential extinction.
Another effect that may contribute to widen the theoretical nominal sequences of single stars is
the presence of binary stars.
As is well known, the incidence of binarity in stars is high. <cit.> analysed the sample of stars from the Hipparcos catalogue, suggesting 33% of solar-type stars in the solar neighbourhood are binaries. Moreover, young massive stars are believed more likely to be binaries. <cit.> studied the optical spectra of O-type stars in NGC 6611 and derived the minimal binary fraction to be 0.44, but it could be increased up to 0.67 if all binary candidates are confirmed.
In order to estimate the effect of binaries on synthetic CMDs, we run a model with a percentage of binaries of 50%.
To reproduce the assumed 50% contamination, we randomly combine the sample stars in the mock catalogue without assuming a particular value of the mass ratio,
until we reach the total number of observed objects with the required binary fraction.
Since for binaries consisting of equal-mass components the apparent magnitude may increase by up to 0.75 mag, to obtain a complete estimate of their effect above $m_{F606W}$=24, we
first work on stars brighter than $m_{F606W}$=25. After taking into account the luminosity increase due to binarity, we further select objects brighter than $m_{F606W}$=24 and bluer than the line ($m_{F606W}$-$m_{F814W}$)=(29.95-$m_{F606W}$)/7, for stars redder than this line are considered as RGB stars.
This procedure is repeated one hundred times to obtain the best-fit model, the average luminosity function and its standard deviation at the selected magnitude bins.
In this case we adopt a fixed extinction, as discussed in the previous section.
Figure <ref> shows the simulated CMD obtained by the models with EO=2and a binary fraction of 50%. The inset shows the distribution of the mass ratios $q=M_{2}/M_{1}$ between the primary and secondary components, which is consistent with a flat distribution <cit.>.
Compared to the middle panel of Figure <ref>, we see the effect of binaries is to broaden the main sequence and also the BHeB sequence. The number of stars falling between these two sequences
is larger than that in the case computed without considering binaries.
However the BHeB sequence is clearly split into two parallel sequences
between $23 \leq m_{F606W} \leq 24$, and the red He-burning sequence seems also too broad, as part of these stars move to the location of yellow giant/supergiant. In particular the splitting of the BHeB seqence is not seen in the observed diagram, perhaps indicating that either the assumed binary fraction or the resulting mass ratio are too high for this galaxy of low metallicity.
Furthermore, because of the asymmetric behaviour of the superposition of star pairs in the CMD,
the fiducial lines shift toward the red side and,
in order to reconcile the model with the observations, one should make use of a lower
attenuation, by a factor $\delta(m_{F606W}-m_{F814W})\sim$0.02.
The best model obtained by PARSEC V1.2 with EO=2and binary fraction F=0.5. The insert shows the mass ratio distribution of binaries.
§ DISCUSSION AND CONCLUSION
In <cit.> we had obtained strong indications that,
in presence of sizable overshooting from the convective core during central H burning,
the standard PARSEC models with envelope overshooting EO=0.7fail to reproduce the width of the observed blue loops of intermediate- and high-mass stars
in three star forming dwarf galaxies in the Local Group, Sextans A, WLM and NGC6822, characterized by low metallicity, 0.001$\leq$Z$\leq$0.004.
Meanwhile we found that a significant envelope overshooting at the bottom of the convective envelope, EO=2.0-4.0,
must be used to overcome this discrepancy in models with metallicity typical of the aforementioned galaxies, 0.001$\leq$Z$\leq$0.004.
We stressed that this discrepancy could be cured by adopting a metallicity significantly lower than the observed one,
which however indicates that using the extension of the blue loops predicted by models to infer the metallicity
of the galaxies could be risky.
In this paper we continue this investigation by considering the case of SagDIG, a nearby star forming dwarf irregular whose metallicity is estimated to be
even lower than that of the aforementioned galaxies, Z=0.0005.
This galaxy is an ideal workbench to test the performance of models of intermediate- and high-mass stars because it harbours a recent burst of star formation, and
it is sufficiently nearby that intermediate-mass stars with masses as low as M=2can be detected. The foreground contamination has been eliminated using proper motions of individual stars <cit.>.
From a preliminary superposition of the standard PARSEC evolutionary tracks with EO=0.7and the observed CMD of SagDIG, we already see that the models are not able to reproduce the observed loops. This has already been noticed by <cit.> who, in an attempt to determine the metallicity of the galaxy from the blue loop superposition, were forced to try also the lowest
value of metallicity of the old Padova models, Z=0.0001.
Before performing new calculations with enhanced envelope overshooting, as suggested by <cit.>,
we discuss if the problem could be alleviated by adopting the Ledoux criterion instead of the Schwarzschild
one for the determination of the unstable region. On one hand, it has already been shown in the past that the adoption of the Ledoux criterion may favour
the development of extended blue loops <cit.>.
On the other hand, the fact that the problem of short blue loops is found even in the tracks of intermediate-mass stars, for which
the results should be independent of the adopted instability criterion because they
do not possess intermediate unstable region (within the profile of chemical composition),
suggests that the reason should be different.
By analyzing a model of M=12and Z=0.004 under standard PARSEC assumptions, which evolves without performing a blue loop, we find that the Ledoux criterion tend to suppress the blue loop,
in the sense that the star favours the red-ward evolution after central H burning.
This is apparently in contrast to the results reported by <cit.>, who indicated that the Ledoux criterion favours the blue loop.
However, a thorough inspection of that seminal paper shows that their model with the Schwarzschild criterion does not perform a blue loop only because
it begins central He burning already in the blue loop region, i.e. as a BHeB star.
With a series of new models not reported here for seek of conciseness,
we show that at He ignition, the mass pocket between the H-exhausted core and the H/He discontinuity is
already negligible, fully confirming the finding of <cit.>.
The Schwarzschild criterion allows the formation of a larger intermediate convective shell than
in the case of the Ledoux criterion, and this convective shell pushes the H/He discontinuity deeper
in the star, reducing the above mass pocket.
Thus the case A model in <cit.> begins He burning already in the BHeB equilibrium configuration and not in the RSG stage, without the need of performing a blue loop.
On the contrary, their case B model, with the Ledoux criterion, encounters the condition of a thin mass pocket between the H-exhausted core and the H/He discontinuity,
slightly later during the evolution, after He ignition in the RSG phase, and thus performs a blue loop.
When significant core overshooting is allowed during the H-burning phase, the situation changes critically.
Being the H-exhausted core larger than that in the case without core overshooting,
its contraction after central H-burning is stronger,
pushing the star into the RSG stage, independently of the adopted instability criterion.
These are the two cases shown in Figure <ref> for M=12and Z=0.004.
However, though the differences between the models computed with the two different criteria are minimal,
the track computed with the Schwarzschild criterion develops a larger intermediate
convective region that deepens slightly the H-He discontinuity than in the one computed with the
Ledoux criterion. The former develops the blue loop while in the latter case
the condition of a thin intermediate layer is encountered only
toward the end of central He burning, when the BHeB structure is no more
a possible configuration for the star so that
it burns the entire central He in the RSG stage.
Having definitely excluded the Ledoux criterion as a possible cure of the problem of the loops extension,
we computed additional evolutionary tracks with larger values of envelope overshooting,
EO=2and EO=4. Combining the results with other specified parameters, the IMF and SFR law, the photometric errors and the extinction in our own CMD simulator,
we construct the synthetic CMDs that are compared with the observed one of SagDIG.
In all models the location of the observed fiducial main sequence is well reproduced with a reasonable value of the
attenuation. This is expected since the main sequence phase is not affected by envelope overshooting
and the match is actually used to determine the extinction to be adopted in the simulated CMDs.
As expected, models with larger envelope overshooting perform more extended blue loops, and their BHeBS get closer to the MS.
In order to decide which value of envelope overshooting reproduces better
the observed extension of the blue loops, we also draw and compare the fiducial lines corresponding to the BHeBS of the synthetic and of the observed CMDs.
We find that the model with EO=2matches the observations best, while the blue loops predicted by the case EO=0.7are not hot enough and those predicted
in the case of EO=4are likely hotter than the data indicate.
It is worth stressing that, since the value of the envelope overshooting is not an adjustable parameter in the CMD comparison,
we have no better and more statistically sound method to decide which is the best case, than that of comparing the fiducial main sequences.
We have also tested how the results depend on other additional assumptions concerning the attenuation and the possible effect of binarity.
If we assume a differential attenuation with a reasonable model (Figure <ref>) where the attenuation has a certain dispersion around a mean value
that increases with the intrinsic luminosity of the stars <cit.>, the overall CMD fit looks better but the
preferred value of the envelope overshooting remains unchanged.
However, due to the dispersion introduced in the synthetic CMD, the uncertainty of the fiducial points (the horizontal bars in the CMD figures), corresponding to
the color median absolute deviation in each magnitude bin, becomes slightly larger.
To single out the effect of binarity, we consider
the case with constant attenuation and
a binary fraction of 50%.
The effect of binaries is to broaden both the main sequence and the BHeB sequence, with the consequence that
the number of stars falling between the two sequences
is larger than that in the case computed without considering binaries.
However the BHeB sequence is clearly split into two parallel sequences
between $23 \leq m_{F606W} \leq 24$, and the red He-burning sequence becomes less populated, as part of these stars move into the region populated by yellow giants/supergiants.
The splitting of the BHeB sequence is not seen in the observed diagram, perhaps indicating that either the assumed binary fraction or the resulting mass ratio are too high for this galaxy of low metallicity.
Furthermore, because of the asymmetric behaviour of the superposition of star pairs in the CMD,
the fiducial lines shift toward the red side and, in order to reconcile the model with the observations, one should adopt a slightly lower
attenuation, $\delta(m_{F606W}-m_{F814W})\sim$0.02.
Even in this case the models that perform better are those computed with an envelope overshooting EO=2.
The results of this paper are consistent with those found by <cit.> for three galaxies with slightly higher
metallicities than the one considered here.
Thus the current investigation corroborates the finding that the mixing scale below the formal Schwarzschild border
in the envelopes of intermediate- and high-mass stars must be significantly higher than currently assumed in PARSEC models.
We recall that there are other conditions where
strong mixing below the conventional instability region is invoked
such as, for example, to enhance the efficiency of the carbon dredge up during the thermally pulsing Asymptotic Giant Branch phase <cit.>.
On the other hand, we note that such a high value is likely incompatible with
the location of the Red Giant Branch (RGB) bumps observed in Globular Clusters,
which can be well reproduced by the models using an envelope overshooting not larger than EO=0.5.
Thus, if the interpretation in terms of mixing into the stable regions due to convection is correct, it is likely that the mixing scale below the formal Schwarzschild border is not a fixed
fraction of the pressure scale height, as assumed in usual extensions of the MLT, but instead it depends on the interior structure of the star.
In fact our results highlight a difficulty inherent in the MLT
which, by itself, cannot predict the size of the velocity field.
In this respect it is interesting to compare our finding with the results of a recent 3D implicit large eddy simulation of the turbulent convection in the envelope of a 5red giant star <cit.>.
The simulation refers to the envelope of a 5star at the end of central He burning.
The convective unstable region extends inward from an outer radius R$\sim$4.1$\times$10$^{12}$cm to an inner radius r$\sim$2.2$\times$10$^{12}$cm, where the Schwarzschild
condition of stability is satisfied ($\nabla_{rad}=\nabla_{ad}$, see their Figure 17).
At this layer, the velocity field shows a sudden drop, but does not vanishes; it ceases
to be dominated by the radial components and begins to be dominated by the transversal components (see their Figure 4). This indicates a clear regime of convective turn-over extending
for about 0.6($\sim$0.25$\times$10$^{12}$cm)
below the formally unstable region, which the authors recognize as the lower boundary of the convective envelope. In this region the rms velocities decrease from $\sim$10$^5$cm/s to $\sim$10$^4$cm/s.
Thus the simulation shows that there is a region of about 0.6below the formal Schwarzschild border
where velocities remain significantly high, characterized by the transversal components being about twice the radial ones. From Figure 4 of <cit.> one may also see that the velocity field extends much below this point
and, though it is difficult to evaluate from the figure alone,
the rms velocities are clearly larger than $\sim$10$^3$cm/s, even at several $\sim$10$^{11}$cm
below the Schwarzschild border.
In our track of M=5and Z=0.004, at the stage of maximum penetration
near the tip of the first red giant ascent (not the second as in the <cit.> simulation), the model has the following structure. The stellar radius is R$\sim$7.4$\times$10$^{12}$cm
and the inner Schwarzschild border is at r$\sim$0.8$\times$10$^{12}$cm.
The size of the unstable region is $\sim$6.6$\times$10$^{12}$cm.
The maximum velocities predicted using the MLT are a few $\sim$10$^5$cm/s,
but drop to $\sim$2.4$\times$10$^4$cm/s near the Schwarzschild border.
The pressure scale height at the inner Schwarzschild border is $\sim$4$\times$10$^{11}$cm
and a mixing scale of 0.7corresponds to $\delta_r(0.7)\sim$2.4$\times$10$^{11}$cm while,
one of 4corresponds to $\delta_r(4)\sim$6$\times$10$^{11}$cm.
This non linearity is due to the fact that we measure the overshooting region in terms of
pressure and not in length, and that the pressure scale height below the bottom of the convective region decreases at decreasing radius (while it keeps increasing again in the inner He core).
Our convective region is thus not very different from the one described in <cit.>
simulation. With radial velocities of about $\sim$10$^3$cm/s, convection can span
a distance of $\delta_r(4)$ in about 20yrs which,
since our model spends a few thousand years before convection begins to retreat,
is about 100 times shorter than the evolutionary time of the star in this phase.
Thus, at first glance, the <cit.> simulation indicates that the internal mixing
of about 2-4below the formal convective border are possible
in an evolutionary timescale.
What we have neglected in this simple discussion are the effects of the
molecular weight discontinuity that forms between the outer H rich and
the inner He rich regions, and that could strongly limit or even suppress the penetration of convective motions (e.g. <cit.>).
At face value, our finding indicates that turbulent entrainment into the stably stratified layers <cit.> should be quite efficient in the envelopes of such stars.
Clearly detailed numerical simulations (e.g. <cit.>) especially addressed to the analysis of the mixing efficiency in presence of chemical composition gradient, would be extremely helpful to clarify this issue.
§ ACKNOWLEDGEMENTS
We thank S. Charlot and Y. Chen for helpful discussions.
A.B. acknowledges support from INAF through grant PRIN-2014-14.
P.M. acknowledges support from Progetto di Ateneo 2012, University of Padova, ID: CPDA125588/12.
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1511.00116
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Kummer and gamma laws through independences on trees ]Kummer and gamma laws through independences on trees — another parallel with the Matsumoto–Yor property
Agnieszka Piliszek
Wydział Matematyki i Nauk Informacyjnych
Politechnika Warszawska
Koszykowa 75
00-662 Warszawa, Poland
Jacek Wesołowski
Wydział Matematyki i Nauk Informacyjnych
Politechnika Warszawska
Koszykowa 75
00-662 Warszawa, Poland
The paper develops a rather unexpected parallel to the multivariate Matsumoto–Yor (MY) property on trees considered in <cit.>. The parallel concerns a multivariate version of the Kummer distribution, which is generated by a tree. Given a tree of size $p$, we direct it by choosing a vertex, say $r$, as a root. With such a directed tree we associate a map $\Phi_r$. For a random vector ${\bf S}$ having a $p$-variate tree-Kummer distribution and any root $r$, we prove that $\Phi_r({\bf S})$ has independent components. Moreover, we show that if ${\bf S}$ is a random vector in $(0,\infty)^p$ and for any leaf $r$ of the tree the components of $\Phi_r({\bf S})$ are independent, then one of these components has a Gamma distribution and the remaining $p-1$ components have Kummer distributions. Our point of departure is a relatively simple independence property due to <cit.>. It states that if $X$ and $Y$ are independent random variables having Kummer and Gamma distributions (with suitably related parameters) and $T:(0,\infty)^2\to(0,\infty)^2$ is the involution defined by $T(x,y) =(y/(1+x), x+xy/(1+x))$, then the random vector $T(X,Y)$ has also independent components with Kummer and gamma distributions.
By a method inspired by a proof of a similar result for the MY property, we show that this independence property characterizes the gamma and Kummer laws.
§ INTRODUCTION
Let $X$ and $Y$ be independent random variables. There are several well-known cases where $U=\phi(X,Y)$ and $V=\psi(X,Y)$ are also independent. A number of distributions have actually been characterized this way. Classical results along these lines include Bernstein's characterization <cit.> of the Gaussian distribution through independence of $U=X-Y$ and $V=X+Y$, and Lukacs' characterization <cit.> of the Gamma distribution through independence of $U=X/Y$ and $V=X+Y$.
At the end of the 1990s, a new result of this kind, called the Matsumoto–Yor (MY) property was discovered; see, e.g., <cit.>. It states that if $X$ has a generalized inverse Gaussian (GIG) distribution and $Y$ is Gamma, the random variables $U=1/(X+Y)$ and $V=1/X-1/(X+Y)$ are independent. It arose in studies <cit.> of the conditional structure of some functionals of the geometric Brownian motion. See <cit.> for a related characterization of the GIG and Gamma distributions through the independence of $X$ and $Y$ and of $U$ and $V$.
The MY property is also strongly rooted in classical multivariate analysis. Its matrix-variate version appears naturally in the conditional structure of Wishart matrices; see, e.g., <cit.> as well as <cit.> and <cit.>. A higher-dimensional version of the MY property and related characterization was studied in <cit.>, where a Gamma-type multivariate distribution was obtained by connecting its density shape to a tree. Through a suitable transformation related to directed trees, a random vector having the latter distribution was mapped to independent components with GIG and Gamma distributions. This approach also led to a characterization of the product of GIG and Gamma distributions. In the special case of a chain with two vertices these results are equivalent to the characterization through the original MY property.
The MY property attracted a lot of attention in the last 15 years. In particular, <cit.> tried to identify all possible functions $f$ and distributions of independent $X$ and $Y$ such that $f(X+Y)$ and $f(X)-f(X+Y)$ are also independent. The MY property corresponds to the case $f(x) = 1/x$. Another important case identified in that paper occurs when $f(x)=\ln(1+ 1/x)$. Thus if $X$ and $Y$ are independent with Kummer and Gamma distributions, then
U=X+Y \quad \mbox{and} \quad V = \frac{1+{1}/(X+Y)}{1+{1}/{X}}
are independent Kummer and Beta distributed random variables, respectively.
Characterizations by independence of $X$ and $Y$ and of $U$ and $V$ were obtained in <cit.>. To derive their results, however, the authors needed to impose technical conditions of differentiability or local integrability of logarithms of strictly positive densities. Recently a regression characterization under natural integrability condition was given in <cit.> without any assumptions on the densities.
In the present paper we are interested in an independence property discovered recently in <cit.>. It states that if $X$ and $Y$ are independent random variables with Kummer and Gamma distributions, then
U=Y/(1+X) \quad \mbox{and} \quad V=X\,\frac{1+X+Y}{1+X}
are also independent and have Kummer and Gamma distributions, respectively. In studying this property, we will exploit several of the ideas described above. In particular, we will give a characterization of the Gamma and Kummer distributions through the independence of the components in the pairs $(X,Y)$ and $(U,V)$. In the proof, inspired from <cit.>, we will use the method of functional equations for densities assuming local integrability of their logarithms. This is reported in Section <ref>.
Next, we will introduce and study multivariate versions of the property described above. Our approach parallels the one adopted in <cit.> for a multivariate version of the MY property. We will first define a $p$-variate tree-Kummer distribution from an undirected tree $T$ of size $p$. For each vertex $r$ of $T$, we will define the directed tree by choosing $r$ as its root. To each such directed tree, we will associate a transformation $\Phi_r:(0,\infty)^p\to(0,\infty)^p$ and show that if a random vector ${\bf S}$ has a tree-Kummer distribution, then $\Phi_r(\bf S)$ has independent components with Gamma and Kummer distributions. This analogue of Theorem 3.1 in <cit.> is given in Section <ref>.
In Section <ref> we will derive a characterization of products of Gamma and Kummer distributions (and thus of the tree-Kummer distribution) assuming that for any leaf $r$ of the tree $T$, the components of $\Phi_r(\bf S)$ are independent. This result parallels Theorem 4.1 in <cit.>. Finally, Section <ref> contains some concluding remarks.
§ KUMMER AND GAMMA CHARACTERIZATION
The Kummer distribution $\mathcal{K}(\alpha,\beta,\gamma)$ with parameters $\alpha,\gamma>0$, $\beta\in\R$ has density
f(x)\propto \frac{x^{\alpha-1}}{(1+x)^{\alpha+\beta}}\,e^{-\gamma x}\,\mathbbm{1}_{(0,\infty)}(x).
If $\beta>0$ it is a natural exponential family generated by the second kind Beta distribution. More information on the Kummer distribution, its properties and applications can be found in <cit.> and in the monograph <cit.>.
By the Gamma distribution $\mathcal{G}(\alpha,\gamma)$ with parameters $\alpha,\gamma>0$, we mean the distribution whose density is given by
g(x)\propto x^{\alpha-1}e^{-\gamma x}\mathbbm{1}_{(0,\infty)}(x).
Consider two independent random variables $X$ and $Y$ with respective distributions $X\sim \mathcal{K}(a,b-a,c)$ and $Y\sim \mathcal{G}(b,c)$, where $a,b,c>0$. Define a bijection $T:\; (0,\infty)^2\to (0,\; \infty)^2$ by
T(x,y) = \left( \frac{y}{1+x}, x\left(1 + \frac{y}{1+x}\right)\right).
Let $(U,V)=T(X,Y)$. It has been observed in <cit.> that $U$ and $V$ are independent and $U\sim \mathcal{K}(b, a-b, c)$, $V\sim\mathcal{G}(a,c)$. Our objective in this section is to give a converse of this result, that is a characterization of the Kummer and the Gamma distribution through the independence property mentioned above. Unfortunately, as in the case of the characterization of the Kummer and Gamma distributions obtained by <cit.>, we also need to impose some regularity conditions on densities.
Let $X$ and $Y$ be two independent positive random variables with positive and continuously differentiable densities on $(0,\infty)$. Suppose that
$$ U = \frac{Y}{1+X},\; \; V = X\left(1 + \frac{Y}{1+X}\right),$$
are independent. Then there exist constants $a,b,c>0$, such that $ X\sim \mathcal{K}(a,b-a,c)$, $ Y\sim \mathcal{G}(b, c)$ or, equivalently, $U\sim \mathcal{K}(b, a-b, c)$ and $V\sim\mathcal{G}(a,c)$.
Note that $T$ is an involution. Given that $(U,V)=T(X,Y)$, we also have $(X,Y) = T(U,V)$. Given that the random vectors $(U, V)$ and $(X,Y)$ have independent components with continuous densities $p_U$, $p_V$, $p_X$ and $p_Y$ respectively, the independence property can be rewritten as
p_U(u) p_V(v) = |J(u,v)|p_X(v/1+u)p_Y{u(1+v/1+u)}
for all $x,y,u,v>0$, where $J$ is the Jacobian. Furthermore, given that
J(u,v) = \frac{1}{1+u} \left(1 + \frac{v}{1+u}\right)
and the densities are strictly positive it follows that Eq. (<ref>) can alternatively be written as the functional equation
A(u) + B(v) = C(v/1+u) + D{u(1+v/1+u)}
A(x) = \ln \,p_U(x) +\ln \,x,\qquad B(x) = \ln \,p_V(x)+\ln \,x,
C(x) = \ln \,p_X(x)+\ln \,x, \qquad D(x) = \ln \,p_Y(x)+\ln \,x.
Differentiating both sides of (<ref>) with respect to $u$ gives
A'(u) =-\frac{v}{(1+u)^2} C'\left(\frac{v}{1+u}\right) + D'\left\{u\left(1+\frac{v}{1+u}\right)\right\} \left\{1+\frac{v}{(1+u)^2}\right\}.
Now, we insert $(x,y) = T(u,v)$ in the above equation and get
C'(x)x(1+x) = D'(y) {1+x+y+x(1+x)}-A'(y/1+x)(1+x+y).
Note that the right-hand side of Eq. (<ref>) converges to $\{D'(y)-A'(y)\}(1+y)$ as $x \to 0$. Hence, the left-hand side of Eq. (<ref>) also has a limit, say $-C_0$, when $x\to 0^+$ and $C_0$ does not depend on $y$ since there is no $y$ on the left-hand side of Eq. (<ref>). Therefore,
A'(y) =\frac{C_0}{1+y}+D'(y).
We insert it back into Eq. (<ref>) to arrive at
C_0/x+D'(y/1+x)1+x+y/xyy/1+x =-/ C'(x)+ D'(y){ 1+x+y/x(1+x)+1}.
On the other hand with $u=y$ and $v=x(1+y)$, Eq. (<ref>) reads
ayA(y)+B{x(1+y)}=C(x) +D{y(1+x)}.Differentiation with respect to $x$ yields
b'B'{x(1+y)} (1+y)(1+x) -C'(x)(1+ x) = D'{y(1+x)}y(1+x) .Note that the left-hand side above has a finite limit when $y\to 0^+$. Consequently, denoting $z=y(1+x)$ we conclude that $\lim_{z\to 0^+} D'(z) z=: b$ exists and is finite.
Therefore the left-hand side of Eq. (<ref>) is finite when $x\to\infty$. By comparing it with the right-hand side of (<ref>) we conclude that $-\lim_{x\to\infty}C'(x)=:c$ is finite. Therefore, letting $x\to\infty$ in Eq. (<ref>), we get
$$\frac{b}{y} = c + D'(y).$$
In view of the definition of $D$, we conclude that $p_Y(y)\propto y^{b-1} e^{-cy}$ and $b,c>0$, i.e., $Y\sim\mathcal{G}(b,c)$.
Due to the fact that $T$ is an involution the functional equation (<ref>) is symmetric in $(A,B)$ and $(C,D)$. Consequently, there exists a constant $a$ such that $B$ is of the form
B'(x) =\frac{a}{x} -d .
In order to find $C$ let us note that Eq. (<ref>) yields
B'(x)(1+x) -C'(x)(1+x)=\lim_{y\to 0} D'(y(1+x))y(1+x)=b
and thus
C'(x) = \frac{a}{x}-d-\frac{b}{1+x}.
Finally, we get
C(x) = a\ln x - d x -b\ln (1+x) + c_0.
p_X(x)\propto \frac{x^{a-1}}{(1+x)^{b}}\,e^{-dx}\mathbbm{1}_{(0,\infty)}(x)
and necessarily, $a>0$.
What is left to do is to determine the relationship between the parameters. Referring again to the definitions of $A$, $B$, $C$, $D$ and to the original Eq. (<ref>), we have
\begin{multline*}
p_U(u)\, p_V(v) =\frac{1 + \frac{v}{1+u}}{1+u} p_X\left(\frac{v}{1+u}\right)p_Y\left\{u\left(1+\frac{v}{1+u}\right)\right\}
\\
\propto\frac{1+u+v}{(1+u)^2}\left(\frac{v}{1+u}\right)^{a-1}\left(\frac{1+u+v}{1+u}\right)^{-b} e^{-d\frac{v}{1+u}}\left(u\,\frac{1+u+v}{1+u}\right)^{b-1}e^{-cu\,\frac{1+u+v}{1+u}}$$
\\
\end{multline*}
Given that the function on the right-hand side has to be a product of a function of $u$ and a function of $v$, it follows that $d=c$. Thus, the result follows.
\end{proof}
To weaken the smoothness assumptions imposed on densities in Theorem \ref{tw1}, we will use local integrability instead of continuous differentiability, as proposed in \cite{We02a} for the MY-type functional equation, and then applied in \cite{KV11} in the characterization of the Kummer and Gamma distributions.
\begin{lemma}
It is sufficient to assume that logarithms of all densities are locally integrable in Theorem \ref{tw1}.
\end{lemma}
\begin{proof}
\setlength\arraycolsep{2pt}
Given that the other assumptions of Theorem \ref{tw1} are satisfied, we conclude that Eq.~\eqref{ay} holds for almost all $(x,y)\in (0,\infty)^2$. Given also that the functions $A$, $B$, $C$, $D$ are locally integrable we can take any $0<x_0< x_1<\infty$ and integrate both sides of \eqref{ay} with respect to $x$ from $x_0$ to $x_1$. Then
\begin{eqnarray}
\int_{x_0}^{x_1}B\{x(1+y)\} dx -\int_{x_0}^{x_1}D\{y(1+x)\} \,dx & = & \int_{x_0}^{x_1} C(x) \,dx - A(y)(x_0-x_1).
\nonumber
\end{eqnarray}
We substitute $s = x(1+y)>0$ in the first integral and $t=y(1+x)>0$ in the second integral on the left-hand side above. Consequently, for almost all $y\in(0,\infty)$
\begin{eqnarray}
\int_{x_0(1+y)}^{x_1(1+y)}\frac{B(s)}{1+y} \, ds -\int_{(1+x_0)y}^{(1+x_1)y}\frac{D(t)}{y} \, dt &=& \int_{x_0}^{x_1} C(x) dx - A(y)(x_0-x_1).\label{cg_y}
\end{eqnarray}
The left-hand side of equation \eqref{cg_y} is continuous in $y\in (0,\infty)$. Hence, function $A$ can be extended to a continuous function $\tilde{A}$ on $(0,\infty)$.\\
Similarly, integrating \eqref{ay} with respect to $y$ from $y_0$ to $y_1$, $0<y_0<y_1<\infty $, we get
%\int_{y_0}^{y_1}B(x(1+y)) dy -\int_{y_0}^{y_1}D(y(1+x))dy & = & (y_1-y_0) C(x) - \int_{y_0}^{y_1}A(y)dy, \nonumber\\
%\end{eqnarray}for almost all $x>0$.\\\\
%Kładąc $s = x(1+y)$ ($dy= \frac{ds}{x}$) oraz $t=y(1+x)$ ($dy=\frac{dt}{1+x}$) dostajemy:
\begin{eqnarray}
\int_{(1+y_0)x}^{(1+y_1)x}\frac{B(s)}{x}\,ds -\int_{y_0(1+x)}^{y_1(1+x)}\frac{D(t)}{1+x}\,dt & = & (y_1-y_0) C(x) - \int_{y_0}^{y_1}A(y)dy\label{cg_x}
\end{eqnarray}
for almost all $x>0$. Therefore, $C$ has also a continuous extension $\tilde{C}$ which is a function on $(0,\infty)$.
For any $x,y>0$ let $u= x(1+y)$ and $v = y(1+ x)$. Then
\begin{equation}
\label{dziala}
\left\{ \begin{array}{lll}
x &=& \frac{1}{2}\left(\sqrt{(1+v-u)^2 + 4u}-(1+v-u)\right)>0,\\
y & =& \frac{1}{2}\left(\sqrt{(1+u-v)^2 + 4v}-(1+u-v)\right)>0.
\end{array} \right.
\end{equation}
Plugging these values of $x$ and $y$ into Eq.~\eqref{ay} with $\tilde{A}=A$ and $\tilde{C}=C$, we get
\begin{multline}
B(u) - D(v) = \tilde{C}\left\{\frac{1}{2}\left(\sqrt{(1+v-u)^2 + 4u}-(1+v-u)\right)\right\} +
\\-\tilde{A}\left\{\frac{1}{2}\left(\sqrt{(1+u-v)^2 + 4v}-(1+u-v)\right)\right\}.
\label{cg_bd}
\end{multline}
The right-hand side of Eq.~\eqref{cg_bd} can be extended continuously for $u\in(0,\infty)$, which gives a continuous extension $\tilde{B}$ of $B$ on the left-hand side. Similarly, we can extend $D$ to $\tilde{D}$, which is continuous on $(0,\infty)$.
Consequently, we can assume that Eq.~\eqref{ay} is satisfied for all ${(x,y)\in(0,\infty)^2}$ and that the functions $A$, $B$, $C$ and $D$ appearing in \eqref{ay} are continuous. Now, repeating step by step the above reasoning, we see that $A$, $B$, $C$ and $D$ may be assumed continuously differentiable, i.e., all the assumptions of Theorem \ref{tw1} are satisfied.
\end{proof}
\section{Independence properties of tree-Kummer distribution}
\label{sec:3}
\subsection{A symmetrization of the independence property}
\label{sec:3.1}
The first step in approaching the tree-version of the MY property was to consider a symmetrized version of the classical MY property; see \cite{MW04}, pp. 686--687. Similarly here we start with a derivation of a symmetric version of the property of independence of Kummer and Gamma distributions from \cite{HV15}.
Consider independent random variables $X\sim \mathcal{K}(a,b-a,c)$ and $Y\sim \mathcal{G}(b,c)$ with parameters $a,b,c>0$, and $(U,V)=T(X,Y)$. Define a random vector $$(S_1,S_2)=\left(X,\frac{Y}{1+X}\right)=\left(\frac{V}{1+U},U\right).$$ Then $(X,Y)=(S_1,S_2(1+S_1))$ and $(V,U)=(S_1(1+S_2),S_2)$. For the density $p$ of $(S_1,S_2)$ we get
with $\psi(x,y)=\left(x, y/(1+x)\right)$ being a bijection on $(0,\infty)^2$. Therefore, $\psi^{-1}(s_1,s_2)=(x(s_1,s_2),\,y(s_1,s_2))=(s_1,\,(1+s_1)s_2)$ and thus
p(s_1,s_2)\propto s_1^{a-1}s_2^{b-1}\,e^{-c(s_1+s_2+s_1s_2)}\mathbbm{1}_{(0,\infty)^2}(s_1,s_2).
We will consider a more general distribution of this type by introducing three positive parameters $c_1$, $c_2$ and $c_{1,2}$ and writing
\bel{kt2}
p(s_1,s_2)\propto s_1^{a_1-1}s_2^{a_2-1}\,e^{-(c_1s_1+c_2s_2+c_{1,2}s_1s_2)}\mathbbm{1}_{(0,\infty)^2}(s_1,s_2).
\ee
This is the analogue of the density $f$ in \cite{MW04}, p. 687.
Let us also define two bijective mappings: $\Phi_r: (0,\infty)^2\rightarrow (0,\infty)^2$, $r\in\{1,2\}$, by
\begin{eqnarray*}
\Phi_1(s_1,s_2) &=& (s_{1,(1)},\,s_{2,(1)}) = \left(s_1\left(1 + \frac{c_{1,2}}{c_1}s_2 \right),\, s_2\right)=\left(s_1\left(1 + \frac{c_{1,2}}{c_1} s_{2,(1)}\right) ,\,s_2\right).
\\
\Phi_2(s_1,s_2) &=&(s_{1,(2)},s_{2,(2)}) = \left(s_1,\, s_2\left(1+\frac{c_{1,2}}{c_2}s_1\right)\right)= \left(s_1,\, s_2\left(1+\frac{c_{1,2}}{c_2}s_{1,(2)}\right)\right).
\end{eqnarray*}
They are analogs of the mappings $\psi_1$ and $\psi_2$ in \cite{MW04}, p. 686.
Assume that a random vector $(S_1,S_2)$ has the density $p$ as in \CG{Eq.}~\eqref{kt2}. Define $(X_{1,(1)}, X_{2,(1)})= \Phi_1(S_1,S_2)$ and $(X_{1,(2)}, X_{2,(2)})= \Phi_2(S_1,S_2)$. Then standard computations involving density transformation yield
\bel{x1}\left(\frac{c_{1,2}}{c_1}X_{1,(1)},\,\frac{c_{1,2}}{c_2}\,X_{2,(1)}\right)\sim \mathcal{G}(a_1,c')\otimes \mathcal{K}(a_2,a_1-a_2,c')\ee and
\bel{x2}\left(\frac{c_{1,2}}{c_2}X_{1,(2)},\,\frac{c_{1,2}}{c_2} X_{2,(2)}\right)\sim \mathcal{K}(a_1,a_2-a_1,c')\otimes\mathcal{G}(a_2,c'),\ee
where $c'= c_1c_2/{c_{1,2}}$ and $\mu\otimes\nu$ denotes a distribution which is a product of distributions $\mu$ and $\nu$.
Our aim in this section is to extend the above construction to any dimension. As in \cite{MW04} the language of undirected and directed trees will be very helpful in this context. Similarly to \cite{MW04}, Section 2, we need to provide background information regarding the tree language and certain facts about transformations.
\subsection{Trees and transformations}
\label{sec:3.2}
Let us recall that a graph $G=(V,E)$, where $V$ is the set of nodes and $E\subseteq \{\{u,v\}: u,v\in V, u\neq v\}$ is the set of edges, is called a \textit{tree}, if it is connected and acyclic. The symbol $\deg_G(i)$ stands for the degree of vertex $i\in V$, i.e., $\deg _G(i) = |\{\ell:\; \{\ell,i\}\in E\}|$, where $|B|$ denotes a number of elements in a finite set $B$. A vertex of degree 1 is a \emph{leaf}.
We define a \emph{subtree} $S$ of a tree $T=(V, E)$ as the graph $S=(V(S), E(S))$, where $V(S)\subseteq V$ and $E(S) = \left\{ \{i,j\} \in E: i, j\in V(S)\right\}$. If $S$ is a subtree of $T$ we write $S\subseteq T$.
Let $T=(V,E)$ be a tree of size $p\geq 2$. For a fixed root $r\in V$, we direct $T$ from the root towards leaves and denote such a directed tree by $T_r$. Having the tree directed, we can say that node $i$ is a \textit{child} of vertex $j$ (or $j$ is a \textit{parent} of $i$) if and only if $\{i,j\}\in E$ and the tree is directed from $j$ to $i$ (note that every node, unless it is a leaf, has at least one child and every node but the root has exactly one parent). The set of all children of $i$ in $T_r$ will be denoted by $\mathfrak{C}_r(i)$ and the parent of vertex $i$ by $\mathfrak{p}_r(i)$. We say that the undirected tree $T$ is the \emph{skeleton} of $T_r$.
\begin{remark}
Note that when drawing parallels between the result presented here and those in \cite{MW04}, one has to keep in mind that there, contrary to the tradition in the graph theory, the tree is directed in the opposite way: from leaves toward a chosen root. Therefore, the parents and children notions here and in \cite{MW04} have to be swapped.
\end{remark}
For any $r\in V$ (and hence $T_r$) and fixed set of parameters $c_{i,j}>0$, $i,j\in V$ ($c_{i,i}=:c_i$) we define \CG{a} transformation $\Phi^T_r: \mathbb{R}_+^p \rightarrow \mathbb{R}_+^p $ by
\bel{frt}\Phi_r^T (s_i, i\in V) = ( s_{i,(r)}, i\in V ),\ee where
\bel{gw}s_{i,(r)}=s_i\,\displaystyle{\prod_{j\in \mathfrak{C}_r(i)}\,\left(1+\frac{c_{i,j}}{c_i} s_{j,(r)}\right)},
\ee
where by convention an empty product is equal to $1$, i.e., if $V\ni i\ne r$ is a leaf then $s_{i,(r)}=s_i$. The definition \eqref{gw} is inverse recursive with a starting point being any vertex with maximal distance from the root $r$. Usually we will write $\Phi_r$ instead of $\Phi_r^T$. Compare these mappings with the mappings $\psi_r$, $r\in V$, defined in (2.5) and (2.6) of \cite{MW04}.
It is easy to see that $\Phi_r$ is a bijection for any $r\in V$. The transformation $\Phi_r^{-1}: \; \mathbb{R}_+^p \rightarrow \mathbb{R}_+^p$ is given by ${\Phi_r^{-1}(y_j, j\in V)=(s_i, i\in V)}$, where
$$s_i = \frac{y_i}{\prod_{j\in \mathfrak{C}_r(i)}(1+\frac{c_{i,j}}{c_i}y_j)},\quad i\in V,$$
and again the product over the empty set is considered to be equal to $1$.
First we compute the Jacobian $J_r$ of $\Phi_r^{-1}$. Note, that if we enumerate the nodes in such a way that each child has a number greater than its parents, the Jacobi matrix is upper triangular. Thus, we only need to find the elements on its diagonal which are partial derivatives
\frac{\partial s_i}{\partial y_i}=\left\{\prod_{j\in \mathfrak{C}_r(i)}\left(1+\frac{c_{i,j}}{c_i}y_j\right)\right\}^{-1}, \quad i\in V.
Considering that every node but the root has exactly one parent, it appears in exactly one partial derivative. Hence, we have
\bel{jakob}
J_r(y_i, i\in V)=\left\{\prod_{i\in V\setminus\{r\}}\,\left(1+\frac{c_{\mathfrak{p}_r(i),i}}{c_{\mathfrak{p}_r(i)}}y_i\right)\right\}^{-1}.
\ee
We will also need the identity given in the following proposition.
%Udowodnimy za chwilę twierdzenie wyrażające pewną własność $\Phi_r$ w języku drzew.
\begin{proposition}\label{tw_graf_og}
For any $r\in V$ if $(s_{i,(r)})_{i\in V}$ and $(s_i)_{i\in V}$ are related by \eqref{gw} then
\bel{summ}\sum_{m\in V}\, c_m s_{m,(r)} =\sum_{S\subseteq T}\prod_{i\in V(S)} \frac{s_i}{c_i^{\deg _S (i)-1}}\prod_{\{j,k\}\in E(S)} c_{j,k} .\ee
\end{proposition}
\begin{proof}
%Przypomnijmy oznaczenie $ k_{i,(r)}=\Phi _r^i (\mathbf{k})$.\\
Fix a root $r\in V$ in the tree $T$. Let $\mathcal{S}^m_r$ denote the family of directed subtrees of $T_r$ with the root $m\in V$. By $\mathcal{S}(r,m)$ we denote the family of undirected trees which are skeletons (undirected trees) of (directed) trees from $\mathcal{S}^m_r$.
% \fill (1,0) circle(2pt)node[anchor=west] {$m$};
% %\fill (1.5,1) circle(2pt) node[anchor=west] {$v_3$};
% %\fill (0.5,1) circle(2pt)node[anchor=west] {$v_1$};
% \fill (1,-1) circle(2pt) node[anchor=west] {$r$};
% \draw[->] (1,0) -- (0.5,0.5);
% \draw[->] (1,0) edge(1.5,0.5);
% \draw[->] (1,0)--(1,0.5) ;
% \draw[densely dashed] (1,-1)--(1,0);
% \draw [color=red] (1,0.3) ellipse(0.7cm and 0.5cm);
We will show, using mathematical induction, that for every $m\in V$
\bel{row_og}
\sum_{S\in \mathcal{S}(r,m)}\prod_{\{j,k\}\in E(S)} c_{j,k}\prod_{i \in V(S)} \frac{s_i}{c_i^{\deg _{S}(i)-1}} =c_m s_{m,(r)}.
\ee
Given that
\sum_{S\subseteq T}\prod_{i\in V(S)} \frac{s_i}{c_i^{\deg _S (i)-1}}\prod_{\{j,k\}\in E(S)} c_{j,k} =\sum_{m\in V}\sum_{S\in\mathcal{S}(r,m)}\prod_{\{j,k\}\in E(S)} c_{j,k}\prod_{i \in V(S)} \frac{s_i}{c_i^{\deg _{S} (i)-1}},
the identity \eqref{summ} follows from Eq.~\eqref{row_og}.
In order to prove \eqref{row_og} we will rely on the definition \eqref{gw}. The proof will be by induction with respect to the distance of the vertex $m$ from its farthest descendant.
We first consider the case $\mathfrak{C}_r(m)=\emptyset$. Then the left-hand side of \eqref{row_og} equals
$s_mc_m$ as the only element of $\mathcal{S}(r,m)$ is a trivial tree $(\{m\},\emptyset)$. In this case $s_{m,(r)}=s_m$ due to \eqref{gw}, and thus the identity \eqref{row_og} follows.
Now we need to consider the case $\mathfrak{C}_r(m)\ne \emptyset$ assuming that for all $n\in \mathfrak{C}_r(m)$ the identity \eqref{row_og} holds, i.e.,
\bel{indu}\sum_{S\in\mathcal{S}(r,n)}\prod_{\{j,k\}\in E(S)} c_{j,k}\prod_{i \in V(S)} \frac{s_i}{c_i^{\deg _{S} (i)-1}} =c_n s_{n,(r)}.\ee
\begin{figure}[h]\begin{tikzpicture}
\tikzstyle{male} = [ font={\large}, shape=circle, minimum size=0.1cm,text=black,
very thick, draw=black!80,bottom color=black!50, top color=white, text width=0.25cm, align=left]
\tikzstyle{male1} = [ font={\large}, shape=circle, minimum size=0.1cm,text=black,
very thick, draw=black!80, top color=white, text width=0.25cm, align=left]
\tikzstyle{mynodestyle} = [ font={\Large\bfseries}, shape=circle, minimum size=0.2cm,text=black,
very thick, draw=black!80, top color=white, text width=0.5cm, align=center]
\tikzstyle{duzy} = [ font={\Large\bfseries}, shape=circle, minimum size=0.2cm,text=black,
very thick, draw=black!80, top color=white,bottom color=black!50, text width=0.5cm, align=center]
\node[ mynodestyle] (v1) at (-4,3) {$r$};
\node[ duzy] (v3) at (-2,3) {$m$};
\node[male] (v2) at (0,4) {$n_1$};
\draw[dashed, ->] (v1) -- (v3);
\node [male] (v4) at (0,3) {$n_2$};
\draw[dashed, ->] (v1) -- (-2,2);
\draw[dashed, ->] (v3) -- (0,2);
\draw[->] (v2) -- (1,5);
\draw[ ->] (v2) -- (1,4);
\draw[->] (v4) -- (1,3);
\draw[->] (v4) -> (1,2);
\draw[->] (v4)-> (0.8,3.5);
\node[male1] (vk) at (-1,2) {$n_k$};
\draw[->] (v3) -- (v4);
\draw[->] (v3) -- (v2);
\draw[->] (v3) -- (vk);
\end{tikzpicture}\caption{Tree $T_r$ with one of possible $S\in \mathcal{S}(r,m)$ colored grey.}
\label{fig1}\end{figure}
%For a given $S=(V_S,E_S)\in\mathcal{S}(r,n)$, $n\in\mathfrak{C}_r(m)$, we denote by $S^{(m)}$ a tree obtained by connecting the vertex $m$ with the %vertex $n$ of $S$, that is $S^{(m)}=(V_S\cup\{m\},\,E_S\cup\{\{m,n\}\})$. Note that the $\deg_S(i)=\deg_{S^{(m)}}(i)$ for $i\in V_S\setminus\{n\}$ %and
%$\deg_S(n)=\deg_{S^{(m)}}(n)-1$. Therefore the last equation can be rewritten as follows
%Ponieważ dodanie wierzchołka $m$ do wybranego poddrzewa $S(r,n)$, $n\in D_r(m)$, zmieni, zwiększając o $1$, tylko stopień wierzchołka $n$, to %równoważnie możemy napisać:
%$$\sum_{S\subset\mathcal{S}(r,n)}\prod_{\{j,k\}\in E(S)} c_{j,k}\prod_{i \in V(S)} \frac{s_i}{c_i^{\deg_{S^{(m)}}(i)-1}} =s_{n,(r)}.$$
Note that any element of $\mathcal{S}(r,m)$ is a union of subtrees from $\mathcal{S}(r,n)$, $n\in \mathfrak{C}_r(m)$, extended by attaching to such a union the vertex $m$ with suitably chosen edges (see Figure \ref{fig1} for illustration). For any $B\subset\mathfrak{C}_r(m)$ (ordered by numbers of vertices) we denote $\mathcal{S}(r,B)=\times_{w\in B}\,\mathcal{S}(r,w)$, a Cartesian product of families of subtrees. Denote also $S_B=(S_w)_{w\in B}\in\mathcal{S}(r,B)$ and note that it is a vector of subtrees. Using this notation we can write the family $\mathcal{S}(r,m)$ as follows
\mathcal{S}(r,m) =
\left\{\left( V(S_B)\cup\{m\},\; E(S_B)\cup\bigcup_{w\in B}\,\{\{m,w\}\}\right),\; S_B\in\mathcal{S}(r,B),\;B\subset\mathfrak{C}_r(m) \right\},
V(S_B)=\bigcup_{w\in B}\, V(S_w) \quad \mbox{and} \quad E(S_B)=\bigcup_{w\in B}\,E(S_w).
In particular, the element in the above family associated to $B=\emptyset\subset\mathfrak{C}_r(m)$ is the trivial tree $(\{m\},\emptyset)$. Then
\begin{multline}\label{aaa}
\sum_{S\in\mathcal{S}(r,m)}\prod_{\{j,k\}\in E(S)} c_{j,k}\prod_{i \in V(S)} \frac{s_i}{c_i^{\deg_S(i)-1}} \\
=c_ms_m +
\sum_{\emptyset\ne B\subset\mathfrak{C}_r(m)}\,\sum_{S_B\in\mathcal{S}(r,B)}\,\prod_{\{j,k\}\in\,E(S_B(m))}\,c_{j,k}\,\prod_{i\in V(S_B(m))}\, \frac{s_i}{c_i^{\deg_{S_B(m)}(i)-1}},
\end{multline}
S_B(m)=\left(V(S_B(m)),\,E(S_B(m))\right)=\left(V(S_B)\cup\{m\},\; E(S_B)\cup\bigcup_{w\in B}\,\{\{m,w\}\}\right).
Therefore, $$\deg_{S_B(m)}(i)=\left\{\begin{array}{lll}|B| & \mbox{for} & i=m,\\ \deg_{S_w}(i) & \mbox{for} & i\in V(S_w)\setminus\{w\},\;w\in B,\\ \deg_{S_w}(w)+1 & \mbox{for} & i=w,\;w\in B.\end{array}\right.$$ Consequently, the double sum in \eqref{aaa} can be written as
c_ms_m\sum_{\emptyset\ne B\subset\mathfrak{C}_r(m)}\,\frac{1}{c_m^{|B|}}\,\left(\prod_{w\in B}\,\frac{c_{m,w}}{c_w}\right)\,\sum_{S_B\in\mathcal{S}(r,B)}\,\prod_{w\in B}\,\left(\prod_{\{j,k\}\in E(S_w)}\,c_{j,k}\,\prod_{i\in V(S_w)}\, \frac{s_i}{c_i^{\deg_{S_w}(i)-1}}\right)
=c_ms_m\sum_{\emptyset\ne B\subset\mathfrak{C}_r(m)}\,\frac{1}{c_m^{|B|}}\,\left(\prod_{w\in B}\,\frac{c_{m,w}}{c_w}\right)\,\prod_{w\in B}\,\left(\sum_{S_w\in\mathcal{S}(r,w)}\,\prod_{\{j,k\}\in\,E(S_w)}\,c_{j,k}\,\prod_{i\in V(S_w)}\,\frac{s_i}{c_i^{\deg_{S_w}(i)-1}}\right)
$$$$=c_ms_m\sum_{\emptyset\ne B\subset\mathfrak{C}_r(m)}\,\frac{1}{c_m^{|B|}}\,\prod_{w\in B}\,c_{m,w}\,s_{w,(r)},$$
where the last equality follows form the induction assumption \eqref{indu} applied to every sum under the product $\prod_{w\in B}$ in the previous expression. Summing up we have
\sum_{S\in\mathcal{S}(r,m)}\prod_{\{j,k\}\in E(S)} c_{j,k}\prod_{i \in V(S)} \frac{s_i}{c_i^{d_{(r,m)}(i)}} =c_ms_m\left[1+\sum_{\emptyset\ne B\subset\mathfrak{C}_r(m)}\,\frac{1}{c_m^{|B|}}\,\prod_{w\in B}\,c_{m,w}\,s_{w,(r)}\right]
where the last equation is a consequence of \eqref{gw}.
\end{proof}
Note that the right-hand side of \eqref{summ} does not depend on $r$, i.e., on the direction of the tree. Therefore the following result is an immediate consequence of Proposition~\ref{tw_graf_og}.
\begin{corollary}For any $r_1$, $r_2\in V$
$$\sum_{i\in V}c_i s_{i, (r_1)} = \sum_{i\in V}c_i s_{i, (r_2)}.$$
\end{corollary}
\subsection{Independence property of tree-Kummer distribution}\label{TKum}
Let $T=(V,E)$, as above, be a tree of size $p$. We define the set $C_T$ as the set of all symmetric matrices ${\bf C}=[c_{i,j}]_{i,j\in V}$ such that $c_{i,j}>0$, when $\{i,j\}\in E$, $c_{i,i}=c_i>0$ and $c_{i,j}=0$ otherwise. We say that a random vector $\mathbf{X}$ has \emph{tree-Kummer} distribution, $\mathrm{TK}(\textbf{a}, \textbf{C})$, where $\mathbf{a}=(a_i, i\in V)\in \mathbb{R}^p_+$, and $\textbf{C}\in C_T$ if its density is of the form
f(\mathbf{x}) \propto\prod _{i\in V} x_i ^{a_i-1} \exp\left(- \sum_{S\subseteq T}\,C(S)\,\prod_{\{j,k\}\in E(S)} c_{j,k} \right)\mathbbm{1}_{(0;\infty)^p}(\mathbf{x}),
C(S)=\prod_{i\in V(S)} \frac{x_i}{c_i^{\deg_S(i)-1}}.
As it will be seen later, if $v\in V$ is a leaf then the law of $X_v$ is Kummer. On the other hand for a degenerate tree of size $1$, the $\mathrm{TK}$ distribution is just a Gamma distribution. This distribution is an analogue of the $W_G^c$ distribution introduced in \cite{MW04}, Section~3, which is also Gamma for the degenerate tree of size $1$, and if ${\bf X}$ is a random vector with $W_G^c$ distribution, then for any leaf $v\in V$ the random variable $X_v$ has a GIG distribution.
% Celem kolejnych rozważań będzie udowodnienie twierdzenia o niezależności składowych wektora $\Phi^T_r(\mathbf{X})$ przy założeniu, że $X\sim TK(\textbf{a}, \textbf{C}_T)$.
% Theorem \ref{tw_graf_og} możemy wreszcie powiedzieć coś o rozkładach zmiennych losowych powstałych przez przekształcenie, za pomocą $\Phi$, %odpowiedniego wektora losowego.
Note that in the case of the tree $T$ which is a 1--2, chain the above density agrees with \eqref{kt2}. Thus, as observed in Section~\ref{sec:3.1}, the transformations $\Phi_1$ and $\Phi_2$ applied to a bivariate random vector with such density produce random vectors with independent Kummer and Gamma components. We will extend this observation to a tree-Kummer law generated by any tree. It will lead to a multivariate version of the independence property from \cite{HV15}.
\begin{theorem}\label{tw_og}
Let $T=(V, E)$ be a tree, $|V|=p\geq 2$, and $\mathbf{S}=(S_i, i\in V)$ be a random vector
following the tree-Kummer $\mathrm{TK}(\mathbf{a}, \mathbf{C})$ distribution with $\textbf{a}\in \mathbb{R}^p_+$, $\mathbf{C}\in C_T$.
Denote $\mathbf{X}_r = \Phi_r(\mathbf{K})$, where $\Phi_r$ is the transformation defined in \eqref{frt} and \eqref{gw} with parameters $c_{i,j}$ and $c_i$, where $i,j,r\in V$.
Then, for any $r\in V$ the components of the random vector ${\mathbf{X}_r=(X_{i,(r)},i\in V)}$ are independent. Moreover,
$${X_{r,(r)}\sim \mathcal{G}(a_r, c_r)}\quad\mbox{and} \quad \frac{c_{\mathfrak{p}_r(i),i}}{c_{\mathfrak{p}_r(i)}}\,X_{i,(r)}\sim \mathcal{K}\left(a_i,a_{\mathfrak{p}_r(i)}-a_i,\frac{c_{\mathfrak{p}_r(i)}\,c_i}{c_{\mathfrak{p}_r(i),i}}\right),\quad i\in V\setminus\{r\}.$$
\end{theorem}
This theorem may be viewed as an analogue of Theorem~3.1 in \cite{MW04}.
\begin{proof}
As for any $r\in V$ the map $\Phi_r$ is a diffeomorphism from $(0,\infty)^p$ onto $(0,\infty)^p$, the random vector $\mathbf{X}_r$ has a density $f_r$ of the form $$f_r(x_i, i\in V) = |J_r(x_i, i\in V)|f_T\left(\Phi^{-1}_r(x_i, i\in V)\right)\mathbbm{1}_{(0;\infty)^p}(\mathbf{x}).
% is as follows $$J_r(x_1,\ldots,x_p) = \prod_{i\in V\setminus\{r\}}\left(1+\frac{c_{R_r(i)i}}{c_{R_r(i)}}x_i\right)^{-1}.$$
%to pozostaje jedynie zająć się gęstością $f_T$ w punkcie $\Phi_r^{-1}(x_1, \ldots, x_p)$.
Proposition \ref{tw_graf_og} implies that for every vertex $r \in V$,
\sum_{S\subseteq T}\prod_{i\in V(S)} \frac{x_i}{c_i^{\deg _S (i)-1}}\prod_{\{l,j\}\in E(S)} c_{l,j} =\sum_{i\in V} c_i \Phi _r^{(i)} (\mathbf{x}),
where by $\Phi_r^{(i)}(\mathbf{x})$ we mean $x_{i,(r)}$ as defined in Eq.~\eqref{gw}, so ${\Phi_r(\cdot)=(\Phi_r^{(i)}(\cdot), i\in V)}$).
Thus, for $x_i>0$ and $i\in V$
\begin{eqnarray*}
f_T \{ \Phi_r^{-1}(x_i, i\in V)\} & \propto & \prod_{i\in V}
\left\{ \left(\Phi_r^{(-1)}\right)^{(i)}(\mathbf{x})\right\}^{a_i-1} \, \exp\left[ - \sum_{i\in V} c_i \Phi _r^{(i)} \{\Phi_r^{-1}(\mathbf x)\}\right]\\
&=& \prod_{i\in V} \left(\frac{x_i}{\prod_{j\in \mathfrak{C}_r(i)}(1+\frac{c_{i,j}}{c_i} x_j)}\right)^{a_i-1}\, \exp\left(-\sum_{i\in V} c_i x_i\right).\end{eqnarray*}
To compute the denominator of the above expression, note that for any set of numbers $(\alpha_{i,j},\,\{i,j\}\in E)$,
\prod_{i\in V}\prod_{j\in \mathfrak{C}_r(i)} \alpha _{i,j} = \prod_{i\in V\setminus \{r\}} \alpha_{\mathfrak{p}_r(i),i}.
f_T \{ \Phi_r^{-1}(x_i, i\in V) \} \propto x_r^{a_r-1}e^{-c_r x_r} \prod_{i\in V\setminus\{r\}} e^{ c_i x_i} x_i^{a_i -1}\left(1+\frac{c_{\mathfrak{p}_r(i),i}}{c_{\mathfrak{p}_r(i)}} x_i\right)^{-a_{\mathfrak{p}_r(i)}+1}.
Calling also on the formula \eqref{jakob} for the Jacobian $J_r$, we see that the density of $\Phi_r(\mathbf{X})$ at $\mathbf{x}\in (0,\infty)^p$ is such that
f_r(\mathbf{x}) \propto \,x_r^{a_r-1}e^{-c_r x_r}\mathbbm{1}_{(0;\infty)}(x_r)\prod_{i\in V\setminus\{r\}} x_i^{a_i-1} \left(1+\frac{c_{\mathfrak{p}_r(i), i}}{c_{\mathfrak{p}_r(i)}}x_i \right)^{-a_{\mathfrak{p}_r(i)}}e^{-c_i x_i}\mathbbm{1}_{(0;\infty)}(x_i).
To conclude the proof, it suffices to note that the right-hand side above is a product of densities of distributions of $X_{i, (r)}$, $i\in V$, as stated in the theorem.
\end{proof}
\begin{remark}
Anticipating the characterization result of the next section, we remark that if in the above theorem we take ${\bf S}\sim\mathrm{TK}(\mathbf{a}, c\mathbf{C})$ for a positive number $c>0$, then with $\Phi_r$ defined by $\mathbf{C}$ as above we will get independent $X_{i,(r)}$, $i\in V$, with
{X_{r,(r)}\sim \mathcal{G}(a_r, cc_r)}\quad\mbox{and} \quad \frac{c_{\mathfrak{p}_r(i),i}}{c_{\mathfrak{p}_r(i)}}\,X_{i,(r)}\sim \mathcal{K}\left(a_i,a_{\mathfrak{p}_r(i)}-a_i,c\frac{c_{\mathfrak{p}_r(i)}\,c_i}{c_{\mathfrak{p}_r(i),i}}\right),\quad i\in V\setminus\{r\}.
\end{remark}
As in \cite{MW04}, we illustrate Theorem \ref{tw_og} with two examples.
\begin{example} Let $T$ be 1--2--3 chain and $c_1=c_2=c_3=c_{1,2}=c_{2,3}=c$. Let ${\bf S}=(S_1,S_2,S_3)$ have the respective tree-Kummer distribution and $\Phi_1, \Phi_2, \Phi_3$ are defined by Eq.~\eqref{frt} and \eqref{gw}. Then from Theorem~\ref{tw_og} we conclude that
\begin{multline*}
\Phi_1({\bf S})=\left(S_1(1+S_2(1+S_3)),\,S_2(1+S_3),\,S_3\right) \\
\sim\mathcal{G}(a_1,c)\otimes\mathcal{K}(a_2,a_1-a_2,c)\otimes\mathcal{K}(a_3,a_2-a_3,c),
\end{multline*}
\Phi_2({\bf S})=\left(S_1,\,S_2(1+S_1)(1+S_3),\,S_3\right)\sim\mathcal{K}(a_1,a_2-a_1,c)\otimes\mathcal{G}(a_2,c)\otimes\mathcal{K}(a_3,a_2-a_3,c).
\end{example}
\begin{example}
Let $T$ be a ``daisy'' on four vertices, $V=\{1,2,3,4\}$ with edges $E=\{$1--4, 2--4, 3--4$\}$ and $c_i=c_{i,j}=c$ for all $i\in V$. Assume that ${\bf S}=(S_1,S_2,S_3,S_4)$ has a respective tree-Kummer distribution and $\Phi_r$, $r\in V$, are defined by Eq.~\eqref{frt} and \eqref{gw}. Then from Theorem~\ref{tw_og} we conclude that
\begin{multline*}
\Phi_1({\bf S})=\left(S_1(1+S_4(1+S_2)(1+S_3)),\,S_2,\,S_3,\,S_4(1+S_2)(1+S_3)\right) \\ \sim\mathcal{G}(a_1,c)\otimes\mathcal{K}(a_2,a_4-a_2,c)\otimes
\mathcal{K}(a_3,a_4-a_3,c)\otimes\mathcal{K}(a_4,a_1-a_4,c)
\end{multline*}
\begin{multline*}
\Phi_4({\bf S})=\left(S_1,\,S_2,\,S_3,\,S_4(1+S_1)(1+S_2)(1+S_3)\right) \\ \sim\mathcal{K}(a_1,a_4-a_1,c)\otimes\mathcal{K}(a_2,a_4-a_2,c)\otimes
\mathcal{K}(a_3,a_4-a_3,c)\otimes\mathcal{G}(a_4,c).
\end{multline*}
\end{example}
\section{Multivariate characterization of Kummer and Gamma laws}
\label{sec:4}
Let us consider first a symmetrized and slightly generalized version of Theorem~\ref{tw1}, which can be viewed as a characterization of the simplest tree-Kummer distribution for the tree being a 1--2 chain.
\begin{theorem}\label{sym2} Let $(S_1,S_2)$ be a random vector having a positive and continuously differentiable density $f$ on $(0,\infty)^2$. For positive numbers $c_1$, $c_2$ and $c_{1,2}$ define the bijections $\Phi_1$ and $\Phi_2$ as in Section~\ref{sec:3.1}. Assume that random vectors $\mathbf{X}_{(1)} = (X_{1,(1)}, X_{2,(1)})=\Phi_1(S_1,S_2)$ and $\mathbf{X}_{(2)}=(X_{1,(2)}, X_{2,(2)})=\Phi_2(S_1,S_2)$ have independent components. Then there exist positive numbers $a_1$, $a_2$ and $c$ such that
f(s_1,s_2)\propto s_1^{a_1-1}s_2^{a_2-1}\,e^{-c(c_1s_1+c_2s_2+c_{1,2}s_1s_2)}\,\mathbbm{1}_{(0;\infty)^2}(s_1,s_2).
\frac{c_{1,2}}{c_2}X_{2,(1)}\sim\mathcal{K}\left(a_2,a_1-a_2,c\frac{c_1c_2}{c_{1,2}}\right),\qquad\,X_{2,(2)}\sim\mathcal{G}(a_1,cc_2).
\end{theorem}
\begin{proof}
Note that $X= c_{1,2}X_{2,(1)}/c_1$ and $Y=c_{1,2}X_{1,(1)}/c_2$ are independent. Define $U={Y}/(1+X)$ and $V=X(1+U)$. Then $U={c_{1,2}}X_{1,(2)}/c_1$ and $V={c_{1,2}}X_{2,(2)}/c_2$. Consequently, they are also independent. Now by Theorem~\ref{tw1} it follows that there exist positive numbers $a_1$, $a_2$ and $c'$ such that $X\sim \mathcal{K}(a_1,a_2-a_1,c')$ and $Y\sim\mathcal{G}(a_2,c')$; furthermore, $U\sim\mathcal{G}(a_1,c')$ and $V\sim \mathcal{K}(a_2,a_1-a_2,c')$. Then it is easily seen that the joint density $f$ of $(S_1,S_2)$ has the form
f(s_1,s_2)\propto s_1^{a_1-1}s_2^{a_2-1}e^{-c'\frac{c_{1,2}}{c_1c_2}\left(c_1s_1+c_2s_2+c_{1,2}s_1s_2\right)}\,\mathbbm{1}_{(0;\infty)^2}(s_1,s_2),
and the result follows upon taking $c=c'{c_{1,2}}/(c_1c_2)$.
\end{proof}
The above result provides a characterization of the bivariate tree-Kummer distribution of ${\bf S}=(S_1,S_2)$ for a tree being the 1--2 chain. Equivalently, this result can be interpreted as a characterization of univariate Kummer and Gamma distributions by considering the (properly scaled) components of $\Phi_1({\bf S})$ and $\Phi_2({\bf S})$.
The aim of this section is to extend the above characterization to any tree. The result we obtain is analogous to the characterization of GIG and Gamma distribution obtained in a tree setting in \cite{MW04}. The analogue of the tree-Kummer distribution used here is a tree-GIG distribution called the $W^c_G$ distribution in \cite{MW04}.
\begin{theorem}\label{tw_char}
Let $T=(V,E)$ be a tree of size $p$. Let $\mathbf{C}\in C_T$ and $\Phi_r$ be defined by \eqref{frt} and \eqref{gw}. Let $\mathbf{X}=(X_i, i\in V)$ be a $p$-dimensional random vector with positive density, which is continuously differentiable on its support $(0,\infty)^p$. Suppose that for every $r\in V$, which is a leaf of $T$, the components of the random vector $\mathbf{X}_{(r)}=\Phi_r(\mathbf{X})= (X_{i,(r)},i\in V)$ are independent. Then there exist $\textbf{a}=(a_i, i\in V)\in (0,\infty )^p$ and $c>0$ such that $X_{r,(r)}\sim \mathcal{G}(a_r, cc_r)$ and
\frac{c_{\mathfrak{p}_r(i),i}}{c_{\mathfrak{p}_r(i)}}\,X_{i,(r)}\sim \mathcal{K}\left(a_i,a_{\mathfrak{p}_r(i)}-a_i,c\frac{c_{\mathfrak{p}_r(i)}\,c_i}{c_{\mathfrak{p}_r(i),i}}\right)\; \mbox{for } i\in V\backslash \{r\}.
$$ Consequently, $\textbf{X}\sim \mathrm{TK}(\mathbf{a}, c \mathbf{C})$.
\end{theorem}
\begin{proof}
The proof will be by induction with respect to $p=|V|$. The case $p=2$ is given in Theorem~\ref{sym2}. For fixed $p>2$, we assume the assertion of the theorem to be true for every tree of size up to $p-1$ and we consider an arbitrary tree of size $p$.
Let $n$ and $m$ be vertices from the set of leaves $L\subset V$ of tree $T$ (of course, $L$ consists of at least two elements). Given that $n$ and $m$ are leaves, each of them has precisely one neighbor, $n_1$ and $m_1$, respectively.
\begin{figure}[h]
\begin{tikzpicture}
\tikzstyle{male} = [ shape=circle, minimum size=0.05cm,text=black,
very thick, draw=black!80, top color=white,bottom color=black!50, text width=0.3cm, align=center]
\tikzstyle{male1} = [ shape=circle, minimum size=0.05cm,text=black,
very thick, draw=black!80, top color=white,bottom color=black!50, text width=0.05cm, align=center]
\tikzstyle{mynodestyle} = [ font={\Large\bfseries}, shape=circle, minimum size=0.1cm,text=black,
very thick, draw=black!80,text width=0.5cm, align=center]
\tikzstyle{duzy} = [ font={\Large\bfseries}, shape=circle, minimum size=0.2cm,text=black,
very thick, draw=black!80, top color=white,bottom color=black!50, text width=0.5cm, align=center]
\node[ mynodestyle] (v1) at (-4,3) {$n$};
\node[ duzy] (v3) at (-2,3) {$n_1$};
\node[male1] (v2) at (0,4) {};
\draw[ ->] (v1) -- (v3);
\node [male1] at (0,3) (v4){};
\node (m1) [male] at (0,1.5) {$m_1$};
\node (m) [male] at (1.5,1.5){$m$};
\draw[->, dashed] (v3)--(m1);
\draw[->] (v2) -- (1,5);
\draw[ ->] (v2) -- (1,4);
\draw[->] (v4) -- (1,3);
\draw[->] (v4) -> (1,2.5);
\draw[->] (v4)-> (0.8,3.5);
\draw[->] (v3) -- (v4);
\draw[->] (v3) -- (v2);
\draw[->] (m1) -- (m);
%\node [above=1cm, align=flush left,text width=2cm] at (v1)
% {
% $T_n$
% };
\end{tikzpicture}\caption{$T_n$ with the subtree $\widetilde{T}_{n_1}$ colored grey.}\label{fig.2}
\end{figure}
In order to make use of the induction assumption we will remove one vertex, $n$, from $T$ together with the edge $\{n,n_1\}$ and denote such a reduced tree of size $p-1$ by $\widetilde{T}$. Obviously, $\widetilde{T}$ is a tree. Let us introduce a random vector $\mathbf{\widetilde X}= (\widetilde{X}_{j})_{j\in V\setminus\{n\}}$, where $\widetilde X_j = X_j$ for $j\in V\setminus\{n,n_1\}$ and $ \widetilde X_{n_1} = X_{n_1} (1 + {c_{n_1 n}}{} X_n/c_{n_1})$. Note that $\mathbf{\widetilde X}$ takes on values in $(0,\infty)^{p-1}$.
Let $\widetilde{\Phi}_r$ , $r\in V\setminus \{n\}$, be a transformation defined by $\widetilde T$ according to \eqref{gw} with the same parameters $c_{i,j}$, $c_i$ but for $i,j\in V\setminus \{n\}$. We set
\widetilde {\textbf{X}}_{ (r)}=( \widetilde X_{1, (r)},\ldots, \widetilde X_{p-1, (r)})= \widetilde{\Phi}_r(\widetilde{\textbf{X}}).
Let $n_1$ be the root in $\widetilde{T}$. Clearly, $\widetilde T_{n_1}$ is a directed subtree of $T_n$; see Figure \ref{fig.2}. Therefore $\mathfrak{C}_{n_1}^{\widetilde{T}}(n_1) = \mathfrak{C}_n^T(n_1)$ and $\widetilde X_{j, (n_1)}=X_{j, (n)} $ for $j\neq n_1$. Using these facts and \eqref{gw} we deduce that
\begin{multline*}
\widetilde X_{n_1, (n_1)} =X_{n_1}\left(1+\frac{c_{n_1,n}}{c_{n_1}} X_n\right)\prod_{j\in \mathfrak{C}^{\widetilde T}_{n_1}(n_1)} \left(1+\frac{c_{n_1,j}}{c_{n_1}}\widetilde X_{j, (n_1)}\right) \\ =\left(1+\frac{c_{n_1 n}}{c_{n_1}} X_n\right)X_{n_1}\prod_{j\in \mathfrak{C}^T_n(n_1)} \left(1+\frac{c_{n_1,j}}{c_{n_1}}X_{j, (n_1)}\right)
=X_{n_1, (n)} \left(1+\frac{c_{n_1 n}}{c_{n_1}} X_n\right).
\end{multline*}
$$\widetilde X_{n_1, (n_1)} = \left(1+\frac{c_{n_1,n}}{c_{n_1}} \frac{X_{n, (n)}}{1+\frac{c_{n_1,n}}{c_n} X_{n_1, (n)}}\right)\,X_{n_1, (n)} .
The latter equality, together with the independence of the components of $\textbf{X}_{(n)}$, entails the mutual independence of the components of $\widetilde {\mathbf{X}}_{(n_1)}$.
We also note that for any $\ell \in L\setminus \{n\}$, $X_{j, (\ell)} = \widetilde X_{j, (\ell)}$ for $j\in V\setminus\{n\}$, i.e., $\widetilde{X}_\ell$ is a subvector of ${\bf X}_\ell$ and thus
the components of $\widetilde{\mathbf{X}}_{(\ell)}$ are also independent.
The set of leaves of $\widetilde{T}$ is a subset of $L\cup \{n_1\}$, so eventually we get a $(p-1)$-dimensional random vector $\widetilde{\textbf{X}}$, for which the assumptions of Theorem~\ref{tw_char} are satisfied. Hence, in particular, there exists a vector of positive components ${\bf a}^{(n)}=(a_i^{(n)},\,i\in V\setminus\{n\})$ and $c^{(n)}>0$ such that for $j\notin \{m,n\}$,
\bel{jeden}
\frac{c_{\mathfrak{p}_m(j),j}}{c_{\mathfrak{p}_m(j)}}\,\widetilde X_{j, (m)} = \frac{c_{\mathfrak{p}_m(j),j}}{c_{\mathfrak{p}_m(j)}}\,X_{j, (m)}\sim \mathcal{K}\left(a_j^{(n)}, a_{\mathfrak{p}_m(j)}^{(n)} - a_j^{(n)}, c^{(n)}\frac{c_{\mathfrak{p}_m(j)}\,c_j}{c_{\mathfrak{p}_m(j),j}}\right),
\ee
(here we used the fact that $\mathfrak{p}^{\widetilde T}_m(j)=\mathfrak{p}^T_m(j)=:\mathfrak{p}_m(j)$ for $j\notin \{m,n\}$),
\bel{dwa}
\widetilde X_{m, (m)} = X_{m, (m)}\sim \mathcal{G}(a_m^{(n)},c^{(n)} c_m)
\ee
and for $j\notin \{n, n_1\}$,
\bel{trzy}
\frac{c_{\mathfrak{p}_n(j),j}}{c_{\mathfrak{p}_n(j)}}\,\widetilde X_{j, (n_1)} =\frac{c_{\mathfrak{p}_n(j),j}}{c_{\mathfrak{p}_n(j)}}\, X_{j, (n)}\sim \mathcal{K}\left(a_j^{(n)}, a_{\mathfrak{p}_n(j)}^{(n)} - a_j^{(n)}, c^{(n)}\frac{c_{\mathfrak{p}_n(j)}\,c_j}{c_{\mathfrak{p}_n(j),j}}\right)
\ee
(here we used the fact that $\mathfrak{p}^{\widetilde T}_{n_1}(j)=\mathfrak{p}^T_n(j)=:\mathfrak{p}_n(j)$ for $j\notin \{n_1,n\}$ and that it does not matter if $n_1$ is a leaf in $\widetilde{T}$).
Changing roles of nodes $n$ and $m$ (i.e., removing vertex $m$ instead of $n$ from tree $T$) and then proceeding as before for the random vector
$\widetilde{ \textbf{X}} =(\widetilde X_j)_{j\in V\setminus\{m\}}$, where $\widetilde X_j=X_j$ for $j\notin \{m, m_1\}$ and $\widetilde X_{m_1}=X_{m_1} (1+ {c_{m_1,m}}X_m/c_{m_1})$ for $j\notin \{ m,n\}$, we get
\bel{cztery}
\frac{c_{\mathfrak{p}_n(j),j}}{c_{\mathfrak{p}_n(j)}}\,\widetilde X_{j, (n)} =\frac{c_{\mathfrak{p}_n(j),j}}{c_{\mathfrak{p}_n(j)}}\, X_{j, (n)}\sim \mathcal{K}\left(a_j^{(m)}, a_{\mathfrak{p}_n(j)}^{(m)} - a_j^{(m)}, c^{(m)}\frac{c_{\mathfrak{p}_n(j)}\,c_j}{c_{\mathfrak{p}_n(j),j}}\right),
\ee
\bel{piec}
\widetilde X_{n, (n)} = X_{n, (n)}\sim \mathcal{G}(a_n^{(m)}, c^{(m)}c_n),
\ee
and for $j\notin \{m, m_1\}$,
\bel{szesc}
\frac{c_{\mathfrak{p}_m(j),j}}{c_{\mathfrak{p}_m(j)}}\,\widetilde X_{j, (m_1)} = \frac{c_{\mathfrak{p}_m(j),j}}{c_{\mathfrak{p}_m(j)}}\,X_{j, (m)}\sim \mathcal{K}\left(a_j^{(m)}, a_{\mathfrak{p}_m(j)}^{(m)} - a_j^{(m)}, c^{(m)}\frac{c_{\mathfrak{p}_m(j)}\,c_j}{c_{\mathfrak{p}_m(j),j}}\right).\ee
In the next step we are going to show that $a_j^{(n)}=a_j^{(m)}$ for every $j\in V$ and that $c^{(n)}=c^{(m)}$ for any leaves $m$, $n$.
To see that $c^{(n)}=c^{(m)}$ for any leaves $m$, $n$, it suffices to compare third parameters in \eqref{jeden} and \eqref{szesc} --- or \eqref{trzy} and \eqref{cztery} --- since we chose above arbitrary leaves $m,n$. Furthermore, it follows from \eqref{trzy} and \eqref{cztery} that $a_j^{(n)} = a_j^{(m)}$ for any $j\notin \{m, n, n_1\}$ and from \eqref{jeden} and \eqref{szesc} that $a_j^{(n)} = a_j^{(m)}$ for $j\notin \{m, n, m_1\}$ for any choice of leaves $m$, $n$.
Note that in any tree except the "daisy" one can choose leaves $m$, $n$ in such a way that $n_1\ne m_1$. Then $a_j^{(n)}=a_j^{(m)}=:a_j$ for any $j\notin \{m,n\}$. For $j=n$, $m$ we just write $a_m:=a_m^{(n)}$ and $a_n:=a_n^{(m)}$.
If $n_1=m_1$ for any leaves $n,m$, we show the equality $a_{n_1}^{n}=a_{n_1}^m$ using the assumption of independence of components of $\mathbf{X}_{(r)}$ for $r\in L$. We have
\bel{eq_ind}
|J_{\Phi_m}(s_i, i\in V)| \prod_{i\in V}f_{i,(m)}(s_{i,(m)}) =|J_{\Phi_n}(s_i, i\in V)| \prod_{i\in V}f_{i,(n)}(s_{i,(n)})
\ee
where $f_{i,(\ell)}$ is the density of $X_{i,(\ell)}$, $i\in V$, $\ell = m,n$.
Now we compare the powers of $s_{n_1}$ on the left and the right-hand side of \eqref{eq_ind}. Note that $s_{n_1}$ appears only in
f_{n_1, (\ell)}\left(s_{n_1}\prod_{j\in\C_\ell (n_1)}(1+\frac{c_{n_1,j}}{c_{n_1}}s_{j,(\ell)})\right),
for all $\ell \in \{m,n\}$. In particular, see \eqref{cztery} and \eqref{jeden}, it is raised to the power $a_{n_1}^{(m)}$ in $f_{n_1, (m)}$ and to the power $a_{n_1}^{(n)}$ in $f_{n_1, (n)}$. Consequently, \eqref{eq_ind} yields $a_{n_1}^{(m)}=a_{n_1}^{(n)}=:a_{n_1}$. Now if there are at least three leaves we conclude that $a_j^{(m)}=a_j$ for any $j$ and for any leaf $m$.
In the case of the chain 1--2--3 no comparisons between the distributions from \eqref{jeden}--\eqref{szesc} is possible. Therefore to identify all the parameters we plug the densities from \eqref{jeden}--\eqref{szesc} directly into the identity \eqref{eq_ind} with $m=1$ and $n=3$. In this fashion, we can see that all the unknown parameters in \eqref{jeden}--\eqref{szesc} are derived from a single collection $(a_i)_{i\in V}$ and a number $c$. Therefore, due to Theorem \ref{tw_og}, we can write that, for any $r\in V$,
X_{r,(r)}\sim G (a_r, cc_r),\;X_{i,(r)}\sim K\left(a_i,a_{\mathfrak{p}_r(i)}-a_i,cc_i, \frac{c_{\mathfrak{p}_r(i)}, i}{c_{\mathfrak{p}_r(i)}}\right) \textit{ for } i\in V\setminus \{r\}
$$ where $c>0$, $a_i>0$, $i\in V$.
\end{proof}
Again, as in \cite{MW04}, we give two examples to illustrate Theorem \ref{tw_char}.
\begin{example}
Consider chain 1--2--3 and related maps $\Phi_1$, $\Phi_3$ with all $c_i$'s and $c_{i,j}$'s equal to $1$. Let ${\bf S}$ be a random vector valued in $(0,\infty)^3$. Assume that the components of
$$\Phi_1({\bf S})=\left(S_1(1+S_2(1+S_3)),\,S_2(1+S_3),\,S_3\right)$$
are independent and that the components of
$$\Phi_3({\bf S})=\left(S_1,\,S_2(1+S_1),\,S_3(1+S_2(1+S_1)\right)$$
are also independent. Then from Theorem~\ref{tw_char}, the density of ${\bf S}$ is of the form
f({\bf s})\propto s_1^{a_1-1}s_2^{a_2-1}s_3^{a_3-1}\,e^{-c(s_1+s_2+s_3+s_1s_2+s_2s_3+s_1s_2s_3)}\mathbbm{1}_{(0;\infty)^3}(\mathbf{s}).
\end{example}
\begin{example}
Consider a ``three petal daisy,'' that is the tree with vertices $V=\{1,2,3,4\}$ and edges $E=\{$1--4, 2--4, 3--4$\}$ and related maps $\Phi_1$, $\Phi_2$, $\Phi_3$ with all $c_i$'s and $c_{i,j}$'s equal to 1. Let ${\bf S}$ be a random vector valued in $(0,\infty)^4$ with sufficiently smooth density. Assume that the components of
\Phi_1({\bf S})=\left(S_1(1+S_4(1+S_2)(1+S_3)),\,S_2,\,S_3,\,S_4(1+S_2)(1+S_3)\right)
are independent, the components of
\Phi_2({\bf S})=\left(S_1,\,S_2(1+S_4(1+S_1)(1+S_3)),\,S_3,\,S_4(1+S_1)(1+S_3)\right)
are independent, and the components of
\Phi_3({\bf S})=\left(S_1,\,S_2,\,S_3(1+S_4(1+S_1)(1+S_2),\,S_4(1+S_1)(1+S_2)\right)
are also independent. Then, from Theorem~\ref{tw_char}, the density of ${\bf S}$ is of the form
\begin{multline*}
f({\bf s})\propto s_1^{a_1-1}s_2^{a_2-1}s_3^{a_3-1}s_4^{a_4-1} \\
\times \,e^{-c(s_1+s_2+s_3+s_4+s_1s_4+s_2s_4+s_3s_4+ s_1s_2s_4+s_1s_3s_4+s_2s_3s_4+s_1s_2s_3s_4)}\mathbbm{1}_{(0;\infty)^4}(\mathbf{s}).
\end{multline*}
\end{example}
\section{Concluding remarks}
\label{sec:5}
In closing, we would like to remark that recently the approach of \cite{MW04} to the MY property on trees was extended to the matrix-variate case in \cite{Bo15}. To a large extent, this work was possible due to existing characterizations of the Wishart (the matrix analogue of the Gamma distribution) and the matrix GIG distributions based on the matrix version of MY property; see, e.g., \cite{Ko15, LW00, MW06, We02b}. In the case of Kummer and Gamma distributions, a matrix version of the independence property from \cite{KV12} was obtained in \cite{Kou12} and related characterization remains an open problem. A matrix version of the property and an extension of the characterization result from Section~\ref{sec:2} are currently under study.
It would also be interesting to know whether the property from \cite{HV15} can be embedded in the context of stochastic processes as was the case for the original MY property. As mentioned before, this was done in \cite{MY01} and \cite{MY03}. Similarly, the MY property on trees was related to the hitting times of the Brownian motion and conditional structures of the geometric Brownian motion in \cite{WW07} and \cite{MWW09}. Therefore it is also of some interest to determine whether the tree version of the property from \cite{HV15} discussed here has also connections to stochastic processes.
\vspace{3mm}
{\bf Acknowledgement.} The authors are greatly indebted to Christian Genest and Pawe{\l} Hitczenko for their considerable help regarding presentation of the material in this paper.
\vspace{2mm}
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\end{document}
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1511.00311
|
Let $k$ be a field of characteristic not $2$. We give a positive answer to Serre's injectivity question for any smooth connected reductive $k$-group whose Dynkin diagram contains connected components only of type $A_n$, $B_n$ or $C_n$. We do this by relating Serre's question to the norm principles proved by Barquero and Merkurjev. We give a scalar obstruction defined up to spinor norms whose vanishing will imply the norm principle for the non-trialitarian $D_{n}$ case and yield a positive answer to Serre's question for connected reductive $k$-groups whose Dynkin diagrams contain components of non-trialitarian type $D_n$ also. We also investigate Serre's question for reductive $k$-groups whose derived subgroups admit quasi-split simply connected covers.
§ INTRODUCTION
Let $k$ be a field. Then the following question of Serre, which is open in general, asks
[Serre, <cit.>, p. 233]
Let $G$ be any connected linear algebraic group over a field $k$. Let $L_1,L_2,\ldots,L_r$ be finite field extensions of $k$ of degree $d_1,d_2,\ldots,d_r$ respectively such that $\gcd_i(d_i)=1$. Then is the following sequence exact ?
\[1\to \HH^1(k,G)\to \prod_{i=1}^{r}\HH^1(L_i,G)\]
The classical result that the index of a central simple algebra divides the degrees of its splitting fields answers Serre's question affirmatively for the group $\PGL_n$. Springer's theorem for quadratic forms answers it affirmatively for the (albeit sometimes disconnected) group $O(q)$ and Bayer-Lenstra's theorem (<cit.>) for the groups of isometries of algebras with involutions. Jodi Black (<cit.>) answers Serre's question positively for absolutely simple simply connected and adjoint $k$-groups of classical type. In this paper, we use and extend Jodi's result to connected reductive $k$-groups whose Dynkin diagram contains connected components only of type $A_n$, $B_n$ or $C_n$.
Let $k$ be a field of characteristic not $2$. Let $G$ be a connected reductive $k$-group whose Dynkin diagram contains connected components only of type $A_n$, $B_n$ or $C_n$. Then Serre's question has a positive answer for $G$.
We also investigate Serre's question for reductive $k$-groups whose derived subgroups admit quasi-split simply connected covers. More precisely, we give a uniform proof for the following :
Let $k$ be a field of characteristic not $2$. Let $G$ be a connected reductive $k$-group whose Dynkin diagram does not contain connected components of type $E_8$. Assume further that its derived subgroup admits a quasi-split simply connected cover. Then Serre's question has a positive answer for $G$.
We relate Serre's question for $G$ with the norm principles of other closely related groups following a series of reductions used previously by Barquero and Merkurjev to prove the norm principles for reductive groups whose Dynkin diagrams do not contain connected components of type $D_n, E_6$ or $E_7$ (<cit.>). We also give a scalar obstruction defined up to spinor norms whose vanishing will imply the norm principle for the non-trialitarian $D_{n}$ case and yield a positive answer to Serre's question for connected reductive $k$-groups whose Dynkin diagrams contain components of this type also.
In the next section, we begin with preliminary reductions to restrict ourselves to the case of characteristic $0$ fields and reductive groups $G$ with $G'$ simply connected. In Section 3, we introduce the intermediate groups $\hat{G}$ and $\tilde{G}$ and relate Serre's question for $G$ to Serre's question for $\hat{G}$ and $\tilde{G}$ via the norm principle. In Section 4, we investigate the norm principle for (non-trialitarian) type $D_n$ groups and find the scalar obstruction whose vanishing will imply the norm principle for the non-trialitarian $D_{n}$ case. In the final section, we use the reduction techniques used in Sections <ref> and <ref> to discuss Serre's question for connected reductive $k$-groups whose derived subgroups admit quasi-split simply connected covers.
§ PRELIMINARIES
We work over the base field $k$ of characteristic not $2$ (which we show can be restricted to characteristic $0$). By a $k$-group, we mean a smooth connected linear algebraic group defined over $k$. And mostly, we will restrict ourselves to reductive groups. We say that a $k$-group $G$ satisfies $SQ$ if Serre's question has a positive answer for $G$.
§.§ Reduction to characteristic $0$
Let $G$ be a connected reductive $k$-group whose Dynkin diagram contains connected components only of type $A_n$, $B_n$, $C_n$ or non-trialitarian $D_n$. Without loss of generality we may assume that $k$ is of characteristic $0$ (<cit.>, Pg 47). We give a sketch of the reduction argument for the sake of completeness.
Suppose that the characteristic of $k$ is $p >0$. Let $L_1,L_2,\ldots,L_r$ be finite field extensions of $k$ degree $d_1,d_2,\ldots,d_r$ respectively such that $\gcd_i(d_i)=1$ and let $\xi$ be an element in the kernel of
\[\HH^1(k,G)\to \prod_{i=1}^{r}\HH^1(L_i,G).\]
By a theorem of Gabber, Liu and Lorenzini (<cit.>, Thm 9.2) which was pointed out to us by O. Wittenberg, we note that any torsor under a smooth group scheme $G/k$ which admits a zero-cycle of degree $1$ also admits a zero-cycle of degree $1$ whose support is étale over $k$. Thus without loss of generality we can assume that the given coprime extensions $L_i/k$ are in fact separable.
By (<cit.>, Thm 1 & 2), there exists a complete discrete valuation ring $R$ with residue field $k$ and fraction field $K$ of characteristic zero. Let $S_i$ denote corresponding étale extensions of $R$ with residue fields $L_i$ and fraction fields $K_i$.
There exists a smooth $R$-group scheme $\tilde{G}$ with special fiber $G$ and connected reductive generic fiber $\tilde{G}_K$. Now given any torsor $t\in \HH^1(k,G)$, there exists a torsor $\tilde{t} \in \HH^1_{et}(R, \tilde{G})$ specializing to $t$ which is unique upto isomorphism. This in turn gives a torsor $\tilde{t}_K$ in $\HH^1(K, \tilde{G}_K)$ by base change, thus defining a map $i_k : \HH^1(k, G)\to \HH^1(K, \tilde{G}_K)$ (<cit.>, Pg 29). It clearly sends the trivial element to the trivial element. The map $i$ also behaves well with the natural restriction maps, i.e., it fits into the following commutative diagram :
\[\begin{tikzcd}
\HH^1(k, G) \arrow{r}{i_k} \arrow{d} & \HH^1(K, \tilde{G}_K)\arrow{d} \\
\prod \HH^1(L_i, G) \arrow{r}{\prod i_{L_i}} & \prod \HH^1(K_i, \tilde{G}_{K}).
\end{tikzcd}\]
Let $\tilde{\xi}$ denote the torsor in $\HH^1_{et}(R, \tilde{G})$ corresponding to $\xi$ as above. And let $i_k(\xi) = \tilde{\xi}_K$. This, therefore, is in the kernel of
\[\HH^1(K,\tilde{G}_K)\to \prod_{i=1}^{r}\HH^1(K_i,\tilde{G}_K).\]
Suppose that $\tilde{G}_K$ satisfies $SQ$. Then $\tilde{\xi}_K$ is trivial. However by (<cit.>), the natural map $\HH^1_{et}(R, \tilde{G})\to \HH^1(K, \tilde{G}_K)$ is injective and hence $\tilde{\xi}$ is trivial in $\HH^1_{et}(R, \tilde{G})$. This implies that its specialization, $\xi$, is trivial in $ \HH^1(k, G)$.
Thus from here on, we assume that the base field $k$ has characteristic $0$.
§.§ Lemmata
Let $k$-groups $G$ and $H$ satisfy $SQ$. Then $G\times_k H$ also satisfies $SQ$.
Consider the exact sequence $1\to H\to G\times_k H\xrightarrow{\pi} G\to 1$ of algebraic groups. Note that the projection map is surjective at all field points, ie, $\pi(L) : G\times_k H(L)\to G(L)$ is surjective for all fields $L/k$. Thus $1\to \HH^1(L,H)\to \HH^1(L_i,G\times_k H)$ is exact. Then a chase of the following diagram yields a proof of the lemma.
\[\begin{tikzcd}
1\arrow{r} & \HH^1(k, H)\arrow{r} \arrow{d} & \HH^1(k,G\times_k H) \arrow{r} \arrow{d} &\HH^1(k, G) \arrow{d} \\
1 \arrow{r} &\prod \HH^1(L_i, H) \arrow{r} & \prod \HH^1(L_i, G\times_k H) \arrow{r}{\prod \delta_{L_i}} & \prod \HH^1(L_i, G)
\end{tikzcd}\]
Let $1\to Q\to H\to G\to 1$ be a central extension of a $k$-group $G$ by a quasi-trivial torus $Q$. Then $H$ satisfies $SQ$ if and only if $G$ satisfies $SQ$.
Since $Q$ is quasi-trivial, $\HH^1(L,Q)=\{1\}\ \forall \ L/k$. From the long exact sequence in cohomology, we have the following commutative diagram.
\[\begin{tikzcd}
1\arrow{r} & \HH^1(k, H)\arrow{r} \arrow{d} & \HH^1(k,G) \arrow{r}{\delta_k} \arrow{d} & H^2(k,Q) \arrow{d} \\
1 \arrow{r} &\prod \HH^1(L_i, H) \arrow{r} & \prod \HH^1(L_i, G) \arrow{r}{\prod \delta_{L_i}} & \prod H^2(L_i, Q)\end{tikzcd}\]
From the above diagram, it is clear that if $G$ satisfies $SQ$, so does $H$.
Conversely assume that $H$ satisfies $SQ$. Let $a\in \HH^1(k,G)$ become trivial in $\prod \HH^1(L_i,G)$. Then $\delta_k(a)$ becomes trivial in each $H^2(L_i,Q)$. Hence the corestriction $\Cor_{L_i/k}\left(\delta_k(a)\right)={\delta_k(a)}^{d_i}$ becomes trivial in $H^2(k,Q)$ which implies that $\delta_k(a)$ is itself trivial in $H^2(k,Q)$. Therefore $a$ comes from an element $b\in \HH^1(k,H)$ which is trivial in $\prod \HH^1(L_i,H)$. (The fact that $\HH^1(L_i,Q)=\{1\}$ guarentees $b$ is trivial in $\HH^1(L_i,H)$). Since $H$ satisfies $SQ$ by assumption, $b$ is trivial in $\HH^1(k,H)$ which implies the triviality of $a$ in $\HH^1(k,G)$.
§.§ Further reductions : z-extensions
Recall that there is a central extension (called a $z$-extension) $1\to Q\to H\to G\to 1$ of $G$ by a quasitrivial torus $Q$ such that $H'$ is semisimple and simply connected (<cit.>, Prop 3.1 and <cit.>, Lem 1.1.4). Thus by Lemma <ref>, our given reductive group $G$ satisfies $SQ$ if it's $z$-extension $H$ does. That is,
Let $G_1$ be a connected reductive $k$-group such that that $G_1'=R_{E/k}(H')$ where $E/k$ is a separable field extension and $H'$ is an absolutely simple simply connected group (whose Dynkin diagram contains only connected components of classical type $A_n$, $B_n$, $C_n$ or non-trialitarian $D_n$) over $E$. If every such $G_1$ satisfies $SQ$, then so does any connected reductive $k$-group $G$ whose Dynkin diagram contains only connected components of classical type $A_n$, $B_n$, $C_n$ or non-trialitarian $D_n$.
Thus, without loss of generality, we assume from now on that $G$ is a connected reductive $k$-group with $G' = R_{E/k}(H')$ where $H'$ is one of the following groups :
$\phantom{.}^1A_{n-1}$ :
The special linear group $\SL_1(A)$ where $A$ is a central simple algebra of degree $n$ over $E$.
$\phantom{.}^2A_{n-1}$ :
The special unitary group $\SU (B,\tau)$ where $B$ is a central simple algebra of degree $n$ over a quadratic extension $F$ of $E$ with a unitary involution $\tau$.
$B_n$ :
The spinor group $\Spin (V,q)$ where $(V,q)$ is a non-degenerate quadratic space over $E$ of dimension $2n+1$.
$C_n$ :
The symplectic group $\Sp (A,\sigma)$ where $A$ is a central simple algebra of degree $2n$ over $E$ with symplectic involution $\sigma$.
$D_n$ :
(non-trialitarian) The spinor group $\Spin (A,\sigma)$ where $A$ is a central simple algebra of degree $2n$ over $E$ and $\sigma$ is an orthogonal involution.
§ SERRE'S QUESTION AND NORM PRINCIPLES
§.§ Intermediate groups $\hat{G}$ and $\tilde{G}$
Notations are as in Section 5 of (<cit.>)
In this section we work with the reductive $k$-group $G$ as assumed after Lemma <ref> in the previous section, further assuming[Note that this condition is more restrictive than what was deduced in Lemma <ref>. These restrictions will be removed in the Section <ref>] that its semisimple part $G'$ is an absolutely simple simply connected group of classical type $A_n$, $B_n$, $C_n$ or $D_n$. Let $Z(G)=T$ and $Z(G')=\mu$.
Let $\rho : \mu\hookrightarrow S$ be an embedding of $\mu$ into a quasi-trivial torus $S$. Let $e(G',\rho)$ denote the cofibre product $\hat{G}= \frac{G'\times S}{\mu}$. We call $e(G',\rho)$ to be an envelope of $G'$.
\[
\begin{tikzcd}
\mu \arrow{r}{\delta} \arrow{d}{\rho} & G' \arrow{d} \\
S \arrow{r}{\gamma} & \hat{G}
\end{tikzcd}
\]
Depending on the type of $G'$, we choose envelopes $\hat{G}=e(G', \rho)$ given by the list below :
$\phantom{,}^1 A_{n-1}$ :
$S={G}_m$, $G'=\SL_1(A)$ and $\hat{G}=\GL_1(A)$ where $A$ is a central simple algebra of degree $n$ over $k$.
$\phantom{,}^2 A_{n-1}$ :
$S=R_{K/k}\mbb{G}_m$, $G'=\SU(B,\tau)$ and $\hat{G}=\GU (B,\tau)$ where $B$ is a central simple algebra of degree $n$ over a quadratic extension $K$ of $k$ with a unitary involution $\tau$.
$B_n$ :
$S=\mbb{G}_m$, $G'=\Spin (V,q)$ and $\hat{G}=\Gamma^{+}(V,q)$ where $(V,q)$ is a non-degenerate quadratic space over $k$ of dimension $2n+1$.
$C_n$ :
$S=\mbb{G}_m$, $G'=\Sp (A,\sigma)$ and $\hat{G}=\GSp (A,\sigma)$ where $A$ is a central simple algebra of degree $2n$ over $k$ with symplectic involution $\sigma$.
$D_n$ :
(non-trialitarian) $S=R_{Z/k}\mbb{G}_m$, $G'=\Spin (A,\sigma)$ and $\hat{G} = \Omega(A,\sigma)$ where $A$ is a central simple algebra of degree $2n$ over $k$, $Z/k$ the discriminant quadratic extension and $\sigma$ is an orthogonal involution.
For each of the above cases, $S=Z(\hat{G})$ and $\hat{G}$ fit into an exact sequence as follows :
\[1\to S\to \hat{G} \to G'^{\ ad}\to 1.\]
Here $G'^{\ ad}$ corresponds to the adjoint group of $G'$. By the following theorem (<cit.>, Thm 0.2) we know that $G'^{\ ad}$ satisfies $SQ$ for $G'$ as above.
Let $k$ be a field of characteristic different from 2 and let $G''$ be an absolutely simple algebraic $k$- group which is not of type $E_8$ and which is either a simply connected or adjoint classical group or a quasisplit exceptional group. Then Serre's question has a positive answer for $G''$.
Thus, for connected reductive groups $G$, with $G'$ absolutely simple and simply connected and for envelopes $\hat{G}$ chosen as above, Lemma <ref> implies that the envelopes $\hat{G}$ satisfy $SQ$.
Define an intermediate abelian group $\tilde{T}$ to be the cofibre product $\frac{T\times S}{\mu}$.
\[
\begin{tikzcd}
\mu \arrow{r} \arrow{d}{\rho} & T \arrow{d}{\alpha} \\
S \arrow{r}{\nu} & \tilde{T}
\end{tikzcd}
\]
Let the algebraic group $\tilde{G}$ be the cofibre product defined by the following diagram :
\[\begin{tikzcd}
G'\times T \arrow{r}{m} \arrow{d}{id\ \times \alpha} & G \arrow{d}{\beta} \\
G'\times \tilde{T} \arrow{r}{\epsilon} & \tilde{G}.
\end{tikzcd}
\]
Then we have the following commutative diagram with exact rows (Prop 5.1, <cit.>) . Note that each row is a central extension of $\tilde{G}$.
\[\begin{tikzcd}
1 \arrow{r} & \mu \arrow{r}{\delta, \nu\rho} \arrow{d}{\rho} & G'\times \tilde{T}\arrow{r}{\epsilon}\arrow{d} & \tilde{G}\arrow{r}\arrow{d}{id} & 1 & \hspace*{5mm} (*) \\
1 \arrow{r} & S \arrow{r}{\gamma,\nu} & \hat{G}\times \tilde{T} \arrow{r} & \tilde{G} \arrow{r} & 1 & \hspace*{5mm} (**)\end{tikzcd} \]
Since $\tilde{T}$ is abelian, the existence of the co-restriction map shows that $\tilde{T}$ satisfies $SQ$. Since $\hat{G}$ satisfies $SQ$, we can apply Lemmas <ref> and <ref> to (**) to see that $\tilde{G}$ satisfies $SQ$.
§.§ Norm principle and weak norm principle
Let $f:G\to T$ be a map of $k$-groups where $T$ is an abelian $k$-group. Then we have norm maps $N_{L,k} : T(L)\to T(k)$ for any separable field extension $L/k$.
\[\begin{tikzcd}
G(L) \arrow{r}{f(L)} & T(L) \arrow{d}{{N_{L/k}}} \\
G(k) \arrow{r}{f(k)} & T(k)
\end{tikzcd}
\]
We say that the norm principle holds for $f:G\to T$ if for all separable field extensions $L/k$,
\[N_{L/k}(\Image f(L)) \subseteq \Image f(k).\]
Note that the norm principle holds for any algebraic group homomorphism between abelian groups.
That is, we say that the norm principle holds for $f: G\to T$ if given any separable field extension $L/k$ and any $t\in T(L)$ such that $t\in \brac{\Image f(L) : G(L)\to T(L)}$, then $N_{L/k}(t)\in \brac{\Image f(k) : G(k)\to T(k)}$.
We say that the weak norm principle holds for $f: G\to T$ if given any $t\in T(k)$ such that $t\in \brac{\Image f(L) : G(L)\to T(L)}$, then $t^{[L:k]}=N_{L/k}(t)\in \brac{\Image f(k) : G(k)\to T(k)}$.
It is clear that if the norm principle holds for $f$, then so does the weak norm principle.
Let $G,T,S$ be $k$-groups with $S,T$ abelian and $f:G\to T$, $h: T\hookrightarrow S$ be two $k$-group maps with $h$ injective. Then the (weak) norm principle holds for $f: G\to T$ if the (weak) norm principle holds for $h\circ f : G\to S$.
Let us show the statement for the norm principles. Let $t\in T(L)$ such that $f(L): G(L)\to T(L)$ maps $g\leadsto t$. Let $h(L)(t)=s\in S(L)$. Thus $h\circ f(L)(g) = s$. Since the norm principle holds for $h\circ f$, there exists a $g'\in G(k)$ so that $h\circ f(k)(g') = \N_{L/k}(s)$.
Let $\theta = f(k)(g')\in T(k)$. Then $h: T(k)\to S(k)$ maps both $\N_{L/k}(t)$ and $\theta$ to $\N_{L/k}(s)$. As $h$ is injective, we get that $\N_{L/k}(t) = \theta\in \brac{\Image f(k): G(k)\to T(k)}$. The corresponding statement for the weak norm principles follows from a similar proof.
§.§ Relating Serre's question and norm principle
The deduction of SQ for $G$ from $\hat{G}$ and $\tilde{G}$ follows via the (weak) norm principles.
Let $\beta : G\to \tilde{G}$ be the embedding of $k$-groups with the cokernel $P$ isomorphic to the torus $\frac{S}{\mu}$ where $\tilde{G}$ and $G$ are as in Section <ref>. Thus we have the following exact sequence :
\[ 1\to G\xrightarrow{\beta} \tilde{G} \xrightarrow{\pi} P\to 1.\]
If the weak norm principle holds for $\pi : \tilde{G} \to P$, then $G$ satisfies $SQ$.
From the long exact sequence of cohomology, we have the following commutative diagram :
1 & \to & G(k) & \to & \tilde{G}(k) & \xrightarrow{\pi_k} & P(k) & \xrightarrow{\delta_k} & \HH^1(k,G) & \xrightarrow{\beta_k} & \HH^1(k, \tilde{G}) \\
& & \downarrow & & \downarrow & & \downarrow & & \downarrow & & \downarrow \\
1 & \to & \prod G(L_i) & \to & \prod \tilde{G}(L_i) & \xrightarrow{\prod \pi_{L_i}} & \prod P(L_i) & \xrightarrow{\prod \delta_{L_i}} & \prod \HH^1(L_i,G) & \to & \prod \HH^1(L_i, \tilde{G}).
\end{array}$
Recall that $\tilde{G}$ satisfies $SQ$. Let $a\in \HH^1(k,G)$ become trivial in $\prod \HH^1(L_i, G)$. As $\tilde{G}$ satisfies $SQ$, $\beta_k(a)$ becomes trivial in $\HH^1(k,\tilde{G})$. Hence $a=\delta_k(b)$ for some $b\in P(k)$ and $\delta_{L_i}(b)$ is trivial in $\HH^1(L_i,G)$.
Therefore, there exist $c_i\in \tilde{G}(L_i)$ such that $\pi_{L_i}(c_i)=b$. Showing that $G$ satisfies $SQ$, ie, that $a$ is trivial, is equivalent to showing $b\in \brac{\Image \pi_k: \tilde{G}(k)\to P(k)}$. However $b\in \brac{\Image \pi_{L_i} : \tilde{G}(L_i)\to P(L_i)}$. Since the weak norm principle holds for $\pi : \tilde{G}\to P$, $b^{d_i}\in \Image \brac{\pi_k: \tilde{G}(k)\to P(k)}$ where $[L_i:k]=d_i$ for each $i$. As $\gcd_i(d_i)=1$, this means $b\in \Image \brac{\pi_{k}: \tilde{G}(k)\to P(k)}$.
§.§ Serre's question for $G$ with $G'$ not absolutely simple
As assumed after Lemma <ref>, let $G$ now be reductive with $G'=R_{E/k}(H')$ where $H'$ is an absolutely simple simply connected group of classical type over $E$. Let $H$ be an envelope listed before of $H'$ . Observe that $R_{E/k}(H)$ is an envelope of $G'$ (<cit.>).
$H$ satisfies $SQ$ because it is fits into an exact sequence
\[1\to \mathrm{quasi-trivial\ torus}\to H\to H'^{\ ad}\to 1\]
Hence $R_{E/k}(H)$ also satisfies $SQ$ because $\HH^1(L, R_{E/k}H) = \HH^1(L\otimes_k E, H)$.
The proof of Lemma <ref> shows that if some envelope $\hat{G}$ satisfies $SQ$ (which in turn shows that the corresponding $\tilde{G}$ satsifies $SQ$) and the weak norm principle holds for $\tilde{G}\to P$, then $G$ satisfies $SQ$.
Thus, using the envelope $R_{E/k}(H)$ for $G$, we have the following :
Let $G$ be any connected reductive $k$-group with $G'$ simply connected whose Dynkin diagram contains only connected components of classical type $A_n$, $B_n$, $C_n$ or non-trialitarian $D_n$ (as assumed after Lemma <ref>). If the weak norm principle holds for $\pi : \tilde{G} \to P$, then $G$ satisfies $SQ$.
We recall now the norm principle of Merkurjev and Barquero for reductive groups of classical type.
Let $G$ be a reductive group over a field $k$. Assume that the Dynkin diagram of $G$ does not contain connected components $D_n, n \geq 4, E_6$ or $E_7$. Let $T$ be any commutative $k$-group. Then the norm principle holds for any group homomorphism $G\to T$.
This shows that the norm principle and hence the weak norm principle holds for the map $\pi : \tilde{G}\to P$ for reductive $k$-groups $G$ as in the main theorem (Thm $\ref{mainthm}$), concluding the proof for it.
Theorem 1.2
Let $k$ be a field of characteristic not $2$. Let $G$ be a connected reductive $k$-group whose Dynkin diagram contains connected components only of type $A_n$, $B_n$ or $C_n$. Then Serre's question has a positive answer for $G$.
§ OBSTRUCTION TO NORM PRINCIPLE FOR (NON-TRIALITARIAN) $D_N$
§.§ Preliminaries
Let $\As$ be a central simple algebra of degree $2n$ over $k$ and let $\sigma$ be an orthogonal involution. Let $\C\As$ denote its Clifford algebra which is a central simple algebra over its center, $Z/k$, the discriminant extension. Let $i$ denote the non-trivial automorphism of $Z/k$ and let $\su$ denote the canonical involution of $\C\As$.
Recall that, depending on the parity of $n$, $\su$ is either an involution of the second kind (when $n$ is odd) or of the first kind (when $n$ is even). Let $\muu :\Sim\left(\C\As, \su\right)\to R_{Z/k}\mathbb{G}_m $ denote the multiplier map sending similitude $c$ to $\su(c)c$.
Let $\Omega\As$ be the extended Clifford group, which is an envelope of $\Spin\As$ as mentioned before. We recall below the map $\xx : \Omega\As(k)\to Z^*/k^*$ as defined in (<cit.>, Pg 182).
Given $\omega\in \Omega\As(k)$, let $g\in \GO^{+}\As(k)$ be some similitude such that $\omega\leadsto gk^*$ under the natural surjection $\Omega\As(k)\to \PGO^+\As(k)$.
Let $h = \mu(g)^{-1}g^2 \in \Oo^+\As(k)$ and let $\gamma \in \Gamma\As(k)$ be some element in the special Clifford group which maps to $h$ under the vector representation $\chi' : \Gamma\As(k)\to \Oo^+\As(k)$. Then $\omega^2 = \gamma z$ for some $z\in Z^*$ and $\xx\left(\omega\right) = zk^*$.
Note that the map $\xx$ has $\Gamma\As(k)$ as kernel. Also if $z\in Z^*$, then $\xx(z)=z^2k^*$.
By following the reductions in (<cit.>), it is easy to see that one needs to investigate whether the norm principle holds for the canonical map
\[\Omega\As\to \frac{\Omega\As}{\left[\Omega\As,\Omega\As\right]}.\]
We will need to investigate the norm principle for two different maps depending on the parity of $n$.
§.§.§ The map $\mu_*$ for $n$ odd
Let $U\subset \mathbb{G}_m \times R_{Z/k}\mathbb{G}_m$ be the algebraic subgroup defined by
\[U(k) = \{(f,z)\in k^*\times Z^*| f^4 = \N_{Z/k}(z)\}.\]
Recall the map $\mu_{*} : \Omega\As\to U$ defined in (<cit.>, Pg 188) which sends
\[\omega \leadsto \left(\muu(\omega), ai(a)^{-1}\muu(\omega)^2\right),\]
where $\omega\in \Omega\As(k) $ and $\xx(\omega)=a \ \ k^*$. This induces the following exact sequence (<cit.>, Pg 190)
\[1\to \Spin\As\to \Omega\As\xrightarrow{\mu_*} U\to 1.\]
Since the semisimple part of $\Omega\As$ is $\Spin\As$, the above exact sequence
shows that it suffices to check the norm principle for the map ${\mu}_{*}$.
§.§.§ The map $\muu$ for $n$ even
Recall the following exact sequence induced by restricting $\muu$ to $\Omega\As$ (<cit.>, Pg 187)
\[1\to \Spin\As\to \Omega\As \xrightarrow{\muu} R_{Z/k}\mathbb{G}_m\to 1.\]
Since the semisimple part of $\Omega\As$ is $\Spin\As$, the above exact sequence
shows that it suffices to check the norm principle for the map $\muu$.
§.§ An obstruction to being in the image of $\mu_{*}$ for $n$ odd
Given $(f,z)\in U(k)$, we would like to formulate an obstruction which prevents $(f,z)$ from being in the image $\mu_*\left(\Omega\As(k)\right)$. Note that for $z\in Z^*$, $\mu_*(z) = (\N_{Z/k}(z), z^4)$ and hence the algebraic subgroup $U_0\subseteq U$ defined by
\[U_0(k) = \{(N_{Z/k}(z), z^4) | z\in Z^*\}\]
has its $k$-points in the image $\mu_*\left(\Omega\As(k)\right)$.
Let $\mu_{n[Z]}$ denote the kernel of the norm map $R_{K/k}\mu_n \xrightarrow{N} \mu_n$ where $K/k$ is a quadratic extension. Note that $\mu_{4[Z]}$ is the center of $\Spin \As$ as $n$ is odd. Also recall that (<cit.>, Prop 30.13, Pg 418)
\[\HH^1\left(k, \mu_{4[Z]}\right) \cong \frac{U(k)}{U_0(k)}.\]
Thus, we can construct the map $S : \PGO^{+}\As(k)\to \HH^1\left(k, \mu_{4[Z]}\right)$ induced by the following commutative diagram with exact rows :
\[\begin{tikzcd}
1 \arrow{r} & Z^*\arrow{r} \arrow{d}{\mu_*} & \Omega\As(k) \arrow{r}{\chi'} \arrow{d}{\mu_*} & \PGO^+\As(k) \arrow{r} \arrow{d}{S}& 1 \\
1 \arrow{r} & U_0(k) \arrow{r} & U(k) \arrow{r} & \HH^1\left(k, \mu_{4[Z]}\right) \arrow{r} & 1
\end{tikzcd}\]
The map $S$ also turns out to be the connecting map from $\PGO^+\As(k)\to \HH^1\left(k, \mu_{4[Z]}\right)$ (<cit.>, Prop 13.37, Pg 190) in the long exact sequence of cohomology corresponding to the exact sequence
\[1\to \mu_{4[Z]}\to \Spin \As \to \PGO^+\As\to 1.\]
Since the maps $\mu_* : Z^*\to U_0(k)$ and $\chi' : \Omega\As(k) \to \PGO^+\As(k)$ are surjective, an element $(f,z)\in U(k)$ is in the image $\mu_*\left(\Omega\As(k)\right)$ if and only if its image $[f,z]\in \HH^1\left(k, \mu_{4[Z]}\right)$ is in the image $S\left(\PGO^+\As(k)\right)$.
Therefore we look for an obstruction preventing $[f,z]$ from being in the image $S(\PGO^+\As(k))$. Recall the following commutative diagram with exact rows and columns :
\[\begin{tikzcd}
& & & 1\arrow{d} & \\
& & & \mu_2\arrow{d} & \\
1\arrow{r} & \mu_2 \arrow{r}\arrow{d} & \Spin\As \arrow{r}{\chi}\arrow{d}{id} & \Oo^+\As \arrow{r}\arrow{d}{\pi} & 1 \\
1 \arrow{r} & \mu_{4[Z]}\arrow{r} & \Spin \As \arrow{r}{\chi'} & \PGO^+\As\arrow{r}\arrow{d} & 1\\
& & & 1 & \\
\end{tikzcd}\]
The long exact sequence of cohomology induces the following commutative diagram with exact columns (<cit.>, Prop 13.36, Pg 189)
\[
\begin{tikzcd}
\Oo^+\As(k) \arrow{rr}{Sn} \arrow{d}{\pi} & & \frac{k^*}{k^{*2}} \arrow{d}{i} \\
\PGO^+\As(k) \arrow{rr}{\mathrm{S}} \arrow{d}{\mu} & & \HH^1\left(k, \mu_{4[Z]}\right)\arrow{d}{j}\\
\frac{k^*}{k^{*2}} & = & \frac{k^*}{k^{*2}}
\end{tikzcd}\]
Spinor norms and S for $n$ odd
$\mu : \PGO^+\As(k)\to \frac{k^*}{k^{*2}}$ is induced by the multiplier map $\mu : \GO^+\As\to \mathbb{G}_m$
$i : \frac{k^*}{k^{*2}} \to \HH^1\left(k, \mu_{4[Z]}\right) = \frac{U(k)}{U_0(k)}$ is the map sending $f k^{*2}\leadsto [f, f^2]$
$j : \frac{U(k)}{U_0(k)} = \HH^1\left(k, \mu_{4[Z]}\right)\to \frac{k^*}{k^{*2}}$ is the map sending $[f,z]\leadsto \N(z_0)k^{*2}$ where $z_0\in Z^*$
is such that $z_0i(z_0)^{-1} = f^{-2}z$.
We call an element $(f,z)\in U(k)$ to be special if there exists a $[g]\in \PGO^{+}\As(k)$ such that $j([f,z])=\mu([g])$.
Let $(f,z)\in U(k)$ be a special element and let $[g]\in \PGO^{+}\As(k)$ be such that $j([f,z])=\mu([g])$. From the discussion above, it is clear that $(f,z)$ is in the image $\mu_*\left(\Omega\As(k)\right)$ if and only if $[f,z]$ is in the image $S\left(\PGO^+\As(k)\right)$.
Thus $S([g])[f,z]^{-1}$ is in $\kernel j = \Image i$ and hence there exists some $\alpha\in k^*$ such that
\[[f,z] = S([g])[\alpha, \alpha^2] \in \frac{U(k)}{U_0(k)}.\]
Note that if $g$ is changed by an element in $\Oo^+\As(k)$, then $\alpha$ changes by a spinor norm by Figure 1 above. Thus given a special element, we have produced a scalar $\alpha \in k^*$ which is well defined upto spinor norms.
\begin{align*}
[f,z]\in S\left(\PGO^+\As(k)\right) & \iff [\alpha, \alpha^2]\in S\left(\PGO^+\As(k)\right) \\
& \iff (\alpha, \alpha^2) \in \mu_*\left(\Omega\As(k)\right).
\end{align*}
This happens if and only if there exists $w\in \Omega\As(k)$ such that
\begin{align*}
\alpha &= \muu(w)\\
\alpha^2 &= \xx(w)i(\xx(w))^{-1}\muu(w)^2
\end{align*}
This implies $\xx(w)\in k^*$ and hence $w\in \Gamma\As(k)$. Thus $\alpha$ is a spinor norm, being the similarity of an element in the special Clifford group. Also note if $\alpha$ is a spinor norm, then $\alpha = \muu(\gamma)$ for some $\gamma\in \Gamma\As(k)$ and $\mu_*(\gamma) = \left(\muu(\gamma), \muu(\gamma)^2\right)$.
Thus a special element $(f,z)$ is in the image of $\mu_*$ if and only if the produced scalar $\alpha$ is a spinor norm. We call the class of $\alpha$ in $\frac{k^*}{\Sn}$ to be the scalar obstruction preventing the special element $(f,z)\in U(k)$ from being in the image $\mu_*\left(\Omega\As(k)\right)$.
§.§ An obstruction to being in the image of $\muu$ for $n$ even
Given $z\in Z^*$, we would like to formulate an obstruction which prevents $z$ from being in the image $\muu\left(\Omega\As(k)\right)$ . Note that for $z\in Z^*$, $\muu(z) = z^2$ and hence the subgroup $Z^{*2}$ is in the image $\muu\left(\Omega\As(k)\right)$.
Like in the case of odd $n$, we can construct the map $S : \PGO^{+}\As(k)\to \frac{Z^*}{Z^{*2}}$ induced by the following commutative diagram with exact rows (<cit.>, Definition 13.32, Pg 187) :
\[\begin{tikzcd}
1 \arrow{r} & Z^* \arrow{r} \arrow{d}{\muu} & \Omega\As(k) \arrow{d}{\muu} \arrow{r}{\chi'} & \PGO^+\As(k) \arrow{d}{S} \arrow{r} & 1 \\
1 \arrow{r} & Z^{*2} \arrow{r} & Z^{*} \arrow{r} & \frac{Z^*}{Z^{*2}} \arrow{r}& 1
\end{tikzcd}\]
Again by the surjectivity of the maps, $\muu: Z^*\to Z^{*2}$ and $\chi' : \Omega\As(k) \to \PGO^+\As(k)$, an element $z\in Z^*$ is in the image $\muu\left(\Omega\As(k)\right)$ if and only if its image $[z]\in \frac{Z^*}{Z^{*2}}$ is in the image $S\left(\PGO^+\As(k)\right)$. Therefore we look for an obstruction preventing $[z]$ from being in the image $S(\PGO^+\As(k))$. And as before, we arrive at the the following commutative diagram with exact rows and columns (<cit.>, Prop 13.33, Pg 188)
\[
\begin{tikzcd}
\Oo^+\As(k) \arrow{rr}{Sn} \arrow{d}{\pi} & & \frac{k^*}{k^{*2}} \arrow{d}{i} \\
\PGO^+\As(k) \arrow{rr}{\mathrm{S}} \arrow{d}{\mu} & & \frac{Z^*}{Z^{*2}}\arrow{d}{j}\\
\frac{k^*}{k^{*2}} & = & \frac{k^*}{k^{*2}}
\end{tikzcd}\]
Spinor norms and S for $n$ even
$\mu : \PGO^+\As(k)\to \frac{k^*}{k^{*2}}$ is induced by the multiplier map $\mu : \GO^+\As\to \mathbb{G}_m$
$i : \frac{k^*}{k^{*2}} \to \frac{Z^*}{Z^{*2}}$ is the inclusion map
$j : \frac{Z^*}{Z^{*2}}\to \frac{k^*}{k^{*2}}$ is induced by the norm map from $Z^*\to k^*$.
We call an element $z\in Z^*$ to be special if there exists a $[g]\in \PGO^{+}\As(k)$ such that $j([z])=\mu([g])$.
Let $z\in Z^*$ be a special element and let $[g]\in \PGO^{+}\As(k)$ be such that $j([z])=\mu([g])$. As before a special element $z\in Z^*$ is in the image $\muu\left(\Omega\As(k)\right)$ if and only if $[z]$ is in the image $S\left(\PGO^+\As(k)\right)$.
Thus $S([g])[z]^{-1}$ is in $\kernel j= \Image i$ and hence there exists some $\alpha\in k^*$ such that
\[[z] = S([g])[\alpha] \in \frac{Z^*}{Z^{*2}}.\]
Note that if $g$ is changed by an element in $\Oo^+\As(k)$, then $\alpha$ changes by a spinor norm by Figure 2 above. Thus given a special element, we have produced a scalar $\alpha \in k^*$ which is well defined upto spinor norms.
\begin{align*}
[z]\in S\left(\PGO^+\As(k)\right) & \iff [\alpha]\in S\left(\PGO^+\As(k)\right) \\
& \iff (\alpha)\in \muu\left(\Omega\As(k)\right).
\end{align*}
Since $\alpha\in k^*$ also, this is equivalent to $\alpha$ being a spinor norm (<cit.>, Prop 13.25, Pg 184).
We call the class of $\alpha$ in $\frac{k^*}{\Sn}$ to be the scalar obstruction preventing the special element $z\in Z^*$ from being in the image $\muu\left(\Omega\As(k)\right)$.
§.§ Scharlau's norm principle for $\mu : \GO^+\As \to \mathbb{G}_m$
Let $\mu : \GO^+\As\to \mathbb{G}_m$ denote the multiplier map and let $L/k$ be a separable field extension of finite degree. Let $g_1\in GO^+\As(L)$ be such that $\mu\left(g_1\right)=f_1\in L^*$. Let $f$ denote $\N_{L/k}\left(f_1\right)$. We would like to show that $f$ is in the image $\mu\left(\GO^+\As(k)\right)$.
Note that by a generalization of Scharlau's norm principle (<cit.>, Prop 12.21; <cit.>, Lemma 4.3) there exists a $\tilde{g}\in \GO\As(k)$ such that $f = \mu(\tilde{g})$ . However we would like to find a proper similitude $g\in \GO^+\As(k)$ such that $\mu(g)=f$.
We investigate the cases when the algebra $A$ is non-split and split separately.
§.§.§ Case I : $A$ is non-split
Note that $g_1\in \GO^+\As(L)$. If $\tilde{g}\in \GO^+\As(k)$, we are done. Hence assume $\tilde{g}\not\in \GO^+\As(k)$. By a generalization of Dieudonné's theorem (<cit.>, Thm 13.38, Pg 190), we see that the quaternion algebras
\begin{align*}
B_1 = \left(Z, f_1\right) & = 0 \in \Br(L), \\
B_2 = \left(Z, f \right) & = A \in Br(k).
\end{align*}
Since $A$ is non-split, $B_2\neq 0\in \Br(k)$. However co-restriction of $B_1$ from $L$ to $k$ gives a contradiction, because
\[0 = \Cor B_1 = \left(Z, \N_{L/k}(f_1) \right) = B_2 \in \Br(k).\]
Hence $\tilde{g}\in \GO^+\As(k)$.
§.§.§ Case II : $A$ is split
Since $A$ is split, $A = \End V$ where $(V,q)$ is a quadratic space and $\sigma$ is the adjoint involution for the quadratic form $q$. Again, if $\tilde{g}\in \GO^+\As(k)$, we are done. Hence assume $\tilde{g}\not\in \GO^+\As(k)$. That is
\[\det(\tilde{g})= -f^{2n/2} = -(f^n).\]
Since $A$ is of even degree ($2n$) and split, there exists an isometry[Since $V$ is of even dimension $2n$, $h$ can be chosen to be a hyperplane reflection for instance] $h$ of determinant $-1$. Set $g = \tilde{g}h$. Then $\det(g)=f^n$ where $\mu(g)=f$. Thus we have found a suitable $g\in \GO^+\As(k)$ which concludes the proof of the following :
The norm principle holds for the map $\mu : \GO^+\As \to \mathbb{G}_m$.
§.§ Spinor obstruction to norm principle for non-trialitarian $D_n$
Let $L/k$ be a separable field extension of finite degree. And let $w_1\in \Omega\As(L)$ be such that for
$n$ odd : $\mu_{*}(w_1) = \theta$ which is equal to $(f_1,z_1) \in U(L)$,
$n$ even : $\muu(w_1) = \theta$ which is equal to $z_1 \in \left(R_{Z/k}\mathbb{G}_m\right)(L)$.
We would like to investigate whether $\N_{L/k}(\theta)$ is in the image of $\mu_*\left(\Omega\As(k)\right)$ (resp $\muu\left(\Omega\As(k)\right)$ ) when $n$ is odd (resp. even) in order to check if the norm principle holds for the map $\mu_* : \Omega\As\to U$ (resp. $\muu : \Omega\As\to R_{Z/k}\mathbb{G}_m$).
Let $[g_1]\in \PGO^+\As(L)$ be the image of $w_1$ under the canonical map $\chi' : \Omega\As(L)\to \PGO^+\As(L)$. Clearly $\theta$ is special and let $g_1\in \GO^+\As(L)$ be such that $\mu([g_1])=j([\theta])$.
By Theorem <ref>, there exists a $g\in \GO^+\As(k)$ such that[The map $j$ commutes with $\N_{L/k}$ in both cases.]
\[\mu([g])=\N_{L/k}\left(j[\theta]\right)=j\left([\N_{L/k}\theta]\right).\]
Hence $\N_{L/k}(\theta)$ is special.
By Subsection <ref> (resp. <ref>) , $\N_{L/k}(\theta)$ is in the image of $\mu_*$ (resp $\muu$) if and only if the scalar obstruction $\alpha\in \frac{k^*}{\Sn}$ defined for $\N_{L/k}(\theta)$ vanishes. Thus we have a spinor norm obstruction given below.
Let $L/k$ be a finite separable extension of fields. Let $f$ denote the map $\mu_*$ (resp $\muu$) in the case when $n$ is odd (resp. even). Given $\theta\in f\left(\Omega\As(L)\right)$, there exists scalar obstruction $\alpha\in k^*$ such that
\[N_{L/k}(\theta) \in f\left(\Omega\As(k)\right)\iff \alpha=1\in \frac{k*}{\Sn}.\]
Thus the norm principle for the canonical map
\[\Omega\As\to \frac{\Omega\As}{\left[\Omega\As,\Omega\As\right]}\]
and hence for non-trialitarian $D_n$ holds if and only if the scalar obstructions are spinor norms.
§ GROUPS WITH QUASI-SPLIT SIMPLY CONNECTED COVERS
Let $G$ be a connected reductive $k$-group and let $G'$ denote its derived subgroup. Let $G^{sc}$ denote the simply connected cover of $G'$. Then one has the exact sequence $1\to C\to G^{sc}\to G'\to 1$, where $C$ is a finite $k$-group of multiplicative type, central in $G^{sc}$. The group $C$ is also sometimes termed the fundamental group of $G'$.
Let $G$ be any connected reductive $k$-group whose Dynkin diagram does not contain connected components of type $E_8$. Assume further that $G^{sc}$ is quasi-split. We would like to show that $G$ satisfies $SQ$ by following the reduction techniques used in Sections <ref> and <ref>.
Let $G$ be a connected reductive $k$-group. If $G^{sc}$ is quasi-split, then there exists a $z$-extension $1\to Q\to H\xrightarrow{\psi} G\to 1$, where $Q$ is a quasi-trivial $k$-torus, central in reductive $k$-group $H$ with $H'$ simply connected and quasi-split.
This is simply because $\psi|_{H'}: H'\to G$ yields the simply connected cover of $G'$.
Lemmata <ref> and <ref> imply that we can restrict ourselves to connected reductive $k$-groups $G$ such that $G'$ is simply connected and quasi-split.
If $k$ is a finite field, Steinberg's theorem tells us that $\HH^1(k, G)=1$. Hence $G$ satisfies $SQ$ vacaously. Therefore, let us assume that $k$ is an infinite field from here on.
Let $H$ be any reductive $k$-group such that its derived subgroup $H'$ is semi-simple simply connected and quasi-split. Let $T$ denote the $k$-torus $H/H'$. Then the natural exact sequence $1\to H'\to H\xrightarrow{\phi} T\to 1$ induces surjective maps $\phi(L) : H(L)\to T(L)$ for all field extensions $L/k$. In particular, the norm principle holds for $\phi : H\to T$.
There exists a quasi trivial maximal torus $Q_1$ of $H'$ defined over $k$ (<cit.>, Lem 6.7). Let $Q_1\subset Q_2$, where $Q_2$ is a maximal torus of $H$ defined over $k$. The proof of (<cit.>, Lem 6.6) shows that $\phi |_{Q_2} : Q_2\to T$ is surjective and that $Q_2\cap H'$ is a maximal torus of $H'$. Since $Q_2\cap H'\subseteq Q_1$, we get the following extension of $k$-tori
\[1\to Q_1\to Q_2\to T\to 1\]
Since $Q_1$ is quasitrivial, $\HH^1\brac{L,Q_1}=0$ for any field extension $L/k$ which gives surjectivity of $\phi(L) : Q_2(L)\to T(L)$ and hence of $\phi(L) : H(L)\to T(L)$.
Let $\hat{G}$ be an envelope of $G'$ defined using an embedding of $\mu=Z(G')$ into a quasi-trivial torus $S$. Note that $G'$ is assumed to be simply connected and quasi-split and is also the derived subgroup of $\hat{G}$ by construction.
\[
\begin{tikzcd}
\mu \arrow{r}{\delta} \arrow{d}{\rho} & G'\arrow{d} \\
S \arrow{r}{\gamma} & \hat{G}
\end{tikzcd}
\]
Thus, we get an exact sequence $1\to G'\to \hat{G} \to \hat{G}/G' \to 1$ to which we can apply Lemma <ref> to conclude that the norm principle holds for the canonical map $\hat{G}\to \frac{\hat{G}}{\sqbrac{\hat{G},\hat{G}}}$.
Constructing the intermediate group $\tilde{G}$ as in Section <ref>, we see that the norm principle also holds for the natural map $\tilde{G}\to \tilde{G}/G$ (<cit.>, Prop 5.1). Then using Thm <ref> (<cit.>) and Lemma <ref>, we can conclude that Thm $\ref{quasisplit-serre}$ (restated below) holds.
Theorem 1.3
Let $k$ be a field of characteristic not $2$. Let $G$ be a connected reductive $k$-group whose Dynkin diagram does not contain connected components of type $E_8$. Assume further that $G^{sc}$ is quasi-split. Then Serre's question has a positive answer for $G$.
Acknowledgements : The author acknowledges support from the NSF-FRG grant 1463882. She also thanks Professors A.S Merkurjev, R. Parimala and O.Wittenberg for their many valuable suggestions and critical comments.
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|
1511.00299
|
label1]Roman-Pascal Riwar
label2]Driss M. Badiane
label1]Manuel Houzet
label1]Julia S. Meyer
label3]Yuli V. Nazarov
[label1]Univ. Grenoble Alpes, INAC-SPSMS, F-38000 Grenoble, France;
CEA, INAC-SPSMS, F-38000 Grenoble, France
[label2]Department of Physics, College of William and Mary, Williamsburg, Virginia 23187, USA
[label3]Kavli Institute of NanoScience, Delft University of Technology,
Lorentzweg 1, NL-2628 CJ, Delft, The Netherlands
We use the Landauer-Büttiker scattering theory for electronic transport to calculate the current cross-correlations in a voltage-biased three-terminal junction with all superconducting leads. At low bias voltage, when charge transport is due to coherent multiple Andreev reflections, we find large cross-correlations compared with their normal-state value. Furthermore, depending on the parameters that characterize the properties of the scattering region between the leads, the cross-correlations can reverse their sign with respect to the case of non-interacting fermionic systems.
Contribution for the special issue of Physica E in memory of Markus Büttiker.
multiterminal Josephson junctions
current noise and cross-correlations
multiple Andreev reflections
§ INTRODUCTION
Multiple Andreev reflections are the processes that explain how a dissipative charge transport can take place in a junction between superconducting leads at subgap voltages, $eV<2\Delta$, where $\Delta$ is the superconducting gap, and low temperatures <cit.>. Indeed, due to the energy gap in the excitation spectrum of conventional superconductors, the direct transfer of a quasiparticle between the leads is not possible in that voltage range. However, a subgap electron incident on a superconducting lead can be Andreev reflected as a hole, while a Cooper pair is created in the lead <cit.>. By performing $n$ successive Andreev reflections, it is possible to transfer $\sim n/2$ Cooper pairs – and one quasiparticle – between two superconductors at voltage bias $eV>2\Delta/n$. This highly correlated process produces a rich subgap structure in the current-voltage characteristics, $I(V)$. Theoretically, this was predicted in incoherent <cit.> and coherent <cit.> ballistic junctions, as well as in diffusive junctions <cit.>. Experimentally, subgap structures in the $I(V)$-characteristics have been observed in a variety of systems, namely Josephson junctions based on exotic materials and nanoscale systems, such as semiconductor nanowires <cit.>, carbon nanotubes <cit.>, and graphene flakes <cit.>. Multiple Andreev reflections also result in a large shot noise, $S\sim q^* I$, both in the incoherent <cit.> and coherent <cit.> regime. This effect can be ascribed to the divergence of the effective charge transferred in that process, $q^* =n e$ with $n\sim 2\Delta/(eV)$, as the voltage decreases. The enhancement of the shot noise at low voltage bias was observed in tunnel <cit.>, metallic <cit.>, and atomic point contact junctions <cit.>. In the context of topological superconductivity, multiple Andreev reflections were recently discussed as a parity-changing process for the Majorana bound state that is formed in a topological Josephson junction <cit.>.
Further insight in the multiple Andreev reflection processes may be acquired through current cross-correlations in a multi-terminal geometry. As Markus Büttiker demonstrated in his seminal paper on shot noise, the cross-correlations in non-interacting fermionic systems are always negative due to the Pauli principle <cit.>. Later, several scenarios for a sign-reversal of the cross-correlations in the presence of interactions were proposed (see Ref. <cit.> for a review). For instance, the cross-correlations of the currents through two normal leads weakly contacted to a superconductor can be positive <cit.>, due to a crossed Andreev reflection process in which an electron incident from one of the normal leads is Andreev-reflected to the other one <cit.>. The possibility to use such cross-correlations for a signature of entanglement, due to the singlet-state of the crossed Andreev pair, was also discussed <cit.>. Maximally positive cross-correlations – meaning that they are exactly opposite to the autocorrelation – in a topological superconductor in contact with two normal leads were predicted to be a signature of Majorana edge states <cit.>, in the regime where the applied voltage exceeds the energy splitting between them. (By contrast, the cross-correlations vanish in the limit $V\to0$ <cit.>.) Recently, positive cross-correlations were measured in hybrid structures with tunnel junctions <cit.> and semiconducting nanowires <cit.>.
Motivated by these results, one of us studied the cross-correlations of multiple Andreev reflections in a normal chaotic dot attached to three superconducting leads <cit.>. Depending on the coupling parameters, it was found that the cross-correlations acquire the same amplification factor as the shot noise. Furthermore, a sign reversal at low voltage was predicted under certain conditions. However, this study was restricted to the incoherent regime, when the junction does not carry a supercurrent. This regime may occur when a small magnetic flux is applied to the junction to suppress the Josephson coupling between the leads. An incoherent regime is also expected at a temperature or an applied bias smaller than the gap, but larger than the energy scale that would characterize the induced minigap in the density of states of the dot in equilibrium. Cross-correlations in junctions with three superconducting terminals were measured recently <cit.>. However, in that experiment only negative cross-correlations were observed.
The present work addresses the complementary coherent regime, where both an a.c. Josephson effect and dissipative quasiparticle transport take place.
Phase-dependent multiple Andreev reflections in the $I(V)$-characteristics were investigated experimentally in a diffusive conductor <cit.> and studied theoretically both in a single mode <cit.> and a diffusive <cit.> junction. Our aim is to demonstrate that large positive cross-correlations may also occur in the coherent regime. The outline of the article is the following: in section <ref> we introduce the scattering theory of multiple Andreev reflections. Then we present the results for the current in section <ref> and for the noise and cross-correlations in section <ref>. Section <ref> contains the conclusions and outlook.
§ SCATTERING THEORY OF MULTIPARTICLE ANDREEV REFLECTION
We consider a junction consisting of a normal scattering region connected to three superconducting leads, see Fig. <ref>. Two leads (with labels $\alpha=1,2$) are grounded, while the third lead ($\alpha=3$) is biased with the voltage $V$. Furthermore, the superconducting loop geometry between leads 1 and 2 allows for imposing a superconducting phase difference $\phi$ that is tunable with the application of a magnetic flux through the loop.
Setup of a three-terminal superconducting junction. The superconducting leads are connected through a normal scattering region. A magnetic flux $\Phi$ applied through the loop formed between leads 1 and 2 allows controlling their superconducting phase difference, $\phi=2\pi \Phi/\Phi_0$, where $\Phi_0=hc/(2e)$ is the superconducting flux quantum. A voltage bias $V$ is applied to lead 3 while the leads 1 and 2 are grounded.
For simplicity, we assume that there is one channel per terminal, that the normal scattering region does not break time-reversal symmetry, and that there is no spin-orbit coupling in the system. Then the normal region can be characterized by a $3\times3$ scattering matrix, $\hat S(\varepsilon)=\{S_{\alpha\gamma}(\varepsilon)\}_{\alpha,\gamma=1,2,3}$. If it is shorter than the superconducting coherence length, the energy-dependence of $\hat S(\varepsilon)$ can be neglected. Assuming that leads 1 and 2 are symmetrically coupled to lead 3, the scattering matrix can be parametrized as
\begin{equation}
\label{eq:scatt}
\hat S=\left(\begin{array}{ccc}
\sqrt{R}e^{ia}&\sqrt{D}&\sqrt{D_2}\\
\sqrt{D}&\sqrt{R}e^{ia}&\sqrt{D_2}\\
\sqrt{D_2}&\sqrt{D_2}&\sqrt{R_2}e^{ib}
\end{array}\right)\ ,
\end{equation}
up to irrelevant phases. Here $R=1-D-D_2$, $R_2=1-2D_2$, $a=\arccos[-D_2/(2\sqrt{R D})]$, and $b=\arccos[(D_2-2D)/(2\sqrt{R_2D})]$. The scattering matrix is thus parametrized by two real parameters: the transparency $D$ between leads 1 and 2 and the transparency $2D_2$ between lead 3 and the other leads, with the constraints $D\leq1$ and $D_2\leq 2\sqrt{D}\,(1-\!\sqrt{D})$.
When $\phi=0$, the junction is equivalent to a two-terminal junction with transparency $2D_2$. However, the symmetry between leads 1 and 2 is broken at finite $\phi$. This can be related to the formation of a doubly-degenerate Andreev bound state with energy
\begin{equation}
E_A(\varphi_1,\varphi_2,\varphi_3)=\Delta\sqrt{1-D\sin^2\frac{\varphi_1-\varphi_2}2-D_2\left(\sin^2\frac{\varphi_2-\varphi_3}2+\sin^2\frac{\varphi_3-\varphi_1}2\right)}\ ,
\label{eq-EA}
\end{equation}
when the junction is in equilibrium <cit.>. Here $\varphi_\alpha$ is the superconducting phase of terminal $\alpha$ and $\phi=\varphi_1-\varphi_2$. For $D_2\ll1$ and finite voltage, a quasi-bound state with energy $E_{\rm qb}(\phi)=\Delta\sqrt{1-D\sin^2(\phi/2)}$ between leads 1 and 2 remains and affects the current as well as the noise and cross-correlations. The phase dependence of $I_3$, where $I_\alpha$ is the current going to contact $\alpha=1,2,3$, was studied in Ref. <cit.>. Below we extend their results by computing the effect of multiple Andreev reflections on the current flowing between leads 1 and 2, $(I_1-I_2)/2$, as well as the current noise, $S_{33}$, and the cross-correlations, $S_{12}$.
To calculate the transport properties of the junction, we make use of the Landauer-Büttiker theory, extended to describe hybrid junctions with superconducting leads <cit.>.
For this, we first derive the wavefunctions associated with scattering states, which solve the time-dependent Bogoliubov-de Gennes equations describing the junction. In particular, the incoming and outgoing wavefunctions associated with an incoming electron-like state from lead $\beta$ at energy $E$ can be decomposed into their electron (e) and hole (h) amplitudes on the normal side of the interface between the junction and lead $\alpha$,
\begin{equation}
\hat \psi^{\rm in/out}_{{\rm e}\beta E}(t)=\left\{\psi^{\rm in/out,e}_{{\rm e}\beta E,1}(t),\psi^{\rm in/out,e}_{{\rm e}\beta E,2}(t),\psi^{\rm in/out,e}_{{\rm e}\beta E,3}(t),\psi^{\rm in/out,h}_{{\rm e}\beta E,1}(t),\psi^{\rm in/out,h}_{{\rm e}\beta E,2}(t),\psi^{\rm in/out,h}_{{\rm e}\beta E,3}(t)\right\}^T\ .
\end{equation}
Using their Fourier transform in energy space, $\hat \psi^{\rm in/out}_{{\rm e}\beta E}(t)=\int d\varepsilon/(2\pi)\, \hat \psi^{\rm in/out}_{{\rm e}\beta E}(\varepsilon)e^{-i\varepsilon t}$, we can relate the incoming and outgoing components of the electron part of the wave function through the scattering matrix $\hat S^{\rm e}$ for electrons. Namely,
\begin{equation}
\label{eq:WFARe}
\psi^{\rm out,e}_{{\rm e}\beta E,\alpha}(\varepsilon)=\sum_{\gamma=1,2,3} S^{\rm e}_{\alpha\gamma}
\psi^{\rm in,e}_{{\rm e}\beta E,\gamma}(\varepsilon+eV_\alpha-eV_\gamma)\ ,
\end{equation}
where $S^{\rm e}_{\alpha\gamma}=e^{i(\varphi_\alpha-\varphi_\gamma)/2}S_{\alpha\gamma}$, and $V_\alpha$ is the voltage at terminal $\alpha$. As specified above, $V_1=V_2=0$ and $V_3=V$. Note that we use units where $\hbar=1$.
Similarly, we can relate the incoming and outgoing components of the hole part of the wave function,
\begin{equation}
\label{eq:WFARh}
\psi^{\rm out,h}_{{\rm e}\beta E,\alpha}(\varepsilon)=\sum_{\gamma=1,2,3} S_{\alpha\gamma}
^{\rm h}\psi^{\rm in,h}_{{\rm e}\beta E,\gamma}(\varepsilon-eV_\alpha+eV_\gamma)\ ,
\end{equation}
where $\hat S^{\rm h}=(\hat S^{\rm e})^*$ is the scattering matrix for holes.
Furthermore, the electron and hole components of the wavefunction at terminal $\alpha$ are related via Andreev reflections,
\begin{eqnarray}
\psi^{\rm in,e}_{{\rm e}\beta E,\alpha}(\varepsilon)&=&a(\varepsilon)
\psi^{\rm out,h}_{{\rm e}\beta E,\alpha}(\varepsilon)
+2\pi J(E) \delta_{\alpha\beta}\delta(\varepsilon-E)\ ,
\label{eq:WFSe}
\\
\psi^{\rm in,h}_{{\rm e}\beta E,\alpha}(\varepsilon)&=&a(\varepsilon)
\psi^{\rm out,e}_{{\rm e}\beta E,\alpha}(\varepsilon)\ .
\label{eq:WFSh}
\end{eqnarray}
Here the Andreev reflection amplitude is given as
\begin{equation}
\varepsilon/\Delta-i\sqrt{1-\varepsilon^2/\Delta^2}\ , & |\varepsilon|<\Delta\ ,\\
\varepsilon/\Delta-{\rm sign}(\varepsilon)\sqrt{\varepsilon/\Delta^2-1}\ , & |\varepsilon|\geq\Delta\ .
\end{array}\right.
\end{equation}
The source term, $J(E)=\sqrt{1-|a(E)|^2}$, on the r.h.s. of Eq. (<ref>) is a normalization coefficient, which ensures that the current carried by such a scattering state is $e/(2\pi)$.
Similarly, the incoming and outgoing wavefunctions, $\hat \psi^{\rm in/out}_{{\rm h}\beta E}(t)$, associated with an incoming hole-like state from lead $\beta$ at energy $E$ solve the same Equations (<ref>)-(<ref>), except for the source term which appears in Eq. (<ref>) rather than Eq. (<ref>). In particular, introducing the wavevector $\psi_{\nu,\alpha}=\left(\psi^{{\rm in, e}}_{\nu,\alpha},\psi^{{\rm out, h}}_{\nu,\alpha},\psi^{{\rm out, e}}_{\nu,\alpha},\psi^{{\rm in, h}}_{\nu,\alpha}\right)^T$, where $\nu=\{{\rm p},\beta, E\}$ for an incident particle of type ${\rm p}={\rm e/h}$, from lead $\beta$, and with energy $E$, one readily checks the particle/hole symmetry relation
$\psi_{\bar\nu,\alpha}=\left(-\psi^{{\rm in, h}*}_{\nu,\alpha},\psi^{{\rm out, e}*}_{\nu,\alpha},-\psi^{{\rm out, h}*}_{\nu,\alpha},\psi^{{\rm in, e}*}_{\nu,\alpha}\right)^T$,
where $\bar\nu=\{\bar {\rm p},\beta,-E\}$ with $\bar{\rm e}={\rm h}$ and $\bar{\rm h}={\rm e}$.
Using a Floquet decomposition, a solution of Eqs. (<ref>)-(<ref>) may be written in the form
\begin{equation}
\label{eq:floquet}
\psi^{\rm in/out,{\rm p}}_{{\rm e}\beta E,\alpha}(\varepsilon)=2\pi \sum_{n=-\infty}^\infty\Psi^{\rm in/out,{\rm p}}_{{\rm e}\beta E,\alpha}(n)\delta(\varepsilon-E-neV)
\ .
\end{equation}
This reflects the periodic time-dependence of the Bogoliubov-de Gennes equations that describe the junction. Using the decomposition of Eq. (<ref>), we may write Eqs. (<ref>) and (<ref>) as
\begin{equation}
\Psi^{\rm in,{\rm p}}_{{\rm e}\beta E,\alpha}(n)=a_{n}(E) \Psi^{\rm out,\bar {\rm p}}_{{\rm e}\beta E,\alpha}(n)+J(E)\delta_{\alpha\beta}\delta_{n,0}\delta_{{\rm p},{\rm e}}\ ,
\end{equation}
with $a_{n}(E)=a(E+neV)$, and Eqs. (<ref>) and (<ref>) become
\begin{equation}
\Psi^{\rm out,{\rm e/h}}_{{\rm e}\beta E,\alpha}(n)=\sum_\gamma S_{\alpha\gamma}^{\rm e/h}
\Psi^{\rm in,{\rm e/h}}_{{\rm e}\beta E,\gamma}(n\pm\delta_{\alpha,3}\mp\delta_{\gamma,3})\ .
\end{equation}
To obtain the current, noise, and cross-correlations, we need the operator for the current flowing to lead $\alpha$,
\begin{equation}
\label{eq:Iop}
I_\alpha(t)=ev_F {\cal C}^\dagger_\alpha(t)\sigma_z{\cal C}_\alpha(t)\ .
\end{equation}
Here $v_F$ is the Fermi velocity,
\begin{equation}
\sigma_z=\left(\begin{array}{cccc}
\end{array}\right)\ ,
\end{equation}
and ${\cal C}_\alpha$ is a Nambu spinor in particle-hole and in/out space,
\begin{equation}
{\cal C}_\alpha(t)=\left(c_{{\rm in},\alpha\uparrow}(t),
c_{{\rm out},\alpha\downarrow}^\dagger(t),
c_{{\rm out},\alpha\uparrow}(t),
c_{{\rm in},\alpha\downarrow}^\dagger(t)\right)^T\ ,
\end{equation}
where $c^\dagger_{{\rm in/out},\alpha\sigma}$ is a creation operator for an incoming/outgoing electron in lead $\alpha$ with spin $\sigma=\uparrow,\downarrow$. Using a Bogoliubov transformation, ${\cal C}_\alpha$ can be expressed in terms of the wavefunctions introduced above and the annihilation and creation operators $\gamma_{\nu,\sigma},\gamma^\dagger_{\nu,\sigma}$ for a Bogoliubov quasiparticle $\nu$ with spin $\sigma$,
\begin{equation}
\label{eq:Bogo}
{\cal C}_\alpha(t)
\psi_{\nu,\alpha}(t)
\gamma_{\nu,\uparrow}
\gamma_{\nu,\downarrow}^\dagger
\ .
\end{equation}
\begin{equation}
\gamma_\nu=\left\{\begin{array}{ll}
\gamma_{\nu ,\uparrow} \quad & {\rm if}\enspace E>0\ ,\\
\gamma_{\bar \nu,\downarrow}^\dagger & {\rm if}\enspace E<0\ ,\\
\end{array}\right.
\end{equation}
we may write Eq. (<ref>) as
\begin{equation}
I_\alpha(t)=\frac e{2\pi}\sum_{\nu,\mu}M_{\nu\mu,\alpha}(t)\gamma^\dagger_\nu\gamma_\mu\ ,\qquad {\rm where} \qquad M_{\nu\mu,\alpha}(t)=\psi^\dagger_{\nu,\alpha}(t)\sigma_z\psi_{\mu,\alpha}(t)\ .
\label{eq-bogoliubov}
\end{equation}
Note that $\gamma_\nu$ obeys fermionic anticommutation relations, $\{\gamma_\nu,\gamma_\mu\}=0$ and $\{\gamma_\nu,\gamma^\dagger_\mu\}=\delta_{\nu\mu}$.
Assuming an equilibrium occupation of the scattering states, $\langle\gamma_\nu^\dagger\gamma_\nu\rangle=f(E)$, where $f(E)$ is the Fermi distribution function, we can compute the expectation value of the current, $\langle I_\alpha(t) \rangle$. In particular, the d.c. current reads
\begin{equation}
\label{eq:Idc}
\bar I_\alpha =I_\alpha^N-\frac e{2\pi}\int\limits_{-\infty}^\infty dE \tanh\frac{E}{2T}
\left\{
2J(E)\Re\left[a(E)\Psi^{\rm out,h}_{{\rm e}\alpha E,\alpha}(0)\right]
+\sum_{\beta=1,2,3} \sum_{n=-\infty}^{\infty}
\left[\left(|a_{n}(E)|^2+1\right)\left( |\Psi^{\rm out,h}_{{\rm e}\beta E,\alpha} (n)|^2-|\Psi^{\rm out,e}_{{\rm e}\beta E,\alpha} (n)|^2\right)
\right]
\right\}
\ ,
\end{equation}
\begin{equation}
I_\alpha^N=\frac {2e^2} h \sum_\beta |S_{\alpha\beta}|^2(V_\beta-V_\alpha)
\end{equation}
is the current flowing through the structure in the normal state. In particular, one finds $I_3^N=-I_1^N/2=-I_2^N/2=2D_2(2e^2/h)V$.
The current noise is given as
\begin{equation}
S_{\alpha\beta}(\omega)= \lim_{{\cal T}\to \infty}\frac1{2{\cal T}}\int_{-\cal T}^{\cal T} dt\int_{-\infty}^\infty d\tau\,e^{i\omega\tau}\langle \{\delta I_\alpha(t),\delta I_\beta(t+\tau)\}\rangle \ ,
\end{equation}
where $\delta I_\alpha(t)= I_\alpha(t)-\langle I_\alpha(t)\rangle.$
Using Eq. (<ref>), we find
\begin{equation}
\label{eq:S}
S_{\alpha\beta}(\omega)=\left(\frac{e}{2\pi}\right)^2 \lim_{{\cal T}\to \infty}\frac1{2{\cal T}}\int_{-\cal T}^{\cal T} dt\int_{-\infty}^\infty d\tau\,
\sum_{\nu\mu}M_{\nu\mu,\alpha}(t)M_{\mu\nu,\beta}(t+\tau)e^{i(\omega-E_\nu+E_\mu)\tau}\left[f_\nu(1-f_\mu)+f_\mu(1-f_\nu)\right]\ ,
\end{equation}
where $E_\nu$ is the energy of state $\nu$ and $f_\nu=f(E_\nu)$.
By performing the time integrations in Eq. (<ref>), one may express $S_{\alpha\beta}(\omega)$ explicitly in terms of the scattering wavefunctions, $\Psi^{\rm in/out,p}_{{\rm e}\gamma E,\delta}$, so that its numerical evaluation is straightforward. However, the expression is quite lengthy and we do not provide it in this article.
Note that, in the normal case and at zero temperature, one obtains the noise
\begin{equation}
S_{33}^N=\frac{4e^3}h 2D_2(1-2D_2)V,
\end{equation}
in accordance with the result for a two-terminal junction with transmission $2D_2$, and negative cross-correlations
\begin{equation}
S_{12}^N=-\frac{4e^3}h D_2^2V<0.
\label{eq-S_N}
\end{equation}
In the following sections, we compute the d.c. current (<ref>) and its correlators (<ref>) for the superconducting three-terminal junction at zero temperature, associated with the scattering matrix (<ref>).
§ CURRENT
Let us first discuss the current $I_3$ to the voltage biased lead 3. Panels (a) in Figs. <ref>-<ref> show $I_3(V)$ for different values of the transmissions $D$ and $D_2$. The current displays multiple Andreev reflection features at voltages $eV_n=2\Delta/n$ as in the two-terminal case.
In addition, in the tunneling regime at $D_2\ll1$ shown in Fig. <ref>, it has resonances at the voltages $eV_{\rm Rabi} =E_{\rm qb}(\phi)$ and $eV_{\rm DoS} =\Delta+E_{\rm qb}(\phi)$, as discussed in Ref. <cit.>. The two resonances have different origins. Namely, at the voltage $V_{\rm Rabi}$, the Josephson frequency $\omega_J=2eV_{\rm Rabi}$ matches the energy difference, $2E_{\rm qb}(\phi)$, between the situation where the quasi-bound state is occupied or empty. On the other hand, the voltage $V_{\rm DoS}$ corresponds to the situation where the gap edge of the lead 3 aligns with the energy of the empty quasi-bound state at energy $E_{\rm qb}(\phi)$. The values $E_{\rm qb}(\phi)$ for the phases $\phi$ shown in Fig. <ref> are given in Table <ref>.
In Figs. <ref> and <ref>, we show the results outside of the tunneling regime. As $D_2$ increases, the above features get washed out for most values of the phases $\phi$ as the variation of the Andreev bound state energy (<ref>) with $\varphi_3$ increases. Namely, as a functions of $\varphi_3$, the bound state energy takes values in the interval $[E_A^-,E_A^+]$,
\begin{equation}
E_A^\pm(\phi) =\Delta\sqrt{1-D\sin^2\frac\phi2-D_2\left(1\pm\left|\cos\frac\phi2\right|\right)}\ .
\end{equation}
Note that phases $\phi$ close to $\pi$ are special. Namely, at $\phi=\pi$, one finds $E_A^-(\pi)=E_A^+(\pi)$, and the adiabatic current between leads 1 and 2 is zero. Thus, for phases $\phi$ sufficiently close to $\pi$, one still finds a quasi-bound state at energy $\tilde E_{\rm qb}\approx\sqrt{1-D-D_2}$. The values $E_A^\pm$ for the transmissions $D$ and $D_2$ and phases $\phi$ chosen in Figs. <ref> and <ref> are given in Table <ref>. In Fig. <ref>, additional structure away from the voltages $eV_n$ appears for values of $\phi$ such that the quasi-bound state carries an adiabatic current. However, it is difficult to give a clear interpretion in terms of the processes discussed above. For Fig. <ref>, the interval is too large to see any sharp features.
Fig. <ref>: $D=0.7,D_2=0.005$ Fig. <ref>: $D=0.7, D_2=0.25$ Fig. <ref>: $D=0.12,D_2=0.45$
$\phi=0$ $\frac{E_{\rm qb}}{\Delta}=1,\; \frac{I_{\rm ad}}{e\Delta}=0$ $\frac{E_A^-}{\Delta}\approx 0.71,\; \frac{E_A^+}{\Delta}=1,\; \frac{\bar I_{\rm ad}}{e\Delta}=0$ $\frac{E_A^-}{\Delta}\approx 0.32,\; \frac{E_A^+}{\Delta}=1,\; \frac{\bar I_{\rm ad}}{e\Delta}=0$
$\phi=0.5\pi$ $\frac{E_{\rm qb}}{\Delta}\approx0.81,\; \frac{I_{\rm ad}}{e\Delta}\approx0.43$ $\frac{E_A^-}{\Delta}\approx 0.47,\; \frac{E_A^+}{\Delta}\approx0.76,\; \frac{\bar I_{\rm ad}}{e\Delta}\approx0.56$ $\frac{E_A^-}{\Delta}\approx 0.41,\; \frac{E_A^+}{\Delta}\approx0.90,\; \frac{\bar I_{\rm ad}}{e\Delta}\approx0.05$
$\phi=0.9\pi$ $\frac{E_{\rm qb}}{\Delta}\approx0.56,\; \frac{I_{\rm ad}}{e\Delta}\approx0.19$ $\frac{E_A^-}{\Delta}\approx 0.17,\; \frac{E_A^+}{\Delta}\approx0.36,\; \frac{\bar I_{\rm ad}}{e\Delta}\approx0.37$ $\frac{E_A^-}{\Delta}\approx 0.60,\; \frac{E_A^+}{\Delta}\approx0.71,\; \frac{\bar I_{\rm ad}}{e\Delta}\approx0.01$
$\phi=\pi$ $\frac{E_{\rm qb}}{\Delta}\approx0.55,\; \frac{I_{\rm ad}}{e\Delta}=0$ $\frac{E_A^-}{\Delta}=\frac{E_A^+}{\Delta}\approx0.22,\; \frac{\bar I_{\rm ad}}{e\Delta}=0$ $\frac{E_A^-}{\Delta}=\frac{E_A^+}{\Delta}\approx0.66,\; \frac{\bar I_{\rm ad}}{e\Delta}=0$
Energies of the quasi-bound state and adiabatic current for the values of transmissions and phases shown in Figs. <ref>-<ref>.
Three-terminal junction characterized by the transparencies $D=0.7$ and $D_2=0.005$. We show (a) the current $I_3$, (b) the current $(I_1-I_2)/2$, (c) the noise $S_{33}$, and (d) the cross-correlations $S_{12}$ as a function of the bias voltage $V$ for different phases $\phi=0,0.5\pi,0.9\pi,\pi$ in black ($\filledmedsquare$), blue ($\medbullet$), red ($\blacktriangle$), and green ($\Diamondblack$), respectively. For $\phi=0.5\pi, 0.9\pi$, the solid vertical lines indicate the voltages $eV_{\rm Rabi}(\phi)$ while the dashed vertical lines inidcate the voltages $eV_{\rm DoS}(\phi)/n$ ($n=1,2$). In panels (a) and (c), in addition, $eV=2\Delta$ is shown. Note that the current $(I_1-I_2)/2$ in (b) is zero both for $\phi=0$ and $\phi=\pi$. In (d) we also show a zoom on the low-voltage regime to highlight the features due to multiple Andreev reflections, plotted on a logarithmic scale, for $\phi=0,\pi$. The vertical lines indicate the voltages $eV_n=2\Delta/n$ ($n\leq5$) for $\phi=0$ and $eV_{\rm DoS}(\phi)/n$ ($n\leq3$) for $\phi=\pi$.
Three-terminal junction characterized by the transparencies $D=0.7$ and $D_2=0.25$. We show (a) the current $I_3$, (b) the current $(I_1-I_2)/2$, (c) the noise $S_{33}$, and (d) the cross-correlations $S_{12}$ as a function of the bias voltage $V$ for different phases $\phi=0,0.5\pi,0.9\pi,\pi$ in black ($\filledmedsquare$), blue ($\medbullet$), red ($\blacktriangle$), and green ($\Diamondblack$), respectively. The vertical lines indicate the voltages $eV_n=2\Delta/n$ for $n\leq6$. Note that the current $(I_1-I_2)/2$ in (b) is zero both for $\phi=0$ and $\phi=\pi$. In (d) we also show a zoom on the low-voltage regime to highlight the positive cross-correlations at $\phi=0$ (top) as well as the large negative cross-correlations, plotted on a logarithmic scale, for $\phi=0.5\pi,0.9\pi$ (bottom).
Three-terminal junction characterized by the transparencies $D=0.12$ and $D_2=0.45$. We show (a) the current $I_3$, (b) the current $(I_1-I_2)/2$, (c) the noise $S_{33}$, and (d) the cross-correlations $S_{12}$ as a function of the bias voltage $V$ for different phases $\phi=0,0.5\pi,0.9\pi,\pi$ in black ($\filledmedsquare$), blue ($\medbullet$), red ($\blacktriangle$), and green ($\Diamondblack$), respectively. The vertical lines indicate the voltages $eV_n=2\Delta/n$ for $n\leq6$. Note that the current $(I_1-I_2)/2$ in (b) is zero both for $\phi=0$ and $\phi=\pi$.
Current $(I_1-I_2)/2$ as a function of the phase difference $\phi$ for different voltage biases, $eV/\Delta=0.16,0.7,1.16,1.7$ (black, blue, red, and green, respectively). The thin dashed line corresponds to the adiabatic current $\bar I_{\rm ad}$. The transparencies of the junction are given as $D=0.7$ whereas $D_2=0.005$ and $D_2=0.25$ in (a) and (b), respectively. The vertical lines in (a) indicate the phases $\phi_c^{\rm Rabi}(eV/\Delta=0.7)=\pi -\phi_c^{\rm DoS}(eV/\Delta=1.7)$ and $2\pi-\phi_c^{\rm Rabi}(eV/\Delta=0.7)=\pi+\phi_c^{\rm DoS}(eV/\Delta=1.7)$. Note that the curve for $eV/\Delta=0.16$ in (a) is not displayed as it overlaps with the adiabatic current.
Next we turn to the current $(I_1-I_2)/2$, shown in panels (b) of Figs. <ref>-<ref> as a function of $V$ and in Fig. <ref> as a function of $\phi$. At small voltages and $D_2\ll1$, the current is given by the adiabatic current,
\begin{equation}
I_{\rm ad}(\phi)=-2e\frac{\partial}{\partial\phi}E_{\rm qb}(\phi).
\end{equation}
As voltage is increased, two different features are observed. These features are linked with the resonances in the current $I_3$ discussed above. At $eV_{\rm Rabi} =E_{\rm qb}(\phi)$, one finds a suppression of the current, corresponding to an average occupation of the doubly-degenerate quasi-bound state with one quasiparticle. At $eV >eV_{\rm DoS}=\Delta+E_{\rm qb}(\phi)$, the injection of quasiparticles from lead 3 leads to a reversal of the current, corresponding to a double occupation of the quasi-bound state. Due to the phase dependence of $E_{\rm qb}(\phi)$, the reversal appears first around the phase $\pi$ and then spreads over the entire phase interval as voltage is further increased.
In summary, there are several different voltage regimes: at $eV<E_{\rm qb}(\pi)=\Delta\sqrt{1-D}$ the current is adiabatic. At voltages $E_{\rm qb}(\pi) <eV<\Delta$, one observes resonances at $\phi_c^{\rm Rabi}$ and $2\pi-\phi_c^{\rm Rabi}$ with $\phi_c^{\rm Rabi}=2\arcsin\sqrt{[1-(eV/\Delta)^2]/D}$. At voltages $\Delta<eV<\Delta+E_{\rm qb}(\pi)$, the current is again adiabatic. At voltages $\Delta+E_{\rm qb}(\pi)<eV<2\Delta$, the current is reversed in the phase interval $\phi\in[\pi-\phi_c^{\rm DoS},\pi+\phi_c^{\rm DoS}]$ with $\phi_c^{\rm DoS}=\pi-2\arcsin\sqrt{[1-(eV/\Delta-1)^2]/D}$. Finally, at voltages $eV>2\Delta$, the current is reversed for all phases.
As can be seen in Fig. <ref>(a), for $D_2\ll1$, the current $(I_1-I_2)/2$ is well decribed by the above considerations, except for the sign reversal in a narrow interval around $\phi=\pi$ observed in the regime $\Delta<eV<\Delta+E_{\rm qb}(\pi)$. This can be traced to the fact that ionization processes, where a quasiparticle escapes the bound state, are very weak for phases close to $\pi$ because the dependence of the bound state energy on the applied voltage vanishes at $\phi=\pi$. As a consequence, higher order processes with a threshold at $E_{\rm DoS}(\phi)/n$ are sufficient to establish a significant population of the quasi-bound state. The sign reversal seen here occurs at $eV=E_{\rm DoS}(\phi)/2$.
As discussed above, the above features get washed out for most values of the phases $\phi$ when $D_2$ increases. The adiabatic current in that case is given as
\begin{equation}
\bar I_{\rm ad}(\phi)=-2e\frac{\partial}{\partial\phi}\bar E_A(\phi),
\end{equation}
where $\bar E_A(\phi)=\int_0^{2\pi}d\varphi_3 \,E_A(\varphi_1,\varphi_2,\varphi_3)/(2\pi)$. The current outside of the tunneling regime is shown in Fig. <ref>(b). Here the current $(I_1-I_2)/2$ follows the adiabatic approximation only for small voltages and phases close to zero. Facilitated by multiple Andreev reflections, a sign reversal for phases close to $\pi$ is observed at all voltages. Furthermore, we see resonances at voltages smaller than $\Delta$. As the energy of the bound state $E_A$ strongly varies with $\varphi_3$, it is not obvious why the resonances at low voltage are so narrow.
In Figs. <ref>-<ref>, panels (b), we note that for voltages $eV>2\Delta$ the current $(I_1-I_2)/2$ flattens out at a value that is opposite in sign as compared to the adiabatic current and smaller in magnitude.
§ NOISE AND CROSS-CORRELATIONS
We now turn to the noise and cross-correlations. The noise is shown in Figs. <ref>-<ref>, panels (c). When $D_2$ is large and multiple Andreev reflections are important, the noise is strongly enhanced at low voltages, see Fig. <ref>(c).
As shown in Fig. <ref>(d), the cross-correlations at $\phi=0$ may be large and positive at low voltages. In particular, this is the case when $D_2$ is large. It is due to crossed Andreev reflections, where Cooper pairs are split between lead 1 and 2. At large voltages, when superconducting correlations are less important, the cross-correlations are always negative. The large-voltage asymptotic approaches the normal state result, Eq. (<ref>).
In addition to this main result, we notice that the noise and cross-correlations display additional features that are related with the features discussed above for the current. Namely, the multiple Andreev reflection features at voltages $eV_n=2\Delta/n$ are visible in the noise and cross-correlations as well. While their position is independent of $\phi$, their visibility varies. In particular, they become more and more pronounced as $\phi$ increases from $0$ to $\pi$. In the tunneling regime, additional features due to the presence of a quasi-bound state are present, see Fig. <ref>(d). The Rabi oscillations occurring near voltages $eV_{\rm Rabi}$ lead to large negative cross-correlations. This feature is similar to the large supercurrent noise that was predicted in equilibrium Josephson junctions due to the thermal fluctuation in the Andreev bound state occupation <cit.>. Large negative cross-correlations are also observed in the interval $[V_{\rm DoS}(\phi)/2,V_{\rm DoS}(\phi)]$, where the average occupation of the quasi-bound state change from $0$ to a value close to $2$. (The associated change of sign of $I_1-I_2$ can be seen in panel (b) of Fig. <ref>.) When $D_2$ increases, the negative cross-correlations at finite phase difference become less pronounced as seen in Fig. <ref>(d). Note that, even at large voltages, a dependence of the noise and cross-correlations on the phase $\phi$ remains in the form of a voltage-independent excess contribution.
§ CONCLUSION
Three-terminal Josephson junctions realize positive cross-correlations at low voltages. These correlations are strongly enhanced compared to the case where only one of the leads is superconducting due to the process of multiple Andreev reflections. Here we studied the coherent regime where supercurrents and dissipative quasiparticle currents lead to an interesting interplay that manifests itself in the current as well as in the noise and cross-correlations.
As a next step, it would be interesting to address the crossover between coherent and incoherent multiple Andreev reflection regimes in multichannel hybrid junctions. The methods introduced in our work could also be helpful to test recent predictions related with topological aspects of the Andreev subgap spectrum in multiterminal Josephson junctions <cit.>.
We dedicate this article to the memory of Markus Büttiker. His groundbreaking contributions to mesoscopic physics have shaped the field and remain an inspiration for our research.
This work was supported by ANR,
through grants ANR-11-JS04-003-01 and ANR-12-BS04-0016-03, and by the Nanosciences Foundation in Grenoble, in the frame of its Chair of Excellence program.
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|
1511.00226
|
${}^1$Bogoliubov Laboratory for Theoretical Physics, Joint Institute of Nuclear Research,
Joliot-Curie str. 6, Dubna, 141980, Russia
$^{2}$Institute of Mechanics and Energetics, Russian State Agrarian University, Timiryazevskaya, 49,
Moscow 127550, Russia
An intrinsic time of homogeneous models is global. The Friedmann equation by its sense ties time intervals.
Exact solutions of the Friedmann equation in Standard cosmology and Conformal cosmology are presented.
Theoretical curves interpolated the Hubble diagram on latest supernovae are expressed in analytical form.
The class of functions in which the concordance model is described is Weierstrass meromorphic functions.
The Standard cosmological model and Conformal one fit the modern Hubble diagram equivalently.
However, the physical interpretation of the modern data from concepts of the Conformal cosmology is simpler, so is preferable.
§ INTRODUCTION
The supernovae type Ia are used as standard candles to test cosmological models.
Recent observations of the supernovae have led cosmologists to conclusion of the Universe filled with dust and
mysterious dark energy in frame of Standard cosmology <cit.>.
Recent cosmological data on expanding Universe challenge cosmologists in insight of Einstein's gravitation.
To explain a reason of the Universe's acceleration the significant efforts have been applied (see, for example, <cit.>).
The Conformal cosmological model <cit.> allows us to describe the supernova data without Lambda term.
The evolution of the lengths in the Standard cosmology is replaced by the evolution of the masses in the Conformal cosmology.
It allows to hope for solving chronic problems accumulated in the Standard cosmology.
Solutions of the Friedmann differential equation belong to a class of Weierstrass meromorphic functions.
Thus, it is natural to use them for comparison predictions of these two approaches.
The paper presents a continuation of the article on intrinsic time in Geometrodynamics <cit.>.
§ FRIEDMANN EQUATION IN CLASSICAL COSMOLOGY
A global time exists in homogeneous cosmological models (see, for example, papers <cit.>).
The conformal metric $(\tilde\gamma_{ij})$ <cit.> for three-dimensional sphere in spherical coordinates
$(\chi, \theta, \varphi)$ is defined via the first quadratic form
\begin{equation}\label{back}
a_0^2\left[d\chi^2+\sin^2\chi (d\theta^2+\sin^2\theta d\varphi^2)\right].
\end{equation}
Here $a_0$ is a modern value of the Universe's scale.
For a pseudosphere in (<ref>) instead of $\sin\chi$ one should take $\sinh\chi$, and for a flat space
one should take $\chi.$ The intrinsic time $D$ is defined with minus as logarithm of ratio of scales
In the Standard cosmological model the Friedmann equation is used for fitting SNe Ia data. It
ties the intrinsic intervals time with the coordinate time one
\begin{equation}\label{Friedmannclass}
\left(\frac{dD}{dt}\right)^2\equiv
\left(\frac{\dot{a}}{a}\right)^2=H_0^2\left[\Omega_{\rm M}\left(\frac{a_0}{a}\right)^3+\Omega_{\Lambda}\right].
\end{equation}
Three cosmological parameters favor for modern astronomical observations
$$H_0=h\cdot 10^5 m/s/Mpc,\quad h=0.72\pm 0.08$$
– Hubble constant,
$$\Omega_\Lambda = 0.72,\quad \Omega_{\rm M}=0.28$$
– partial densities. Here $\Omega_{\rm M}$ is the baryonic density parameter, $\Omega_\Lambda$
is the density parameter corresponding to $\Lambda$-term, constrained with $\Omega_{\rm M}+\Omega_\Lambda =1.$
The solution of the Friedmann equation (<ref>) is presented in analytical form
\begin{equation}\label{classicalsolution}
a(t)=a_0\sqrt[3]{\frac{\Omega_{\rm M}}{\Omega_\Lambda}}
\left[{\rm sinh}\left(\frac{3}{2}\sqrt{\Omega_\Lambda}H_0 t)\right)\right]^{2/3}.
\end{equation}
Here $a(t)$ is a scale of the model, $a_0=1$ is its modern value.
The second derivative of the scale factor is
ä=H_0^2 a_0/2[2Ω_Λ(a/a_0)-Ω_M(a_0/a)^2].
In the modern epoch the Universe expands with acceleration, because $2\Omega_\Lambda >\Omega_{\rm M};$
in the past, its acceleration is negative $\ddot{a}<0.$
This change of sign of the acceleration without clear physical reason puzzles researchers.
From the solution (<ref>), if one puts
the age – redshift relation is followed
The age $t_0$ of the modern Universe is able to be obtained by taking $z=0$ in (<ref>)
\begin{equation}
t_0=\frac{2}{3\sqrt{\Omega_\Lambda}}\frac{1}{H_0}{\rm Arcsinh}\sqrt{\frac{\Omega_\Lambda}{\Omega_{\rm M}}}.
\end{equation}
Since for light
$$ds^2=-c^2dt^2+a^2(t)dr^2=0,\quad cdt=-a(t)dr,$$
we have, denoting $x\equiv a/a_0$,
-a_0 r=c∫dt/x=c∫dx/x1/dx/dt.
Rewriting the Friedmann equation (<ref>), one obtains a quadrature
Substituting the derivative (<ref>) into (<ref>), we get the integral
H_0 r=c/√(Ω_Λ)∫_1/(1+z)^1dx/√(x^4+ 4a_3 x),
where we denoted a ratio as
$$4a_3\equiv \frac{\Omega_{\rm M}}{\Omega_\Lambda}.$$
Then we introduce a new variable $y$ by the following substitution
√(x^4+4a_3 x)≡x^2-2y.
Raising both sides of this equation in square, we get
a_3 x=-x^2 y+y^2.
Differentials of both sides of the equality (<ref>) can de expressed in the form:
Utilizing the equality (<ref>), one can rewrite it
Then, we take the expression from the equation (<ref>)
where a sign plus if
$$0\le x\le\sqrt[3]{\frac{a_3}{2}}=\frac{1}{2}\sqrt[3]{\frac{\Omega_{\rm M}}{\Omega_\Lambda}},$$
and a sign minus if
$$\sqrt[3]{\frac{a_3}{2}}=\frac{1}{2}\sqrt[3]{\frac{\Omega_{\rm M}}{\Omega_\Lambda}}\le x\le 1,$$
and substitute it into the right hand of the differential equation (<ref>).
The equation takes the
following form
dx/√(x^4+4a_3 x)=∓dy/√(4y^3+a_3^2)≡
≡ ∓dy/2√((y-e_1)(y-e_2)(y-e_3)),
$$y\equiv\frac{1}{2}\left(x^2-\sqrt{x^4+4a_3 x}\right),$$
with three roots:
e_1 ≡ 1/8(Ω_M/Ω_Λ)^2/3(1+√(3)), e_2≡-1/4(Ω_M/Ω_Λ)^2/3,
e_3 ≡ 1/8(Ω_M/Ω_Λ)^2/3(1-√(3)).
The integral (<ref>) for the interval $\sqrt[3]{a_3/2}\le x\le 1$, corresponding to
\begin{equation}\label{interval}
0\le z\le 2\sqrt[3]{\frac{\Omega_\Lambda}{\Omega_{\rm M}}}\approx 1.74,
\end{equation}
gives the coordinate distance - redshift relation in integral form
H_0 r=
= c/√(Ω_Λ)∫_[1-√(1+4a_3(1+z)^3)]/(2(1+z)^2)^∞dy/√(4y^3+a_3^2)-
- c/√(Ω_Λ)∫_(1-√(1+4a_3))/2^∞dy/√(4y^3+a_3^2).
The interval considered in (<ref>) covers the modern cosmological observations one <cit.>
up to the right latest achieved redshift limit $z\sim 1.7$.
The integrals in (<ref>) are expressed with use of inverse Weierstrass $\wp$-function <cit.>
H_0 r = -c/√(Ω_Λ)℘^-1
+ c/√(Ω_Λ)
The invariants of the Weierstrass functions are
$$g_2=0,\qquad g_3=-a_3^2=-\left(\frac{\Omega_{\rm M}}{4\Omega_\Lambda}\right)^2;$$
the discriminant is negative
$$\Delta\equiv g_2^3-27g_3^2<0.$$
Let us rewrite the relation (<ref>) in implicit form between the variables with use of $\wp$-function
$$\wp u=\frac{1-\sqrt{1+\Omega_{\rm M}/\Omega_\Lambda (1+z)^3}}{2(1+z)^2},$$
$$u\equiv \frac{1}{c}\sqrt{\Omega_\Lambda}H_0 r-\wp^{-1}\left(\frac{1-\sqrt{1+\Omega_{\rm M}/\Omega_\Lambda}}{2}\right),$$
The Weierstrass $\wp$-function can be expressed through an elliptic Jacobi cosine function <cit.>
℘u=e_2+H1+cn (2√(H)u)/1-cn(2√(H)u),
where, the roots from (<ref>) are presented in the form
$$e_1=m+\imath n,\quad e_2=-2m,\quad e_3=m-\imath n,$$
$$m\equiv \frac{1}{8}\left(\frac{\Omega_{\rm M}}{\Omega_\Lambda}\right)^{2/3},\quad
n\equiv\frac{\sqrt{3}}{8}\left(\frac{\Omega_{\rm M}}{\Omega_\Lambda}\right)^{2/3},$$
and $H$ is calculated according to the rule
$$H\equiv\sqrt{9m^2+n^2}=\frac{\sqrt{3}}{4}\left(\frac{\Omega_{\rm M}}{\Omega_\Lambda}\right)^{2/3}.$$
Then, from (<ref>) we obtain an implicit dependence between the variables, using Jacobi cosine function
cn [√(3)(Ω_M/Ω_Λ)^1/3u]=f(z)-1/f(z)+1,
where we introduced the function of redshift
≡ 2/√(3)(Ω_Λ/Ω_M)^2/3
1-√(1+Ω_M/Ω_Λ(1+z)^3)/(1+z)^2+1/√(3).The modulo of the elliptic function (<ref>) is obtained by the following rule <cit.>:
Claudio Ptolemy classified the stars visible to the naked eye into six classes according to their brightness.
The magnitude scale is a logarithmic scale, so that a difference of 5
magnitudes corresponds to a factor of 100 in luminosity. The absolute magnitude $M$ and the apparent magnitude
$m$ of an object are defined as
\begin{eqnarray}
M&\equiv& -\frac{5}{2} {\rm lg} \frac{L}{L_0},\nonumber\\
m&\equiv& -\frac{5}{2} {\rm lg} \frac{l}{l_0},\nonumber
\end{eqnarray}
where $L_0$ and $l_0$ are reference luminosities. In astronomy, the radiated power $L$ of a star or a galaxy,
is called its absolute luminosity. The flux density $l$ is called its apparent luminosity. In Euclidean
geometry these are related as
$$l=\frac{L}{4\pi d^2},$$
where $d$ is our distance to the object. Thus one defines the luminosity distance $d_L$ of an object as
\begin{equation}\label{dL}
d_L\equiv\sqrt{\frac{L}{4\pi l}}.
\end{equation}
In Friedmann – Robertson – Walker cosmology the absolute luminosity
\begin{equation}\label{L}
L=\frac{N_\gamma E_{\rm em}}{t_{\rm em}},
\end{equation}
where $N_\gamma$ is a number of photons emitted, $E_{\rm em}$ is their average energy,
$t_{\rm em}$ is emission time. The apparent luminosity is expressed as
\begin{equation}\label{l}
l=\frac{N_\gamma E_{\rm abs}}{t_{\rm abs}A},
\end{equation}
where $E_{\rm abs}$ is their average energy, and
$$A=4\pi a_0^2 r^2$$
is an area of the sphere around a star. The number of photons is conserved, but their energy is redshifted,
\begin{equation}\label{Eabs}
E_{\rm abs}=\frac{E_{\rm em}}{1+z}.
\end{equation}
The times are connected by the relation
\begin{equation}\label{tabs}
t_{\rm abs}=(1+z)t_{\rm em}.
\end{equation}
Then, with use of (<ref>), (<ref>), the apparent luminosity (<ref>)
can be presented via the absolute luminosity (<ref>) as
$$l=\frac{N_\gamma E_{\rm em}}{t_{\rm em}}
\frac{1}{(1+z)^2}\frac{1}{4\pi a_0^2 r^2}=\frac{1}{(1+z)^2}\frac{L}{4\pi a_0^2 r^2}.$$
From here, the formula for luminosity distance (<ref>) is obtained
\begin{equation}\label{dLs}
d_L (z)_{SC}=(1+z)a_0 r.
\end{equation}
Substituting the formula for coordinate distance (<ref>) into (<ref>),
we obtain the analytical expression for the luminosity distance
- .℘^-1[1-√(1+Ω_M/Ω_Λ(1+z)^3)/2(1+z)^2]).The modern observational cosmology is based on the Hubble diagram.
The effective magnitude – redshift relation
\begin{equation}\label{mMcl}
m(z)-M=5{\rm lg} [d_L(z)_{SC}]+{\cal M},
\end{equation}
is used to test cosmological theories ($d_L$ in units of megaparsecs) <cit.>.
Here $m(z)$ is an observed magnitude, $M$ is the absolute magnitude,
and ${\cal M}=25$ is a constant.
§ FRIEDMANN EQUATION IN CONFORMAL COSMOLOGY
The fit of Conformal cosmological model with $\Omega_{\rm rigid} = 0.755,$ $\Omega_{\rm M}=0.245$
is the same quality approximation
as the fit of the Standard cosmological model with $\Omega_\Lambda = 0.72,$ $\Omega_{\rm M}=0.28$,
constrained with $\Omega_{\rm rigid}+\Omega_\Lambda =1$ <cit.>.
The parameter $\Omega_{\rm rigid}$ corresponds to a rigid state, where the energy density coincides with the pressure $p=\rho$ <cit.>.
The energy continuity equation follows from the Einstein equations
So, for the equation of state $\rho=p$, one is obtained the dependence $\rho\sim a^{-6}.$
The rigid state of matter can be formed by a free massless scalar field <cit.>.
Including executing fitting, we write the conformal Friedmann equation <cit.>
with use of significant conformal partial
parameters, discarding all other insignificant contributions
\begin{eqnarray}\label{Friedmannconf}
\left(\frac{dD}{d\eta}\right)^2&\equiv&
\left(\frac{{a}'}{a}\right)^2=\\
&=&\left(\frac{{\cal H}_0}{c}\right)^2\left[\Omega_{\rm rigid}\left(\frac{a_0}{a^4}\right)+
\Omega_{\rm M}\left(\frac{a_0}{a}\right)\right].\nonumber
\end{eqnarray}
In the right side of (<ref>) there are densities $\rho (a)$ with corresponding conformal weights;
in the left side a comma denotes a derivative with respect to conformal time. The conformal Friedmann equation
ties intrinsic time interval with conformal time one. If we have accepted the intrinsic York's time <cit.>
in Friedmann equations (<ref>), (<ref>), we should have lost the connection between
temporal intervals[“The time is out of joint”. William Shakespeare. Hamlet. Act 1. Scene V. Longman, London (1970).].
After introducing new dimensionless variable $x\equiv {a}/{a_0},$
the conformal Friedmann equation (<ref>) takes a form
(2c/√(Ω_M) H_0)^2x^2(dx/dη)^2=
= 4x^3-g_3≡4(x-e_1)(x-e_2)(x-e_3),
where one root of the cubic polynomial in the right hand side (<ref>)
is real, other are complex conjugated
e_1 ≡ √(Ω_rigid/Ω_M)1+√(3)/2, e_2≡-√(Ω_rigid/Ω_M),
e_3 ≡ √(Ω_rigid/Ω_M)1-√(3)/2.The invariants are the following
$$g_2=0,\qquad g_3= -\frac{4\Omega_{\rm rigid}}{\Omega_{\rm M}}.$$
where ${\cal H}_0$ is the conformal Hubble constant. The conformal Hubble parameter is defined via the Hubble parameter as
${\cal H}\equiv (a/a_0)H$.
The differential equation (<ref>) describes an effective problem of classical mechanics –
a falling of a particle with mass $8c^2/(\Omega_M{\cal H}_0^2)$ and
zero total energy in a central field with repulsive potential
Starting from an initial point $x=0$ it reaches a point $x=1$ in a finite time $\eta_0$.
We get an integral from the differential equation (<ref>)
\begin{equation}
\int\limits_{1/(1+z)}^1\frac{x dx}{\sqrt{4x^3-g_3}}=-\frac{\sqrt{\Omega_{\rm M}}{\cal H}_0}{2c}\eta.
\end{equation}
Then, we introduce a new variable $u$ by a rule
\begin{equation}\label{xpitau}
\end{equation}
Weierstrass function $\wp (u)$ <cit.> satisfies to the differential equation
\left[\frac{d\wp(u)}{du}\right]^2=4\left[\wp (u)-e_1\right]\left[\wp (u)-e_2\right]\left[\wp (u)-e_3\right],
$$\wp (\omega_\alpha)=e_\alpha,\qquad \wp'(\omega_\alpha)=0,\qquad \alpha=1,2,3.$$
The discriminant is negative
$$\Delta\equiv g_2^3-27g_3^2<0.$$
The Weierstrass $\zeta$-function satisfies to conditions of quasi-periodicity
$$\zeta (\tau+2\omega)=\zeta (\tau)+2\eta,\qquad
\zeta (\tau+2\omega')=\zeta (\tau)+2\eta',$$
$$\eta\equiv\zeta (\omega),\qquad \eta'\equiv \zeta (\omega').$$
The conformal age – redshift relationship is obtained in explicit form
\begin{equation}\label{age}
\eta=\frac{2c}{\sqrt{\Omega_{\rm M}}{\cal H}_0}\left(\zeta\left[\wp^{-1}\left(\frac{1}{1+z}\right)\right]-
\zeta\left[\wp^{-1}(1)\right]\right).
\end{equation}
Rewritten in the integral form the Friedmann equation is known in cosmology as the Hubble law.
The explicit formula for the age of the Universe can be obtained
\begin{equation}
\eta_0=\frac{2c}{\sqrt{\Omega_{\rm M}}{\cal H}_0}
\left(\zeta \left[\wp^{-1}(0)\right]-\zeta \left[\wp^{-1}(1)\right]\right).
\end{equation}
An interval of coordinate conformal distance is equal to an interval of conformal time $dr=d\eta$, so we can
rewrite (<ref>) as conformal distance – redshift relation.
A relative changing of wavelength of an emitted photon corresponds to a relative changing of the scale
where $\lambda$ is a wavelength of an emitted photon, $\lambda_0$ is a wavelength of absorbed photon.
The Weyl treatment <cit.> suggests also a possibility to consider
\begin{equation}\label{WeylCC}
1+z=\frac{m_0 a_0}{[a(\eta) m_0]},
\end{equation}
where $m_0$ is an atom original mass.
Masses of elementary particles, according to Conformal cosmology interpretation (<ref>), become running
$$m(\eta)=m_0 a(\eta).$$
The photons emitted by atoms of the distant stars billions of years ago, remember the size of atoms.
The same atoms were determined by their masses in that long time. Astronomers now compare the spectrum of
radiation with the spectrum of the same atoms on Earth, but with increased since that time.
The result is a redshift $z$ of Fraunhofer spectral lines.
In conformal coordinates photons behave exactly as in Minkowski space.
The time intervals $dt= - a dr$ used in Standard cosmology and the time interval used in Conformal
cosmology $d\eta = - dr$ are different.
The conformal luminosity distance $d_L(z)_{CC}$ is related to the standard luminosity one $d_L(z)_{SC}$ as
$$d_L(z)_{CC}= (1+z)d_L(z)_{SC}=(1+z)^2r (z),$$
where $r(z)$ is a coordinate distance.
For photons $dr/d\eta=-1,$ so we obtain the explicit dependence:
luminosity distance – redshift relationship
= 2c(1+z)^2/√(Ω_M)H_0
ζ[℘^-1(1)]).The effective magnitude – redshift relation in Conformal cosmology has a form
\begin{equation}\label{mMConf}
m(z)-M=5{\rm lg} [d_L(z)_{CC}]+{\cal M}.
\end{equation}
§ COMPARISONS OF APPROACHES
The Conformal cosmological model states that conformal quantities are observable magnitudes.
The Pearson $\chi^2$-criterium was applied in <cit.>
to select from a statistical point of view the best fitting of Type Ia supernovae data
The rigid matter component $\rho_{\rm rigid}$ in the Conformal model substitutes the $\Lambda$-term of the Standard model.
It corresponds to a rigid state of matter, when the energy density is equal to its pressure.
The result of the treatment is: the best-fit of the Conformal model is almost the same quality approximation as
the best-fit of the Standard model.
Curves: the effective magnitude – redshift relation of the two models.
Curves of the two models are shown in Fig.<ref>.
A fine difference between predictions of the models (<ref>) and (<ref>):
effective magnitude – redshift relation
$$\Delta (m(z)-M)=5{\rm lg}[d_L(z)_{CC}]-5{\rm lg}[d_L(z)_{SC}]$$
is depicted in Fig.<ref>.
The differences between the curves are observed in the early and in the past stages of the Universe's evolution.
In Standard cosmology the Hubble, deceleration, jerk parameters are defined as <cit.>
\begin{eqnarray}
H(t)&\equiv&+\left(\frac{\dot{a}}{a}\right)=H_0\sqrt{\frac{\Omega_{\rm M}}{a^3}+\Omega_\Lambda},\nonumber\\
\frac{\Omega_{\rm M}/2-\Omega_\Lambda a^3}{\Omega_{\rm M}+\Omega_\Lambda a^3},\nonumber\\
\end{eqnarray}
As we have seen, the $q$-parameter changes its sign during the Universe's evolution at an inflection point
$$a^{*}=\sqrt[3]{\frac{\Omega_{\rm M}}{2\Omega_\Lambda}},$$
the $j$-parameter is a constant.
Difference between curves of the two models: The effective magnitude – redshift relation.
We can define analogous parameters in Conformal cosmology also
\begin{eqnarray}
{\cal H}(\eta)&\equiv&+\left(\frac{{a}'}{a}\right),\label{Hparameter}\\
\end{eqnarray}
Let us calculate the conformal parameters with use of the conformal Friedmann equation (<ref>).
The Hubble parameter
$${\cal H}(\eta)=\frac{{\cal H}_0}{a^2}\sqrt{\Omega_{\rm rigid}+\Omega_{\rm M} a^3}>0;$$
the deceleration parameter
$$q(\eta)=\left(\frac{\Omega_{\rm rigid}-(\Omega_{\rm M}/2)a^3}{\Omega_{\rm rigid}+
\Omega_{\rm M}a^3}\right)>0,$$
so the scale factor grows with deceleration;
the jerk parameter
$$j(\eta)=\frac{3\Omega_{\rm rigid}}{\Omega_{\rm rigid}+\Omega_{\rm M}a^3}>0$$
changes from 3 to $3\Omega_{\rm rigid}$.
The dimensionless parameter $q(\eta)$ and $j(\eta)$ are positive during all evolution.
The Universe has not been undergone a jerk.
§ CONCLUSIONS
Weierstrass and Jacobi functions traditionally used for a long time in classical mechanics and astronomy,
are in demand in theoretical cosmology also. The conformal age – redshift relation,
and the effective magnitude – redshift relations, that are basis formulae for observable cosmology,
are expressed explicitly in meromorphic functions. Instead of integral relations, which are used to in cosmology,
the derived formulae are expressed through higher transcendental functions, easy to use,
because they are built-in analytical software package MATHEMATICA.
The Hubble Space Telescope cosmological supernovae Ia team presented data of high redshifts.
Classical cosmological and Conformal cosmological approaches fit the Hubble diagram with equal accuracy.
According to concepts of Conformal gravitation, conformal quantities of General Relativity are interpreted
as physical observables. The conformal cosmological interpretation is preferable because of explaining the
resent data without adding the $\Lambda$-term.
It is appropriate to remind the correct statement of the Nobel laureate in Physics Steven Weinberg
<cit.> about interpretation of experimental data on redshift. “I do not want to give the impression
that everyone agrees with this interpretation of the red shift.
We do not actually observe galaxies rushing away from us; all we are sure of is that the lines in their spectra
are shifted to the red, i. e. towards longer wavelengths. There are eminent astronomers who doubt that the
red shifts have anything to do with Doppler shifts or with expansion of the universe”.
§ ACKNOWLEDGMENT
For fruitful discussions I would like to thank Profs. A.B. Arbuzov, R.G. Nazmitdinov, and V.N. Pervushin.
A.G. Riess et al. The Astrophys. J. 607, 665 (2004).
A. Borowiec, W. Godłowski, and M. Szydłowski, Phys. Rev. D 74, 043502 (2006).
M. Szydłowski, A. Stachowski. Cosmological models with running cosmological term and decaying dark matter.
arXiv:1508.05637 [astro-ph.CO].
V. Pervushin, A. Pavlov,
Principles of Quantum Universe (Saarbrücken, Lambert Academic Publishing, 2014).
A. Pavlov, Intrinsic time in Wheeler – DeWitt conformal superspace.
Gravitation & Cosmology, to be published.
E. Kasner. Am. J. Math. 43, 217 (1921).
C.W. Misner, Phys. Rev. 186, 1319 (1969).
E.T. Whittaker, G.N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, 1927).
A.F. Zakharov, V.N. Pervushin, Int. J. Mod. Phys. D19, 1875 (2010).
Ya. B. Zel'dovich, Soviet Physics JETP. 14, 1143 (1962).
J.W. York, Phys. Rev. Lett. 28, 1082 (1972).
S. Weinberg, The First Three Minutes. A Modern View of the Origin of the Universe
(Basic Books, New York, 1977).
|
1511.00346
|
POPL 2016...
Omitted for Submission
Ichiro Hasuo Shunsuke Shimizu
University of Tokyo, Japan
Corina Cîrstea
University of Southampton, UK
§ RESPONSE TO THE REVIEWS
Thank you all again for your extensive reading and suggestions. They
gave us a great opportunity for us to think again about the paper's presentation. We believe it has improved a lot, and we are happy about it.
In the revision below where major additions are in blue, we made the following changes.
§.§ On the value of our progress measure
* To demonstrate the value of our results in Section 2—a
lattice-theoretic extension of Jurdzinski's progress measure—we
now present a fresh view in Table 1 (p.2). It presents progress
measures as combination of invariants and ranking functions,
notions widely known to the POPL community (not only to hardcore
theoreticians, but also to practical-minded people working on
program verification). Together with some text that we added in
the third last paragraph, c1, p2, we believe this will interest
many (especially many program verification people) and might lead to extension of some synthesis methods to nested/alternating gfp & lfp properties.
* Accordingly we added some technical discussions and examples in Section
2 (like Def. 2.2, Ex. 2.3, Rem. 2.9 and Ex. 2.16); they present
our point in Table 1 more concretely.
* It is now stated explicitly that Section 2 does not use any
categorical preliminaries (the second last paragraph, c1, p3).
§.§ On motivations and appeal to the POPL community in general
* To improve on the previous insufficient description of motivations and practical use, we added some paragraphs in Section 1.2, for each of the three contributions. For the first contribution (lattice-theoretic progress measure) we use Table 1 and emphasize possible relationship to the automatic synthesis techniques (for invariants and ranking functions) studied extensively in the PL community; for the second and third contributions (coalgebraic model checking) we copy from our rebuttal and argue about general merits of coalgebraic abstraction, as well as about specific practical implications that our result would have. In particular, we emphasize that our matrix-based presentation in Section 4.2 will make actual implementation fairly straightforward.
* Implementation is now emphasized as an important direction of future work (see Section 1.3 and also the beginning of Section 4.2). We somehow took it for granted and forgot to mention it explicitly!
* We also made it sure that technical developments are preceded by what
they are aiming at, i.e. our main results, to aid reading. See the beginning of Section 4 and that of Section 5.
§.§ Intuitions behind more specific technical notions
* We are grateful that you pointed out some technical definitions are lacking intuitions. We believe we improved about this. An example is Definition 4.3, where the composite in (12) is identified as an extension of the widely-adopted formalization in coalgebraic modal logic.
* Another point that we were lacking intuitions is the translation between our presentation and Jurdzinski's. Now we are explicit about its essence in Rem. 2.5—which we identify as a "Stone-duality-like" translation—and used throughout the paper.
§.§ On generality of results in Section 5: probabilistic
branching and Büchi automata
* On both of the issues of probabilistic branching and (simulations
for) Büchi automata, we have indeed obtained—in the
time between the rebuttal and now—a concrete result that characterizes
the language of a Büchi automaton coalgebraically. (The
result works also in the probabilistic setting.) This
allowed us to be more specific about future work in the current paper (see the
second last paragraph, c1, p4). See also the second paragraph,
c1, p4.
We believe the other issues raised in the reviews (e.g. illustration of
technical definitions in Section 2.3) are adequately addressed (see
Example 2.11). Thank you again for your extensive comments—we
believe they made the paper a lot better.
In the context of formal verification in general and model checking in
particular, parity games serve as a mighty vehicle: many
problems are encoded as parity games, which are then solved by the
seminal algorithm by Jurdzinski. In this paper we identify the essence
of this workflow to be the notion of progress measure, and
formalize it in general, possibly infinitary, lattice-theoretic
terms. Our view on progress measures is that they are to
nested/alternating fixed points what invariants are to
safety/greatest fixed points, and what ranking functions are to
liveness/least fixed points. That is, progress measures are
combination of the
latter two notions (invariant and ranking function) that have been
extensively studied in the context of (program) verification.
We then apply our theory of progress measures to a general model-checking
framework, where systems are categorically presented as coalgebras.
The framework's
theoretical robustness is witnessed by a smooth
transfer from the branching-time setting to the linear-time one.
Although the framework can be used to
derive some decision procedures for finite settings, we also
expect the proposed framework to form a basis for sound proof methods
for some undecidable/infinitary problems.
D.2.4Software/Program VerificationModel checking
F.4.1Mathematical LogicModal logic
ixed-point logic, model checking, coalgebra
After finishing the paper, some thoughts...
* In <ref>, why don't we say
$u_{0},u_{2},\dotsc$ are $\nu$-variables and
$u_{1},u_{3},\dotsc$ are $\mu$-variables? Justified by the Bekic
* This is OK in <ref>, but for the
coalgebraic model checking the current presentation is
much more useful. Otherwise translation of formulas into
equational systems takes more efforts, grouping variables
of the same polarity
§ INTRODUCTION
§.§ Backgrounds
Parity Games and Fixed-Point Logics
For the purpose of formal verification where one aims at
establishing that a system satisfies a certain property (called a
specification), it is common to express: a model of the system as
a state-based transition system such as an automaton or a Kripke structure;
and a specification as a formula in some modal logic. For the latter, in particular,
logics with fixed-point operators—such as LTL and CTL—serve
well thanks to their remarkable expressivity <cit.>. The
modal $\mu$-calculus (see e.g. <cit.>)
provides a clean syntax that incorporates the least and greatest fixed-point
operators ($\mu$ and $\nu$) in a systematic manner.
Dealing with such fixed
points is however a nontrivial task—this is especially the case when $\mu$'s
and $\nu$'s are nested and they alternate. Many engineers find it challenging
to express their intuition as a fixed-point formula; furthermore, many
algorithms are first introduced for an alternation-free
fragment and then later extended to the full fragment (see
e.g. <cit.> and <cit.>).
For the purpose of analyses of fixed-point logics and designing
algorithms for them, parity games have emerged as a very useful
tool in the last decade or so. A parity game is played by two players
$\even$ and $\odd$, on a board each position $x$ of which has a natural
number $\pri(x)\in\omega$ called its priority. Notably its
winning condition is the parity condition: the player $\even$
wins if the largest priority that occurs infinitely often in a given
play (an infinite sequence of positions) is an even number. This
condition—that may seem ad-hoc at first sight—turns out to be
extremely useful for modeling nested and alternating $\mu$'s and
$\nu$'s. It is in a sense a combinatorial presentation of
an alternation between
$\mu$'s and $\nu$'s.
The use of parity games has been boosted further by Jurdzinski's
algorithm that efficiently determines the winner at each position of a
parity game <cit.>. It exhibits a practical complexity that
is exponential only in so-called the alternation depth of a
parity game. It has then become a norm, in the context of
fixed-point logics and algorithmic formal verification, to take the
following parity-game workflow: it reduces a
problem in question to the decision problem of some parity game, and then
solves the latter by Jurdzinski's algorithm. A
notable example is the model-checking problem for the modal
$\mu$-calculus (see e.g. <cit.>).
The key ingredient of
Jurdzinski's algorithm is what is called a progress measure (a
notion originally from <cit.>)—it can be understood as an
extension of a ranking function (used e.g. for
termination proofs) to a setting with nested $\mu$'s and $\nu$'s.
Coalgebras and Coalgebraic Modal Logics
On the other side of formal verification (namely system models),
coalgebra has attracted attention as a categorical abstraction
of state-based systems <cit.> for more
than a decade. An $F$-coalgebra is an arrow $c\colon X\to FX$
in some category $\C$, where $F\colon \C\to \C$ is an endofunctor. By
changing $\C$ and $F$ a coalgebra instantiates to a variety of
transition systems, such as Kripke structures, LTSs, Markov chains,
tree automata, processes in the $\pi$-calculus, and so on. Abstracting
away from specific choices of $\C$ and $F$ allows us to develop a
uniform theory that applies to various systems.
One notable success is a uniform definition of bisimulation
that is independent from $\C$ and
$F$. See <cit.>.
Along with the development of the theory of coalgebras,
coalgebraic modal logics have been developed as
languages suited for specifying about
coalgebras (see e.g. <cit.>).
Besides the approaches with Moss' cover modality <cit.>
and Stone-like dualities <cit.>, the one with
predicate liftings <cit.> is widely adopted in
the literature. The theory has since produced many uniform results
about coalgebraic modal logics as specification languages. They
are on:
expressivity (i.e. that bisimilarity is
captured) <cit.>; sound and complete
axiomatizations <cit.>;
satisfiability complexity <cit.>; cut elimination and
interpolation <cit.>; and so on.
Fixed-point operators in coalgebraic modal logics have been actively
too. See
e.g. <cit.>,
and also <cit.> where coalgebraic
automata are studied as translations of $\mu$-calculus formulas.
In particular, in <cit.>, algorithms for the model-checking
and satisfiability problems of a coalgebraic $\mu$-calculus
are presented. These algorithms reduce the
problems to parity games—this follows the common parity-game
workflow that we already discussed. For
satisfiability they also need a tableau system devised for this purpose.
§.§ Contributions
In this paper we scrutinize the aforementioned parity-game
workflow of: reducing to a parity game, and solving by Jurdzinski's
algorithm. We identify its essence in progress measures—a key
notion in Jurdzinski's algorithm <cit.>—rather than in
parity games themselves. This leads us to a lattice-theoretic
transfinite notion of progress measure that works without any
finiteness assumption, a restriction that is inevitable in the
combinatorial notion of parity game. We then go on to develop a generic
(and not necessarily finitary) framework for model checking, where
system models and specifications also have generic presentations in the language of
coalgebras and coalgebraic modal logics.
More specifically, our technical contributions are as follows.
Lattice-Theoretic Progress Measure
Taking an arbitrary complete lattice $L$ as a value domain
(instead of a finite power $\Bool^{m}$ of $\Bool=\{\ttrue,\ffalse\}$),
we present a lattice-theoretic characterization of solutions of
recursive equations
with (nested and alternating) greatest and least
fixed-points. The characterization is by the notions of
prioritized ordinal and progress
measure—notions that are essentially generalization of what
are in Jurdzinski's work <cit.>.
Our general formalization allows one to use
progress measures also in infinitary settings where we deal with infinite-state systems,
quantitative verification (i.e. the set of truth values is
infinite), or both.
One can also think of our progress measure as the combination of the
common proof methods by: invariants for safety/gfp
properties, and ranking functions for liveness/lfp
properties (Table <ref>).
These methods have been extensively studied especially
in the field of program verification—where problems
are inherently infinitary due to the $\mathtt{Integer}$ datatype—with an emphasis on
automatic synthesis of invariants and ranking functions (see
e.g. recent <cit.>).
properties witnessed by
safety, gfp invariants
liveness, lfp ranking functions
nested gfp's
winning strategies for parity games
(if finitary),
and lfp's
progress measures (in general)
Progress measures $=$ (invariants $+$ ranking functions)
Our current results therefore
open the way to combining these automated synthesis techniques, and to
obtaining automated proof methods for nested lfp/gfp
properties (like the response formula $\mathsf{G}(p\to
\mathsf{F}q)$ but much, much more). Once done its impact will be significant, since
currently most automation attempts in the field focus on only safety
or liveness, and not their combination.
We note that these results (in <ref>)
are formulated solely in (rather elementary) lattice-theoretic terms,
without any category theory. While their principal use in the current
paper is in coalgebraic model checking, their application areas are
expected to be widespread, in quantitative verification, program
verification, and so on—by model checking and deductive methods alike.
Progress Measure for Coalgebraic $\mu$-Calculus
Model Checking We apply the notion of progress measure to model
checking of a coalgebraic modal $\mu$-calculus
$\CmuGL$. Specifically, given a coalgebra $c\colon X\to FX$ (as a
system model), a $\CmuGL$-formula $\varphi$ (as a specification)
and the domain $\Omega$ of truth values, we characterize the
semantics $\sem{\varphi}_{c}\colon X\to \Omega$ of $\varphi$ over $c$
in terms of progress measures. Unlike the original definition of
the semantics $\sem{\varphi}_{c}$ (that is highly nonlocal due to
fixed-point operators), it can be checked locally whether given
data constitute a progress measure.
The lattice-theoretic
generality of our progress measure allows: a
state space $X$ that is infinite; a domain $\Omega$ of truth
values that is other than
$\Bool=\{\ttrue,\ffalse\}$ (such as the unit interval $[0,1]$);
and so on. Furthermore, for its finitary special case, we derive
a model-checking algorithm that is based on progress measures.
We expect our theoretical framework
(general, possibly infinitary, in <ref>)
to be a foundation on which various verification
techniques—a candidate being an extension of the simulation-based
method in <cit.>—can be formulated and proved sound.
Besides, our generic model-checking algorithm
(in <ref>, as a finitary
special case of the framework
in <ref>) is a
uniform algorithm that works for a variety of
endofunctors $F$ and modalities over $F$ (normal modal logic over
Kripke models, neighborhood frames, graded modal logic, coalition
logic, and so on; see Example <ref>). Moreover,
thanks to its concrete presentation with matrices, our algorithm
should be easy to implement.
Currently it is not clear whether our algorithm in <ref> competes with
tailor-developed ones for a specific modal logic. However we believe
our generic algorithm is at least worthwhile—much like
a big part of the coalgebraic attempts towards abstraction and
genericity, see <ref>—for the following reasons: 1) among the examples covered
by our generic algorithm, not all enjoy
tailor-developed algorithms; and 2) we believe our algorithm,
though currently basic, can expose further “handles” for
optimization. The latter means: in many parity game-based
algorithms, the part of solving parity games is left as a
blackbox; and in principle opening up a blackbox (like we do)
should be good for optimization, possibly allowing for “shortcut
fusion”-like optimization.
Coalgebraic $\mu$-Calculus as a Linear-Time Logic
In order to further demonstrate the theoretical robustness of our
framework, we present an
adaptation of the framework to linear-time model checking.
In this case a system is a coalgebra $c\colon X\to \pow FX$ (with additional nondeterministic
represented by the powerset monad $\pow$); and the question is
whether there is an infinitary trace
$z$ of $c$ starting from $x$ such that $z$ satisfies
a $\CmuGL$-formula $\varphi$.
It turns out that the combination with coalgebraic theory of
traces and simulations (developed e.g. in <cit.>) allows a
smooth transfer from the previous “branching-time” setting to the
current linear-time one.
The outcome is a uniform treatment of branching and
(nondeterministic) linear-time logics—which does not seem to be
achieved before despite the obvious efforts by the coalgebra community. This venture also needs a technical piece, namely the “pumping”-like result (Thm. <ref>) by Zorn's lemma.
Our technical contributions are: a progress
measure-based characterization of linear-time model checking
(where, again, whether given data is a valid progress measure or not
can be checked locally); and a decision procedure for linear-time
model checking (with the restriction that the state space $X$ is
finite and the truth values are Boolean). The former solves the
challenge, presented in <cit.>, of a local characterization
of linear-time semantics
(called “step-wise semantics” in <cit.>)
for coalgebraic fixed-point logics.
§.§ Future Work
There are a lot of further topics to study in
our current venture to coalgebraic $\mu$-calculus. They include:
implementation of our model-checking algorithms
in <ref> and <ref>
(the one in <ref> should
especially be easy because of the presentation by matrices);
experiments, comparison with tailor-made algorithms and further optimization;
satisfiability and small-model property; universal linear-time model
checking (in this paper we study the existential one); synthesis; and
$\CmuGL$ as linear-time logic for systems with probabilistic
branching. In particular we expect the last to be not hard, given the
lattice-theoretic generality of the current results.
It should also
help that
the coalgebraic theory of traces and simulations has been recently
extended to the probabilistic setting <cit.> (using the Giry monad over the
category of measurable spaces). We can say we understand the
mathematical structures therein fairly well:
these studies suggest that the probabilistic setting is
better-behaved than the nondeterministic setting, from a
coalgebraic point of view. See <cit.> for further details.
Besides, our lattice-theoretic theory of nested fixed points allows
progress measures (which we identify as the essence of parity games)
to be applied to infinitary settings.
We believe it will be useful for the following purposes.
Working out
these further applications is future work.
Establishing an Alternating Fixed-Point in Theorem
Proving In an infinitary setting (such as the state space
$|X|$ and/or the truth domain $\Omega$ are infinite), the search
space for our (infinitary) progress measures will be infinite,
and hence is not amenable to algorithmic search. Even so, one could
resort to human ingenuity to find one.
An advantage of a progress measure-based characterization of the
semantics $\sem{\varphi}_{c}$ is, as we mentioned earlier, the
validity of a progress measure can be checked locally in a
straightforward manner. This is unlike the original definition of
the semantics $\sem{\varphi}_{c}$ (see Def. <ref>)
that involves highly nonlocal information like $V\colon X\to
\Omega$. We believe this advantage will be especially useful
when one works with fixed-point specifications in a proof
Due to the same advantage,
our progress measure-based characterization might also form a
basis of sound (but not necessarily complete) model-checking algorithms
that rely e.g. on mathematical programming. This is much like
in <cit.> where Kleisli simulations (whose existence is
checked by linear programming and hence is PTIME) give a sound proof
for weighted language inclusion (an undecidable property).
As a Tool in a Meta-Theory In higher-order model
checking (see e.g. <cit.>), a
higher-order recursion scheme (HORS)
generates an infinite tree that is then model-checked against a
modal $\mu$-formula. The generated tree is in general
irrational—hence cannot be identified with a finite-state
automaton. However it is shown <cit.> that the
model-checking is decidable; an algorithm operates directly with
the HORS that generates the tree, but not with the tree itself.
In this setting (and similar ones), we expect our infinitary
progress measure to be a useful tool on the level of
meta-theory, e.g. for showing the correctness of
an algorithm.
We also envisage the use of our current results
in lifting (bi)simulation notions for Büchi and parity
automata (see e.g. <cit.>) to the coalgebraic level
of abstraction and generality.
In this direction we have obtained
some preliminary results that characterize the accepted languages
of Büchi/parity automata via coalgebras in a Kleisli
category—results that will hopefully enable us to extend our
coalgebraic theory of traces and simulations
in <cit.> to
Büchi/parity acceptance conditions.
We also intend to study the
relationship between our current work and quantitative
extensions of parity games, a topic of extensive research efforts <cit.>.
§.§ Notations
Throughout the paper, the domain of truth
values is denoted by $\Omega$ and is assumed to be a complete lattice,
with its order denoted by $\sqsubseteq$, and its supremums and infimums
denoted by $\bigsqcup$ and $\bigsqcap$.
Typical examples of $\Omega$ are the set $\Bool=\{\ttrue,\ffalse\}$ of Boolean truth
values, and the unit interval $\unitInt$ for a quantitative notion of truth.
In <ref> we will use another complete lattice
$L$; this will be instantiated by $L=\Omega^{X}$—where $X$ is the
state space of the system in question—for the use in later sections.
Since $\Omega$ is a complete lattice, any monotonic
endofunction $f$ on $\Omega$ has the greatest and
least fixed points $\nu f,\mu f$. The same holds for $L$ in place of $\Omega$.
We fix a countable set $\Var$ of
(fixed-point) variables. It is ranged by $u,v,w,\dotsc$.
We let $\eta$ designate fixed-point operators in general; it is either
$\mu$ or $\nu$. Confusion with a monad unit is unlikely.
The set of natural numbers is identified with the smallest infinite
ordinal and denoted by $\omega$.
§.§ Organization of the Paper
In <ref>
we present our lattice-theoretic notion of progress measure and prove
that it characterizes the solution of a system of fixed-point equations.
In <ref> we introduce our logic $\CmuGL$—it is a
coalgebraic modal logic with both greatest and least fixed-point
operators ($\nu$, $\mu$); it is parametrized not only by the set
$\Lambda$ of predicate liftings (i.e. modalities) for a functor $F$, but also by the set
$\Gamma$ of propositional connectives.
In <ref> we adapt progress measures
in <ref> to the purpose of $\CmuGL$ model
checking (against $F$-coalgebras), derive a model-checking algorithm and
analyze its complexity. This framework is further adapted
in <ref> to (existential) linear-time
model checking—where a system has additional nondeterministic
branching. We present a decision procedure there.
Appendices to the current paper are found in the extended
version <cit.>. Omitted proofs are there, too.
Omitted proofs are found in Appendix <ref>.
§ PROGRESS MEASURES FOR EQUATIONAL
§.§ Prelude: (Unnested) Fixed Points, Invariants and Ranking Functions
In general, there are two different ways for characterizing (not nested)
least/greatest fixed points (lfp's and gfp's). The first is the Knaster-Tarski one:
the lfp is the least prefixed point; and the gfp is the greatest
postfixed point. The second is the Cousot-Cousot
one <cit.>: the lfp $\mu f$ of a monotone function $f\colon
\CompLat\to \CompLat$ over a complete lattice $\CompLat$ is the (possibly
transfinite) supremum of the chain $\bot\sqsubseteq f(\bot)\sqsubseteq
f^{2}(\bot)\sqsubseteq\cdots$; similarly the gfp $\nu f$ is the infimum of
$\top\sqsupseteq f(\top)\sqsupseteq\cdots$. Sometimes these chains
are guaranteed to stabilize
after $\omega$ steps, for example when $f$ satisfies suitable continuity
conditions (the Kleene fixed-point theorem).
In this paper our principal interests will be finding lower bounds for
fixed points; see Rem. <ref> for system
verification motivations.
Among the last four characterizations (Knaster-Tarski and Cousot-Cousot,
for each of lfp and gfp), what are suited for this purpose of
ours are: the Cousot-Cousot one for lfp's; and the
Knaster-Tarski one for gfp's (the other two only give us upper bounds).
We explicitly note this fact for the record:
[lower bounds for fixed points]
Let $\CompLat$ be a complete lattice and $f\colon \CompLat\to\CompLat$
be a monotone function.
For each ordinal $\alpha$ we have $f^{\alpha}(\bot)\sqsubseteq
\mu f$. Here $f^{\alpha}(\bot)$ is defined by obvious induction:
$f^{\alpha+1}(\bot)=f(f^{\alpha}(\bot))$ for a successor
ordinal; and
for a limit ordinal.
For any $l\in \CompLat$, $l\sqsubseteq f(l)$ implies
$l\sqsubseteq \nu f$.
We emphasize that this simple theoretical observation is what underlies the
difference between the common proof methods for safety/gfp
properties and for liveness/lfp properties (Table <ref>). For the
former (gfp's) one would seek for an invariant, that is, a
postfixed point $l$ such that $l\sqsubseteq f(l)$. For the latter
(lfp's) one would typically synthesize a
ranking function, an $\omega$-valued function that strictly
decreases in each step. We formulate—also for the sake of some intuitions—the general principle
behind the latter, focusing on $L=\Bool^{X}$.
Let $f\colon \Bool^{X}\to\Bool^{X}$ be a monotone function.
A ranking
function for $f$ is an ordinal- (or $\NoGood$, indicating “failure”)
valued function $\rk\colon X\to \Ord\amalg\{\NoGood\}$
such that: 1)
$\rk(x)\neq 0$ for each $x\in X$; 2) for each ordinal $\alpha$,
$\{x\mid \rk(x)\le \alpha+1\}\subseteq
f\bigl(\{x\mid \rk(x)\le \alpha\}\bigr)
$; and 3) for each limit ordinal $\alpha$,
$\{x\mid \rk(x)\le \alpha\}= \bigcup_{\beta<\alpha}
\{x\mid \rk(x)\le \beta\}
Assume that $X$ is equipped with a transition relation $R\subseteq
X\times X$ and we are interested in reachability to a subset $U\subseteq
X$. We would then define $f$ by: $f(X'):=U\cup \{x\mid \exists x'.\,
xRx'\land x'\in X'\}$; this yields $f^{\alpha}(\bot)$ to be the set of
states from which $U$ is reachable within $\alpha-1$ steps.
A prototypical ranking function is given by $\rk(x):=(\text{the distance
from $x$ to $U$})+1$.
In Def. <ref>,
a ranking function $\rk$ for $f$ witnesses $\mu f$, the least fixed
point of $f$. That is,
$\rk(x)\neq\NoGood$ implies $x\in \mu f$.
The following is easily shown by induction on an ordinal $\alpha$:
for any $x\in X$ such that $\rk(x)=\alpha$, we have
$x\in f^{\alpha}(\bot)$. The claim then follows from Lem. <ref>.<ref>.
Implicit in the above is a bijective correspondence—not unlike in Stone-like
* a ranking function $\rk\colon X\to
\Ord\amalg\{\NoGood\}$; and
* an approximating sequence
$U_{0}\subseteq U_{1}\subseteq\cdots$ such that: 1)
$U_{0}=\bot=\emptyset$, 2)
$U_{\alpha+1}\subseteq f(U_{\alpha})$, and 3)
$U_{\alpha}=\bigcup_{\beta<\alpha}U_{\beta}$ for any limit ordinal
From the former to the latter we let $U_{\alpha}:=\{x\mid \rk(x)\le
\alpha\}$; conversely we let $\rk(x):=\inf\{\alpha\mid x\in
§.§ Equational Systems
With the preparations
in <ref> for unnested fixed
points, we set out to study nested and alternating ones.
As a formalism of expressing them
we prefer equational systems, to the (probably more common)
modal $\mu$-calculus-like notations.
Here we shall follow the accounts of similar notions
in <cit.> and <cit.>.
[equational system]
Let $L$ be a complete lattice. An equational system $E$
over $L$ is an expression of the form
\begin{equation}\label{eq:sysOfEq}
\begin{array}{c}
u_{1}=_{\eta_{1}}f_{1}(u_{1},\dotsc, u_{m}),\quad
\dotsc,\quad
u_{m}=_{\eta_{m}} f_{m}(u_{1},\dotsc, u_{m})
\end{array}
\end{equation}
where: $\seq{u}{m}$ are variables,
$\eta_{1},\dotsc,\eta_{m}\in\{\mu,\nu\}$, and $\seq{f}{m}\colon
L^{m}\to L$ are monotone functions.
A variable $u_{j}$ is said to be a $\mu$-variable if
$\eta_{j}=\mu$; it is a $\nu$-variable if $\eta_{j}=\nu$.
We say $u_{i}$ has a bigger priority than $u_{j}$ if $j<i$.
Note that, in the last definition, we have been vague about the distinction between
a function $f_{i}$ as a semantical object and a syntactic symbol that denotes
It is straightforward to generalize
the definition and allow different variables to take values in different
complete lattices $\seq{L}{m}$, and extend accordingly our technical
developments below.
We assume $L_{1}=\cdots=L_{m}=L$ for
ease of presentation.
The order of equations matters in an equational system
like (<ref>).[Here we follow the ordering convention
in <cit.>. In <cit.> the order is reversed, and
the rightmost equation is solved first.] Intuitively, the system (<ref>) is
solved starting from the leftmost equation, where the remaining
variables $u_{2},\dotsc, u_{m}$ are left as undetermined parameters.
The interim solution of the leftmost equation (for $u_{1}$, in terms of
$u_{2},\dotsc, u_{m}$) is then used in the second equation
$u_{2}=_{\eta_{2}}f_{2}(u_{1},\dotsc, u_{m})$ to eliminate
the occurrences of $u_{1}$ in its right-hand side. We continue this way;
then solving the last
equation would give us a closed (i.e. without any
variables occurring in it) solution for $u_{m}$. Such closed solutions are then propagated
from right to left in (<ref>), finally giving a closed
solution to each variable $u_{i}$.
The above intuitions can be put in the following precise terms.
The solution of an equational system (<ref>) is defined
as follows.
For each $i\in[1,m]$ and $j\in[1,i]$, we define monotone functions
f^_iL^m-i+1 →L
and l^(i)_jL^m-i→L
as follows, inductively on $i$. For the base case $i=1$:
f^_1(l_1,…,l_m) := f_1(l_1,…,l_m),
l^(1)_1(l_2,…, l_m) :=
η_1[f^_1(,l_2,…, l_m)L→L].
In the last line we take the lfp or gfp (according to
$\eta_{1}\in\{\mu,\nu\}$) of the (monotone) function
$f^{\ddagger}_{1}(\place,l_{2},\dotsc, l_{m})\colon L\to
For the step case, the function $f^{\ddagger}_{i+1}$ makes use of the
$i$-th interim solutions $l^{(i)}_{1},\dotsc,l^{(i)}_{i}$ for the variables
$u_{1},\dotsc, u_{i}$ obtained so far:
f^_i+1(l_i+1,…, l_m):=
f_i+1( l^(i)_1(l_i+1,…, l_m), …,
l^(i)_i(l_i+1,…, l_m),
l_i+1,…, l_m
We then let
l^(i+1)_i+1(l_i+2,…, l_m)
f^_i+1(,l_i+2,…, l_m)L→L
and use it to obtain the $(i+1)$-th interim solutions
$l^{(i+1)}_{1},\dotsc,l^{(i+1)}_{i}$. That is, for each $j\in [1,i]$,
\begin{equation}\label{eq:defInterimSolutionForSmallerJ}
l^{(i+1)}_{j}(l_{i+2},\dotsc, l_{m})
l^{(i)}_{j}\bigl(\,l^{(i+1)}_{i+1}(l_{i+2},\dotsc, l_{m}),\; l_{i+2},\dotsc,l_{m}\,\bigr)
\end{equation}
Finally, the solution
\dotsc,
\in L^{m}$ of the equational system (<ref>) is defined by
, where we identify a function $l^{(m)}_{j}\colon 1\to L$ with an element of $L$.
It is easy to see that all the functions $f^{\ddagger}_{i}$ and
$l^{(i)}_{j}$ involved here are monotone.
That the solution uniquely exists is then guaranteed by the
Knaster-Tarski theorem.
As a simple example, consider an equational system
u_1=_μ u_2,
u_2=_ν u_1.
Solving the first equation yields $u_{1}=u_{2}$ (i.e. $l^{(1)}_{1}(l_{2})=l_{2}$); using it
to eliminate $u_{1}$ in the second equation, we obtain $u_{2}=_{\nu}
u_{2}$ (i.e. $f^{\ddagger}_{2}(l_{2})=l_{2}$). We conclude
is the solution.
It is not hard to see that, if we change the order of the equations,
the resulting system
u_2=_ν u_1,
u_1=_μ u_2
has a different solution $u_{1}=u_{2}=\bot$.
It is not hard to give a precise correspondence between equational
systems and their modal $\mu$-calculus-like presentations.
Each equation $u_{j}=_{\eta_{j}} f_{j}(u_{1},\dotsc, u_{m})$ corresponds
to a fixed-point formula $\eta_{j} u_{j}.\, f_{j}(u_{1},\dotsc, u_{m})$;
since an
system like (<ref>) is solved from left to right, the formula
corresponds to an equation on the left occurs inside the formula
for an equation on the right. For example, if $m=2$, the equational
system (<ref>) is presented as
η_2 u_2. f_2(
(η_1 u_1. f_1(u_1, u_2)) ,
u_2 )
In the light of such a correspondence to $\mu$-calculus-like formulas, the definition
of bigger/smaller priorities in Def. <ref> coincides with what
is customary (an outside fixed-point operator has a
bigger priority).
A precise translation can be defined following <cit.>;
see also Def. <ref> later, in the special case of
coalgebraic fixed-point logic.
[aiming at lower bounds]
Assume that an equational system $E$ is given. For the purpose of
system verification, one is typically not so much interested
in its
itself, as in a suitable lower bound of it.
For a simple example consider the setting of
Example <ref>, and assume that $X$, $R$ and $U$ are
given as follows.
A common question would be if $U$ is reachable from a specific state of
our interest,
say $x_{3}$. To verify it the ranking function
(x_0)=1, (x_i)=i+1 for each $i\ge 1$,
(x_i)= for each $i<0$
suffices. This choice of a ranking function—while it gives a lower bound
$\{x_{0},x_{1},\dotsc\}\subseteq\mu f$ of $\mu f$—does not witness e.g.$x_{-3}\in \mu f$ (that actually holds). This is not a problem
because we are interested only in $x_{3}$.
This phenomenon (of only
giving a lower bound) is the case with
verification algorithms in general: they conduct “directed” searches from
the states in question. Therefore in this paper we focus on
characterizing lower bounds of the solution of an equational
system. Upper bounds, in contrast, are useful in
refuting that certain states have certain properties.
§.§ Progress Measures
We shall now characterize lower bounds of (nested and alternating) fixed
points specified by an equational system.
We use the technical notion of progress measure;
it is a
lattice-theoretic generalization of the notion of parity progress measure
in <cit.>, and hence is seen as a generalization of
winning strategies for parity games, too.
Roughly speaking, these are how one combines
invariants (for gfp's) and ranking functions (for lfp's,
see Table <ref> and <ref>)
in an intricate way so that priorities in alternation are respected.
Lem. <ref> we approximate least fixed
points by transfinite sequences starting from $\bot$. In general there are
multiple $\mu$-variables in an equational system—we have one “counter”
for each of them, and use their tuple that we call a prioritized
In particular, the definition of the preorder $\preceq_{i}$ between
prioritized ordinals—derived from the one in <cit.> and
defined for each variable $u_{i}$—lies in the technical core.
[prioritized ordinal, $\preceq_{i}$]
Let $E$ be the equational system in (<ref>), over a complete
lattice $L$. Let us collect all those indices $i\in [1,m]$ for which $u_{i}$ is a
$\mu$-variable in the equational system $E$, and arrange them so that
$i_{1}<\cdots <i_{k}$. That is,
{i∈[1,m]|η_i=μ in (<ref>)}.
Then a prioritized ordinal for $E$ is a $k$-tuple
$(\seq{\alpha}{k})$ of ordinals. Note that $k$ is the number of
$\mu$-variables in $E$.
For each $i\in [1,m]$ we define a preorder $\preceq_{i}$ between
prioritized ordinals—we call $\preceq_{i}$ the $i$-th truncated
lexicographic order—as follows.
Let $a\in [1,k]$ be such that
i_1<⋯<i_a-1<i≤i_a<⋯< i_k,
that is, $u_{i_{a}}$ is the $\mu$-variable with the smallest priority
that is at least as big as that of $i$.
Then we define
if, between the $i$-truncations
$ (\alpha_{a},\dotsc,\alpha_{k})$
$ (\alpha'_{a},\dotsc,\alpha'_{k})$
of the prioritized ordinals,
we have
. Here the last $\preceq$ denotes the lexicographic extension of
the usual order $\le$ between ordinals, with the latter elements being
the more significant. Note here that the $i$-truncation
is obtained by dropping the first elements that correspond to the
$\mu$-variables with priorities smaller than that of $u_{i}$.
In case $\preceq_{i}$ holds in both ways we write $=_{i}$. Note that
is in general coarser than the equality between prioritized ordinals
(see Example <ref>).
We define
holds but
Let us consider the following example $E_{0}$ of an equational system:
\begin{multline*}
u_{1}=_{\mu} f_{1}(\vec{u}),
\quad
u_{2}=_{\nu} f_{2}(\vec{u}),
\quad
u_{3}=_{\mu} f_{3}(\vec{u}),
\quad
\\
u_{4}=_{\mu} f_{4}(\vec{u}),
\quad
u_{5}=_{\nu} f_{5}(\vec{u}),
\end{multline*}
where $\vec{u}$ stands for $u_{1},\dotsc,u_{5}$. A prioritized
ordinal for this $E_{0}$ is a tuple $(\alpha_{1},\alpha_{2},\alpha_{3})$
of ordinals, where the ordinals $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$
correspond to
the $\mu$-variables $u_{1}$, $u_{3}$ and $u_{4}$, respectively.
It holds that $(\omega,2,2)\preceq_{1}(0,3,2)$. To see that, since $u_{1}$ is
with the smallest priority, we have to check $(\omega,2,2)\preceq (0,3,2)$.
This holds; recall that $\preceq$ is the lexicographic order with
the latter being the more significant. We can similarly see that:
\begin{align*}
\quad
\\
&(\omega,2,2)=_{4}(0,3,2), \quad\text{and}
\end{align*}
Note here that the $3$-, $4$- and $5$-truncations of $(\omega,2,2)$ and
$(0,3,2)$ are: $(2,2)$ and $(3,2)$; $(2)$ and $(2)$; and $()$ and
$()$, respectively.
In the following definition,
the element
$\approximant_{i}(\alpha_{1},\dotsc,\alpha_{k})\in \CompLat$
is understood as the “$(\alpha_{1},\dotsc,\alpha_{k})$-th
of the solution for the variable $u_{i}$ in the equational
system (<ref>).
[progress measure for an equational system]
Assume the same setting as in Def. <ref>, with
$E$ being the equational system (<ref>) and $i_{1}<\cdots
enumerating the indices of all the $\mu$-variables.
A progress measure $p$ for $E$ is given by a tuple
p =
( _i(α_1,…,α_k) )_i,αk
consists of:
* the maximum prioritized ordinal
$(\overline{\alpha_{1}},\dotsc, \overline{\alpha_{k}})$; and
* the approximants $\approximant_{i}(\alpha_{1},\dotsc,\alpha_{k})\in \CompLat$, defined for
$i\in[1,m]$ and each
prioritized ordinal
such that
\alpha_{1}\le\overline{\alpha_{1}},\dotsc,
\alpha_{k}\le\overline{\alpha_{k}}
The approximants $\approximant_{i}(\alpha_{1},\dotsc,\alpha_{k})$
are subject to:
Let $i\in[1,m]$ (hence $u_{i}$ is either a $\mu$- or
$\nu$-variable). Then
\begin{multline*}
\preceq_{i}
\;\text{implies}
\\
\approximant_{i}(\alpha_{1},\dotsc,\alpha_{k})
\sqsubseteq
\approximant_{i}(\alpha'_{1},\dotsc,\alpha'_{k}).
\end{multline*}
($\mu$-variables, base case)
Let $a\in [1,k]$. Then $\alpha_{a}=0$ implies
$\approximant_{i_{a}}(\alpha_{1},\dotsc, \alpha_{a},\dotsc,\alpha_{k})=\bot$.
(Note the correspondence between: the subscript $i_{a}$ of
$\approximant_{i_{a}}$; and the counter $\alpha_{a}$ that
is assumed to be $0$.)
($\mu$-variables, step case)
Let $a\in [1,k]$, and let
\alpha_{a}+1,\dotsc,\alpha_{k})$ be
a prioritized ordinal
such that its $a$-th counter
$\alpha_{a}+1$ is a successor ordinal. Then, regarding
the approximant $\approximant_{i_{a}} (\alpha_{1},\dotsc,\alpha_{a-1},
\alpha_{a}+1,\alpha_{a+1},\dotsc,\alpha_{k})$,
there exist ordinals
\beta_{a-1}$ such that
\begin{equation}\label{eq:progressMeasDefMuVarStepCase}
\begin{aligned}
\approximant_{i_{a}} (\alpha_{1},\dotsc,\alpha_{a-1},
\alpha_{a}+1,\alpha_{a+1},\dotsc,\alpha_{k})
\\
\left(\,
\begin{array}{c}
\approximant_{1} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\alpha_{a+1},\dotsc,\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\alpha_{a+1},\dotsc,\alpha_{k})
\end{array}
\,\right)
\end{aligned}
\end{equation}
and $\beta_{1}\le\overline{\alpha_{1}},\dotsc,
\beta_{a-1}\le \overline{\alpha_{a-1}}$.
Recall here that $f_{i_{a}}$ is a function in the
system (<ref>).
Cond. (<ref>) originates
from the definition
in Lem. <ref>.<ref>; a notable difference
here is
that the counters with smaller priorities (i.e. from the
first to the $(a-1)$-th) can be modified arbitrarily.
($\mu$-variables, limit case)
Let $a\in [1,k]$, and
\alpha_{k})$
be a prioritized ordinal
such that its $a$-th counter
is a limit ordinal. Then, regarding the approximant
$\approximant_{i_{a}} (\alpha_{1},\dotsc,
\alpha_{k})$, we have
\begin{equation}\label{eq:progressMeasDefMuVarLimitCase}
\approximant_{i_{a}} (\alpha_{1},\dotsc,
\alpha_{a},\dotsc,\alpha_{k})
% \\
% &\qquad\sqsubseteq\;
\sqsubseteq
\bigsqcup_{\beta<\alpha_{a}}
\approximant_{i_{a}} (\alpha_{1},\dotsc,
\beta,\dotsc,\alpha_{k})\enspace.
\end{equation}
Let $i\in [1,m]\setminus\{i_{1},\dotsc, i_{k}\}$ (i.e. $u_{i}$ is a $\nu$-variable in the
system (<ref>)); let $a\in [1,k]$ such that
i_1<⋯<i_a-1<i<i_a<⋯< i_k.
Let $(\alpha_{1},\dotsc,\alpha_{k})$ be a prioritized ordinal.
Then, regarding the approximant
$\approximant_{i} (\alpha_{1},\dotsc,\alpha_{k})$, there exist
$\beta_{1},\dotsc,\beta_{a-1}$ such that
\begin{equation}\label{eq:progressMeasDefNuVar}
\begin{aligned}
\approximant_{i} (\alpha_{1},\dotsc,
\alpha_{a-1},\alpha_{a},\dotsc,\alpha_{k})
\\
\left(\,
\begin{array}{c}
\approximant_{1} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k})
\end{array}
\,\right)
\end{aligned}
\end{equation}
and $\beta_{1}\le \overline{\alpha_{1}},\dotsc,
\beta_{a-1}\le \overline{\alpha_{a-1}}$.
This condition is somewhat similar to
Cond. <ref> above: it
comes from the condition $l\sqsubseteq f(l)$ in
Lem. <ref>.<ref>;
and much like in
Cond. <ref>, the
counters with smaller priorities can be modified arbitrarily.
[correctness of progress measures]
Let $E$ be the equational system in (<ref>) over $L$,
and $(
\dotsc,
)$ be its solution (Def. <ref>).
* (Soundness)
A progress measure gives a lower bound of the solution. That is,
( _i(α_1,…,α_k) )_i,α
is a progress measure.
Then for each $i\in
[1,m]$ we have
There exists a progress measure
that achieves the
optimal, that is,
( _i(α_1,…,α_k) )_i,α
such that
for each $i\in[1,m]$.
I have to be precise about the
correspondence between cardinals and ordinals.
Moreover, such an “optimal” progress measure can be chosen
so that the ordinals in its maximum prioritized ordinal
are suitably bounded, in the following sense.
Let $\ascCL(L)$ be the ordinal defined by the supremum of the length
of any (possibly transfinite) strictly ascending chain in $L$. Then
$\overline{\alpha_{a}}\le \ascCL(L)$ for
each $a\in [1,k]$.
In the item <ref>, the bound $\ascCL(L)$
is generally better than the bound by the size $|L|$ of the complete lattice
$L$. For example, in case $L=\Bool^{X}$ (where $\Bool=\{\ttrue,\ffalse\}$
and $X$ is a set), $\ascCL(L)=|X|$ while $|L|=2^{|X|}$.
We will need the following relaxation in establishing a correspondence
to Jurdzinski's notion of parity progress measure <cit.>.
[extended progress measure for equational systems]
Assume the setting of Def. <ref>. An
extended progress measure $p$ for $E$ is the same as a progress
measure, except that Cond. <ref> of
Def. <ref> is replaced by the following:
Let $a\in [1,k]$. Then $\alpha_{a}=0$ implies either
$p_{i_{a}}(\alpha_{1},\dotsc,\alpha_{k}) =\bot$, or
there exists a
ordinal $(\alpha'_{1},\dotsc,\alpha'_{k})$ such that
(\alpha_{1},\dotsc,\alpha_{k})$ and
$p_{i_{a}}(\alpha_{1},\dotsc,\alpha_{k}) \sqsubseteq
An extended progress measure is still sound in the sense of
Thm. <ref>.<ref>.
In Appendix <ref>,
in the extended version <cit.>,
as a sanity check, we
present a
correspondence between our notion of progress measure
(Def. <ref>) and
Jurdzinski's parity progress measure <cit.>.
Jurdzinski's formalization follows that of ranking functions,
while ours here is based on approximation sequences $p(0)\sqsubseteq
p(1)\sqsubseteq \cdots$ in the lattice $L=\Bool^{X}$. The relationship
between the two is much like in
Rem. <ref>.
For a simple example following the spirit of
Example <ref>, let us consider a set $X$ and a
transition relation $R\subseteq X\times X$, and introduce
a “modal operator” $\Box\colon \Bool^{X}\to\Bool^{X}$ by
$\Box(X'):=\{x\in X\mid \forall x'.\, xRx' \Rightarrow x'\in X'\}$.
We now
fix a subset $F\subseteq X$, and consider the following equational
over $L=\Bool^{X}$.
\begin{equation}\label{eq:10272034}
\;=_{\mu}\;
(F\cap u_{2})\cup
\Box u_{1},
\qquad
\;=_{\nu}\;
\end{equation}
The system corresponds to the $\mu$-calculus formula
$\nu u_{2}. \mu u_{1}.\, (F\cap u_{2})\cup \Box u_{1}$, and
it is not hard to see—possibly relying on the Knaster-Tarski and
Cousot-Cousot characterizations,
see <ref>—that the solution
for $u_{2}$ is the set of states any infinite path from which visits $F$ infinitely often.
For this specific system (<ref>), a progress measure
(Def. <ref>) is given by data
( α,
(p_2(α))_α≤α )
subject to suitable conditions.
Some simplifications are possible, exploiting that
in (<ref>) (and elsewhere) counters with smaller
priorities can be modified arbitrarily. We see, after this
simplification, that
a progress measure for the equational system (<ref>)
is given by
and p_2,
all being subsets of $X$, such that: 1) $p_{1}(0)=\emptyset$; 2)
$p_{1}(\alpha+1)\subseteq (F\cap p_{2})\cup \Box p_{1}(\alpha)$;
3) $p_{1}(\alpha)=\bigcup_{\beta<\alpha}p_{1}(\beta)$ for a limit
ordinal $\alpha$; and 4) $p_{2}\subseteq p_{1}(\overline{\alpha})$.
This “witnesses” the solution of (<ref>),
i.e. $x\in p_{2}$ implies that any infinite path from $x$ visits $F$
infinitely often.
§ COALGEBRAIC $\MU$-CALCULUS $\CMU_{\GAMMA,\LAMBDA}$
From this section on we apply the theory developed
in <ref> to a coalgebraic $\mu$-calculus
In the current section, as a
preparation, we introduce the logic $\CmuGL$: its syntax, semantics,
and a translation to equational systems (so that the results
in <ref> apply).
§.§ Coalgebraic Preliminaries
We start with a minimal set of coalgebraic preliminaries. For further backgrounds on coalgebras
see e.g. <cit.>; and see
e.g. <cit.> for categorical preliminaries.
From now to <ref> we fix the base category to be the one $\Sets$ of
sets and functions.
Let $F$ be an endofunctor on $\Sets$. An $F$-coalgebra is a
function $c\colon X\to FX$, where $X$, $F$ and $c$ are intuitively
understood as a state space, a behavior type and a transition structure,
respectively. Therefore an $F$-coalgebra is “a transition system of the
behavior type $F$.” Some examples are presented later in
Example <ref>.
\begin{equation}\label{eq:coalgHom}
\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!
\vcenter{\xymatrix@R=.4em@C+1em{
\ar@{->}[r]^-{Ff}
\\
\ar@{->}[r]^-{f}
\ar[u]^{c}
\ar[u]^{d}
\end{equation}
Given two coalgebras $c\colon X\to FX$ and $d\colon Y\to FY$ for the
same functor, a coalgebra homomorphism from $c$ to $d$ is a function
$f\colon X\to Y$ such that the above diagram (<ref>) commutes.
In many examples of $F$, the notion of homomorphism expresses a natural
definition of behavior-preserving map. Conversely, it is common
the theory of coalgebras that the notion of $F$-behavioral
equivalence is defined using homomorphism (namely via cospans,
see <cit.>).
\begin{equation}\label{eq:behq}
\hspace{-3em}
\vcenter{\xymatrix@R=.8em@C+2em{
\ar@{-->}[r]^-{F(\beh(c))}
\\
\ar@{-->}[r]_-{\beh(c)}
\ar[u]^{c}
\ar[u]^{\text{final}}_{\zeta}
% \quad\text{in $\Sets$.}
\end{equation}
Furthermore, many functors $F$ allow a “classifying coalgebra”—one
that contains every possible $F$-behavior without
redundancy. This is categorically captured by finality of a
coalgebra. Precisely, a coalgebra $\zeta\colon Z\to FZ$ is final
if, for any coalgebra $c\colon X\to FX$ there exists a unique
homomorphism $\beh(c)\colon X\to Z$, as shown in (<ref>).
This way we understand the carrier $Z$ of a final coalgebra to be the
set of all $F$-behaviors; and the map $\beh(c)$ induced by finality (as
in (<ref>)) as the behavior map.
§.§ $\Cmu_{\Gamma,\Lambda}$: Syntax
It is common in the study of coalgebraic modal logics that the set of
modalities is parametrized.
In our logic $\Cmu_{\Gamma,\Lambda}$, moreover, we parametrize
propositional connectives too. This allows us to accommodate
unconventional connectives that occur in quantitative setting, like the truncated
sum in the Łukasiewicz
$\mu$-calculus <cit.> and the
average operator in e.g. <cit.>.
[$\Lambda$, $\Gamma$]
A modal signature $\Lambda$ over $F$ is a ranked alphabet
$\Lambda=(\Lambda_{n})_{n\in\omega}$. An element
$\lambda\in\Lambda_{n}$ is an $n$-ary modality,
and we write $|\lambda|$ for its arity $n$.
We assume that a modal signature comes with its interpretation.
Assigned to each $\lambda\in\Lambda$ is a natural transformation
$\sem{\lambda}$, whose components are functions
natural in $X$,
and a component $\sem{\lambda}_{X}$ must be monotone with respect to
(pointwise extensions of)
the order $\sqsubseteq$ of the domain $\Omega$ of truth values. Such $\sem{\lambda}$ is commonly called
(monotone) predicate lifting <cit.>.
(Do we need this?)
We also identify $\Lambda$ with a polynomial functor, i.e. $\Lambda
X=\coprod_{\lambda\in\Lambda} X^{|\lambda|}$.
a propositional signature is a ranked alphabet $\Gamma$
where each $\gamma\in\Gamma$ is called a propositional
connective. Unlike a modal signature, each $\gamma\in\Gamma$
is interpreted by a function $\sem{\gamma}\colon
\Omega^{|\gamma|}\to\Omega$; we require that $\sem{\gamma}$ be monotone.
That, in turn, induces
a natural transformation
( Ω^X,Ω )^|γ|
by $\lambda f_{1}.\,\dotsc\lambda f_{|\gamma|}.\; \lambda v.\;
\sem{\gamma}(f_{1}v,\dotsc, f_{|\gamma|v})$.
In the case $\Omega=\unitInt$ of continuous truth values, we
additionally assume that $\sem{\lambda}$ and $\sem{\gamma}$ are
continuous, in the sense that they preserve the supremum $\bigsqcup_{i\in\omega}t_{i}$ of
an ascending $\omega$-chain $a_{0}\sqsubseteq a_{1}\sqsubseteq\cdots$.
In what follows we will often write $\lambda$ and $\gamma$ for
$\sem{\lambda}$ and $\sem{\gamma}$.
The language of our coalgebraic modal logic
$\Cmu_{\Gamma,\Lambda}$ over $\Gamma$ and $\Lambda$ is given by the
following set of formulas.
\begin{align*}
\varphi,\varphi_{i}
\,::=\,
\mid
\boxdot_{\gamma}(\varphi_{1},\dotsc,\varphi_{|\gamma|})
\mid
\heartsuit_{\lambda}(\varphi_{1},\dotsc,\varphi_{|\lambda|})
\mid
\\
\mu u.\, \varphi
\mid
\nu u.\, \varphi
\end{align*}
Here $u\in\Var$ is a (fixed-point) variable. The notations
$\boxdot_{\gamma}$ (for $\gamma\in\Gamma$) and $\heartsuit_{\lambda}$
(for $\lambda\in\Lambda$) are to distinguish propositional connectives
(the former) from modalities (the latter).
Examples of predicate lifting-based coalgebraic logics abound.
* Standard (normal) modal logic is obtained by taking
$F=\pow(\AP)\times\pow(\place)$ (with $\pow$ the covariant
powerset functor and $\AP$ a set of atomic propositions), $\Omega
= \Bool$, $\Gamma=\{\ttrue,\ffalse,\land,\lor\}$ with the usual
interpretations, and $\Lambda=\AP\cup\{\Box,\Diamond\}$ with
\begin{align*}
&\sem{p}_{X}\colon 1\to \Bool^{FX},~~ *\mapsto \bigl\{\,(U,Y) \mid
p \in U\,\bigr\}\;\text{(where $p\in \AP$),}
\\
&\sem{\Box}_{X}\colon \Bool^{X}\to \Bool^{FX},~~
(V\subseteq X)\mapsto \bigl\{\,(U,Y)
\,\bigl|\bigr.\, Y \subseteq V \,\bigr\},
\\
&\sem{\Diamond}_{X}\colon \Bool^{X}\to \Bool^{FX},~~
(V\subseteq X)\mapsto \bigl\{\,(U,Y)
\,\bigl|\bigr.\, Y \cap V \ne \emptyset \,\bigr\}.
\end{align*}
Atomic propositions are thus identified with $0$-ary modalities, as is standard in coalgebraic modal logic (see e.g. <cit.>).
* Hennessy-Milner logic is obtained by taking $F = (\pow(\place))^A$ (with $A$ a set of labels), $\Omega$ and $\Gamma$ as before, and $\Lambda = \{[a],\langle a \rangle\}$ with associated predicate liftings
\begin{align*}
&\sem{[a]}_X : \Bool^X \to \Bool^{F X},~~
%(Y \subseteq X)
Y\mapsto \{f \in \pow(X)^A \mid f(a) \subseteq Y\},
\\
& \sem{\langle a\rangle}_X : \Bool^X \to \Bool^{F X},~~
% \\
% &\qquad\qquad\qquad
% (Y \subseteq X)
Y\mapsto \{f \in \pow(X)^A \mid f(a) \cap Y \ne \emptyset \}.
\end{align*}
* Monotone neighborhood logic <cit.> is
obtained by taking $F X = \{ Y \in \pow (\pow (X)) \mid Y
\mbox{ is upward-closed} \}$, $\Omega$ and $\Gamma$ as before and
$\Lambda = \{\Box \}$, with an associated predicate lifting
\begin{align*}
&\sem{\Box} : \Bool^X \to \Bool^{ FX},~~(U \subseteq X) \mapsto \{ Y \in \pow(\pow (X)) \mid U \in Y\}.
\end{align*}
* Graded modal logic <cit.> is obtained by taking $F X = (\omega + 1)^X$, $\Omega$ and $\Gamma$ as before, and $\Lambda$ consisting of graded modalities $\Box_k$ ("for all but $k$ successors") and $\Diamond_k$ ("for more than $k$ successors") for $k \in \omega$, with associated predicate liftings
\begin{align*}
&\sem{\Box_k} : \Bool^X \to \Bool^{ FX},~~
%(Y \subseteq X)
Y\mapsto \{f \in F X \mid\textstyle \sum_{x \notin Y} f(x) \le k\},\\
&\sem{\Diamond_k} : \Bool^X \to \Bool^{ FX} ,~~
%(Y \subseteq X)
Y\mapsto \{f \in F X \mid \textstyle\sum_{x \in Y} f(x) > k\}.
\end{align*}
* Our approach also covers the coalition logic <cit.>, interpreted over game frames—these
are coalgebras of the (class-valued) functor
\begin{align*}
F X = \{(S_1,\ldots,S_N,f) \mid \emptyset \ne S_i \in \Sets, f : \prod_{i \in N}S_i \to X\}
\end{align*}
with $N$ a set of agents, tuples $(S_1,\ldots,S_N)$ capturing agent strategies, and functions $f : \prod_{i \in N}S_i \to X$ modeling the outcomes of strategy choices for the agents. The modalities $[C]$, with $C \subseteq N$ a coalition, arise from predicate liftings $\sem{[C]}_X : \Bool^X \to \Bool^{F X}$ given by
$\sem{[C]}_X(Y) = \{ (S_1,\ldots,S_N,f) \in F X \mid \exists \sigma_C \in S_C.\forall \sigma_{\overline{C}} \in S_{\overline{C}}. f(\sigma_C,\sigma_{\overline{C}}) \in Y\}$,
where $S_C = \prod_{i \in C}S_i$, $\overline{C} = N \setminus C$, and
$(\sigma_C,\sigma_{\overline{C}})$ is defined as expected.
* Here is an example that would yield the usual “linear-time
logic” like LTL (i.e. formulas are interpreted over infinite words),
in the setting of <ref>.
$F=\pow(\AP)\times(\place)$, $\Omega = \Bool$, $\Gamma=\{\ttrue,
\ffalse, \land,\lor\}$ and $\Lambda=\AP\cup\{\X\}$, with the
predicate liftings
p_X1→^FX, *↦{ (U,x)∈()×X |. p∈U },
(V⊆X)↦{ (U,x)
|. x∈V }.
In addition to the above $\Bool$-valued logics—that are also accounted
for by other coalgebraic approaches to modal logic (see
e.g. <cit.>)—our approach additionally covers
many-valued logics. For example, the Łukasiewicz logic of
<cit.> can be recovered by taking $F = \pow \dist$, where
$\dist : \Sets \to \Sets$ is the probability distribution
defined by $\dist X = \{\mu : X \to \unitInt \mid \sum_{x} \mu(x) = 1\}$, $\Omega = \unitInt$, $\Gamma = \{\sqcup,\oplus\}$ and $\Lambda = \{ \Diamond \}$, with interpretations $\sem{\sqcup}, \sem{\oplus}: \unitInt \times \unitInt \to \unitInt$ given by
\begin{align*}
& \sem{\sqcup}_X(p,q) = \max(p,q), &
& \sem{\oplus}_X(p,q) = \min(1,p+q)
\end{align*}
and predicate lifting $\sem{\Diamond}_X : \unitInt^X \to \unitInt^{FX}$ given by
\begin{align*}
& \sem{\Diamond}_X(p)(f) = \max_{\mu \in f}\sum_{x}\mu(x) p(x).
\end{align*}
Another many-valued logic that is covered by our framework is
logics with future discounting <cit.>.
A basic fragment is given as follows:
$F=\pow(\AP)\times(\place)$, $\Omega = [0,1]$, $\Gamma=\{\ttrue,
\ffalse, \land,\lor\}$ and $\Lambda=\AP\cup\{\X\}$, with the
predicate liftings
p_X(*)(U,x) =
1 if $p\in U$
0 otherwise,
d↦[ (U,x)↦1/2·d(x) ].
Note the factor $\frac{1}{2}$ that discounts the value of truth in the next step.
§.§ Equational Presentation
In this paper we
favor working with equational
presentations of $\mu$-calculus formulas.
Furthermore, for simplicity,
we shall present $\mu$-calculus formulas as simple equational
systems, meaning that each right-hand side is of depth at most $1$.
[simple $\CmuGL$-equational system]
A simple $\CmuGL$-equational system
is an expression of the form
\begin{equation}\label{eq:CmuGLEqSys}
\dotsc,\quad
u_{m}=_{\eta_{m}} \varphi_{m}
\end{equation}
where: $\eta_{i}\in\{\mu,\nu\}$;
$\seq{u}{m}\in\Var$ are fixed-point variables; and
$\seq{\varphi}{m}$ are simple $\CmuGL$-formulas
of the form
u_i, ⊡_γ(u_i_1,…,u_i_|γ|)
or _λ(u_i_1,…,u_i_|λ|).
We make a further requirement that, in case $\eta_{i}=\mu$,
the corresponding equation is of the form $u_{i}=_{\mu} u_{j}$ for
some $j\in [1,m]$. This inessential requirement simplifies our
subsequent exposition.
A simple $\CmuGL$-equational system (<ref>) is
closed if
all the variables that occur in $\seq{\varphi}{m}$
are among $u_{1},\dotsc,u_{m}$.
Note that, much like in <ref>,
the order of equations in (<ref>) matters—the equations
are solved from left to right, i.e. priorities increases as one goes from left
to right.
Translation of $\mu$-calculus formulas into equational systems is
standard; so is translation in the other direction.
For each $\CmuGL$-formula $\varphi$, its equational presentation
$E_{\varphi}$ is defined by the following
induction. Here $u_{\varphi}\in\Var$ denotes the variable on the
left-hand side of the last equation
in the equational system $E_{\varphi}$,
$E_{1};\dotsc;E_{k}$ denotes the concatenation of equational systems
$E_{1},\dotsc, E_{k}$,
and the variable $v$ in each
clause is chosen to be a fresh one.
\begin{align*}
\;&:=\; \bigl(\,v=_{\nu} u\,\bigr)
\\
\;&:=\;
\bigl(\,
v=_{\nu} \boxdot_{\gamma}(u_{\varphi_{1}},\dotsc, u_{\varphi_{n}})
\,\bigr)
\\
\;&:=\;
\bigl(\,
v=_{\nu} \heartsuit_{\lambda}(u_{\varphi_{1}},\dotsc, u_{\varphi_{n}})
\,\bigr)
\\
E_{\eta u.\,\varphi}
\;&:=\;
\bigl(\,
u =_{\eta} u_{\varphi}
\,\bigr)
\qquad\text{where $\eta\in\{\mu,\nu\}$}
\end{align*}
The choice of $=_{\nu}$ in the first three clauses is arbitrary from
the semantical viewpoint:
changing it into $=_{\mu}$ yields the same semantics. It is however
beneficial from the algorithmic and presentational viewpoints—in
particular the resulting system enjoys the
requirement in Def. <ref> (that a $\mu$-equation
is of the form $u_{i}=_{\mu}u_{j}$).
It is straightforward to see that a closed formula $\varphi$ yields a
closed equational system $E_{\varphi}$.
Conversely, given a simple $\CmuGL$-equational system $E$ like
in (<ref>), we define its formulaic presentation
$\varphi_{E}$ by induction on the number $m$ of equations. If $m=1$
then an equation $u_{1}=_{\eta_{1}}\varphi_{1}$ becomes the formula
$\eta_{1}u_{1}.\, \varphi_{1}$. For the step case,
let $E'$ be obtained by dropping the first equation, that is,
u_2=_η_2φ_2, …, u_m=_η_m φ_m
Then we define $\varphi_{E}$ to be the result of replacing $u_{1}$ in
$\varphi_{E'}$ with $\eta_{1}u_{1}.\, \varphi_{1}$. That is,
φ_E'[ η_1u_1.φ_1 / u_1 ].
The two translations are mutually inverse—not necessarily
syntactically, but the semantics is preserved. See
Prop. <ref> later. Therefore, in what
we do not distinguish a $\CmuGL$-formula $\varphi$
and its equational presentation $E_{\varphi}$. Both will be denoted by
Let $\Gamma=\{\land,\lor\}$ and $\Lambda=\AP\cup\{\X\}$ (from
Example <ref>). The $\CmuGL$-formula
νu. μv. ((p v)u)
gets translated into the simple $\CmuGL$-equational system
\begin{multline*}
u_{1}=_{\nu} p,
\;\;
u_{2}=_{\nu} v,
\;\;
u_{3}=_{\nu} \X u_{2},
\;\;
u_{4}=_{\nu} u_{1}\lor u_{3},
\;\;
\\
u_{5}=_{\nu} u,
\;\;
u_{6}=_{\nu} \X u_{5},
\;\;
u_{7}=_{\nu} u_{4}\land u_{6},
\;\;
v=_{\mu} u_{7},
\;\;
u=_{\nu} v,
\end{multline*}
Def. <ref>.
The translation in the other direction
gives rise to a complicated formula which, however,
is easily seen to be equivalent to the original formula
under (obviously sound) simplifications like
νu_1. p
into $p$.
§.§ $\Cmu_{\Gamma,\Lambda}$: Semantics
Formulas of $\Cmu_{\Gamma,\Lambda}$ are interpreted over
$F$-coalgebras (see <ref>). The following
inductive interpretation is a standard one; it follows the tradition of
coalgebraic modal
logic <cit.>
as well as that of fixed-point logics <cit.>.
[semantics of $\Cmu_{\Gamma,\Lambda}$ formulas]
Let $\Gamma$ and $\Lambda$ be propositional and modal signatures in
Def. <ref>, and
Let $c\colon X\to FX$ be a coalgebra. A formula
$\varphi$ of $\CmuGL$—with free variables $\seq{u}{m}$—is
assigned its denotation over $c$; it is given by a function
that is defined inductively in the following way. Here $\vec{V}$ is short for
$V_{1},\dotsc, V_{m}$, where $V_{i}\colon X\to \Omega$.
\begin{align*}
\sem{u_{i}}_{c}
&:= V_{i}(x),
\\
\sem{
\boxdot_{\gamma}\bigl(\seq{\varphi}{n}\bigr)}_{c}
\\
\gamma
\bigl(\,
\,\dotsc,\,
\sem{\varphi_{n}}_{c}(\vec{V})(x)\,\bigr),
\\
\sem{
\heartsuit_{\lambda}\bigl(\seq{\varphi}{n}\bigr)}_{c}
&:= \\
\Bigl(\,\lambda_{X}
\bigl(\,
\sem{\varphi_{n}}_{c}(\vec{V})\,\bigr)
\,\Bigr)\bigl(c(x)\bigr),
\\
\sem{
\mu u.\, \varphi
\bigl(\,\mu\bigl(\,\sem{ \varphi}_{c}(\vec{V}, \place)\colon
\Omega^{X}\to \Omega^{X}\,\bigr)\,\bigr)(x),
\\
\sem{
\nu u.\, \varphi
\bigl(\,\nu\bigl(\,\sem{ \varphi}_{c}(\vec{V}, \place)\colon
\Omega^{X}\to \Omega^{X}\,\bigr)\,\bigr)(x).
\end{align*}
Recall that $\gamma\colon \Omega^{n}\to\Omega$ and
$\lambda_{X}\colon (\Omega^{X})^{n}\to\Omega^{FX}$
are assumed to be given (Def. <ref>).
In the last two clauses it is assumed, by suitably rearranging
that the bound variable $u$
is the last one $u_{m}$ among the free variables $\seq{u}{m}$ of
$\varphi$. The necessary fixed points of the function
$\sem{ \varphi}_{c}(\vec{V}, \place)\colon
\Omega^{X}\to \Omega^{X}$ are guaranteed by the Knaster-Tarski
theorem, since $\Omega$ (and hence $\Omega^{X}$) is a complete lattice
and the function $\sem{ \varphi}_{c}(\vec{V}, \place)$ is easily seen to
be monotone.
Let $f$ be a coalgebra homomorphism
from $c\colon X\to FX$ to $d\colon Y\to FY$, as in (<ref>).
For each closed $\CmuGL$-formula $\varphi$ and each $x\in X$, we have
As discussed in <ref> we favor working with
equational presentation of formulas. We shall therefore define their
semantics, too.
[semantics of simple
$\Cmu_{\Gamma,\Lambda}$-equational systems]
Let $E$ be
a simple
$\Cmu_{\Gamma,\Lambda}$-equational system
\begin{equation}\label{eq:201506191457}
\dotsc,\;
u_{m}=_{\eta_{m}} \varphi_{m}
\end{equation}
from Def. <ref>; assume that it is closed.
Let $c\colon X\to FX$ be an $F$-coalgebra.
Then $E$ and $c$ together induce an equational system $E_{c}$
(in the sense of Def. <ref>)
over the complete lattice $L=\Omega^{X}$—this is by identifying
a simple formula $\varphi_{i}$ on a right-hand side with
the function $\sem{\varphi_{i}}_{c}\colon (\Omega^{X})^{m}\to \Omega^{X}$
defined in Def. <ref>.
Finally, solving $E_{c}$ as in Def. <ref> yields a
solution $(
\dotsc,
)$ that is an element of $(\Omega^{X})^{m}$. The last component
$l^{\sol}_{m}$ is referred to as the semantics
of the simple
$\Cmu_{\Gamma,\Lambda}$-equational system $E$ over the coalgebra $c$.
The two semantics—the direct one, and the one via equational
presentation—coincide, as expected.
Let $\varphi$ be a closed $\CmuGL$-formula, and $c\colon X\to FX$
be a coalgebra. Consider its equational presentation $E_{\varphi}$
(a simple
$\Cmu_{\Gamma,\Lambda}$-equational system,
Def. <ref>).
Then the semantics of $E_{\varphi}$ over $c$—in the sense of
Def. <ref>, i.e. the solution of
the equational system $E_{\varphi,c}$ over $\Omega^{X}$—coincides with
$\sem{\varphi}_{c}$ from Def. <ref>.
Straightforward by induction.
§ $\CMU_{\GAMMA,\LAMBDA}$ MODEL CHECKING AGAINST $F$-COALGEBRAS
Let us turn to the model-checking problem of the modal logic
$\CmuGL$ against $F$-coalgebras. Later
in <ref> we study
model checking against coalgebras with additional nondeterministic
branching—i.e. there the logic $\CmuGL$ is thought of as a
“linear-time” logic. In contrast, here $\CmuGL$ is
a “branching-time” logic, in the sense that there is no additional branching to
be abstracted away.
Prop. <ref>, together
with Thm. <ref>, already
gives us a characterization of the semantics $\sem{\varphi}_{c}$ in
terms of progress measures. In this section
we shall rephrase it to yet another form, called MC progress
measure, that is easier to manipulate. Using it we present our main
technical results, namely a generic model-checking algorithm
(Algorithm <ref>) and its complexity (Thm. <ref>).
The following correspondence for (polyadic) modalities—that is not unlike in the Yoneda lemma—will
be used in the following developments.
Let $\lambda$ be a natural transformation,
given by arrows
$\lambda_{X}\colon (\Omega^{X})^{n}\to
\Omega^{FX}$ that are natural in $X$. (This is the setting in
Def. <ref>, where $\lambda$ is an $n$-ary
modality). Let $m\in \omega$ and $j_{1},\dotsc, j_{n}\in [1,m]$.
These data induce an arrow
λ^j_1,…, j_nF(Ω^m)→Ω
j_{n}}}:=\lambda_{\Omega^{m}}(\pi_{j_{1}},\dotsc,\pi_{j_{n}})$. Recall
$\lambda_{\Omega^{m}}$ is of type $(\Omega^{\Omega^{m}})^{n}\to
\Omega^{F(\Omega^{m})}$, and $\pi_{j}\colon \Omega^{m}\to \Omega$.
Moreover, let us define $\tilde{\lambda}\colon F(\Omega^{n})\to \Omega$
by $\tilde{\lambda}:=\lambda^{\tuple{1,2,\dotsc,n}}=\lambda_{\Omega^{n}}(\pi_{1},\dotsc,\pi_{n})$,
where $\pi_{1},\dotsc,\pi_{n}\colon \Omega^{n}\to \Omega$. Then we have
[rd]^- λ^j_1,…,j_n
§.§ MC Progress Measure
We start with customizing the lattice-theoretic notion of progress
measure (Def. <ref>) to one that is
tailored to $\CmuGL$ model checking. For reuse in
later sections, the definition is separated into the
transition-irrelevant part (which we call pre-progress measure),
and the full definition.
[pre-progress measure, pPM]
Let $F\colon \Sets\to\Sets$ be a functor.
Let $\varphi$ be a
$\CmuGL$-formula—where $\Lambda$ is a modal signature over
$F$—that is identified with a simple equational system
u_m=_η_m φ_m
in <ref>.
Let $i_{1}<\cdots
enumerate the indices of all the $\mu$-variables.
A pre-progress measure (pPM) $p$ for $\varphi$
is given by a tuple
p =
( _i(α_1,…,α_k) )_i,αk
consists of:
* the maximum prioritized ordinal
$(\overline{\alpha_{1}},\dotsc, \overline{\alpha_{k}})$; and
* the approximants $\approximant_{i}(\alpha_{1},\dotsc,\alpha_{k})\in \Omega$, defined for
$i\in[1,m]$ and each
prioritized ordinal
such that
\alpha_{1}\le\overline{\alpha_{1}},\dotsc,
\alpha_{k}\le\overline{\alpha_{k}}
The approximants $\approximant_{i}(\alpha_{1},\dotsc,\alpha_{k})$
are subject to:
Let $i\in[1,m]$ (hence $u_{i}$ is either a $\mu$- or
$\nu$-variable). Then
($\mu$-variables, base case)
Let $a\in [1,k]$. Then $\alpha_{a}=0$ implies
$p_{i_{a}}(\alpha_{1},\dotsc,\alpha_{k}) =\bot$, or
there exists a
ordinal $(\alpha'_{1},\dotsc,\alpha'_{k})$ such that
(\alpha_{1},\dotsc,\alpha_{k})$ and
$p_{i_{a}}(\alpha_{1},\dotsc,\alpha_{k}) \sqsubseteq
(Note here that this condition mirrors Cond. 2' of
Def. <ref>, rather than
Cond. 2 of
Def. <ref>. Prop. <ref>
justifies doing so.)
($\mu$-variables, step case)
Let $a\in [1,k]$, and let
\alpha_{a}+1,\dotsc,\alpha_{k})$ be
a prioritized ordinal
such that its $a$-th counter
$\alpha_{a}+1$ is a successor ordinal.
the approximant $\approximant_{i_{a}} (\alpha_{1},\dotsc,
\alpha_{a}+1,\dotsc,\alpha_{k})$. Since $u_{i_{a}}$ is a
$\mu$-variable, by a requirement in
Def. <ref> the corresponding equation
is of the form $u_{i_{a}}=_{\mu} u_{i'}$ for some $i'\in
We require that
there exist ordinals
$\beta_{1},\dotsc,\beta_{a-1}$ such that
\begin{multline*}
\approximant_{i_{a}} (\alpha_{1},\dotsc,
\alpha_{a}+1,\dotsc,\alpha_{k})
\\
\sqsubseteq
\approximant_{i'} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k})
\end{multline*}
\beta_{a-1}\le\overline{\alpha_{a-1}}$.
($\mu$-variables, limit case)
Let $a\in [1,k]$, and
\alpha_{k})$
be a prioritized ordinal
such that its $a$-th counter
is a limit ordinal.
We require
_i_a (α_1,…,
_i_a (α_1,…,
Let $i\in [1,m]\setminus\{i_{1},\dotsc, i_{k}\}$ (i.e. $u_{i}$ is a $\nu$-variable in the
system (<ref>)); let $a\in [1,k]$ such that
i_1<⋯<i_a-1<i<i_a<⋯< i_k.
Let $(\alpha_{1},\dotsc,\alpha_{k})$ be a prioritized ordinal.
We require the following on the approximant
$\approximant_{i} (\alpha_{1},\dotsc,\alpha_{k})$:
* (RHS is a variable) If the formula
$\varphi_{i}$ in the
$i$-th equation $u_{i}=_{\nu}\varphi_{i}$
is a variable $u_{i'}$ (for some $i'\in [1,m]$), then
there exist ordinals
$\beta_{1},\dotsc,\beta_{a-1}$ such that
\begin{multline*}
\approximant_{i_{a}} (\alpha_{1},\dotsc,
\alpha_{a},\dotsc,\alpha_{k})
\\
\sqsubseteq
\approximant_{i'} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k})
\end{multline*}
\beta_{a-1}\le\overline{\alpha_{a-1}}$.
* (RHS is a propositional formula) If the
formula $\varphi_{i}$ is a propositional formula
there exist ordinals
$\beta_{1},\dotsc,\beta_{a-1}$ such that
\begin{align*}
&\approximant_{i} (\alpha_{1},\dotsc,
\alpha_{a},\dotsc,\alpha_{k})
\\
\sem{\gamma}
\left(\,
\begin{array}{c}
\approximant_{j_{1}} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k}),
\\
\dotsc,
\\
\approximant_{j_{n}} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k})
\end{array}
\,\right)
\end{align*}
\beta_{a-1}\le\overline{\alpha_{a-1}}$.
Let $\alpha$ be an ordinal.
The collection of all pre-progress measures for a formula $\varphi$,
whose maximum prioritized ordinal $(
\overline{\alpha_{1}},\dotsc,
\overline{\alpha_{k}})
$ satisfies $\overline{\alpha_{i}}= \alpha$ for each $i\in [1,k]$,
shall be
denoted by $\pPM_{\varphi,\alpha}$.
Recall that $\Omega$ is the complete lattice of truth values. In the
definition of $\pPM_{\varphi,\alpha}$, the explicit bound by $\alpha$ is
there so that the collection $\pPM_{\varphi,\alpha}$ is a (small) set.
Comparing the previous definition with Def. <ref>
of progress measures, what are missing here are the treatment
of modal formulas $\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})$
in Cond.
<ref>—this is precisely the
case where the transition structure of the coalgebra in question
becomes relevant. In the current setting of $\CmuGL$ as a “branching-time”
logic, this case is taken care of in the following way. MC stands for
“model checking.”
[MC progress measure]
Assume the setting of Def. <ref>, and let
$c\colon X\to FX$ be a coalgebra in $\Sets$. An MC progress
measure for $\varphi$ over $c$
is given by:
* some ordinal $\alpha$, called the maximum ordinal, and
* a function $Q\colon X\to \pPM_{\varphi,\alpha}$,
that are subject to the following condition.
($\nu$-variables, RHS is a modal formula)
Let $x\in X$ and $p:=Q(x)$ be a pre-progress
measure for $\varphi$. Let $i\in [1,m]$ and
assume the setting of
Cond. <ref> of
Def. <ref> (i.e. $u_{i}$ is a $\nu$-variable),
and further that the formula $\varphi_{i}$ is a modal
Now consider
the approximant $p_{i}(\alpha_{1},\dotsc,
\alpha_{a},\dotsc,\alpha_{k})\in \Omega$ of $p$.
We require there exist ordinals
$\beta_{1},\dotsc,\beta_{a-1}$ such that
\begin{equation}\label{eq:10260954}
\begin{aligned}
\alpha_{a},\dotsc,\alpha_{k})\sqsubseteq
\\&
\PT_{\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})}
\alpha_{a},\dotsc, \alpha_{k})\bigl((FQ\co c)(x)\bigr),
\end{aligned}
\end{equation}
and $\beta_{1}\le\alpha,\dotsc,
\beta_{a-1}\le\alpha$.
Note that
$(FQ\co c)(x)\in F(\pPM_{\varphi,\alpha})$ since
$X\stackrel{c}{\to} FX\stackrel{FQ}{\to} F(\pPMpa)$.
For each
$(\alpha'_{1},\dotsc,\alpha'_{k})$, the function
\begin{align*}
% & \PT_{\varphi_{i_{a}}}(\alpha'_{1},\dotsc,\alpha'_{k})(q)
% \\&=
\PT_{\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})}
\;\colon\; F(\pPM_{\varphi,\alpha})\to \Omega
\end{align*}
in (<ref>) is defined as follows.
(The name $\PT$ comes from “predicate
\begin{equation}\label{eq:branchingTimePredTransf}
\begin{aligned}
\\
\Bigl[
\stackrel{F(\ev(\alpha'_{1},\dotsc,\alpha'_{k}))}{\longrightarrow}
\stackrel{\lambda^{\tuple{j_{1},\dotsc, j_{n}}}}{\longrightarrow}
\Omega
\Bigr],
\end{aligned}
\end{equation}
where $\lambda^{\tuple{j_{1},\dotsc, j_{n}}}$ is from
Lem. <ref>, and the
is defined by “fixing a prioritized
ordinal,” that is,
) ∈Ω^m.
The composite in the definition of
\overrightarrow{\alpha'})$
in (<ref>)
might seem exotic,
but the definition here is in fact a straightforward adaptation of the common
interpretation of modal formulas in coalgebraic logics. Recall the interpretation of a modal formula
$\heartsuit_{\lambda}(\varphi_{1},\dotsc,\varphi_{n})$ in
Def. <ref>, that is also the standard one in the literature
(see e.g. <cit.>). Then it is not hard—by naturality of
$\lambda$, much like in the proof of
Thm. <ref>—that this
standard definition of
$\sem{\heartsuit_{\lambda}(\vec{\varphi})}_{c}$ is equivalent to the following,
$\sem{\varphi_{i}}_{c}\colon X\to \Omega$ are the interpretations of the
subformulas (for $i\in[1,n]$), and $\tilde{\lambda}$ is from
Lem. <ref>.
Xc→ FX
Ω ).
This indeed resembles the right-hand side of (<ref>),
Ω )(x).
and, since
$\lambda^{\tuple{1,\dotsc,n}}=\lambda_{\Omega^{n}}(\pi_{1},\dotsc,\pi_{n})$ (Lem. <ref>),
( FX
FΩ )
[correctness of MC progress measure]
Assume the setting of Def. <ref>. In
particular, the formula $\varphi$ is translated to an equational
system with $m$ variables.
Let $Q$ be an MC
progress measure (with the maximum ordinal $\alpha$), $x\in X$ and $p:=Q(x)$.
where $\sem{\varphi}_{c}\colon X\to \Omega$
is from Def. <ref>.
There exists an MC progress measure $Q$
that achieves
the optimal. That is, an MC progress measure $Q$ such that
=\sem{\varphi}_{c}(x)$ for each $x\in X$.
Moreover, $Q$ can be chosen so that its maximum ordinal $\alpha$
is $\alpha=\ascCL(\Omega^{X})$, where $\ascCL(\Omega^{X})$ is the
length of the longest strictly ascending chain in $\Omega^{X}$
(see Thm. <ref>.2).
§.§ Algorithms
Here we shall further translate the notion of MC progress measure
(Def. <ref>) to a Jurdzinski-style
presentation; the latter shall be called a matrix progress
measure. The correspondence is an extension of the one in
Appendix <ref>;
(in the extended version <cit.>);
see also Rem. <ref>. We shall then devise a
model-checking algorithm based on matrix progress measures. Thanks to
the concrete presentation with matrices, we believe its implementation
is a fairly straightforward task.
Throughout <ref> we focus on the Boolean
setting (i.e. $\Omega=\Bool$), and restrict the state space $X$ of the
coalgebra $c\colon X\to FX$ (as a system model) to be finite. This is a
reasonable assumption because we aim at a concrete algorithm.
In view of Thm. <ref>.2, in
employing the theoretical machinery developed so far, all the ordinals
that occur can be restricted to finite (since $\ascCL(\Bool^{X})=|X|$
is finite).
Furthermore, we restrict the propositional signature $\Gamma$ to
$\Gamma_{n}:=\{\bigwedge_{n}, \bigvee_{n}\}$, where $\bigwedge_{n}$ and
$\bigvee_{n}$ are the $n$-ary conjunction and disjunction operators with
obvious interpretations. This signature of $\Gamma$ is functionally
in the current monotonic Boolean setting: any other propositional
connective $\gamma\colon \Bool^{n}\to \Bool$ can be encoded by
\begin{align*}\small
% &\gamma(l_{1},\dotsc,l_{n})
% =
% \\
\begin{array}{l}
\textstyle\bigvee
\bigl\{
\textstyle\bigwedge \{l_{i_{1}},\dotsc,l_{i_{k}}\}
\,\bigl|\bigr.\,
l_{i_{1}}=\cdots =l_{i_{k}}=\ttrue \,\Rightarrow\,\gamma(l_{1},\dotsc,l_{n})=\ttrue
\,\bigr\}.
\end{array}
\end{align*}
[prioritized ordinal matrix, POM]
Assume the setting of Def. <ref>.
A prioritized ordinal matrix is an $m\times k$ matrix
α^(1)_1 ⋯
⋮ ⋱ ⋮
α^(m)_1 ⋯
where each entry $\alpha^{(i)}_{a}$ is either
* an ordinal, or
* the symbol $\NoGood$ for “failure.”
It is required that, if any entry $\alpha^{(i)}_{a}$ is $\NoGood$
then all entries on the same row is $\NoGood$, that is,
The set of all POMs, such that all the ordinals therein are no
bigger than $\alpha$, is denoted by $\POM_{\alpha}$.
3 5
1 0
* 4
* *
A POM is therefore an $m$-tuple of prioritized ordinals, where some
prioritized ordinals can be replaced by $\NoGood$.
Its $i$-th row will be a prioritized ordinal for
the $i$-th variable $u_{i}$.
In view of the monotonicity conditions (in
Def. <ref> and <ref>) and the
definition of $\preceq_{i}$ (Def. <ref>), we
can see that some first elements in a row (precisely: those which
correspond to $\mu$-variables with a smaller priority than $u_{i}$)
do not make any difference. Such entries can safely be denoted by $*$
(“arbitrary”). An example is shown in the above: it is a POM
for an equational system with 5 variables, in which
$u_{1}, u_{3}, u_{4}$ are $\mu$-variables and $u_{2}, u_{5}$ are
$\nu$-variables. We shall however restrict use of $*$ for providing
intuitions; it does not appear in the technical
In the current section (<ref>) where $X$ is
assumed to be a finite set, it is not needed to allow any ordinal as an
entry of a POM (Def. <ref>). Natural numbers
will just suffice.
[matrix progress measure, MPM]
Assume the setting of Def. <ref>. A
matrix progress measure (MPM) for $\varphi$ over $c$, with a
maximum ordinal $\alpha$, is a function $R\colon X\to \POM_{\alpha}$
that satisfies the following conditions. Let $x\in X$ be arbitrary,
and consider $R(x)\in \POM_{\alpha}$.
2. ($\mu$-variables, base case)
Let $a\in [1,k]$ and consider the corresponding
$\mu$-variable $u_{i_{a}}$.
Assume $\alpha^{(i_{a})}_{1}\neq\NoGood.$
Then we must have
$(R(x))^{(i_{a})}\succ_{i_{a}}(0,0,\dotsc,0)$. Note
that the $i_{a}$-th row $(R(x))^{(i_{a})}$ of $R(x)$ is
a prioritized ordinal, and recall $\succ_{i_{a}}$ from
Def. <ref>. Note also that the required
inequality is strict.
(That is, a row in $R(x)$ that corresponds to a $\mu$-variable must
not be $(*,\dotsc,*,0,\dotsc,0)$.)
3. ($\mu$-variables, step case)
Let $a\in [1,k]$ and consider the corresponding
$\mu$-variable $u_{i_{a}}$.
Assume $\alpha^{(i_{a})}_{1}\neq\NoGood$.
Let $u_{i_{a}}=_{\mu}
(where $i'\in [1,m]$) be the corresponding equation in
$\varphi$. If $i'\le i_{a}$, then we must have
$(R(x))^{(i_{a})}\succ_{i}(R(x))^{(i')}$. Note the
inequality is strict.
($\mu$-variables, limit case)
Let $a\in [1,k]$, and consider the corresponding
$\mu$-variable $u_{i_{a}}$. Then
$(R(x))^{(i_{a})}_{a}$ must not be a limit ordinal.
(This condition is vacuous when $\alpha$ is finite.)
$i\in [1,m] \setminus\{i_{1},\dotsc, i_{k}\}$ (i.e. $u_{i}$ is a $\nu$-variable). Assume that $\alpha^{(i)}_{1}\neq\NoGood$.
Let $u_{i}=_{\nu}\varphi_{i}$ be the
equation in $\varphi$.
* (RHS is a variable)
If the formula
is a variable $u_{i'}$ (for some $i'\in [1,m]$).
Then $(R(x))^{(i)}\succeq_{i}(R(x))^{(i')}$.
* (RHS is a propositional formula) Recall
that we have restricted propositional connectives
to $\bigwedge$ and $\bigvee$
(Assumption <ref>).
If $\varphi_{i}=\bigwedge(u_{i_{1}},\dotsc,
then we require all of
\begin{equation}\label{eq:201507081932}
\dotsc,
\end{equation}
to hold.
If $\varphi_{i}=\bigvee(u_{i_{1}},\dotsc,
then we require at least one
of (<ref>) to hold.
* (RHS is a modal formula)
Assume that the formula
$\varphi_{i}$ is a modal formula
$\varphi_{i}=\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})$. Let
Consider the following composite $h\colon X\to \Bool$:
\begin{equation}\label{eq:201507082123}
\begin{aligned}
\biggl(\,
\begin{array}{r}
X\stackrel{c}{\rightarrow} FX\stackrel{FR}{\rightarrow}F(\POM_{\alpha})
\stackrel{F(\ev'(\vec{\alpha}))}{\longrightarrow}\quad
\\
\stackrel{\lambda^{\tuple{j_{1},\dotsc, j_{n}}}}{\longrightarrow}
\Bool\,
\end{array}
\biggr),
\end{aligned}
\end{equation}
where $\ev'(\vec{\alpha})\colon \POM_{\alpha}\to
\Bool^{m}$
is defined by
\begin{align*}
\Bigl(\ev'(\vec{\alpha})\bigl(\,(\beta^{(i)}_{j})_{i,j}\,\bigr)\Bigr)_{i'}=\ttrue
\,\defiff\,
\vec{\alpha}\succeq_{i'} (\beta^{(i')}_{1},\dotsc,
\beta^{(i')}_{k})
\end{align*}
for each $i'\in [1,m]$. We require that
Again, much like for Cond. 5(c) in
Def. <ref>, the composite in (<ref>) is
understood as an analogue of the usual interpretation of modal
formulas in coalgebraic logics (cf. Def. <ref>).
[correctness of MPM]
Assume the setting of Def. <ref>.
* (Soundness)
If there exists an MPM
$R\colon X\to \POM_{\alpha}$ such that
$(R(x))^{(m)}_{k}\neq\NoGood$, then
* (Completeness) There is an optimal MPM
$R_{0}\colon X\to \POM_{|X|}$ such that:
$\sem{\varphi}_{c}(x)=\ttrue$ if and only if
We follow <cit.> and present an algorithm that looks
for the optimal MPM. See Algorithm <ref>. There
we use the following functions.
An algorithm for $\CmuGL$ model checking, in the setting
of Def. <ref> and
Assumption <ref>.
Here $R(x,i)$ denotes the prioritized ordinal
$\bigl(R(x,i,1),\dotsc,R(x,i,k)\bigr)$. Note that on lines 16, 19
and 23, $u_{i}$ is necessarily a $\nu$-variable.
A $\CmuGL$-formula $\varphi$ presented as an equational system
u_m=_η_m φ_m
where $u_{i_{1}},\dotsc,u_{i_{k}}$ are $\mu$-variables, and a coalgebra $c\colon X\to FX$
$\sem{\varphi}_{c}\in \Bool^{X}$
each $x\in X$, $i\in[1,m]$ and $j\in [1,k]$
each $a\in [1,k]$
Cond. 2
the main loop
each $x\in X$ and $i\in [1,m]$
$u_{i}$ is a $\mu$-variable, $i=i_{a}$ and $\varphi_{i}=u_{i'}$
Cond. 3
$R(x,i):=\max_{\preceq_{i}} \bigl\{\,
cf. Def. <ref>
\bigl.
\bigl(
\dotsc,
\bigr)
\,\bigr\}
$u_{i}$ is a $\nu$-variable and $\varphi_{i}=u_{i'}$
Cond. 5(a)
$R(x,i):=\max_{\preceq_{i}} \bigl\{\,
Cond. 5(b), the $\bigwedge$-case
$R(x,i):=\max_{\preceq_{i}} \bigl\{\,
Cond. 5(b), the $\bigvee$-case
$R(x,i):=\max_{\preceq_{i}} \bigl\{\,
R(x,i), \,\bigr.\,\bigr.$
\min_{\preceq_{i}} \bigl\{\,\,R(x,j_{1}),\dotsc,R(x,j_{n})
\,\bigr\}\,\bigr\}$
Cond. 5(c)
$R(x,i):=\max_{\preceq_{i}} \bigl\{\,
\PTMPM_{i}(x)\,\bigr\}$
cf. Def. <ref>
each $j\in [1,k]$
$u_{i}$ has seen to be false at $x$
no change is made
$\{x\in X\mid
[$\max_{\preceq_{i}}, \min_{\preceq_{i}}$]
In Algorithm <ref>,
the function
$\max_{\preceq_{i}}$ takes a set of prioritized
ordinals (and possibly $(\NoGood,\dotsc,\NoGood)$) and returns a prioritized
ordinal such that: the first irrelevant entries (due to priorities
smaller than that of $u_{i}$) are set to $0$; and the
rest is the maximum (with the lexicographic order, the latter the
more significant) among the corresponding suffixes of the prioritized
ordinals given as input. In case
the input set contains
$(\NoGood,\dotsc,\NoGood)$, then the output is $(\NoGood,\dotsc,\NoGood)$ too.
For example, in the setting of
Example <ref>,
max_≼_3{(1,2,3), (3,4,1)}=(0,2,3)
where the first element of each sequence is irrelevant.
The function $\min_{\preceq_{i}}$ is defined similarly, by: truncating the
first irrelevant elements, choosing the smallest one in the
lexicographic order, and padding the missing elements with $0$. The
output is $(\NoGood,\dotsc,\NoGood)$ in case the input set
contains nothing other than $(\NoGood,\dotsc,\NoGood)$.
The functions $\max_{\preceq_{i}}$ and
$\min_{\preceq_{i}}$ can be efficiently implemented: if
the input is the set of $N$ prioritized ordinals then the time
complexity is $O(Nk)$.
In Algorithm <ref>,
the function $\PTMPM_{i}$ takes a state $x\in X$ and
\begin{equation}\label{eq:201507082125}
\begin{aligned}
\\
& \textstyle\min_{\preceq_{i}}
\bigl\{\,
\vec{\alpha}\in |X|^{k}
\,\bigl|\bigr.\,
\bigl(\,\lambda^{\tuple{j_{1},\dotsc,j_{n}}}\co F\bigl(\ev'(\vec{\alpha})\co
R\bigr)\co c\,\bigr)(x)=\ttrue
\,\bigr\}
\end{aligned}
\end{equation}
where the composite is from (<ref>) and
$R\colon X\to \POM_{\alpha}$ is given by the current values of
$\bigl(R(x,i,j)\bigr)_{x,i,j}$ in the algorithm.
The complexity of $\PTMPM_{i}$ depends greatly on the choice of a functor
$F$ and a predicate lifting $\lambda$. A uniform and brute-force
algorithm for $\PTMPM_{i}$ is possible, however, by enumerating
all $\vec{\alpha}\in |X|^{k}$ from the smaller ones with respect to
$\preceq_{i}$, and checking for each $\vec{\alpha}$ whether the
condition in (<ref>) is satisfied. The worst-case
is $O(km^{2}|X|^{k+1}+C|X|^{k})$ with some constant $C$, on the assumption that the value
( λ^j_1,…,j_nF('(α⃗))FRc )(x)
that appear in (<ref>)
is computed in time $O(km^{2}|X| +C)$.
The last assumption is derived as follows:
the computation of
$\ev'(\vec{\alpha})$ is in $O(km)$; hence
the computation of
$F(\ev'(\vec{\alpha}))$ is
in $O(km|X|)$; that of
$\lambda^{\tuple{j_{1},\dotsc,j_{n}}}$ is in $O(m)$ (exploiting Lem. <ref>); and the other
components like $c$ and application of $F$ have
only a constant contribution $C$ to the complexity.
Most $F$ and $\Lambda$ allow much better complexity of $\PTMPM_{i}$.
For example, the choice $F=\pow(\AP)\times(\place)$ and
$\lambda=\mathsf{X}$ (the next-time modality) in
Example <ref>.6 (that will yield a logic like LTL in <ref>),
the function
$\PTMPM_{i}$ picks up the prioritized ordinal $R(x',i)$ of the successor
$x$ and truncates its first irrelevant elements to $0$. This can be
done in time $O(k)$. More generally, often it is possible to
“propagate backwards” by computing
, for which a one-step complete set of deduction rules
can be used (see e.g. <cit.>). Such optimizations by
deduction rules are left as future work.
Algorithm <ref> indeed returns
(*** The following improvement turns out to be not necessary. ***)
The “naive” algorithm in Algorithm <ref> has an
advantage that it mirrors the conditions in Def. <ref>
and hence its correctness is straightforward. Its complexity is not
optimal, however, because the number of iterations of the main loop
(lines 7–32) is bounded only by the number of changes made, that
$|X|^{km|X|}$ (recall that $R(x,i,j)\le |X|$ for each $x\in X,i\in
[1,m],j\in [1,k]$). This complexity is worse than the ones
commonly appear in the model-checking literature (see e.g. <cit.>), where a
is exponential only in the alternation depth of
a formula $\varphi$. The latter is bounded by $k$ in our
current setting.
By further inspection of the proof of
Thm. <ref> we obtain an improved
algorithm. See Algorithm <ref>. The key is
to specify more explicitly the order of application of the operations in the main loop
(lines 8–31 of Algorithm <ref>) to different $x,
i$ and $j$. For this purpose we use a counter ($A$ in Algorithm <ref>).
The following complexity result is derived from an analysis of
Algorithm <ref>. Recall that it assumes a
brute-force algorithm for $\PTMPM_{i}$ (Def. <ref>); fixing
$F$ and $\Lambda$ will allow further
optimization. See Rem. <ref>.
It nevertheless achieves a complexity that is exponential only in $k$.
This is much like the most known complexity results for model-checking
(see e.g. <cit.>)—note that $k$ bounds the
alternation depth of a formula $\varphi$.
In the setting
of Def. <ref> and
Assumption <ref>, the
model-checking problem can be decided in time
\begin{equation}\label{eq:complexity}
\tag*{\myqed}
\end{equation}
A straightforward optimization is possible:
each iteration of the inner loop
(lines 8–32) tests all $(x,i)$; this is unnecessary.
Algorithm <ref> is presented as it is, however,
since the correspondence to Def. <ref> is clearer.
It should be possible also to improve the complexity so that it is
exponential to the alternation depth, instead of to the number $k$ of
$\mu$-operators, of the given formula $\varphi$.
§ SMALL-MODEL PROPERTY AND SATISFIABILITY, TRIAL 2 (THE MOST
PROMISING ONE)
After all, satisfiability is much harder a problem than
model-checking. For example, <cit.> needed to devise a
tableau system to go from the former to the latter.
What we have done in <ref> can be
easily adapted to satisfiability checking. We focus on the Boolean
setting in which $\Omega=\Bool$ and
$\Gamma=\{\bigwedge_{n},\bigvee_{n}\mid n\in\omega\}$, as in
Assumption <ref>.
The difficulty in the quantitative case seems to be that the number of
equations can grow
arbitrarily. Cf. <cit.>.
[$\CmuGL$-satisfiability equation]
Let $\varphi$ be a $\CmuGL$-formula, and
consider the simple $\CmuGL$-equational system
\begin{equation}\label{eq:201507092256}
\dotsc,\;
u_{m}=_{\eta_{m}} \varphi_{m}
\end{equation}
that arises as the translation of $\varphi$
(<ref>). Let $i_{1}<\cdots<i_{k}$ be
the indices of all the $\mu$-variables, as in
Def. <ref>.
We turn the $\CmuGL$-equational system (<ref>) into
an equational system (in the sense of Def. <ref>) over the
complete lattice $L=\Bool$, in the following way. The resulting
equational system is denoted by $\ESatisPhi$.
* An equation of the form $u_{i}=_{\eta_{i}}u_{i'}$ is left as it
* An equation of the form $u_{i}=_{\nu}
\boxdot_{\gamma}\bigl(u_{j_{1}},\dotsc,u_{j_{n}}\bigr)$
is turned into the equation
$\sem{\gamma}\colon \Omega^{n}\to \Omega$ is a monotone function
from Def. <ref>.
* An equation of the form $u_{i}=_{\nu}
\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})$
is turned into
f_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})$, where
the function $f_{\lambda}\colon \Omega^{n}\to \Omega$ is defined by:
\begin{equation}\label{eq:fLambdaDef}
\begin{aligned}
&f_{\lambda}(l_{1},\dotsc,l_{n}) =\ttrue
\\
\exists t\in F1.\; (\tilde{\lambda}\co F\tuple{\seq{l}{n}})(t)=\ttrue,
\end{aligned}
\end{equation}
where $\tilde{\lambda}\colon F(\Omega^{n})\to \Omega$ is from
Lem. <ref>. Note that
1ln→ Ω^n
, hence
Monotonicity of $f_{\lambda}$ follows easily from that of
(*** ... The definition must be modified to take proper
care of conjunction. They must take the same $t\in F1$ ***)
[correctness of satisfiability equations]
Let $(
\dotsc,
)\in \Omega^{m}$
be the solution of $\ESatisPhi$ in the sense of
Def. <ref>. We have: $l^{\sol}_{m}=\ttrue$ if and only
if there exists a coalgebra $c\colon X\to FX$ and its state $x\in X$
such that
For the `only if' direction, let $X:=\{u_{1},\dotsc,u_{m}\}/\sim$
where $\sim$ is the equivalence relation generated by:
u_i∼u_j if there exists ...
§ SMALL-MODEL PROPERTY AND SATISFIABILITY, TRIAL 1 (ONE THAT WILL
NOT PROBABLY WORK)
*** No, this shouldn't work! consider $\tuple{a}\top\land\tuple{b}\top$
over a stream automaton. ***
* probably entries from $\{0,1,\NoGood\}$ are enough
* the state space: $\{u_{1},\dotsc,u_{m}\}$, or possibly
$\{u_{1},\dotsc,u_{m}\}\times F1$
* Of course we cannot use an arbitrary satisfying coalgebra
$c$ as a guidance!!
Note that, in the completeness claim, a satisfiability progress
measure is not uniform over $c$ and $x$. Indeed there are
logics in
which the truth value of a formula can take an arbitrary number below $1$, but
not exactly $1$. See e.g. <cit.>.
Therefore in a quantitative case the satisfiability problem is
Input: $\varphi$ and $l$
Output: if there is $c$ and $x$ such that $l\sqsubseteq\sem{\varphi}_{c}(x)$
and not
Input: $\varphi$
Output: the supremum of $\sem{\varphi}_{c}(x)$ over all $c$ and $x$
§ SMALL-MODEL PROPERTY AND SATISFIABILITY
What we have done in <ref> can be
easily adapted to satisfiability checking.
The following is a counterpart of Def. <ref>.
[satisfiability progress measure]
Assume the setting of Def. <ref>.
A $\CmuGL$ satisfiability progress measure for the formula
is a pre-progress measure $p\in \pPM_{\varphi,\alpha}$ (Def. <ref>) that is
subject to the following condition.
($\nu$-variables, RHS is a modal formula) Let
$i\in [1,m]\setminus\{i_{1},\dotsc,i_{k}\}$ (i.e. $u_{i}$ is a $\nu$-variable) and that the
corresponding equation $u_{i}=_{\nu}\varphi_{i}$ has
as $\varphi_{i}$ a modal formula
We require that
there exists an element $t_{i}\in F1$ for each $i\in [1,m]$ such that,
for each approximant $p_{i}(\alpha_{1},\dotsc,
\alpha_{a},\dotsc,\alpha_{k})\in \Omega$, there are ordinals
$\beta_{1},\dotsc,\beta_{a-1}$ such that
\begin{align*}
\alpha_{a},\dotsc,\alpha_{k})\sqsubseteq
\\&
\PT_{\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})}
\alpha_{a},\dotsc, \alpha_{k})\bigl(\,(Fp)(t)\,\bigr),
\end{align*}
where $\PT_{\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})}$
is from (<ref>) and
$(Fp)(t)\in F(\pPM_{\varphi,\alpha})$ because
1p→ _φ,α
, hence
, and
[correctness of $\CmuGL$ satisfiability progress measure]
Assume the setting of Def. <ref> (hence
of Def. <ref>).
* (Soundness)
If $p\in \pPMpa$ is a satisfiability progress measure,
then there exists a coalgebra $c\colon X\to FX$ and its
state $x\in X$ such that
* (Completeness)
For any coalgebra $c\colon X\to FX$ and its state $x\in
X$, there exists an optimal satisfiability progress measure
$p\in \pPMpa$ (over some $\alpha$) such that
$ p_{m}(\alpha,\dotsc,\alpha)
Note that, in the completeness claim, a satisfiability progress
measure is not uniform over $c$ and $x$. Indeed there are
logics in
which the truth value of a formula can take an arbitrary number below $1$, but
not exactly $1$. See e.g. <cit.>.
For soundness, we take define a coalgebra $c\colon 1\to F1$—whose
carrier is a singleton $1=\{*\}$—by $c(*):=t$, and let $p\colon
1\to \pPMpa$ be an MC progress measure
(Def. <ref>). It is straightforward
that $p$ is indeed an MC progress measure. We have
$p_{m}(\alpha,\dotsc,\alpha)\sqsubseteq\sem{\varphi}_{c}(*)$ by
Thm. <ref>.1.
Consider the simple $\CmuGL$-equational system
\begin{equation}\label{eq:201507092256}
\dotsc,\;
u_{m}=_{\eta_{m}} \varphi_{m}
\end{equation}
that arises as the translation of $\varphi$
In this section we establish the small-model property of
$\CmuGL$—that a satisfiable formula $\varphi$ has a model whose size
is suitably bounded—and present
an algorithm for satisfiability
check. Continuing <ref> we fix the domain
of truth values to be $\Omega=\Bool$; and restrict propositional
connectives to $\bigwedge$ and $\bigvee$. This is much like in
Assumption <ref>. Our developments
will be based on the notion of matrix progress measure (MPM) from
Def. <ref>.
[the collapsing function $\collapse$]
Assume the setting of Def. <ref>.
We define a function $\collapse\colon \POM_{\omega}\to \POM_{mk}$—that collapses a prioritized ordinal matrix
(Def. <ref>) with arbitrary natural numbers
as its entries into one with entries from $[0,mk]\amalg\{\NoGood\}$—as
follows. Given a POM $\bigl(\alpha^{(i)}_{j}\bigr)_{i\in [1,m],j\in
[1,k]}$, the $(i,j)$-entry $\widetilde{\alpha}^{(i)}_{j}$ of
the POM $\collapse\bigl(\alpha^{(i)}_{j}\bigr)_{i,j}$ is defined as
* $\widetilde{\alpha}^{(i)}_{j}:=\NoGood$ if
* $\widetilde{\alpha}^{(i)}_{j}:=0$ if
For the other entries,
we define
$\widetilde{\alpha}^{(i)}_{j}:=N$, for $N\in [1,mk]$, if
$\alpha^{(i)}_{j}$ is the $N$-th smallest number among the set
i∈[1,m], j∈[1,k]
Here we do not count duplicates.
For example:
3 6
8 1 1
8 0 5
3 5
6 1 1
6 0 4
Note also that the size of the set $\POM_{mk}$ is bounded by
$(mk+2)^{mk}$—each entry is an element of $[0,mk]\amalg\{\NoGood\}$.
[small model property of
Assume the setting of Def. <ref>, and assume
further that $\Omega=\Bool$ and that
$\Gamma=\{\bigwedge_{n},\bigvee_{n}\mid n\in\omega\}$. If there exists
a coalgebra $c\colon X\to FX$ and $x\in X$ such that
$\sem{\varphi}_{c}(x)=\ttrue$, then there exists one such that $|X|\le
Let us consider the following (monotone and decreasing) function
$\Psi\colon \pow(\POM_{mk})\to \pow(\POM_{mk})$. For $Y\subseteq
\POM_{mk}$,
\begin{align*}
\\
\left\{
P\in Y
\,\left|\,
\begin{minipage}{.38\textwidth}
$P$ satisfies Cond.~2, 3, 4, 5(a) and 5(b) of Def.~\ref{def:MPM}
(where we put $P$ in place of $R(x)$), and, considering Cond.~5(c), there exists $t\in FY$
such that
\newline
\end{minipage}
\right.\right\}.
\end{align*}
Here $P^{(i)}$ denotes the $i$-th row of a POM $P$; note that Cond. 4
is vacuous. Monotonicity of $\Psi$ is obvious because $Y\supseteq
Y'\neq \emptyset$ implies $FY\supseteq FY'$ (a well-known property
of a $\Sets$-functor, see e.g. <cit.>).
Now we repeatedly apply $\Psi$ to the set $\POM_{mk}$; since
is finite the chain
\begin{equation}\label{eq:201507091941}
\POM_{mk}\supseteq \Psi(\POM_{mk})\supseteq
\Psi^{2}(\POM_{mk})\supseteq\cdots
\end{equation}
eventually stabilizes. Let its limit be denoted by $Y_{0}$.
By the definition of $\Psi$, for each $P\in Y_{0}$ we can find $t\in
FY_{0}$ such that
F(\ev'(P^{(i)}))\bigr)(t)=\ttrue$. We define a function $d\colon
Y_{0}\to FY_{0}$ by assigning such $t$ to each $P\in Y_{0}$.
It is straightforward that the inclusion map $\iota\colon
\widetilde{X}\hookrightarrow\POM_{m}$ constitutes an MPM for $\varphi$
(Def. <ref>). Therefore by Thm. <ref>.1
(soundness of MPMs), we have that
\begin{equation}\label{eq:201507091934}
P^{(m)}_{k}\neq \NoGood \;\text{for some $P\in Y_{0}$}
\;\Longrightarrow\; \varphi \text{ is satisfiable.}
\end{equation}
We shall now prove the converse direction
of (<ref>). Assume $\varphi$ is satisfiable, that
is, there exists a coalgebra $c\colon X\to FX$ and $x\in X$ such
that $\sem{\varphi}_{c}(x)=\ttrue$.
By Thm. <ref>.2 (completeness of MPMs), there exists
an MPM $R\colon X\to \POM_{|X|}$ such that
$\bigl(R(x)\bigr)^{(m)}_{k}\neq \NoGood$. Let $P:=\collapse(R(x))\in
\POM_{mk}$. We shall prove that $P\in Y_{0}$, that is,
$P$ survives along the chain (<ref>).
It is straightforward that $P$ satisfies Cond. 2, 3, 4, 5(a) and
of Def. <ref>
(where we put $P$ in place of $R(x)$)—the collapsing function
$\collapse$ preserves the order ($\le$) between different entries of
hence preserves the order $\preceq_{i}$ between different rows.
Now it suffices to show the following claim:
Y_1:={ (R(x')) |. x'∈X },
where $X$ is the carrier of the coalgebra that witnesses
satisfiability of $\varphi$. Then for each $P'\in Y_{1}$, there
exists $t'\in FY_{1}$ such that
By Thm. <ref>.2 (completeness of MPM), there exists
an MPM $R\colon X\to \POM_{|X|}$ such that
$\bigl(R(x)\bigr)^{m}_{k}\neq \NoGood$. Let us consider the composite
$\widetilde{R}:= \collapse\co R\colon X\to \POM_{m}$, and define
X:={R(x')|x'∈X} ⊆_m.
We shall equip a coalgebraic structure $\widetilde{c}$ to $\widetilde{X}$ and claim that its
$\widetilde{x}:=\widetilde{R}(x)$ satisfies
To define $\widetilde{c}\colon
\widetilde{X}\to F\widetilde{X}$, let $P\in \widetilde{X}$ and choose
$x'\in X$ such that $\widetilde{R}(x')=P$. By the definition of
$\widetilde{X}$, such $x'$ necessarily exists. Then we define
( Xc→
FX )(x')
note here that $\widetilde{R}\colon X\to \POM_{m}$ factorizes as $X\to
\widetilde{X}\hookrightarrow \POM_{m}$ by the definition of
$\widetilde{X}$ as an image.
We shall now prove that the inclusion map $\iota\colon
\widetilde{X}\hookrightarrow\POM_{m}$ constitutes an MPM for $\varphi$
(Def. <ref>). The crucial point is that, by the definition,
the order between different rows—the pointwise extension of $\le$,
hence $\preceq_{i}$ for any $i$—is preserved by $\collapse$, including whether some entries are
$0$ or $\NoGood$. This immediately
derives Cond. 2, 3, 5(a) and 5(b) in Def. <ref>. Cond. 4 is trivial.
It remains to show that $\iota$ satisfies Cond. 5(c) in Def. <ref>.
§ COALGEBRAIC $\MU$-CALCULUS $\CMU_{\GAMMA,\LAMBDA}$
AS A NONDETERMINISTIC LINEAR-TIME LOGIC
In this section we adapt the previous results to the setting where we
think of $\CmuGL$ as a (nondeterministic) linear-time logic, that is, where a system
in question exhibits nondeterministic branching over transitions of
type $F$. Such a system is represented
as a
function $c\colon X\to \pow FX$.
Our main results here are: 1) categorical characterization of
the truth value of a linear-time logic formula using progress measures
(Thm. <ref>);
2) a “smallness” result that cuts down the search spaces for
linear-time model checking (Thm. <ref>);
and 3) a decision procedure (Thm. <ref>) that depends on the smallness result.
§.§ Coalgebraic Preliminaries
In what follows we will be dealing with coalgebras of the type
$c\colon X\to \pow FX$, where $F\colon \Sets\to\Sets$
(that is like in <ref>) is understood as
the type of linear-time behaviors, and $\pow$ is the powerset
This is a common setting taken in the coalgebraic studies of trace
semantics. The use of a monad $T$ in a coalgebra $c\colon X\to TFX$
with “$T$-branching over $F$-linear time behaviors” originates
in <cit.>, and is subsequently adopted e.g.in <cit.>.[Another
common coalgebraic formalization of linear-time semantics is via
determinization, and uses Eilenberg-Moore categories (as opposed
to Kleisli) as base categories. See
e.g. <cit.>.
Adapting the current model-checking
framework to this Eilenberg-Moore approach
seems hard: fixed-point specifications are usually interpreted
over infinitary traces such as infinite words; and
this makes determinization,
the core of the Eilenberg-Moore approach, much more complicated (like Büchi word
automata become Rabin automata, see e.g. <cit.>). ] The
formalization in the current paper most closely follows that
in <cit.>. We shall again present minimal
preliminaries to this Kleisli approach to coalgebraic trace
semantics. See e.g. <cit.> for further
details; for monads and Kleisli categories see <cit.>.
A monad $T$ on $\Sets$ is an endofunctor equipped with natural
transformations $\eta^{T}\colon \id\Rightarrow T$ (unit) and $\mu^{T}\colon T\co
T\Rightarrow T$ (multiplication) that are subject to certain
“monoid” commutative diagrams. In our current example of the powerset
monad $\pow$, its unit $\eta^{\pow}$ is the singleton map and its
multiplication $\mu^{\pow}$ is given by union. In the class of examples
of $T$ that are
relevant to us, the unit turns an element into “a branching with a
unique choice”; and the multiplication “suppresses” two
transitions into one (see <cit.>).
The Kleisli category $\Kleisli{T}$ has sets as its objects, and
an arrow $X\relto Y$ in $\Kleisli{T}$ is given by a function $X\to TY$.
It becomes a category using the monad structure of $T$. For example,
given two successive arrows $f\colon X\relto Y$ and $g\colon Y\relto Z$
in $\Kleisli{T}$, its composition $g\Kco f\colon X\relto Z$ is given by
the composite
Xf→ TYTg→ T(TZ) μ_Z→ TZ
of functions. It is also easy to see that we have the so-called
Kleisli inclusion functor $J\colon\Sets\to\Kleisli{T}$ by
$JX=X$ and $Jf=\eta^{T}\co f$.
Note that we used the symbols $\relto$ and $\Kco$ (as
opposed to $\to$ and $\co$) for constructs in $\Kleisli{T}$, for
distinction. In what follows we stick to this convention.
Note that for our example of $T=\pow$, the Kleisli category
is nothing but the category $\Rel$ of sets and binary relations. We will
however stick to $\Kleisli{\pow}$, hoping that the theory will be
transported to other monads (such as the Giry monad on $\Meas$, for
probabilistic branching).
The following is our current notion of system model.
For technical reasons, we impose certain conditions on $F$. These
conditions are
common ones and imposed also
in <cit.>.
[nondeterministic $F$-coalgebra]
Let $F\colon \Sets\to\Sets$ be a functor, such that the following hold.
A final
$\zeta\colon Z\iso FZ$ exists in $\Sets$.
The functor $F$ comes with a
distributive law $\xi\colon F\pow\Rightarrow\pow F$ over the powerset
monad $\pow$ (which, as is well-known <cit.>, induces a lifting $\oF\colon
\Kleisli{\pow}\to\Kleisli{\pow}$ of $F$).
nondeterministic $F$-coalgebra is $c\colon X\to
\pow FX$ in $\Sets$, that is, an arrow $c\colon
X\relto \oF X$ in the Kleisli category $\Kleisli{\pow}$.
Examples of such functors are
polynomial functors inductively generated by
F,F_i ::= 𝕀|A|F_1×F_2|∐_i∈IF_i
where $A$ is a constant functor that takes any set to $A\in\Sets$.
e.g. <cit.> for further details on
Cond. <ref>–<ref>.
In view of <ref>, in the current setting, we can identify
a state $z$
of a final coalgebra $\zeta\colon Z\iso FZ$ with a (possibly infinite,
long-term) linear-time behavior of the type $F$. For example, when
$F=\pow(\AP)\times(\place)$ (Example <ref>.6), a final coalgebra is carried by the set
$Z=(\pow(\AP))^{\omega}$ of infinite streams of subsets of $\AP$. Such streams
are commonly called computations in the context of model
The following result <cit.> allows us to characterize, in categorical terms, the set of possible
(linear-time) $F$-behaviors of a nondeterministic
$F$-coalgebra.[In papers like <cit.> coalgebraic
finite trace semantics is studied. Here “finite” means
linear-time behaviors that eventually come to halt within finitely many
steps; and the set of finite $F$-behaviors is identified with the
carrier of an initial $F$-algebra in $\Sets$ (as opposed to a final
$F$-coalgebra).] The same holds in a probabilistic setting, too; see e.g. <cit.>.
[coalgebraic infinitary[Note that
“infinitary” does not mean that a behavior is necessarily of an infinite length. For
example, if $F=\{\checkmark\}+A\times (\place)$, a final $F$-coalgebra
is carried by the set $Z=A^{*}+A^{\omega}$ of all words over $A$ of
finite or infinite length. All words (finite or infinite) are deemed to be “infinitary”
traces. ] trace semantics <cit.>]
Let $F\colon \Sets\to\Sets$ be a functor that satisfies the conditions
in Def. <ref>; and $c\colon X\to \pow FX$ be a
nondeterministic $F$-coalgebra.
Consider the diagram
\begin{equation}\label{eq:nondetCoalgTraceSem}
\vcenter{
\xymatrix@C+1em@R=.8em{
{\oF X}
\ar@{->}[r]^{\oF f}|-*\dir{|}
{\oF Z}
\\
\rar[u]^{c}
\ar@{->}[r]_{f}|-*\dir{|}
\rar[u]_{J\zeta}^{\cong}
\qquad\text{in $\Kleisli{\pow}$;}
\end{equation}
1) there exists at least one function $f\colon X\to \pow Z$ that
the diagram commute; and 2) among such $f$, there exists the greatest
one with respect to (the pointwise extension of) the inclusion order in
$\pow Z$.
The greatest one shall be denoted by $\tr(c)\colon X\to \pow
Z$ and called the (infinitary) trace semantics of $c$. Moreover,
an element $z\in \tr(c)(x)$—identified with a single linear-time
behavior over time—is referred to as an infinitary trace of
$c$ from $x$.
We note that the definition of $\tr(c)$ in
Prop. <ref> amounts to the following:
$\tr(c)$ is the greatest fixed point of the monotone function
\begin{equation}\label{eq:monotoneFuncForWhichNondetTraceIsGreatestFixedPoint}
\Psi\colon \Kleisli{\pow}(X,Z)\to\Kleisli{\pow}(X,Z),
\quad
f\mapsto (J\zeta)^{-1}\odot \oF f\odot c
\end{equation}
where $\odot$ denotes composition of arrows in $\Kleisli{\pow}$.
It has been observed that, for many examples of the functor
$F$, the greatest homomorphism $\tr(c)$ in Prop. <ref> indeed captures the set of all
possible linear-time behaviors. See e.g. <cit.>
and <cit.>.
§.§ $\CmuGL$ as a Linear-Time Logic
We take a modal language $\CmuGL$ whose modal signature $\Lambda$ is
over $F$. Hence a $\CmuGL$-formula $\varphi$ specifies a property of
$F$-behaviors, where the latter are identified with elements $z\in Z$
of a final coalgebra $\zeta\colon Z\iso FZ$. See <ref>.
[semantics of the logic $\CmuGL$ over nondeterministic
Let $\varphi$ be a closed $\CmuGL$-formula, and $c\colon X\to \pow FX$
be a nondeterministic $F$-coalgebra. The denotation of $\varphi$
over $c$ is given by a function
defined by
X(c) Z
J(φ_ζ) Ω )
where: $\tr(c)$ is the infinitary trace semantics of $c$
(Prop. <ref>);
$\sem{\varphi}_{\zeta}$ is the denotation of $\varphi$ over the
(proper) $F$-coalgebra $\zeta\colon Z\to FZ$ defined in
Def. <ref>; and $J\colon \Sets\to\Kleisli{\pow}$ is the
Kleisli inclusion functor (<ref>).
Given a nondeterministic $F$-coalgebra $c$ and its state
$x$, a typical question is
whether some (or all) of its linear-time behaviors satisfy
a formula $\varphi$. This problem is the existential (or
universal) model-checking problem, respectively.
In the current paper we focus on existential model checking.
Take the combination of $F,\Omega,\Gamma$ and $\Lambda$ in
Example <ref>.6. A Kripke structure
can then be thought of
as a nondeterministic $F$-coalgebra.[A Kripke structure is
most naturally modeled by a function $c'\colon X\to \pow (\AP)\times \pow
X$. This gives rise to a function $c\colon X\to \pow\bigl(
\,\pow(\AP)\times X\,\bigr)$ in an obvious way that turns state-labels
into transition-labels, namely
$c(x)=\{((\pi_{1}\co c')(x), x')\mid x'\in (\pi_{2}\co c')(x)\}$.]
Recall that a final coalgebra is carried by the set
$(\pow(\AP))^{\omega}$ of computations;
in this case the infinitary trace semantics $\tr(c)\colon X\to
\pow\bigl((\pow(\AP))^{\omega}\bigr)$ is precisely the map that
carries each state $x\in X$ to the set of computations that arise from
the paths from $x$.
A $\CmuGL$-formula $\varphi$ is interpreted over elements of a
final coalgebra, i.e. computations. Overall,
Def. <ref> in this setting yields the set of
truth values that $\varphi$ can take, ranging over all the possible
computations $z\in \tr(c)(x)$
that start from the given state $x\in X$.
(*** Already introduced***)
It turns out that the two variants (universal vs. existential) call for
different semantical frameworks. However they share the following notion
in common. Its definition follows the pattern of that of progress
measure (Def. <ref>), but misses some conditions
in Def. <ref>.
[pre-progress measure, PPM]
Let $F\colon \Sets\to\Sets$ be a functor as described in
Def. <ref>. Let $\varphi$ be a
$\CmuGL$-formula—where $\Lambda$ is a modal signature over
$F$—that is identified with a simple equational system
u_m=_η_m φ_m
in <ref>.
Let $i_{1}<\cdots
enumerate the indices of all the $\mu$-variables.
A pre-progress measure (PPM) $p$ for $\varphi$
is given by a tuple
p =
( _i(α_1,…,α_k) )_i,αk
consists of:
* the maximum prioritized ordinal
$(\overline{\alpha_{1}},\dotsc, \overline{\alpha_{k}})$; and
* the approximants $\approximant_{i}(\alpha_{1},\dotsc,\alpha_{k})\in \Omega$, defined for
$i\in[1,m]$ and each
prioritized ordinal
such that
\alpha_{1}\le\overline{\alpha_{1}},\dotsc,
\alpha_{k}\le\overline{\alpha_{k}}
The approximants $\approximant_{i}(\alpha_{1},\dotsc,\alpha_{k})$
are subject to:
Let $i\in[1,m]$ (hence $u_{i}$ is either a $\mu$- or
$\nu$-variable). Then
($\mu$-variables, base case)
Let $a\in [1,k]$. Then $\alpha_{a}=0$ implies
\alpha_{a},\dotsc,\alpha_{k})=\bot$.
($\mu$-variables, step case)
Let $a\in [1,k]$, and let
\alpha_{a}+1,\dotsc,\alpha_{k})$ be
a prioritized ordinal
such that its $a$-th counter
$\alpha_{a}+1$ is a successor ordinal. Then, regarding
the approximant $\approximant_{i_{a}} (\alpha_{1},\dotsc,
\alpha_{a}+1,\dotsc,\alpha_{k})$:
* (RHS is a variable) If the formula
$\varphi_{i_{a}}$ on the right-hand side of the
$i_{a}$-th equation $u_{i_{a}}=_{\mu}\varphi_{i_{a}}$
is a variable $u_{i'}$ (for some $i'\in [1,m]$), then
there exist ordinals
$\beta_{1},\dotsc,\beta_{a-1}$ such that
\begin{multline*}
\approximant_{i_{a}} (\alpha_{1},\dotsc,
\alpha_{a}+1,\dotsc,\alpha_{k})
\\
\sqsubseteq
\approximant_{i'} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k})
\end{multline*}
\beta_{a-1}\le\overline{\alpha_{a-1}}$.
* (RHS is a propositional formula) If the
formula $\varphi_{i_{a}}$ is a propositional formula
there exist ordinals
$\beta_{1},\dotsc,\beta_{a-1}$ such that
\begin{align*}
&\approximant_{i_{a}} (\alpha_{1},\dotsc,
\alpha_{a}+1,\dotsc,\alpha_{k})
\\
\sem{\gamma}
\left(\,
\begin{array}{c}
\approximant_{j_{1}} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k}),
\\
\dotsc,
\\
\approximant_{j_{n}} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k})
\end{array}
\,\right)
\end{align*}
\beta_{a-1}\le\overline{\alpha_{a-1}}$.
($\mu$-variables, limit case)
Let $a\in [1,k]$, and
\alpha_{k})$
be a prioritized ordinals
such that its $a$-th counter
is a limit ordinal. Then
we have
_i_a (α_1,…,
_i_a (α_1,…,
Let $i\in [1,m]\setminus\{i_{1},\dotsc, i_{k}\}$ (i.e. $u_{i}$ is a $\nu$-variable in the
system (<ref>)); let $a\in [1,k]$ such that
i_1<⋯<i_a-1<i<i_a<⋯< i_k.
Let $(\alpha_{1},\dotsc,\alpha_{k})$ be a prioritized ordinal.
Then, regarding the approximant
$\approximant_{i} (\alpha_{1},\dotsc,\alpha_{k})$:
* (RHS is a variable) If the formula
$\varphi_{i}$ on the right-hand side of the
$i$-th equation $u_{i}=_{\nu}\varphi_{i}$
is a variable $u_{i'}$ (for some $i'\in [1,m]$), then
there exist ordinals
$\beta_{1},\dotsc,\beta_{a-1}$ such that
\begin{multline*}
\approximant_{i_{a}} (\alpha_{1},\dotsc,
\alpha_{a},\dotsc,\alpha_{k})
\\
\sqsubseteq
\approximant_{i'} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k})
\end{multline*}
\beta_{a-1}\le\overline{\alpha_{a-1}}$.
* (RHS is a propositional formula) If the
formula $\varphi_{i}$ is a propositional formula
there exist ordinals
$\beta_{1},\dotsc,\beta_{a-1}$ such that
\begin{align*}
&\approximant_{i} (\alpha_{1},\dotsc,
\alpha_{a},\dotsc,\alpha_{k})
\\
\sem{\gamma}
\left(\,
\begin{array}{c}
\approximant_{j_{1}} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k}),
\\
\dotsc,
\\
\approximant_{j_{n}} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k})
\end{array}
\,\right)
\end{align*}
\beta_{a-1}\le\overline{\alpha_{a-1}}$.
Let $\alpha$ be an ordinal.
The collection of all pre-progress measures for a formula $\varphi$,
whose maximum prioritized ordinal $(
\overline{\alpha_{1}},\dotsc,
\overline{\alpha_{k}})
$ satisfies $\overline{\alpha_{i}}= \alpha$ for each $i\in [1,k]$
shall be
denoted by $\pPM_{\varphi,\alpha}$.
Recall that $\Omega$ is the complete lattice of truth values. In the
definition of $\pPM_{\varphi,\alpha}$, the explicit bound by $\alpha$ is
there so that the collection $\pPM_{\varphi,\alpha}$ is a (small) set.
Comparing the previous definition with Def. <ref>
of progress measures, what are missing here are the treatment
of modal formulas $\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})$
in Cond. <ref>
and <ref>. These missing cases are
precisely when we need to use the dynamics of a coalgebra; they will be
suitably filled in
in <ref>–<ref>,
differently for existential and universal model checking.
§.§ (Existential) Linear-Time Model-Checking for $\CmuGL$
We shall follow essentially the same path as
in <ref>. We shall use precisely
the same notion of pre-progress measure
(Def. <ref>). The additional compatibility condition
with the dynamic structure of the system in question is
different reflecting the difference between the
systems in question ($X\to FX$ in $\Sets$, or $X\relto FX$ in $\Kleisli{\pow}$).
The following is a counterpart of Def. <ref>;
LT is for linear-time.
[LTMC progress measure]
Let $\varphi$ be a $\CmuGL$-formula, identified with
a simple $\CmuGL$-equational system
u_m=_η_m φ_m.
Let $c\colon X\to \pow FX$ be a nondeterministic $F$-coalgebra (with
some conditions on $F$; see Def. <ref>).
An LTMC progress measure for $\varphi$ over $c$ is given by a tuple
$(\alpha,Y\stackrel{q}{\to} FY,r,s)$ of:
* some ordinal $\alpha$,
* an $F$-coalgebra $q\colon Y\to FY$, and
* functions $r\colon Y\to \pPMpa$ and $s\colon Y\to X$
that are subject to the following condition.
Let $y\in Y$.
($\nu$-variables, RHS is a modal formula)
In the setting of
Cond. <ref> of
Def. <ref>,
assume further that the formula $\varphi_{i}$ is a modal formula:
the approximant $p_{i}(\alpha_{1},\dotsc,
\alpha_{a},\dotsc,\alpha_{k})\in \Omega$ of $p:=r(y)$. There
must exist ordinals $\beta_{1},\dotsc,\beta_{a-1}$ such that
\begin{equation}\label{eq:201507051642}
\begin{aligned}
\alpha_{a},\dotsc,\alpha_{k})
\sqsubseteq
\\
\PT_{\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})}
\alpha_{a},\dotsc, \alpha_{k})
\bigl((Fr\co q)(y)\bigr),
\end{aligned}
\end{equation}
and $\beta_{1}\le\alpha,\dotsc,
\beta_{a-1}\le\alpha$.
(Compatibility with $c$) For each
$y\in Y$ we have $(Fs\co
q)(y)\in c(x)$. That is
\begin{equation}\label{eq:existentialProgMeasBwdSim}
\vcenter{\xymatrix@R=.6em{
{\oF Y}
\rar[r]^-{\oF Js}
{\oF X}
\\
\rar[u]^{Jq}
\rar[r]_-{Js}
\ar@{}[ur]|{\subseteq}
\rar[u]_{c}
\quad\text{in $\Kleisli{\pow}$,}
\end{equation}
where $Jq\colon Y\to \pow FY$ is given by
[correctness of LTMC progress measure]
Assume the setting of Def. <ref>.
particular, the formula $\varphi$ is translated to an equational
system with $m$ variables.
$(\alpha,Y\stackrel{q}{\to} FY,r,s)$
be an LTMC
progress measure. Let $y\in Y$ be an arbitrary state, $x:=s(y)$ (a state of the
coalgebra $c$) and $p:=r(y)$ (a
pre-progress measure). Then there exists an
infinitary trace $z\in \tr(c)(x)$ of $x$
such that
\sem{\varphi}_{\zeta}(z)$. Here $\sem{\varphi}_{\zeta}\colon Z\to \Omega$
is from Def. <ref>.
(Completeness) Let $x\in X$, and
$z\in \tr(c)(x)$ be an
infinitary trace from $x$.
There is an LTMC progress measure
$(\alpha,Y\stackrel{q}{\to} FY,r,s)$
and some $y\in Y$ such that $s(y)=x$, $\beh(q)(y)=z$
=\sem{\varphi}_{\zeta}(z)$ where $p:=r(y)$.
$\beh(q)$ is the behavior map induced by
finality (<ref>).
The completeness result in the last theorem is not totally
satisfactory, especially from an algorithmic point of view.
The question is the size of an
LTMC progress measure: in the proof we used $Y\subseteq X\times Z$, but
this can be very large—$Z$ is an uncountable set for most common
functors $F$.
Fortunately we have the
following theorem
that cuts down the set $Y$ from $X\times Z$ to
$X\times \pPMpa$ (that is potentially much smaller, especially when
The result thus
opens up a way to a generic coalgebraic
model-checking algorithm, and is one of our main technical contributions.
[small LTMC progress measure]
Assume the setting of Def. <ref>, and let
$x\in X$. For any infinitary trace $z\in
\tr(c)(x)$,
there exists
an LTMC progress measure
$(\alpha,Y\stackrel{q}{\to} FY,r,s)$ and some $y\in Y$ such that:
$s(y)=x$, and $p_{m}(\alpha,\dotsc,\alpha)=\sem{\varphi}_{\zeta}(z)$ where
$p:=r(y)$. Moreover
$(\alpha,Y\stackrel{q}{\to} FY,r,s)$ can be chosen so that:
$Y\subseteq X\times \pPMpa$;
and $r=\pi_{2}$ and $s=\pi_{1}$.
Our proof of the last theorem comes in a pumping
flavor. In it, since the relevant set is possibly infinite, we resort to
Zorn's lemma.
Failure of positionality is already in the simple setting with a Kripke
structure and LTL. Consider a Kripke structure with three states $x_{0},
x_{1}, x_{2}$ with
$x_{0}\to x_{1}$,
$x_{0}\to x_{2}$,
$x_{1}\to x_{0}$,
and $x_{2}\to x_{0}$, and $x_{1}\models P$, $x_{2}\models Q$. Consider
an LTL specification $\mathsf{F}P\land \mathsf{F}Q$. Then a successful
scheduler must not be positional!
§.§ Decision Procedure
We exploit the previous results and derive a
decision procedure for linear-time $\CmuGL$-model checking.
We make the following assumption; its justification is discussed shortly.
In what follows we assume that the
satisfiability problem of $\CmuGL$ (against $F$-coalgebras) is decidable.
Moreover we
assume the small model property: for each satisfiable
$\CmuGL$-formula $\varphi$, we can compute a natural number
$N_{\varphi}\in \omega$ such that there exists an $F$-coalgebra
that satisfies $\varphi$ the size of whose state space is no
greater than $N_{\varphi}$. That is: there exists a coalgebra
$\varepsilon\colon E\to FE$, its state $e\in E$ and an MC
progress measure $Q\colon E\to \pPM_{\varphi,\alpha}$
(Def. <ref>) such that
$Q(e)(\alpha,\dotsc,\alpha)=\ttrue$ and $|E|\le N_{\varphi}$. It
is moreover guaranteed by
Thm. <ref>.2 that we can
take $\alpha:=N_{\varphi}$.
Finally, we assume that $F$ preserves finiteness, that is, $FB$
is finite if $B$ is finite.
Assumption <ref> is a mild one. For example, <cit.> shows that
the assumption holds when the logic $\CmuGL$ comes with a one-step complete, contraction-closed
and exponentially-tractable set of deductive
rules. These conditions hold in well-known modal
logics, including (the fixed-point extensions of) the normal modal logic
$\mathsf{K}$, and monotone modal logic
(Example <ref>).
Graded and coalition frames do not satisfy preservation of
Of more relevance here is the fact that the assumption holds for
(coalgebras of) polynomial functors $F$ (with suitable finiteness requirements), which are the ones typically used to specify linear time behavior; modalities and deductive rules for such functors can be modularly derived from their structure, using an approach similar to that of <cit.>, and proving the tractability of the set of rules is straightforward in this case.
It also seems that the
framework in <ref> can be adapted
to satisfiability check and hence to the small-model property. Due
to lack of space we do not do so in the current paper and just
assume the small model property.
[linear-time $\CmuGL$-model checking is decidable]
Assume the setting of Def. <ref>. Assume
further that: $\Omega=\Bool$; and
$X$ is a finite set. Then it is decidable whether there exists an
infinitary trace $z\in \tr(c)(x)$ such that
§.§ Universal Model-Checking
* Universal model checking vs. existential model checking
* Must use different order.
It is crucial that, for universal and existential model-checking, we
use two different preorders $\SmythLeq$ and $\HoareLeq$, respectively.
[the Smyth $\SmythLeq$ and Hoare $\HoareLeq$ preorders]
We define preorders
$\SmythLeq$ and
$\HoareLeq$ over $\pow(\Omega^{m})$, as follows.
\begin{equation}\label{eq:smythHoareDefChar}
\begin{aligned}
\alpha\SmythLeq\alpha'\;
\defiff\;
\forall V'\in\alpha'.\, \exists V\in \alpha.\, V\sqsubseteq V'
\\
\upcl\alpha\supseteq\alpha'
\;\Longleftrightarrow\;
\upcl\alpha\supseteq\upcl\alpha';
\\
\alpha\HoareLeq\alpha'\;
\defiff\;
\forall V\in\alpha.\, \exists V'\in \alpha'.\, V\sqsubseteq V'
\\
\alpha\subseteq\dwcl\alpha'
\;\Longleftrightarrow\;
\dwcl\alpha\subseteq\dwcl\alpha'.
\end{aligned}
\end{equation}
Here $\upcl \alpha:=\{V\mid \exists V'\in \alpha.\, V'\sqsubseteq V\}$
and $\dwcl \alpha:=\{V\mid \exists V'\in \alpha.\, V\sqsubseteq V'\}$
are the upward and downward closures, respectively.
The least and greatest elements are $\{\bot\}$ (that is equivalent to
$\Omega^{m}$) and
$\emptyset$, for $\SmythLeq$; and $\emptyset$ and $\{\top\}$
(that is equivalent to
The preorders $\SmythLeq$ and $\HoareLeq$ are not partial orders—for example, $\{\ffalse, \ttrue\}$ and $\{\ffalse\}$ are
equivalent with respect to $\SmythLeq$, for $\Omega=\Bool$.
(This example suggests use of $\SmythLeq$ for universal model
checking, too.)
The following fact, however, allows us to apply the observations
in <ref>. It is derived immediately by
the characterization in (<ref>).
Let $\Omega$ be a complete lattice.
Then the poset
induced by the preorder $\SmythLeq$
is a complete lattice. Indeed,
given a family $(\alpha_{i})_{i\in I}$ such that $\alpha_{i}\subseteq \Omega^{m}$,
\begin{align*}\textstyle
\bigsqcap_{\mathsf{S}}
\bigl\{\,[\alpha_{i}]\,\bigl|\bigr.\,i\in I\bigr\}
\bigl[\,\textstyle\bigcup_{i} (\upcl\alpha_{i})\,\bigr]
\bigl[\,\textstyle\bigcup_{i} \alpha_{i}\,\bigr]
\; \text{and}\;
\\
\textstyle
\bigsqcup_{\mathsf{S}}
\bigl\{\,[\alpha_{i}]\,\bigl|\bigr.\,i\in I\bigr\}
\bigl[\,\textstyle\bigcap_{i} (\upcl\alpha_{i})\,\bigr]
\end{align*}
provide the desired infimum and supremum, respectively.
The same holds for the Hoare preorder $\HoareLeq$. The infimums and
supremums are given by
\begin{align*}\textstyle
\bigsqcap_{\mathsf{H}}
\bigl\{\,[\alpha_{i}]\,\bigl|\bigr.\,i\in I\bigr\}
\bigl[\,\textstyle\bigcap_{i} (\dwcl\alpha_{i})\,\bigr]
\; \text{and}\;
\\
\textstyle
\bigsqcup_{\mathsf{H}}
\bigl\{\,[\alpha_{i}]\,\bigl|\bigr.\,i\in I\bigr\}
\bigl[\,\textstyle\bigcup_{i} (\dwcl\alpha_{i})\,\bigr]
\bigl[\,\textstyle\bigcup_{i} \alpha_{i}\,\bigr].
\end{align*}
For further details on those preorders
see e.g. <cit.> and <cit.>.
In the current setting where $\CmuGL$ is thought of as a linear-time
logic, unlike in <ref>, it is not the case that
a formula $\varphi$ (or its equational presentation) and a coalgebra $c$
directly give rise to an equational system over some complete lattice.
Our characterization of the (linear-time) semantics of $\varphi$ over
instead, relies on the following notion.
[predicate transformer $\PT^{E,c}_{i}$]
Let $E=(u_{i}=_{\eta_{i}}\varphi_{i})_{i\in[1,m]}$ be a simple $\CmuGL$-equational system, and $c\colon X\to
\pow FX$ be a nondeterministic $F$-coalgebra. For each $i\in [1,m]$ we
define the $i$-th predicate transformer induced by $E,c$
as follows. Let $p\colon X\to \pow(\Omega^{m})$, that is, an arrow
$p\colon X\relto \Omega^{m}$ in $\Kleisli{\pow}$.
When $\varphi_{i}$ is a variable $u_{j}$ ,
\begin{align*}
\PT^{E,c}_{i}(p)
\;:=\;\bigl(\,
\stackrel{p}{\relto}
\Omega^{m}
\stackrel{J\pi_{i,j}}{\relto}
\Omega^{m}
\,\bigr)
\end{align*}
where $J$ is the Kleisli inclusion functor and
$\pi_{i,j}\colon \Omega^{m}\to\Omega^{m}$ is
the function that replaces the $i$-th component of the input
with its $j$-th, i.e.
the unique arrow that makes the following diagram commute.
where $i'$ ranges $i'\in[1,m]\setminus\{i\}$.
When $\varphi_{i}$ is a propositional formula
$\boxdot_{\gamma}(u_{i_{1}},\dotsc, u_{i_{k}})$,
\begin{align*}
\PT^{E,c}_{i}(p)
\;:=\;\bigl(\,
\stackrel{p}{\relto}
\Omega^{m}
\stackrel{J\gamma_{i,i_{1},\dotsc,i_{k}}}{\relto}
\Omega^{m}
\,\bigr)
\end{align*}
$\gamma_{i,i_{1},\dotsc,i_{k}}\colon \Omega^{m}\to\Omega^{m}$ is
the unique arrow that makes the following diagram commute.
where $i'$ ranges $i'\in[1,m]\setminus\{i\}$.
When $\varphi_{i}$ is a modal formula
$\heartsuit_{\lambda}(u_{i_{1}},\dotsc, u_{i_{k}})$,
\begin{align*}
\PT^{E,c}_{i}(p)
\;:=\;\bigl(\,
\stackrel{c}{\relto}
\overline{F}X
\stackrel{p}{\relto}
\overline{F}(\Omega^{m})
\stackrel{J\sigma}{\relto}
\Omega^{m}
\,\bigr)
\end{align*}
where $\sigma$ is...
The equational system $E_{c}$ on the preorder $\pow(\Omega^{m})$,
induced by a simple $\CmuGL$-equational system $E$ and a
$F$-coalgebra $c\colon X\to \pow FX$.
Existential model-checking of “an infinite path.” Sometimes one gets
a useless witness.
§ COALGEBRAIC $\MU$-CALCULUS $\CMU_{\GAMMA,\LAMBDA}$
AS A PROBABILISTIC LINEAR-TIME LOGIC
The sub-Giry monad $\Giry$ is an adaptation of the one
from <cit.>—with subdistributions that can assign to the
whole space a value that is
strictly less than $1$.
[the sub-Giry monad $\Giry$]
Let $\Meas$ denote the (usual) category of measurable sets and
measurable functions.
The sub-Giry monad is the monad $(\Giry,\eta^{\Giry},\mu^{\Giry})$ on $\Meas$ such that
* $\Giry(X,\sigalg_X)=(\Giry X, \sigalg_{\Giry X})$, where
the underling set $\Giry X$ is the set of all subprobability
measures on $(X,\sigalg_X)$. The latter means those measures which assign to the whole
space $X$ a value in the unit interval $[0,1]$.
The $\sigma$-algebra $\sigalg_{\Giry X}$ on $\Giry X$ is the smallest $\sigma$-algebra such that, for all $S\in\sigalg_X$,
the function $\text{ev}_S:\Giry X\to[0,1]$ defined by $\text{ev}_S(P)=P(S)$ is measurable.
* $\Giry f(\nu)(S)=\nu(f^{-1}(S))$ where $f:(X,\sigalg_X)\to(Y,\sigalg_Y)$ is measurable, $\nu\in\Giry X$, and $S\in\sigalg_Y$.
* $\eta^{\Giry}_{(X,\sigalg_X)(x)}$ is given by the Dirac measure: $\eta^{\Giry}_{(X,\sigalg_X)}(x)(S)$ is $1$ if $x\in S$ and $0$ otherwise.
* $\mu^{\Giry}_{(X,\sigalg_X)}(\Psi)(S)=\int_{\Giry (X,\sigalg_X)} \text{ev}_S \,d\Psi$ where
$\Psi\in\Giry^2 X$, $S\in\sigalg_X$ and $\text{ev}_S$ is defined as above.
We use a functor $F\colon \Meas\to\Meas$ for designating the type of
linear-time behaviors; on it we impose the following mild
The same condition has been exploited
in <cit.>; and it
allows us to work in the realm of standard Borel spaces—a class of
“well-behaved” measurable spaces that arise from Polish spaces
(see e.g. <cit.>).
[(standard Borel) polynomial functor]
A standard Borel polynomial functor $F$ on $\Meas$ is defined by the following BNF notation:
\begin{equation*}\textstyle
F\;::=\; \id\,\mid\, (A,\sigalg_A) \,\mid\, F_1\times F_2 \,\mid\, \coprod_{i\in I} F_i\, ,
\end{equation*}
$I$ is a countable set and
$(A,\sigalg_A)\in\Meas$ is a standard Borel
(see e.g. <cit.>).
The set $FX$ has an obvious $\sigma$-algebra $\sigalg_{FX}$ associated
to it,
$\sigalg_{F_1 X\times F_2 X}$ is the smallest $\sigma$-algebra that contains $A_1\times A_2$ for
each $A_1\in\sigalg_{F_1X}$ and $A_2\in\sigalg_{F_2X}$;
$\sigalg_{\coprod_{i\in I} F_i}=\{\coprod_{i\in I} A_i\mid A_i\in
\sigalg_{F_iX}\}$. The action of $F$ on arrows is obvious.
In what follows, a standard Borel polynomial functor is often referred
simply as a polynomial functor.
A (standard Borel) polynomial functor $F$ comes with a canonical distributive
$\lambda\colon F\Giry\Rightarrow\Giry F$ that is defined inductively on
the construction of $F$ <cit.>. It follows by a standard
argument that $F$ has a lifting $\overline{F}\colon
\Kleisli{\Giry}
\to \Kleisli{\Giry}$, in the sense that $\overline{F}$ and $F$ are
compatible via the Kleisli inclusion $J$:
[coalgebra with probabilistic branching]
Let $F$ be a (standard Borel) polynomial functor. An
$F$-coalgebra with probabilistic branching is an arrow $c\colon X\to
\Giry FX$ in $\Meas$, that is equivalently, an arrow $c\colon
X\relto FX$ in the Kleisli category $\Kleisli{\Giry}$.
Note that we in fact allow sub-probabilistic branching due to
the choice of $\Giry$ to be the sub-Giry monad.
The following result is from <cit.> and is closely
related to the observations in <cit.>.
[probabilistic infinite trace semantics $\tr(c)$]
Let $F$ be a (standard Borel) polynomial functor. Then a final
$F$-coalgebra $\zeta\colon Z\to FZ$ exists in $\Meas$. Furthermore,
the lifting $J\zeta\colon Z\relto FZ$—where $J\colon\Meas\to\Kleisli{\Giry}$ is
the Kleisli inclusion functor—exhibits
weak maximal finality. The latter means: for each
$F$-coalgebra $c\colon X\relto FX$ with probabilistic branching, there exists
a coalgebra morphism $f$ from $c$ to $J\zeta$ as in
in $\Kleisli{\Giry}$;
and furthermore, there
exists a maximum one among such $f$ with respect to the pointwise
order between Kleisli arrows $X\relto Z$. Such a maximum morphism from
$c$ to $J\zeta$ shall be denoted by $\tr(c)$.
It turns out that for many examples of the functor $F$ the maximal
homomorphism $\tr(c)$ in Prop. <ref> captures the usual
notion of infinite trace. This is in particular the case for
probabilistic tree automata; see <cit.>
where the correspondence is established in concrete terms.
Let $\varphi$ be a $\CmuGL$-formula, identified with
a simple $\CmuGL$-equational system
\begin{equation}\label{eq:CmuGLEqSysLinearTimeProb}
\dotsc,\quad
u_{m}=_{\eta_{m}} \varphi_{m}.
\end{equation}
Let $c\colon X\to \Giry FX$ be a probabilistic $F$-coalgebra.
In what follows we shall think of the logic $\CmuGL$
for the functor
$F\colon \Meas\to\Meas$ as a “linear-time
logic,” and interpret it
over an $F$-coalgebra $c\colon X\relto
\overline{F}X$ with additional probabilistic branching. Roughly:
a linear-time behavior is an element $z$ of the carrier $Z$
of a final $F$-coalgebra in $\Meas$; and, given a
$\CmuGL$-formula $\varphi$ and a state $x$,
we aim at defining (and computing)
the probability with which a randomly chosen infinite trace $z\in Z$
from $x$ satisfies the formula $\varphi$.
We assume $\Omega$ to be a complete lattice with a measurable structure
$\sigalg_{\Omega}$, and that any subset of $\Omega$ of the form
Ω_> b_0{b∈Ω|b > b_0}
$\Omega_{\ge b_{0}}$,
$\Omega_{< b_{0}}$ or
$\Omega_{\le b_{0}}$ (defined similarly) are measurable.
The semantics
\dotsc,
)\in (\Omega^{X})^{m}$ are all measurable. ...
Not yet. See <cit.>, or references for probabilistic $\mu$-calculus.
Next: use downward closed order
We thank
Kenta Cho,
Tetsuri Moriya,
Shota Nakagawa,
Jurriaan Rot,
Natsuki Urabe
for useful discussions.
I.H. and S.S. are
supported by Grants-in-Aid No. 24680001 & 15KT0012,
C.C. was supported by a Royal Society International Exchanges Grant (IE131642).
§ IN CASE OF PARITY GAMES:
CORRESPONDENCE TO JURDZINSKI'S
Here, as a sanity check, we shall show that our notion of progress
measure (Def. <ref>) instantiates to
Jurdzinski's parity progress measure <cit.>,
in the special case where an
equational system is induced by a parity game.
The following definitions are all standard. See e.g. <cit.>.
[parity game]
A parity game is a quadruple
$G=(X_{\even},X_{\odd},E,\pri)$ of:
a finite set $X_{\even}$ of the
player $\even$'s positions;
a finite set $X_{\odd}$ of the
player $\odd$'s positions;
a transition relation $E\subseteq X\times X$ where
$X:=X_{\even}\cup X_{\odd}$ is the set of all the positions; and
a priority function $\pri\colon X\to\{1,2,\dotsc,d\}$
for some $d\in \omega$. The following are additionally assumed,
mostly for simplicity:
the sets $X_{\even}$ and $X_{\odd}$ are disjoint;
$X$ is nonempty; each position has at least one $E$-successor;
$d$ is an even number. Note, however, that whether $x\in X$ is
$\even$'s position or $\odd$'s is independent from if $\pri(x)\in [1,d]$ is even
or odd.
A play of $G$ is an infinite sequence $x_{0}x_{1}\dotsc$ of
positions such that $(x_{i},x_{i+1})\in E$ for each $i\in \omega$.
A play $x_{0}x_{1}\dotsc$ is winning for the player $\even$
sup{k∈{1,2,…,d} |. k=(x_i) for infinitely
many $i\in\omega$}
is an even number.
A strategy $\sigma$ of the player $\even$ is a function
$\sigma\colon X^{*}\times X_{\even}\to X$,
with the intuition that
$\even$ chooses his move $\sigma(\vec{x}, x)$ depending on the history
$\vec{x}$ of the positions already visited, and the current position
$x\in X_{\even}$.
A play $x_{0}x_{1}\dotsc$ conforms to a strategy $\sigma$ of
$\even$ if, for each $i\in\omega$ such that $x_{i}\in X_{\even}$ we
$x_{i+1}=\sigma(x_{0}x_{1}\dotsc x_{i-1},x_{i})$.
A winning strategy for the player $\even$ is a strategy $\sigma$
such that every play that conforms to $\sigma$ is winning for $\even$.
Finally, a position $x\in X$ is winning
for $\even$ if there exists a winning strategy for $\even$.
The following notion is precisely the one in <cit.>, modulo
some minor modifications that are made for the fit to the current context.
[parity progress measure]
Let $G$ be a parity game $G=(X_{\even},X_{\odd},E,\pri)$; let
$X=X_{\even}\amalg X_{\odd}$. A parity progress
measure for $G$ is a function
, where $\NoGood$ is a fresh symbol,
such that:
* The $a$-th component of the tuple $q(x)$ is never bigger
than the number $n_{2a-1}$ of the positions of the priority $2a-1$. That is,
$\bigl(q(x)\bigr)_{a}\le n_{2a-1}:= |\pri^{-1}(2a-1)|$ (or $q(x)=\NoGood$) for each $a\in
* If $x\in X_{\even}$, then there exists a successor $y$ of $x$
such that:
\begin{align*}
&q(x)\succ_{\pri(x)} q(y) &&\text{if $\pri(x)$ is odd;}
\\
&q(x)\succeq_{\pri(x)} q(y) &&\text{if $\pri(x)$ is even.}
\end{align*}
Here the order
$\succ_{\pri(x)}$ is the same as in
Def. <ref>, except that $\NoGood$ is assumed
to be the greatest element.
* If $x\in X_{\odd}$, then for any successor $y$ of $x$ we have
\begin{align*}
&q(x)\succ_{\pri(x)} q(y) &&\text{if $\pri(x)$ is odd;}
\\
&q(x)\succeq_{\pri(x)} q(y) &&\text{if $\pri(x)$ is even.}
\end{align*}
A parity game gives rise to an equational system. The latter is over
(a product of) the
Boolean lattice $\Bool=\{\ttrue,\ffalse\}$—$\ttrue$ means “$\even$ is winning.”
[equaltional system $E_{G}$ from a game $G$]
Let $G=(X_{\even},X_{\odd},E,\pri)$ be a parity game. For each priority
$i\in [1,d]$, let $n_{i}\in \omega$ be the number of positions with a
priority $i$, that is, $n_{i}=|\{x\in X\mid \pri(x)=i\}|$. Furthermore,
let us fix an enumeration
\begin{equation}\label{eq:201506211434}
\;
\;\dotsc,\;
\end{equation}
of all positions of $G$; it is arranged so that $\pri(x_{i,j})=i$.
The equational system $E_{G}$ induced by $G$ is
with variables $u_{1},\dotsc,u_{d}$—we have one variable for each priority
$i\in [1,d]$—where each variable $u_{i}$ takes its value in the complete
lattice $\Bool^{n_{i}}$.[Therefore we shall use
the extension of the theory developed in the above that allows
different variables $u_{i}$ to take values in different lattices
$L_{i}$. As mentioned just after Def. <ref>, such extension is easy. ]
Concretely, the system $E_{G}$ is of the form
u_1=_η_1 f_1(u⃗) , …,
u_d=_η_d f_d(u⃗)
$\eta_{i}$ and $f_{i}$ are defined as follows.
* The fixed-point symbol $\eta_{i}$ is $\nu$ if $i$ is even; it is $\mu$ if $i$ is odd.
* The monotone function
% \Bool^{|X|}\cong
\Bool^{n_{1}}\times\cdots\times\Bool^{n_{d}}\to \Bool^{n_{i}}$
is defined, for each $j\in [1,n_{i}]$, by:
\begin{align*}
& \bigl(\,
\,\bigr)_{j}
\;:=\;
\\
\begin{cases}
\;\bigsqcup \bigl\{\pi_{j'}(u_{i'})\,\bigl|\bigr.\,
(x_{i,j},x_{i',j'})\in E\bigr\}
&\text{if $x_{i,j}\in X_{\even}$,}
\\
\;\bigsqcap \bigl\{\pi_{j'}(u_{i'})\,\bigl|\bigr.\,
(x_{i,j},x_{i',j'})\in E\bigr\}
&\text{if $x_{i,j}\in X_{\odd}$.}
\end{cases}
\end{align*}
Here $\bigsqcup$ and $\bigsqcap$ denotes a supremum and an infimum, respectively,
in the complete lattice $\Bool$.
(*** Version 1: this is not convenient for the purpose of complexity
bound ***)
Let $G=(X_{\even},X_{\odd},E,\pri)$ be a parity game
\begin{equation}
\;
\;\dotsc,\;
\end{equation}
be an enumeration of all positions of $G$ such that $\pri(x_{i,j})=i$
(therefore the positions are organized in the increasing order with
respect to priorities).
The equational system $E_{G}$ induced by $G$ is one
over the complete lattice $\Bool=\{\ttrue,\ffalse\}$, defined as follows.
Its variables are precisely the positions $x_{1,1},\dotsc,x_{d,m_{d}}$
of $G$,
and it is of the form
\begin{multline*}
x_{1,1}=_{\eta_{1,1}} f_{1,1}(\vec{x})
x_{1,m_{1}}=_{\eta_{1,m_{1}}} f_{1,m_{1}}(\vec{x}),
\; \dotsc,
\\
x_{d,1}=_{\eta_{d,1}} f_{d,1}(\vec{x})
x_{d,m_{d}}=_{\eta_{d,m_{d}}} f_{d,m_{d}}(\vec{x}),
\end{multline*}
where $\vec{x}$ stands for $x_{1,1},\dotsc,x_{d,m_{d}}$. We define
$\eta_{i,j}$ and $f_{i,j}$ by:
* If $i$ is even, then
$\eta_{i,j}=\nu$ and $f_{i,j}(\vec{x}):=\bigsqcup {\{x\mid
(x_{i,j},x)\in E\}} $. Here $\bigsqcup$ denotes a supremum
in the complete lattice $\Bool$.
* If $i$ is odd, then
$\eta_{i,j}=\mu$ and $f_{i,j}(\vec{x}):=\bigsqcap {\{x\mid
(x_{i,j},x)\in E\}} $. Here, similarly, $\bigsqcap$ is an infimum
in $\Bool$.
We are ready to show that our notion of progress measure generalizes
Jurdzinski's <cit.>. We use the extension in Def. <ref>.
Let $G$ be a parity game; let $E_{G}$ be the equational system that
arises from $G$ (Def. <ref>).
Let $x$ be an arbitrary position of $G$, and assume that
$x=x_{i,j}$ in the enumeration (<ref>) (in particular $\pri(x)=i$).
$q\colon X\to \omega^{d/2}$ be a parity
progress measure for $G$
(Def. <ref>) such that $q(x)\neq
\NoGood$. Then this $q$
gives rise to an extended progress
measure $p$
(in the sense of Def. <ref>) such that:
regarding the maximum prioritized ordinal we have
$\overline{\alpha_{a}}\le n_{2a-1}$ for each $a\in [1,d/2]$; and
\,\bigr)_{j}=\ttrue$.
Conversely, let $p$ be
a progress measure
for $E_{G}$ (in the sense of Def. <ref>)
that satisfies
( p_i(α_1,…,α_d/2)
)_j=. It
gives rise to
a parity progress measure $q$ for the parity game $G$ such that
\NoGood$.
In what follows we shall identify an element $p_{i}\in \Bool^{n_{i}}$
with a subset $p_{i}\subseteq \{x_{i,1},\dotsc,x_{i,n_{i}}\}$, and
furthermore identify a tuple $(p_{1},\dotsc,p_{d})\in
\Bool^{n_{1}}\times\cdots\times
\Bool^{n_{d}}$ with a subset $p\subseteq X$.
For the former direction, assume that $q\colon X\to \omega^{d/2}$ is
such a parity progress measure. We define a progress measure $p$ by:
$\overline{\alpha_{a}}:= n_{2a-1}$ for each $a\in [1,d/2]$, and
\begin{align*}
x_{i,j}\in p_{i}(\alpha_{1},\dotsc,\alpha_{d/2})
\;\defiff\;
\end{align*}
Checking that thus defined $p$ is indeed an (extended) progress measure is not
hard. Monotonicity (Cond. <ref> of
Def. <ref>) follows from the transitivity of
Cond. <ref>' (see
Def. <ref>) is easy, too.
To check
Cond. <ref>, let us first consider
the case when
$x_{2a-1,j}\in X_{\odd}$.
\begin{equation}\label{eq:201506211744}
\begin{aligned}
& x_{2a-1,j}\in
\\
\\
\text{for each successor $y$ of $x_{2a-1,j}$,}\;
\\
\\
\text{(by $q(x_{2a-1,j}) \succ_{2a-1}q(y)$)}
\\
\text{for each successor $y$ of $x_{2a-1,j}$,}\;
\\
\\
\text{for each successor $y$ of $x_{2a-1,j}$,}\;
\\
y\in p (\alpha_{1},\dotsc,\alpha_{a},\dotsc,\alpha_{d/2})
\quad\text{(by def.\ of $p$)}
\\
x_{2a-1,j}\in f_{2a-1}
\left(
\begin{array}{c}
p_{1} (\alpha_{1},\dotsc,\alpha_{a},\dotsc,\alpha_{d/2}),
\\
\dotsc,
\\
p_{d} (\alpha_{1},\dotsc,\alpha_{a},\dotsc,\alpha_{d/2})
\end{array}
\right),
\end{aligned}
\end{equation}
as required. Here $(\dagger)$ holds since
$(\alpha_{1},\dotsc,\alpha_{a},\dotsc,\alpha_{d/2})$ is the greatest
among those which are strictly $\prec_{2a-1}$-smaller than the
prioritized ordinal
The case when $x_{2a-1,j}\in X_{\even}$ is similar, replacing the
three occurrences of “each” by “some” in (<ref>).
There is no need of showing
Cond. <ref> since every $\alpha_{a}$
in question is finite.
Cond. <ref> is shown much like
in (<ref>). Finally, that $\bigl(\,p_{i}(\overline{\alpha_{1}},\dotsc,\overline{\alpha_{d/2}})
\,\bigr)_{j}=\ttrue$, that is,
p_{i}(\overline{\alpha_{1}},\dotsc,\overline{\alpha_{d/2}})$, follows
immediately by $q(x)\neq\NoGood$.
For the opposite direction, we first use
Thm. <ref>—soundness, and then
completeness—to obtain a “small” progress measure, that is, one
such that $\overline{\alpha_{a}}\le \ascCL(L_{2a-1})$ for each
$a\in[1,d/2]$. (Recall that we are using an immediate generalization of
Thm. <ref> where different variables
$u_{i}$ are allowed to take values in different lattices $L_{i}$). Now
$L_{2a-1}=\Bool^{n_{2a-1}}$ yields $\ascCL(L_{2a-1})=n_{2a-1}$: in a
strictly ascending chain, each subset is at least one element bigger than
its previous step. This gives us a small progress measure $p$ such that
$\overline{\alpha_{a}}\le n_{2a-1}$.
This $p$ is used to define a parity progress measure
$q\colon X\to \omega^{d/2}$ by:
\begin{align*}
\min\bigl\{\,
\,\bigl|\bigr.\,
x\in p(\seq{\alpha}{d})
\,\bigr\},
\end{align*}
where the minimum is taken with respect to the lexicographic order
$\preceq$ (the latter an element is, the more significant it is), and the minimum of the
empty set is defined to be $\NoGood$.
We still have to check this!
It is easy to check that the $q$ is indeed what is desired.
A position $x=x_{i,j}$ of a parity game $G$ is winning for the player $\even$
if and only if the solution (Def. <ref>) of $E_{G}$
has it that $l^{\sol}_{i}(j)=\ttrue$.
Immediately from
Prop. <ref>,
Thm. <ref> and
the correctness of parity progress measures <cit.>.
§ AN IMPROVED ALGORITHM FOR $\CMUGL$ MODEL CHECKING
Improved algorithm for $\CmuGL$ model checking, in the setting
of Def. <ref> and
Assumption <ref>.
We use a counter $A=\bigl(A(1),\dotsc,A(k)\bigr)\in [0,|X|]^{k}$.
the same as in Algorithm <ref>
the same as in Algorithm <ref>
lines 1–6 of Algorithm <ref>
initialization and Cond. 2
initialize a counter $A=\bigl(A(1),\dotsc,A(k)\bigr)\in [0,|X|]^{k}$
the main loop, executed for each $A$
each $x\in X$ and $i\in [1,m]$
$ R(x,i)=A$
use the counter $A$
$u_{i}$ is a $\mu$-variable and $\varphi_{i}=u_{i'}$
Cond. 3
lines 9–10 of Algorithm <ref>
$u_{i}$ is a $\nu$-variable
Cond. 5
lines 13–25 of Algorithm <ref>
lines 28–30 of Algorithm <ref>
no change is made
update $A$; see Def. <ref>
$\{x\in X\mid
Algorithm <ref> is one that improves
Algorithm <ref>. In it we additionally use the following
In Algorithm <ref>,
the function $\nextPO$ takes a prioritized ordinal
$(\alpha_{1},\dotsc,\alpha_{k})\in [0,|X|]^{k}$ and returns the
smallest prioritized ordinal
(with respect to $\preceq$, the
lexicographic order in Def. <ref>) such that
(α_1,…,α_k) ≺(α'_1,…,α'_k).
We define $\nextPO(|X|,\dotsc,|X|)=(|X|,\dotsc,|X|)$ for convenience.
For example, when $k=3$, $\nextPO(2,|X|,0)=(0,0,1)$.
[correctness of
Algorithm <ref>]
Algorithm <ref> indeed returns
By establishing that Algorithms <ref>
and <ref> yield the same output. Let
$R_{1}(x,i,j)$ denote the value of $R(x,i,j)$ after line 34 of
Algorithm <ref>. It is obvious that,
throughout the execution of
Algorithm <ref>, we have
$R(x,i) \preceq_{i}R_{1}(x,i)$.
The following is not hard to see by induction on $A$. In each iteration of the
main loop, after
line 17,
we have
\begin{equation}\label{eq:201507091337}
R(x,i)\preceq A
\;\Longleftrightarrow\;
\end{equation}
To see it, we proceed in the following way.
* In Algorithm <ref>, every
time we come to line 4,
the set $\{R(x,i)\mid x\in X,i\in [1,m]\}$ is classified
* those which are strictly smaller than $A$ with respect to
$\preceq$, in which case $R(x,i)=R_{1}(x,i)$ by
the induction hypothesis;
* those which are equal to $A$, and
* those which are strictly greater than $A$.
* It suffices to show the following. Assume that $x,i$ are
so that $R(x,i)$ is
in the second
category in the above (i.e. $R(x,i)=A$). If it is
unchanged after execution of lines 5–17, then
* The last claim is not hard to see, by examining how it is
determined whether $R(x,i)$ should be updated or not, in
Algorithms <ref>
and <ref>. Namely, it is
determined based on whether the relevant prioritized
ordinals are strictly greater than $R(x,i)$, and it does
not matter how greater they are.
Since the main loop ranges all possible $A$'s (from
$(0,\dotsc,0)$ to $(|X|,\dotsc,|X|)$),
by (<ref>), after line 19
of Algorithm <ref> we have
$R(x,i)=R_{1}(x,i)$ for all $x,i$. This proves the claim.
In the setting
of Def. <ref> and
Assumption <ref>, the
model-checking problem can be decided in time
\begin{equation}
\left(
\begin{array}{r}
\\
\end{array}
\right) \,\right),
\end{equation}
where $C$ is the constant from Def. <ref>.
In Algorithm <ref> the main loop (lines 4-19)
executes $|X|^{k}$ times. Now consider the for-loop in lines 6–16.
In one execution of the for-loop:
* Lines 8–10 are executed at most $|X|\cdot k$ times and each
takes $O(k(k+1))$ times (see
Def. <ref>), therefore taking overall
|X|k\cdot k (k+1))=
* Lines 11–13 are executed at most $|X|\cdot (m-k)$ times and
each takes time $O(km^{2}|X|^{k+1}+C|X|^{k})$
(see Def. <ref>). Therefore
lines 11–13 incur time consumption
* Line 14 takes time $O(km|X|)$.
Overall, one execution of the for-loop (lines 6–16) takes
km|X|)$ time. This is repeated at most $|X|\cdot m$ times: the
for-loop in lines 6–16 is repeated until no change is made (lines
5–17); and as long as it does not terminate, each execution of the
for-loop makes at least one $R(x,i)$ from $A$ to something strictly
greater (with respect to $\preceq$). Line 18 takes $O(k)$ time and
this is overwhelmed by the other parts.
Combining all these yields the claimed complexity.
§ OMITTED PROOFS
§.§ Proof of Thm. <ref>
The item <ref> (soundness). We shall prove the
following claim $(*)$
by induction on $i$. This obviously suffices, by the definition of $(
\dotsc,
)$ (Def. <ref>).
For each $i\in[1,m]$ the following holds:
for each $j\in [1,i]$, and
for each prioritized ordinal $(\alpha_{1},\dotsc,\alpha_{k})$
such that
\begin{equation}\label{eq:201506161832}
\approximant_{j}(
\alpha_{1}
% \overline{\alpha_{k}}
\alpha_{k}
\sqsubseteq
\left(\,
\begin{array}{c}
\approximant_{i+1} (
\alpha_{1}
\dotsc,
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m} (
\alpha_{1}
\alpha_{k}
\end{array}
\,\right),
\end{equation}
where $l^{(i)}_{j}\colon L^{m-i}\to L$ is the $i$-th interim solution
in Def. <ref>.
In the following proof by induction on $i$,
we distinguish cases, according to whether $u_{i}$ is a $\mu$-variable
or a $\nu$-variable. There is no need of distinguishing the
base case ($i=1$) from the step case: it is easy to take proper care of
the occurrences of $i-1$ in the proof below.
Case: $u_{i}$ is a $\mu$-variable.
Let us choose $a\in [1,k]$ so
that $i=i_{a}$, that is, the variable $u_{i}$ currently in question
has the $a$-th smallest priority among all the $\mu$-variables.
Our proof of the claim $(*)$ shall follow the following path. We will first
consider the special case $(*)_{j=i}$ of the claim $(*)$ where $j$ is
fixed to be $j=i$.
Then we will show the inequality (<ref>) by
(transfinite) induction. The general claim $(*)$ for the other $j\in
[0,i-1]$ will be
derived from this special case $j=i$.
Let us fix $j=i$ in the claim $(*)$; thus we set out to prove
the following claim $(*)_{j=i}$.
$(*)_{j=i}$ For each prioritized ordinal $(\alpha_{1},\dotsc,\alpha_{k})$
such that
\begin{equation}\label{eq:201506162111}
\approximant_{i}(
\alpha_{1}
% \overline{\alpha_{k}}
\alpha_{k}
\sqsubseteq
\left(\,
\begin{array}{c}
\approximant_{i+1} (
\alpha_{1}
\dotsc,
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m} (
\alpha_{1}
\alpha_{k}
\end{array}
\,\right).
\end{equation}
We first note a
that follows from the definition of $l^{(i)}_{i}$
(Def. <ref>) and
Lem. <ref>.<ref>:
the right-hand side $l^{(i)}_{i}(\dotsc)$ of the claimed inequality (<ref>)
is given a lower bound by
where $\alpha$ is an arbitrary ordinal and $\Phi\colon L\to L$ is
defined by
\begin{equation}\label{eq:201506162145}
\begin{aligned}
\Phi(l)
\left(\,
\begin{array}{c}
\\
\approximant_{i+1} (
\alpha_{1}
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m} (
\alpha_{1}
\alpha_{k}
\end{array}
\,\right)
\\
\left(\,
\begin{array}{c}
\\
\approximant_{i+1} (
\alpha_{a+1},
\dotsc,
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m} (
\alpha_{a+1},
\dotsc,
\alpha_{k}
\end{array}
\,\right)
\\
\left(
\begin{array}{c}
\left(
\begin{array}{c}
\\
\approximant_{i+1} (
\alpha_{a+1},
\dotsc,
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m} (
\alpha_{a+1},
\dotsc,
\alpha_{k}
\end{array}
\right),
% \mathrlap{\qquad {\footnotesize \text{1st}}}
\\
\dotsc,
\\
\left(
\begin{array}{c}
\\
\approximant_{i+1} (
\alpha_{a+1},
\dotsc,
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m} (
\alpha_{a+1},
\dotsc,
\alpha_{k}
\end{array}
\right),
% \mathrlap{\qquad {\footnotesize \text{$(i-1)$-th}}}
\\
\\
\approximant_{i+1} (
\alpha_{a+1},
\dotsc,
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m} (
\alpha_{a+1},
\dotsc,
\alpha_{k}
\end{array}
\right),
\end{aligned}
\end{equation}
where the second equality $(\dagger)$ holds due to
Cond. <ref> (monotonicity) of
Def. <ref>, and that
for each $b\in [i+1,m]$ (cf. Def. <ref>).
In particular, the definition of $\Phi$ relies on
but not on $\alpha_{a}$.
We shall show, by
(transfinite) induction on the ordinal $\alpha_{a}$, that
\begin{equation}\label{eq:201506161816}
\approximant_{i}(
\alpha_{1}
\alpha_{a}
\alpha_{k}
\;\sqsubseteq \;
\Phi^{\alpha_{a}}(\bot)
\end{equation}
for each ordinal $\alpha_{a}$ such that $\alpha_{a}\le \overline{\alpha_{a}}$.
If $\alpha_{a}=0$, then the left-hand side
of (<ref>) is $\bot$—this is
Cond. <ref> of
Def. <ref>.
If $\alpha_{a}$ is a successor ordinal
$\alpha'_{a}+1$, then
\begin{align*}
\alpha_{1},\dotsc,
\alpha'_{a}+1,
\alpha_{a+1}
\dotsc,
\alpha_{k}
\\
\sqsubseteq
\left(\,
\begin{array}{c}
\approximant_{1} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,\alpha_{k}),
\\
\dotsc,
\\
\approximant_{i-1} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\approximant_{i} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\approximant_{i+1} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,\alpha_{k})
\end{array}
\,\right)
\\
&\qquad\text{for some ordinals $\beta_{1},\dotsc, \beta_{a-1}$, by
Cond.~\ref{item:progressMeasDefMuVarStepCase} of
\\
\sqsubseteq
\left(\,
\begin{array}{c}
\left(
\begin{array}{c}
\approximant_{i} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\approximant_{i+1} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k})
\end{array}
\right),
\\
\dotsc,
\\
\left(
\begin{array}{c}
\approximant_{i} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\approximant_{i+1} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k})
\end{array}
\right),
\\
\approximant_{i} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\approximant_{i+1} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,\alpha_{k})
\end{array}
\,\right)
\\
&\qquad\text{by the induction hypothesis (the claim $(*)$ for $i-1$)}
\\
\left(\,
\begin{array}{c}
\left(
\begin{array}{c}
\approximant_{i} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\approximant_{i+1} (0,\dotsc,0,
\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (0,\dotsc,0,\alpha_{a+1},\dotsc,
\alpha_{k})
\end{array}
\right),
\\
\dotsc,
\\
\left(
\begin{array}{c}
\approximant_{i} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\approximant_{i+1} (0,\dotsc,0,
\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (0,\dotsc,0,\alpha_{a+1},\dotsc,
\alpha_{k})
\end{array}
\right),
\\
\approximant_{i} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\approximant_{i+1} (0,\dotsc,0,
\alpha_{a+1},\dotsc,
\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (0,\dotsc,0,\alpha_{a+1},\dotsc,
\alpha_{k})
\end{array}
\,\right)
\\
&\qquad\text{by monotonicity
(Cond.~\ref{item:progressMeasDefMonotonicity} of
Def.~\ref{def:progressMeasForEqSys}), much like in $(\dagger)$
\\
\Phi\bigl(\,
\approximant_{i} (\beta_{1},\dotsc,\beta_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k})
\,\bigr)
\\
&\qquad\text{by def.\ of $\Phi$, see~(\ref{eq:201506162145})}
\\
\Phi\bigl(\,
\approximant_{i} (\alpha_{1},\dotsc,\alpha_{a-1},
\alpha'_{a},\alpha_{a+1},\dotsc,
\alpha_{k})
\,\bigr)
\\
&\qquad\text{by monotonicity, much like the second last equality}
\\
\Phi^{\alpha'_{a}+1}(\bot)
\qquad\text{by the induction hypothesis~(\ref{eq:201506161816}).}
\end{align*}
Finally, if $\alpha_{a}$ in (<ref>) is a limit
ordinal, then
\begin{align*}
& \approximant_{i}(
\alpha_{1},\dotsc,
\alpha_{a},
\alpha_{a+1}
\dotsc,
\alpha_{k}
\\
\sqsubseteq
\bigsqcup_{\beta < \alpha_{a}}
\approximant_{i}(
\alpha_{1},\dotsc,
\beta,
\alpha_{a+1}
\dotsc,
\alpha_{k}
\quad
\text{by Cond.~\ref{item:progressMeasDefMuVarLimitCase} of
\\
\sqsubseteq
\bigsqcup_{\beta < \alpha_{a}}
\Phi^{\beta}(\bot)
\quad
\text{by the induction hypothesis~(\ref{eq:201506161816})}
\\
\Phi^{\alpha_{a}}(\bot)
\quad
\text{by def.\ of $\Phi^{\alpha_{a}}$. }
\end{align*}
This concludes the proof that the inequality (<ref>)
holds for any ordinal $\alpha_{a}$. By the fact
that $\Phi^{\alpha_{a}}(\bot)$ is a lower bound for the right-hand
side of (<ref>) (that is
argued after the inequality (<ref>)), we have now shown
that the claim $(*)_{j=i}$ holds.
Finally, the general claim $(*)$ for any $j\in [1,i-1]$ (other than $j=i$) is shown as follows.
\begin{align*}
& \approximant_{j}(
\alpha_{1},\dotsc,
\alpha_{k}
\\
\sqsubseteq
\left(
\begin{array}{c}
\approximant_{i}(
\alpha_{1},\dotsc,
\alpha_{k}
\\
\approximant_{i+1}(
\alpha_{1},\dotsc,
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m}(
\alpha_{1},\dotsc,
\alpha_{k}
\end{array}
\right)
\\
&\qquad\text{by the induction hypothesis (the claim $(*)$ for $i-1$)}
\\
\sqsubseteq
\left(
\begin{array}{c}
\begin{array}{c}
\approximant_{i+1}(\alpha_{1},\dotsc,
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m}(\alpha_{1},\dotsc,
\alpha_{k}
\end{array}
\right),
\\
\approximant_{i+1}(
\alpha_{1},\dotsc,
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m}(
\alpha_{1},\dotsc,
\alpha_{k}
\end{array}
\right)
\\
&\quad\text{by the claim $(*)_{j=i}$ (that we have
already shown for the current $i$)}
\\
\left(
\begin{array}{c}
\approximant_{i+1}(
\alpha_{1},\dotsc,
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m}(
\alpha_{1},\dotsc,
\alpha_{k}
\end{array}
\right)
\\
&\quad\text{by the
of $l^{(i)}_{j}$; see Def.~\ref{def:solOfEqSys}.}
\end{align*}
This concludes one case of our proof of the claim $(*)$
(by induction on $i$), where $u_{i}$ is a $\mu$-variable.
Case: $u_{i}$ is a $\nu$-variable.
Let us choose $a\in [1,k]$ so
i_1<⋯<i_a-1<i<i_a<⋯< i_k.
We shall prove the special case of $(*)$ where $j=i$. The claim $(*)$
for general $j\in [1,i]$ follows from this special case, much like in
the above for the case where $u_{i}$ is a $\mu$-variable.
By the definition of $l^{(i)}_{i}$
(Def. <ref>) and
Lem. <ref>.<ref>,
showing the following (i.e.
$ \approximant_{i}(
\alpha_{1}
\alpha_{k}
is a suitable postfixed point) suffices for establishing
the desired inequality (<ref>):
\begin{equation}\label{eq:201506162250}
\approximant_{i}(
\alpha_{1}
\alpha_{k}
\sqsubseteq
\left(
\begin{array}{c}
\approximant_{i}(
\alpha_{1}
\alpha_{k}
\\
\approximant_{i+1}(
\alpha_{1}
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m}(
\alpha_{1}
\alpha_{k}
\end{array}
\right).
\end{equation}
We proceed as follows.
\begin{align*}
\approximant_{i}(
\alpha_{1}
\alpha_{k}
\\
\left(\,
\begin{array}{c}
\approximant_{1} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a},\dotsc,\alpha_{k})
\end{array}
\,\right)
\\
&\qquad\text{for some ordinals $\beta_{1},\dotsc, \beta_{a-1}$, by
Cond.~\ref{item:progressMeasDefNuVar} of
\\
\sqsubseteq
\left(
\begin{array}{c}
\left(
\begin{array}{c}
\approximant_{i} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a}\dotsc,
\alpha_{k}),
\\
\approximant_{i+1} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a}\dotsc,
\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a}\dotsc,
\alpha_{k})
\end{array}
\right),
\\
\dotsc,
\\
\left(
\begin{array}{c}
\approximant_{i} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a}\dotsc,
\alpha_{k}),
\\
\approximant_{i+1} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a}\dotsc,
\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a}\dotsc,
\alpha_{k})
\end{array}
\right),
\\
\approximant_{i} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a}\dotsc,
\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a}\dotsc,\alpha_{k})
\end{array}
\right)
\\
&\qquad\text{by the induction hypothesis (the claim $(*)$ for $i-1$)}
\\
\left(
\begin{array}{c}
\approximant_{i} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a}\dotsc,
\alpha_{k}),
\\
\dotsc,
\\
\approximant_{m} (\beta_{1},\dotsc,\beta_{a-1},
\alpha_{a}\dotsc,\alpha_{k})
\end{array}
\right)
\\
&\qquad\text{by def.\ of $f^{\ddagger}_{i}$ (Def.~\ref{def:solOfEqSys})}
\\
\left(
\begin{array}{c}
\approximant_{i}(
\alpha_{1}
\alpha_{k}
\\
\dotsc,
\\
\approximant_{m}(
\alpha_{1}
\alpha_{k}
\end{array}
\right),
\end{align*}
where the last equality is, once again, a consequence of
(Cond. <ref> of
Def. <ref>) and that
. This concludes the proof of
Thm. <ref>.<ref>.
The item <ref> (completeness).
The proof is by induction on the number $m$ of equations in the equational
system $E$.
For each $l\in L$, let $E^{(l)}$ be
the equational system obtained from
$E$ in (<ref>) by removing the last equation and
substituting $l$ for the last variable $u_{m}$. That is,
\begin{equation}\label{eq:truncatedEquation}
\begin{aligned}
& E^{(l)}
\;:=\;
\left[
\begin{array}{c}
u_{1} =_{\eta_{1}} f_{1}(u_{1},\dotsc, u_{m-1}, l),
\\
\dotsc,
\\
u_{m-1} =_{\eta_{m-1}} f_{m-1}(u_{1},\dotsc, u_{m-1}, l)
\end{array}
\right].
\end{aligned}
\end{equation}
For the rest of the proof we distinguish cases, depending on whether the last variable $u_{m}$
is a $\mu$-variable or a $\nu$-variable.
Case: $u_{m}$ is a $\mu$-variable.
By the induction hypothesis there exists a progress measure that
achieves the exact solution of the system $E^{(l)}$, for each $l\in L$.
Such a progress measure shall be denoted by
\begin{align*}
& p^{(l)}\;=\;
\\
\bigl(\,
\overline{\alpha^{(l)}_{1}},\dotsc,
\overline{\alpha^{(l)}_{k-1}}),
\,
\bigl(\,\approximant^{(l)}_{i}(\alpha_{1},\dotsc,\alpha_{k-1})\,\bigr)_{i\in
\,\bigr).
\end{align*}
(Note here that $E^{(l)}$ has $k-1$ $\mu$-variables, since $E$ has $k$ of
such and we assumed that $u_{m}$ is a $\mu$-variable.)
It is easily seen to satisfy, for each $i\in [1,m-1]$:
\begin{equation}\label{eq:201506181551}
\bigl(\,
\overline{\alpha^{(l)}_{1}},\dotsc,
\overline{\alpha^{(l)}_{k-1}}\,\bigr)
\;= \;
\end{equation}
where $l^{(m-1)}_{i}(l)$ is (a component of) the $(m-1)$-th
interim solution of the original
equational system $E$ (Def. <ref>).
Furthermore, by induction, we assume that
α^(l)_a ≤(L)
for each $a\in[1,k-1]$.
Using the above data, we shall construct a desired progress measure $p$
for the system $E$. We define its approximants
$p_{i}(\alpha_{1},\dotsc,\alpha_{k})$ by induction on the ordinal
$\alpha_{k}$. For the base case:
\begin{align*}
\\
p^{( p_{m}
\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\qquad\text{for each $i\in[1,m-1]$.}
\end{align*}
For the step case we
the function $f^{\ddagger}_{m-1}$ from Def. <ref>
and define
\begin{align*}
\,\bigr)
\\
% p^{(\,p_{m}
% (\alpha_{1},\dotsc,\alpha_{k-1},\alpha_{k}+1)
% \,)}_{i}
% (\alpha_{1},\dotsc,\alpha_{k-1})
\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!
\,)}_{i}
\quad\text{for each $i\in[1,m-1]$.}
\end{align*}
For the limit case, similarly, we
\begin{equation}\label{eq:201506181814}
\begin{aligned}
\bigsqcup_{\beta<\alpha_{k}}
\\
\,)}_{i}
\\
&\qquad\text{for each $i\in[1,m-1]$.}
\end{aligned}
\end{equation}
In the above definitions it may happen that
$p^{(l)}(\alpha_{1},\dotsc,\alpha_{k-1})$ is not defined because
the ordinals $\alpha_{1},\dotsc,\alpha_{k-1}$ exceed the domain of
$p^{(l)}$, that is,
$\overline{\alpha^{(l)}_{i}}< \alpha_{i}$ for some $i\in[1,k-1]$.
In such a case we use, in place of $(\alpha_{1},\dotsc,\alpha_{k-1})$,
the greatest prioritized ordinal that is smaller than it (with respect
the lexicographic order $\preceq$ in Def. <ref>).
Regarding the maximum prioritized ordinal
$(\overline{\alpha_{1}},\dotsc, \overline{\alpha_{k}})$ of $p$:
* On the ordinal $\overline{\alpha_{k}}$ for the last
$\mu$-variable $u_{m}$, the approximants for $u_{m}$ form a (transfinite) ascending chain
\begin{equation}\label{eq:201506181609}
\;\sqsubseteq\;
\;\sqsubseteq\;\cdots\enspace;
\end{equation}
because the chain is nothing but $\bot\sqsubseteq f^{\ddagger}_{m}(\bot)
\sqsubseteq f^{\ddagger}_{m}(f^{\ddagger}_{m}(\bot))
\sqsubseteq\cdots$.
This chain in the complete lattice $L$ eventually stabilizes (bounded
by the ordinal $\ascCL(L)$, by the definition of the latter); we let $\overline{\alpha_{k}}$ to be
an ordinal, such that $\alpha_{k}\le \ascCL(L)$, at which the chain (<ref>) has
* On the other ordinals $\overline{\alpha_{a}}$ for $a\in [1,k-1]$,
we define
\begin{equation}\label{eq:201506181759}
\overline{\alpha_{a}}
\bigvee_{\beta\le \overline{\alpha_{k}}}
\overline{\alpha^{( p_{m}
\end{equation}
it is obvious, from the induction hypothesis that
α^(l)_a ≤(L)
for each $l\in L$ and $a\in[1,k-1]$, that $\overline{\alpha_{a}}$
in (<ref>) is no bigger than $\ascCL(L)$.
We have to check that the data $p$ thus defined is indeed a progress
measure (Def. <ref>). On
Cond. <ref> (monotonicity): it holds if
$i=m$ because (<ref>) is an ascending chain;
for $i\in[1,m-1]$ the claim follows from the induction hypothesis
that $p^{(l)}$ is a progress measure for $E^{(l)}$.
Cond. <ref> is easy, distinguishing
cases for $a=k$ (obvious by definition) and $a\in[1,k-1]$ (by the
induction hypothesis).
On Cond. <ref>, let $a=k$ (hence $i_{a}=m$).
\begin{align*}
& p_{m}
\\
\,\bigr)
\quad\text{by definition}
\\
\left(\,
\begin{array}{c}
\left(\,
\approximant_{m}
\,\right),
\\
\dotsc,
\\
\left(\,
\approximant_{m}
\,\right),
\\
\approximant_{m}
\end{array}
\,\right)
\\
\left(\,
\begin{array}{c}
p^{( \approximant_{m}
\\
\dotsc,
\\
p^{( \approximant_{m}
\\
\approximant_{m}
\end{array}
\,\right)
\\
\text{for some ordinals $\beta_{1},\dotsc,\beta_{k-1}$, by~(\ref{eq:201506181551})}
\\
\left(\,
\begin{array}{c}
p^{( \approximant_{m}
\\
\dotsc,
\\
p^{( \approximant_{m}
\\
\approximant_{m}
\end{array}
\,\right)
\\
\left(\,
\begin{array}{c}
\\
\dotsc,
\\
\\
\approximant_{m}
\end{array}
\,\right)
\\
&\qquad\text{by def.\ of $p_{1},\dotsc,p_{m-1}$,}
\end{align*}
as required. Here the equality $(\dagger)$ holds since the first $k-1$ arguments do
not matter in the definition of $\approximant_{m}$, that is,
(this is easily seen by induction on $\alpha_{k}$).
On Cond. <ref>, if $a\in[1,k-1]$, the
claim follows immediately by the induction hypothesis.
Cond. <ref> is easy, by definition
for $a=k$ and by the induction hypothesis for $a\in
[1,k-1]$. Cond. <ref> is easy, too, by the
induction hypothesis (we have assumed that $u_{m}$ is a
Finally, we have to show that the progress measure $p$ defined in the
indeed achieves the exact solution, that is,
for each $i\in[1,m]$. For $i=m$ this is easy: $ l^{\sol}_{m}$
is characterized as the supremum of the chain
$\bot\sqsubseteq f^{\ddagger}_{m}(\bot)
\sqsubseteq f^{\ddagger}_{m}(f^{\ddagger}_{m}(\bot))
\sqsubseteq\cdots$ (Def. <ref> and
Lem. <ref>.<ref>); and the
last chain coincides, by definition, with the one
in (<ref>).
For the other $i$ (i.e. $i\in [1,m-1]$):
\begin{align*}
\quad\text{by Def.~\ref{def:solOfEqSys}}
\\
\bigl(\,
\overline{\alpha^{(l^{\sol}_{m})}_{1}},\dotsc,
\overline{\alpha^{(l^{\sol}_{m})}_{k-1}}\,\bigr)
\quad\text{by~(\ref{eq:201506181551})}
\\
p^{( \approximant_{m}(
\overline{\alpha_{1}},\dotsc,
\overline{\alpha_{k}})
\bigl(\,
\overline{\alpha^{(\approximant_{m}(
\overline{\alpha_{1}},\dotsc,
\overline{\alpha_{k}}))}_{1}},\dotsc,
\overline{\alpha^{(\approximant_{m}(
\overline{\alpha_{1}},\dotsc,
\overline{\alpha_{k}}))}_{k-1}}\,\bigr)
\\
p^{( \approximant_{m}(
\overline{\alpha_{1}},\dotsc,
\overline{\alpha_{k}})
\bigl(\,
\overline{\alpha_{1}},\dotsc,
\overline{\alpha_{k-1}}\,\bigr)
\\
&\qquad\text{by the convention we adopted just after (\ref{eq:201506181814})}
\\
\overline{\alpha_{1}},\dotsc,
\overline{\alpha_{k}})
\quad\text{by def.\ of $p_{i}$.}
\end{align*}
This concludes the case where
$u_{m}$ is a $\mu$-variable.
Case: $u_{m}$ is a $\nu$-variable.
The proof in this case is similar and simpler. We shall therefore
only its main points; further details can be easily filled out.
By the induction hypothesis there exists a progress measure $p^{(l)}$ that
achieves the exact solution of the system $E^{(l)}$, for each $l\in L$:
\begin{align*}
& p^{(l)}\;=\;
\\
\bigl(\,
\overline{\alpha^{(l)}_{1}},\dotsc,
\overline{\alpha^{(l)}_{k}}),
\,
\bigl(\,\approximant^{(l)}_{i}(\alpha_{1},\dotsc,\alpha_{k})\,\bigr)_{i\in
\,\bigr).
\end{align*}
(Note that $E^{(l)}$ has $k$ $\mu$-variables.)
We use such $p^{(l)}$ to define a desired progress measure $p$ of $E$.
Its approximants are defined by:
\begin{align*}
\\
\,)}_{i}
\quad\text{for each $i\in[1,m-1]$.}
\end{align*}
the maximum prioritized ordinal
$(\overline{\alpha_{1}},\dotsc, \overline{\alpha_{k}})$, we define
\overline{\alpha^{(l^{\sol}_{m})}_{a}}
$ for each $a\in [1,k]$. Obviously, from the induction hypothesis,
$\overline{\alpha_{a}}$ can be chosen so that
$\overline{\alpha_{a}}\le \ascCL(L)$.
In seeing that $p$ is indeed a progress measure, checking the conditions
of Def. <ref> is mostly straightforward by the
induction hypothesis.
The only nontrivial point is Cond. <ref>, in
case $i=m$. We have to show that
\begin{equation}\label{eq:201506181840}
\;\sqsubseteq\;
\left(\,
\begin{array}{c}
\approximant_{1} (\beta_{1},\dotsc,\beta_{k}),
\\
\dotsc,
\\
\approximant_{m-1} (\beta_{1},\dotsc,\beta_{k}),
\\
\end{array}
\,\right)
\end{equation}
for some ordinals $\seq{\beta}{k}$ (note that all the $\mu$-variables
have priorities smaller than that of $u_{m}$).
We set $\beta_{a}:=\overline{\alpha^{(l^{\sol}_{m})}_{a}} $ for each
$a\in [1,k]$. Then the right-hand side of (<ref>)
\begin{align*}
\left(\,
\begin{array}{c}
\approximant^{(l^{\sol}_{m})}_{1} (
\overline{\alpha^{(l^{\sol}_{m})}_{1}}
\overline{\alpha^{(l^{\sol}_{m})}_{k}}
\\
\dotsc,
\\
\approximant^{(l^{\sol}_{m})}_{m-1} (
\overline{\alpha^{(l^{\sol}_{m})}_{1}}
\overline{\alpha^{(l^{\sol}_{m})}_{k}}
\\
\end{array}
\,\right)
\\
\bigl(\,
\dotsc,\,
\,\bigr)
\\
f^{\ddagger}_{m}(l^{\sol}_{m} )
\quad\text{by Def.~\ref{def:solOfEqSys}}
\\
\end{align*}
where the last equality is because of the definition of
$l^{\sol}_{m} = l^{(m)}_{m}$ as a greatest fixed point
(Def. <ref>). This proves (<ref>).
It is straightforward to show that the progress measure $p$ indeed
achieves the exact solution. This completes the proof.
§.§ Proof of Prop. <ref>
The structure of the proof remains the same; we shall focus on what is
Let us denote the function $\Phi$ in (<ref>) by
$\Phi_{\alpha_{a+1},\dotsc,\alpha_{k}}$, so that its dependence on
$\alpha_{a+1},\dotsc,\alpha_{k}$ becomes explicit.
We shall prove, instead of (<ref>), the following claim:
\begin{equation}\label{eq:201506211659}
\approximant_{i}(
\alpha_{1}
\alpha_{a}
\alpha_{k}
\;\sqsubseteq \;
\quad\text{for some $\alpha$}
\end{equation}
by transfinite induction on a tuple
$\alpha_{a},\alpha_{a+1},\dotsc,\alpha_{k}$ (ordered lexicographically
with $\preceq$,
with the latter being the more significant).
For one base case, assume $\alpha_{a}=0$ and
$p_{i_{a}}(\alpha_{1},\dotsc,\alpha_{k})=\bot$. For example this must be
the case, by Def. <ref>, when
$\alpha_{a}=\alpha_{a+1}=\cdots=\alpha_{k}=0$—this is because
there is no
$(\alpha'_{1},\dotsc,\alpha'_{k})$ such that
In this case (<ref>)
holds with $\alpha=0$.
For the other base case, assume that $\alpha_{a}=0$
and there exists
$(\alpha'_{1},\dotsc,\alpha'_{k})$ such that
(\alpha_{1},\dotsc,\alpha_{k})$ and
$p_{i_{a}}(\alpha_{1},\dotsc,\alpha_{k}) \sqsubseteq
The former condition means
(\alpha_{a},\dotsc,\alpha_{k})$ with the lexicographic order $\prec$;
by the induction hypothesis, therefore, we have
\alpha'_{1}
\alpha'_{a}
\alpha'_{k}
\;\sqsubseteq \;
$ for some $\alpha'$. It follows easily from Cond. <ref> that
$\Phi_{\alpha_{a+1},\dotsc,\alpha_{k}}$ is monotonic with respect to
the ordinals $\alpha_{a+1},\dotsc,\alpha_{k}$. Summarizing, we have
\begin{multline*}
p_{i_{a}}(\alpha_{1},\dotsc,\alpha_{k}) \sqsubseteq
\\
\sqsubseteq
\sqsubseteq
\end{multline*}
showing the claim.
The other cases (when $\alpha_{a}$ is a successor or limit ordinal) are
the same as in the proof of Thm. <ref>.
This concludes the proof.
§.§ Proof of Lem. <ref>
We prove the following (more general) statement by induction on $\varphi$:
for any formula $\varphi$ and for each $V'_{1},\dotsc,V_{m}\colon Y\to \Omega$, we
\begin{equation}\label{eq:201507061247}
\sem{\varphi}_{c}(f^{*}(V'_{1}),\dotsc,f^{*}(V'_{m}))
= f^{*}\bigl(\,
\sem{\varphi}_{d}(V'_{1},\dotsc,V'_{m})
\,\bigr)
\end{equation}
where $f^{*}(V')$ is defined by $X\stackrel{f}{\to}
Y\stackrel{V'}{\to} \Omega$.
For the cases other than fixed-point operators the proof is as usual
in coalgebraic modal logic. For the modal formula case we exploit the
naturality of $\lambda$.
For the case of a $\mu$-formula $\mu u.\,\varphi$, we shall rely on the Cousot-Cousot
characterization of least fixed points and prove the following
by transfinite induction on the ordinal $\alpha$:
\begin{equation}\label{eq:201507061251}
\begin{aligned}
\place)\,\bigr]^{\alpha}(\bot_{X\to \Omega})
\bigl[\,\sem{\varphi}_{d}(\vec{V'},
\place)\,\bigr]^{\alpha}(\bot_{Y\to \Omega})
\,\bigr)
\quad
\text{for each ordinal $\alpha$.}
\end{aligned}
\end{equation}
The base case follows from the induction hypothesis (for the proof
of (<ref>)), since $f^{*}(\bot_{Y\to
\Omega})=\bot_{X\to \Omega}$. For the step case,
\begin{align*}
& \bigl[\,\sem{\varphi}_{c}(\overrightarrow{f^{*}(V')},
\place)\,\bigr]^{\alpha+1}(\bot_{X\to \Omega})
\\
\bigl(\,\sem{\varphi}_{c}(\overrightarrow{f^{*}(V')},
\place)\,\bigr)
\Bigl(\, f^{*}\bigl(\,
\bigl[\,\sem{\varphi}_{d}(\vec{V'},
\place)\,\bigr]^{\alpha}(\bot_{Y\to \Omega})
\,\bigr)
\,\Bigr)
\\&
\quad\text{by ind.\ hyp. (for~(\ref{eq:201507061251}))}
\\
\sem{\varphi}_{d}
\bigl(\,\vec{V'},\,
\bigl[\,\sem{\varphi}_{d}(\vec{V'},
\place)\,\bigr]^{\alpha}(\bot_{Y\to \Omega})
\,\bigr)
\;\text{by ind.\ hyp. (for~(\ref{eq:201507061247}))}
\\
\bigl[\,\sem{\varphi}_{d}(\vec{V'},
\place)\,\bigr]^{\alpha+1}(\bot_{Y\to \Omega})
\,\bigr),
\end{align*}
as required. For the limit case (when $\alpha$ is a limit ordinal)
the claim follows from the fact that $f^{*}$ preserves supremums.
The case of a $\nu$-formula is symmetric to the last case. This
concludes the proof.
§.§ Proof of Lem. <ref>
Straightforward from the naturality of $\lambda$ along the arrow
\Omega^{m}\to \Omega^{n}$. Specifically, consider the diagram
where $f$ stands for $\tuple{\pi_{j_{1}},\dotsc,\pi_{j_{n}}}$.
Starting from the element $\tuple{\pi_{1},\dotsc,\pi_{n}}\in
(\Omega^{\Omega^{n}})^{n}$ on the top-left corner proves the claim.
§.§ Proof of Thm. <ref>
In view of Prop. <ref> and
Thm. <ref>, it suffices to show that:
* an MC progress measure
(Def. <ref>) gives rise to
* a progress
measure (in the sense of Def. <ref>) for the
equational system $E_{\varphi,c}$ over $\Omega^{X}$ that arises from
$\varphi$ and $c$,
and vice versa.
The correspondence between the two notions is straightforward by
(un)Currying. In particular,
Thm. <ref>.2 gives the bound for a
maximal ordinal $\alpha$ by $\ascCL(\Omega^{X})$.
We must check that Cond. 1–5
(in each notion) are suitably transferred to each other;
we shall focus on the cases handled in
Cond. 5(c) of Def. <ref>. The other cases are straightforward.
It is not hard to see that, for this case, what needs to be shown is
the following claim that informally reads
“$\PT_{\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})}$ properly
imitates the semantics of
\begin{equation}\label{eq:201507051408}
\begin{aligned}
FQ\co c\,\bigr)(x)
\\&\qquad
\lambda_{X}\bigl(
\pi_{j_{1}}\co\ev(\vec{\alpha'})\co Q,\dotsc, \pi_{j_{n}}\co\ev(\vec{\alpha'})\co Q
\bigr) (c(x))
% \colon X\to \Omega
\end{aligned}
\end{equation}
for each $x\in X$. Here recall that $\pi_{j}\co\ev(\vec{\alpha'})\co
Q$ is of type $X \to \Omega$, and $\lambda_{X}\colon (\Omega^{X})^{n}\to
\Omega^{FX}$; the right-hand side of (<ref>) therefore coincides with
Q)$ (see Def. <ref>).
Now let us prove the equality (<ref>). First notice
the naturality of $\lambda$, where we write $Q'$ for $\ev(\vec{\alpha'})\co
\begin{equation}\label{eq:201507051429}
\vcenter{\xymatrix@R=1em@C+2em{
\ar[r]^-{\lambda_{\Omega^{m}}}
\ar[d]_{(\Omega^{Q'})^{n}}
\ar[d]^{\Omega^{FQ'}}
\\
\ar[u]_{Q'}
\ar[r]_-{\lambda_{X}}
% \mathrlap{\enspace,}
\end{equation}
that is used in:
\begin{align*}
FQ\co c\,\bigr)(x)
\\
\bigl(\,\lambda^{\tuple{j_{1},\dotsc, j_{n}}}\co F(\ev(\vec{\alpha'}))\co
FQ\co c\,\bigr)(x)
\\
&\qquad\quad\text{by def.\ of $\PT_{\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})}(\vec{\alpha'})$}
\\
\bigl(\,\lambda^{\tuple{j_{1},\dotsc, j_{n}}}\co FQ'\co c\,\bigr)(x)
\quad\text{by $Q'=\ev(\vec{\alpha})\co Q$}
\\
\bigl(\,\lambda_{\Omega^{m}}(\pi_{j_{1}},\dotsc, \pi_{j_{n}})\co FQ'\co c\,\bigr)(x)
\quad\text{by $Q'=\ev(\vec{\alpha})\co Q$}
\\
\bigl(\,\lambda_{X}(\pi_{j_{1}}\co Q',\dotsc, \pi_{j_{n}}\co Q')\co c\,\bigr)(x)
\quad\text{by~(\ref{eq:201507051429}),}
\end{align*}
as required. This proves (<ref>) and hence the
§.§ Proof of Thm. <ref>
The proof, much like in Appendix <ref>, is by showing that MPMs and MC
progress measures induce each other. We then appeal to
Thm. <ref> to obtain the
The mutual construction between an MPM $R$ and an MC
progress measure $Q$ is much like in
Prop. <ref>. The essence
is: for any prioritized ordinal $(\alpha_{1},\dotsc,\alpha_{k})$,
and a row $(\NoGood,\dotsc,\NoGood)$ in an MPM handles an
exceptional case that
$(Q(x))_{i}(\alpha_{1},\dotsc,\alpha_{k})=\ffalse$ for every
Then Cond. 1–5 of both notions are easily seen to be mutually
transferred. Note that, for MPMs, Cond. 1 is not needed.
§.§ Proof of Thm. <ref>
Let $R_{0}\colon X\to \POM_{|X|}$ be the optimal MPM guaranteed in
Thm. <ref>.2. It is easy to see that, at any time
during the execution of the algorithm, we have $R(x,i,j)\le
(R_{0}(x))^{(i)}_{j}$ for any $x,i,j$. Here $\le$ is the usual
inequality between natural numbers, where $\NoGood$ is deemed to be
the greatest.
Therefore we have
\begin{equation}\label{eq:201507082245}
\{x\in X\mid
\supseteq
\{x\in X\mid
\end{equation}
It is also easy to see that, once the algorithm
the data $\bigl(R(x,i,j)\bigr)_{x,i,j}$ defines an MPM. By
Thm. <ref>.1 (soundness) we have
the opposite inclusion $\subseteq$
in (<ref>). This proves the claim.
§.§ Proof of
Thm. <ref>
It can be easily seen that each iteration of the main loop (lines
7–33) strictly increases $R(x,i)$ for at least one $(x,i)$ with respect to the
preorder $\preceq_i$ (except for the last iteration). Since each
$R(x,i)$ belongs to $\bigl[0,|X|\,\bigr]^{k}\amalg\{(\NoGood,\dotsc,\NoGood)\}$, each $R(x,i)$ increases at most $(|X|+1)^k$
times. There are $m|X|$ of $(x,i)$'s; therefore the main loop iterates at most
$m|X|(|X|+1)^k$ times. It is obvious that inner loop (lines 8–32) iterates $m|X|$
The complexities of lines 9–12 and lines 13–15 are $O(k)$, and those
of lines 16–18 and lines 19–22 are $O(km)$, by bounding $n$ by
$m$. The complexity of lines 23–25 is $O(km^{2}|X|^{k+1}+C|X|^{k})$,
as noted in Definition <ref>; it dominates the overall
complexity of the inner loop. From these
we derive the claimed complexity.
§.§ Proof of
Thm. <ref>
For the item <ref> (soundness),
the desired infinitary trace $z$ is obtained by $z:=\beh(q)(y)\in
Z$, where $\beh(q)$ is from (<ref>).
We shall first establish that $z$ is indeed an
infinitary trace of $c$ from $x$, that is, $z\in \tr(c)(x)$.
* Cond. 6 of
Def. <ref> asserts that $Js$ is
a backward Kleisli simulation, a notion
from <cit.>. Its soundness
against infinitary trace semantics—the latter being coalgebraically formalized
in Prop. <ref>—has been
established in <cit.>, under the
conditions of nonemptiness and
image-finiteness. These conditions are obviously
by the arrow $Js\colon Y\relto X$ in $\Kleisli{\pow}$, since
it is the graph relation of a function $s\colon Y\to X$.
From this we conclude that the inequality
holds; see <cit.>.
* Next we compare two arrows $\tr(Jq)$ and $J(\beh(q))$ of the type
$Y\relto Z$. We aim at $\tr(Jq)\supseteq J(\beh(q))$.
By the characterization of $\tr(Jq)$ as a greatest fixed point
(Prop. <ref>), it suffices to show
that $J(\beh(q))$ is a fixed point
of the function $\Psi$
in (<ref>). This
is shown as follows.
\begin{align*}
\\
(J\zeta)^{-1}\odot \oF J\beh(q)\odot Jq
\\
(J\zeta)^{-1}\odot J(F\beh(q) \co q)
\quad\text{by $\oF J=JF$, see
\\
(J\zeta)^{-1}\odot J(\zeta\co \beh(q))
\quad\text{by def.~(\ref{eq:behq}) of $\beh(q)$}
\\
\end{align*}
Combining the two items in the above, we conclude
(c)⊙Js ⊇ J(q)
and equivalently $\beh(q)(y)\in \tr(c)(s(y))=\tr(c)(x)$ because $(J\beh(q))(y)=\bigl\{\beh(q)(y)\bigr\}$.
It remains to be shown that
It is crucial here that $r\colon Y\to \pPMpa$ forms an
MC progress measure
(Def. <ref>) for $\varphi$ over
$q\colon Y\to FY$. This fact is obvious when one
compares Cond 5(c) in
Def. <ref>
and <ref>. It follows
from Thm. <ref> that
Then our goal follows from the fact that
$\sem{\varphi}_{q}(y)=\sem{\varphi}_{\zeta}(z)$; the latter is a
consequence of
Lem. <ref>. This concludes the proof of
the item 1 (soundness).
For the item 2 (completeness),
let us first fix an optimal
progress measure $Q^\zeta\colon Z\to \pPMpa$, i.e. one such that
for each $z'\in Z$. By
Thm. <ref>.<ref>
such $Q^{\zeta}$ exists.
We define an LTMC progress
as follows.
For each infinitary
trace $z\in Z$, there exists an MC progress
measure $p^{(z)}$ such that
(by completeness,
Thm. <ref>).
An ordinal $\alpha$ is chosen so that $\alpha^{(z)}\le
\alpha$ for each $z\in Z$; this is possible since $Z$ is a
(small) set.
We define $Y$ by
Y:={ (x',z')∈X×Z |. z'∈(c)(x') }.
The functions $r$ and $s$ are defined by $r(x',z'):=
Q^{\zeta}(z')$ and $s(x',z'):=x'$.
The construction of the coalgebra structure $q\colon Y\to FY$ is as follows.
Let $y'=(x',z')\in Y$ be an arbitrary element of $Y$, so that $z'\in \tr(c)(x')$. We can pick $t\in FX$
such that: $t\in c(x')$ and
\begin{equation}\label{eq:201507061746}
\zeta(z')\in\bigl(\xi_{Z}\co F(\tr(c))\bigr)(t)
\quad\text{where }
FX\stackrel{F(\tr(c))}{\to} F\pow Z \stackrel{\xi_{Z}}{\to} \pow FZ.
\end{equation}
Recall that $\xi\colon F\pow\Rightarrow \pow F$ is a distributive
law (Def. <ref>).
Indeed, $z'\in \tr(c)(x')$ implies
\begin{equation}\label{eq:201507061748}
\begin{aligned}
\zeta(z')&\in
(\mu^{\pow}_{FZ}\co \pow \xi_{Z}\co \pow F\tr(c)\co
\\
\text{by def.~(\ref{eq:nondetCoalgTraceSem}) of $\tr(c)$, expanded
in $\Sets$}
\\
&= \bigl(\,\mu^{\pow}_{FZ}\co \pow (\xi_{Z}\co
\\
&= \textstyle\bigcup_{t\in c(x')} (\xi_{Z}\co F\tr(c))(t)
\\
\text{by def.\ of $\mu^{\pow}$ (union) and $\pow$'s
action on arrows (direct image);}
\end{aligned}
\end{equation}
hence there must be some $t\in c(x')$ such that (<ref>)
Now consider the following diagram:
\begin{equation}\label{eq:201507062152}
\vcenter{\xymatrix@R=.8em{
\ar[r]^-{F\tuple{X,\tr(c)}}
\ar@/_/@{-->}[rrd]
{F(X\times \pow Z)}
\ar[r]^-{F\str}
{F\pow (X\times Z)}
\ar[r]^-{\xi_{X\times Z}}
{\pow F(X\times Z)}
\\
{F\pow Y}
\ar[u]_{F\pow \iota}
\ar[r]_-{\xi_{Y}}
{\pow F Y}
\ar[u]^{\pow F\iota}
% \\
% FX\xrightarrow{F\tuple{X,\tr(c)}}
% F(X\times \pow Z)
% \xrightarrow{F\str}
% F\pow(X\times Z)
% \xrightarrow{\xi_{X\times Z}}
% \pow F(X\times Z),
\end{equation}
where $\str(x',U):=\{(x',z')\mid z'\in U\}$ equips the monad $\pow$ with
a strength <cit.>, and $\iota\colon Y\hookrightarrow
X\times Z$ is the
inclusion function. Factorization via the dashed arrow can be seen
the restriction
which obviously follows from
the definition of $Y$ and $\str$. The arrow $FX\to
\pow FY$ that arises in (<ref>) shall be denoted by
Let us now note that the following diagrams commute—we use naturality
of $\xi$ and compatibility of $\xi$ and $\str$ with the
monad structure of $\pow$.
\begin{equation}\label{eq:201507062141}
\begin{aligned}
\vcenter{\xymatrix@R=.8em{
\ar[r]^-{F\eta^{\pow}_{X}}
\ar`u[rr] `/9.99pt[rr]^-{\eta^{\pow}_{FX}} [rr]
{F\pow X}
\ar[r]^{\xi_{X}}
{\pow FX}
\\
\ar[r]^-{F\tuple{X,\tr(c)}}
\ar@/_/@{-->}[rrd]
\ar@/^/@{=} [ru]
\ar `d[rrrd]+/d1.8em/ `[rrrd]_{h} [rrrd]
{F(X\times \pow Z)}
\ar[r]^-{F\str}
\ar[u]^{F\pi_{1}}
{F\pow (X\times Z)}
\ar[r]^-{\xi_{X\times Z}}
\ar[u]^{F\pow \pi_{1}}
{\pow F(X\times Z)}
\ar[u]^{\pow F \pi_{1}}
\\
{F\pow Y}
\ar[u]_{F\pow \iota}
\ar[r]_-{\xi_{Y}}
{\pow F Y}
\ar[u]^{\pow F\iota}
\\
\vcenter{\xymatrix@R=.8em{
{F\pow Z}
\ar[r]^-{\xi_{Z}}
{\pow FZ}
\\
\ar[r]^-{F\tuple{X,\tr(c)}}
\ar@/_/@{-->}[rrd]
\ar@/^.7pc/[rru]^-{F(\tr(c))}
\ar `d[rrrd]+/d1.8em/ `[rrrd]_{h} [rrrd]
{F(X\times \pow Z)}
\ar[r]^-{F\str}
\ar[ru]^{F\pi_{2}}
{F\pow (X\times Z)}
\ar[r]^-{\xi_{X\times Z}}
\ar[u]^{F\pow \pi_{2}}
{\pow F(X\times Z)}
\ar[u]^{\pow F \pi_{2}}
\\
{F\pow Y}
\ar[u]_{F\pow \iota}
\ar[r]_-{\xi_{Y}}
{\pow F Y}
\ar[u]^{\pow F\iota}
\\
% \vcenter{\xymatrix@R=.8em{
% {}
% &
% {FX}
% \ar[r]^-{F\eta^{\pow}_{X}}
% \ar`u[rr] `/9.99pt[rr]^-{\eta^{\pow}_{FX}} [rr]
% &
% {F\pow X}
% \ar[r]^{\xi_{X}}
% &
% {\pow FX}
% \\
% {FX}
% \ar[r]^-{F\tuple{X,\tr(c)}}
% \ar@/^/@{=} [ru]
% \ar@/_.7pc/[rrd]_-{F(\tr(c))}
% &
% {F(X\times \pow Z)}
% \ar[r]^-{F\str}
% \ar[u]^{F\pi_{1}}
% \ar[rd]_-{F\pi_{2}}
% &
% {F\pow(X\times Z)}
% \ar[r]^-{\xi_{X\times Z}}
% \ar[u]^{F\pow \pi_{1}}
% \ar[d]^-{F\pow\pi_{2}}
% &
% {\pow F(X\times Z)}
% \ar[u]^{\pow F \pi_{1}}
% \ar[d]_-{\pow F\pi_{2}}
% \\
% &&
% {F\pow Z}
% \ar[r]^-{\xi_{Z}}
% &
% {\pow FZ}
% }}
\end{aligned}
\end{equation}
We claim that the set $h(t)\subseteq FY$,
$t\in FX$ is
the one we chose so that (<ref>) holds, contains an
element $t'$ such that $(F\pi_{1})(t')=t$ and
$(F\pi_{2})(t')=\zeta(z')$. Indeed, we have
\begin{align*}
&(\pow F\pi_{2}\co h)(t)
% & \bigl(\,\pow F\pi_{2}\co\xi_{X\times Z}\co F\str\co
% F\tuple{X,\tr(c)}\,\bigr)(t)
\\
\bigl(\xi_{Z}\co F(\tr(c))\bigr)(t)
\quad\text{by the second diagram in~(\ref{eq:201507062141})}
\\
&\ni \zeta(z')
\quad\text{by~(\ref{eq:201507061746});}
\end{align*}
therefore there exists an element $t'$ of the set
$h(t)$ such that $(F\pi_{2})(t')=\zeta(z')$.
We also have $(F\pi_{1})(t')=t$—in fact $(F\pi_{1})(t'')=t$ holds
for any element $t''$ of the set $h(t)$, since $(\pow F\pi_{1}\co
h)(t)=\{t\}$ by the first diagram in (<ref>).
Finally, we define $q(y')\in FY$ by $q(y')=q(x',z'):=t'\in h(t)\subseteq
FY$. It is not hard to see that the data
$(\alpha,Y\stackrel{q}{\to} FY,r,s)$ thus
satisfies the conditions in
Def. <ref>. Specifically, Cond. 5(c) follows immediately from the fact that $Q^\zeta\colon Z\to
\pPMpa$ is an MC
progress measure (Def. <ref>) over the
final coalgebra $\zeta$. In its course we use the fact that, for
each $y'=(x',z')\in Y$,
\begin{align*}
(Fr\co q)(y') &= \bigl(F(Q^{\zeta}\co \pi_{2})\bigr)(q(y'))
\quad\text{by def.\ of $r$}
\\
&= (FQ^{\zeta}\co \zeta)(z')
\quad\text{by def.\ of $q$.}
\end{align*}
Cond. 6 follows from $(F\pi_{1})(q(y'))=t\in c(x')$ (see (<ref>)).
Thus we have obtained an LTMC progress measure $q\colon
Y\to FY$. Let $y\in Y$ required in the statement to be $y:=(x,z)$.
Now we shall check that the data thus obtained indeed satisfies the
conditions in the statement. That $s(y)=x$ is by definition of $s$
and $y$. For the condition that $\beh(q)(y)=z$, we observe that
commutes—recall that $t'=q(x',z')$ is chosen so that
$(F\pi_{2})(t')=\zeta(z')$ (see above). Therefore $\pi_{2}$ in the
diagram is a
coalgebra homomorphism and we have
\begin{align*}
\beh(q)(y)&=\beh(\zeta)(\pi_{2}(y))\quad\text{by finality}
\\&=\beh(\zeta)(z)=z,
\end{align*}
as required. To see that
=\sem{\varphi}_{\zeta}(z)$ where $p:=r(y)$, we have
\begin{align*}
\\
\bigl(Q^{\zeta}(z)\bigr)_{m}(\alpha,\dotsc,\alpha)
\quad\text{by def.\ of $r$}
\\
\sem{\varphi}_{\zeta}(z)
\quad\text{by the choice of $Q^{\zeta}$.}
\end{align*}
This concludes the proof.
§.§ Proof of Thm. <ref>
Let $(\alpha,Y_{0}\stackrel{q_{0}}{\to} FY_{0},r_{0},s_{0})$ denote, for the sake of
distinction, the LTMC
progress measure that we constructed in the proof of
Thm. <ref>.2.
So note that $Y,q,\dotsc$ in the proof of
Thm. <ref>.2, and those
which are
here, are different. Recall that
$Y_{0}\subseteq X\times Z$. We shall define $Y\subseteq X\times
\pPMpa$
as (an image of) a subset of $Y_{0}$.
Let $\mathcal{Y}$ be the family of subsets
$U\subseteq Y_{0}$ (hence $U\subseteq X\times Z$) that satisfy the following conditions.
* (Initial state)
$(x,z)\in U$
(where $x\in X$ and $z\in\tr(c)(x)$ are both from the statement);
* (No redundancy)
In case $(x',z'_{1})\in U$, $(x',z'_{2})\in U$ and
$Q^{\zeta}(z'_{1})=Q^{\zeta}(z'_{2})$ hold, then
On the family $\mathcal{Y}$ we define an order $\unlhd$ by:
$U\unlhd U'$ if
* $U= U'$, or
* $U\subsetneq U'$, and
\begin{equation}\label{eq:201507071341}
\begin{aligned}
\text{ for any $(x',z')\in U$, there exists $t''\in FU'$ such that}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\\
(F\pi_{1})(t'')&=(F\pi_{1}\co q_{0})(x',z'),
\quad\text{and}\;
\\
(FQ^{\zeta}\co F\pi_{2})(t'')&=
\end{aligned}
\end{equation}
Reflexivity, antisymmetry and transitivity of $\unlhd$ are straightforward.
We aim at applying Zorn's lemma to obtain a maximal element of
$\mathcal{Y}$ with respect to $\unlhd$.
Note first that $\mathcal{Y}$ is nonempty; indeed
$\{(x,z)\}\in\mathcal{Y}$. Let $\{U_{i}\}_{i\in I}$ be a totally
ordered subset of $\mathcal{Y}$. Then $\bigcup_{i}U_{i}$—where
$\bigcup$ is the set-theoretic union—is
an upper bound of $\{U_{i}\}_{i}$ in $\mathcal{Y}$. Indeed,
$\bigcup_{i}U_{i}$ belongs to $\mathcal{Y}$: Cond. 1 is obvious;
and Cond. 2 is easy too, because we can find $i_{1},i_{2}\in I$
such that
$(x',z'_{1})\in U_{i_{1}}$
$(x',z'_{2})\in U_{i_{2}}$, and $U_{i_{1}}\subseteq U_{i_{2}}$
(without loss of generality) because $\{U_{i}\}_{i\in I}$ is
totally ordered. It remains to be shown that
for each $i\in I$. In case $U_{i}$ is the maximum (with respect to
$\unlhd$) in $\{U_{i}\}_{i\in I}$, then it is so with respect to
the inclusion order $\subseteq$; therefore
Assume otherwise, in which case there exists $j\in I$ such that
$U_{i}\unlhd U_{j}$ and $U_{i}\neq U_{j}$. Now $U_{j}\subseteq
\bigcup_{i}U_{i}$ and hence $FU_{j}\subseteq F(\bigcup_{i}U_{i})$
(as subsets of $FY_{0}$,
since any $\Sets$-functor preserves monos with a nonempty domain).
It is now easy to check the condition (<ref>), and to see that
What we have shown so far allows us to appeal to Zorn's lemma and
to conclude that $\mathcal{Y}$ has a maximal element with respect
to $\unlhd$. We pick
and that is denoted by $Y'$.
On such a maximal element $Y'$ we shall show that the
condition (<ref>) holds with $U=U'=Y'$.
Assume otherwise. By the proof of
Thm. <ref>.2, there
does exist an element $t'':=q_{0}(x',z')\in FY_{0}$ such that
the two equalities in (<ref>) hold. Hence the
condition (<ref>) holds for $U=Y',U'=Y_{0}$.
Now let us define
\begin{align*}
Y'_{\rm r}&:=\bigl\{\,(x',z')\in Y_{0}\,\bigl|\bigr.\, \exists (x',z'')\in Y'
\text{ s.t.\ } Q^{\zeta}(z')=Q^{\zeta}(z'')\,\bigr\}\setminus
\\
Y'_{\rm n}&:=
Y_{0}\setminus (Y'\cup Y'_{\rm r}).
\end{align*}
We shall show that $Y'_{\rm n}\neq\emptyset$, and that the
condition (<ref>) holds with $U=Y'$ and
$U'=Y'\cup Y'_{\rm n}$. This will yield contradiction with
the maximality of $Y'$ with respect to $\unlhd$. Note here that
$Y_{0}=Y'\amalg Y'_{\rm r}\amalg Y'_{\rm n}$.
By the definition of $Y'_{\rm r}$, we can choose a function $f\colon
r}\to Y'$ such that $f(x',z')$ is $(x',z'')$ such that
$Q^{\zeta}(z')=Q^{\zeta}(z'')$. This means that the following diagram commutes.
By (essentially) applying $F$ to the diagram we obtain the
following, where $g$ denotes the arrow
$\tuple{F\pi_{1}, F(Q^{\zeta}\co \pi_{2})}$. The images
$g[FY_{0}]$ and $g[F(Y'+Y'_{\rm n})]$ are characterized by epi-mono
@/^1pc/@ >->[rd]
@/_1pc/@ >->[ruu]
The dashed arrow arises as a diagonal fill-in. This proves
\begin{equation}\label{eq:201507071724}
\bigl\langle\, F\pi_{1},\, F(Q^{\zeta}\co \pi_{2})\bigr\rangle
\;=\;
\bigl\langle\, F\pi_{1},\, F(Q^{\zeta}\co \pi_{2})\bigr\rangle
\bigl[F(Y'+Y'_{\rm n})\bigr],
\end{equation}
since the other direction $\supseteq$ is straightforward from
$Y_{0}\supseteq Y'+Y'_{\rm n}$.
It is an immediate corollary of (<ref>) that $Y'_{\rm
n}\neq\emptyset$—otherwise we have $g[FY_{0}]=g[FY']$ that
contradicts with the assumption that the
condition (<ref>)
holds with $U=Y', U'=Y_{0}$ but fails with $U=U'=Y'$. It
follows too that the
condition (<ref>) holds with
$U=Y', U'=Y'+Y'_{\rm n}$ since (<ref>) asserts that
$Y_{0}$ and $Y'+Y'_{\rm n}$ “have the same strength” when it comes
to the
condition (<ref>). It is also obvious from the
definition of $Y'_{\rm n}$—it is defined by excluding “redundant”
elements in $Y'_{\rm r}$—that $Y'+Y'_{\rm n}$ satisfies Cond. 2 of
the set $\mathcal{Y}$. Summarizing, we have shown that $Y'+Y'_{\rm
n}\in \mathcal{Y}$ is such that $Y'\unlhd Y'+Y'_{\rm n}$, with a
strict inequality. This contradicts with the maximality of $Y'$.
Thus we have shown that the
condition (<ref>) holds with $U=U'=Y'$.
Now let us go back to the
construction of the “small” LTMC
progress measure required in the statement.
We define
Y:={ (x',Q^ζ(z'))∈X×|(x',z')∈Y'
The functions $r,s$ are defined by
projections: $r:=\pi_{2}$ and $s:=\pi_{1}$.
The coalgebraic structure $q\colon Y\to FY$ is defined using the above
fact that the
condition (<ref>) holds for
$U=U'=Y'$. Specifically, let $(x',p)\in Y$; then by Cond. 2 of
$Y'\in \mathcal{Y}$, there exists a unique $z'\in Z$ such that
$p=Q^{\zeta}(z')$ and $(x',z')\in Y'$. We use the
condition (<ref>) (for $U=U'=Y'$) to find $t''\in
that satisfies the two equalities
in (<ref>). Finally we define
q(x',p):= F(𝕀_X×Q^ζ)(t”).
Note that the types match up: $t''\in FY'\subseteq F(X\times Z)$
and $F(\id_{X}\times Q^{\zeta})\colon F(X\times Z)\to F(X\times
\pPMpa)$, and the latter obviously factors through
$FY\hookrightarrow F(X\times
\pPMpa)$.
It is straightforward to check that
$(\alpha,Y\stackrel{q}{\to} FY,r,s)$
thus obtained indeed constitutes an LTMC progress measure.
Let us turn to
Cond. 5(c) of Def. <ref>,
and in particular (<ref>), for example.
Let $(x',p)\in Y$ and $z'\in Z$ be the unique one
such that
$p=Q^{\zeta}(z')$ and $(x',z')\in Y'$.
We have
\begin{align*}
\\
\\
\PT_{\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})}
\alpha_{a},\dotsc, \alpha_{k})
\bigl(\,(FQ^{\zeta}\co \zeta)(z')\,\bigr)
\\
&\quad\text{since $Q^{\zeta}$ is an MC progress measure for
$\zeta\colon Z\to FZ$ (Def.~\ref{def:branchingTimeProgMeas})}
\\
\PT_{\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})}
\alpha_{a},\dotsc, \alpha_{k})
\bigl(\,(F\pi_{2})(q(x',p))\,\bigr)
\\
&\quad\text{by def.\ of $q$; see~(\ref{eq:201507071341})}
\\
\PT_{\heartsuit_{\lambda}(u_{j_{1}},\dotsc,u_{j_{n}})}
\alpha_{a},\dotsc, \alpha_{k})
\bigl(\,(Fr\co q)(x',p)\,\bigr).
\end{align*}
for suitable $\beta_{1},\dotsc,\beta_{a-1}$, as required.
Cond. 6, let $z'$ be chosen in the same way as above. Then we have
\begin{align*}
&(Fs\co q)(x',p) = (F\pi_{1})(q(x',p))
\\
&= F\pi_{1} (q_{0}(x',z'))
\\
&\quad\text{by def.\ of $q$; see~(\ref{eq:201507071341})}
\\
&\in c(x'),
% \quad\text{since }
\end{align*}
where the last membership is because
$(\alpha,Y_{0}\stackrel{q_{0}}{\to} FY_{0},r_{0},s_{0})$
is an LTMC progress measure for $c$.
It remains to show that the LTMC progress measure that we
have constructed indeed realizes $\sem{\varphi}_{\zeta}(z)$. Let
$y:=(x,Q^{\zeta}(z))$ be the element $y\in Y$ required in the
statement. We have
\begin{align*}
\\
&= (Q^{\zeta}(z))_{m}(\alpha,\dotsc,\alpha)
\quad\text{by definition}
\\
&= \sem{\varphi}_{\zeta}(z)
\quad\text{by the choice of $Q^{\zeta}$, see the proof of Thm.~\ref{thm:correctnessOfNondetExistProgMeasEqSys}.}
\end{align*}
This concludes the proof.
§.§ Proof of
Thm. <ref>
We first check if $\varphi$ is satisfiable by $F$-coalgebras.
If $\varphi$ not, then obviously the answer to the problem in
the statement is false. Assume otherwise.
We claim that the maximum ordinal $\alpha$ in
Theorem <ref> can be bounded by
$N_{\varphi}$ from Assumption <ref>. Indeed,
by the small model property
(Assumption <ref>), we can assume that
there exists a coalgebra
$\varepsilon\colon E\to FE$, its state $e\in E$ and an
MC progress measure $Q'\colon E\to
\pPM_{\varphi,\alpha'}$ (Def. <ref>) such
that $Q'(e)(\alpha',\dotsc,\alpha')=\ttrue$ and $|E|\le N_{\varphi}$.
Moreover, by
Thm. <ref>.1–2, we can
the maximum ordinal $\alpha'$ of $Q'$ to be $\alpha'=|E|\le
N_{\varphi}$. It is then straightforward to adapt the proof of
Thm. <ref> in the following way.
* The final coalgebra $\zeta\colon Z\to FZ$ is replaced by
$\varepsilon\colon E\to FE$.
* The optimal MC progress measure $Q^{\zeta}$ is replaced by
the MC progress measure $Q'$ in the above.
Going through the adapted proof proves the
statement of Thm. <ref> with the
ordinal $\alpha$ in it bounded by $\alpha'$, hence by
Since $X$ is assumed to be finite and so is
$N_{\varphi}\in\omega$, we see that the set $X\times
\pPM_{\varphi,N_{\varphi}}$ is a finite set.
By the last assumption in Assumption <ref>,
the set $FY$ is finite for any (necessarily finite) subset $Y\in X\times
\pPM_{\varphi,N_{\varphi}}$. Therefore there are only finitely
many functions $q\colon Y\to FY$. We enumerate all such, for each
$Y\in X\times
\pPM_{\varphi,N_{\varphi}}$, and check if there is any that
$(\alpha,q,\pi_{2},\pi_{1})$ an LTMC progress measure. If there
is, then the answer to the problem in
the statement is true (by
Thm. <ref>.1). If
there is none, then the answer is false by the above arguments.
|
1511.00522
|
§ INTRODUCTION
The Thirring model was introduced in 1958 by Walter Thirring as a two-dimensional quantum field theory, when he was looking for an exactly solvable theory with interacting fermions <cit.>.
Its main feature is a four fermion interaction with a current term.
While its original formulation in two dimensions is quite well known <cit.>, there are a lot of open questions in a three-dimensional spacetime with a variable number of $\Nf$ fermion flavours.
The Lagrangian in Euclidean spacetime is given by
\begin{equation}
\mathcal{L}=\bar\psi_j \ii \slashed{\partial} \psi_j-\frac{g^2}{2\Nf}\left(\bar\psi_j \gamma^\mu \psi_j\right)^2 \qquad j=1,\dots,\Nf.\label{e:th_orig_lagrangian}
\end{equation}
We employ the usual Hubbard-Stratonovich transformation to introduce an auxiliary field $V_\mu$ and obtain the Lagrangian
\begin{equation}
\mathcal{L}=\bar\psi_j\ii\gamma^\mu \left(\partial_\mu-\ii V_\mu\right) \psi_j + \lambda_\mathrm{Th} V^\mu V_\mu
\end{equation}
with $\lambda_\mathrm{Th}=\nicefrac{\Nf}{2g^2}$.
We use a four-dimensional representation of the Clifford algebra, which is reducible in a three-dimensional spacetime.
This theory was shown to be renormalizable in a large $\Nf$ expansion for $2<d<4$ <cit.>.
The 3D formulation is useful to study chiral symmetry breaking and is motivated by a strong similarity to QED$\srm{3}$ <cit.>.
These models can be found in the literature to describe the electronic properties of graphene <cit.> or high temperature superconductors <cit.>.
The most interesting feature of the Thirring model is its chiral symmetry.
The action is invariant under transformations generated by the matrices $\left\{\mathbbm{1}, \gamma_4, \gamma_5, \ii\gamma_4\gamma_5\right\}$.
In addition, there is a flavour symmetry, which leads to a full symmetry group of $U(\Nf,\Nf)$ <cit.>.
This symmetry can be spontaneously broken to
\begin{equation}
U(\Nf,\Nf)\rightarrow U(\Nf)\otimes U(\Nf). \label{e:th_breaking_pattern}
\end{equation}
The large $\Nf$ expansion shows, that no symmetry breaking occurs for $\Nf\rightarrow\infty$.
On the other hand, there is a correspondence due to a Fierz identity of the irreducible Thirring model with $\Nf=1$ to a Gross-Neveu model.
The latter is known to always show chiral symmetry breaking, see for example <cit.>.
Since $\Nf=1$ in the irreducible representation can be thought of as “$\Nf=0.5$” in the reducible representation, this can be taken as a hint, that our model shows chiral symmetry breaking for small $\Nf$.
Thus, there must exist a critical flavour number $\Nfc$, such that chiral symmetry breaking is possible for $\Nf<\Nfc$ but not for $\Nf>\Nfc$.
The main motivation of our work is to determine the value of $\Nfc$ from lattice simulations.
There have been numerous other approaches to determine $\Nfc$, which found a variety of values:
The earliest works used Schwinger-Dyson equations and found values of $\Nfc=3.24$ <cit.>, $\Nfc=4.32$ <cit.> or even $\Nfc=\infty$ <cit.>[Although the model is considered in the irreducible representation in this paper.].
There is a paper <cit.> constructing an effective potential from the large $\Nf$ expansion, which found $\Nf=2$ and an extensive study <cit.> of four fermion theories with methods of the functional renormalization group, where $\Nfc\approx 5.1$ is found.
Lattice studies were performed with staggered fermions only.
Earlier simulations <cit.> used a HMC algorithm, allowing integer values of staggered fermion flavours, where each lattice flavour corresponds to two flavours of the continuum model.
These studies report results for simulations with $\Nf=2,4,6$.
They used a non-vanishing mass, which breaks the chiral symmetry explicitly, and obtained fits to the equation of state, that relates the chiral condensate to the mass.
The authors conclude, that there must be a change in the chiral behaviour between $\Nf=4$ and $\Nf=6$.
Moreover simulations with a Hybrid Molecular Dynamics algorithm were performed with non-integer values of $\Nf$.
The first study <cit.> agrees well with the other lattice results, while a more recent investigation <cit.> found $\Nfc\approx 6.6$.
As pointed out in <cit.>, there is a difficulty in using staggered fermions:
The staggered lattice model shows a chiral symmetry breaking pattern different from (<ref>) and it is not clear, if the continuum limit of the staggered lattice model leads to the correct breaking pattern.
Another point that may be important to correctly interpret our results, is made in <cit.>.
It is shown there, that the renormalizability of the Thirring model in large $\Nf$ expansion requires the vacuum polarization tensor to be transversal (as in QED).
This can be achieved by a coupling constant renormalization with renormalized coupling
\begin{equation}
g_\mathrm{R}^2=\frac{g^2}{1-g^2 J(m)}, \label{e:renorm_coupling}
\end{equation}
where $J(m)$ is an integral depending on the bare mass $m$ with a value of $\frac{2}{3}$ for $m\rightarrow 0$ in first order of the large $\Nf$ expansion.
Remarkably, the renormalized coupling can get negative, if the inverse of the bare coupling squared gets smaller than $J(m)$.
Then the model is believed to be in an unphysical phase where it is not unitary.
The authors of <cit.> interpret a decrease in the chiral condensate for increasing bare coupling as a transition to this unphysical phase and assume that the position of the maximum corresponds to $g\srm{R}^2\rightarrow\infty$ in equation (<ref>).
This interpretation is criticized in <cit.>, since in the functional renormalization group approach, the transversality of the vacuum polarization is not necessary to ensure renormalizability.
§ CURRENT APPROACH
In order to overcome the disadvantages of the staggered fermions, we use the SLAC fermion formulation <cit.>.
This is possible here, since the Thirring model is not a gauge theory and the argument considering the non-covariance of the vacuum polarization of <cit.> does not apply here.
It was shown <cit.> that Wess-Zumino models with SLAC derivative are renormalizable and have the correct continuum limit.
In recent years, the nonlocal SLAC fermions have been successfully used in many simulations of Yukawa-type models, see <cit.> and references therein.
We take this as a strong indication that the SLAC derivative works here, too.
In position space, it is given by
\begin{equation}
\partial^{\mathrm{SLAC}}_{xy}=
\begin{cases}
0 & x=y\\
\frac{\pi}{L}(-1)^{x-y}\frac{1}{\sin\mleft(\frac{\pi}{L}(x-y)\mright)} & x\neq y
\end{cases}
\end{equation}
while the application in momentum space is per definition just a multiplication with the lattice momenta.
There are no doublers and we have the exact chiral symmetry.
We can perform simulations with this setup using a rHMC algorithm, which allows us to simulate the model for non-integer $\Nf$.
We are mostly interested in the chiral condensate, which should be non-zero in the broken phase.
Due to the exactly implemented chiral symmetry with the SLAC fermions, it is always zero on every single configuration.
To induce a condensate, we introduced a coupling to a global $U(1)$-symmetric Gross-Neveu term as follows.
We include a new action preserving $U(1)$ symmetry, which reduces to a trivial factor of $1$ in the limit of vanishing coupling.
With $\Sigma\coloneqq\frac{1}{V} \sum_x \bar\psi_x \psi_x$ and $\Pi\coloneqq\frac{1}{V} \sum_x \bar\psi_x \gamma_5\psi_x$ it is given by
\begin{equation*}
1 =\lim_{g_\mathrm{g}\rightarrow 0} \ex{-S_\mathrm{g}} = \lim_{g_\mathrm{g}\rightarrow 0} \ex{\frac{g_\mathrm{g}}{2\Nf} \left(\Sigma^2 - \Pi^2\right)}
\end{equation*}
We perform another Hubbard-Stratonovich transformation, which introduces global fields $\sigma, \pi$ via
\begin{equation*}
\ex{-S_\mathrm{g}} = \frac{\Nf}{2\pi g_\mathrm{g}} \int_{-\infty}^{\infty}\dx\sigma\int_{-\infty}^{\infty}\dx\pi \, \ex{-\frac{\Nf}{2 g_\mathrm{g}}\left(\sigma^2+\pi^2\right)+\Sigma\sigma+\ii\Pi\pi}.
\end{equation*}
We rescale the fields by $\sqrt{V}$ and define $\lambda_\mathrm{g}\coloneqq\nicefrac{V\Nf}{2g_\mathrm{g}}$.
Now the terms $\Sigma\sigma$ and $\Pi \pi$ can be viewed as part of the Dirac operator.
The resulting full action of the simulated model and its partition sum are
\begin{align*}
S&=\sum_x \bar\psi\left( \ii \slashed \partial + \slashed V -\frac{1}{\sqrt{V}}\left( \sigma + \ii \gamma_5 \pi\right) \right)\psi+\lambda_\mathrm{Th}\sum_x V_\mu V^\mu + \lambda_\mathrm{g}\left(\sigma^2+\pi^2\right)\coloneqq S\srm{D}+S\srm{Th}+S\srm{g}\\
Z&=\int \mathcal{D} \bar\psi \mathcal{D}\psi \mathcal{D} V_\mu \dx \sigma \dx \pi\; \ex{-S}\coloneqq \int_0^\infty \dx r\;r \ex{-S\srm{g}(r)} Z(r).
\end{align*}
In the last equation, we defined transformed global fields in a polar coordinate form by $\sigma=r\cos\phi$ and $\pi=r\sin\phi$.
We get a formulation for the absolute value of the chiral condensate by
\begin{equation}
\chi=\frac{1}{V}\sum_x\left<\bar\psi\ex{\ii\gamma_5\phi}\psi\right>=\int_0^\infty\dx r\;r \ex{-\lambda\srm{g}r^2} \chi_r\label{e:rot_integral}
\end{equation}
Here, $\chi_r$ is defined as
\begin{align}
\chi_r&=\frac{1}{\sqrt{V}}\frac{\partial \ln Z(r)}{\partial r}= \frac{1}{V Z(r)}\int\Dx\bar\psi\Dx\psi\Dx V_\mu\int_0^{2\pi}\dx\phi\; \sum_x\bar\psi\ex{\ii\gamma_5\phi}\psi \ex{-S\srm{D}-S\srm{Th}}.
\label{e:rotated_condensate}
\end{align}
In addition, we can define a corresponding susceptibility by inserting $\frac{\partial^2\ln Z(r)}{\partial r^2}$ into (<ref>).
§ RESULTS AND CURRENT WORK
Histograms of the chiral condensate for $\Nf=2$, $\lambda\srm{g}=0.5$ and lattice size 12. The first figure shows the chiral condensate in the unphysical regime, the second and third show the ring shapes in the broken phase and the last is close to the physical phase transition.
The method described in section <ref> allows us to obtain histograms of real and imaginary parts of the chiral condensate like in figure <ref>.
For $\Nf=2$ and $3$ we can see ring shapes in these histograms, indicating a chirally broken phase.
We take the radius of these rings as an estimate for the absolute value of the chiral condensate as in figure <ref>.
Absolute value of the chiral condensate for $\lambda\srm{g}=0.5$ and size 12. For $\Nf\geq 4$ no non-zero condensate can be observed.
Critical values of the Thirring coupling for physical and artefact transition with linear interpolations.
For each $\Nf$ the physical coupling lies at larger $\lambda\srm{Th}$ than the artificial one.
There is a maximal radius for these rings at small inverse couplings with fixed value of $\lambda\srm{Th}\approx 0.17\Nf$.
For increasing $\lambda\srm{Th}$, the size of the rings decreases up to a point, where the rings melt to disks.
This indicates a transition to a chirally symmetric phase.
To the left of the maximal chiral condensate, there is a very sharp decrease in radius and the histograms become very sharply centred points.
Thus, we have a second transition to a phase of zero chiral condensate.
Following the interpretation of <cit.>, this transition is a sign of the renormalized coupling (<ref>) getting imaginary and thus leading to an unphysical theory.
Therefore, we will call this point near the maximal chiral condensate an artefact transition.
We found evidence in the time series of the chiral condensate, that this transition is of first order.
Simulations on lattices of size[The lattice size is chosen such that the SLAC derivative gives antiperiodic boundary conditions in the time direction and periodic in the space directions. This requires an even number of points in time direction and an odd number in space directions, see <cit.>. We always took $L\srm{s}=L\srm{t}-1$, such that size 12 refers to a lattice with $12\times11\times11$ points.] 12 and 16 with a fixed global coupling of $\lambda\srm{g}=0.5$ and non-integer values of $\Nf$ between 1.8 and 4 showed, that the two points of phase transitions get closer with increasing $\Nf$.
Figure <ref> shows the positions of both phase transitions depending on $\Nf$ and $\lambda\srm{g}$.
For $\lambda\srm{g}=0.5$ the two critical couplings get very close to each other around $\Nf\approx 3.5$ and we cannot observe any non-vanishing chiral condensate for larger $\Nf$.
Increasing $\lambda\srm{g}$ in order to recover the Thirring model, the point of coincidence of the two phase transitions moves to smaller $\Nf$.
An extrapolation of the intersections of the extrapolating lines in figure <ref> with a fit function $a x^{-b}+c$ yields a limiting value of $\Nfc\mleft(\lambda\srm{g}\rightarrow\infty\mright)=2.05\pm0.05$.
Susceptibility for $\Nf=1$ and $\lambda\srm{g}=0.5$. The sharp peak on the left corresponds to the artefact transition, which is of first order.
The right peak is at the position of the physical transition.
Position of peaks of the susceptibility for $\Nf=1$ with extrapolations in the global coupling for lattice size 12. Note, that there is no second peak for $\lambda\srm{g}=2.0$ at lattice size 12.
Thus, simulations of our coupled model for $\Nf=1$ should show chiral symmetry breaking for all values of the global coupling.
But also in this case, the radius of the histograms gets too small to extract a reliable estimate at $\lambda\srm{g}\approx 1.6$.
Using maxima of the susceptibility (see figure <ref>) as another indicator of the phase transitions, we were able to determine their positions in $\lambda\srm{Th}$ up to $\lambda\srm{g}=1.8$ as shown in figure <ref>.
For $\lambda\srm{g}=2.0$ the peak at the position of the physical transition cannot be resolved any more at lattice size 12, but it is visible at size 16.
Thus, to determine if the two phase transitions stay separate as suggested by the extrapolations in figure <ref>, we have to increase the lattice size, if we want to further increase the global coupling.
At the moment, we cannot draw a reliable conclusion regarding the critical flavour number of the Thirring model.
§ ACKNOWLEDGEMENTS
D.S. and B.W. were supported by the DFG Research Training Group 1523/2 “Quantum and Gravitational Fields”. B.W. was supported by the Helmholtz International Center for FAIR within the LOEWE initiative of the State of Hesse.
|
1511.00189
|
We consider a sorting machine consisting of two stacks in series where the first stack has the added restriction that entries in the stack must be in decreasing order from top to bottom.
The class of permutations sortable by this machine is known to be enumerated by the Schröder numbers.
In this paper, we give a bijection between these sortable permutations of length $n$ and Schröder paths of order $n-1$: the lattice paths from $(0,0)$ to $(n-1,n-1)$ composed of East steps $(1,0)$, North steps $(0,1)$, and Diagonal steps $(1,1)$ that travel weakly below the line $y=x$.
§ INTRODUCTION
A stack is a sorting device that works by a sequence of push and pop operations. This last-in, first-out machine was shown by Knuth <cit.> to sort a permutation if and only if that permutation avoids the pattern $231$. That is, if there are not three indices $i<j<k$ with $\pi_k<\pi_i<\pi_j$, then it is possible to run $\pi$ through a stack and output the identity permutation. The class of stack-sortable permutations is enumerated by the Catalan numbers.
In the language of permutation patterns, any downset of permutations in the permutation containment ordering is a class, and every class has a basis, which consists of the minimal permutations not in the class.
Given that the basis for the class of permutations sortable by one stack contains only a single pattern of length three, considering two stacks in series is quite natural. However, the problem becomes rather unwieldy. In the case of two stacks in series, Murphy <cit.> showed that the class of sortable permutations has an infinite basis. The enumeration of this class also appears to be difficult. The best known bounds are given by Albert, Atkinson, and Linton <cit.>.
We note that to sort a permutation by two stacks in series, the push and pop operations are such that when an entry is popped out of the first stack, it is immediately pushed into the second stack.
To get a better handle on this problem, many have considered different types of weaker sorting machines. One such weaker machine is a stack in which the entries must increase when read from top to bottom.
Atkinson, Murphy, and Ruškuc <cit.> found an optimal algorithm for sorting permutations which are sortable using two increasing stacks in series.
Note that to obtain the identity permutation, the last stack will be an increasing stack even without declaring this restriction.
Their left-greedy algorithm sorts all permutations sortable by this machine.
Interestingly enough, the basis for these sortable permutations is still infinite, but the permutation class was found to be in bijection with the permutations that avoid $1342$ as enumerated by Bóna <cit.>. Both enumerations were found by using a bijection with $\beta(0,1)$ trees.
One can analogously define a decreasing stack as a stack in which the entries must decrease when read from top to bottom. Smith <cit.> considered sorting with a decreasing stack followed by an increasing stack, a machine called DI (we refer to the decreasing stack as D and the increasing stack as I). We illustrate how a permutation can be sorted with the DI sorting machine in Figure <ref>. Notice that this permutation contains the pattern $2341$, and as such cannot be sorted by two increasing stacks in series.
The class of DI-sortable permutations was shown to have a finite basis, $\{3142, 3241\}$. Kremer <cit.> has shown previously that this class is enumerated by the large Schröder numbers.
Sorting the permutation $23514$.
The Schröder numbers were introduced in somewhat modern times by Schröder <cit.> as legal bracketing of variables, though at least the initial terms of this sequence were known to Hipparchus <cit.>.
Rogers and Shapiro <cit.> found bijections showing certain classes of lattice paths are enumerated by the Schröder numbers.
While there are several such classes of lattice paths, we use the one given in the definition below.
Let $n\geq 1$.
A lattice path from $(0,0)$ to $(n,n)$ taking only East $(1,0)$, North $(0,1)$, and Diagonal $(1,1)$ steps while staying weakly below the main diagonal will be referred to as a Schröder path.
We illustrate all of the Schröder paths from $(0,0)$ to $(2,2)$ in Figure <ref>.
Schröder paths from $(0,0)$ to $(2,2)$.
In Section <ref>, we state the algorithm from <cit.> for producing a word from a DI-sortable permutation, then classify when a sorting word is the result of the algorithm.
In Section <ref>, we give an algorithm which produces a Schröder path from a word produced by the algorithm in Section <ref>, then show there is a bijective correspondence between algorithmic DI words and Schröder paths.
In Section <ref>, we discuss some properties of DI-sortable permutations which can be ascertained from aspects of their corresponding Schröder paths, and we conclude with some open questions.
On a related note, Ferrari <cit.> gave bijections between permutations sorted by restricted deques and a different class of lattice paths also enumerated by the Schröder numbers.
Also, Bandlow, Egge, and Killpatrick <cit.> gave a bijection between a different permutation class and Schröder paths. Additionally, more background on stacks in general was consolidated by Bona <cit.>.
§ THE SORTING ALGORITHM
Let $W$ be a word, $i$ be a positive integer, and L be a letter.
In what follows, we denote the letter in the $i$th position of $W$ as $W(i)$,
$\#_\DI{L}(W)$ as the number of occurrences of L in $W$, and
$W^i$ as the first $i$ characters in $W$.
We often refer to $W^i$ as a prefix of $W$.
We use $\DI{L}^i$ to denote $i$ repetitions of L.
In <cit.>, Smith showed the following algorithm was an optimal way to sort permutations using the DI machine in the sense that this algorithm would sort any DI-sortable permutation.
* If the top entry of the second (increasing) stack is the next entry of the output, then pop the entry to the output.
* If all of the $m$ entries in the first (decreasing) stack make up the next $m$ entries of the output, then push those entries to the second stack.
* Otherwise, if the next entry of the input is smaller than the top entry of the second stack and larger than the top entry of the first stack, then push it onto the first stack.
We will apply the convention that the next entry of the input satisfies each of the aforementioned properties if there is no entry in the corresponding stack with which to compare it.
Thus, this step can be thought of as pushing the next entry from the input to the top of the first stack if that entry can legally sit atop each of the two stacks at this stage.
* Finally if neither of those moves are available, push the top entry from the first stack to the second.
It was noted in <cit.> that Step 2 of this algorithm is not needed to produce an optimal algorithm.
However, we continue to utilize it as outputting entries earlier in the process is useful in constructing a bijection between the DI-sortable permutations and Schröder paths.
Let $\DI{E}$ represent a push step from the input to the first stack, let $\DI{N}$ represent pop/push from the first stack to the second stack, and let $\DI{C}$ represent the pop from the second stack to the output.
See Figure $\ref{fig:defenc}$.
A sorting word of a permutation is a word representing steps that can be taken to sort that permutation using two stacks in series (without restriction).
We use $\DI{L}_i$ to denote the occurrence of the letter L in $W$ which corresponds to the movement of symbol $i$.
The operations E, N, and C corresponding to the DI machine from Definition <ref>.
A word $W$ is a sorting word of a sortable permutation of length $n$ if and only if the following two conditions are met:
* The length of $W$ is $3n$ and $W$ contains exactly $n$ of each of the letters $\DI{E}$, $\DI{N}$, and $\DI{C}$.
* For all $x\in[3n]$, $\#_\DI{E}(W^x) \geq \#_\DI{N}(W^x) \geq \#_\DI{C}(W^x)$.
That is, in any prefix of $W$, the number of Cs does not exceed the number of Ns, which does not exceed the number of Es.
Note that for every sorting word, there exists a permutation $\pi$ which it sorts.
To find $\pi$, apply $W$ to the identity permutation, and let $\sigma$ be the output (which is not necessarily sorted).
Then $W$ sorts $\pi=\sigma^{-1}$.
A DI word of a permutation $\pi$ is a sorting word representing steps that can be taken to sort $\pi$ using the DI machine.
The permutation $23514$ has four DI words: EEENNNENCCCENCC (illustrated in Figure <ref>), EEENNNENCCECNCC, EEENNNENCECCNCC, and EEENNNENECCCNCC.
The algorithmic DI word (ADI word for short) of a permutation $\pi$ is the unique word representing steps that will be taken when applying Algorithm <ref> to sort $\pi$ by the DI machine.
Algorithm <ref> was used in the process illustrated by Figure <ref>.
So the ADI word of $23514$ is EEENNNENCCCENCC.
To use the indices outlined in Definition <ref>, the ADI word of $23514$ can also be given as
\DI{E}_3
\DI{E}_5
\DI{N}_5
\DI{N}_3
\DI{N}_2
\DI{E}_1
\DI{N}_1
\DI{C}_1
\DI{C}_2
\DI{C}_3
\DI{E}_4
\DI{N}_4
\DI{C}_4
\DI{C}_5
For the duration of the paper, we say that $\DI{X} \lew \DI{Y}$ in a word $W$ if $\DI{X}$ appears before $\DI{Y}$ in $W$.
Similarly we say $\DI{X}\precw\DI{Y}$ if $\DI{X}$ appears immediately before $\DI{Y}$ in $W$.
We begin with some observations about the relative locations of symbols in a DI-word, then follow with a classification of when a sorting word is a DI-word and an ADI-word.
Let $\pi$ be a DI-sortable permutation and let $W$ be a DI word.
Let $i,j\in[n]$.
* $\DI{E}_{\pi_i}\lew\DI{E}_{\pi_j}$ if and only if $i < j$; symbols appearing in $\pi$ earlier enter the first stack earlier.
* $\DI{C}_{\pi_i}\lew\DI{C}_{\pi_j}$ if and only if $\pi_i < \pi_j$; symbols must exit the second stack in increasing order.
If $\pi_i < \pi_j$ and $\DI{N}_{\pi_i}\lew\DI{N}_{\pi_j}$, then $\DI{C}_{\pi_i}\lew\DI{N}_{\pi_j}$;
if a smaller number enters the second stack before a larger number, it must be output before the larger number can enter the second stack – otherwise the increasing condition for the second stack is not satisfied.
Hence if $\pi_i < \pi_j$ and $\DI{N}_{\pi_i}\lew\DI{E}_{\pi_j}$, then $\DI{C}_{\pi_i}\lew\DI{N}_{\pi_j}$.
We highlight this specifically as it will be used in later proofs.
* If $\DI{N}_{\pi_i} \lew \DI{N}_{\pi_j} \lew \DI{C}_{\pi_i}$, then $\DI{C}_{\pi_j}\lew\DI{C}_{\pi_i}$;
if $\pi_j$ enters the second stack after $\pi_i$ and before $\pi_i$ exits, then $\pi_j < \pi_i$ by the increasing condition of the second stack. So $\pi_j$ must exit before $\pi_i$.
* If $\DI{E}_{\pi_j} \lew \DI{C}_{\pi_i} \lew \DI{N}_{\pi_j}$ (which implies $\pi_i < \pi_j$), then $\DI{N}_{\pi_i} \lew \DI{E}_{\pi_j}$;
if a smaller number is pushed to the output while a larger number is in the first stack, the smaller number was popped from the first stack before the larger number entered the first stack – otherwise the larger number must be pushed to the second stack before the smaller number is pushed to the second stack.
* If $\pi_i < \pi_j$ and $\DI{E}_{\pi_j}\lew\DI{E}_{\pi_i}$, then $\DI{E}_{\pi_j} \lew \DI{N}_{\pi_j} \lew\DI{E}_{\pi_i}$;
if a larger number enters the first stack before a smaller number does, the larger number must move to the second stack before the smaller number enters the first stack – otherwise the decreasing condition for the first stack is violated.
We now give a classification of when a sorting word is a DI word.
A sorting word $W$ of a permutation $\pi$ of length $n$ is a DI word if and only if
for all $i\in[n]$, if no Ns appear in $W$ between $\DI{N}_{\pi_i}$ and $\DI{C}_{\pi_i}$, then $\#_\DI{E}(W^x) = \#_\DI{N}(W^x)$ where $x$ satisfies $W(x) = \DI{N}_{\pi_i}$; that is at step $x$, the first stack is empty.
Alternatively, we can say a sorting word $W$ is a DI word of some permutation $\pi$ of length $n$ if and only if every entry in $\pi$ that preceded the entries output by a sequence of $\DI{C}$s must have moved to the second stack (and possibly the output) prior to this exodus.
Suppose that $W$ is a DI word of a permutation $\pi$ and for some $i\in[n]$, no Ns appear in $W$ between $\DI{N}_{\pi_i}$ and $\DI{C}_{\pi_i}$, and $W(x) = \DI{N}_{\pi_i}$.
Assume that the first stack is nonempty after stage $x$, and let $\pi_j$ be a symbol in the first stack.
Then $\DI{E}_{\pi_j}\lew\DI{N}_{\pi_i}\lew\DI{N}_{\pi_j}$, and since no Ns appear between $\DI{N}_{\pi_i}$ and $\DI{C}_{\pi_i}$, it follows that $\DI{N}_{\pi_i} \lew \DI{C}_{\pi_i}\lew\DI{N}_{\pi_j}$.
By Observation <ref> (<ref>), we have that $\DI{N}_{\pi_i}\lew\DI{E}_{\pi_j}$, which gives a contradiction.
So at stage $x$, the first stack must be empty.
Conversely, suppose $W$ is a sorting word of a permutation $\pi$ of length $n$ which is not a DI word.
Recall that since $W$ represents the sorting of $\pi$ through two stacks in series, any entries in the second stack will obey the increasing condition at all times.
Consequently, the movements corresponding to $W$ when applied to $\pi$ must cause a violation of the decreasing condition on the first stack.
In such a case, we have two entries $\pi_i$ and $\pi_j$ where $\pi_i > \pi_j$ and at some point $\pi_i$ is below $\pi_j$ in the first stack.
That is, $\DI{E}_{\pi_i} \lew \DI{E}_{\pi_j} \lew \DI{N}_{\pi_j} \lew \DI{N}_{\pi_i}$.
Since the end result of the sorting is the identity permutation, we must have $\DI{C}_{\pi_j}\lew\DI{C}_{\pi_i}$.
Therefore $\DI{E}_{\pi_i} \lew \DI{E}_{\pi_j} \lew \DI{N}_{\pi_j} \lew \DI{C}_{\pi_j} \lew \DI{N}_{\pi_i} \lew \DI{C}_{\pi_i}$.
Let $\DI{N}_{\pi_k}$ be the last $\DI{N}$ in $W$ to appear before $\DI{C}_{\pi_j}$.
Since the second stack is increasing and $W$ is a sorting word, $\pi_k\leq\pi_j$ and so $\DI{N}_{\pi_k}\lew \DI{C}_{\pi_k}\leqw \DI{C}_{\pi_j}$.
By the selection of $k$, no Ns appear in $W$ between $\DI{N}_{\pi_k}$ and $\DI{C}_{\pi_k}$.
Observe that $\pi_i$ is in the first stack (and hence the first stack is nonempty) at the step corresponding to $\DI{N}_{\pi_k}$.
To similarly characterize our ADI words, we first introduce a lemma showing how “output-greedy" Algorithm <ref> is.
Let $n\geq 1$ and $i\in[n]$ such that $\pi_i\neq n$.
Let $\pi$ be a DI-sortable permutation of length $n$ with ADI word $W$.
If $\DI{C}_{\pi_i}\not\precw\DI{C}_{\pi_i+1}$, then $\DI{C}_{\pi_i}\lew\DI{E}_{\pi_i+1}$.
That is, if two consecutive values do not exit in two consecutive stages, then the larger value does not enter the machine until after the smaller value exits.
Assume by way of contradiction that $\DI{C}_{\pi_i}\not\precw\DI{C}_{\pi_i+1}$ and $\DI{E}_{\pi_i+1}\lew \DI{C}_{\pi_i}$.
First suppose that $\DI{N}_{\pi_i+1} \lew \DI{C}_{\pi_i}$ and let $x$ satisfy $W(x) = \DI{C}_{\pi_i}$.
Then necessarily $\pi_i+1$ must be immediately below $\pi_i$ in the second stack after stage $x-1$.
So Step 1 of Algorithm <ref> applies at stage $x+1$, giving that $W(x+1)=\DI{C}_{\pi_i+1}$.
Hence $\DI{C}_{\pi_i}\precw \DI{C}_{\pi_i+1}$, a contradiction.
Alternatively, suppose $\DI{E}_{\pi_i+1} \lew \DI{C}_{\pi_i} \lew\DI{N}_{\pi_i+1}$, and let $x$ satisfy $W(x) = \DI{E}_{\pi_i+1}$.
Then $\DI{N}_{\pi_i} \lew\DI{E}_{\pi_i+1}$
by Observation <ref> (<ref>).
Therefore $\DI{N}_{\pi_i} \lew\DI{E}_{\pi_i+1} \lew\DI{C}_{\pi_i}$, meaning that at stage $x$, $\pi_i+1$ is pushed into the first stack while some symbol in the second stack is smaller than $\pi_i+1$ (namely $\pi_i$).
Therefore Step 3 of Algorithm <ref> should not be applied at stage $x$, which is a contradiction.
See Figure <ref> and notice when entries $1,2,3$ are output, $4$ is still in the input.
This can also be seen in Example <ref>, where $\DI{C}_1\precw\DI{C}_2\precw\DI{C}_3\lew\DI{E}_4$.
We now conclude this section with a classification of when a DI word is the ADI word produced by Algorithm <ref> for some DI-sortable permutation.
A DI word $W$ of a DI-sortable permutation $\pi$ of length $n$ is its ADI word if and only if $\DI{E}_1\precw\DI{N}_1\precw\DI{C}_1$ and for each $i\in[n]$ such that $\pi_i\neq 1$, either $\DI{C}_{\pi_i-1}\precw\DI{C}_{\pi_i}$ or $\DI{E}_{\pi_i}\precw\DI{N}_{\pi_i}\precw\DI{C}_{\pi_i}$.
That is, a DI word $W$ of a DI-sortable permutation $\pi$ is its ADI word if and only if any maximal sequence of consecutive copies of $\DI{C}$s in $W$ is immediately preceded by $\DI{EN}$.
Suppose $W$ is the ADI word of $\pi$ and $i\in[n]$.
It follows by Algorithm <ref> that $\DI{E}_1\precw\DI{N}_1\precw\DI{C}_1$, so we may assume going forward that $\pi_i\neq 1$.
Additionally any references to Steps in the following argument refer to Algorithm <ref>.
Assume that $\DI{C}_{\pi_{i}-1}\not\precw\DI{C}_{\pi_i}$.
By Lemma <ref>, $\DI{C}_{\pi_i-1}\lew\DI{E}_{\pi_i}$.
Define $x$ so that $W(x)=\DI{E}_{\pi_i}$.
Therefore after stage $x-1$, $\pi_i$ is smaller than all symbols in the second stack and larger than all symbols in the first stack – otherwise Step 3 would not be applied at stage $x$.
In fact, all symbols smaller than $\pi_i$ have been output by stage $x-1$ since $\DI{C}_{\pi_i-1}\lew\DI{E}_{\pi_i}$.
Therefore after stage $x$, $\pi_i$ is the only symbol in the first stack.
Hence Step 2 applies at stage $x+1$, giving $W(x+1)=\DI{N}_{\pi_i}$.
Then Step 1 applies at stage $x+2$ giving that $W(x+2)=\DI{C}_{\pi_i}$ and therefore $\DI{E}_{\pi_i}\precw\DI{N}_{\pi_i}\precw\DI{C}_{\pi_i}$.
Conversely, suppose the DI word $W$ is not the ADI word of $\pi$, and let $A$ denote the ADI word of $\pi$.
Let $t\in[3n]$ be the largest integer such that $A^{t-1}=W^{t-1}$, that is $t$ is the smallest integer so that $A(t)\neq W(t)$.
We consider the state of the DI machine after stage $t-1$.
Define $i\in[n]$ so that $\DI{C}_{\pi_i-1}\leqw W(t-1)\lew\DI{C}_{\pi_i}$ (and also $\DI{C}_{\pi_i-1}\leqw[A] W(t-1)\lew[A]\DI{C}_{\pi_i}$).
By convention, we say that $\DI{C}_0$ occurs at position 0, and since $W(t)<\DI{C}_n$, the index $i$ is well-defined.
Therefore all symbols smaller than $\pi_i$ have been output by stage $t-1$ and no Cs appear between $W(t-1)$ and $\DI{C}_{\pi_i}$ in $W$ and $A$.
Hence by stage $t-1$, $\pi_i$ is either at
* the top of the second stack ($\DI{E}_{\pi_i}\lew[] \DI{N}_{\pi_i}\leqw[] W(t-1)\lew[]\DI{C}_{\pi_i}$ in $W$ and $A$),
* at the bottom of the first stack ($\DI{E}_{\pi_i}\leqw[] W(t-1)\lew[] \DI{N}_{\pi_i}\lew[]\DI{C}_{\pi_i}$ in $W$ and $A$), or
* part of the input yet to enter the stacks ($W(t-1)\lew[]\DI{E}_{\pi_i}\lew[] \DI{N}_{\pi_i}\lew[]\DI{C}_{\pi_i}$ in $W$ and $A$).
Suppose first that $\pi_i$ is located at the top of the second stack by stage $t-1$.
See Figure <ref>(a).
Observe that Step 1 of Algorithm <ref> outputs $\pi_i$ at stage $t$, so $A(t)=\DI{C}_{\pi_i}$.
Since $\pi_i$ is smaller than all symbols in the first stack (possibly vacuously), $W(t)\neq\DI{N}$.
Thus $W(t)=\DI{E}$.
Since $\pi_i$ is smaller than any symbol in the first stack and in the remaining input, no N can appear in $W$ between $W(t)$ and $\DI{C}_{\pi_i}$; otherwise the second stack is no longer increasing.
Therefore $\DI{E}\precw\DI{C}_{\pi_i}$, and hence neither $\DI{C}_{\pi_i-1}\precw\DI{C}_{\pi_i}$ nor $\DI{N}_{\pi_i}\precw\DI{C}_{\pi_i}$.
[.75]32 [.75]33 [.75]34
(a) (b) (c)
Some states of the DI machine from the first two cases of the proof of Theorem <ref>.
Next suppose that $\pi_i$ is at the bottom of the first stack by stage $t-1$.
Then $W(t-1)\lew[A]\DI{N}_{\pi_i}$ and hence $\DI{C}_{\pi_i-1}\leqw[A] W(t-1)\lew[A]\DI{N}_{\pi_i}\lew[A] \DI{C}_{\pi_i}$.
So $\DI{C}_{\pi_i-1}\not\precw[A]\DI{C}_{\pi_i}$, and hence $\DI{C}_{\pi_i-1}\lew[A]\DI{E}_{\pi_i}\leqw[A]W(t-1)$ by Lemma <ref>.
Since $W^{t-1}=A^{t-1}$, we have that $\DI{C}_{\pi_i-1}\lew[W]\DI{E}_{\pi_i}$ as well.
Let $W(x)=\DI{E}_{\pi_i}$ (so $x\leq t-1$ and thus $A(x)=\DI{E}_{\pi_i}$).
Then Step 3 applies at stage $x$, meaning that $\pi_i$ is larger than all symbols in the first stack by stage $x-1$, but since $\DI{C}_{\pi_i-1}\lew[A]\DI{E}_{\pi_i}$ all symbols smaller than $\pi_i$ have been output.
So $\pi_i$ is alone in the first stack after stage $x$.
Since $\pi_i$ is smaller than any symbol (if any) in the second stack, it follows that $A(x)=\DI{E}_{\pi_i}\precw[A]\DI{N}_{\pi_i}\precw[A]\DI{C}_{\pi_i}$.
So $A(x+1)=\DI{N}_{\pi_i}$.
Since $A(t-1)\lew[A] \DI{N}_{\pi_i} = A(x+1)$,
$t-1 < x+1$.
So $x=t-1$ and hence $A(t)=\DI{N}_{\pi_i}$ and $A(t-1)=W(t-1)=\DI{E}_{\pi_i}$.
Observe that $\pi_i$ is alone in the first stack by stage $t-1$ and
$W(t)\neq\DI{C}$ (because $\DI{C}_{\pi_i-1}\leqw W(t-1) = \DI{E}_{\pi_i} \precw W(t)\lew\DI{C}_{\pi_i}$),
so $W(t)=\DI{E}_{\pi_k}$ for some $\pi_k>\pi_i$.
Figures <ref> (b) and (c) show the states of the DI-machine using $W$ on $\pi$ after stages $t-1$ and $t$, respectively.
Since $\DI{C}_{\pi_i-1}\leqw W(t-1) = \DI{E}_{\pi_i}$, no Cs appear between $\DI{E}_{\pi_i}$ and $\DI{C}_{\pi_i}$ in $W$.
Hence $\DI{C}\not\precw\DI{N}_{\pi_i}$ and there are no Cs between $\DI{N}_{\pi_i}$ and $\DI{C}_{\pi_i}$.
Furthermore $\DI{E}\not\precw\DI{N}_{\pi_i}$ since $\DI{E}_{\pi_i}\not\precw\DI{N}_{\pi_i}$ and otherwise $W$ is not a sorting word.
Therefore $\DI{N}\precw \DI{N}_{\pi_i}$.
Furthermore, there are no Ns between $\DI{N}_{\pi_i}$ and $\DI{C}_{\pi_i}$, otherwise Observation <ref> (<ref>) is violated.
So either $\DI{N}\precw\DI{N}_{\pi_i}\precw\DI{C}_{\pi_i}$ or $\DI{E}\precw\DI{C}_{\pi_i}$.
Last, we show that $\pi_i$ must be in one of the stacks by stage $t-1$, which will conclude the proof.
Assume to the contrary that $\pi_i$ is not in either stack by stage $t-1$.
Clearly both $A(t)$ and $W(t)$ are not $\DI{C}$ (necessarily $\DI{C}_{\pi_i}$), so either
$A(t)=\DI{N}$ and $W(t)=\DI{E}$ or $A(t)=\DI{E}$ and $W(t)=\DI{N}$.
Suppose at stage $t-1$, $\pi_k$ is the next symbol to enter the stacks, and if the stacks are nonempty, $\pi_q$ sits atop the first stack and $\pi_m$ sits atop the second stack.
See Figure <ref>(a).
[.75]35 [.75]36 [.75]37
(a) (b) (c)
Some stages of the DI machine from the final cases of the proof of Theorem <ref>.
First assume that $A(t)=\DI{N}_{\pi_q}$ and $W(t)=\DI{E}_{\pi_k}$ (and therefore the first stack is nonempty by stage $t-1$).
Then either Step 2 or Step 4 of Algorithm <ref> applies at stage $t$.
However Step 2 cannot be satisfied because $\pi_i$ is the next symbol to be output, so Step 4 is applied, meaning the conditions of Step 3 are not satisfied. Hence either the second stack is nonempty and $\pi_k > \pi_m$, or $\pi_k < \pi_q$.
Since $W$ is a DI word and $W(t)=\DI{E}_{\pi_k}$, it follows that $\pi_k > \pi_q$ (and thus $\pi_k\neq \pi_i)$, which implies that the second stack must be nonempty and $\pi_k>\pi_m$; hence $\DI{N}_{\pi_m}\lew\DI{E}_{\pi_k}$.
Since $\DI{E}_{\pi_k}\lew\DI{E}_{\pi_i}$ we have that
$W(t) =\DI{E}_{\pi_k} \lew \DI{N}_{\pi_k} \lew\DI{E}_{\pi_i}$ by Observation <ref> (<ref>).
So $\DI{C}_{\pi_i} \lew\DI{C}_{\pi_m}$ (since $\pi_i<\pi_m$),
$\DI{C}_{\pi_m}\lew\DI{N}_{\pi_k}$ (follows from $\DI{N}_{\pi_m}\lew\DI{E}_{\pi_k}$, $\pi_m<\pi_k$, and Observation <ref> (<ref>)), and $\DI{N}_{\pi_k}\lew\DI{E}_{\pi_i}$ (because $\pi_i<\pi_k$, $\DI{E}_{\pi_k}\lew\DI{E}_{\pi_i}$, and Observation <ref> (<ref>)).
Therefore $\DI{C}_{\pi_i}\lew \DI{E}_{\pi_i}$, a contradiction.
Hence we cannot have $A(t)=\DI{N}$ and $W(t)=\DI{E}$.
Therefore $A(t)=\DI{E}_{\pi_k}$ and $W(t)=\DI{N}_{\pi_q}$.
This implies that $\pi_k > \pi_q$ (and thus $\pi_k\neq\pi_i$) and the first stack is nonempty at stage $t-1$, respectively.
This also gives that $W(t)=\DI{N}_{\pi_q}\lew\DI{E}_{\pi_k}$.
Furthermore $\pi_q>\pi_i$ since all other symbols smaller than $\pi_i$ have been output by stage $t-1$.
So $\DI{C}_{\pi_q}\lew\DI{N}_{\pi_k}$ because from $\DI{N}_{\pi_q}\lew\DI{E}_{\pi_k}$, $\pi_q<\pi_k$, and Observation <ref> (<ref>).
Furthermore $\DI{N}_{\pi_k}\lew\DI{E}_{\pi_i}$ since $\pi_i<\pi_k$, $\DI{E}_{\pi_k}\lew\DI{E}_{\pi_i}$, and Observation <ref> (<ref>).
Since $\pi_i<\pi_q$, it follows that $\DI{C}_{\pi_i}\lew\DI{C}_{\pi_q}$.
So $\DI{C}_{\pi_q}\lew\DI{N}_{\pi_k}\lew\DI{E}_{\pi_i}\lew\DI{C}_{\pi_i}\lew\DI{C}_{\pi_q}$, which is a contradiction.
We can now strengthen Theorem <ref>.
This result will be used in an argument in Section <ref>, specifically the proof of Theorem <ref>.
Consider the ADI word $W$ of a DI-sortable permutation $\pi$.
Let $x\in[3n-1]$ and $\pi_i = \min\{\pi_k\mid W(x)\lew\DI{C}_{\pi_k}\}$.
If $\DI{E}_{\pi_i}\lew W(x)$, then $W(x)\neq \DI{E}$.
That is, Algorithm <ref> will output all available entries in the stacks before another push from the input is allowed.
If $W(x)=\DI{C}_{\pi_i-1}$, then clearly $W(x)\neq\DI{E}$.
If $\DI{C}_{\pi_i-1}\lew W(x)\lew \DI{C}_{\pi_i}$, then by Theorem <ref>, $\DI{E}_{\pi_i}\precw \DI{N}_{\pi_i}\precw \DI{C}_{\pi_i}$.
So $W(x)=\DI{N}_{\pi_i}\neq \DI{E}$.
§ THE BIJECTION
In this section we give a bijection that takes DI-sortable permutations to their corresponding Schröder paths via their ADI words.
Let $\pi$ be a DI-sortable permutation of length $n$ with corresponding ADI word $W$.
To produce a Schröder path, we use the following constructions.
Suppose that $W$ contains $k$ maximal, consecutive substrings consisting only of Cs.
Let $\tau=(\tau_1,\tau_2,\dots,\tau_k)$ be the partition of $n$ such that $\tau_i$ is the length of the $i$th such substring.
Define a $k$-tuple $\ell$ so that $\ell_1=\tau_1$ and $\ell_i = \tau_i+\ell_{i-1}$,
when $2\leq i \leq k$. Note that $\ell_k =n$ since it counts all the Cs of $W$.
Finally, define an increasing $k$-tuple $\rho$ so that for $x\in[3n]$ we have $\DI{E}\precw W(x)\precw\DI{C}$ if and only if $W(x)=\DI{N}$ and $\#_{\DI{N}}(W^x) = \rho_i$.
Note that $\rho$ is well-defined by Theorem <ref>.
In other words, we have:
* $\ell_i$ is the number of Cs in the first $i$ maximal subsequences of Cs in $W$.
* $\rho_i$ is the number of Ns in $W$ before the $i\th$ maximal subsequence of Cs.
By Theorem <ref>, the $\rho_i\th$ N in $W$ is preceded by an E and succeeded by a C.
Let $\pi=81736245$.
Then $W = \DI{EN\,\textcolor{red}{ENC}\,ENEENN\,\textcolor{red}{ENCC}\,\textcolor{red}{ENC}\,\textcolor{red}{ENCCCC}}$, which contains $k=4$ maximal subsequences of consecutive Cs having lengths $1$, $2$, $1$, and $4$, respectively.
So $\tau = (1,2,1,4)$ and hence $\ell = (1,3,4,8)$, and $\rho = (2,6,7,8)$.
The maximal substrings of the form $\DI{ENC}^{\tau_i}$ are given in red.
Using this, we give an algorithm for producing a Schröder path from a DI-sortable permutation.
Let $\pi$ be a DI-sortable permutation of length $n$.
Use Algorithm <ref> to find the ADI word $W$ of $\pi$. From $W$, obtain $\tau$ and $\ell$.
Construct a word $T$ by replacing each maximal consecutive substring in $W$ of the form ENC$^a$ with an N.
Observe that $\DI{N}$ is the last letter of $T$, and $\#_{\DI{N}}(T)=n$.
* Construct a word $S_\DI{D}$ from $T$ by replacing the $\ell_i$th N in $T$ with a D for each $i\in[k]$.
Observe that since $\ell_k=n$, the final character in this word is D.
Remove this D to produce $S$.
Let $\pi = 81736245$ as given in Example <ref>, in which we give $W$, $\tau$, and $\ell$.
We now apply Algorithm <ref> to $\pi$.
We first produce $T= \DI{E\textcolor{blue}{N}NE\textcolor{blue}{N}EE\textcolor{blue}{N}NNN\textcolor{blue}{N}}$ by replacing the maximal consecutive substrings ENC$^{\tau_i}$ ($1\leq i \leq 4$) with an N.
Next, we replace the $\ell_i$th Ns in $T$ with D (in blue) to produce $S_\DI{D}=\DI{EDNEDEEDNNND}$, then remove the final D to produce $S = \DI{EDNEDEEDNNN}$.
See Figure <ref>.
The Schröder path EDNEDEEDNNN corresponding to $\pi=81736245$.
We now give an observation, then prove two lemmas which collectively show that Algorithm <ref> provides a one-to-one correspondence between ADI words and Schröder paths.
Let $W$ be an ADI word for a permutation $\pi$ of length $n$ with $k$ substrings of the form $\DI{ENC}^{\tau_i}$ and $k$-tuples $\rho$ and $\ell$ as defined earlier, and let $\ell_0=0$.
Observe that $\DI{C}_m$ is the $m$th occurrence of C in $W$.
Let $i\in[k]$ and $\hat{x},x \in[3n]$ such that $W(\hat{x})\precw\DI{C}_{\ell_{i-1}+1}$ and $W(x)=\DI{C}_{\ell_i}$; in other words $W(\hat{x})$ is the N which immediately precedes the $i$th maximal subsequence of Cs and $W(x)$ is the last C in the $i$th maximal substring of Cs.
See Figure <ref>.
Then, $\ell_i=\#_\DI{C}(W^x) \leq \#_\DI{N}(W^x) = \#_\DI{N}(W^{\hat{x}}) = \rho_i$.
Observe that in Example <ref>, $\ell_i \leq \rho_i$ for each $i\in[4]$.
Location number:
\overbrace{\DI{C}_{\ell_{i-1}+1}\ \DI{C}_{\ell_{i-1}+2}\ \cdots\cdots\ \DI{C}_{\ell_{i-1}+\tau_i-1}\ \begin{array}[t]{@{}c@{}}\DI{C}_{\ell_i}\\\uparrow\\x\end{array}}^{\text{i\th sequence of $\DI{C}$s}}\DI{E}\cdots$
An illustration of the indexing used for a word $W$ in Observation <ref>.
The ADI word for a DI-sortable permutation of length $n>0$ yields a Schröder path from $(0,0)$ to $(n-1,n-1)$ via Algorithm <ref>.
Let $W$ be the ADI word for a DI-sortable permutation $\pi$ of length $n$.
Suppose that $W$ contains $k$ disjoint maximal consecutive substrings of Cs.
Let $T$, $S_\DI{D}$, and $S$ be the words produced in Algorithm <ref>.
Then the length of $T$ is $2n-k$ with $\#_\DI{N}(T) = n$ and $\#_\DI{E}(T)=n-k$.
Therefore the length of $S$ is $2n-k-1$ with $\#_\DI{N}(S) = \#_\DI{E}(S)=n-k$ and $\#_\DI{D}(S)=k-1$.
It follows that $S$ corresponds to a path from $(0,0)$ to $(n-1,n-1)$.
Let $x\in[2n-k-1]$ and suppose that $S(x)=\DI{N}$.
It is sufficient to show that $\#_\DI{E}(S^x) \geq \#_\DI{N}(S^x)$.
We have $T(x)=\DI{N}$.
Let $i\in[n]$ such that $\rho_{i-1} < \#_\DI{N}(T^x) \leq \rho_i$, with the convention that $\ell_0=0, \rho_0=0$.
Since $\ell_{i-1}\leq \rho_{i-1}$ and
in producing $S$ we replace the $\ell_j$th N in $T$ with a D
whenever $1 \leq j < k$, we have $\#_\DI{D}(S^x) \geq i-1$.
Let $\hat{x}\in[3n]$ so that $W(\hat{x})=\DI{N}$ and corresponds to the N at $T(x)$.
Then $\#_\DI{N}(T^x)=\#_\DI{N}(W^{\hat{x}})$.
Furthermore $\DI{C}_{\ell_{i-1}}\lew W(\hat{x}) \lew \DI{C}_{\ell_{i-1}+1}$, and since, in producing $T$, we remove $i-1$ instances of $\DI{E}$ that precede the $\DI{N}$ at location $\hat{x}$ in $W$, it follows that $\#_\DI{E}(W^{\hat{x}}) = \#_\DI{E}(T^x)+(i-1)$.
In addition, observe that $\#_\DI{E}(S^x) = \#_\DI{E}(T^x)$ and $\#_\DI{N}(T^x) = \#_\DI{N}(S^x)+\#_\DI{D}(S^x)$.
\[\begin{array}{rcl}
\#_\DI{E}(S^x) &=& \#_\DI{E}(T^x) = \#_\DI{E}(W^{\hat{x}})-(i-1) \\
&\geq& \#_\DI{N}(W^{\hat{x}})-(i-1) = \#_\DI{N}(T^x) - (i-1) = \#_\DI{N}(S^x)+\#_\DI{D}(S^x)-(i-1) \geq \#_\DI{N}(S^x).
\end{array}\]
Hence $S$ is a Schröder path from $(0,0)$ to $(n-1,n-1)$.
For every Schröder path $S$ from $(0, 0)$ to $(n -1, n-1)$, there is an ADI word $W$ of a DI-sortable permutation of length $n$ such that Algorithm <ref> applied to $W$ gives $S$.
We begin with a Schröder path from $(0,0)$ to $(n-1,n-1)$ and consider it as a word $S$.
Let $k=\#_\DI{D}(S) +1$.
Then $\#_\DI{E}(S) = \#_\DI{N}(S) = n-k$.
Furthermore, we have $\#_\DI{E}(S') \geq \#_\DI{N}(S')$ for any prefix $S'$ of $S$, since the Schröder path stays weakly below the main diagonal.
Following Algorithm <ref> backwards, append a D to the end of $S$ to create the “extended” Schröder path $S_{\DI{D}}$.
For each $i\in[k]$, let $t_i$ be the location of the $i\th$ D in $S_\DI{D}$, and let $\ell_i$ be the location of the $i\th$ D in the substring of $S_{\DI{D}}$ consisting of all of the Ns and Ds of $S_\DI{D}$.
Since the substring of $S_{\DI{D}}$ consisting of all of the Ns and Ds of $S_\DI{D}$ has exactly $n$ letters, observe that $\ell_k=n$ and for convention, we set $\ell_0=0$.
Next create $T$ by replacing each D with an N in $S_{\DI{D}}$.
Then $\#_\DI{N}(T) = n = k + \#_\DI{E}(T)$, and
observe that if $t_i \leq x < t_{i+1}$ for some $i\in[k-1]$, then
$\#_\DI{D}(S_\DI{D}^x) = i$ and
$\#_\DI{N}(S_\DI{D}^x) \leq \#_\DI{E}(S_\DI{D}^x) = \#_\DI{E}(T^x)$.
\[\#_\DI{N}(T^x) =\#_\DI{N}(S_\DI{D}^x) + \#_\DI{D}(S_\DI{D}^x) \leq \#_\DI{E}(T^x)+i.\]
Define $r_i=\min\{ x\mid \#_\DI{N}(T^x) = \#_\DI{E}(T^x)+i\}$ for each $i\in[k]$.
In other words, $r_i$ is the minimal length for a prefix of $T$ which has exactly $i$ more Ns than Es.
By convention, let $r_0=0$.
Observe that $T(r_i) = \DI{N}$ and $\#_\DI{N}(T^{r_i-1}) = \#_\DI{E}(T^{r_i-1}) + i-1$ for each $i\in[k]$ and that $r_k \leq 2n-k$ since $\#_\DI{N} (T^{2n-k}) = \#_\DI{E} (T^{2n-k}) +k$.
Furthermore, note that $r_i \geq t_i$ for each $i\in[k]$.
\[ \#_\DI{D}(S^{r_i}_\DI{D}) =
\#_\DI{N}(T^{r_i}) - \#_\DI{N}(S^{r_i}_\DI{D}) =
\#_\DI{E}(T^{r_i}) + i - \#_\DI{N}(S^{r_i}_\DI{D}) =
\#_\DI{E}(S^{r_i}_\DI{D}) + i - \#_\DI{N}(S^{r_i}_\DI{D}) \geq i,\]
since $S_{\DI{D}}$ is a Schröder path from $(0,0)$ to $(n,n)$.
In particular, $r_k \geq t_k = 2n-k$ hence $r_k = 2n-k$.
Now create $\hat{T}$ by inserting an E before the $k$ Ns at locations $r_1,r_2,\dots,r_k$ of $T$.
Let $\hat{r}_i$ be the new location of these Ns in $\hat{T}$, that is $\hat{r}_i = r_i + i$, meaning the inserted Es are at locations $\hat{r}_i-1$ in $\hat{T}$.
Again by convention, let $\hat{r}_0=0$.
We have $\#_\DI{E}(\hat{T}) = n = \#_\DI{N}(\hat{T})$ and $\hat{r}_k = 2n$.
We now establish that $\#_\DI{N}(\hat{T}^x)\leq \#_\DI{E}(\hat{T}^x)$ for each $x\in[2n]$.
First suppose $x=\hat{r}_i-1$ for some $i\in[k]$.
Then $x-i = r_i-1$.
Furthermore $\#_\DI{N}(\hat{T}^x)=\#_\DI{N}(T^{x-i})=\#_\DI{N}(T^{r_i-1})=\#_\DI{E}(T^{r_i-1})+i-1$ and
\[\#_\DI{E}(\hat{T}^x)=\#_\DI{E}(T^{r_i-1})+i=\#_\DI{N}(T^{r_i-1})+1 = \#_\DI{N}(\hat{T}^x)+1.\]
Observe that since $\hat{T}(\hat{r}_i)=\DI{N}$, we have $\#_\DI{E}(\hat{T}^{\hat{r}_i})=\#_\DI{N}(\hat{T}^{\hat{r}_i})$.
This will be used later.
Now suppose $\hat{r}_i \leq x < \hat{r}_{i+1}-1$ where $0\leq i\leq k-1$.
Then $r_i \leq x-i < r_{i+1}$, so $\#_\DI{N}(\hat{T}^x) = \#_\DI{N}(T^{x-i})$.
So by definition of $r_{i+1}$,
\[
\#_\DI{N}(\hat{T}^x)
\leq
\#_\DI{E}(T^{x-i})+i
Additionally since $\hat{r}_k = 2n$, we have $\#_\DI{N}(\hat{T}^{\hat{r}_k})=n=\#_\DI{E}(\hat{T}^{\hat{r}_k})$.
Thus $\#_\DI{N}(\hat{T}^{x})\leq\#_\DI{E}(\hat{T}^{x})$ for all $x\in[2n]$.
Now, create the partition $\tau=(\tau_1,\tau_2,\dots,\tau_k)$ of $n$ where $\tau_i = \ell_i-\ell_{i-1}$ for each $i\in[k]$.
Next, create $W$ by inserting $\tau_i$ copies of C after the Ns in $\hat{T}$ located at $\hat{r}_i$ for each $i\in[k]$.
Let $\doublehat{r}_i$ be the new location of these Ns in $W$, that is $\doublehat{r}_i = \hat{r}_i + \ell_{i-1}$, meaning the inserted Cs are at locations $\doublehat{r}_i+1,\dots,\doublehat{r}_i+\tau_i$.
By its creation, it follows from the properties of $\hat{T}$ that $\#_\DI{E}(W)=\#_\DI{N}(W)=\#_\DI{C}(W)=n$ and $\#_\DI{N}(W^x) \leq \#_\DI{E}(W^x)$ for all $x\in[3n]$.
To show that $\#_\DI{C}(W^x) \leq \#_\DI{N}(W^x)$ for all $x\in[3n]$, we need only to show the inequality holds when $x=\doublehat{r}_i+\tau_i$ for each $i\in[k]$.
So let $x=\doublehat{r}_i+\tau_i$ for some $i\in[k]$. Recall $t_i$ was defined to be the location of the $i\th$ D in $S_\DI{D}$, and it was established that $t_i \leq r_i$ for all $i \in [k]$.
\[\#_\DI{N}(W^x)
= \#_\DI{N}(\hat{T}^{\hat{r}_i})
= \#_\DI{N}(T^{r_i})
\geq \#_\DI{N}(T^{t_i})
= \#_\DI{N}(S^{t_i}) + \#_\DI{D}(S^{t_i})
=\ell_i = \#_\DI{C}(W^x).
\]
Therefore $\#_\DI{C}(W^x) \leq \#_\DI{N}(W^x) \leq \#_\DI{E}(W^x)$ for each $x\in[3n]$.
So by Observation <ref>, $W$ is a sorting word for a permutation $\pi$ of length $n$.
Moreover, $\#_\DI{E}(W^{\doubleExphat{r}_i})=\#_\DI{N}(W^{\doubleExphat{r}_i})$ for each $i\in[k]$ since $\#_\DI{E}(\hat{T}^{\hat{r}_i})=\#_\DI{N}(\hat{T}^{\hat{r}_i})$.
From Lemma <ref>, $W$ is a DI word of $\pi$.
Finally, one can see by the construction of $W$ that $\DI{E}_{\ell_{i-1}+1}\precw\DI{N}_{\ell_{i-1}+1}\precw\DI{C}_{\ell_{i-1}+1}$ for each $i\in[k]$ and $\DI{C}_{\pi_i-1}\precw\DI{C}_{\pi_i}$ if $\pi_i\notin\{\ell_{i-1}+1\mid i\in[k]\}$.
Hence from
Theorem <ref>, $W$ is an ADI word for some DI-sortable permutation $\pi$.
Moreover, applying Algorithm <ref> to $W$ gives $S$.
By Lemmas <ref> and <ref>, our main result follows:
For any $n>0$, Algorithm <ref> produces a bijection between the set of ADI words from DI-sortable permutations of length $n$ and Schröder paths from $(0,0)$ to $(n-1,n-1)$.
Suppose we start with the Schröder path $S=\DI{EDNEDEEDNNN}$ illustrated in Figure <ref>.
We demonstrate the inverse of Algorithm <ref> described in Lemma <ref>.
$S_\DI{D}$ $T$ $\hat{T}$ $W$
The process by which the Schröder path in Figure <ref> becomes an ADI-word.
Circled numbers indicate consecutive copies of C.
We begin by producing $S_\DI{D}=\DI{EDNEDEEDNNND}$ by appending a diagonal (Figure <ref>(a)).
We identify $\ell=(1,3,4,8)$ and $\tau=(1,2,1,4)$ based on the relative locations of the diagonal edges with respect to the north edges.
We construct $T=\DI{E\textcolor{blue}{N}NE\textcolor{blue}{N}EE\textcolor{blue}{N}NNN\textcolor{blue}{N}}$ by converting each diagonal edge to a north edge (Figure <ref>(b)).
Next we find $r=(3,10,11,12)$ and construct $\hat{T}=\DI{EN\textcolor{blue}{EN}ENEENN\textcolor{blue}{EN}\textcolor{blue}{EN}\textcolor{blue}{EN}}$ by inserting E before the N at each of the locations in $T$ specified by $r$.
Note that $\hat{r}=(4,12,14,16)$, and with this we construct $W=\DI{ENE\textcolor{blue}{NC}ENEENNE\textcolor{blue}{NCC}E\textcolor{blue}{NC}E\textcolor{blue}{NCCCC}}$.
Observe that $W$ is the ADI word of 81736245, and Algorithm <ref> applied to $W$ gives $S$ (see Example <ref>).
§ PERMUTATION PROPERTIES OBSERVABLE FROM THEIR SCHRÖDER PATHS
While the bijection obtained from Algorithm <ref> can be used to show exactly which Schröder path corresponds to a given DI-sortable permutation, there are several properties that translate prominently between permutation and path.
A right-to-left minimum of a permutation is an element that is the least element seen thus far when reading the permutation from right to left.
The permutation $\pi = 81736245$ has four right-to-left minima, namely $1,2,4,5$.
If $W$ is the ADI word of a DI-sortable permutation $\pi$, then $\DI{E}_{\pi_i}\precw\DI{N}_{\pi_i}\precw\DI{C}_{\pi_i}$ if and only if $\pi_i$ is a right-to-left minimum.
Furthermore, the number of right-to-left minima of a DI-sortable permutation $\pi$ is one more than the number of diagonal steps in its corresponding Schröder path.
Suppose $\pi_i$ is a right-to-left minimum of $\pi$ with ADI word $W$.
There must be some $k\in[n]$ that satisfies $\DI{E}_{\pi_k}\precw\DI{N}_{\pi_k}\precw\DI{C}_{\pi_k}\precw\cdots\precw\DI{C}_{\pi_i}$ by Theorem <ref>. Assume by way of contradiction that $k \neq i$.
Then $\DI{E}_{\pi_i}\lew\DI{E}_{\pi_k}$ and $\DI{C}_{\pi_k}\lew\DI{C}_{\pi_i}$.
Hence $i < k$ and $\pi_k < \pi_i$, which is a contradiction to $\pi_i$ being a right-to-left minimum.
So $\DI{E}_{\pi_i}\precw\DI{N}_{\pi_i}\precw\DI{C}_{\pi_i}$.
Now suppose that $\DI{E}_{\pi_i}\precw\DI{N}_{\pi_i}\precw\DI{C}_{\pi_i}$ for some $i\in[n]$ and let $j\in[n]$ such that $i < j$.
Then $\DI{E}_{\pi_i} \lew \DI{E}_{\pi_j}$.
Therefore $\DI{C}_{\pi_i} \lew \DI{E}_{\pi_j}$, so $\DI{C}_{\pi_i}\lew \DI{C}_{\pi_j}$.
Hence $\pi_i < \pi_j$.
So $\pi_i$ is a right-to-left minimum.
Hence there is a one-to-one correspondence between consecutive sequences ENC in $W$ and right-to-left minima in $\pi$. And as seen in Algorithm <ref>, these sequences give rise to one fewer diagonal in the corresponding Schröder path of $\pi$.
Lemma <ref> leads to a visual proof to the Corollary <ref>.
The DI-sortable permutations with one right-to-left minimum, that is the DI-sortable permutations that end in $1$, are enumerated by the Catalan numbers. In particular, the number of DI-sortable permutations of length $n+1$ ending in $1$ is $\displaystyle{C_{n}=\frac{1}{n+1}{\binom{2n}{n}}}$.
The Schröder paths from $(0,0)$ to $(n,n)$ without diagonal steps are Dyck paths, known to be enumerated by the Catalan numbers.
Alternatively, recall that the basis for our DI-sortable permutations is $\{3142, 3241\}$. Thus, the DI-sortable permutations of length $n+1$ ending in $1$ are exactly the permutations whose first $n$ entries must avoid $213$, which are known to be enumerated by Catalan numbers, $C_n$ <cit.>.
For a given diagonal in a Schröder path $S$ and prefix $S'$ of $S$ terminating after the given diagonal, the we say the height of the diagonal is $\#_\DI{N}(S')+\#_\DI{D}(S')$.
In other words, its height is the $y$-coordinate of the point in the path where the diagonal terminates.
Let $\pi=81736245$ with corresponding Schröder path $S$ given in Figure <ref>.
Then the height of its diagonals are $1$, $3$, and $4$.
The heights of the diagonals determine the values of the right-to-left minima.
In particular, if the heights of the diagonals in the Schröder path corresponding to the permutation $\pi$ are $h_1, h_2, \ldots, h_k$, the right-to-left minima of $\pi$ are $1, h_1+1, h_2+1,\ldots, h_k+1$.
Suppose that $S$ has $k$ diagonals (and hence $\pi$ has $k+1$ right-to-left minima), and let $W$ be the ADI word of $\pi$.
Notice since $\ell_i$ counts the Cs in the first $i$ maximal substrings of Cs, we have $\ell_i$ counts $\DI{C}_1,\DI{C}_2,\ldots, \DI{C}_{\ell_i}$.
Thus $\DI{E}_{a}\precw\DI{N}_{a}\precw\DI{C}_{a}$ if and only if $a=\ell_i+1$ for some $i$ where $0\leq i \leq k$.
(Recall $\ell_0=0$ by convention.)
From Lemma <ref> the right-to-left minima are exactly $\ell_i+1$ for $0\leq i \leq k$.
Let $x_1,x_2,\dots,x_k$ be locations in $S$ so that $x_i$ is the location of the $i\th$ D; that is $S(x_i)=\DI{D}$ and $\#_\DI{D}(S^{x_i})=i$.
From Step 3 in Algorithm <ref>, we observe that $\#_\DI{N}(S^{x_i})+\#_\DI{D}(S^{x_i})=\ell_i$ is the height of the $i\th$ diagonal for each $i\in[k]$.
So the heights of the diagonals of $S$ are one less than the right-to-left minima of $\pi$ (excluding 1).
An interval of a permutation $\pi$ is a consecutive subsequence of $\pi$ that contains consecutive values.
The permutation $\sigma = 685712943$ contains intervals $6857, 12, 9, 43$.
A permutation $\pi$ is said to be plus-decomposable if $\pi$ is the concatenation of two non-empty intervals $\omega$ and $\tau'$ where the values of $\omega$ are less than those of $\tau'$. Further, if we rescale the entries of $\tau'$ by subtracting the length of $\omega$ from each entry of $\tau'$ to get a permutation $\tau$, we denote $\pi = \omega \oplus \tau$. If a permutation is not plus-decomposable, we say the permutation is plus-indecomposable.
The permutation $\pi=43126758$ is plus-decomposable and can be written as $\pi = 4312 \oplus 231\oplus 1$. The permutation $\sigma = 685712943$ is plus-indecomposable.
The Schröder path corresponding to a plus-decomposable DI-sortable permutation $\pi = \omega \oplus \tau$ is the Schröder path for $\omega$ followed by a diagonal step (on the main diagonal) followed by the Schröder path for $\tau$. Furthermore, this means the Schröder paths with diagonal steps on the main diagonal correspond exactly to the plus-decomposable DI-sortable permutations.
Consider the plus-decomposable permutation $\pi = \omega \oplus \tau$ and let $W$ be the ADI word of $\pi$.
Let $x\in[3n-1]$ satisfy $W(x)=\DI{E}_{\pi_k}$ with $k=|\omega|+1$ and $\pi_i = \min\{\pi_\ell\mid W(x)\lew C_{\pi_\ell}\}$.
Then by Corollary <ref>, we have $\DI{E}_{\pi_k} = W(x)\leqw\DI{E}_{\pi_i}$, so $k\leq i$.
Since $\pi = \omega\oplus\tau$ and $i\geq k = |\omega|+1$, we have $\pi_i>|\omega|$.
Hence by the definition of $\pi_i$, we have $\DI{C}_{|\omega|}\lew W(x) = \DI{E}_{\pi_k}$.
Thus we know Algorithm <ref> will force the exit of all the entries of $\omega$ before any entry of $\tau'$ (entries of $\pi$ corresponding to $\tau$) enters the stacks.
Therefore the ADI word for $\pi$ will be the ADI word for $\omega$ followed by the ADI word for $\tau$.
When Algorithm <ref> acts on the ADI word for $\pi$, it will convert the first portion of the ADI word to the Schröder path for $\omega$, but followed by $\DI{D}$ as this $\DI{D}$ is no longer the last $\DI{D}$ of the word. At the same time, it converts the remaining portion of the ADI word to the Schröder path for $\tau$.
In Figure <ref>, we see that this Schröder path for $\pi=43126758=4312\oplus231\oplus1$ is the concatenation of the Schröder paths for $4312$, $231$, and $1$ (a degenerate path), connected by diagonal edges.
The plus-decomposition of the Schröder path of $43126758$ as given in Example <ref>.
Aguiar and Moreira <cit.> showed that the number of Schröder paths from $(0,0)$ to $(n,n)$ that do not have diagonal steps on the main diagonal equals the number of Schröder paths from $(0,0)$ to $(n,n)$ that do have diagonal steps on the main diagonal as a particular case of a larger result. That is, they showed the number of Schröder paths that do not have diagonal steps on the main diagonal are enumerated by the small Schröder numbers, which gives us the following corollary.
The plus-decomposable DI-sortable permutations are in bijection with the plus-indecomposable DI-sortable permutations. In particular, both classes are enumerated by the small Schröder numbers.
We conclude with a few open questions:
* Are there any other properties of the DI-sortable permutations that are easily seen by looking at the corresponding Schröder paths?
* Is there another bijection between the DI-sortable permutations and the Schröder paths that conserves more or different properties of the DI-sortable permutations in the Schröder paths?
We would like to thank the referees for their careful reading of this paper as well as their multiple suggestions for improvement and clarity.
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1511.00566
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Rotationally fissioned asteroids produce unbound daughter asteroids that have very similar heliocentric orbits. Backward integration of their current heliocentric orbits provides an age of closest proximity that can be used to date the rotational fission event. Most asteroid pairs follow a predicted theoretical relationship between the primary spin period and the mass ratio of the two pair members that is a direct consequence of the YORP-induced rotational fission hypothesis. If the progenitor asteroid has strength, asteroid pairs may have high mass ratios with possibly fast rotating primaries. However, secondary fission leaves the originally predicted trend unaltered. We also describe the characteristics of pair members produced by four alternative routes from a rotational fission event to an asteroid pair. Unlike direct formation from the event itself, the age of closest proximity of these pairs cannot generally be used to date the rotational fission event since considerable time may have passed.
Asteroid pairs are a population of main belt asteroids that have nearly identical heliocentric orbits. These occur at a frequency in excess of that expected by random fluctuations of background asteroid orbit density <cit.>. Furthermore, when the orbits of these asteroid pairs are carefully integrated backwards, many have close encounters in phase space (small distance and low velocity) in the recent past <cit.>. This suggested a common origin, and the YORP-induced rotational fission hypothesis was immediately proposed <cit.>. Matching spectral types between pair members complements a common origin hypothesis <cit.>. If a common origin hypothesis is accepted than these pairs provide powerful tools to test many theories including space weathering, mutual body tides and binary evolution.
§ YORP-INDUCED ROTATIONAL FISSION HYPOTHESIS
The YORP-induced rotational fission hypothesis is a widely explored set of arguments that endeavors to explain the existence and properties of asteroid pairs, binaries and other multi-member systems from the rotational fission of rubble pile asteroids due to the rotational acceleration of the YORP effect <cit.>. The YORP effect is the spin and orbit averaged rotational torque on an asteroid due to thermal radiation <cit.>, and its effects have been directly observed in nature <cit.>. The theory for this effect is well-developed for principal axis rotating asteroids <cit.>, and the rotational acceleration is:
\begin{equation}
\dot{\omega}_Y = \frac{3 Y}{4 \pi \rho R^2} \left( \frac{G_1}{a_\odot^2 \sqrt{1-e_\odot^2}} \right)
\end{equation}
where $G_1 = 10^{14}$ kg km s$^{-2}$ is the solar constant at an astronomical unit divided by the speed of light, $a_\odot$ and $e_\odot$ are the heliocentric semi-major axis and eccentricity, respectively, $\rho$ and $R$ are the density and radius of the asteroid, respectively, and $Y$ is the YORP coefficient <cit.>. The YORP coefficient is mostly dependent on the shape and orientation of the body but explicitly not size-dependent and observed values lay between 10$^{-3}$ and 10$^{-2}$ <cit.>. While the YORP effect is prevalent on all small bodies in the near-Earth region, the strong dependence of the rotational acceleration from the YORP effect on size and distance limits the consequences of this phenomena to small asteroids with radii $R \lesssim 6$ km in the main belt <cit.>. However, there is a less-developed so-called tangential component that can increase the strength of the YORP torque for prograde rotators by a factor of two and so may extend this size domain by a few tens of percent; perhaps more importantly, this tangential component has a prograde bias <cit.>.
For asteroids undergoing spin up due to the YORP effect, centrifugal accelerations increase throughout the body, but they are resisted by gravity, cohesive forces and mechanical strength. Centrifugal accelerations match gravitational accelerations for a planar two sphere approximation, when the spin rate $\omega$ reaches a critical value:
\begin{equation}
\label{eqn:disruptionlimit}
\omega_c = \omega_d q_d \qquad q_d = \sqrt{\frac{1 - q^{1/3} + q^{2/3}}{\left(1+q^{1/3} \right)^2}}
\end{equation}
where $\omega_d = \sqrt{4 \pi G \rho / 3}$ is the critical disruption spin rate for a test particle on the surface of a constant density sphere, $G$ is the gravitational constant, $\rho$ is the density of the asteroid, $q_d$ is a function of the mass ratio $q$, which is the mass of the smaller, secondary component $m_s$ divided by the mass of the larger, primary component $m_p$ <cit.>. If the connection between the two components does not possess any strength, then when the spin rate reaches this critical value, the two components enter into orbit about each other.
If the internal structure of the asteroid has strength, then the asteroid goes into tension as the YORP effect increases the spin rate. For the planar two sphere approximation, the expression for the critical spin rate can be expressed simply as:
\begin{equation}
\label{eqn:strengthlimit}
\omega_c = \sqrt{ \left( \omega_d q_d \right)^2 + \left( \omega_t q_t \right)^2} \qquad q_t = \sqrt{\frac{ \left(1 - q^{1/3} + q^{2/3} \right) \left( 1 + q \right)^{2/3}}{q^{1/3}}}
\end{equation}
where $\omega_t = \sqrt{3 \sigma_c / \left( 4 \pi \rho R^2 \right)}$ is a simple prescription for the critical rotation needed to disrupt a progenitor asteroid of effective radius $R$ with a critical strength $\sigma_c$ and no gravity <cit.>, and the factor $q_t$ scales the area of the stressed internal surface with the size of the smaller, secondary fissioned mass. Naturally, failure occurs along the weakest internal surface in the body and this determines the mass ratio $q$ of the orbiting components. Note, while this equation is dimensionally correct and acceptable for order of magnitude estimates, more detailed and accurate formulae are available that are derived directly from the application of the Drucker-Prager criterion on the von Mises stress and incorporate the role of cohesion as well <cit.>. Furthermore, we note the similarity between our simple critical spin limit calculated above and those calculated by <cit.>. They consider the role of cohesion as characterized by a material friction angle in detail and find that variations of the friction angle from 0$^\circ$ to 90$^\circ$ produce only a factor of two change in the critical tensile strength spin limit.
There are only a few estimates of the critical strength of asteroids available. Examining the spin rate distribution of near-Earth and small main belt asteroids as a function of radius, there is a clear $\sim$2.3 hr spin period limit for asteroids larger than $\sim$0.2 km <cit.>; some smaller asteroids are observed to rotate much faster <cit.>. This pattern and the lack of binaries with primary radii less than $200$ m can be understood if a typical asteroid critical strength is less than $\sim$25–100 Pa <cit.>. The rapidly rotating asteroid 29075 1950 DA requires a lower limit on the critical strength of $\gtrsim$75 Pa <cit.>. The only direct estimate is derived from the disintegration of P/2013 R3 and suggests a critical strength of 40–210 Pa for this $165$ m radius asteroid <cit.>. For the ensuing simple calculations, an asteroid's critical strength is assumed to have an inverse dependence on the square root of the size of the body: $\sigma_c = \sigma_{c,0} \sqrt{ R_0 / R}$ where $\sigma_{c,0} = 125$ Pa and $R_0 = 165$ m. This relation follows the general understanding that mechanical strength is determined by flaws, which grow in size with the body, in analogy to Griffith's crack theory assuming a Weibull distribution of cracks.
§ FREE ENERGY AFTER A ROTATIONAL FISSION EVENT
The critical spin rate determines the free energy of a newly formed binary system after a YORP-induced rotational fission event. The free energy is the sum of the rotational kinetic energy of each body, their relative translational kinetic energy, and their mutual gravitational potential energy. For two spheres in contact, which is the simplest approximation for a body undergoing rotational fission at the moment of fission <cit.>, the initial free energy $E_i$ of a rotationally fissioned binary is:
\begin{equation}
\frac{E_i}{E_c} = q_1 q_d^2 - q_2 \qquad q_1 = \frac{2 + q \left( 7 + q^\frac{1}{3} \left( 10 + q^\frac{1}{3} \left( 7 + 2 q \right) \right) \right)}{q \left( 1 + q \right)^\frac{2}{3}} \qquad q_2 = \frac{10 \left(1 + q \right)^\frac{1}{3}}{1 + q^\frac{1}{3}}
\end{equation}
where $E_c = 2 \pi \rho q \omega_d^2 R^5 / 15 \left( 1 + q \right)^2$ is an energy normalization constant <cit.>.
This free energy is the energy accessible to each of the energy reservoirs: spin of each body and their relative motion. The spin-orbit coupling of the higher-order non-Keplerian gravitational potentials of the two components with orbit (and, to a lesser extent, tides) transfers energy between these reservoirs. If the energy in the orbital motion ever exceeds that in the gravitational potential then the system is on a disruption trajectory and will become unbound once the two components reach their mutual Hill sphere. This can only occur in the spherical approximation for asteroids with mass ratios $q \lesssim 0.2$, because only for these systems is the critical spin rate high enough that the free energy is positive and disruption trajectories are available <cit.>. As shown in Figure <ref>, the observed asteroid pairs obey the relationship between the rotation rate of the larger, primary member of the pair and the mass ratio predicted by considerations of the free energy of a YORP-induced rotational fission event, this is clearly the primary mechanism for the formation of asteroid pairs <cit.>. We briefly review the theory here and then expand the theory to consider the roles of strength and secondary fission on the relationship between mass ratio and primary spin rate. These may be the principal mechanisms for the production of outliers to the theory as presented in <cit.>.
The final spin period of the larger, primary component of an asteroid pair after rotational fission as a function of mass ratio. The dots are measured spin periods and mass ratios of observed asteroid pairs from <cit.>. The lines show the spin period-mass ratio relationship for a planar, two sphere approximation for a rotationally fissioned progenitor with different strengths: strengthless ($\omega_t = 0$; bold, solid), while the others are $\omega_t = 0.243 \omega_d$ (dashed), $\omega_t = 0.324 \omega_d$ (dot-dashed), $\omega_t = 0.432 \omega_d$ (dotted), and $\omega_t = 0.648 \omega_d$ (thin, solid). The critical tensile strength spin rate is $\omega_t = \sqrt{3 \sigma_c / \left( 4 \pi \rho R^2 \right)}$, where the absolute tensile strength is $\sigma_c = \sigma_{c,0} \sqrt{ R_0 / R}$. With the caveat that these parameters ($\sigma_{c,0} = 125$ Pa and $R_0 = 165$) are poorly characterized as discussed in the text, the lines representing bodies with strength correspond to a range of sizes: 500 m, 400 m, 320 m and 230 m, respectively.
§.§ Asteroid pair formation directly from a strengthless asteroid
Within the closed system of a binary formed after YORP-induced rotational fission, the free energy must be conserved. If the two components scatter each other onto a disruption trajectory due to higher-order non-Keplerian gravitational terms, i.e. spin-orbit coupling, then the free energy available to the spin states of each component is reduced since some energy is permanently stored in the disruption orbit. Also, the ratio of the rotational energy in the secondary to the primary is $\propto q^{5/3} \omega_s / \omega_p$, which means for similar spin rates and low mass ratios $q \lesssim 0.2$, the free energy stored in the secondary is relatively insignificant, so we assert as a first-order approximation that the secondary is rotating at the critical spin rate. Again, we use a planar two-sphere approximation, however it must be recognized that some asphericity is necessary for the two components to have found a disruption trajectory from their initially binary state after rotational fission, although the level of asphericity needed is very low for the binary system to explore phase space and find an escape trajectory since the initial semi-major axis of the system is very small compared to the radius of the primary <cit.>. Then, given these assumptions, the free energy of the disrupted system is only the energy in the two spin states:
\begin{equation}
\frac{E_d}{E_c} = q_3 \left( q_d^2 + \frac{\omega_p^2}{\omega_d^2} q^{-5/3} \right) \qquad q_3 = 2 \left( q^2 \left( 1 + q \right) \right)^{1/3}
\end{equation}
where $\omega_p$ is the spin rate of the primary. Setting this energy equal to the free energy available after fission, the final spin rate of the primary is:
$\omega_p / \omega_d = q^{5/3} \left( \left( q_1 - q_3 \right) q_d^2 -q_2 \right) / q_3 $. This spin rate as a rotation period is plotted (bold, solid line) as a function of mass ratio in Figure <ref>.
This simple relationship provides an easy to test hypothesis for the asteroid pair population, and the data fit the prediction well <cit.>, as shown. Some of the systems stray from the predicted relation, and <cit.> explored how changes in the spherical approximation may account for these discrepancies. In particular, asserting that the components have different shapes changes the initial free energy system since the critical spin rate for rotational fission is a function of the distance between the mass centers of the components. We can imagine at least two other modifications to the scenario above that may create asteroid pairs that do not follow the above relation.
§.§ Asteroid pair formation directly from an asteroid with strength
As discussed above, asteroids likely have significant tensile strength as their sizes decrease. If this is the case, then the free energy $E_i$ available to the binary system will be greater than in the case of only gravity, and using the same approximations as above for the free energy of the disrupted asteroid pair system, the final spin rate of the primary is:
\begin{equation}
\frac{E_i}{E_c} = q_1 \left( q_d^2 + \frac{\omega_t^2}{\omega_d^2} q_t^2 \right) - q_2 \qquad \frac{\omega_p}{\omega_d} = \sqrt{ \frac{ q^{5/3} \left( \left( q_1 - q_3 \right) \left( q_d^2 + q_t^2 \left(\frac{\omega_t}{\omega_d}\right)^2 \right) - q_2 \right)}{ q_3} }
\end{equation}
where the strength of the progenitor asteroid is characterized by the ratio of the critical tensile strength spin limit to the critical gravitational spin rate $\omega_t / \omega_d$. When this ratio is zero, the strengthless solution is obtained, but as this ratio increases, the primary spin rate after disruption increases for a given mass ratio. Because the spin rate necessary to fission the progenitor increases with $\omega_t$, increasingly higher mass ratio systems have positive free energies and can disrupt. For instance, the minimum necessary strength of the progenitor asteroid in order for it to spin fission in half (i.e. $q = 1$) is $\omega_t / \omega_d \approx 0.324$. Given the nominal asteroid strength relationship between critical tensile strength spin limit and size as determined above, this occurs for an asteroid with a radius of 400 m. A larger asteroid is unlikely to fission in half and form an unbound asteroid pair, while a smaller asteroid may fission in half and even have a rapidly rotating primary. In general, asteroids may span a wide variety of critical strengths due to variations in size and strength laws, and so a number of different $\omega_t / \omega_d$ ratios are plotted in Figure <ref>.
In summary, if the progenitor asteroid has strength, then it is possible for the primary to rotate at low periods even when the mass ratio is relatively high. Moreover, the critical mass ratio is no longer $0.2$, but a function of the ratio $\left(\omega_t / \omega_d \right)$, which is directly related to the critical strength and therefore, the size of the progenitor asteroid.
Similar to Figure <ref>, the final spin period of the larger, primary component of an asteroid pair after both rotational fission and secondary fission as a function of mass ratio $q$. The dots are measured spin periods and mass ratios of observed asteroid pairs from <cit.>. The lines show a planar, three sphere approximation for a paired binary system with semi-major axes of infinite (bold, solid), 6 (dashed), 3 (dot-dashed), 2 (dotted) and 1.5 (thin, solid) primary radii assuming an equal mass secondary fission event, so a mass ratio $p = 1$.
§.§ Asteroid pair formation after a secondary fission event
The asteroid pair production process was found to be incredibly efficient, and secondary fission was proposed as mechanism to stabilize rotationally fissioned asteroids to create long-lasting binary systems <cit.>. Secondary fission occurs when the smaller, secondary binary member is torqued due to spin-orbit couping to a critical spin rate and thus, the secondary itself undergoes rotational fission. For simplicity, we only consider strengthless asteroids, so the initial free energy is the same as in that case. Now however, there is a binary system paired with the disrupted tertiary member, so the free energy includes terms for the spin states of all three bodies, their relative translational energies, and the mutual potential energy of the binary:
\begin{equation}
\frac{E_b}{E_c} = q_3 \left( p_1 + \frac{\omega_p^2}{\omega_d^2} q^{-5/3} \right) - q_2 \frac{1+q^{1/3}}{2 a \left( 1 + p \right) } \qquad p_1 = \frac{1 - \left(1 - p^{1/3} \right) \left(1 + p^{2/3} \right) p^{1/3} }{\left(1 + p^{1/3} \right)^2 \left( 1 + p \right)^{2/3} }
\end{equation}
where $p$ is the mass ratio of the smaller, tertiary component to the larger, secondary component after secondary fission. The final spin rate of the primary is then a function of both relevant mass ratios $q$ and $p$ as well as the semi-major axis $a$ measured in primary radii $R_p$ of the bound system:
\begin{equation}
\frac{\omega_p}{\omega_d} = \sqrt{ \frac{ q^{5/3} \left( \left( 1 + q^{1/3} \right) q_2 - 2 \left(1 + p \right) \left( q_2 - q_1 q_d^2 + q_3 p_1 \right) a \right)}{ 2 q_3 \left( 1 + p \right) a} }
\end{equation}
This relationship between primary spin rate and the mass ratio of the bound system to the unbound pair member is shown in Figure <ref>. While there is a dependence on the semi-major axis of the bound system, it's remarkable how unchanged the expected spin period of the primary in a paired binary compared to a lone primary. This is important, because the secondary fission hypothesis suggests that most low mass ratio binaries formed via this process <cit.>. Indeed, two such systems have already been discovered: 3749 Balam is a triple system with an associated pair member <cit.> and 8306 Shoko is a binary system with an associated pair member <cit.>. The creation of these binary systems simultaneously with their pair members means that binary evolution can be tied to a well estimated timescale. Thus, this enables a powerful technique to learn about tidal parameters and asteroid geophysics.
Flowchart depicting the evolution of an asteroid after a YORP-induced rotational fission event (reproduced from <cit.>). Arrows indicate direction of evolution between cartoons and labels of the different evolutionary states. Solid and dashed underlines indicate long- and short-term stability, respectively. Colors indicate the dominant evolutionary process: black (purely dynamical), blue (tidal), red (BYORP effect), green (YORP effect), and yellow (planetary flyby).
§ ALTERNATIVE ASTEROID PAIR FORMATION MECHANISMS
The binary evolution model that has been developed to tie together the observed binary systems is reviewed in detail in <cit.>, and it is best summarized by the flowchart in Figure <ref>. If after YORP-induced rotational fission the asteroid is bound (negative free energy), then it follows the upper path, whereas if the asteroid is unbound (positive free energy) it follows the lower path. Asteroid pairs formed directly from rotational fission or secondary fission events are indicated by the middle path. This schematic shows four other paths to asteroid pair formation, which we discuss in turn below after introducing mutual body tides and the BYORP effect.
Mutual body tides are gravitational torques that arise from the delayed response of each binary component to the changing gravitational field of the system. If the bodies were inviscid fluids that reacted instantly to gravity, then these tides would be non-existent. However, real asteroids react viscously to tides dissipating energy in the form of heat and tidally torquing the orbit <cit.>. For the binary systems created by YORP-induced rotational fission, these tidal torques typically expand the semi-major axis and damp the eccentricity of the mutual orbit <cit.>.
The binary YORP (BYORP) effect is similar to the YORP effect, but instead of an averaged thermal radiation torque on the spin state, it is an average thermal radiation torque on the mutual orbit. Typically this effect averages to zero, since the relative orientation of the binary components is effectively random with respect to the mutual orbit, but when either or both of the binary components is in a spin-orbit resonance, then the BYORP effect becomes non-negligible <cit.>. The direction of this torque on the orbit is dependent on the shape and orientation of the bodies in the spin-orbit resonance, and so it can either contract or expand the orbit <cit.>.
§.§ Asteroid pair formation from singly synchronous mutual orbit expansion
Binary asteroids formed from YORP-induced rotational fission with low mass ratios typically tidally synchronize only the secondary since the tidal synchronization timescales are so different <cit.>. Once synchronized, the secondary begins to evolved due to the BYORP effect <cit.>. If it evolves inward, it likely reaches a tidal-BYORP equilibrium, which matches the properties of most observed small binary asteroid systems <cit.>. However, if the BYORP effect expands the mutual orbit, then it may expand to the Hill radius. Given estimates of the strength of the BYORP effect <cit.>, semi-major axis growth may take $\sim$10$^5$–10$^7$ years before it reaches the Hill radius of the system. Once it reaches the Hill radius, the binary will disrupt.
Asteroid pairs created this route take significantly longer to form after the rotational fission event than asteroid pairs created directly from rotational fission, which may be formed only $\sim$0–10$^3$ years after the rotational fission event. This has significant implications for interpreting the dynamical ages of asteroid pairs determined from backwards integration of the heliocentric orbits. Furthermore, the secondaries of these asteroid pairs have been tidally decelerated before the system is disrupted. This is very different than the secondaries of asteroid pairs formed directly from rotational fission are as likely to still be rapidly rotating as not since the spin-orbit coupling during the temporary binary phase after YORP-induced rotational fission is best approximated by a random walk <cit.>.
§.§ Asteroid pair formation from doubly synchronous mutual orbit expansion
Binary asteroids formed from YORP-induced rotational fission with negative free energy typically have high mass ratios and so tidally synchronize both binary components. Tidal synchronization timescales for small ($R <10$ km) asteroids in this configuration are typically less than or similar to near-Earth asteroid lifetimes and so significantly shorter than main belt collisional lifetimes <cit.>. Once tidally locked, these binaries undergo BYORP effect driven evolution. The direction of this mutual orbit evolution is dependent on the shapes and orientations of both binary members; it's possible that the torque on each member add in the same direction or subtract from each other, but it's unlikely that they perfectly cancel. Like above, semi-major axis growth may take $\sim$10$^5$–10$^7$ years before the mutual orbit expands to the Hill radius of the system. Once it reaches the Hill radius, the binary disrupts.
Asteroid pairs created via this mechanism will have very different characteristics than those formed directly from YORP-induced rotational fission or after a secondary fission event. These asteroid pairs are likely to have high mass ratios and low spin rates. Furthermore, the dynamical age determined from backwards integrating the heliocentric orbits of the pair members, again like above, does not simply correspond with the date of the rotational fission events.
§.§ Asteroid pair formation from mutual orbit expansion after de-synchronization of one member
The most complicated scenario envisioned for the formation of asteroid pairs occurs when both members of doubly synchronous binary asteroids have expansive BYORP torques and that during this expansion, once of the binary components desynchronizes and begins to circulate due to the YORP effect. This is very similar to the process that forms the wide asynchronous binary population <cit.>, but one of the components is still synchronous. Thus the BYORP effect continues to expand the system to the Hill radius and once it reaches the Hill radius, the binary disrupts.
Asteroid pairs created by this chain of events would appear strange. They would have a high mass ratio, but one could be rotating quite rapidly (it could be either the primary or the secondary) while the other rotates very slowly. Like the previous two mechanisms, the dynamical age determined from backwards integration of the asteroid pair is not indicative of the timing of the rotational fission event.
§.§ Asteroid pair formation from planetary flybys
The simplest formation mechanism for asteroid pairs can only occur when binary asteroids are on planet crossing orbits. Then it is quite possible for planetary flybys to disrupt these binaries <cit.>, however the perturbations from the very planet that caused the disruption may make the determination of an asteroid pair quite difficult. This process could occur to any binary asteroid system morphology, so the observed asteroid pair properties would span a large range.
§ CONCLUSIONS
In this proceedings, we have summarized the evidence that most asteroid pairs form either directly from YORP-induced rotational fission or subsequent secondary fission events. This is important because asteroid pair formation is then intimately tied to binary formation. Indeed, the discovery that some binary asteroids like 8306 Shoko and 3749 Balam have unbound pair members is likely to create stringent constraints on binary evolution in the near future. We have also summarized the different asteroid pair formation mechanisms due to the BYORP effect and planetary flybys. These asteroid pairs can appear very different but also similarly to those that formed directly following a rotational fission event. Importantly, the interpretation of the dynamical age of the asteroid pair system is very different.
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1511.00022
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DESY 15-183
KEK Preprint 2015-1869
Department of Physics,
University of Siegen,
D–57068 Siegen, Germany
Faculty of Physics and Astronomy,
Würzburg University,
D–97074 Würzburg, Germany
DESY Theory Group,
D–22603 Hamburg, Germany
Department of Physics,
University of Siegen,
D–57068 Siegen, Germany
High Energy Accelerator Research Organization (KEK),
Tsukuba, Ibaraki 305-0801, Japan
Institute for Theoretical Physics,
Karlsruhe Institute of Technology,
D–76128 Karlsruhe, Germany
We study in a bottom-up approach the
theoretically consistent description of additional
resonances in the electroweak sector beyond the discovered Higgs
boson as simplified models. We focus on scalar
and tensor resonances. Our formalism is suited for strongly
coupled models, but can also be applied to weakly interacting
theories. The
spurious degrees of freedom of tensor resonances that would lead to
bad high-energy behavior are treated using a generalization of the
Stückelberg formalism. We calculate scattering
amplitudes for vector-boson and Higgs boson pairs. The high-energy
region is regulated by the T-matrix unitarization procedure, leading
to amplitudes that are well behaved on the whole phase
space. We
present numerical results for complete partonic processes that involve
resonant vector-boson scattering, for the current and
upcoming runs of LHC.
§ INTRODUCTION
After the discovery of a $125\;\GeV$ Higgs boson, phenomenological
high-energy physics has entered a new era. The new particle fits the
expectation of the minimal Standard Model (SM). This model is thus
established as an effective field theory (EFT) that correctly
describes all current particle data (except for still missing possible
particle signals for dark matter and additional CP violation). We
know about high-energy scales where
the effective theory eventually breaks down — the scale of neutrino
mass generation, the Planck scale — but those are far outside the
reach of collider physics. The hierarchy between those scales and the
electroweak symmetry breaking scale, combined with the fact that all
known elementary particles are weakly interacting, puzzles us due to
the apparent
fine-tuning in perturbative renormalization. However, the hierarchy
puzzle as such has no phenomenological consequences. In principle,
the SM may provide a complete description of all present and future
collider data, limited just by our ability to do calculations.
Nevertheless, the apparent success of the SM does not imply that we
have full control over the spectrum at presently accessible energies,
say between 100 GeV as the electroweak mass scale and a few TeV.
First of all, there is the possibility of extra light weakly
interacting particles which escape detection at the LHC.
We will not consider this in the present work but investigate new
physics above the mass scale of $W$, $Z$, and Higgs.
The SM is complete as a renormalizable theory and weakly interacting.
Hence, it provides a mechanism for suppressing the impact
of new physics on observables. This fact is generally expressed by the
decoupling theorem <cit.>: All heavy particles (heavy
compared to the masses of $W,Z$, Higgs) can be integrated out, and
their physical effects are suppressed by powers of $m/M$ or $E/M$,
where $E$ is the effective energy of the measured elementary
interaction, and $M$ is the mass scale associated with new physics.
The EFT approach, which has been widely adopted for precision LHC
analyses, encodes this in a Lagrangian which contains operators of
dimension six and, in some cases, eight or even
higher <cit.>. Decoupling of scalar particles in the
case of Two-Higgs doublet models (2HDM) has been considered
in <cit.>, as well as in <cit.>.
For a new particle with a mass of $1\;\TeV$, the leading corrections to SM
particle properties are at the percent level and below. This is a
challenge for LHC analyses. On the other hand, in scattering
processes at the LHC, the partonic energy $E$ can easily enter the
$\TeV$ range, so direct detection is favored. Various classes of
new-physics models with extended fermion and gauge sectors can be
excluded up to several $\TeV$. However, the current experimental
sensitivity on details of the Higgs/Nambu-Goldstone sector is still
marginal. This is due to the fact that the effective energy available
for vector-boson scattering in LHC collisions, for instance, is
severely suppressed by steeply falling quark and $W/Z$ structure
In this paper we study new physics that is coupled to the
Higgs/Nambu-Goldstone sector and manifests itself in scattering
processes of $W$, $Z$, and Higgs particles. The Higgs particle does
not occur in the initial state and has its own experimental issues, so
we restrict the discussion to Nambu-Goldstone
bosons <cit.>, which the
Nambu-Goldstone boson equivalence
theorem <cit.>
relates to longitudinally
polarized $W$ and $Z$ bosons. That is, we investigate processes of
the class $V^*V^*\to VV$ ($V=W^\pm,Z,H$), where the initial vector
bosons are radiated almost on-shell and collinear off initial
energetic quarks in the colliding protons.
§.§ New effects in vector-boson scattering
Vector-boson scattering (VBS) as a physical process in hadronic
collisions has been observed recently by the ATLAS and CMS
collaborations <cit.>.
The SM prediction has been confirmed, but the initial limits on extra
interactions are still rather weak, probing an energy scale close to
the pair-production threshold of $\sim 200\;\GeV$. With higher energy
and better precision becoming available at the LHC, and at future
lepton and hadron colliders, data will become much more sensitive to
new effects in this sector. There is no reason to
restrict the modelling to weak interactions. In fact, the initially
limited experimental resolution and energy reach encourages us to consider
new strong interactions, as such deviations from the SM are
experimentally most accessible.
For decades, the theory of VBS processes has been the subject of a
vast literature, first in the disguise of the low-energy
theorem <cit.>, for questions of
unitarity <cit.>
and as a means of phenomenological
studies <cit.>. A
review of recent work can be found
in <cit.>. Most of those studies were
tailored to the Higgs-less case, which is by now excluded. In the
presence of a light Higgs, in the SM, all VBS processes are
perturbative and
respect unitarity at all energies. This situation changes drastically
once non-SM interactions are present.
Regarding the possible scenarios of new physics affecting VBS, there
are no significant restrictions from low-energy data or from the
absence of LHC discoveries. Asymptotically, the process is
determined by the amplitudes of Nambu-Goldstone boson scattering,
where the initial state contains an even number of Nambu-Goldstone bosons
and thus no half-integer representations of $SU(2)_L$. Any bosonic
excitation coupling to this state also has integer $SU(2)_L$ quantum numbers
and thus cannot couple left-handed with right-handed SM fermions.
In the limit of exact electroweak symmetry,
VBS processes and ordinary SM (fermionic) processes thus probe
distinct areas of new physics.
Electroweak symmetry breaking mixes those
sectors, but the mixing terms are again suppressed by the electroweak
scale (in operators, by additional factors of the Higgs doublet), and
are therefore subleading.
The only important constraint is quantum-mechanical unitarity, which
is severely violated in a perturbative calculation if we naively
insert the dimension-eight operators of the EFT. We have
discussed this fact in detail in
Ref. <cit.> and proposed a
framework of unitarization which allows us to augment the SM in an
arbitrary way, while maintaining high-energy unitarity and
simultaneously matching the new effects to the low-energy EFT. We
will adopt this framework, the T-matrix scheme, for the concrete
models below.
§.§ Outline of the present paper
Extending the work of <cit.>, in the present paper we
consider a wider class of scenarios beyond the SM and beyond the
electroweak mass scale. Instead of just extrapolating the EFT, which
generically leads to asymptotic saturation of amplitudes, we add new
states. The quantum numbers of the new states are chosen such that
they retain unsuppressed interactions with the VBS system in the limit
of vanishing gauge couplings. As mentioned above, this implies a certain set
of quantum-number assignments and, incidentally, suppresses their
couplings to the SM fermion sector. We may consider strongly coupled
states, which we would classify as resonances in analogy with mesons
in QCD, or weakly coupled states which we would call new elementary
particles. There is a continuous transition between these
extremes, such that we can cover all cases on equal footing.
We defer the discussion of vector resonances to a future publication,
since those states mix, after EWSB, with $W$ and $Z$ bosons and thus
exhibit a possibly different phenomenology. This limits the model to
four distinct cases, namely scalar and tensor resonances with two
different assignments of electroweak quantum numbers, respectively.
We embed these states in an extended EFT and match this to the
low-energy EFT where the resonances are integrated out. For the
high-energy limit, we apply the T-matrix scheme which keeps the model
within unitarity bounds when it eventually becomes strongly
interacting at energies above the resonance.
The case of a tensor resonance requires special considerations. While
renormalizable weakly interacting theories cannot include elementary
tensor particles, it is nevertheless possible to set up an effective
theory which contains a tensor particle and remains weakly interacting
over a considerable range of energies. This has been observed in the
context of gravity in extra
dimensions <cit.>, where massive tensor particles arise in the low-energy effective theory.
Massive gravitons provide a very specific pattern of couplings
to the Higgs doublet, gauge bosons and fermions. We will set up a
more generic model where such relations are absent, and construct a
Lagrangian description of Stückelberg type, where we can separate
the genuine tensor resonance with a controlled high-energy behavior
from unrelated higher-dimensional operators that become relevant
asymptotically. The massive-graviton model emerges as a special case.
(Massive) higher-spin fields have been discussed e.g. in <cit.>.
Given the observation that new resonances cannot necessarily be
distinguished from asymptotic saturation if the resonance energy is high and
event rates are low, we may ask the question whether the two cases
are distinguishable, i.e., whether a resonance model yields a
different prediction from a EFT extrapolation with specific
coefficients. We will discuss this issue in an exemplary way for
specific parameter sets. Furthermore, the new model allows for weakly
coupled resonances that do not leave a significant trace in the
low-energy EFT, but could nevertheless lead to a visible signal in
collider data.
To obtain numerical results, we take the unitarized model, which is
originally formulated in the gaugeless limit, re-insert gauge
couplings and continue the amplitudes off-shell along the lines
of <cit.>. This allows us to set up a model definition for a
Monte-Carlo integrator and event generator, which we use to generate
partonic event samples for the LHC, cross sections and physical
distributions. A more detailed elaboration of the
calculations can be found in <cit.>.
§ EXTENDED EFFECTIVE FIELD THEORY (EFT)
§.§ Low-Energy EFT
We are going to develop models for the high-energy behavior of
scattering amplitudes of SM particles. This cannot be done without
precisely stating the assumptions that go into those models, and to
cast them into convenient notation and parameterization.
First of all, we assume that the SM is a reasonable
low-energy effective theory. That is, a weakly interacting
(Lagrangian) gauge field theory with spontaneous $SU(2)_L\times
U(1)_Y\to U(1)_{EM}$ symmetry breaking mediated by a complex Higgs
doublet, supplemented by the standard sets of quarks and leptons,
describes all particle-physics data at and below the electroweak scale
to a good approximation.
Regarding the interactions of fermions and vector bosons, this
conclusion can be drawn from the impressive success in fitting
electroweak and flavour data to the SM. We cannot yet be so sure in
the Higgs sector proper. While the Higgs boson was discovered in
accordance with the mass range that the precision analysis of
electroweak observables suggests, there is still room for sizable
deviations from the SM predictions for its couplings. In particular,
the Higgs
self-couplings have not been measured at all. Nevertheless, we will
assume that those couplings are close to their SM values, such that
deviations can be attributed to higher-dimensional terms in the EFT.
Future data from LHC and beyond will tell whether this is true. If
not, we may generalize our findings to a nonlinearly realized Higgs
sector. We have set up our parameterization such that this would
cause few modifications in the calculations.
A second assumption regards the low-energy spectrum: we assume that
there are no additional light particles, such as Higgs singlets or extra
doublets, below the EW scale. If this was not true, it would not
invalidate the extended-EFT approach, but require the low-energy EFT
to be revised in order to include extra particles as building blocks.
Again, the model extensions discussed here would remain unchanged, but
we could expect a richer phenomenology of final states that emerge
from couplings to the extra light particles.
§.§ Including Resonances
We want to describe massive tensor and scalar resonances as extensions
of the SM, coupled to the scattering channels accessible in VBS. We
start from the low-energy EFT, the SM with higher-dimensional
operators included, and add a resonance with appropriate spin and
gauge quantum numbers to the Lagrangian. Requiring the assumed
symmetries to be manifest, uniquely determines the form of the
couplings, again in an EFT sense, i.e. as an power series expansion of
operators in some inverse mass scale $\Lambda$.
It is tempting to identify $\Lambda$ with the resonance mass $M$.
This would imply
arbitrary strong interactions at the mass scale of the resonance. The
form of couplings would be arbitrary since for $E\approx M=\Lambda$,
there is no
viable power expansion, and there are no reliable predictions. While
this is a conceivable scenario, we rather consider a more economical
setup where the resonance at mass $M$ can be separated from other
effects which are attributed to an even higher scale $\Lambda$. As we
will show below, it is possible and consistent to choose $\Lambda\gg M$,
both for scalar and tensor states. $\Lambda$ is then the appropriate
scale for all higher-dimensional operators in the extended EFT. In
the low-energy EFT, integrating out the resonance yields well-defined
higher-dimensional couplings suppressed by powers of $M$, which
combine with the undetermined $\Lambda$-suppressed coefficients
inherited from the extended EFT. Depending on their relative
magnitude, we may — or may not — be able to relate the operator
coefficients in the low-energy EFT to the resonance couplings of the
extended EFT.
§ RESONANCES: SPIN CLASSIFICATION
§.§ Scalar Resonances
A new massive spin-zero state might appear as another Higgs boson.
Indeed, a new Higgs singlet $\phi$ can couple to the SM
Higgs doublet
$\vH$ via the renormalizable
operators $\tr{\vH^\dagger \vH}\phi$ and $\tr{\vH^\dagger \vH}\phi^2$,
while a new Higgs doublet $\vH'$ can couple via $\tr{\vH^\dagger\vH'}^2$ and
For notational conventions, cf. appendix <ref>.]
These terms contribute
to Higgs mixing and self-interactions, but not directly to VBS. In
the EFT formalism, the observed Higgs boson is the only light scalar
by definition, and in the renormalizable part of the Lagrangian it saturates
the vector-boson couplings. Coupling an extra scalar to VBS then
requires two Higgs-field derivatives $\vD_\mu\vH$ and thus introduces
an effective dimension-five operator.
In a renormalizable extension of the SM Higgs sector, after
diagonalization new Higgses may eventually appear in VBS processes.
However, we have just noted that in the EFT formalism, their couplings are
higher-dimensional and thus power-suppressed. This is an incarnation
of the Higgs decoupling theorem <cit.>.
Renormalizability corresponds to the existence of special trajectories
in parameter space, where all irrelevant (i. e. higher-dimensional)
operators can be removed simultaneously from the Lagrangian by a
nonlinear field redefinition. Without a good reason a priori for
allowing only points on these trajectories,
we consider the renormalizable (possibly weakly interacting) case as a
special case that is included in the general framework. This applies,
in particular, to Higgs sector extensions by singlets and doublets, as
long as the extra scalars can be considered heavy in the sense of the
EFT formalism.
For our purposes, the phenomenology of generic scalar resonances is
then very similar to tensor resonances (see below), namely
breaking the renormalizability of the SM and inducing
higher-dimensional operators both in the low-energy EFT where they are
integrated out, and in the high-energy model where they appear
explicitly in the phenomenological Lagrangian. We will have to apply
a unitarization framework in the energy range at and beyond the resonance.
§.§ Tensor Resonances: Fierz-Pauli formalism
We now turn to massive spin-two particles, postponing spin-one for
later investigations, as stated above.
The physical particle corresponds to an irreducible
representation of the rotation group in its own rest frame and thus
consists of five component fields, mixing under rotation.
Strictly speaking, there is no reason to develop a relativistic field
theory for a generic interacting spin-two particle. If there is no UV
completion of the interacting model, it is not possible to construct a
complete Hilbert space and unitary scattering matrix. However, for
convenience of calculation, it is clearly advantageous to embed the
tensor particle in the usual relativistic field-theory context of the
EFT for the SM. We therefore introduce extra
fields, coupled to currents built from SM fields in a Lorentz- and
gauge-invariant way, in a Lagrangian formalism.
For the scalar case, this is straightforward since a spin-zero
particle is represented by a Lorentz scalar field that also has a
single component. In the tensor case, we have to deal
with the fact that the appropriate Lorentz representation has more
than five components.
In the rest frame, the Lorentz symmetry (or its universal cover
$SL(2,\mathbb{C})$) is kinematically broken down to its $SU(2)$
subgroup, the universal cover of
the rotation symmetry. The Lorentz decuplet decomposes into the
irreducible spin states
\begin{equation}
\label{eq:decomposition}
\text{symmetric tensor} \to \text{spin states}\ (2) + (1) + (0) +
(0) \quad .
\end{equation}
Looking at the symmetric rest-frame polarization tensor
$\varepsilon^{\mu\nu}$, the irreducible parts correspond to the
components $\varepsilon^{ij}$ (traceless), $\varepsilon^{0i}$,
$\varepsilon^{00}$, and $\sum\varepsilon^{ii}$ (trace), respectively. Under
the full Lorentz group, $\varepsilon^{\mu\nu}$ is also reducible and
decomposes into the traceless and trace parts. However, in the
presence of interactions it is not straightforward to maintain this
decomposition for off-shell
amplitudes <cit.>.
Our model setup requires that, on-shell, only the pure spin-two state
propagates. If we represent the resonance by a single field, the
tensor-field propagator must reduce to the form <cit.>
\begin{equation}
\label{eq:prop-FP}
G_f^{\mu\nu,\rho\sigma}(k) =
\frac{\ii\sum_\lambda\bar\varepsilon_{(\lambda)}^{\mu\nu} (k,m)
\; \varepsilon_{(\lambda)}^{\rho\sigma} (k,m)}
{k^2 - m_f^2 + i\epsilon} + \text{non-resonant}
\end{equation}
Here, $\lambda$ sums over a basis of five real-symmetric, mutually
orthogonal polarization tensors that satisfy the
\begin{equation}
\label{eq:epsilon-constraints}
k_\mu \varepsilon_{(\lambda)}^{\mu\nu} (k,m) = 0,
\qquad
\varepsilon^{\;\;\;\mu}_{(\lambda)\;\mu} (k,m) = 0,
\end{equation}
as long as $k$ is an on-shell momentum vector, $k^2=m^2$.
The solution to this problem is unique up to the non-resonant part
\begin{align}
\label{eq:propagatortensor}
&= \ii \frac{P^{\mu_1\mu_2,\nu_1\nu_2}(k,m)}{k^2-m^2+\ii \epsilon}
+ \text{non-resonant},
\end{align}
where the projection operator of spin-two can be written in terms of the
spin-one projection operator,
\begin{align}
\sum_\lambda \bar{\varepsilon}_{(\lambda)}^{\mu_1\mu_2}(k,m) \,
\varepsilon_{(\lambda)}^{\nu_1\nu_2}(k,m)
\notag \\
\frac{1}{2}\biggl[ P^{\mu_1\nu_1}(k,m)P^{\mu_2\nu_2}(k,m) +
\notag \\
\qquad\qquad - \frac{1}{3} P^{\mu_1\mu_2}(k,m)P^{\nu_1\nu_2}(k,m),
\end{align}
\begin{align}
\sum_\lambda \bar{\varepsilon}_{(\lambda)}^{\mu}(k,m)
\varepsilon_{(\lambda)}^{\nu}(k,m)
= g^{\mu\nu}-\frac{k^\mu k^\nu}{m^2}.
\end{align}
This propagator, with vanishing non-resonant part, can be obtained from the
free Fierz-Pauli Lagrangian <cit.> coupled
to a tensor source $J_f^{\mu\nu}$
\begin{align}
\label{eq:lagrangiantensor}
\LL =&\frac{1}{2} \partial_\alpha f_{\mu\nu}\partial^\alpha f^{\mu\nu}
- \frac{1}{2} m^2 f_{\mu \nu}f^{\mu \nu} \notag\\
& \;- \partial^\alpha f_{\alpha\mu} \partial_\beta f^{\beta\mu}
- f^\alpha_{\;\alpha}\partial^\mu\partial^\nu f_{\mu\nu}
- \frac{1}{2} \partial_\alpha f^\mu_{\;\mu}\partial^\alpha f^\nu_{\;\nu}
+ \frac{1}{2} m^2 f^\mu_{\;\mu} f^\nu_{\;\nu}
+ f_{\mu\nu}J_f^{\mu\nu}.
\end{align}
In the classical theory, the Lagrangian (<ref>)
enforces the conditions
\begin{equation}
\partial_\mu f^{\mu\nu} = 0
\quad\text{and}\quad
f^\mu_{\;\mu} = 0\,.
\end{equation}
This is, in principle, a valid Lagrangian description of a tensor
resonance. However, since we have to deal with off-shell amplitudes for an
effective theory, it will be useful to investigate the role of
various terms in more detail. Returning to the
propagator (<ref>), there are momentum factors
$k^\mu$ in different combinations that project out the proper spin-two
part on the pole. Going to lower energies, these factors vanish
more rapidly than the $g^{\mu\nu}$ terms and therefore reduce to
operators of higher dimension. Beyond the resonance, they will rise
more rapidly and therefore potentially provide the dominant part that
enters the unitarization prescription.
§.§ Tensor Resonances: Stückelberg formulation
As discussed above,
the extra momentum factors in the spin-two propagator represent the
mismatch between the $SO(3)$ little group representation of massive
on-shell particles and the full Lorentz-group off-shell
representations in a relativistic description. This is in analogy
with a massive spin-one boson, which in the relativistic case acquires an
extra zero component. In the following, we identify the extra degrees
of freedom for a propagating spin-two object and separate them for the
purpose of power-counting in an actual calculation.
To this end, inspired by the spin-one case, we will use the
so-called Stückelberg formulation for
tensor resonances. This has been studied in the context of effective field
theories for massive
gravity <cit.>,
<cit.> and <cit.>. The work along
these lines has been nicely reviewed in <cit.>.
Given an arbitrary symmetric polarization tensor
$\varepsilon^{\mu\nu}$ that is not restricted by auxiliary conditions,
we can subtract terms constructed from momenta, vector and scalar
\begin{equation}
\varepsilon^{\prime\;\mu\nu} = \varepsilon^{\mu\nu}
\frac1m(k^\mu\varepsilon_V^\nu + k^\nu\varepsilon_V^\mu)
\frac{k^\mu k^\nu}{m^2} \varepsilon_S
\end{equation}
and demand that (i) the Fierz-Pauli polarization tensor
$\varepsilon'{}^{\mu\nu}$ satisfies the on-shell
constraints (<ref>), and (ii) the vector
polarization is transversal
$k_\mu\varepsilon_V^\mu=0$. The resulting vector and scalar polarizations
$\varepsilon_V$, $\varepsilon_S$, $\varepsilon_T$ can be expressed as
contractions of the original $\varepsilon^{\mu\nu}$,
\begin{align}
\varepsilon_V^\mu
\frac{1}{m}\left(k_\nu\varepsilon^{\mu\nu}
- \frac{1}{m^2}k^\mu k_\nu k_\rho\varepsilon^{\nu\rho}\right),
\\
\varepsilon_S
\frac13\left(4\frac{k_\mu k_\nu}{m^2} - g_{\mu\nu}\right)
\varepsilon^{\mu\nu},
\\
\varepsilon_T
\frac13\left(g_{\mu\nu} - \frac{k_\mu k_\nu}{m^2}\right)
\varepsilon^{\mu\nu}.
\end{align}
Formally, this subtraction removes the extra representations in the
decomposition (<ref>). We note that this
prescription naturally extends to off-shell wave functions.
For the purpose of calculation, we can reproduce the
effect of the propagator (<ref>) if we remove all
$k^\mu$ factors from the tensor-field propagator but add a vector and
two scalar fields with their respective propagators. To enforce the on-shell
relations (<ref>) for their polarization (i.e.,
wave function) factors, their interactions must be prescribed
by the original tensor interactions. In field theory, such relations
can be enforced by demanding a gauge invariance. Since the momenta
have been banished from the numerators of the propagators this way, the
power-counting in the resulting Feynman rules will be explicit, in
analogy with the 't Hooft-Feynman gauge of a gauge
Stückelberg <cit.>
originally formulated the algorithm
that systematically introduces the compensating fields together with the
extra gauge invariance in the Lagrangian formalism. Applying the
algorithm to the massive tensor case, we start with the Fierz-Pauli
Lagrangian which corresponds to the minimal single-field propagator of
the pure spin-two tensor. After removing any explicit constraints from
the tensor field, we introduce first the Stückelberg vector $A^\mu$
that cancels the $f^{0\mu}$ components, by the replacement
\begin{equation}
f^{\mu \nu} \rightarrow
f^{\mu \nu} + \frac{1}{m}\partial^\mu A^\nu + \frac{1}{m}\partial^\nu A^\mu,
\end{equation}
and then cancel the extra unwanted $A^0$ components that this field
introduces, together with $f^{00}$, by a Stückelberg
scalar $\sigma$,
\begin{equation}
A^\mu \rightarrow A^\mu + \frac{1}{m}\partial^\mu \sigma
\end{equation}
Finally, we introduce another Stückelberg scalar $\phi$ for
cancelling the trace by
\begin{equation}
f^{\mu \nu} \rightarrow f^{\mu \nu} + g^{\mu \nu} \phi
\end{equation}
This scheme guarantees that the interactions of
the new fields in the Lagrangian are correctly related to the original
interactions of the tensor field. The resulting Lagrangian exhibits
the gauge invariances that reflect the redundancy of the
Stückelberg fields and there is a gauge (called unitary gauge) in which
all Stückelberg fields vanish and the original Fierz-Pauli Lagrangian is
recovered. The new Fierz-Pauli Lagrangian with the additional scalar and
vector modes reads
\begin{equation}
\label{eq:FullStueckelberg}
\begin{aligned}
\LL \; =
& \quad \frac{1}{2} \partial_\alpha f_{\mu\nu}\partial^\alpha f^{\mu\nu}
- \frac{1}{2} m^2 f_{\mu \nu}f^{\mu \nu}
- \partial^\alpha f_{\alpha\mu} \partial_\beta f^{\beta\mu}
- f^\alpha_{\;\alpha}\partial^\mu\partial^\nu f_{\mu\nu}\\
& \; - \frac{1}{2} \partial_\alpha f^\mu_{\;\mu}\partial^\alpha f^\nu_{\;\nu}
+ \frac{1}{2} m^2 f^\mu_{\;\mu} f^\nu_{\;\nu}
- \partial_\mu A_\nu \partial^\mu A^\nu
+ \partial_\mu A^\mu \partial_\nu A^\nu \\
& \; - 2 m f_{\mu \nu} \partial^\mu A^\nu
+ 2 m f^\mu_{\;\mu} \partial_\nu A^\nu
+ 6 m \phi \partial_\mu A^\mu \\
& \; - 2 f_{\mu \nu} \partial^\mu \partial^\nu \sigma
+ 2 f^\mu_{\;\mu} \partial^2 \sigma
- 2 f_{\mu \nu} \partial^\mu \partial^\nu \phi
+ 2 f^\mu_{\;\mu} \partial^2 \phi \\
& \; -3 \partial_\mu \phi \partial^\mu \phi
+ 6 m^2 \phi \phi
+ 3 m^2 f^\mu_{\;\mu} \phi
\\
& \; + \left ( f_{\mu \nu}+g_{\mu \nu} \phi+
\frac{2}{m}\partial_\mu A_\nu +
\frac{2}{m^2}\partial_\mu \partial_\nu \sigma
\right )J_f^{\mu \nu} \qquad .
\end{aligned}
\end{equation}
The scheme simplifies slightly since both scalars are related to the
original tensor, so their interactions are not independent. We can
choose the gauge
\begin{equation}
\phi = -\sigma
\end{equation}
and arrive at a minimal Stückelberg Lagrangian <cit.>
(adjusted by partial integration and simplified),
\begin{equation}
\begin{aligned}
\LL \; =
& \quad \frac{1}{2} \partial_\alpha f_{\mu\nu}\partial^\alpha f^{\mu\nu}
- \frac{1}{2} m^2 f_{\mu \nu}f^{\mu \nu} \\
& \; - \left (\partial^\alpha f_{\alpha \mu} -
\frac{1}{2} \partial_\mu f^\rho_{\;\rho}
- m A_\mu \right ) ^2 \\
& \; - \frac{1}{4} \partial_\alpha f^\mu_{\;\mu}\partial^\alpha f^\nu_{\;\nu}
+ \frac{1}{4} m^2 f^\mu_{\;\mu} f^\nu_{\;\nu}
- \partial_\mu A_\nu \partial^\mu A^\nu
+ m^2 A_\mu A^\mu\\
& \; + \left (
\partial_\mu A^\mu - 3 m \sigma + \frac{1}{2}m f^\mu_{\;\mu}
\right )^2 \\
& \; +3 \partial_\mu \sigma \partial^\mu \sigma
- 3 m^2 \sigma \sigma
\\
& \; + \left ( f_{\mu \nu} - g_{\mu \nu} \sigma +
\frac{2}{m}\partial_\mu A_\nu + \frac{2}{m^2}\partial_\mu
\partial_\nu \sigma
\right )J_f^{\mu \nu} \qquad .
\end{aligned}
\end{equation}
For perturbative calculations we have to fix the gauge up to
residual gauge transformations $\lambda(x)$ that decouple on-shell,
i.e. satisfy the harmonic condition $(\partial^2+m^2)\lambda=0$. To
this end, we choose linear gauge conditions,
\begin{align}
\partial_\mu A^\mu - 3 m \sigma + \frac{1}{2}m f^\mu_{\;\mu} &= 0 \\
\partial^\alpha f_{\alpha \mu} - \frac{1}{2} \partial_\mu f^\rho_{\;\rho}
- m A_\mu &= 0
\end{align}
and end with a diagonalized Lagrangian,
\begin{equation}
\begin{aligned}
\LL \; =
& \quad \frac{1}{2} f_{\mu\nu}\left (-\partial^2 - m^2 \right
)f^{\mu \nu} \\
& \; + \frac{1}{2} f^\mu_{\;\mu}\left (-\frac{1}{2} \left
(-\partial^2 - m^2 \right ) \right )f^\nu_{\;\nu} \\
& \; + \frac{1}{2} A_\mu \left ( -2 \left (-\partial^2 - m^2 \right )
\right) A^\mu\\
& \; +\frac{1}{2} \sigma \left ( 6 \left
( -\partial^2 - m^2 \right ) \right )\sigma
\\
& \; + \left ( f_{\mu \nu} - g_{\mu \nu} \sigma +
\frac{1}{m}\left (\partial_\mu A_\nu + \partial_\nu A_\mu \right )
+ \frac{2}{m^2} \partial_\mu \partial_\nu \sigma
\right )J_f^{\mu \nu} .
\end{aligned}
\end{equation}
Next, we normalize the fields canonically
\begin{equation}
\begin{aligned}
\LL \; =
& \quad \frac{1}{2} f_{\mu\nu}\left (-\partial^2 - m^2 \right
)f^{\mu \nu} \\
& \; + \frac{1}{2} f^\mu_{\;\mu}\left (-\frac{1}{2} \left
(-\partial^2 - m^2 \right ) \right )f^\nu_{\;\nu} \\
& \; + \frac{1}{2} A_\mu \left ( \partial^2 + m^2 \right ) A^\mu\\
& \; +\frac{1}{2} \sigma \left ( -\partial^2 - m^2 \right ) \sigma
\\
& \; + \left ( f_{\mu \nu} - \frac{1}{\sqrt{6}} g_{\mu \nu} \sigma +
\frac{1}{\sqrt{2}m}\left (\partial_\mu A_\nu + \partial_\nu A_\mu \right )
+ \frac{\sqrt{2}}{\sqrt{3}m^2} \partial_\mu \partial_\nu \sigma
\right )J_f^{\mu \nu}
\end{aligned}
\end{equation}
and find the canonical propagators
\begin{align}
\label{eq:proptensor}
\Delta_{\mu\nu,\rho\sigma} (f)
&= \frac{\ii}{k^2 - m^2}
\left(\frac{1}{2} g_{\mu\rho}g_{\nu\sigma}
-\frac{1}{2} g_{\mu\nu}g_{\rho\sigma}
\right )
\\
\Delta_{\mu\nu} (A)
&= \frac{-\ii}{k^2 - m^2} g_{\mu\nu}
\\
\Delta (\sigma)
&= \frac{\ii}{k^2 - m^2}
\end{align}
for the resulting unconstrained tensor, vector, and scalar fields,
For a complete formulation at the quantum level, the gauge-fixed
Lagrangian has to be embedded in a BRST formalism. Introducing
appropriate Faddeev-Popov ghosts and auxiliary Nakanishi-Lautrup
fields, the classical action can be rendered BRST invariant. The
quantum effective action with resonance exchange is then defined as the
solution to a Slavnov-Taylor equation, to all orders in the EW
perturbative expansion. The gauge-fixing terms become BRST
variations which do not contribute to physical amplitudes, and the
Stückelberg fields combine with the ghosts and auxiliary fields to
BRST representations that can be consistently eliminated from the Hilbert
space. For free fields, this procedure is detailed
in <cit.>.
As desired, these propagators do not contain any
momentum factors. This fact turns out to be essential for a
Monte-Carlo calculation for physical processes, where all bosons are
off-shell in a generic momentum configuration.
§.§ Tensor Resonances: Summary
Given this lengthy derivation, we may ask again whether the
Stückelberg formulation has any advantage over the original
Fierz-Pauli Lagrangian. Algebraically, both are equivalent and result
in identical on-shell amplitudes.
This should be viewed in analogy with massive vector bosons, for which
the Stückelberg approach reproduces the usual reformulation as a
spontaneously broken gauge theory. Again, this is mathematically
equivalent to the original model, as has been pointed out
repeatedly <cit.>. However, once the accessible
energy in a process exceeds the resonance mass, there is a conceptual
difference. In the gauge-theory version, there is no
higher-dimensional operator with a $1/M$ coefficient. Any additional
effects would come with a new cutoff $1/\Lambda$. Scattering
amplitudes are bounded beyond the resonance as long as $\Lambda$ is
considered large. By contrast, in the formulation with massive vector
bosons, there are $k^\mu/M$ terms in the propagator which a priori
require the inclusion of a whole series of operators with $1/M$
factors. The model is strongly interacting from the onset and has no
predictivity. If actual data show that interactions are indeed weak,
this fact would be interpreted as a fine-tuned cancellation among
Turning this argument around, if a vector boson is observed to
interact weakly over a significant range of scales above its mass, it
is natural to describe it as a gauge boson,
which in turn determines the allowed interaction pattern.
Analogously, if we assume that a tensor resonance interacts
weakly over a significant range of scales above its mass, it is
natural to describe it by the Stückelberg approach. We will
therefore adopt the Stückelberg Lagrangian as the basis of a
tensor-EFT with a minimum set of free parameters.
Clearly, we can always add extra interactions with further free
parameters. Those interactions take the form of higher-dimensional
operators which do not contribute on the resonance. They describe
unrelated new-physics effects.
§ LAGRANGIAN FOR THE EXTENDED EFT
We now combine the findings of the previous section in order to set up a
Lagrangian description of the resonances, as an extension of the low-energy
EFT which already (implicitly) includes the complete set of higher-dimensional
operators. Apart from the Lorentz representations as scalar or tensor, we have
to consider the representation of the internal symmetry group. As we will
argue in detail below, we take this as the Higgs-sector global symmetry
$SU(2)_L\times SU(2)_R$, where only the $SU(2)_L\times U(1)_Y$ subgroup is
gauged. $SU(2)_R$ breaking terms can be systematically included, but we do
not consider those in the present work.
§.§ Isospin
In the literature on VBS, resonances have traditionally been
categorized in terms of weak isospin, i.e., custodial $SU(2)_C$
multiplets. This is appropriate for a Higgsless scenario, where the
actual scale of EWSB is given by its natural value $4\pi v\approx
3\;\TeV$ (cf. e.g. <cit.>). Without a light Higgs boson, VBS
scattering at the LHC would probe the physics at energies below the true EWSB
scale, so the (approximate) low-energy symmetry applies.
However, since the discovery of the Higgs boson, we know that VBS
processes probe a scale above the masses of the physical Higgs and the
electroweak gauge bosons. We have to impose the unbroken high-energy
symmetry on the theoretical description. Neglecting hypercharge, this
is $SU(2)_L\times SU(2)_R$. We therefore describe new resonances
coupled to the SM Higgs sector in terms of
$SU(2)_L\times SU(2)_R$ multiplets.
It is not obvious that new physics coupled to the Higgs sector
actually has this symmetry. $SU(2)_L\times SU(2)_R$ is, first of all,
an accidental approximate symmetry of the SM EWSB sector. There are no possible
terms in the dimension-four Higgs potential that break $SU(2)_R$, so
EWSB leaves the diagonal custodial $SU(2)$ symmetry untouched.
However, hypercharge and top-quark couplings are not consistent with
$SU(2)_R$. Nevertheless,
in the gaugeless limit the hypercharge coupling vanishes,
and top quarks are irrelevant for VBS anyway, so $SU(2)_R$ remains a
good symmetry of VBS (at high $p_T$) in the SM. Beyond the SM, new
effects in VBS are transmitted only via the Higgs doublet. In the
low-energy EFT, they require higher-dimensional operators.
These would cause power corrections to the $\rho$ parameter and are therefore
constrained by the observed agreement of the measured $\rho$ parameter
with the pure SM prediction. For our purposes, we thus adopt
$SU(2)_R$ as a symmetry of new physics in the Higgs sector, to keep
things simple. We have to keep in mind that this need not be the
case, and leave the discussion of $SU(2)_R$ breaking in this context
to future work.
Resonances of even spin with unsuppressed couplings to a pair of
Higgs/Nambu-Goldstone bosons, must reside in the symmetric part of the
decomposition of the product representation of the $SU(2)_L\times
SU(2)_R$ symmetry, $(\frac{1}{2},\frac{1}{2})\otimes
(\frac{1}{2},\frac{1}{2})$. In the effective interaction operator,
this representation appears as a $H\otimes H^\dagger$ factor. There
are only two possibilities:
* $(0,0)$: a neutral singlet (isoscalar).
* $(1,1)$: a $3\times 3$ matrix, which contains nine components.
After EWSB, the multiplet decomposes into an isotensor (five
components), an isovector (three components), and an isoscalar (one
component). In terms of the gauged $SU(2)_L \times U(1)_Y$ subgroup, the
nonet decomposes into a complex $SU(2)_L$ triplet with a doubly
charged component and a real $SU(2)_L$ triplet, as described
in <cit.>. The relative mass splitting between these
states is of order $(m_W/M)^2$, where $M$ is the average resonance
mass. For our purposes where we assume $M\gg m_W$, we ignore that
splitting and thus deal with a nonet of degenerate resonance
We note that due to the existence of the light Higgs, the close
analogy between spin and isospin is broken at this point: tensor
states have just five physical degrees of freedom, but an isotensor
resonance in VBS, given the symmetry assumptions of the present paper, does not
exist in isolation. The distinction comes into play
once physical Higgs bosons are involved in a process. In VBS
amplitudes, the symmetry relates, for any given resonance multiplet,
Nambu-Goldstone pairs with Higgs pairs, i.e., $VV (V=W,Z)$ to $HH$ production.
For simplicity of notation, we will continue to denote the $(0,0)$
case as isoscalar and the $(1,1)$ as isotensor, respectively, keeping
in mind that the latter case actually is always accompanied by
isovector and isoscalar components.
For a scalar isoscalar resonance $\sigma$, we may consider couplings
of the form
\begin{equation}
\label{sHH}
\sigma\tr{\vH^\dagger \vH}
\end{equation}
\begin{equation}
\label{sDHDH}
\sigma\tr{(\vD_\mu \vH)^\dagger (\vD^\mu \vH)}.
\end{equation}
The former operator is of lower dimension and might therefore be
considered the dominant contribution. It is part of the Higgs
potential and influences Higgs mixing and production processes. In
the present work, we assume that the scalar state has been broken down
in terms of the SM Higgs doublet and further states, which themselves
arrange as multiplets. Since the SM Higgs couplings in the
lowest-order EFT, the pure SM, saturate the Higgs couplings to SM
particles and are fixed by definition, residual mixing and potential
terms arrange into higher-dimensional operators. In particular, a
resonance coupled to Nambu-Goldstone bosons is represented by the
term (<ref>), while the lower-dimensional term (<ref>) does
not enter. We therefore do not consider (<ref>) and concentrate
on the dimension-five coupling (<ref>).
This leads to a current for the scalar isoscalar resonance of the form
\begin{align}
J_{\sigma} &= F_\sigma
\tr{ \left ( \vD_\mu \vH \right )^\dagger \vD^\mu \vH} \; .
\end{align}
§.§ The Isotensor Representation
While the description of an isoscalar is simple, we have to look at the
interactions of the isotensor more carefully. For simplicity, we will
first restrict ourselves to a scalar field multiplet.
A resonance with chiral $SU(2)_L\times SU(2)_R$ quantum numbers $(1,1)$ has
nine scalar degrees of freedoms. In the chiral representation these nine
degrees of freedom can be represented as the tensor $\Phi^{ab}$ with
the indices $a,b \in \{1,2,3\}$. Therefore, the Lagrangian
describing an isotensor resonance in the Nambu-Goldstone/Higgs
boson sector can be written as
\begin{align}
\label{eq:Lagrangian_chiraltensor}
\LL_{\Phi}= \frac{1}{2} \partial_\mu \Phi^{ab} \partial^{\mu}\Phi^{ab}
- \frac{m_\Phi^2}{2} \Phi^{ab}\Phi^{ab}
+ J^{ab}_{\Phi}\Phi^{ab} \,
\end{align}
where the current has a $SU(2)_L$ and a $SU(2)_R$ index
\begin{align}
\label{eq:Isotensor-current}
J^{ab}_\Phi =
F_\phi \tr{\left(\vD_\mu \vH\right)^\dagger \tau^a \vD^\mu \vH \tau^b}\, .
\end{align}
Analogously to the isoscalar case, the coupling $F_\phi$ is suppressed
by a new physics scale $\Lambda$. To expose the coupling structure to the
Nambu-Goldstone/Higgs boson sector, the current can be expanded in
the gaugeless limit
\begin{align}
\label{eq:isotensor_current_expanded}
\begin{aligned}
\tr{\left(\vD_\mu \vH\right)^\dagger \tau^a \vD_\nu \vH \tau^b}
=& \frac{1}{2} \left(\partial_\mu h \partial_\nu h - \partial_\mu
w^i \partial_\nu w^i \right) \delta^{ab}
- \frac{1}{2}
\left (\partial_\mu w^i \partial_\nu h+\partial_\nu
w^i \partial_\mu h \right)\varepsilon^{abi} \\
&+\frac{1}{2} \left(\partial_\mu w^a \partial_\nu w^b
+ \partial_\mu w^b \partial_\nu w^a \right)
\end{aligned}
\end{align}
Here, the decomposition into isotensor, isovector and isoscalar
is already manifest. The resonance $\Phi^{ab}$ can be represented in
a basis constructed from tensor products of $SU(2)$ generators by the
Clebsch-Gordon decomposition
\begin{align}
\mathbf{1} \otimes \mathbf{1} = \mathbf{2} + \mathbf{1} + \mathbf{0} \, .
\end{align}
Using the basis in the appendix <ref>,
the resonance $\Phi^{ab}$ is
rewritten into its $SU(2)_C$ components
\begin{align}
\Phi^{ab} \rightarrow \Phi_t + \Phi_v + \Phi_s
\end{align}
\begin{align}
\Phi_t &= \phi_t^{++}\tau_t^{++}+\phi_t^{+}\tau_t^{+}+
\phi_t^{0}\tau_t^{0}+\phi_t^{-}\tau_t^{-}+\phi_t^{--}\tau_t^{--} \, ,\\
\Phi_v &= \phi_v^{+}\tau_v^{+} +
\phi_v^{0}\tau_v^{0}+\phi_v^{-}\tau_v^{-} \, ,\\
\Phi_s &= \phi_s\tau_s \, .
\end{align}
The Lagrangian (<ref>)
can be written in terms of the $SU(2)_C$ basis
\begin{align}
\LL_\phi =& \frac{1}{2}\sum_{i=s,v,t} \tr{(\partial_\mu \Phi_i)^\dagger
\partial^\mu \Phi_i
-m_\Phi^2 \Phi_i^2}
+\tr{\left(\Phi_t + \frac{1}{2}\Phi_v - \frac{2}{5}\Phi_s \right)J_\phi }\\
J_{\phi} =& F_\phi
\left (
\left ( \vD_\mu \vH \right )^\dagger \otimes \vD^\mu \vH
+\frac {1}{8} \tr{\left ( \vD_\mu \vH \right )^\dagger \vD^\mu \vH }
\right ) (\tau^a\otimes \tau^a)
\end{align}
In absence of the Higgs boson, the coefficient of the second term
is chosen in such a way, that the trace of the current vanishes.
In this scenario, the isovector and isoscalar degree of freedoms
decouple from the model and only the isotensor is needed to describe
this resonance. However, including a Higgs the Lagrangian
(<ref>) guarantees the amplitude
relation between the Higgs and Nambu-Goldstone bosons that will be
introduced in section <ref>.
The crossing relations are manifest in the
scattering amplitudes for the Nambu-Goldstone/Higgs boson,
which can be determined most easily in the gaugeless limit.
One prominent example for such scalar isotensor resonances appears in
the context of composite Higgs models of the type Little Higgs,
particularly in the so called Littlest Higgs model <cit.>.
These resonances predominantly couple to the (electro)weak gauge sector of
the SM.
§.§ The Tensor Current
We now construct the effective current that is coupled to a tensor
resonance multiplet. By assumption, the resonance should be produced
in VBS processes. We have to consider independent couplings to the
gauge and Higgs/Nambu-Goldstone sectors. The gauge-sector couplings
should vanish in the gaugeless limit, so we are led to consider
the Higgs-sector coupling.
For a tensor isoscalar resonance, the
lowest-dimensional current consists of two terms,
\begin{align}
J^{\mu \nu}_f&=
F_f \left (
\tr{ \left ( \vD^\mu \vH \right )^\dagger \vD^\nu \vH}
- \frac{c_f}{4} g^{\mu \nu}
\tr{ \left ( \vD_\rho \vH \right )^\dagger \vD^\rho \vH}
\right) .
\end{align}
The second term actually couples to the trace of the tensor field,
which vanishes on-shell. It is therefore part of the non-resonant
continuum and can alternatively be replaced by higher-dimensional
operators in the EFT. Nevertheless, it is required if, for instance,
we want to construct a traceless current. For now, we leave the
coefficient $c_f$ undetermined.
The tensor-field coupling then reads
\begin{equation}
f_{\mu \nu}J_f^{\mu \nu}
\end{equation}
in the Fierz-Pauli formulation (section <ref>), and
\begin{equation}
f_{\mu \nu} J_f^{\mu \nu}
- \sigma {J_f}^{\mu}_{\, \mu}
- \frac{1}{m}A_\mu \partial_\nu J_f^{\mu \nu}
+ \frac{2}{m^2} \sigma \partial_\mu \partial_\nu J_f^{\mu \nu}
\end{equation}
in the Stückelberg formulation (section <ref>). In the
second version, the momentum factors in the propagator have been
turned into derivatives that act on the current. There is also a
coupling to the trace of the current.
The formally dominant high-energy ($s\to\infty$) behavior of the
amplitude thus is given by the exchange of Stückelberg vector and
scalar. The contribution would vanish if the current was conserved.
Evaluating the divergence of first and second order, using
(<ref>) and (<ref>) in the
\begin{align}
\label{eq:derivative_Tensor_current}
\begin{aligned}
\partial_\mu J_f^{\mu \nu}
F_f \tr{ \left ( \vD^2 \vH \right )^\dagger \vD^\nu \vH}
+\frac{F_f}{4} \left ( c_f +2 \right )
\tr{ \left ( \vD_\mu \vH \right )^\dagger\left [ \vD^\mu,\vD^\nu
\right ] \vH} \\
&-\frac{F_f}{4} \left (c_f -2 \right )
\tr{ \left ( \vD_\mu \vH \right )^\dagger\left \{ \vD^\mu,\vD^\nu
\right \} \vH} \\
- F_f \lambda \tr{\widehat{\vH^\dagger \vH}} \tr{ \vH ^\dagger \vD^\nu \vH} \\
&-\ii g F_f
\tr{ \left ( \vD_\mu \vH \right )^\dagger\vW^{\mu\nu} \vH}
-\ii g^\prime F_f
\tr{\vH \vB^{\mu\nu} \left(\vD_\mu \vH\right)^\dagger }\, ,
\end{aligned}
\end{align}
\begin{align}
\label{eq:2derivative_Tensor_current}
\begin{aligned}
\partial_{\nu}\partial_\mu J_f^{\mu \nu}
=& F_f \tr{ \left(\vD_\mu \vH \right)^\dagger
\left(
\vD_\nu \vD^\mu \vD^\nu \vH + \vD^\mu \vD^2 \vH
- \frac{c_f}{2} \vD^2 \vD^\mu \vH
\right)}\\
&+F_f \tr{ \left(\vD^2 \vH \right)^\dagger \vD^2 \vH}
+ F_f \tr{ \left(\vD_\mu \vD_\nu \vH \right)^\dagger \vD^\nu \vD^\mu \vH}\\
&- \frac{c_f}{2} F_f \tr{ \left(\vD_\mu \vD_\nu \vH \right)^\dagger \vD^\mu \vD^\nu \vH}\\
- F_f \lambda \tr{\widehat{\vH^\dagger \vH}} \tr{\vD_\mu \vH
^\dagger \vD^\mu \vH}
- F_f \lambda \tr{\widehat{\vH^\dagger \vH}} \tr{ \vH ^\dagger \vD^2
\vH} \\
&- 2F_f \lambda \tr{{\vH^\dagger \vD_\mu \vH}} \tr{ \vH ^\dagger \vD^\mu \vH}\\
&+\frac{g^2 F_f}{2}
\left(\tr{ \left ( \vD_\mu \vH \right )^\dagger \vH \left(\vD^{ \mu}
\vH \right )^\dagger\vH}
-\tr{ \left ( \vD_\mu \vH \right )^\dagger \left(\vD^{ \mu} \vH \right )
\vH^\dagger\vH}
\right)\\
&+\frac{{g^\prime}^2 F_f}{2}
\left(\tr{ \left ( \vD_\mu \vH \right )^\dagger \vH \left(\vD^{ \mu}
\vH \right )^\dagger\vH}
-\tr{ \left ( \vD_\mu \vH \right )^\dagger \left(\vD^{ \mu} \vH \right )
\vH^\dagger\vH}
\right)\\
&+\frac{ g^2 F_f}{2}
\tr{ \vH^\dagger\vW_{\mu\nu} \vW^{\mu\nu} \vH}
+\frac{ {g^\prime}^2 F_f}{2}
\tr{ \vH \vB_{\mu\nu} \vB^{\mu\nu} \vH^\dagger} \\
&+ g{g^\prime} F_f
\tr{ \vH^\dagger \vW_{\mu\nu} \vH \vB^{\mu\nu}} \\
&-\ii g F_f
\tr{ \left ( \vD_\mu \vH \right )^\dagger\vW^{\mu\nu} \vD_\nu \vH }\\
&-\ii g F_f
\tr{ \left(\vD_\mu \vH\right) \vB^{\mu\nu} \left(\vD_\mu
\vH\right)^\dagger } \, ,
\end{aligned}
\end{align}
we observe that the current is not conserved.
However, none of the nonvanishing
terms contributes to the $VV\to VV$ process at high energy. The
Stückelberg fields effectively decouple, and the high-energy
behavior can be calculated from the propagator (<ref>).
If we take EWSB into account, we do get a nonvanishing divergence also
at the two-particle level. New terms arise that are proportional to
powers of $v$, and thus to the $W$, $Z$, and Higgs masses. The
Stückelberg vector transmits, via EWSB mixing, a coupling to
transversal vector bosons. In amplitudes, these factors are
accompanied by factors of $1/m$. In the limit of a heavy resonance,
the Stückelberg terms are thus parametrically suppressed and become
relevant only for energies significantly beyond the resonance mass.
Conversely, if the resonance mass is comparable to the electroweak
scale, the Stückelberg terms are significant.
The remainder of the amplitude that corresponds to the genuine tensor
propagator (<ref>) does not contain momentum factors.
Nevertheless, the interaction is of dimension five, so we expect
contributions that rise with energy. This occurs for external
longitudinally polarized vector bosons which carry a momentum factor.
We obtain a factor $s^2$ in the numerator that asymptotically cancels
with the denominator, so the effective rise is proportional to
$s/m^2$. Qualitatively, this is the same result as for the case of a
scalar resonance, or for a Higgs-less theory.
We conclude that we can unitarize the amplitude uniformly for all
spin-isospin channels, starting from the gaugeless Nambu-Goldstone boson
limit, without having to account for transversal gauge bosons or
higher powers of $s$ beyond the resonance. The algorithm can be taken
unchanged from the pure-EFT case <cit.>. However, we
have to restrict the allowed values of resonance masses and couplings
such that the Stückelberg terms discussed above remain numerically
small within some finite energy range. Outside this range, we can no
longer separate the Higgs/Nambu-Goldstone sector of the theory but are
sensitive to unknown strong interactions that involve all channels of
longitudinal, transversal, and Higgs exchange simultaneously. While
the unitarization scheme of <cit.> is also applicable in
that situation, it becomes technically more involved; we defer this
case to future work.
§.§ Complete model definition
We now list the effective Lagrangians that we consider in the
subsequent calculations. In all cases, the basic theory is the SM
EFT, i.e., the SM with the observed light Higgs boson in linear
representation, extended by higher-dimensional operators. We add
four different resonance multiplets, corresponding to all combinations
of spin and isospin 0 and 2, respectively. The Lagrangians can be
The spin-two Lagrangian is presented in the Stückelberg gauge.
Regarding the resonance fields, we should further select electroweak
quantum numbers, as discussed in section <ref>, by
defining the precise form of the covariant derivative acting on the
resonance field in the kinetic operator. However, as long as we are
not interested in EW radiative corrections, we may work with a simple
partial derivative and omit the gauge couplings to $W$, $Z$, and photon.
The Lagrangian for the isoscalar-scalar $\sigma$,
the isotensor-scalar $\phi$, the isoscalar-tensor $f$
and the isotensor-tensor $X$ are given by
\begin{align}
\LL_\sigma =&
\frac{1}{2} \partial_\mu \sigma \partial^\mu \sigma
-\frac{1}{2} m_\sigma^2 \sigma^2
+ \sigma J_\sigma \, \\
\LL_\phi =& \frac{1}{2}\sum_{i=s,v,t} \tr{\partial_\mu \Phi_i \partial^\mu \Phi_i
-m_\Phi^2 \Phi_i^2}
+\tr{\left(\Phi_t + \frac{1}{2}\Phi_v - \frac{2}{5}\Phi_s \right)J_\phi }\, ,\\
\LL^{}_{f} =
& \frac{1}{2} f_{f\mu\nu}\left (-\partial^2 - m_f^2 \right )f_f^{ \mu \nu}
+ \frac{1}{2} f^{ \mu}_{f\mu}\left (-\frac{1}{2} \left (-\partial^2
- m_f^2 \right ) \right )f^{\nu}_{f\nu} \notag \\
&+ \frac{1}{2} A_{f\mu} \left ( -\partial^2 - m_f^2 \right ) A_f^\mu
+\frac{1}{2} \sigma_f \left ( -\partial^2 - m_f^2 \right ) \sigma_f
\notag \\
& + \left ( f_{f\mu \nu} - \frac{1}{\sqrt{6}} g_{\mu \nu} \sigma +
\frac{1}{\sqrt{2}m_f}\left (\partial_\mu A_\nu + \partial_\nu A_\mu \right )
+ \frac{\sqrt{2}}{\sqrt{3}m_f^2} \partial_\mu \partial_\nu \sigma
\right )J_f^{\mu \nu} \, , \\
\LL^{}_{X} =
& \frac{1}{2}\sum_{i=s,v,t} \operatorname{tr} \Big[
X^{}_{i\mu\nu}\left (-\partial^2 - m_X^2 \right )X_i^{{} \mu \nu}
+ X^{{} \mu}_{i\mu}\left (-\frac{1}{2} \left (-\partial^2 - m_X^2
\right ) \right )X^{{} \nu}_{i\nu} \notag \\
& \phantom{ \frac{1}{2}\sum_{i=s,v,t} \operatorname{tr} \Big[ }
+ A_{i\mu} \left ( -\partial^2 - m_X^2 \right ) A_i^\mu
+ \sigma_i \left ( -\partial^2 - m_X^2 \right ) \sigma_i
\Big]
\notag \\
& + \operatorname{tr} \Bigg[
\left ( X^{}_{t\mu \nu} - \frac{g_{\mu \nu}}{\sqrt{6}} \sigma_t +
\frac{\partial_\mu A_{t\nu} + \partial_\nu A_{t\mu} }{\sqrt{2}m_X}
+ \frac{\sqrt{2}}{\sqrt{3}m_X^2} \partial_\mu
\partial_\nu \sigma_t
\right )J_X^{\mu \nu} \notag \\
&\phantom{+ \operatorname{tr} \Big[ }
+\frac{1}{2}{ \left ( X^{}_{v\mu \nu} - \frac{g_{\mu
\nu}}{\sqrt{6}} \sigma_v +
\frac{\partial_\mu A_{v\nu} + \partial_\nu A_{v\mu} }{\sqrt{2}m_X}
+ \frac{\sqrt{2}}{\sqrt{3}m_X^2} \partial_\mu
\partial_\nu \sigma_v
\right )J_X^{\mu \nu} } \notag \\
&\phantom{+ \operatorname{tr} \Big[ } -\frac{2}{5} { \left (
X^{}_{s\mu \nu} - \frac{g_{\mu \nu}}{\sqrt{6}} \sigma_s
\frac{\partial_\mu A_{s\nu} + \partial_\nu A_{s\mu} }{\sqrt{2}m_X}
+ \frac{\sqrt{2}}{\sqrt{3}m_X^2} \partial_\mu
\partial_\nu \sigma_s
\right )J_X^{\mu \nu} } \Bigg] \, ,
\end{align}
respectively, where the tensor resonances are formulated in the
Stückelberg formalism with associated fields $\sigma_f$, $A_f$ and $f_f$
denoting the scalar, vector and tensor degrees of freedom,
The corresponding Stückelberg fields for the isotensor-tensor
receive extra indices $\{s,v,t\}$
which represent the isoscalar, isovector and isotensor
fields of the $SU(2)_C$ multiplet, respectively.
The couplings to the Nambu-Goldstone boson current in each
case is given by
J_σ = F_σ
( _μ)^†^μ ,
J_ϕ = F_ϕ
( _μ)^†⊗^μ+1/8 ( _μ)^†^μ
)τ^aa ,
J^μν_f =
F_f (
( ^μ)^†^ν
- c_f/4 g^μν
( _ρ)^†^ρ
) ,
J^μν_X =
F_X [
1/2 (
( ^μ)^†⊗^ν+ ( ^ν)^†⊗^μ)
- c_X/4 g^μν
( _ρ)^†⊗^ρ
+1/8 (
( ^μ)^†^ν
- c_X/4 g^μν
( _ρ)^†^ρ
] τ^aa .
§ UNITARY AMPLITUDES FOR VBS AT THE LHC
§.§ Gaugeless limit
For a first estimate of the impact of generic resonances
to vector-boson scattering processes at the LHC, we
study the on-shell Nambu-Goldstone boson scattering amplitudes.
When treating vector-boson scattering as $2 \rightarrow 2$
process of massless scalars at high energies, it is convenient to
describe kinematic dependencies using Mandelstam variables $s,t,u$.
Using custodial symmetry and crossing symmetries, the different
$2\rightarrow 2$ Nambu-Goldstone boson scattering amplitudes are
determined by the master amplitudes $\mathcal{A} \left (w^+w^-
\rightarrow zz \right )$. In the gaugeless
limit, the amplitudes for the resonance multiplets $\sigma$, $\phi$,
$f$, and $X$ are calculated in the gaugeless limit via the Feynman
rules given in appendix <ref>.
§.§.§ Isoscalar-Scalar
\begin{align}
\mathcal{A}_\sigma \left (w^\pm w^\pm \rightarrow w^\pm w^\pm \right )&=
- \frac{1}{4} {F_\sigma}^2 \left(\frac{t^2}{t-m_\sigma^2}+
\frac{u^2}{u-m_\sigma^2} \right) \, ,\\
\left. \begin{aligned}
\mathcal{A}_\sigma& \left (w^\pm z \rightarrow w^\pm z \right )\\
\mathcal{A}_\sigma& \left (hw^\pm \rightarrow hw^\pm \right ) \\
\mathcal{A}_\sigma& \left (hz \rightarrow hz \right )\\
\end{aligned} \right \} &=
- \frac{1}{4} {F_\sigma}^2 \frac{t^2}{t-m_\sigma^2} \, ,\\
\mathcal{A}_\sigma \left (w^\pm w^\mp \rightarrow w^\pm w^\mp \right )&=
- \frac{1}{4} {F_\sigma}^2 \left (\frac{s^2}{s-m_\sigma^2}+ \frac{t^2}{t-m_\sigma^2} \right) \, ,\\
\left. \begin{aligned}
\mathcal{A}_\sigma& \left (w^\pm w^\mp \rightarrow zz \right )\\
\mathcal{A}_\sigma& \left (hh \rightarrow w^\pm w^\mp \right ) \\
\mathcal{A}_\sigma& \left (hh \rightarrow zz \right )\\
\end{aligned} \right \}&=
- \frac{1}{4} {F_\sigma}^2 \frac{s^2}{s-m_\sigma^2} \, ,\\
\left. \begin{aligned}
\mathcal{A}_\sigma& \left (zz \rightarrow zz \right )\\
\mathcal{A}_\sigma& \left (hh \rightarrow hh \right )
\end{aligned} \right \} &=
- \frac{1}{4} {F_\sigma}^2 \left( \frac{s^2}{s-m_\sigma^2}+ \frac{t^2}{t-m_\sigma^2} +\frac{u^2}{u-m_\sigma^2} \right) \, .
\end{align}
§.§.§ Isotensor-Scalar
\begin{align}
\mathcal{A}_\phi \left (w^\pm w^\pm \rightarrow w^\pm w^\pm \right )&=
-\frac{{F_\phi}^2}{8} \left(2 \frac{s^2}{s-m_\phi^2}
+ \frac{1}{2} \frac{u^2}{u-m_\phi^2}
+ \frac{1}{2} \frac{t^2}{t-m_\phi^2} \right)\, ,\\
\left. \begin{aligned}
\mathcal{A}_\phi& \left (w^\pm z \rightarrow w^\pm z \right )\\
\mathcal{A}_\phi& \left (hw^\pm \rightarrow hw^\pm \right ) \\
\mathcal{A}_\phi& \left (hz \rightarrow hz \right )\\
\end{aligned} \right \} &=
\frac{{F_\phi}^2}{8} \left( \frac{1}{2} \frac{t^2}{t-m_\phi^2}
- \frac{u^2}{u-m_\phi^2}
- \frac{s^2}{s-m_\phi^2} \right) \, ,\\
\mathcal{A}_\phi \left (w^\pm w^\mp \rightarrow w^\pm w^\mp \right )&=
-\frac{{F_\phi}^2}{8} \left( \frac{1}{2} \frac{s^2}{s-m_\phi^2}
+ 2 \frac{u^2}{u-m_\phi^2}
+ \frac{1}{2} \frac{t^2}{t-m_\phi^2} \right) \, ,\\
\left. \begin{aligned}
\mathcal{A}_\phi& \left (w^\pm w^\mp \rightarrow zz \right )\\
\mathcal{A}_\phi& \left (hh \rightarrow w^\pm w^\mp \right ) \\
\mathcal{A}_\phi& \left (hh \rightarrow zz \right )\\
\end{aligned} \right \}&=
\frac{{F_\phi}^2}{8} \left( \frac{1}{2} \frac{s^2}{s-m_\phi^2}
- \frac{u^2}{u-m_\phi^2}
- \frac{t^2}{t-m_\phi^2} \right)\, ,\\
\left. \begin{aligned}
\mathcal{A}_\phi& \left (zz \rightarrow zz \right )\\
\mathcal{A}_\phi& \left (hh \rightarrow hh \right )
\end{aligned} \right \} &=
-\frac{3{F_\phi}^2}{16} \left( \frac{s^2}{s-m_\phi^2}
+ \frac{u^2}{u-m_\phi^2}
+ \frac{t^2}{t-m_\phi^2} \right) \, .
\end{align}
§.§.§ Isoscalar-Tensor
\begin{align}
\mathcal{A}_f \left (w^\pm w^\pm \rightarrow w^\pm w^\pm \right )=&
- \frac{1}{24} {F_f}^2 \left(\frac{t^2}{t-m_f^2}P_2(t,s,u)+ \frac{u^2}{u-m_f^2}P_2(u,s,t) \right) \, ,\\
\left. \begin{aligned}
\mathcal{A}_f& \left (w^\pm z \rightarrow w^\pm z \right )\\
\mathcal{A}_f& \left (hw^\pm \rightarrow hw^\pm \right ) \\
\mathcal{A}_f& \left (hz \rightarrow hz \right )\\
\end{aligned} \right \} =&
- \frac{1}{24} {F_f}^2 \frac{t^2}{t-m_f^2}P_2 \left(t,s,u \right) \, ,\\
\mathcal{A}_f \left (w^\pm w^\mp \rightarrow w^\pm w^\mp \right )=&
- \frac{1}{24} {F_f}^2 \left (\frac{s^2}{s-m_f^2}P_2(s,t,u)+
\frac{t^2}{t-m_f}P_2(t,s,u) \right) \, ,\\
\left. \begin{aligned}
\mathcal{A}_f& \left (w^\pm w^\mp \rightarrow zz \right )\\
\mathcal{A}_f& \left (hh \rightarrow w^\pm w^\mp \right ) \\
\mathcal{A}_f& \left (hh \rightarrow zz \right )\\
\end{aligned} \right \}=&
- \frac{1}{24} {F_f}^2 \frac{s^2}{s-m_f^2}P_2 \left(s,t,u \right) \, ,\\
\left. \begin{aligned}
\mathcal{A}_f& \left (zz \rightarrow zz \right )\\
\mathcal{A}_f& \left (hh \rightarrow hh \right )
\end{aligned} \right \} =&
- \frac{1}{24} {F_f}^2 \Bigg(
\frac{s^2}{s-m_f^2}P_2(s,t,u)
&\phantom{- \frac{1}{24} {F_f}^2 \Bigg(} +\frac{u^2}{u-m_f^2}P_2(u,s,t) \Bigg) \, .
\end{align}
Here and in the following, $P_2(s,t,u) = [3(t^2 + u^2) - 2
s^2]/s^2$ is the second order Legendre polynomial in terms of the
Mandelstam variables.
§.§.§ Isotensor-Tensor
\begin{align}
\mathcal{A}_X \left (w^\pm w^\pm \rightarrow w^\pm w^\pm \right )=&
-\frac{{F_X}^2}{96} \Bigg(
\frac{4s^2}{s-m_X^2}P_2 \left(s,t,u \right)
+\frac{t^2}{t-m_X^2}P_2 \left(t,s,u \right) \notag\\
&\phantom{-\frac{{F_X}^2}{96} \Bigg(}
+\frac{u^2}{u-m_X^2}P_2 \left(u,s,t \right)
\Bigg)\, ,\\
\left. \begin{aligned}
\mathcal{A}_X& \left (w^\pm z \rightarrow w^\pm z \right )\\
\mathcal{A}_X& \left (hw^\pm \rightarrow hw^\pm \right ) \\
\mathcal{A}_X& \left (hz \rightarrow hz \right )\\
\end{aligned} \right \} =&
\frac{{F_X}^2}{96} \Bigg (
-\frac{2s^2}{s-m_X^2}P_2 \left(s,t,u \right)
+\frac{t^2}{t-m_X^2}P_2 \left(t,s,u \right)\notag\\
&\phantom{\frac{{F_X}^2}{96} \Bigg (}
-\frac{2u^2}{u-m_X^2}P_2 \left(u,s,t \right)
\Bigg)\, ,
\\
\mathcal{A}_X \left (w^\pm w^\mp \rightarrow w^\pm w^\mp \right )=&
-\frac{{F_X}^2}{96} \Bigg(
\frac{s^2}{s-m_X^2}P_2 \left(s,t,u \right)
+\frac{t^2}{t-m_X^2}P_2 \left(t,s,u \right) \notag\\
&\phantom{-\frac{{F_X}^2}{96} \Bigg(}
+\frac{4u^2}{u-m_X^2}P_2 \left(u,s,t \right)
\Bigg)\,
\\
\left. \begin{aligned}
\mathcal{A}_X& \left (w^\pm w^\mp \rightarrow zz \right )\\
\mathcal{A}_X& \left (hh \rightarrow w^\pm w^\mp \right ) \\
\mathcal{A}_X& \left (hh \rightarrow zz \right )\\
\end{aligned} \right \}=&
\frac{{F_X}^2}{96} \Bigg(
\frac{s^2}{s-m_X^2}P_2 \left(s,t,u \right)
-\frac{2t^2}{t-m_X^2}P_2 \left(t,s,u \right)\notag \\
&\phantom{\frac{{F_X}^2}{96} \Bigg(}
-\frac{2u^2}{u-m_X^2}P_2 \left(u,s,t \right)
\Bigg)\, ,\\
\left. \begin{aligned}
\mathcal{A}_X& \left (zz \rightarrow zz \right )\\
\mathcal{A}_X& \left (hh \rightarrow hh \right )
\end{aligned} \right \} =&
-\frac{1}{32} {F_X}^2 \Bigg(
\frac{s^2}{s-m_X^2}P_2 \left(s,t,u \right)
+\frac{t^2}{t-m_X^2}P_2 \left(t,s,u \right) \notag \\
&\phantom{-\frac{1}{32} {F_X}^2 \Bigg(} +
\frac{u^2}{u-m_X^2}P_2 \left(u,s,t \right)
\Bigg) \, .
\end{align}
§.§ Decomposition of eigenamplitudes
Since the leading-order amplitudes as listed above are unbounded both at
the pole and at high energy, we use the T-matrix
scheme <cit.> to restore unitarity. In order to
implement the scheme in <cit.>, we decompose the
amplitudes into isospin-spin eigenamplitudes (the $S$-wave, $P$-wave
and $D$-wave kinematic functions $\mathcal{S}_i$, $\mathcal{P}_i$ and
$\mathcal{D}_i$ can be found in appendix <ref>):
§.§.§ Isoscalar-Scalar
\begin{align}
\amp_{00}&=
\left (
-\frac{3}{4} \frac{s^2}{s-m_\sigma^2}
-\frac{1}{2} \mathcal{S}_0
\right)
\amp_{02}&=
-\frac{1}{2} F_\sigma^2 \mathcal{S}_2
\amp_{11}&=
-\frac{1}{2} F_\sigma^2 \mathcal{S}_1
\amp_{13}&=
-\frac{1}{2} F_\sigma^2 \mathcal{S}_3
\amp_{20}&=
-\frac{1}{2} F_\sigma^2 \mathcal{S}_0
\amp_{22}&=
-\frac{1}{2} F_\sigma^2 \mathcal{S}_2
\end{align}
§.§.§ Isotensor-Scalar
\begin{align}
\amp_{00}&=
\left (
-\frac{1}{16} \frac{s^2}{s-m_\phi^2}
-\frac{7}{8} \mathcal{S}_0
\right)
\amp_{02}&=
-\frac{7}{8} F_\phi^2 \mathcal{S}_2
\amp_{11}&=
\frac{3}{8} F_\phi^2 \mathcal{S}_1
\amp_{13}&=
\frac{3}{8} F_\phi^2 \mathcal{S}_3
\amp_{20}&=
\left (
-\frac{1}{4} \frac{s^2}{s-m_\phi^2}
-\frac{1}{8} \mathcal{S}_0
\right)
\amp_{22}&=
-\frac{1}{8} F_\phi^2 \mathcal{S}_2
\end{align}
§.§.§ Isoscalar-Tensor
\begin{align}
\amp_{00}&=
\amp_{02}&=
-\frac{1}{40}F_f^2 \frac{s^2}{s-m_f^2}
\left (1+ 6 \frac{s}{m_f^2}+ 6 \frac{s^2}{m_f^4} \right )
\mathcal{S}_2
\amp_{11}&=
-\frac{1}{12}F_f^2 \mathcal{D}_1
\amp_{13}&=
\left (1+ 6 \frac{s}{m_f^2}+ 6 \frac{s^2}{m_f^4} \right )
\mathcal{S}_3
\amp_{20}&=
-\frac{1}{12}F_f^2 \mathcal{D}_0
\amp_{22}&=
\left (1+ 6 \frac{s}{m_f^2}+ 6 \frac{s^2}{m_f^4} \right )
\mathcal{S}_2
\end{align}
§.§.§ Isotensor-Tensor
\begin{align}
\amp_{00}&=
-\frac{7}{48} F_X^2 \mathcal{D}_0
\amp_{02}&=
-\frac{1}{480}F_X^2 \frac{s^2}{s-m_X^2}
-\frac{7}{48} F_X^2\left(1 + 6 \frac{s}{m_X^2}+ 6 \frac{s^2}{m_X^4}
\right) \mathcal{S}_2
\amp_{11}&=
\frac{1}{16} F_X^2 \mathcal{D}_1
\amp_{13}&=
\frac{1}{16} F_X^2 \left(1 + 6 \frac{s}{m_X^2}+ 6 \frac{s^2}{m_X^4}
\right) \mathcal{S}_3
\amp_{20}&=
-\frac{1}{48} F_X^2 \mathcal{D}_0
\amp_{22}&=
-\frac{1}{120}F_X^2 \frac{s^2}{s-m_X^2}
-\frac{1}{48} F_X^2\left(1 + 6 \frac{s}{m_X^2}+ 6 \frac{s^2}{m_X^4}
\right) \mathcal{S}_2 \, .
\end{align}
§.§ Width
As argued below in section <ref>, for the numerical
off-shell calculation of scattering processes we will need approximate
values for the resonance decay widths. If suffices to compute those
in the gaugeless limit. Contributions
proportional to the masses of the vector bosons and the Higgs boson are
assumed to be small at high resonance masses and are therefore neglected.
\begin{align}
\Gamma_\sigma
&= \frac{m_\sigma^3}{32\pi}F_\sigma^2 \, ,
\\
\Gamma_\phi
&= \frac{m_\phi^3}{128\pi}F_\phi^2 \, ,
\\
\Gamma_f
&= \frac{m_f^3}{960\pi}F_f^2 \, ,
\\
\Gamma_X
&= \frac{m_X^3}{3840\pi}F_X^2 \, .
\end{align}
§.§ Matching to the low-energy EFT
For later convenience, we compute the coefficients of the effective
dimension-eight operators $\LL_{S,0}$ and $\LL_{S,1}$ <cit.>,
_S,0 =
( _μ)^†_ν
( ^μ)^†^ν,
_S,1 =
( _μ)^†^μ
( _ν)^†^ν.
which result
from integrating out the resonances $\sigma,\phi,f,X$, one at a time.
F_S,1 = F_σ^2/2 m_σ^2 ,
F_S,0 = F_ϕ^2/2m_ϕ^2 , F_S,1 =-
F_ϕ^2/8 m_ϕ^2 ,
F_S,0 = F_f^2/2 m_f^2 , F_S,1 = - F_f^2/6 m_f^2 ,
F_S,0 = F_X^2/24 m_X^2 , F_S,1 = - 7 F_X^2/24 m_X^2 .
§.§ Tensor exchange in unitary gauge
Beyond the resonance, the Nambu-Goldstone bosons scattering amplitudes
rise proportional to powers of the
invariant mass of the scattering system. They eventually violate
unitarity at a certain energy, depending on the resonance coupling.
Computing the $w^+w^-\to zz$ amplitude in the presence of an
isoscalar tensor resonance, for instance,
\begin{align}
\amp_f \left (w^+w^- \rightarrow zz \right ) =&
-\frac{F_f^2}{96}\left (c_f-2 \right )^2 \frac{s^3}{ m_f^4}
-\frac{F_f^2}{48} \left (c_f-2 \right )c_f \frac{s^2}{ m_f^2} \notag\\
& -\frac{F_f^2}{24} \left (3 \left (t^2+u^2 \right )-2s^2 \right )
\frac{1}{s-m_f^2} \, ,
\end{align}
we observe that choosing
$c_f\neq 2$ results in a high degree of divergence.
This is due to contributions of the vector and scalar degree of freedoms
in the Stückelberg parameterization for the tensor
coupled to the derivatives of the current (<ref>)
and (<ref>). As discussed above, such
terms can be written in a non-resonant form and should be interpreted as
coefficients of undetermined higher-dimensional local operators. Setting
thus $c_f =2$, we
obtain an amplitude $\amp_f(s)$ which rises proportional
to $s$ beyond the resonance.
However, the scalar and
vector degree of freedoms provide additional contributions which
are not manifest in the gaugeless limit. A calculation
of the tensor scattering amplitude in the unitary
gauge is necessary. The longitudinal on-shell
$WW \rightarrow ZZ$ amplitude for $c_f=2$ is given by
\begin{align}
\label{eq:amp_wwzz_tensor-scalar}
\amp_f \left(W_LW_L \rightarrow Z_LZ_L \right)
=&-\frac{1}{24}\frac{F_f^2}{s-m_f^2} \Bigg[
\left(P_2\left[\cos (\theta) \right] -1 \right)s^2
+12 m_W^2 m_Z^2 \notag \\
\quad -12\frac{ m_W^2 m_Z^2}{m_f^2} \notag\\
&\phantom{-\frac{1}{24}\frac{F_f^2}{s-m_f^2}} \notag
\quad +\left(s-2 m_W^2\right) \left(s-2 m_Z^2\right) \\
&\phantom{-\frac{1}{24}\frac{F_f^2}{s-m_f^2}} \quad \notag
+4\frac{ m_W^2 m_Z^2}{m_f^4} s^2
+2\frac{\left( m_W^2+ m_Z^2 \right)s^2 -4 m_Z^2m_W^2 s}{m_f^2}
\Bigg] \, .
\end{align}
The first line represents the tensor contribution in the Stückelberg
parameterization. Due to its suppression by a power of $s$, the vector
part in the second line can be neglected
for the longitudinal scattering amplitude. Besides the scalar contribution
originating from the trace of the current, additional contributions
related to the double derivative of the current and its mixing
with the trace part written in the fourth line will rise with energy.
However, they are suppressed by ${m_W^2}/{m_f^2}$ or ${m_W^4}/{m_f^4}$
and can be neglected if the mass of the tensor resonance is large
in comparison to the vector boson masses. In this case,
the longitudinal amplitude of the vector bosons calculated in the
unitary gauge coincides with the amplitude in gaugeless limit.
Furthermore, due to the coupling to the derivatives of the
scalar and vector degrees of freedom, also amplitudes in channels with
transverse polarization
rise with the energy of the vector-boson scattering system.
A full list of these channels in the high-energy limit is displayed in
Table <ref>. We observe that all channels
which include at least one transversally polarized vector boson
are suppressed by ${m_W^2}/{m_f^2}$. Therefore, a calculation
within the gaugeless limit is sufficient to estimate the
high-energy behavior for high masses of the tensor resonance.
(+, +, +,+ )
(+, +, -,- )
(-, -, +,+ )
(-, -, -,- )
- c_f^2m_W^2 m_Z^2/24m_f^2 F_f^2 s
(+, 0, 0,+ )
(0, +, +,0 )
(0, -, -,0 )
(-, 0, 0,- )
m_W m_Z/8m_f^2 F_f^2 t
(+, 0, +,0 )
(0, +, 0,+ )
(0, -, 0,- )
(-, 0, -,0 )
m_W m_Z/8m_f^2 F_f^2 u
(+, 0, -,0 )
(0, +, 0,- )
(0, -, 0,+ )
(-, 0, +,0 )
-m_W m_Z/8m_f^2 F_f^2 t
(+, 0, 0,- )
(0, +, -,0 )
(0, -, +,0 )
(-, 0, 0,+ )
-m_W m_Z/8m_f^2 F_f^2 u
(+, +, 0,0 )
(-, -, 0,0 )
m_f^2 + 2 m_Z^2 /12 m_f^4 m_W^2 F_f^2 s
(0, 0, +,+ )
(0, 0, -,- )
m_f^2 +2 m_W^2 /12 m_f^4 m_Z^2 F_f^2 s
(0, 0, 0,0 )
F_f^2/24 2s^2- 3t^2 - 3u^2 /s
+ m_f^2 (m_W^2+m_Z^2 )
+2m_W^2m_Z^2 /12 m_f^4 F_f^2 s
High energy limit
of the $W^+W^-~\rightarrow~ZZ$ amplitude
for each polarization channel that
rises with energy due to a
isoscalar-tensor resonance ($c_f=2$).
For the tensor-isotensor amplitude, the analogous result with $c_X=2$ is
\begin{align}
\amp_X \left (W^\pm_L W^\mp_L \rightarrow Z_LZ_L \right )=&
\frac{{F_X}^2}{96} \Bigg(
\frac{s^2}{s-m_X^2}P_2 \left(s,t,u \right)
-\frac{2t^2}{t-m_X^2}P_2 \left(t,s,u \right) \notag
-\frac{2u^2}{u-m_X^2}P_2 \left(u,s,t \right)
\Big) \notag \\
\left(\frac{s^2}{s-m_X^2}-\frac{2t^2}{t-m_X^2}-\frac{2u^2}{u-m_X^2}
\right)\notag\\
\left(\frac{t^2}{t-m_X^2}+\frac{u^2}{u-m_X^2} \right)
+ \mathcal{O}\left(s^0 \right) \, ,
\end{align}
containing t-channel and u-channel contributions, as expected.
§.§ Unitarized amplitudes
The tree-level exchange amplitudes that directly result from
evaluating Feynman rules, exhibit two distinct sources of unitarity violation.
Firstly, the amplitude develops a pole at the resonance mass, on the
real axis. Secondly, terms that rise with energy asymptotically
violate unitarity bounds.
In principle, the T-matrix unitarization scheme would be sufficient to
regulate both issues simultaneously. At the pole, this boils down to
standard Dyson resummation, introducing the particle width as an
imaginary part in the denominator. It can easily be verified that
this actually happens for the on-shell scattering amplitudes of
external Nambu-Goldstone bosons. We obtain the correct value for the
resonance width in the gaugeless limit.
However, we want to evaluate the amplitudes off-shell for physical $W$
and $Z$ bosons. The simplified unitarization scheme that we describe
above is not exactly accurate as soon as we include finite corrections
due to transversal gauge bosons and finite $W/Z$ mass. As a result,
there are contributions which are not cancelled on the resonance pole,
and a narrow but unbounded peak remains.
To avoid this problem, we simply insert an a priori width in the
resonant propagator. We thus start from a complex model
amplitude. Therefore, we take the T-matrix scheme of <cit.>
at face value, and drop the reference to the usual K-matrix scheme
which implies an intermediate projection onto the real axis. By
construction, in the gaugeless limit, the correct result is invariant
with respect to the introduction of this width, if it has the correct
on-shell value. For finite gauge couplings and masses, the result
acquires a subleading dependence on this initial value since the model
amplitude is neither on the real axis nor exactly on the Argand
circle. However, the amplitude after unitarization is now bounded
near the resonance pole, as required.
In the asymptotic regime, the simplified T-matrix scheme renders the
amplitude unitary at all energies, if the exchanged resonance is
scalar. This enables us to compute cross sections and generate event
samples in this model for complete processes at the LHC
(cf. section <ref>).
For a tensor resonance, in the Stückelberg approach, the genuine
tensor exchange terms are also regulated completely by this (simplified)
scheme. The extra Stückelberg vector and scalar terms, however,
generate higher powers of $s$
which enter when trading Nambu-Goldstone bosons for physical vector
bosons in unitary gauge, suppressed by powers of $m_h,m_W,m_Z$.
Applying the unitarization framework for those extra terms would require a
complete diagonalization of all vector-boson helicity amplitudes in
unitary gauge. In
any case, parameter ranges where these terms play a role correspond to
a regime where all degrees of freedom of the SM interact strongly via
these couplings. We therefore stay away from this range and choose
parameters where those terms are subleading within the accessible
energy range.
Computing the scale where the Stückelberg vector-scalar terms violate the
relevant unitarity bounds, we obtain the energy limit
\begin{align}
\label{eq:completebound_tensor-scalar_width}
\sqrt{s} &\lesssim
\sqrt{\frac{1}{5} \frac{ m_f}{\Gamma_f}}
\frac{m_f^2}{ m_{whz} }\, ,
\end{align}
for the model which contains an isoscalar tensor, and
\begin{align}
\label{eq:completebound_isotensor-tensor-scalar_width}
\sqrt{s} \lesssim \sqrt{\frac{1}{30}\frac{m_x}{\Gamma_x}}
\frac{m_X^2}{m_{whz}} \,.
\end{align}
for the isotensor tensor multiplet. Here, $m_{whz}$ indicates the
common mass scale of electroweak bosons $W,H,Z$. Inserting the
accessible energy
for the LHC collider, we can invert those relations to extract
parameter regions where the simplified models with a tensor resonance
are valid. The numerical results in the following sections have been
obtained for parameter values that satisfy the bounds.
§ SCENARIOS FOR VBS AT THE LHC
§.§ Implementation
In the previous section, we have derived the analytic expressions that
determine the on-shell VBS amplitudes in the presence of a resonance.
The amplitudes include correction terms that enforce
quantum-mechanical unitarity without altering the physical content of
the model.
Ultimately, we are interested in measurable effects in LHC data. For
a complete calculation, the unitarized amplitudes that are originally
defined for on-shell VBS processes, have to be extrapolated off-shell
in a practically meaningful way. As long as the kinematical
conditions are approximately met, we can evaluate the interactions in
unitary gauge, eliminating all explicit references to Nambu-Goldstone
bosons in favor of physical vector fields, and derive the Feynman
rules in that gauge. The effective Feynman rules for the unitarity
corrections become momentum dependent and
involve theta functions that restrict the insertions to the $s$-channel
of VBS where partial-wave projection and unitarization is defined.
In the physical processes at the LHC,
\begin{equation}
pp \to q q \to q q V V
\end{equation}
where $q$ generically denotes a quark and $V$ is either $W$ or $Z$,
the final-state quarks are detected as jets in the forward direction.
With suitable cuts, we can arrange that there is significant
contribution from the subprocess $VV\to VV$ where the initial-state
vector bosons are spacelike but approximately on-shell, in the limit
of high invariant $VV$ mass. This subprocess, i.e., the associated
off-shell amplitude, obtains contributions from resonance exchange and
is affected by unitarization.
We have implemented this prescription as a model in the Monte-Carlo
integration and event generation package
WHIZARD <cit.>. This
is a universal event generator for simulations at hadron and lepton
colliders at leading order and next-to-leading (QCD) <cit.>
order. Though interfaces to automated tools for beyond the SM models
exist <cit.>, they cannot be used for the
implementation of unitarization projections for operators and
resonances. The reason is the global structure of the unitarization
projection. Therefore the models described in the current paper have
been manually added to the framework.
For each resonance type ($\sigma,\phi,f, X$), we can compute the relation of the
resonance width (section <ref>) to the operator coefficients in the
low-energy EFT (section <ref>) which result when the
resonance is integrated out. These relations are listed in
Table <ref>.
$\sigma$ $\phi$ $f$ $X$
$F_{S,0}$ $\frac{1}{2}$ $2$ $15$ 5
$F_{S,1}$ – -$\frac{1}{2}$ -$5$ -35
Relation of resonance width $\Gamma$ and mass $M$ to the
corresponding $D=8$ operator coefficients in the low-energy EFT, for all
resonance types considered in this paper. The
factors listed in the table have to be multiplied by $32 \pi \Gamma/M^5$.
The analysis of LHC run-I data by the ATLAS
experiment <cit.> has been cast into
bounds on the EFT parameters $F_{S,0}$ and $F_{S,1}$, namely
\begin{equation}
|F_{S,0}| < 480\;\TeV^{-4}
\qquad
|F_{S,1}| < 480\;\TeV^{-4} \qquad ,
\end{equation}
where only one parameter was varied at a time. This analysis covered the
same-sign leptonic decay channel of $W^+W^+$ and $W^-W^-$. It was based on
the T-matrix unitarized version of the extrapolated EFT as its reference
model, with the pure SM as the limit for vanishing parameters. A CMS
analysis can be found in <cit.>
§.§ On-shell Invariant Mass Distributions
In the following, we will present results both for on-shell $W/Z$
final states and for complete partonic final states. On-shell vector
bosons cannot be detected directly but their distributions directly
reflect the actual features of the physical model. Observable
distributions of fermions in the final state, which may be quarks
(jets), charged leptons, or neutrinos, are less directly linked to the
physical process and require detailed analysis along the lines
of <cit.>. This concerns, in particular, the separation of
signal and background based on detector data, which is beyond the
scope of the present paper.
We show results for particular parameter sets where we add one resonance at a
time on top of the SM, namely a scalar-isoscalar, tensor-isoscalar, or
scalar-isotensor resonance, respectively. All extra higher-dimensional
operator coefficients are set to zero. By varying the resonance parameters
within reasonable limits, this gives an overview of the expected
For definiteness, we choose to plot the invariant mass of the vector-boson
pair system in the final state, which is the energy scale of the actual VBS
process. The initial state is convoluted with the parton structure functions,
so the results hold for the LHC ($\sqrt{s}=14\;\TeV$), and
we apply standard VBS cuts to enhance the signal. The final-state vector
bosons are taken on-shell. We show the distribution for the $W^+W^+$ and $ZZ$
final states, where the latter case as the golden channel of VBS is
distinguished by the fact that the $ZZ$ invariant mass can be reconstructed
from the leptonic $Z$ decays. This is not possible for $W^+W^+$, but the
corresponding same-sign lepton channel is distinguished by a favorable
signal-to-background ratio. Note that in the on-shell plots, the vector-boson
decay branching ratios have not been included.
In all invariant-mass plots, we display the distribution for the unitarized
resonance model (blue curves) together with the pure SM prediction (black).
We also plot the unitarity bound for the appropriate partial wave,
extrapolated off-shell by the same algorithm, as a dashed curve (black). For
illustrative purposes, we also display, in each case, the unitarized
extrapolation of the low-energy EFT (red, solid), where we choose the operator
coefficients equal to the formal result of integrating out the resonance.
Finally, we also display numerical results for the EFT without unitarization
(red, dashed) and the resonance with correct width but no further
unitarization (blue, dashed).
§.§.§ Isoscalar-Scalar
The simplest case is a scalar-isoscalar resonance. This is a single isolated
resonance, as it could arise, e.g., as the extra scalar particle in a
singlet-doublet Higgs model or as a low-energy signal of a strongly
interacting Higgs sector that is neutral under the SM gauge group.
Differential cross sections for isoscalar scalar
resonances. Upper plots show a weakly coupled isoscalar
scalar with $m_\sigma=800 \, \mathrm{GeV}$ and $\Gamma_\sigma=80\,
\mathrm{GeV}$, for the processes $pp\rightarrow W^+W^+jj$ (left)
and $pp\rightarrow ZZjj$ (right), respectively. In the lower plot,
there is a low lying isoscalar scalar with $m_\sigma=650 \,
\mathrm{GeV}$ and $\Gamma_\sigma=260\, \mathrm{GeV}$ for the
process $pp\rightarrow ZZjj$. Solid line: unitarized results,
dashed lines: naive result, black dashed line: Limit of saturation
of $\mathcal{A}_{20}$ $(W^+W^+)$ or $\mathcal{A}_{00}$ $(ZZ)$,
respectiveluy. Cuts: $M_{jj} > 500$ GeV;
$\Delta\eta_{jj} > 2.4$; $p^j_T > 20$ GeV;
$|\eta_j| > 4.5$.
In Fig. <ref>, upper row, we have selected a
moderate mass of $800\;\GeV$ and a rather narrow width of $80\;\GeV$,
which corresponds to a weak coupling. The isolated resonance is
clearly visible in the $ZZ$ channel, while the $W^+W^+$ channel is
barely affected. For such weak coupling, the operator coefficient in
the EFT is small and more than one order of magnitude below the
current LHC run-I limit. We can draw the conclusion that in this case
the resonance should be detectable for sufficient luminosity, but the
EFT approximation is not useful.
Turning to a stronger coupling, we show the corresponding distribution in the
$ZZ$ channel for $m_\sigma=650\;\GeV$ and $\Gamma_\sigma =260\;\GeV$ in
Fig. <ref>, lower row.
Here, the EFT parameters are within the range that should become accessible at
LHC run II and beyond. The EFT curve (red, solid) appears correctly
as the Taylor expansion of the resonance curve (blue) for low energy.
However, the energy region where the deviation from the SM becomes
sizable, already coincides with the resonance peak region, so the EFT
considerably underestimates the event yield. Beyond the resonance,
the EFT misses the fact that the distribution falls down again,
approaching the SM prediction (black) from above.
The result also demonstrates that the additional unitarization of the scalar
resonance beyond the Breit-Wigner approximation with constant width is
essential, as is seen by comparing the blue and blue-dashed curves. The naive
EFT result without unitarization (red, dashed) grossly overshoots all
conceivable models, which should not cross the unitarity limit (black-dashed).
§.§.§ Isoscalar-Tensor
As can be observed from Table <ref>, a tensor resonance has a
stronger impact on the low-energy EFT than a scalar resonance of equal width.
In Fig. <ref>, upper row, we display the
distributions for a tensor isoscalar resonance with mass
$m_f=1000\;\GeV$ and width $\Gamma_f =100\;\GeV$.
Differential cross sections of an isoscalar tensor
resonance. Upper plots show a resonance with $m_f= 1000 \,
\mathrm{GeV}$ and $\Gamma_f= 100 \, \mathrm{GeV}$ for the
processes $pp\rightarrow W^+W^+jj$ (left), and $pp\rightarrow
ZZjj$ (right), respectively. The lower plot is for a strongly
interacting isoscalar tensor with $m_f= 1200 \, \mathrm{GeV}$ and
$\Gamma_f= 480 \, \mathrm{GeV}$.
Solid line: unitarized results, dashed lines: naive result,
black dashed line: Limit of saturation of $\mathcal{A}_{22}$ $(W^+W^+)$ or
$\mathcal{A}_{02}$ $(ZZ)$, respectively.
Cuts are the same as in Fig. <ref>.
The resonance visibly modifies the distribution already at low energy, such
that the EFT analysis, given sufficient sensitivity, should catch the
deviation from the SM. Nevertheless, the excess at the peak in the $ZZ$
channel is sizable. Beyond the resonance, unitarization is essential in the
tensor case. In the $W^+W^+$ final state the tensor enters only as t-channel
exchange , so there is no resonance but a broad enhancement. This enhancement
is rather well described by the corresponding unitarized
EFT [Tensor resonances resulting in peaks in diboson spectra
to explain a recent excess in ATLAS data around 2 TeV can be found
e.g. in <cit.>.].
As in the scalar case, the curves without unitarization do not provide a
useful phenomenological description.
In Fig. <ref>, lower row, we consider a heavy
tensor-isoscalar with strong coupling, $m_\phi=1200\;\GeV$ and
$\Gamma_\phi=480\;\GeV$. The resonance peak appears as a broad enhancement, which
extends to both low and high energies. The EFT approximation, with sizable
coefficients, is rather accurate in this case. The actual resonance curve
shows a nontrivial threshold structure which corresponds to the interplay of
all partial waves which are excited by s-channel and t-channel exchange
contributions. However, we should keep in mind that the prediction for such a
strong coupling is uncertain in any case and should not be taken too
§.§.§ Isotensor-Scalar
Turning to the isotensor case, we now get a resonance in all final states
including $W^+W^+$. This is illustrated by the plots in
Fig. <ref> for $m_\phi=800\;\GeV$ and
Differential cross sections of an isotensor scalar
resonance. Upper plots show a resonance with $m_\phi=800 \,
\mathrm{GeV}$ and $\Gamma_\phi=80\, \mathrm{GeV}$ for the
processes $pp\rightarrow W^+W^+jj$ (left), and $pp\rightarrow
ZZjj$ (right), respectively. The lower plot shows a low-lying
isotensor scalar with $m_\phi=650 \, \mathrm{GeV}$ and
$\Gamma_\phi=260\, \mathrm{GeV}$ for the process $pp\rightarrow
W^+W^+jj$. Solid line: unitarized results, dashed lines: naive
result, black dashed line: Limit of saturation of
$\mathcal{A}_{20}$ $(W^+W^+)$ or $\mathcal{A}_{00}$ $(ZZ)$,
respectively. Cuts are the same as in
Fig. <ref>.
Due to the large number of degrees of freedom (nine states which are
degenerate in mass), the peak is rather prominent while the low-energy
EFT parameters are again small. We observe that the peak value is slightly
below ($W^+W^+$) and above ($ZZ$) the appropriate unitarity limit,
respectively. This is the effect of $t$-channel exchange which also contributes
and can have either sign.
Contrary to the weakly interacting scenario,
a non-unitarized low-lying and strongly interacting isotensor-scalar with
mass of $m_\phi= 650 \, \mathrm{GeV}$ and width
$\Gamma_\phi = 260 \, \mathrm{GeV}$
violates the $\amp_{20}$ slightly above the resonance
as illustrated in Fig. <ref>.
Therefore, a unitarization is needed
for this strongly interacting resonance.
The low-energy effective field theory approach
does only coincide in the unitarized case
at high energies, because
the eigenamplitudes of the isotensor-scalar
as well as the dimension-eight operators
are already saturated through the T-matrix formalism.
§.§.§ Isotensor-Tensor
Similarly to the isotensor-scalar, every vector-boson scattering channel
receives a resonant contribution from the isotensor-tensor
multiplet. The $W^+W^+$ and $ZZ$ channel distributions of the
isotensor-tensor resonance with mass $m_X=1400 \, \mathrm{GeV}$ width
$\Gamma_X = 140 \, \mathrm{GeV}$ are plotted
in Fig. <ref>, upper row. Due to the bound of
equation (<ref>), the
mass of the isotensor-tensor has to be chosen slightly higher than
the mass of the isoscalar-tensor in Fig. <ref>
when leaving the ratio of width and mass invariant.
Differential cross sections of an isotensor tensor
resonance. Upper plots show a resonance with $m_X= 1400 \,
\mathrm{GeV}$ and $\Gamma_X= 140 \, \mathrm{GeV}$ for the
processes $pp\rightarrow W^+W^+jj$ (left), and $pp\rightarrow
ZZjj$ (right), respectively. The lower plot shows a strongly
interacting isotensor tensor with $m_X= 1800 \, \mathrm{GeV}$ and
$\Gamma_X= 720 \, \mathrm{GeV}$ for the process $pp\rightarrow
W^+W^+jj$. Solid line: unitarized results,
dashed lines: naive result, black dashed line: Limit of saturation of
$\mathcal{A}_{22}$ $(W^+W^+)$ or $\mathcal{A}_{02}$ $(ZZ)$,
respectively. Cuts are the same as in
Fig. <ref>.
The effective field theory with the dimension-eight operators
coincides with the onset of the isotensor-tensor peak.
Starting slightly below the resonance, the
resonant cross section deviates from
the effective field theory description. Analogously to the
isotensor-scalar, the very distinctive peak of the isotensor-tensor
is not captured by the dimension-eight operators.
In the $W^+W^+$- channel, even the non-unitarized
resonance contribution stays within the unitarity bound of
$\amp_{22}$. Contrary to the isotensor-scalar, the isotensor-tensor
needs unitarization
for the $ZZ$ final state due to the large tensor contributions
in the $t$- and $u-$channel. The
non-unitarized amplitudes violate the $\amp_{02}$ unitarity
already below the mass of the resonance.
Even the resonance peak is hardly visible.
The unitarized
resonance curve shows a peak, although it is slightly above the
unitarity bound.
In a strongly interacting scenario ($\Gamma_X = 720 \, \mathrm{GeV}$ ),
the unitarized isotensor-tensor resonance peaks
below its actual mass at $m_X = 1800 \, \mathrm{GeV}$.
This peak originates from the already saturated eigenamplitudes,
which then fall due to the parton distribution functions at high
energies. Besides the resonance peak, the low-energy effective field
theory coincides with the isotensor-tensor for both unitarized and
non-unitarized results. This is shown in the lower plot of
Fig. <ref>.
§.§ Results for Complete Processes
The actual analysis of LHC data will have to exploit cross sections and
distributions for the complete final state which consists of the two tagging
jets and the decay products of the vector bosons. In this paper, we only
investigate the $ZZ$ channel with its decay into four leptons, selecting the
$e^+e^-\mu^+\mu^-$ final state. This process is straightforward to analyze,
but suffers from the low leptonic branching ratio, so for our simulation we
assume the high-luminosity mode of the LHC with integrated luminosity of
$3\;\ab^{-1}$. We anticipate that by including also the leptonic $WW$ final
state and hadronic final states, the results can be considerably improved.
The simulation generates event samples for the complete process with all
Feynman graphs, so there is no restriction on resonant vector bosons
as the origin of the final-state leptons. We apply standard VBS cuts and
compare, in Fig. <ref>, various distributions for the SM (blue),
resonance model with a single isoscalar-scalar (red), and the unitarized
low-energy EFT (purple).
$m \left (Z,Z \right)$
$\Delta\phi$ $\Theta^*$
$pp\rightarrow e^+ e^- \mu^+ \mu^- jj$ at $\sqrt{s} = 14
\, \mathrm{TeV}$ with luminosity of $3000 \, \mathrm{fb}^{-1}$
with isoscalar tensor at $m_f=1000$ GeV and $\Gamma_f $=100 GeV.
Cuts: $M_{jj} > 500$ GeV;
$\Delta\eta_{jj} > 2.4$;
$p^j_T > 20$ GeV;
$|\eta_j| > 4.5$;
$100\; \mathrm{GeV} > M_{e^+e^-} > 80\; \mathrm{GeV}$;
$100\; \mathrm{GeV} > M_{\mu^+\mu^-} > 80\; \mathrm{GeV}$.
The resonance with mass $m=1000\;\GeV$ and width $\Gamma=100\;\GeV$
appears, as expected, in the invariant mass distribution and, more
indirectly, in other plots. Clearly, this parameter set is at the
margin of observability in this single channel. The situation
obviously improves if we consider resonances with lower mass, larger
coupling, in higher representions, and add other analysis channels.
§ CONCLUSIONS
The Higgs sector of the SM, after the discovery of a light Higgs, is a
new field of study for the experiments at the LHC, and beyond. While
the SM yields precise predictions in accordance with the notion of a
weakly coupled theory, a thorough analysis of electroweak data
should be guided by reference simplified models which differ from the SM.
Extending the EFT by higher-dimensional operators is useful for
analyzing observables with bounded energy, but open scattering data
require enforcing unitarity and extrapolating into a region where
perturbation theory in the EFT is insufficient.
Without reference to any particular high-energy model, we have
augmented the EFT by resonances with even spin, namely scalar or
tensor. Assuming exact $SU(2)_L\times U(1)_Y$ gauge invariance and, for
simplicity, approximate custodial symmetry both in the EFT and beyond,
we can distinguish four distinct resonance multiplets with a single
free mass and coupling parameter each. This class of models includes
the decoupling limit of multi-Higgs models and certain aspects of
massive-graviton models.
The models are set up such that we need only take the interaction with
the Higgs sector into account, while couplings to the gauge and
fermion sectors occur only via mixing. This is consistent with the
symmetry assumptions and with our knowledge about electroweak
precision data, although it is clearly not guaranteed. The models
allow for arbitrary higher-dimensional operators in the EFT, unrelated
to resonance exchange, so we do not lose generality.
All amplitude calculations are meaningless unless we enforce
quantum-mechanical unitarity, since naive extrapolations yield event
rates in the high-energy region that can exceed the unitarity bounds
by orders of magnitude. We have consistently implemented the T-matrix
unitarization scheme which works on the complex scattering matrix of
the model directly, simplified for the asymptotic range where
longitudinal and transveral degrees of freedom decouple.
We have studied the case of a tensor resonance in detail. Since we do
not necessarily restrict ourselves to states that are related to
gravity, the model differs from
the various massive-graviton models and studies that can be found in
the literature. To our knowledge, the coupling of a generic tensor
resonance to the Higgs sector and the resulting predictions for the
LHC have not been considered in detail before. We find that by
employing a Stückelberg procedure for the implementation in the
Lagrangian, instead of the classic Fierz-Pauli approach, we are able
to set up the extended EFT for an isolated tensor resonance manifestly
separated from non-resonant effects. Scalar and tensor
resonances can be handled in close analogy. It turns out that it
is possible to extend an effective theory with an isolated tensor
resonance up to a cutoff of order $\Lambda\lesssim M^2/m_H$, where $M$
is the resonance mass, and $m_H$ is the physical Higgs mass.
We have implemented the models in the Monte-Carlo package WHIZARD and
computed exemplary distributions and simulated event samples for the
LHC. The numerical results illustrate that resonances in VBS may be
detected at the LHC within a certain range of mass and coupling
values. For a final verdict, it will be necessary to perform a
complete experimental study and analysis, based on exclusive event
samples in combination with background and detector description. We
also find that the comparison with pure-EFT results can be misleading
if resonance and background cannot be clearly separated, as it is
typical for the situation at the LHC. We conclude that data should be
analyzed on base of resonance models as well as pure-EFT simulations.
This holds, in particular, if limits or values are to be
combined between distinct final states or with data obtained at a
future lepton collider like the
ILC <cit.>. There has
been a first study similar to the one presented here, investigating
resonances of spins and isospins zero, one and two in 1 TeV lepton
collisions <cit.>, where issues of unitarization did not
play a role.
§.§ Acknowledgments
WK and JRR want to thank for the hospitality at the Institute for High Energy
Physics (IHEP) and the Center for Future High Energy Physics (CFHEP)
of the Chinese Academy of Sciences at Beijing, China, where parts
of this work have been completed. MS acknowledges support
by JSPS and DAAD and thanks for the
hospitality of the KEK theory group during his stay over summer 2015.
§ NOTATION AND CONVENTIONS
§.§ Fields
\begin{align}
&&\vH &=
\frac 1 2
\begin{pmatrix}
v+ h -\ii w^3 & -\ii \sqrt{2} w^+ \\
-\ii \sqrt{2} w^- & v + h + \ii w^3 \\
\end{pmatrix}.
\end{align}
To avoid adding terms proportional to the vacuum
expectation value, when adding a Higgs pair,
we introduce
\begin{align}
\tr{\vH^\dagger \vH} \rightarrow
\tr{\widehat{\vH^\dagger \vH}}:= \tr{\vH^\dagger \vH - \frac{v^2}{4}}.
\label{eq:HHhat}
\end{align}
\begin{equation}
\begin{aligned}
\vW^{\mu\nu} &\equiv W^{\mu\nu}_i \frac{\tau_i}{2}&= +
\frac{\ii}{g} \com{D_W^\mu}{D_W^\nu}
&= \left (\partial^\mu W^\nu_k - \partial^\nu W^\mu_k
+ g \varepsilon_{ijk} W^\mu_iW^\nu_j \right ) \frac{\tau_k}{2} \\
&&&=\partial^\mu \vW^\nu - \partial^\nu \vW^\mu
- \ii g \left [ \vW^\mu , \vW^\nu \right ], \\
\vB^{\mu\nu} &\equiv\frac{Y}{2} B^{\mu\nu}& = +
\frac{\ii}{g^\prime} \com{D_B^\mu}{D_B^\nu}
&=\frac{Y}{2} \left (\partial^\mu B^\nu - \partial^\nu B^\mu \right )\\
&&&=\partial^\mu \vB^\nu - \partial^\nu \vB^\mu
\end{aligned}
\label{eq:Def-FieldStrengthTensor}
\end{equation}
The covariant derivative is defined via
\begin{equation}
\label{eq:CD_ME_Higgs}
\vD_\mu \vH = \partial_\mu \vH - i g \vW_\mu \vH
\end{equation}
\begin{equation}
\label{eq:CD_ME_W}
\vD_\mu \vW_\nu = \partial_\mu \vW_\nu - i g \vW_\mu \vW_\nu
\end{equation}
The equations of motion for the Standard Model yield
\begin{align}
\left(\vD^2 \vH \right) &= \mu^2\vH - \lambda \tr{\vH^\dagger \vH}\vH \, ,
\label{eq:eqofmotionH}\\
\left (\vD^2 \vH \right)^\dagger &= \mu^2 \vH^\dagger - \lambda \tr{\vH^\dagger \vH}\vH^\dagger \, ,
\label{eq:eqofmotionHdagger} \\
\partial_\mu \vB^{\mu\nu} &= -\ii \frac{g^\prime}{2} \left(
\vH^\dagger \vD^\nu \vH - \left (\vD^\nu \vH \right )^\dagger \vH
\right ) \, ,
\\
\vD_\mu \vW^{\mu\nu} &= -\ii \frac{g}{2} \left(
\vD^\nu \vH \vH^\dagger -\vH \left(\vD^\nu \vH \right )^\dagger
\right )
\end{align}
§.§ $SU(2)$ Tensor Products
The tensor products of Pauli matrices for the isospin quintet $\tau_t$,
the isospin vector $\tau_v$, and the isospin scalar $\tau_s$ are
defined, respectively, as
τ_t^++ = τ^+ ⊗τ^+,
τ_t^+ = 1/2 (
τ^+ ⊗τ^3 + τ^3 ⊗τ^+
τ_t^0 = 1/√(6) (
τ^3 ⊗τ^3 - τ^+ ⊗τ^- - τ^- ⊗τ^+
τ_t^- = 1/2 (
τ^- ⊗τ^3 + τ^3 ⊗τ^-
τ_t^– = τ^- ⊗τ^- ,
τ_v^+ = /2 (
τ^+ ⊗τ^3 - τ^3 ⊗τ^+
τ_v^0 = /√(2) (
τ^+ ⊗τ^- - τ^- ⊗τ^+
τ_v^- = -/2 (
τ^- ⊗τ^3 - τ^3 ⊗τ^-
τ_s =
1/2√(3) ( τ^3 ⊗τ^3 + 2 τ^+ ⊗τ^- + 2 τ^- ⊗τ^+ ) ,
where the Pauli matrix for the isospin singlet is related to
\begin{align}
\tau^{aa} \equiv \tau^a \otimes \tau^a = {2\sqrt{3}} \tau_s \, .
\end{align}
All nonzero traces of a product of two tensor products are normalized
\begin{align}
\tr{\tau_t^{++}\tau_t^{--}}= \tr{\tau_t^+\tau_t^-}=\tr{\tau_t^0\tau_t^0}=
\tr{\tau_v^+\tau_v^-}=\tr{\tau_v^0\tau_v^0}=
\tr{\tau_s\tau_s}=1 \, .
\end{align}
From the properties of the tensor product
\begin{equation}
(A\otimes B) (C\otimes D) = AC\otimes BD
\end{equation}
and the trace
\begin{equation}
\tr{A\otimes B} = \tr{A} \tr{B}
\end{equation}
we find
\begin{align}
\tr{\left (A\otimes B\right ) \left ( C \otimes D \right) } &= \tr{A
C} \tr {B D}\,.
\end{align}
This reduces the trace of an isospin singlet
\begin{align}
\tr{ \left ( A\otimes B \right ) \tau^{aa}}=2 \tr{A B} - \tr {A} \tr {B}.
\end{align}
Multiplying the two Pauli matrices related to the isospin singlet leads to
\begin{align}
\tau^{aa}\tau^{bb} = 3 \,\cdot\, \mathbf{1}\otimes \mathbf{1} -2 \, \tau^{aa} .
\end{align}
§ FEYNMAN RULES
The Feynman rules which are used to calculate the vector-boson scattering
amplitudes are summarized in this appendix. Focusing only on weak
vector-boson scattering, the Feynman rules are determined from the
Lagrangian, where gluons, photons and fermions are omitted.
§.§ Lagrangian
All Lagrangians are defined within the Higgs matrix realization whose
definition can be found in appendix <ref>.
The Standard Model Lagrangian is given by
\begin{align}
\LL_{\text{SM}}=&-\frac{1}{2}\tr{\vW_{\mu\nu}\vW^{\mu\nu}}
-\frac{1}{2}\tr{\vB_{\mu\nu}\vB^{\mu\nu}} \notag \\
&+\tr{ \left ( \vD_\mu \vH \right )^\dagger
\vD^\mu \vH }
+ \mu^2\tr{\vH^\dagger \vH}
-\frac{\lambda}{2}\left( \tr{\vH^\dagger \vH} \right)^2.
\end{align}
Dimension-six and -eight operators affecting only the
Higgs/Nambu-Goldstone boson sector are discussed in sections
<ref> and are given by
_HD = F_HD ^†- v^2/4 · ( _μ)^†^μ ,
_S,0 =
( _μ)^†_ν
· ( ^μ)^†^ν ,
_S,1 =
( _μ)^†^μ
· ( _ν)^†^ν .
As an extension to model generic new physics, additional resonances
are introduced. The scalar resonance $\sigma$ and the tensor resonance
$f^{\mu\nu}$ represent singlets of the chiral symmetry group, whereas
$\Phi$ has the quantum numbers $1 \otimes 1$ under $SU(2)_L \times
SU(2)_R$. $\Phi$ is referred to as isotensor for historical reasons,
but it actually includes an isovector $\Phi_v$ and isoscalar $\Phi_s$ besides
the isotensor $\Phi_t$. Also the Fierz-Pauli tensor $f$ can be reformulated
into a tensor $f_f$, a vector $A_f$ and a scalar $\sigma_f$ such that
canonical propagators can be used for each degree of freedom
separately instead of the complicated tensor propagator
\begin{align}
\Delta_{\mu\nu,\rho\sigma} (f)
&= \frac{\ii}{k^2-m^2+\ii \epsilon}P_{\mu\nu,\rho\sigma}(k,m) \, ,\\
\Delta_{\mu\nu,\rho\sigma} (f^\prime)
&= \frac{\ii}{k^2 - m^2+\ii \epsilon}
\left(\frac{1}{2} g_{\mu\rho}g_{\nu\sigma}
- \frac{1}{2} g_{\mu\nu}g_{\rho\sigma}
\right ) \, ,
\\
\Delta_{\mu\nu} (A)
&= \frac{-\ii}{k^2 - m^2+\ii \epsilon} g_{\mu\nu} \, ,
\\
\Delta (\sigma)
&= \frac{\ii}{k^2 - m^2+\ii \epsilon} \, ,
\end{align}
where the projection operator of spin-two states can be written in
terms of the spin-one projection operator,
\begin{align}
\begin{aligned}
\frac{1}{2}\Bigl[ P^{\mu_1\nu_1}(k,m)P^{\mu_2\nu_2}(k,m) +
P^{\mu_1\nu_2}(k,m)P^{\mu_1\nu_2}(k,m) \Bigr]
\\
- \frac{1}{3} P^{\mu_1\mu_2}(k,m)P^{\nu_1\nu_2}(k,m),
\end{aligned}
\end{align}
\begin{align}
\sum_\lambda \bar{\varepsilon}_{(\lambda)}^{\mu}(k,m)
\varepsilon_{(\lambda)}^{\nu}(k,m)
= g^{\mu\nu}-\frac{k^\mu k^\nu}{m^2}.
\end{align}
§.§ Unitary Gauge
The Feynman rules in unitary gauge of the Lagrangians defined in this
paper are listed in this section. Only the relevant vertices for the
vector-boson scattering process are shown. In other words, vertices
above four fields for effective operators and above three fields for
resonances are neglected.
§.§.§ Standard Model
A_μ_1W^+_μ_2W^-_μ_3 :
(p_1 μ_3-p_2 μ_3 ) g_μ_1μ_2 .
+ (p_3 μ_2-p_1 μ_2 ) g_μ_1μ_3
. +(p_2 μ_1-p_3 μ_1 ) g_μ_2μ_3
] ,
Z_μ_1W^+_μ_2W^-_μ_3 :
-c_w g
(p_1 μ_3-p_2 μ_3 ) g_μ_1μ_2
+ (p_3 μ_2-p_1 μ_2 ) g_μ_1μ_3 .
. +(p_2 μ_1-p_3 μ_1 ) g_μ_2μ_3] ,
h W^+_μ_2W^-_μ_3 :
m_W g g_μ_2 μ_3 ,
h Z_μ_2Z_μ_3 :
m_Z g g_μ_2 μ_3 ,
W^+_μ_1W^+_μ_2W^-_μ_3W^-_μ_4 :
- g^2 (g_μ_1μ_4g_μ_2μ_3 +g_μ_1μ_3g_μ_2μ_4-2 g_μ_1μ_2g_μ_3μ_4 ) ,
Z_μ_1Z_μ_2W^+_μ_3W^-_μ_4 :
c_w^2 g^2 (g_μ_1μ_4g_μ_2μ_3 +g_μ_1μ_3g_μ_2μ_4-2 g_μ_1μ_2g_μ_3μ_4 )
A_μ_1A_μ_2W^+_μ_3W^-_μ_4 :
e^2 (g_μ_1μ_4g_μ_2μ_3 +g_μ_1μ_3g_μ_2μ_4-2 g_μ_1μ_2g_μ_3μ_4 )
A_μ_1Z_μ_2W^+_μ_3W^-_μ_4 :
e c_wg (g_μ_1μ_4g_μ_2μ_3 +g_μ_1μ_3g_μ_2μ_4-2 g_μ_1μ_2g_μ_3μ_4 )
h h W^+_μ_3W^-_μ_4 :
/2 g^2 g_μ_3μ_4
h h Z_μ_3Z_μ_4 :
/2 g^2/c_w^2 g_μ_3μ_4 .
§.§.§ Lg
g^2 v^3/4 F_HD g_μν
g^2 v^3/4 s_w^2 F_HD g_μν
-v F_HD
( p_1·p_2 + p_1·p_3 + p_2·p_3
5g^2 v^2/4 F_HD g_μν
5g^2 v^2/4 s_w^2 F_HD g_μν
( p_1·p_2 + p_1·p_3 +p_1·p_4 .
. + p_2·p_3 + p_2·p_4 + p_3 ·p_4
) .
§.§.§ Lg
W^+_μ_1W^+_μ_2W^-_μ_3W^-_μ_4 :
g^4 v^4/16 [
(F_S,0 + 2F_S,1 ) (
) .
. + 2 F_S,0 g_μ_1μ_2g_μ_3μ_4
Z_μ_1Z_μ_2W^+_μ_3W^-_μ_4 :
g^4 v^4/16 c_w^2 [
F_S,0 (
) .
. +2 F_S,1 g_μ_1μ_2g_μ_3μ_4 ]
Z_μ_1Z_μ_2Z_μ_3Z_μ_4 :
g^4 v^4/8 c_w^4 ( F_S,0 +F_S,1 ) [
g_μ_1μ_2g_μ_3μ_4 + g_μ_1μ_3g_μ_2μ_4
( F_S,0 +F_S,1 ) . +g_μ_1μ_4g_μ_2μ_3
] ,
h(p_1)h(p_2)W^+_μ_3W^-_μ_4 :
- g^2 v^2/4 [
F_S,0 (
p_1 μ_3p_2 μ_4+ p_1 μ_4p_2 μ_3 )
. +2 F_S,1 g_μ_3μ_4 p_1 ·p_2 ]
h(p_1)h(p_2)Z_μ_3Z_μ_4 :
- g^2 v^2/4 c_w^2 [
F_S,0 (
p_1 μ_3p_2 μ_4+ p_1 μ_4p_2 μ_3 )
. +2 F_S,1 g_μ_3μ_4 p_1 ·p_2 ]
h(p_1)h(p_2)h(p_3)h(p_4) :
2 ( F_S,0 +F_S,1 ) [
( p_1 ·p_2 ) ( p_3 ·p_4 ) .
( F_S,0 +F_S,1 )
+ ( p_1 ·p_3 ) ( p_2 ·p_4 )
( F_S,0 +F_S,1 )
. + ( p_1 ·p_4 ) ( p_2 ·p_3 )
] .
§.§.§ Lg
g^2 v^2/4 F_σ g_μν ,
g^2 v^2/4 c_w^2 F_σ g_μν ,
σh (p_1 ) h(p_2 )
-F_σ p_1 ·p_2 .
§.§.§ Lg
ϕ_t^±± W^∓_μW^∓_ν:
g^2 v^2/4 F_ϕ g_μν ,
ϕ_t^± W^∓_μZ_ν:
g^2 v^2/4 √(2)c_w F_ϕ g_μν ,
ϕ_t^0 W^∓_μW^±_ν:
-g^2 v^2/4 √(6) F_ϕ g_μν ,
ϕ_t^0 Z_μZ_ν:
g^2 v^2/2 √(6)c_w^2 F_ϕ g_μν ,
ϕ_s W^∓_μW^±_ν:
g^2 v^2/8 √(3) F_ϕ g_μν ,
ϕ_s Z_μZ_ν:
g^2 v^2/8 √(3)c_w^2 F_ϕ g_μν ,
ϕ_v^± h (p ) W^∓_μ:
- g v/2 √(2) F_ϕ p_μ ,
ϕ_v^± h (p ) Z_μ:
g v/2 √(2)c_w F_ϕ p_μ ,
ϕ_s h (p_1 ) h (p_2 ):
√(3) /2 F_ϕ p_1 ·p_2 .
§.§.§ Lg
f_μν W^+_ρW^-_σ:
g^2 v^2/8 F_f
g_μσ g_νρ
+ g_μρ g_νσ
-c_f/2 g_μν g_ρσ
] ,
f_μν Z_ρZ_σ:
g^2 v^2/8 c_w^2 F_f
g_μσ g_νρ
+ g_μρ g_νσ
-c_f/2 g_μν g_ρσ
f_μν h (p_1 ) h(p_2 )
-/2 F_f
[p_1 μ p_2 ν + p_1 ν p_2 μ
-c_f/2 g_μνp_1·p_2 ] .
§.§.§ Lg
f_fμν W^+_ρW^-_σ:
g^2 v^2/8 F_f
g_μσ g_νρ
+ g_μρ g_νσ
-c_f/2 g_μν g_ρσ
] ,
f_fμν Z_ρZ_σ:
g^2 v^2/8 c_w^2 F_f
g_μσ g_νρ
+ g_μρ g_νσ
-c_f/2 g_μν g_ρσ
f_fμν h (p_1 ) h(p_2 )
-/2 F_f
[p_1 μ p_2 ν + p_1 ν p_2 μ
-c_f/2 g_μνp_1·p_2 ] .
Because of $\partial_\nu J_f^{\mu \nu} \neq 0$:
A_f μ ( p ) W^+_ρW^-_σ:
g^2 v^2/4 √(2)m_f F_f
(p_ρg_μσ + p_σg_μρ
- c_f/2 p_μg_σρ ) ,
A_f μ( p ) Z_ρZ_σ:
g^2 v^2/4 c_w^2 √(2)m_f F_f
( p_ρg_μσ + p_σg_μρ
- c_f/2 p_μg_σρ ) ,
A_f μ h (p_1 ) h(p_2 )
1/√(2)m_f F_f
p_1^2 p_2 μ +p_2^2 p_1 μ .
.+1/2 ( 2 - c_f )
(p_1 + p_2 )_μ] .
Because of $\partial_\mu \partial_\nu J_f^{\mu \nu} \neq 0$ and $ {J_f}^{\mu}_\mu \neq 0$:
σ_f ( p ) W^+_ρW^-_σ:
g^2 v^2/4√(6) F_f
[( c_f - 1 ) g_ρσ
-1/m_f^2( 2 k_ρk_σ- c_f/2 k^2 g_ρσ )
] ,
σ_f( p ) Z_ρZ_σ:
g^2 v^2/4√(6) c_w^2 F_f
[( c_f - 1 ) g_ρσ
-1/m_f^2( 2 k_ρk_σ- c_f/2 k^2 g_ρσ )
] ,
σ_f h (p_1 ) h(p_2 )
-/√(6) F_f
( c_f - 1 ) ( p_1 ·p_2 )
- 1/m_f^2 ( 2 p_1·(p_1 +p_2 )
p_2 ·(p_1 +p_2 )
. - 1/m_f^2 (
- c_f/2 p_1·p_2 (p_1 +p_2 )^2 )
] .
§.§.§ Lg
X_tμν^±± W^∓_ρW^∓_σ:
g^2 v^2/8 F_X [
g_μσ g_νρ
+ g_μρ g_νσ
-c_X/2 g_μν g_ρσ
] ,
X_tμν^± W^∓_ρZ_σ:
g^2 v^2/8 √(2)c_w F_X [
g_μσ g_νρ
+ g_μρ g_νσ
-c_X/2 g_μν g_ρσ
] ,
X_tμν^0 W^∓_ρW^±_σ:
-g^2 v^2/8 √(6) F_X [
g_μσ g_νρ
+ g_μρ g_νσ
-c_X/2 g_μν g_ρσ
] ,
X_tμν^0 Z_ρZ_σ:
g^2 v^2/4 √(6)c_w^2 F_X [
g_μσ g_νρ
+ g_μρ g_νσ
-c_X/2 g_μν g_ρσ
] ,
X_sμν W^∓_ρW^±_σ:
g^2 v^2/16 √(3) F_X [
g_μσ g_νρ
+ g_μρ g_νσ
-c_X/2 g_μν g_ρσ
] ,
X_sμν Z_ρZ_σ:
g^2 v^2/16 √(3)c_w^2 F_X [
g_μσ g_νρ
+ g_μρ g_νσ
-c_X/2 g_μν g_ρσ
] ,
X_vμν^± h (p ) W^∓_ρ:
- g v/4 √(2) F_X
[ p_μ g_νρ + p_ν g_μρ - c_X/2p_ρ g_μν ] ,
X_vμν h (p ) Z_ρ:
g v/4 √(2)c_w F_X
[ p_μ g_νρ + p_ν g_μρ - c_X/2p_ρ g_μν ] ,
X_sμν h (p_1 ) h (p_2 ):
√(3) /4 F_X [p_1 μ p_2 ν +
p_1 ν p_2 μ
-c_X/2 g_μνp_1·p_2 ] .
§.§ Partial wave functions
In this appendix we collect expressions appearing in the partial-wave
expansion of amplitudes.
\begin{align}
\mathcal{S}_{0}\left(s, m \right) =& m^2+\frac{m^4}{s}\log \left (\frac{m^2}{s + m^2} \right )
-\frac{s}{2}\, , \\
\mathcal{S}_{1}\left(s, m \right) =& 2 \frac{m^4}{s}
+\frac{m^4}{s^2} \left( 2 m^2 +s \right )
\log \left (\frac{m^2}{s + m^2} \right )
+\frac{s}{6}\, , \\
\mathcal{S}_{2}\left(s, m \right) =& \frac{m^4}{s^2}
\left (6 m^2 + 3s \right )
+\frac{m^4}{s^3} \left( 6 m^4 +6m^2 s +s^2 \right )
\log \left (\frac{m^2}{s + m^2} \right )\, , \\
\mathcal{P}_{0}\left(s, m \right) =& 1+\frac{m^2+2s}{s}\log \left (\frac{m^2}{s + m^2} \right )\, , \\
\mathcal{P}_{1}\left(s, m \right) =& \frac{m^2+2s}{s^2}\left(2s + \left({2m^2+s}\right) \log \left (\frac{m^2}{s + m^2} \right ) \right)\, \, , \\
\mathcal{D}_{0}\left(s, m \right) =& m^2 + \frac{11}{2}s + \frac{1}{s}\left(m^4 + 6m^2s +6s^2\right)\log \left (\frac{m^2}{s + m^2} \right )\, , \\
\mathcal{D}_{1}\left(s, m \right) =& 2\frac{m^4}{s}+12m^2+\frac{73}{6}s \notag \\
&+ \frac{1}{s^2}\left(2m^2 +s\right)\left(m^4 + 6m^2s +6s^2\right)\log \left (\frac{m^2}{s + m^2} \right ) \, .
\end{align}
§ T-MATRIX COUNTERTERMS
In the T-matrix unitarization scheme, the
unitarization corrections are expressed as momentum-dependent
counterterms for the use as effective Feynman rules in the complete
amplitude evaluation. Starting from the spin-isospin eigenamplitudes
in the gaugeless limit, section <ref>, the
straightforward application of the algorithm in <cit.>
yields $s$-dependent amplitude corrections $\Delta\amp_{IJ}(s)$. The
insertion as effective Feynman rules proceeds in form of the following
: g^4 v^4/4 [
( Δ_20(s)-10Δ_22(s) )
.+ 15 Δ_22(s)
g_μ_1μ_3g_μ_2μ_4+ g_μ_1μ_4g_μ_2μ_3/s^2 ],
: g^4 v^4/4 c_w^2 [
( 1/3 ( Δ_00(s)-Δ_20(s)
) . .
. . -10/3( Δ_02(s)-Δ_22(s)
) )
. + 5 (Δ_02(s)-Δ_22(s) )
g_μ_1μ_3g_μ_2μ_4+ g_μ_1μ_4g_μ_2μ_3/s^2 ],
: g^4 v^4/4c_w^2 [
(1/2Δ_20(s)-5Δ_22(s) )
+ ( -3/2Δ_11(s)+15/2Δ_22(s) )g_μ_1μ_3g_μ_2μ_4/s^2
.+ (3/2Δ_11(s)+15/2Δ_22(s) )g_μ_1μ_4g_μ_2μ_3/s^2
] ,
: g^4 v^4/4 [
( 1/6 ( 2Δ_00(s)+Δ_20(s)
. .
. .
-5/3(2 Δ_02(s)+Δ_22(s)
) g_μ_1μ_2g_μ_3μ_4/s^2 .
+ ( 5Δ_02(s)-3/2Δ_11(s)+5/2 Δ_22(s) )g_μ_1μ_3g_μ_2μ_4/s^2
. + ( 5Δ_02(s)+3/2Δ_11(s)+5/2 Δ_22(s) )
: g^4 v^4/4c_w^4 [
( 1/3 ( Δ_00(s)+2Δ_20(s)
. .
. .
-10/3( Δ_02(s)+2Δ_22(s) )
) g_μ_1μ_2g_μ_3μ_4/s^2 .
.+ 5 (Δ_02(s)+2Δ_22(s) )
g_μ_1μ_3g_μ_2μ_4+ g_μ_1μ_4g_μ_2μ_3/s^2 ].
These relations are the generalizations of the corresponding formulae
in reference <cit.> for the case of resonances.
Scattering processes involving a Higgs boson have a different
off-shell extrapolation. Therefore, the Higgs momentum is included in
the Feynman rules for the analogous effective vertices given by
W^±^μ_1W^∓^μ_2→h h
: -g^2 v^2 [
( 1/3 ( Δ_00(s)-Δ_20(s)
) . .
. . -10/3( Δ_02(s)-Δ_22(s)
) )
g^μ_1μ_2 (k_3·k_4)/s^2
. + 5 (Δ_02(s)-Δ_22(s) )
k_3^μ_1k_4^μ_2+ k_4^μ_1k_3^μ_2/s^2 ],
Z^μ_1Z^μ_2→h h
: -g^2 v^2/c_w^2 [
( 1/3 ( Δ_00(s)-Δ_20(s)
) . .
. . -10/3( Δ_02(s)-Δ_22(s)
) )
g^μ_1μ_2 (k_3·k_4)/s^2
. + 5 (Δ_02(s)-Δ_22(s) )
k_3^μ_1k_4^μ_2+ k_4^μ_1k_3^μ_2/s^2 ],
: -g^2 v^2 [
(1/2Δ_20(s)-5Δ_22(s) )
+ ( -3/2Δ_11(s)+15/2Δ_22(s) )g^μ_1μ_3(k_2·k_4 )/s^2
.+ (3/2Δ_11(s)+15/2Δ_22(s) )k_4^μ_1k_2^μ_3/s^2
] ,
: -g^2 v^2/c_w^2 [
(1/2Δ_20(s)-5Δ_22(s) )
+ ( -3/2Δ_11(s)+15/2Δ_22(s) )g^μ_1μ_3(k_2·k_4 )/s^2
.+ (3/2Δ_11(s)+15/2Δ_22(s) )k_4^μ_1k_2^μ_3/s^2
] ,
: 4 [
( 1/3 ( Δ_00(s)+2Δ_20(s)
. .
. .
-10/3( Δ_02(s)+2Δ_22(s) )
) ( k_1 ·k_2)( k_3 ·k_4 ) /s^2 .
.+ 5 (Δ_02(s)+2Δ_22(s) )
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|
1511.00406
|
Will Donovan, Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, Kashiwa, Chiba 277-8583, Japan.
Michael Wemyss, The Maxwell Institute, School of Mathematics, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK.
[2010]Primary 14D15; Secondary 14E30, 14F05, 16E45, 16S38
The first author was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and by EPSRC grant EP/G007632/1. The second author was supported by EPSRC grant EP/K021400/1.
Primary 14D15; Secondary 14E30, 14F05, 16E45, 16S38
The first author was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and by EPSRC grant EP/G007632/1. The second author was supported by EPSRC grant EP/K021400/1.
Suppose that $f$ is a projective birational morphism with at most one-dimensional fibres between $d$-dimensional varieties $X$ and $Y$, satisfying $\Rf_*\cO_X=\cO_Y$. Consider the locus $L$ in $Y$ over which $f$ is not an isomorphism. Taking the scheme-theoretic fibre $C$ over any closed point of $L$, we construct algebras $\AB_\fib$ and $\CA$ which prorepresent the functors of commutative deformations of $C$, and noncommutative deformations of the reduced fibre, respectively. Our main theorem is that the algebras $\CA$ recover $L$, and in general the commutative deformations of neither $C$ nor the reduced fibre can do this. As the $d=3$ special case, this proves the following contraction theorem: in a neighbourhood of the point, the morphism $f$ contracts a curve without contracting a divisor if and only if the functor of noncommutative deformations of the reduced fibre is representable.
§ INTRODUCTION
Our setting is a contraction $f\colon X\to X_{\con}$ with at most one-dimensional fibres between $d$-dimensional varieties, satisfying $\Rf_*\cO_X=\cO_{X_{\con}}$. Writing $L \subset X_{\con}$ for the locus over which $f$ is not an isomorphism, it is a fundamental problem to characterise $L$, locally around a closed point in the base. For this, it is natural to study the deformations of the curve(s) above the point, and the question which we answer in this paper is the following.
Which deformation-theoretic framework detects the non-isomorphism locus $L$, Zariski locally around a closed point $p\in X_{\con}$?
The answer turns out to lie in noncommutative deformations, without assumptions on the singularities of $X$, and in arbitrary dimension. This process associates a noncommutative algebra to each point, which should be viewed as an invariant of the contraction $f$. When $d=3$ and the algebra is finite-dimensional, its dimension has a curve-counting interpretation <cit.>, but in all cases the algebra structure gives information about the neighbourhood of the point. This extra information has applications in constructing derived autoequivalences <cit.>, and also in the minimal model program, allowing us to control iteration of flops and count minimal models <cit.>, and produce the first explicit examples of Type $E$ flops <cit.>. It is also conjectured that such algebras classify smooth $3$-fold flops <cit.>, and furthermore it is expected that they control divisor-to-curve contractions.
\[
\begin{array}{ccc}
\begin{array}{c}
\begin{tikzpicture}
\node at (0,0) {\begin{tikzpicture}[scale=1]
\coordinate (T) at (1.9,2);
\coordinate (TM) at (2.12-0.02,1.5-0.1);
\coordinate (BM) at (2.12-0.05,1.5+0.1);
\coordinate (B) at (2.1,1);
\draw[line width=0.5pt]\opt{colordiag}{[red]} (T) to [bend left=25] (TM);
\draw[line width=0.5pt]\opt{colordiag}{[red]} (BM) to [bend left=25] (B);
\foreach \y in {0.1,0.2,...,1}{
\draw[very thin]\opt{colordiag}{[blue]} ($(T)+(\y,0)+(0.02,0)$) to [bend left=25] ($(B)+(\y,0)+(0.02,0)$);
\draw[very thin]\opt{colordiag}{[blue]} ($(T)+(-\y,0)+(-0.02,0)$) to [bend left=25] ($(B)+(-\y,0)+(-0.02,0)$);;}
\draw[rounded corners=15pt,line width=0.5pt] (0.5,0) -- (1.5,0.3)-- (3.6,0) -- (4.3,1.5)-- (4,3.2) -- (2.5,2.7) -- (0.2,3) -- (-0.2,2)-- cycle;
\end{tikzpicture}};
\node at (0,-3.5) {\begin{tikzpicture}[scale=1]
\draw (1.1,0.75) -- (3.1,0.75);
\filldraw\opt{colordiag}{[red]} (2.1,0.75) circle (1pt);
\node at (2.5,0.6) {$\scriptstyle p\phantom{=L}$};
\node at (3,0.6) {$\scriptstyle \phantom{p=}L$};
\draw[rounded corners=12pt,line width=0.5pt] (0.5,0) -- (1.5,0.15)-- (3.6,0) -- (4.3,0.75)-- (4,1.6) -- (2.5,1.35) -- (0.2,1.5) -- (-0.2,0.6)-- cycle;
\end{tikzpicture}};
\draw[->] (0,-1.6) -- node[left] {$\scriptstyle f$} (0,-2.65);
\node at (-2.8,-3.6) {$X_{\con}$};
\node at (-2.8,-0.1) {$X$};
\end{tikzpicture}
\end{array}
& \qquad &
\begin{array}{c}
\begin{tikzpicture}
\node at (0,0) {\begin{tikzpicture}[scale=1]
\coordinate (T) at (1.9,2);
\coordinate (TM) at (2.12-0.02,1.5-0.1);
\coordinate (BM) at (2.12-0.05,1.5+0.1);
\coordinate (B) at (2.1,1);
\draw[line width=0.5pt]\opt{colordiag}{[red]} (T) to [bend left=25] (TM);
\draw[line width=0.5pt]\opt{colordiag}{[red]} (BM) to [bend left=25] (B);
\draw[rounded corners=15pt,line width=0.5pt] (0.5,0) -- (1.5,0.3)-- (3.6,0) -- (4.3,1.5)-- (4,3.2) -- (2.5,2.7) -- (0.2,3) -- (-0.2,2)-- cycle;
\end{tikzpicture}};
\node at (0,-3.5) {\begin{tikzpicture}[scale=1]
\filldraw\opt{colordiag}{[red]} (2.1,0.75) circle (1pt);
\node at (2.5,0.6) {$\scriptstyle p=L$};
\draw[rounded corners=12pt,line width=0.5pt] (0.5,0) -- (1.5,0.15)-- (3.6,0) -- (4.3,0.75)-- (4,1.6) -- (2.5,1.35) -- (0.2,1.5) -- (-0.2,0.6)-- cycle;
\end{tikzpicture}};
\draw[->] (0,-1.6) -- node[left] {$\scriptstyle f$} (0,-2.65);
\end{tikzpicture}
\end{array}
\end{array}
\]
Contractions of $3$-folds: divisor to curve, and curve to point.
§.§ Summary of Results
For a closed point $p\in L$, consider the scheme-theoretic fibre $C=f^{-1}(p)$. Set-theoretically, it is well known that $C$ is a union of $\mathbb{P}^1$s, and we denote these by $C_1,\hdots,C_n$. To specify a deformation problem requires us to provide test objects, and to say what object(s) are being deformed. The test objects for the noncommutative deformation functor are the category $\art_n$ of artinian augmented $\K^n$-algebras, and the objects being deformed are $\{ \cO_{C_1}(-1), \hdots, \cO_{C_n}(-1) \}$. Informally, we wish to control the deformations of the $\cO_{C_i}(-1)$, and also the mutual extensions between them. Formally, as explained in <ref>, this is encoded via the Maurer–Cartan formulation as a functor
\[
\Def^{\cJ}\colon \art_n \rightarrow \Sets.
\]
Given a closed point $p \in X_{\con}$, it is well-known that the formal fibre over $p$ is derived equivalent to a certain noncommutative ring $\AB$. By taking suitable factors, as in <cit.> we obtain the contraction algebra $\CA$, referring the reader to <ref> for full details. Our first main result is then the following. The special case where the locus $L$ is a single point, $d=3$, and $n=1$, was previously shown in <cit.>.
For each closed point $p \in L$, the algebra $\CA$ (depending on $p$) prorepresents the functor of noncommutative deformations $\Def^{\cJ}$ of the reduced fibre over $p$.
It turns out that the geometry of the locus $L$ is controlled by the support of $\CA$.
With the setup above, pick an affine open neighbourhood $\Spec R$ in $X_{\con}$, and consider $L_R:=L\cap \Spec R$.
* (=<ref>, <ref>(<ref>)) There is an $R$-algebra $\Lambda_{\con}$ which satisfies $\Supp_R\Lambda_{\con}=L_R$.
* (=<ref>) For each closed point $p \in L_R$, the completion of $\Lambda_{\con}$ at $p$ is morita equivalent to $\CA$.
This theorem has two main consequences.
The dimension of $L_R$ at $p$ is $\dim_{\mathfrak{R}}\Supp_{\mathfrak{R}}\CA$, where $\mathfrak{R}$ denotes the completion of $R$ at $p$.
When $d=3$, there is a neighbourhood of $p$ over which $f$ does not contract a divisor if and only if $\dim_{\K}\CA<\infty$.
The `only if' direction is easy and is known from our previous work <cit.>. The content is the `if' direction, and this requires significantly more technology.
§.§ Comparing deformation theories
We next show that other natural deformation functors do not control the geometry of $L$. As above, consider the scheme-theoretic fibre $C=f^{-1}(p)$ and the reduced curves $C_1,\hdots,C_n$ therein. To this data, we associate three other deformation problems. Again the details are left to <ref>, but the following table summarises all four functors, giving the test objects and the deformed object in each case. Here $\cart_n$ is the category of commutative artinian augmented $\K^n$-algebras.
\begin{equation*}
\begin{tabular}{l*6c}
\toprule
\multirow{2}{*}{\bf Deformation problem}&&\multirow{2}{*}{\bf Functor}&&{\bf Test}&&{\bf Object(s)}\\
{\bf }&&{\bf }&&{\bf objects}&&{\bf deformed}\\
\midrule
Classical scheme-theoretic && $\cDef^{\cO_C}$&& $\cart_1$&& $\cO_C$ \\
Noncommutative scheme-theoretic && $\phantom{c}\Def^{\cO_C}$ && $\phantom{\mathsf{C}}\art_1$&& $\cO_C$\\
\cmidrule(l){1-7}
Commutative multi-pointed && $\cDef^{\cJ}$&& $\cart_{n}$&& $\oplus_i \cO_{C_i}(-1)$ \\
Noncommutative multi-pointed && $\phantom{c}\Def^{\cJ}$&& $\phantom{\mathsf{C}}\art_n$&& $\{\cO_{C_i}(-1)\}_i$ \\
\bottomrule\\
\end{tabular}
\end{equation*}
The following result drops out of our general construction. It is quite surprising, since it says that noncommutative deformations of the scheme-theoretic fibre give nothing in addition to the classical ones.
The functors $\cDef^{\cO_C}$ and $\Def^{\cO_C}$ are prorepresented by the same object $\CAR$.
In general, however, the prorepresenting objects for the functors $\cDef^{\cO_C}$, $\cDef^{\cJ}$ and $\Def^{\cJ}$ are different. We prove the following.
Neither $\cDef^{\cO_C}$ nor $\cDef^{\cJ}$ detect the dimension of the non-isomorphism locus $L$.
This is clear when the fibre above $p$ has more than one irreducible curve, but is much more surprising when there is only a single irreducible curve in the fibres. We produce in <ref> a contraction, sketched below, with a one-dimensional non-isomorphism locus $L$ in which the central point $0$ is $cD_4$, and all other points are $cA_1$. The commutative deformations of all reduced fibres except the central one are infinite dimensional, whereas the noncommutative deformations are always infinite dimensional.
\[
\begin{tikzpicture}
%The two curves
\draw [bend right,line width=0.75pt]\opt{colordiag}{[red]} (3,-2.1) to (3,-0.1);
\draw [bend right,line width=0.75pt]\opt{colordiag}{[red]} (4,-1.96) to (4,-0.27);
% The exceptional locus
\draw[color=black!70,rounded corners=25pt,line width=0.75pt] (0,0)-- (2,0.2) -- (4,-0.4) -- (6,0) ;
\draw[color=black!70,rounded corners=25pt,line width=0.75pt] (0,-2) -- (2,-2.3) -- (4,-1.9) -- (6,-2);
\draw[color=black!70,bend left,line width=0.75pt] (0,0) to (0,-2);
\draw[color=black!70,bend left,line width=0.75pt] (6,0) to (6,-2);
% The nonisomorphism locus
\draw[color=black!70,rounded corners=25pt,line width=0.75pt] (0,-3.5+\shift) -- (2,-3.8+\shift) -- (4,-3.4+\shift) -- (6,-3.5+\shift);
% The contraction
\draw[->] (3,-2.4) -- node[left] {$\scriptstyle f$} (3,-3.5);
%The two points
\filldraw\opt{colordiag}{[red]} (3,-3.6+\shift) circle (1.5pt);
\filldraw\opt{colordiag}{[red]} (4,-3.47+\shift) circle (1.5pt);
%The labels on points
\node at (3,-4+\shift) {$\scriptstyle 0$};
\node at (4,-3.87+\shift) {$\scriptstyle p$};
%Comm and NC deformations
\node at (-1,-0.5) {$\frac{\K\langle\!\langle x,y\rangle\!\rangle}{x^2,y^2}$};
\node at (-1,-1.5) {$\scriptstyle \K[\![ x]\!]$};
\node at (-3.5,-0.5) {$\frac{\K[\![ x,y]\!]}{x^2,y^2}$};
\node at (-3.5,-1.5) {$\scriptstyle \K[\![ x]\!]$};
\node[align=center, text width=2cm] at (-1,0.75) {\scriptsize Noncommutative};
\node[align=center, text width=2cm] at (-3.5,0.75) {\scriptsize Commutative};
\node[align=center, text width=2cm] at (-1,0.45) {\scriptsize deformations};
\node[align=center, text width=2cm] at (-3.5,0.45) {\scriptsize deformations};
%%arrows into picture
\draw[->,line width=0.75,black!50] (-0.4,-0.5) to (3.1,-0.5);
\draw[->,line width=0.75,black!50] (-0.4,-1.5) to (4.1,-1.5);
\draw[line width=0.75,black!50] (-2.95,-0.5) to (-1.6,-0.5);
\draw[line width=0.75,black!50] (-2.95,-1.5) to (-1.6,-1.5);
% labels for non-iso locus
\node at (6.6,-2) {$\scriptstyle f^{-1}(L)$};
\node at (6.2,-3.5+\shift) {$\scriptstyle L$};
\end{tikzpicture}
\]
Commutative versus noncommutative deformations of $C^{\redu}.$
Thus, it follows that $\Def^{\cJ}$ is the functor that controls the contractibility of curves. For the convenience of the reader, we summarise the above together with the main results of <cit.> in the following table. To discuss autoequivalences, we further assume that $f$ is a flopping contraction, and $X$ is $\mathds{Q}$-factorial with only Gorenstein terminal singularities.
\begin{equation*}
\begin{tabular}{l*6c}
\toprule
\multirow{2}{*}{\bf Deformation problem}&&\multirow{2}{*}{\bf Functor}&&{\bf Detects}&&{\bf Corresponds to}\\
{\bf }&&{\bf }&&{\bf divisor?}&&{\bf autoequivalence?}\\\midrule
Classical && $\cDef^{\cO_C}$ && No&& Yes\\
\cmidrule(l){1-7}
Commutative multi-pointed && $\cDef^{\cJ}$&&No && No\\
Noncommutative multi-pointed && $\phantom{c}\Def^{\cJ}$&& Yes&& Yes\\
\bottomrule\\
\end{tabular}
\end{equation*}
Our method to prove the above theorems involves noncommutative deformation theory associated to DGAs. By passing through various derived equivalences and embeddings, we reduce the geometric deformation problem into an easier problem about simultaneously deforming a collection of simple modules on a complete local ring obtained by tilting.
§.§ Structure of the Paper
In <ref>, we recall the naive noncommutative deformation theory developed by Laudal <cit.>, and subsequently Eriksen <cit.>. We then describe DG deformation theory in <ref>, noting that it is equivalent by work of Segal <cit.> and Efimov–Lunts–Orlov <cit.>, and establish tools for comparing DG deformations which will arise in our construction.
In <ref> we give the geometric setup, before proving the prorepresentability results for noncommutative deformations of the reduced fibre. The key observation, building on <ref>, is that this deformation functor on $X$ is isomorphic to a DG deformation functor associated to a specific locally free resolution on $U$, naturally obtained via tilting. In <ref> we use these results to prove <ref>, <ref>, and the Contraction Theorem.
In <ref> we prove the prorepresentability results for both commutative and noncommutative deformations of the scheme-theoretic fibre, and show that they are prorepresented by the same object. The fact that $\CA$ and $\CAR$ are obtained as factors of a common ring $\AB$ then allows us to relate the different deformation functors, and we do this in <ref>. We conclude in <ref> by giving examples which illustrate the necessity of noncommutative deformations in the above theorems.
§.§ Conventions
Throughout we work over the field of complex numbers $\K$. Unqualified uses of the word `module' refer to left modules, and $\mod A$ denotes the category of finitely generated left $A$-modules. For two $\K$-algebras $A$ and $B$, an $A$-$B$ bimodule is the same thing as an $A\otimes_{\K}B^{\op}$-module. We use the functorial convention for composing arrows, so $f\cdot g$ means $f$ then $g$. In particular, naturally this makes $M\in \mod A$ into an $\End_R(M)$-module. We remark that these conventions are opposite to those in <cit.>, but we do this to match the conventions in <cit.>. Similarly, for quivers, DG category morphisms, and matrix multiplication, we write $ab$ for $a$ then $b$. We reserve the notation $f \circ g$ for the composition $g$ then $f$.
In an abelian category $\cA$, given two objects $a,b\in\cA$ where $a$ is a summand of $b$, we write $[a]$ for the two-sided ideal of $\End(b)$ consisting of all morphisms that factor through a summand of a finite direct sum of $a$'s.
Acknowledgements. The authors would like to thank Jon Pridham, Ed Segal, Olaf Schnürer and Yukinobu Toda for helpful discussions relating to this work.
§ NAIVE AND DG DEFORMATIONS
This section is mainly a review of known material, and is used to set notation. Noncommutative deformations of modules were introduced by Laudal <cit.>, and we review these naive deformation functors in <ref> below. In our geometric setting later, this naive definition is necessary in order to establish the prorepresenting object in <ref>.
However, it is cumbersome to compare two or more naive deformation functors, as is demonstrated in the proofs of <cit.>, and for this the setting of multi-pointed DG deformation functors is much better suited. We review this theory in <ref>, being a slight generalisation of the setting of <cit.>, before in <ref> proving some general DG deformation functor results.
§.§ Naive Deformations of Modules
From the geometric motivation of the introduction, where we want to deform $n$ reduced curves in a fibre simultaneously, the test objects for the naive deformation functors are objects of the category $\art_n$, defined as follows.
An $n$-pointed $\K$-algebra $\Gamma$ is an associative $\K$-algebra, together with $\K$-algebra morphisms $p\colon \Gamma \to \K^n$ and $i: \K^n \to \Gamma$ such that $ip = \Id$. A morphism of $n$-pointed $\K$-algebras $\psi\colon (\Gamma,p,i)\to(\Gamma',p',i')$ is an algebra homomorphism $\psi\colon\Gamma\to\Gamma'$ such that
\[
\begin{tikzpicture}
\node (A) at (0,0) {$\K^n$};
\node (B1) at (1.5,0.75) {$\Gamma$};
\node (B2) at (1.5,-0.75) {$\Gamma'$};
\node (C) at (3,0) {$\K^n$};
\draw[->] (A) -- node[above] {$\scriptstyle i$} (B1);
\draw[->] (A) -- node[below] {$\scriptstyle i'$} (B2);
\draw[->] (B1) -- node[above] {$\scriptstyle p$} (C);
\draw[->] (B2) -- node[below] {$\scriptstyle p'$} (C);
\draw[->] (B1) -- node[right] {$\scriptstyle \psi$}(B2);
\end{tikzpicture}
\]
commutes. We denote the category of $n$-pointed $\K$-algebras by $\alg_n$. We denote the full subcategory consisting of those objects that are commutative rings by $\calg_n$.
We write $\art_n$ for the full subcategory of $\alg_n$ consisting of objects $(\Gamma,p,i)$ for which $\dim_{\K}\Gamma<\infty$ and the augmentation ideal $\n:=\Ker p$ is nilpotent. We write $\cart_n$ for the full subcategory of $\art_n$ consisting of those objects that are commutative rings.
If $\Gamma$ is an associative ring, then by <cit.>, there exists $p$, $i$ such that $(\Gamma,p,i)\in\art_n$ if and only if $\Gamma$ is an artinian $\mathbb{C}$-algebra with precisely $n$ simple modules (up to isomorphism), each of them one-dimensional over $\mathbb{C}$.
If $(\Gamma,p,i)\in\art_n$, the structure morphisms $p$ and $i$ allow us to lift the canonical idempotents $\{\idem{i}, \ldots, \idem{n}\}$ of $\K^n$ to $\Gamma$. We will write
\[
\Gamma_{ij} := \idem{i} \Gamma \idem{j},
\]
and consider $\Gamma$ as a matrix ring $(\Gamma_{ij})$ under standard matrix multiplication. Accordingly, a left $\Gamma$-module $M$ may be described in terms of its vector of summands $M_i := e_i M $, and a right $\Gamma$-module $N$ can be described by its summands $N_i := N e_i $.
Given a $\K$-algebra $\Lambda$, choose a family $\cS=\{S_1,\hdots,S_n\}$ of objects in $\Mod\Lambda$. The deformation functor
\[
\Def_{\Lambda}^{\cS}\colon\art_n\to\Sets
\]
is defined by sending
\[
\left. \left \{ (M,\underline{\delta})
\left|\begin{array}{l}M\in \Mod\Lambda\otimes_{\mathbb{C}}\Gamma^{\op}\\
M \cong (S_i\otimes_{\mathbb{C}}\Gamma_{ij})\mbox{ as right }\Gamma\mbox{-modules}\\ \underline{\delta}=(\delta_i), \mbox{where each }\delta_i\colon M\otimes_\Gamma (\Gamma /\n)e_i\xrightarrow{\sim} S_i\end{array}\right. \right\} \middle/ \sim \right.
\]
* $(S_i\otimes_{\mathbb{C}}\Gamma_{ij})$ refers to the $\mathbb{C}$-vector space $
with the natural right $\Gamma$-module structure coming from the multiplication in $\Gamma$.
* $(M,\underline{\delta})\sim (N,\underline{\delta}')$ iff there exists an isomorphism $\tau\colon M\to N$ of bimodules such that the following diagram commutes for all $i=1,\hdots,n$.
\[
\begin{tikzpicture}
\node (a1) at (0,0) {$M\otimes_\Gamma (\Gamma/\n) e_i$};
\node (a2) at (3,0) {$N\otimes_\Gamma (\Gamma/\n) e_i$};
\node (b) at (1.5,-1) {$S_i$};
\draw[->] (a1) -- node[above] {$\scriptstyle \tau \otimes 1 $} (a2);
\draw[->] (a1) -- node[gap] {$\scriptstyle \delta_i$} (b);
\draw[->] (a2) -- node[gap] {$\scriptstyle \delta'_i$} (b);
\end{tikzpicture}
\]
An important problem in deformation theory is determining when deformation functors are prorepresentable, and also effectively describing the prorepresenting object. We briefly recall these notions, mainly to fix notation.
For any $(\Gamma,p,i)\in\alg_n$, setting $I(\Gamma):=\Ker p$ we consider the $I(\Gamma)$-adic completion $\widehat{\Gamma}$ of $\Gamma$, defined by
\[
\widehat{\Gamma}:=\varprojlim \Gamma/I(\Gamma)^n.
\]
The canonical morphism $\psi_\Gamma\colon \Gamma\to\widehat{\Gamma}$ belongs to $\alg_n$. We say that $\Gamma\in\alg_n$ is complete if $\psi_\Gamma$ is an isomorphism. The pro-category $\proart_n$ is then defined to be the full subcategory of $\alg_n$ consisting of those objects $(\Gamma,p,i)$ for which $\Gamma$ is $I(\Gamma)$-adically complete, and $\Gamma/I(\Gamma)^r\in\art_n$ for all $r\geq 1$. It is clear that $\art_n\subseteq \proart_n$.
For a deformation functor $F\colon\art_n\to\Sets$, recall that
* $F$ is called prorepresentable if $F\cong\Hom_{\proart_n}(\Gamma,-)|_{\art_n}$ for some $\Gamma\in\proart_n$.
* $F$ is called representable if $F\cong\Hom_{\art_n}(\Gamma,-)$ for some $\Gamma\in\art_n$.
It is clear that if $F$ is prorepresented by $\Gamma\in\proart_n$, then $F$ is representable if and only if $\dim_{\mathbb{C}}\Gamma<\infty$.
It is well-known that the functor in <ref> is prorepresentable if $\Ext^t_\Lambda(\bigoplus S_i,\bigoplus S_i)$ is finite dimensional for $t=1,2$ <cit.>: we will not use this fact below, however, instead preferring to establish the prorepresenting object in a much more direct way.
§.§ DG Deformations
With our conventions as in the introduction, recall that a DG category is a graded category $\mathsf{A}$ whose morphism spaces are endowed with a differential $d$, i.e. homogeneous maps of degree one satisfying $d^2=0$, such that
\[
\]
for all $g\in\Hom_{\mathsf{A}}(a,b)$ and all $f\in\Hom_{\mathsf{A}}(b,c)_p$ for $p \in \mathbb{Z}$. In this paper we will be interested in the category $\DG_n$, which has as objects those DG categories with precisely $n$ objects. If $\mathsf{A},\mathsf{B}\in\DG_n$, recall that a DG functor $F\colon\mathsf{A}\to\mathsf{B}$ is a graded functor such that $F(df)=d(Ff)$ for all morphisms $f$. A quasi-isomorphism $F\colon\mathsf{A}\to\mathsf{B}$ is a DG functor inducing a bijection on objects, and quasi-isomorphisms $\Hom_{\mathsf{A}}(a_1,a_2)\to\Hom_{\mathsf{B}}(Fa_1,Fa_2)$ for all $a_1,a_2\in\mathsf{A}$. Two categories $\mathsf{A},\mathsf{B}\in\DG_n$ are called quasi-isomorphic if they are connected through a finite, non-directed chain of quasi-isomorphisms.
Suppose that $\mathsf{A}\in\DG_n$, and $(\Gamma,\n)\in\art_n$, and recall from <ref> that $\n_{ij}:=e_i\n\hspace{0.1em}e_j$. Define $\mathsf{A}\uotimes\n:=\bigoplus_{i,j=1}^n(\mathsf{A}\uotimes\n)_{ij}$, where
\[
(\mathsf{A}\uotimes\n)_{ij}:=\Hom_{\mathsf{A}}(i,j)\otimes_{\K} \n_{ij}.
\]
Observe that $\mathsf{A}\uotimes\n$ has the natural structure of an object in $\DG_n$ (but with no units) with differential $d(a\otimes x):=d(a)\otimes x$. Thus we may consider $\mathsf{A}\uotimes\n$ as a DGLA, with bracket
\[
\quad[a\otimes x,b\otimes y]:=ab\otimes xy-(-1)^{\deg(a)\deg(b)}ba\otimes yx
\]
for homogeneous $a,b\in\mathsf{A}$.
Since $(\Gamma,\n)\in\art_n$, by definition $\n^r=0$ for some $r\geq 1$, hence $\mathsf{A}\uotimes\n$ is a nilpotent DGLA. This being the case, we can consider the standard Maurer–Cartan formulation to obtain a deformation functor.
Given $(\mathsf{A},d)\in\DG_n$, the associated DG deformation functor
\[
\Def^{\mathsf{A}}\colon\art_n\to\Sets
\]
is defined by sending
\[
\left. \left \{
\xi\in \mathsf{A}^1\underline{\otimes}\,\n
\left|
\,
\right. \right\} \middle/ \sim \right.
\]
where as usual the equivalence relation $\sim$ is induced by the gauge action. Explicitly, two elements $\xi_1,\xi_2\in \mathsf{A}^1\uotimes\n$ are said to be gauge equivalent if there exists $x\in\mathsf{A}^0\uotimes\n$ such that
\[
\xi_2=e^x*\xi_1:=\xi_1+\sum_{j=0}^{\infty}\frac{([x,-])^j}{(j+1)!}([x,\xi_1]-d(x)).
\]
The following is a mild extension of the well-known $n=1$ case. The proof is very similar to the known $n=1$ proofs (see e.g. <cit.>, <cit.>, <cit.>), so we do not give it here.
Suppose that $\mathsf{A}\to\mathsf{B}$ is a quasi-equivalence in $\DG_n$. Then the deformation functors $\Def^\mathsf{A}$ and $\Def^\mathsf{B}$ are isomorphic.
§.§ Basic Results
Controlling noncommutative deformations of curves in the next section requires the following two preliminary results, and a corollary. All are well-known in the case $n=1$.
To fix notation, suppose that $\cA$ is an abelian category and that $c,d$ are two chain complexes with objects from $\cA$. Set $\Hom^{\DG}_{\cA}(x,y)$ to be the DG $\K$-module with
\[
\Hom^{\DG}_{\cA}(x,y)_t:=\{ (f_i)_{i\in\mathbb{Z}}\mid f_i\colon x_i\to y_{i+t}\}
\]
and differential $\delta\colon f\mapsto fd_y-(-1)^{\deg(f)}d_xf$.
Now choose a family of objects $a_1,\hdots,a_n\in\cA$, an injective resolution $0\to a_i\to I^i_\bullet$ for each $a_i$, and set $I:=\bigoplus_{i=1}^n I^i_\bullet$. From this, we form $(\End_\cA^{\DG}(I),\delta)$, considered naturally as an object of $\DG_n$.
Suppose that $\cA, \cB$ are abelian categories, and $F\colon \cA\to\cB$ is an additive functor with left adjoint $L$. Choose a family of objects $a_1,\hdots,a_n\in\cA$, and for each choose an injective resolution $0\to a_i\to I^i_\bullet$. If
* $L$ is exact,
* $\mathbf{R}^{t}F(a_i)=0$ for all $t>0$ and all $i=1,\hdots,n$,
* The counit $L \circ F \to \Id$ is an isomorphism on each object $a_i$,
then $\End^{\DG}_{\cA}(\bigoplus I^i_\bullet)$ and $\End^{\DG}_{\cB}(\bigoplus FI^i_\bullet)$ are quasi-isomorphic in $\DG_n$.
Consider the obvious map
\begin{eqnarray}
F\colon \End^{\DG}_{\cA}\left(\bigoplus I^i_\bullet\right)\to\End^{\DG}_{\cB}\left(\bigoplus FI^i_\bullet\right).\label{the obvious q}
\end{eqnarray}
Condition (<ref>) implies that $F$ preserves injective objects, and (<ref>) implies that $F$ preserves the injective resolutions of the $a_i$, hence $0\to Fa_i\to FI^i_\bullet$ are injective resolutions. Hence the cohomologies of the left hand side of (<ref>) compute $\Ext^n_{\cA}(a_i,a_j)$, and the right hand side computes $\Ext^n_{\cB}(Fa_i,Fa_j)$. These are obviously the same, via $F$, since by (<ref>)
\[
\begin{tikzpicture}[xscale=1.1]
\node (a0) at (0,0) {$0$};
\node (a1) at (2,0) {$\Hom_{\cA}(a_i,I^j_0)$};
\node (a2) at (5,0) {$\Hom_{\cA}(a_i,I^j_1)$};
\node (a3) at (8,0) {$\Hom_{\cA}(a_i,I^j_2)$};
\node (a4) at (10,0) {$\hdots$};
\node (b0) at (0,-1) {$0$};
\node (b1) at (2,-1) {$\Hom_{\cA}(LFa_i,I^j_0)$};
\node (b2) at (5,-1) {$\Hom_{\cA}(LFa_i,I^j_1)$};
\node (b3) at (8,-1) {$\Hom_{\cA}(LFa_i,I^j_2)$};
\node (b4) at (10,-1) {$\hdots$};
\node (c0) at (0,-2) {$0$};
\node (c1) at (2,-2) {$\Hom_{\cB}(Fa_i,FI^j_0)$};
\node (c2) at (5,-2) {$\Hom_{\cB}(Fa_i,FI^j_1)$};
\node (c3) at (8,-2) {$\Hom_{\cB}(Fa_i,FI^j_2)$};
\node (c4) at (10,-2) {$\hdots$};
\draw[->] (a0)--(a1);
\draw[->] (a1)--(a2);
\draw[->] (a2)--(a3);
\draw[->] (a3)--(a4);
\draw[->] (b0)--(b1);
\draw[->] (b1)--(b2);
\draw[->] (b2)--(b3);
\draw[->] (b3)--(b4);
\draw[->] (c0)--(c1);
\draw[->] (c1)--(c2);
\draw[->] (c2)--(c3);
\draw[->] (c3)--(c4);
\draw[->] (a1)--node[sloped,left,anchor=south]{$\scriptstyle\sim$}(b1);
\draw[->] (a2)--node[sloped,left,anchor=south]{$\scriptstyle\sim$}(b2);
\draw[->] (a3)--node[sloped,left,anchor=south]{$\scriptstyle\sim$}(b3);
\draw[->] (b1)--node[sloped,left,anchor=south]{$\scriptstyle\sim$}(c1);
\draw[->] (b2)--node[sloped,left,anchor=south]{$\scriptstyle\sim$}(c2);
\draw[->] (b3)--node[sloped,left,anchor=south]{$\scriptstyle\sim$}(c3);
\end{tikzpicture}
\]
commutes and all the vertical morphisms are isomorphisms.
Keeping the notation as above, for each of the objects $a_1,\hdots,a_n\in\cA$, choose a left resolution $Q^\bullet_i\to a_i\to 0$, where for now the $Q$'s are arbitrary. Set $Q:=\bigoplus_{i=1}^n Q^\bullet_i$, and consider
\[
\Delta:=
\begin{pmatrix}
\End^{\DG}_{\cA}(Q[1])&\Hom^{\DG}_{\cA}(Q[1],I)\\[1mm]
0& \End^{\DG}_{\cA}(I)
\end{pmatrix}.
\]
This can be viewed as an object in $\DG_n$ in the obvious way: the homomorphism space between object $i$ and object $j$ is
\[
\begin{pmatrix}
\End^{\DG}_{\cA}(Q_i^\bullet[1])&\Hom^{\DG}_{\cA}(Q_i^\bullet[1],I^j_\bullet)\\[1mm]
0& \End^{\DG}_{\cA}(I^j_\bullet)
\end{pmatrix},
\]
with differential as in <cit.>. There are natural projections $p_1\colon\Delta\to\End_{\cA}^{\DG}(Q[1])$ and $p_2\colon\Delta\to\End_{\cA}^{\DG}(I)$ in $\DG_n$, and splicing the left resolutions with the right resolutions gives an exact complex $Q[1]\to I$.
Suppose that $\cA$ is an abelian category, and choose a family of objects $a_1,\hdots,a_n\in\cA$. With notation as above,
* The projection $p_2$ is a quasi-isomorphism in $\DG_n$.
* If $\Hom^{\DG}_\cA(Q[1],Q[1]\to I)$ is exact, then $p_1$ is a quasi-isomorphism in $\DG_n$.
In particular, provided that $\Hom^{\DG}_\cA(Q[1],Q[1]\to I)$ is exact, $\End_{\cA}^{\DG}(Q)$ and $\End_{\cA}^{\DG}(I)$ are quasi-isomorphic in $\DG_n$.
As above, the complex $Q[1]\to I$ is exact.
(1) By construction of the upper triangular $\Delta$, as in <cit.>
\[
\Ker p_2=\Hom^{\DG}_\cA(I,Q[1]\to I).
\]
Since $I$ is $h$-injective, it follows that $\Ker p_2$ is exact, and thus $p_2$ is a quasi-isomorphism.
(2) Again by construction, $\Ker p_1=\Hom^{\DG}_\cA(Q[1],Q[1]\to I)$, and so if by assumption this is exact, $p_1$ is a quasi-isomorphism.
The final statement follows from (1) and (2), since clearly $\End^{\DG}_\cA(Q)\cong\End^{\DG}_\cA(Q[1])$.
The following is now a direct consequence of <cit.>, and says that the naive deformations and the DG deformations are the same when we deform distinct simples.
Suppose that $\Lambda$ is a $\K$-algebra, and that $\cS=\{S_1,\hdots,S_n\}\subseteq\Mod\Lambda$ are simple and distinct. Choose injective resolutions $0\to S_i\to I^i_\bullet$ and set $\mathsf{A}:=\End^{\DG}_{\Lambda}(I)\in\DG_n$ as above. Then $ \Def^{\mathsf{A}}\cong \Def^{\cS}_{\Lambda}$.
Under the assumption that the $S_i$ are distinct simples, Segal <cit.> shows that $\Def^{\cS}_{\Lambda}$ is isomorphic to the DG deformation functor associated to the bar resolutions of the simples. Since projective resolutions are $h$-projective, the conditions of <ref>(<ref>) are satisfied, so the bar resolution DGA is quasi-isomorphic in $\DG_n$ to $\mathsf{A}$. The result then follows from <ref>.
§ DEFORMATIONS OF REDUCED FIBRES
In this section, in the setting of contractions with at most one-dimensional fibres, we show that the functor of simultaneous noncommutative deformations of the reduced fibre is prorepresented by a naturally defined algebra $\CA$. This algebra is a factor of one obtained by tilting, and this extra control over the prorepresenting object allows us in <ref> to prove that noncommutative deformations recover the contracted locus.
§.§ Setup
This subsection fixes notation. Throughout the paper, we will refer to the three setups in <ref>, <ref> and <ref> below.
(Global) Suppose that $f\colon X\to X_{\con}$ is a projective birational morphism between noetherian integral normal $\mathbb{C}$-schemes, with $\Rf_*\cO_X=\cO_{X_{\con}}$, such that the fibres are at most one-dimensional. Throughout, we write $L$ for the locus of (not necessarily closed) points of $X_{\con}$ above which $f$ is not an isomorphism.
We make no assumptions on the singularities of $X$. Next, for any closed point $p\in L$, we pick an affine neighbourhood $\Spec R$ in $X_{\con}$ containing $p$, and after base change consider the following Zariski local setup.
(Zariski local) Suppose that $f\colon U\to\Spec R$ is a projective birational morphism between noetherian integral normal $\mathbb{C}$-schemes, with $\Rf_*\cO_U=\cO_R$, such that the fibres are at most one-dimensional.
In dimension $3$, an easy example is the blowup of $\mathbb{A}^3$ at the ideal $(x,y)$, but the setup also includes arbitrary flips and flops of multiple curves, as well as divisorial contractions to curves. We make no assumptions on the singularities of $U$.
With the assumptions in <ref>, it is well-known <cit.> that there is a bundle $ \cV:=\cO_U\oplus\cN$ inducing a derived equivalence
\begin{eqnarray}
\begin{array}{c}
\begin{tikzpicture}
\node (a1) at (0,0) {$\phantom{.}\Db(\coh U)$};
\node (a2) at (5,0) {$\Db(\mod \End_U(\cV)).$};
\draw[->] (a1) -- node[above] {$\scriptstyle\RHom_U(\cV,-)$} node [below] {$\scriptstyle\sim$} (a2);
\end{tikzpicture}
\end{array}\label{derived equivalence}
\end{eqnarray}
Throughout we set
\[
\Lambda:=\End_U(\cV) = \End_U(\cO_U\oplus\cN),
\]
and recall from the conventions in <ref> that if $\cF,\cG\in\coh U$ where $\cF$ is a summand of $\cG$, then we define the ideal $[\cF]$ to be the two-sided ideal of $\End_U(\cG)$ consisting of all morphisms factoring through $\add \cF$.
The following is similar to <cit.>, but the definition is now more subtle since in general $\End_{U}(\cV)\ncong \End_R(f_*\cV)$, whereas there is such an isomorphism in the setting of <cit.>. The upshot is that we must work on $U$, and not $\Spec R$.
With notation as above, we define the contraction algebra associated to $\Lambda$ to be $\Lambda_{\con}:=\End_U(\cO_U\oplus\cN)/[\cO_U]$.
The algebra $\Lambda_{\con}$ defined above depends on $\Lambda$ and thus the choice of derived equivalence (<ref>), but this is accounted for in the formal fibre setting below, after passing through morita equivalences. Also, we remark that since $\cN\notin\add \cO_U$ (else $f$ is an isomorphism, e.g. by <ref>), the contraction algebra $\Lambda_{\con}$ is necessarily non-zero.
To obtain well-defined invariants that do not depend on choices, and also to relate to the deformation theory in the following <ref>, we now pass to the formal fibre.
(Complete local) Suppose that $f\colon \mathfrak{U}\to\Spec \mathfrak{R}$ is a projective birational morphism between noetherian integral normal $\mathbb{C}$-schemes, with $\Rf_*\cO_{\mathfrak{U}}=\cO_{\mathfrak{R}}$, where $\mathfrak{R}$ is complete local and the fibres of $f$ are at most one-dimensional.
After passing to this formal fibre, the Zariski local derived equivalence has a particularly nice form, which we now briefly review. Fix a closed point $\m \in L$, and write $\mathfrak{R}:=\widehat{R}$. The above derived equivalence (<ref>) induces an equivalence
\[
\begin{array}{c}
\begin{tikzpicture}
\node (a1) at (0,0) {$\Db(\coh\mathfrak{U})$};
\node (a2) at (4,0) {$ \Db(\mod\widehat{\Lambda}),$};
\draw[->] (a1) -- node[above] {$\scriptstyle\RHom_{\mathfrak{U}}(\widehat{\cV},-)$} node [below] {$\scriptstyle\sim$} (a2);
\end{tikzpicture}
\end{array}
\]
and this can be described much more explicitly. We let $C=\pi^{-1}(\m)$ where $\m$ is the unique closed point of $\Spec \mathfrak{R}$, then giving $C$ the reduced scheme structure, we can write $C^{\redu}=\bigcup _{i=1}^nC_i$ with each $C_i\cong\mathbb{P}^1$. Let $\cL_i$ denote the line bundle on $\mathfrak{U}$ such that $\cL_i\cdot C_j=\delta_{ij}$. If the multiplicity of $C_i$ is equal to one, set $\cM_i:=\cL_i$, else define $\cM_i$ to be given by the maximal extension
\[
0\to\cO_{\mathfrak{U}}^{\oplus(r-1)}\to\cM_i\to\cL_i\to 0
\]
associated to a minimal set of $r-1$ generators of $H^1(\mathfrak{U},\cL_i^{*})$ as an $\mathfrak{R}$-module <cit.>. Then $\cO_{\mathfrak{U}}\oplus \bigoplus_{i=1}^n\cM_i^*$
is a tilting bundle on $\mathfrak{U}$ <cit.> generating $\Per(\mathfrak{U})$. By <cit.> we can write
\[
\cO_{\mathfrak{U}}\oplus\widehat{\cN}\cong \cO_{\mathfrak{U}}^{\oplus a_0}\oplus \bigoplus_{i=1}^n (\cM_i^*)^{\oplus a_i}
\]
for some $a_i\in\mathbb{N}$ and so consequently $\widehat{\Lambda}\cong\End_{\mathfrak{U}}(\cO_{\mathfrak{U}}^{\oplus a_0}\oplus \bigoplus_{i=1}^n (\cM_i^*)^{\oplus a_i})$.
We write $\cM^*:=\bigoplus_{i=1}^n \cM_i^*$ and define
\[
\AB:=\End_{\mathfrak{U}}(\cO_{\mathfrak{U}}\oplus \cM^*),
\]
which is the basic algebra morita equivalent to $\widehat{\Lambda}$. From this, we define the contraction algebra associated to $\clocCon$ to be
\[
\CA:=\End_{\mathfrak{U}}(\cO_{\mathfrak{U}}\oplus \cM^*)/[\cO_{\mathfrak{U}}]\cong \End_{\mathfrak{U}}(\cM^*)/[\cO_{\mathfrak{U}}],
\]
and we define the fibre algebra associated to $\clocCon$ to be
\[
\AB_{\fib}:=\End_{\mathfrak{U}}(\cO_{\mathfrak{U}}\oplus \cM^*)/[\cM^*]\cong \End_{\mathfrak{U}}(\cO_{\mathfrak{U}})/[\cM^*].
\]
Since $\End_{\mathfrak{U}}(\cO_{\mathfrak{U}})\cong\mathfrak{R}$, it follows from the definition that $\AB_{\fib}$ is always commutative, although $\CA$ need not be.
To establish various homological properties we will need to pass through morita equivalences between the algebras $\AB$ and $\widehat{\Lambda}$, and between the algebras $\CA$ and $\widehat{\Lambda}_{\con}$. Here we describe these equivalences, mainly to fix notation. In analogy with <cit.> throughout we write
\[
\cY:=\cO_{\mathfrak{U}}\oplus \bigoplus \cM_i^*, \qquad \cZ:=\cO_{\mathfrak{U}}^{\oplus a_0}\oplus \bigoplus (\cM_i^*)^{\oplus a_i},
\]
so that $\AB=\End_{\mathfrak{U}}(\cY)$ and $\widehat{\Lambda}=\End_{\mathfrak{U}}(\cZ)$. Writing
\[
P:=\Hom_{\mathfrak{U}}(\cY,\cZ), \qquad Q:=\Hom_{\mathfrak{U}}(\cZ,\cY)
\]
it is clear that both $P$ and $Q$ have the structure of bimodules, namely ${}_{\widehat{\Lambda}}P_{\InSpaceOf{\widehat{\Lambda}}{\AB}}$ and ${}_{\InSpaceOf{\widehat{\Lambda}}{\AB}}Q_{\widehat{\Lambda}}$. It is easy to see that $P$ is a progenerator, and that this induces the following result.
With notation as above, there is a morita equivalence
\begin{eqnarray}
\begin{array}{c}
\begin{tikzpicture}[xscale=1]
\node (d1) at (3,0) {$\mod \AB$};
\node (e1) at (7.5,0) {${}_{}\mod\widehat{\Lambda},$};
%equiv and functors
\draw[->,transform canvas={yshift=+0.4ex}] (d1) to node[above] {$\scriptstyle \mathbb{F}:=\Hom_{\AB}(P,-)=-\otimes_{\AB}Q $} (e1);
\draw[<-,transform canvas={yshift=-0.4ex}] (d1) to node [below] {$\scriptstyle \Hom_{\widehat{\Lambda}}(Q,-)=-\otimes_{\widehat{\Lambda}}P $} (e1);
\end{tikzpicture}
\end{array}\label{ME1}
\end{eqnarray}
and a morita equivalence
\begin{eqnarray}
\begin{array}{c}
\begin{tikzpicture}[xscale=1]
\node (d1) at (3,0) {$\mod \CA$};
\node (e1) at (7.5,0) {${}_{}\mod\widehat{\Lambda}_{\con}.$};
%equiv and functors
\draw[->,transform canvas={yshift=+0.4ex}] (d1) to node[above] {$\scriptstyle -\otimes_{\CA}\mathbb{F}\CA$} (e1);
\draw[<-,transform canvas={yshift=-0.4ex}] (d1) to node [below] {$\scriptstyle \Hom_{\widehat{\Lambda}_{\con}}(\mathbb{F}\CA,-) $} (e1);
\end{tikzpicture}
\end{array}\label{ME2}
\end{eqnarray}
§.§ Deformations and Contraction Algebras
In this subsection we will prove that the contraction algebra $\CA$ prorepresents various natural deformation functors. We will translate deformation problems across a variety of different functors, so we now set notation, as in <cit.>.
We pick a closed point $\m\in L$, and as above consider $C:=f^{-1}(\m)$. We write $C^{\redu}=\bigcup _{i=1}^nC_i$ with each $C_i\cong\mathbb{P}^1$, and put $T_i$ for the simple $\Lambda$-modules corresponding to the (perverse) sheaves $\cO_{C_i}(-1)$ across the derived equivalence (<ref>). Further, we denote the simple $\AB$-modules by $S_i:=\mathbb{F}^{-1}\widehat{T}_i$, so that
\[
\begin{array}{c}
\begin{tikzpicture}[xscale=1]
\node (d1) at (2,0.8) {$\Db(\Qcoh U)$};
\node (e1) at (6,0.8) {$\Db(\Mod \Lambda)$};
\node (g1) at (10,0.8) {$\Db(\Mod \AB)$};
%equiv and functors
\draw[->] (d1.5) to node[above] {$\scriptstyle\RHom_U(\cV,-)$} (e1.175);
\draw[<-] (d1.-5) to node [below] {$\scriptstyle-\otimes_{\Lambda}^{\bf L}\cV$} (e1.-175);
\draw[->] (e1.5) to node[above] {$\scriptstyle \mathbb{F}^{-1}\circ\widehat{(-)}$} (g1.175);
\draw[<-] (e1.-5) to node [below] {$\scriptstyle \rest\circ\,\mathbb{F}$} (g1.-175);
\node (d2) at (2,0) {$\cO_{C_i}(-1)$};
\draw[<->] (d1.east |- 3.5,0) -- (e1.west |- 5.5,0);
\node (e2) at (6,0) {$T_i$};
\draw[->] (e1.east |- 3.5,0) -- (g1.west |- 5.5,0);
\node (g2) at (10,0) {$S_i$};
\end{tikzpicture}
\end{array}
\]
For the remainder of this subsection, we will use the following simplified notation:
* $\Def_X$ for $\Def^{\mathsf{A}_1}$, where $\mathsf{A}_1\in\DG_n$ is obtained from the injective resolutions of the $\cO_{C_i}(-1)\in\coh X$.
* $\Def_U$ for $\Def^{\mathsf{A}_2}$, where $\mathsf{A}_2\in\DG_n$ is obtained from the injective resolutions of the $\cO_{C_i}(-1)\in\coh U$.
* $\Def_\Lambda$ for $\Def^{\mathsf{A}_3}$, where $\mathsf{A}_3\in\DG_n$ is obtained from the injective resolutions of the $T_i\in\mod\Lambda$.
* $\Def_\AB$ for $\Def^{\mathsf{A}_4}$, where $\mathsf{A}_4\in\DG_n$ is obtained from the injective resolutions of the $S_i\in\mod\AB$.
The next result is now an easy corollary of the DG results in <ref>, vastly simplifying <cit.>.
With the global setup of <ref>, and notation in <ref>,
* $\Def_X\cong \Def_U$.
* $\Def_U\cong \Def_\Lambda$.
* $\Def_\Lambda\cong\Def_{\AB}$.
* $\Def_{\AB}\cong \Hom_{\proart_n}(\CA,-)$.
In particular, all the deformation functors above are prorepresented by $\CA$.
By <ref>, we just need to show that the DGAs are quasi-isomorphic.
(1) Consider $F:=i_*\colon \Qcoh U\to\Qcoh X$, with exact left adjoint $L=i^*$. Since $C_i$ is closed, $\RDerived i_* \cO_{C_i}(-1)=i_*\cO_{C_i}(-1)$ for each $i = 1,\hdots,n.$ Further, the counit $L \circ F \to \Id$ is an isomorphism for all sheaves, in particular for the sheaves $\cO_{C_i}(-1)$. Hence the result follows from <ref>.
(2) Set $F:=\Hom_U(\cV,-)\colon \Qcoh U\to \Mod\Lambda$, with (non-exact) left adjoint $L:=\cV\otimes_\Lambda-$. To establish the claim we will first use <ref>, so set $E_i:=\cO_{C_i}(-1)$, and for each $i=1,\hdots,n$ choose an injective resolution $0\to E_i\to I_\bullet^i$. On the other hand, choose a projective resolution $P_i^\bullet\to T_i\to 0$ of each of the $T_i$. Since the sheaf $E_i$ corresponds to the module $T_i$ across the equivalence (<ref>), it follows that $\cV\otimes^{\bf L}_\Lambda T_i\cong E_i$. In particular all higher cohomology groups vanish, and so $LP_i^\bullet\to E_i\to 0$ is exact, giving a locally free resolution of $E_i$.
To match the notation of <ref>, set $Q^\bullet_i:=LP_i^\bullet$, and write $P=\bigoplus_{i=1}^nP_i^\bullet$, $Q=\bigoplus_{i=1}^nQ_i^\bullet$, and $I=\bigoplus_{i=1}^nI^i_\bullet$. By adjunction
\begin{eqnarray}
\Hom^{\DG}_U(LP[1],LP[1]\to I)\cong\Hom^{\DG}_\Lambda(P[1],P[1]\to FI).\label{to show exact}
\end{eqnarray}
Since $P$ is $h$-projective, and $P[1]\to FI$ is acyclic, (<ref>) is exact. By <ref>, this then establishes that $\End^{\DG}_U(I)$ and $\End^{\DG}_U(Q)$ are quasi-isomorphic.
Now it is clear that
\[
\End^{\DG}_\Lambda(P)\to \End^{\DG}_U(LP)=\End^{\DG}_U(Q)
\]
is a quasi-isomorphism, and so combining we see that $\End^{\DG}_\Lambda(P)$ is quasi-isomorphic to $\End^{\DG}_U(I)$. Finally, choose an injective resolution $0\to T_i\to J^i_\bullet$ of each $T_i$, and set $J:=\bigoplus_{i=1}^n J^i_\bullet$. It is well known (again using <ref>), that $\End^{\DG}_\Lambda(J)$ is quasi-isomorphic to $\End^{\DG}_\Lambda(P)$, and thus by the above to $\End^{\DG}_U(I)$.
(3) Consider $F:=\mathbb{F}^{-1}\circ\widehat{(-)}\colon\Mod\Lambda\to\Mod\AB$, which has exact left adjoint $L=\rest\circ\,\mathbb{F}$. Being the composition of exact functors, $F$ is also exact, so $\mathbf{R}^{t}F(S_i)=0$ for all $t>0$ and for all $i=1,\hdots,n$. Since $\mathbb{F}$ is an equivalence, and since the adjunction $\rest \dashv \widehat{(-)}$ restricts to an equivalence between finite length $\Lambda$-modules supported at $\m$ and finite length $\widehat{\Lambda}$-modules, it follows that the counit $L \circ F \to \Id$ is an isomorphism on the objects $S_i$. Hence the statement follows from <ref>.
(4) By <ref>, the DG deformation functor $\Def_{\AB}$ is isomorphic to the naive deformation functor in <ref>. Thus, since each $S_i$ is one-dimensional, using the standard argument (see e.g. <cit.>) it follows that
\begin{align*}
\Def_{\AB}(\Gamma)
&=\left. \left \{ \begin{array}{cl} \bullet & \mbox{A left $\AB$-module structure on $M=(S_i\otimes_{\mathbb{C}}\Gamma_{ij})$ such that}\\
&\mbox{$(S_i\otimes_{\mathbb{C}}\Gamma_{ij})$ becomes a $\AB$-$\Gamma$ bimodule.}\\
\bullet &\mbox{A collection of isomorphisms }\delta_i\colon M\otimes_\Gamma (\Gamma /\m)e_i\xrightarrow{\sim} S_i
\end{array} \right\} \middle/ \sim \right. \\
&=\left. \left \{ \begin{array}{cl} \bullet & \mbox{A left $\AB$-module structure on $\Gamma$ such that $\Gamma\in\Mod\AB\otimes\Gamma^{\op}$.}\\
\bullet &\mbox{A collection of isomorphisms }\delta_i\colon \Gamma\otimes_\Gamma (\Gamma /\m)e_i\xrightarrow{\sim} S_i
\end{array} \right\} \middle/ \sim \right. \\
\end{align*}
It remains to show that $\CA\in\proart_n$, but this holds since $\mathfrak{R}$, thus $\AB$, and thus $\CA$, are complete with respect to their augmentation ideals.
The above proof of <ref>(<ref>) establishes that we can compute $\Def_U$ using the DGA associated to the specific locally free resolutions $\cV\otimes_\Lambda P_i^\bullet$ of the $E_i$. It is rare to be able to compute the deformations of a sheaf using the DGA of a locally free resolution, and essentially it is this extra control of the deformation theory that allows us to prove the contraction theorem in <ref> below. Note that even taking the DGA of a different locally free resolution of the $E_i$ may give a different deformation functor.
§ THE CONTRACTION THEOREM
In the global setup of <ref>, by <ref> it follows that simultaneous noncommutative deformations of the reduced fibre is prorepresented by $\CA$, namely
\[
\Def_X\cong\Hom_{\proart_n}(\CA,-).
\]
Further, by <ref>, $\CA$ is morita equivalent to $\widehat{\Lambda}_{\con}$. We are thus motivated to control $\CA$ (and $\Lambda_{\con}$), and we now do this in a sequence of reduction steps, culminating in the contraction theorem of <ref>.
§.§ Behaviour of $\Lambda_{\con}$ under Global Sections and Base Change
We revert to the Zariski local setup of <ref>. The following lemma is important: it is very well-known if $f$ is an isomorphism in codimension two <cit.>, however in our more general setting care is required.
With the setup as in <ref>,
$\End_{U}(\cV^*)\cong \End_R(f_*(\cV^*))$ so
\[
\Lambda\cong\End_R(R\oplus f_*(\cN^*))^{\op}\quad\mbox{and}\quad \Lambda_{\con}\cong\left(\End_R(R\oplus f_*(\cN^*))/[f_*(\cN^*)]\right)^{\op}.
\]
This allows us to reduce many problems to the base $\Spec R$.
Indeed $\End_{U}(\cV)\ncong \End_R(f_*\cV)$ in general, so the statement and proof of <ref> is a little subtle. Since $\cV^*$ is generated by global sections and is tilting, the proof of <ref> follows immediately from <ref> below. This requires the following easy well-known lemma.
Assume that $f\colon Y\to \Spec S$ is a proper birational map between integral schemes, where $S$ is normal.
* Suppose that $\cF\in\coh Y$ is generated by global sections. Then we can find a morphism $\cO_Y^{\oplus a}\twoheadrightarrow \cF$ such that $S^{\oplus a}\twoheadrightarrow f_*\cF$.
* Suppose that $\cW_1,\cW_2$ are finite rank vector bundles on $Y$. Then
\[\Hom_Y(\cW_1,\cW_2)\hookrightarrow\Hom_S(f_*\cW_1,f_*\cW_2).\]
(1) This is a basic consequence of Zariski's main theorem, together with the fact that $f_*$ preserves coherence.
(2) Since $\Hom_Y(\cW_1,\cW_2)=H^0(\cW_1^*\otimes\cW_2)$ and $\cW_1^*\otimes\cW_2$ is a vector bundle, by integrality $\Hom_Y(\cW_1,\cW_2)$ is a torsion-free $S$-module. The statement follows.
Assume that $f\colon Y\to \Spec S$ is a proper birational map between integral schemes, where $S$ is normal. If $\cW$ is a vector bundle on $Y$ of finite rank, which is generated by global sections, such that $\Ext_Y^1(\cW,\cW)=0$, then $\End_Y(\cW)\cong\End_S(f_*\cW)$.
By <ref>(<ref>) there is a short exact sequence
\begin{eqnarray}
0\to\cK\to\cO_Y^{\oplus a}\to\cW\to 0\label{up=down1}
\end{eqnarray}
such that
\begin{eqnarray}
0\to f_*\cK\to S^{\oplus a}\to f_*\cW\to 0\label{up=down2}
\end{eqnarray}
is exact. Applying $\Hom_Y(-,\cW)$ to (<ref>) and applying $\Hom_S(-,f_*\cW)$ to (<ref>), we have an exact commutative diagram
\[
\begin{tikzpicture}[scale=1,node distance=1]
\node (A0) at (-2.25,0) {$0$};
\node (A1) at (0,0) {$\Hom_{Y}(\cW,\cW)$};
\node (A2) at (3.5,0) {$\Hom_{Y}(\cO_Y^{\oplus a},\cW)$};
\node (A3) at (7,0) {$\Hom_{Y}(\cK,\cW)$};
\node (A4) at (10,0) {$\Ext^1_{Y}(\cW,\cW)=0$};
\node (B0) at (-2.25,-1.5) {$0$};
\node (B1) at (0,-1.5) {$\Hom_{S}(f_*\cW,f_*\cW)$};
\node (B2) at (3.5,-1.5) {$\Hom_{S}(S^{\oplus a},f_*\cW)$};
\node (B3) at (7,-1.5) {$\Hom_{S}(f_*\cK,f_*\cW)$};
\draw[->] (A0) -- (A1);
\draw[->] (A1) -- (A2);
\draw[->] (A2) -- (A3);
\draw[->] (A3) -- (A4);
\draw[->] (B0) -- (B1);
\draw[->] (B1) -- (B2);
\draw[->] (B2) -- (B3);
\draw[->] (A1) -- node[right] {$\scriptstyle \alpha$} (B1);
\draw[-,transform canvas={xshift=+0.15ex}] (B2) -- (A2);
\draw[-,transform canvas={xshift=-0.15ex}] (B2) -- (A2);
\draw[->] (A3) -- node[right] {$\scriptstyle \beta$} (B3);
\end{tikzpicture}
\]
where both $\alpha$ and $\beta$ are injective by <ref>(<ref>). By the snake lemma, $\alpha$ is also surjective.
With the setup in <ref>, both $\Lambda$ and $\Lambda_{\con}$ have the structure of an $R$-module. For our purposes later, we need to control this under flat base change. In what follows, for $\p\in\Spec R$ we consider the base change squares
\[
\begin{array}{c}
\begin{tikzpicture}[yscale=1.25]
\node (Xpc) at (-3,0) {$\mathfrak{U}_\p$};
\node (Xp) at (-1,0) {$U_\p$};
\node (X) at (1,0) {$U$};
\node (Rpc) at (-3,-1) {$\Spec \mathfrak{R}_\p$};
\node (Rp) at (-1,-1) {$\Spec R_\p$};
\node (R) at (1,-1) {$\Spec R$};
\draw[->] (Xpc) to node[above] {$\scriptstyle m$} (Xp);
\draw[->] (Xp) to node[above] {$\scriptstyle k$} (X);
\draw[->] (Rpc) to node[above] {$\scriptstyle l$} (Rp);
\draw[->] (Rp) to node[above] {$\scriptstyle j$} (R);
\draw[->] (Xpc) -- node[left] {$\scriptstyle \theta$} (Rpc);
\draw[->] (Xp) -- node[left] {$\scriptstyle \varphi$} (Rp);
\draw[->] (X) -- node[right] {$\scriptstyle f$} (R);
\end{tikzpicture}
\end{array}
\]
where $\mathfrak{R}_\p$ denotes the completion of $R_\p$ at its unique maximal ideal.
With the setup as in <ref>,
* $U_\p$ is derived equivalent to $\Lambda_\p$ via the tilting bundle $k^*\cV=\cO_{U_\p}\oplus k^*\cN$.
* $\mathfrak{U}_\p$ is derived equivalent to $\widehat{\Lambda_\p}$ via the tilting bundle $m^*k^*\cV=\cO_{\mathfrak{U}_\p}\oplus m^*k^*\cN$.
* $(\Lambda_{\con})_\p\cong (\Lambda_\p)_{\con}$. $\widehat{\Lambda}$
* $(\Lambda_{\con})_\p\otimes_{R_\p}\mathfrak{R}_\p\cong (\widehat{\Lambda_\p})_{\con}$.
All statements are elementary, but we give the proof for completeness.
(1) $U$ is derived equivalent to $\Lambda$ via the tilting bundle $\cV:=\cO_U\oplus\cN$. It is well-known that this implies $U_\p$ is derived equivalent to $\Lambda_\p$ via the tilting bundle $k^*\cV=\cO_{U_\p}\oplus k^*\cN$. For example, a proof of the Ext vanishing together with the fact that $\End_{U_\p}(k^*\cV)\cong \Lambda_\p$ can be found in <cit.>. For generation, first observe that $j$ is an affine morphism, hence so is $k$. Then $\RHom_{U_\p}(k^*\cV,x)=0$ implies, by adjunction, that $\RHom_{U}(\cV,k_*x)=0$. Since $\cV$ generates, $k_*x=0$ and so since $k$ is affine $x=0$.
(2) The proof is identical to (1).
(3) Since $f$ is proper, the category $\coh U$ is $R$-linear. In particular $\Hom_U(\cO_U,\cV)$ is a finitely generated $\End_U(\cO_U)$-module, so we can find a surjection
\begin{eqnarray}
\Hom_U(\cO_U,\cO_U)^{\oplus a}\to \Hom_U(\cO_U,\cV)\to 0.\label{geom approx O 1}
\end{eqnarray}
Tracking the images of the identities on the left hand side under the above map gives elements $g_1\hdots,g_a\in\Hom_U(\cO_U,\cV)$, and so we may use these to form a natural morphism
\begin{eqnarray}
\cO_U^{\oplus a}\xrightarrow{h} \cV\label{geom approx O 2}
\end{eqnarray}
such that applying $\Hom_U(\cO_U,-)$ to (<ref>) gives (<ref>). By definition, this means that $h$ is an $\add\cO_U$-approximation of $\cV$. Hence applying $\Hom_U(\cV,-)$ to (<ref>) yields an exact sequence
\begin{eqnarray}
\Hom_U(\cV,\cO_U)^{\oplus a}\to \Hom_U(\cV,\cV)\to \Lambda_\con\to 0.\label{geom approx O 3}
\end{eqnarray}
Interpreting $\Hom_U(-,-)=f_*\sHom(-,-)$, applying the exact functor $j^*(-)=(-)_\p$ to (<ref>) and using flat base change gives the exact sequence
\[
\varphi_*k^*\sHom_{U}(\cV,\cO_U)^{\oplus a}\to \varphi_*k^*\sHom_U(\cV,\cV)\to (\Lambda_\con)_\p\to 0.
\]
Since $\cV$ is coherent we may move the $k^*$ inside $\sHom$, and so the above is simply
\begin{eqnarray}
\Hom_{U_\p}(k^*\cV,\cO_{U_\p})^{\oplus a}\to \Hom_{U_\p}(k^*\cV,k^*\cV)\to (\Lambda_\con)_\p\to 0.\label{geom approx O 5}
\end{eqnarray}
But on the other hand applying the exact functor $k^*$ to (<ref>) gives a morphism
\begin{eqnarray}
\cO_{U_\p}^{\oplus a}\xrightarrow{k^*(h)} k^*\cV\label{geom approx O 6}
\end{eqnarray}
and further applying $j^*$ to (<ref>) and using flat base change shows that
\[
\Hom_{U_\p}(\cO_{U_\p},\cO_{U_\p})^{\oplus a}\to \Hom_{U_\p}(\cO_{U_\p},k^*\cV)\to 0
\]
is exact. Hence $k^*(h)$ is an $\add\cO_{U_\p}$-approximation, and so applying $\Hom_{U_\p}
(k^*\cV,-)$ to (<ref>) gives an exact sequence
\begin{eqnarray}
\Hom_{U_\p}(k^*\cV,\cO_{U_\p})^{\oplus a}\to \Hom_{U_\p}(k^*\cV,k^*\cV)\to (\Lambda_\p)_\con\to 0.\label{geom approx O 8}
\end{eqnarray}
Combining (<ref>) and (<ref>) shows that $(\Lambda_\p)_\con\cong (\Lambda_\con)_\p$, as required.
(4) The proof is identical to (3).
§.§ The Contraction Theorem
In this subsection, so as to be able to work globally in future papers, we first relate the support of $\Lambda_{\con}$ to the locus $L$. We then control the support of $\CA$ to deduce the deformation theory corollaries.
We need the following fact, which is well-known.
Suppose that $f\colon Y\to Z$ is a morphism of noetherian schemes which is not an isomorphism. Then $\RDerived f_*\colon \D(\Qcoh Y)\to\D(\Qcoh Z)$ is not an equivalence.
If $\RDerived f_*$ is an equivalence then its inverse is necessarily given by its adjoint $\LDerived f^*$. Since noetherian schemes are quasi-compact and quasi-separated, both unbounded derived categories are compactly generated triangulated categories (with compact objects the perfect complexes), and so the above equivalence restricts to an equivalence
\[
\Perf(Y)\xleftarrow{\sim}\Perf(Z)\colon \LDerived f^*.
\]
By Balmer <cit.> it follows that $f$ is an isomorphism, which is a contradiction.
Leading up to the next lemma, choose a closed point $\m\in L$, and pick an affine open $\Spec R$ containing $\m$. For any $\p\in\Spec R$ we base change to obtain the following diagram.
\[
\begin{array}{c}
\begin{tikzpicture}[yscale=1.25]
\node (Up) at (-1,0) {$U_\p$};
\node (U) at (1,0) {$U$};
\node (Rp) at (-1,-1) {$\Spec R_\p$};
\node (R) at (1,-1) {$\Spec R$};
\draw[->] (Up) to node[above] {$\scriptstyle k$} (U);
\draw[->] (Rp) to node[above] {$\scriptstyle j$} (R);
\draw[->] (Up) -- node[left] {$\scriptstyle \varphi_\p$} (Rp);
\draw[->] (U) -- node[right] {$\scriptstyle f$} (R);
\end{tikzpicture}
\end{array}
\]
Note that by flat base change the morphism $\varphi_\p$ is still projective birational, and satisfies $\RDerived{\varphi_\p}_*\cO_{U_\p}=\cO_{R_\p}$.
With the global setup of <ref>, and notation as above,
\[
\varphi_\p\mbox{ is not an isomorphism}\iff \p\in\Supp_R\Lambda_{\con}.
\]
As in <cit.>, it is easy to see that the diagram
\begin{equation}
\begin{array}{c}
\begin{tikzpicture}
\node (roof) at (0,0) {$\D(\Qcoh U_\p)$};
\node (U) at (7,0) {$\D(\Mod\Lambda_\p)$};
\node (U') at (0,-1.5) {$\D(\Qcoh \Spec R_\p)$};
\node (base) at (7,-1.5) {$\D(\Mod R_\p)$};
\draw[->] (roof) -- node[above] {$\scriptstyle \Psi:=\RHom_{U_\p}(\cO_{U_\p}\oplus k^*\cN,-)$} node[below] {$\scriptstyle \sim$} (U);
\draw[->] (roof) -- node[left] {$\scriptstyle \RDerived{\varphi_\p}_*$} (U');
\draw[->] (U) -- node[right] {$\scriptstyle e(-)$} (base);
\draw[-,transform canvas={yshift=+0.15ex}] (U') -- (base);
\draw[-,transform canvas={yshift=-0.15ex}] (U') -- (base);
\end{tikzpicture}
\end{array}\label{comm flat Db}
\end{equation}
commutes, where the top functor is an equivalence by <ref>, and by abuse of notation $e$ also denotes the idempotent in $\Lambda_\p$ corresponding to $\cO_{U_\p}$.
($\Rightarrow$) To ease notation, we drop $\p$ and write $\varphi$ for $\varphi_\p$. Since $\RDerived\varphi_*$ is not an equivalence by <ref>, we may find some $x\in\D(\Qcoh U_\p)$ such that the counit
\[
\LDerived\varphi^*\RDerived\varphi_*(x)\xrightarrow{\varepsilon_x} x
\]
is not an isomorphism, and so the object $c := \operatorname{Cone}(\varepsilon_x)$ is non-zero. Now we argue that $\RDerived\varphi_*(c)=0$, which is equivalent to the morphism $\RDerived\varphi_*(\varepsilon_x)$ being an isomorphism. But
\[
\RDerived\varphi_*(x)\xrightarrow{\eta_{\RDerived\varphi_*(x)}} \RDerived\varphi_*\LDerived\varphi^*\RDerived\varphi_*(x)\xrightarrow{\RDerived\varphi_*(\varepsilon_x)} \RDerived\varphi_*(x)
\]
gives the identity map by a triangular identity, and $\eta_{\RDerived\varphi_*(x)}$ is an isomorphism since it is well known that $\eta\colon \Id\to\RDerived\varphi_*\LDerived\varphi^*$ is a functorial isomorphism by the projection formula. Thus indeed $c$ is a non-zero object such that $\RDerived\varphi_*(c)=0$.
Now across the top equivalence in (<ref>), since $c\neq0$ it follows that $\Psi(c)\neq0$ and so $H^j(\Psi(c))\neq 0$ for some $j$. Further since the diagram (<ref>) commutes, $e\Psi(c)=0$, so since $e(-)$ is exact we deduce that $eH^i(\Psi(c))=0$ for all $i$. In particular there exists a non-zero $\Lambda_\p$-module $M:=H^j(\Psi(c))$ such that $eM=0$. It follows that $M$ is a non-zero module for $(\Lambda_\p)_\con$, hence necessarily $(\Lambda_\p)_{\con}\neq 0$. But by <ref> we have $(\Lambda_\p)_{\con}\cong (\Lambda_{\con})_{\p}$, hence $\p\in\Supp_R\Lambda_{\con}$.
($\Leftarrow$) If $\p\in\Supp_R\Lambda_{\con}$, by <ref> $(\Lambda_\p)_{\con}\neq 0$. Hence the right hand functor $e(-)$ in (<ref>) is not an equivalence, and so the left hand functor cannot be an equivalence either. By <ref>, it follows that $\varphi_\p$ is not an isomorphism.
With the global setup of <ref>, choose a closed point $\m\in L$, and pick an affine open $\Spec R$ containing $\m$. Then
* $\Supp_R\Lambda_{\con}=L_R := L \cap \Spec R$.
* $\Supp_{\mathfrak{R}}\CA=\{ \p\in L_R\mid \p\subseteq \m\}$.
(1) It is clear that $L_R=\{\p\in\Spec R\mid \varphi_\p\mbox{ is not an isomorphism}\}$, and so the result is immediate from <ref>.
(2) Applying the above argument to the morphism $\mathfrak{U}\to\Spec \mathfrak{R}$, in an identical manner $\Supp_{\mathfrak{R}}\CA=L_{\mathfrak{R}} := L \cap \Spec \mathfrak{R}$. It is clear that $L_{\mathfrak{R}}=\{ \p\in L_R\mid \p\subseteq \m\}$.
The following is an immediate corollary.
In the Zariski local setup $f\colon U\to\Spec R$ of <ref>, suppose further that $\dim U=3$. Then
\[
\mbox{$f$ contracts curves without contracting a divisor}\iff \dim_{\mathbb{C}}\Lambda_\con<\infty.
\]
In this setting $f$ contracts curves without contracting a divisor if and only if $L_R$ is a zero-dimensional scheme. The result follows from <ref>(<ref>).
It follows from <ref> (or indeed the morita equivalence in <ref>) that the condition $\dim_{\mathbb{C}}\Lambda_\con<\infty$ is independent of the choice of $\Lambda$, and thus the choice of tilting bundle of the form $\cO\oplus\cN$. Hence we may make any choice, and detect the contractibility by calculating the resulting $\dim_{\mathbb{C}}\Lambda_\con$. However, to get well-defined invariants that do not depend on choices, we pass to the formal fibre $\mathfrak{R}$ and use the algebra $\CA$.
In the global setup $f\colon X\to X_{\con}$ of <ref>, suppose further that $\dim X=3$, pick a closed point $\m\in L$ and set $C:=f^{-1}(\m)$. The following are equivalent.
* There is a neighbourhood of $\m$ over which $f$ does not contract a divisor.
* The functor $\Def_X$ of simultaneous noncommutative deformations of the reduced fibre $C^{\redu}$ is representable.
By choosing an affine open $\Spec R$ containing $\m$, this is identical to the proof of <ref>, appealing to <ref>(<ref>) instead of <ref>(<ref>), and using the prorepresentability of $\Def_X$ from <ref>.
§ DEFORMATIONS OF THE SCHEME-THEORETIC FIBRE
In the global setup $f\colon X\to X_{\con}$ of <ref>, we choose a closed point $\m\in L$ and in this section study commutative and noncommutative deformations of the scheme-theoretic fibre $\cO_C$, where $C:=f^{-1}(\m)$. We show that commutative and noncommutative deformations are prorepresented by the same object, and more remarkably that the prorepresenting object can be obtained from the same $\AB$ as $\CA$ can. This allows us to relate deformations of the reduced and scheme-theoretic fibres, in a way that otherwise would not be possible.
§.§ Propresentability
With the Zariski local setup in <ref>, taking the dual bundle in (<ref>) induces a derived equivalence
\begin{eqnarray}
\begin{array}{c}
\begin{tikzpicture}
\node (a1) at (0,0) {\phantom{.}$\Db(\coh U)$};
\node (a2) at (5,0) {$\Db(\mod \End_U(\cV^*))$.};
\draw[->] (a1) -- node[above] {$\scriptstyle\RHom_U(\cV^*,-)$} node [below] {$\scriptstyle\sim$} (a2);
\end{tikzpicture}
\end{array}\label{derived equivalence dual}
\end{eqnarray}
Note that $\End_U(\cV^*) = \Lambda^{\op}$. Also, by <cit.>, under the above equivalence (<ref>) the sheaf $\cO_C$ corresponds to a simple $\Lambda^{\op}$-module, which we denote $T_0^\prime$. This fact is the reason we pass to $\cV^*$, since it will allow us to easily apply <ref> in the proof of <ref> below.
Passing to the formal fibre $\mathfrak{U}\to\Spec\mathfrak{R}$, the dual of the previous bundle in <ref> gives the following natural definition.
We write $\cM:=\bigoplus_{i=1}^n \cM_i$ and define
\[
\BB:=\End_{\mathfrak{U}}(\cO_{\mathfrak{U}}\oplus \cM)=\AB^{\op},
\]
which is the basic algebra morita equivalent to $\widehat{\Lambda}^{\op}$. From this, we define
\[
\BB_{\fib}:=\End_{\mathfrak{U}}(\cO_{\mathfrak{U}}\oplus \cM)/[\cM].
\]
Under this dual setup, the following is obvious.
$\BB^{\phantom \op}_{\fib}\cong\AB_{\fib}^{\op}\cong\AB^{\phantom \op}_{\fib}$, and in particular $\BB_{\fib}$ is commutative.
The first statement is clear since $\BB=\AB^{\op}$. The second statement is <ref>.
In what follows, we let $S_0^\prime$ denote the simple $\BB$-module corresponding to $T_0^\prime$ under the composition of the completion functor and the morita equivalence between $\widehat{\Lambda}^{\op}$ and $\BB$. Similarly to <ref>, for a given scheme-theoretic fibre $C$, below we use the following notation.
* $\Def_X^{\cO_C}$ for the DG deformation functor associated to the injective resolution of $\cO_{C}\in\coh X$.
* $\Def_U^{\cO_C}$ for the DG deformation functor associated to the injective resolution of $\cO_{C}\in\coh U$.
* $\Def_{\Lambda^{\op}}^{T_0^\prime}$ for the DG deformation functor associated to the injective resolution of $T_0^\prime\in\mod \Lambda^{\op}$.
* $\Def_{\BB}^{S_0^\prime}$ for the DG deformation functor associated to the injective resolution of $S_0\in\mod\BB$.
In the global setup $f\colon X\to X_{\con}$ of <ref>, pick a closed point $\m\in L$ and set $C:=f^{-1}(\m)$. Then
\[
\Def^{\cO_C}_X\cong\Hom_{\proart_1}(\AB_{\fib},-),
\]
and further $\AB_{\fib}$ is commutative.
Exactly as in <ref>, we first claim that
\begin{eqnarray}
\Def_X^{\cO_C}\cong\Def_U^{\cO_C}\cong\Def_{\Lambda^{\op}}^{T_0^\prime}\cong \Def_\BB^{S_0^\prime}\cong \Hom_{\proart_1}(\BB_{\fib},-).\label{claimed isos}
\end{eqnarray}
Under $i\colon U\hookrightarrow X$, since $C$ is closed $\RDerived i_*\cO_C=i_*\cO_C$, and so the first claimed isomorphism follows from <ref>. Under the derived equivalence (<ref>) above, the sheaf $\cO_C$ corresponds to the module $T_0^\prime$, so the second claimed isomorphism follows from <ref>, exactly as in the proof of <ref>(<ref>). The third claimed isomorphism follows from the fact that finite length $\Lambda^{\op}$-modules supported at $\m$ are equivalent to finite length $\BB$-modules, and so the result follows from <ref>. For the last claimed isomorphism, since $S_0^\prime$ is the vertex simple, as in <cit.> it is clear that
\[
\Def_\BB^{S_0^\prime}\cong \Hom_{\alg_1}(\BB_{\fib},-).
\]
It remains to show that $\BB_{\fib}\in\proart_1$, but this holds since $\mathfrak{R}$, thus $\BB$, and thus $\BB_{\fib}$, are complete with respect to their augmentation ideals. With (<ref>) established, the remaining statement follows from <ref>.
§.§ Comparison of Deformations
One of the remarkable consequences of <ref> and <ref> is that there is a single $\AB$, of which various factors control different natural geometric deformation functors. Thus proving elementary facts for the ring $\AB$ has strong deformation theory consequences; the following is one such example.
If $\dim_{\mathbb{C}}\CA<\infty$, then
Since $\cO_{\mathfrak{U}}\oplus\cM$ is generated by global sections, by <ref> $\AB\cong \End_{\mathfrak{R}}(\mathfrak{R}\oplus \clocCon_*\cM)^{\op}$. To ease notation we temporarily set $D=\clocCon{}_*\cM$, so $\CA^{\op}=\End_{\mathfrak{R}}(\mathfrak{R}\oplus D)/[\mathfrak{R}]$ and $\AB_{\fib}^{\op}=\End_{\mathfrak{R}}(\mathfrak{R}\oplus D)/[D]$. Since taking opposite rings does not affect dimension, in what follows we can ignore the ops.
To establish the result, we prove the contrapositive. If $\dim_{\mathbb{C}}\AB_{\fib}=\infty$ then there exists a non-maximal prime ideal $\p\in\Spec\mathfrak{R}$ such that $(\AB_{\fib})_\p\neq 0$. As in <ref> we have $(\AB_{\fib})_\p\cong {(\AB_\p)}_{\fib}$, and thus ${(\AB_\p)}_{\fib} = \End_{\mathfrak{R}_\p}(\mathfrak{R}_\p\oplus D_\p)/[D_\p]\neq 0$. Certainly this means that $D_\p$ cannot be free. Now if $\Id\colon D_\p\to D_\p$ factors through $\add\mathfrak{R}_\p$ then $D_\p$ is projective. But since $\mathfrak{R}_\p$ is local, $D_\p$ would then be free, which is a contradiction. Thus $\Id\colon D_\p\to D_\p$ does not factor through $\add\mathfrak{R}_\p$, so
\[
(\CA)_\p\cong {(\AB_\p)}_{\con} = \End_{\mathfrak{R}_\p}(\mathfrak{R}_\p\oplus D_\p)/[\mathfrak{R}_\p]\neq 0.
\]
This implies that $\p\in\Supp_{\mathfrak{R}}\CA$ and so $\dim_{\mathbb{C}}\CA=\infty$.
In the global setup $f\colon X\to X_{\con}$ of <ref>, pick a closed point $\m\in L$ and set $C:=f^{-1}(\m)$. Write $C^{\redu}=\bigcup_{i=1}^nC_i$, then
* If $\Def_X$ is representable, so is $\Def_X^{\cO_C}$.
* Suppose $\dim X=3$. If $\Def_X^{\cO_C}$ is not representable, then $f$ contracts a divisor over a neighbourhood of $\m$.
(1) This is immediate from <ref>, since $\AB_{\fib}$ prorepresents $\Def_X^{\cO_C}$ by <ref>, and $\CA$ prorepresents $\Def_X$ by <ref>.
(2) Follows from (<ref>) and <ref>.
The converse to <ref>(<ref>) is however false; we show this in <ref> below. The following summarises the main results in this paper in the case of $3$-folds.
In the global setup $f\colon X\to X_{\con}$ of <ref>, suppose further that $\dim X=3$, pick a closed point $\m\in L$, set $C:=f^{-1}(\m)$ and write $C^{\redu}=\bigcup_{i=1}^nC_i$.
* Both the noncommutative deformation functor $\Def_X^{\cO_{C}}$ and commutative deformation functor $\cDef_X^{\cO_{C}}$ are prorepresented by $\CAR$.
* The following are equivalent.
(a) The functors $\cDef_X^{\cO_C}$ and $\Def_X^{\cO_C}$ are representable.
(b) $\dim_{\mathbb{C}}\CAR<\infty$.
* The following are equivalent.
(a) The functor $\Def_X$ of simultaneous noncommutative deformations of the reduced fibre $C^{\redu}$ is representable.
(b) $\dim_{\mathbb{C}}\CA<\infty$.
(c) There is a neighbourhood of $\m$ over which $f$ does not contract a divisor.
* The statements in (3) imply the statements in (2), but the statements in (2) do not imply the statements in (3) in general.
Part (1) is <ref>, and part (2) is tautological. Part (3) is <ref> and <ref>, and part (4) is shown by the counterexample in <ref> below.
§ EXAMPLES
In this section, we first illustrate some $\CA$ that can arise in the setting of <ref> for specific $cA_n$ singularities. We then show that the converse to <ref> is false, and also that <ref> fails if we replace noncommutative deformations by commutative ones.
§.§ First Examples
Consider the $cA_n$ singularities $\mathfrak{R}:=\mathbb{C}[[u,v,x,y]]/(uv-f_1\hdots f_n)$ for some $f_i\in\m:=(x,y)\subset \mathbb{C}[[x,y]]$. The algebra $\AB$ in <ref>, and thus the algebras $\CA$ and $\CAR$, can be obtained using the calculation in <cit.>. Here we make this explicit in two examples.
[A $2$-curve flop] Consider the case $f_1=x$, $f_2=y$ and $f_3=x+y$. In this example there are six crepant resolutions of $\Spec\mathfrak{R}$, and each has two curves above the origin. For one such choice, by <cit.>
\[
\AB=\End_{\mathfrak{R}}(\mathfrak{R}\oplus (u,x)\oplus (u,xy)),
\]
which by <cit.> can be presented as the completion of the quiver with relations
\[
\begin{array}{c|c}
\begin{array}{c}
\begin{tikzpicture} [bend angle=45, looseness=1.2]
\node[name=s,regular polygon, regular polygon sides=3, minimum size=2cm] at (0,0) {};
\node (C1) at (s.corner 1) {$\scriptstyle (u,x)$};
\node (C2) at (s.corner 2) {$\scriptstyle \mathfrak{R}$};
\node (C3) at (s.corner 3) {$\scriptstyle (u,xy)$};
\draw[->] (C2) -- node [gap] {$\scriptstyle x$} (C1);
\draw[->] (C1) -- node [gap] {$\scriptstyle y$} (C3);
\draw[->] (C3) -- node [gap,pos=0.4] {$\scriptstyle \frac{x+y}{u}$} (C2);
\draw [->,bend right] (C1) to node [left] {$\scriptstyle { inc}$} (C2);
\draw [->,bend right] (C2) to node [below] {$\scriptstyle u$} (C3);
\draw [->,bend right] (C3) to node [right] {$\scriptstyle { inc}$} (C1);
\end{tikzpicture}\end{array}
\begin{array}{cc}
\begin{array}{c}
\begin{tikzpicture} [bend angle=45,looseness=1.2]
\node[name=s,regular polygon, regular polygon sides=3, minimum size=2cm] at (0,0) {};
\node (C1) at (s.corner 1) [vertex] {};
\node (C2) at (s.corner 2) [cvertex] {};
\node (C3) at (s.corner 3) [vertex] {};
\draw[->] (C2) -- node [gap] {$\scriptstyle c_1$} (C1);
\draw[->] (C1) -- node [gap] {$\scriptstyle c_2$} (C3);
\draw[->] (C3) -- node [gap] {$\scriptstyle c_3$} (C2);
\draw [->,bend right] (C1) to node [left] {$\scriptstyle a_1$} (C2);
\draw [->,bend right] (C2) to node [below] {$\scriptstyle a_3$} (C3);
\draw [->,bend right] (C3) to node [right] {$\scriptstyle a_2$} (C1);
\end{tikzpicture}\end{array}&
\begin{array}{c}
\end{array}}}
\end{array}
\end{array}
\]
given by the superpotential
\[
\]
From this presentation, factoring by the appropriate idempotents it is immediate that $\CAR\cong \mathbb{C}$, and further
\[
\CA\cong
\begin{array}{cc}
\begin{array}{c}
\begin{tikzpicture}[bend angle=20, looseness=1]
\node (a) at (-1.5,0) [vertex] {};
\node (b) at (0,0) [vertex] {};
\node (a1) at (-1.5,-0.2) {$\scriptstyle 1$};
\node (b1) at (0,-0.2) {$\scriptstyle 2$};
\draw[->,bend left] (b) to node[below] {$\scriptstyle a_2$} (a);
\draw[->,bend left] (a) to node[above] {$\scriptstyle c_2$} (b);
\end{tikzpicture}
\end{array}
\begin{array}{c}
\end{array}}
\end{array}
\]
Since $\CA$ is finite dimensional, by <ref>(<ref>) the contraction only contracts curves to a point.
[A divisorial contraction] Consider the case $f_1=f_2=x$ and $f_3=y$. In this case there are three crepant resolutions, and each has two curves above the origin. For one such choice, obtained by blowing up the ideal $u=v=x=0$, the resolution is sketched as follows,
\[
\begin{tikzpicture}
\node at (0,0) {\begin{tikzpicture}[scale=1]
\coordinate (T) at (1.9,2);
\coordinate (TM) at (2.12-0.02,1.5-0.1);
\coordinate (BM) at (2.12-0.05,1.5+0.1);
\coordinate (B) at (2.1,1);
\draw[line width=0.5pt]\opt{colordiag}{[red]} (T) to [bend left=25] (TM);
\draw[line width=0.5pt]\opt{colordiag}{[red]} (BM) to [bend left=25] (B);
\foreach \y in {0.1,0.2,...,1}{
\draw[very thin]\opt{colordiag}{[blue]} ($(T)+(\y,0)+(0.02,0)$) to [bend left=25] ($(B)+(\y,0)+(0.02,0)$);
\draw[very thin]\opt{colordiag}{[blue]} ($(T)+(-\y,0)+(-0.02,0)$) to [bend left=25] ($(B)+(-\y,0)+(-0.02,0)$);;}
\draw[rounded corners=15pt,line width=0.5pt] (0.5,0) -- (1.5,0.3)-- (3.6,0) -- (4.3,1.5)-- (4,3.2) -- (2.5,2.7) -- (0.2,3) -- (-0.2,2)-- cycle;
\end{tikzpicture}};
\node at (0,-3.5) {\begin{tikzpicture}[scale=1]
\draw [->,black] (1.1,0.75) -- (3.1,0.75);
\filldraw\opt{colordiag}{[red]} (2.1,0.75) circle (1pt);
\node at (3.2,0.6) {$\scriptstyle y$};
\draw[rounded corners=12pt,line width=0.5pt] (0.5,0) -- (1.5,0.15)-- (3.6,0) -- (4.3,0.75)-- (4,1.6) -- (2.5,1.35) -- (0.2,1.5) -- (-0.2,0.6)-- cycle;
\end{tikzpicture}};
\draw[->] (0,-1.6) -- (0,-2.65);
\end{tikzpicture}
\]
where above the origin there are two curves, and every other fibre over the $y$-axis contains only one curve. For this resolution
\[
\AB=\End_{\mathfrak{R}}(\mathfrak{R}\oplus (u,x)\oplus(u,xy))
\]
which by <cit.> can be presented as the completion of the quiver with relations
\[
\begin{array}{c|c}
\begin{array}{c}
\begin{tikzpicture} [bend angle=45, looseness=1.2]
\node[name=s,regular polygon, regular polygon sides=3, minimum size=2cm] at (0,0) {};
\node (C1) at (s.corner 1) {$\scriptstyle (u,x)$};
\node (C2) at (s.corner 2) {$\scriptstyle \mathfrak{R}$};
\node (C3) at (s.corner 3) {$\scriptstyle (u,xy)$};
\draw[->] (C2) -- node [gap] {$\scriptstyle x$} (C1);
\draw[->] (C1) -- node [gap] {$\scriptstyle y$} (C3);
\draw[->] (C3) -- node [gap,pos=0.4] {$\scriptstyle \frac{x}{u}$} (C2);
\draw [->,bend right] (C1) to node [left] {$\scriptstyle inc$} (C2);
\draw [->,bend right] (C2) to node [below] {$\scriptstyle u$} (C3);
\draw [->,bend right] (C3) to node [right] {$\scriptstyle inc$} (C1);
\node (C2a) at ($(s.corner 2)+(-135:2pt)$) {};
\draw[<-] (C2a) edge [in=-100,out=-170,loop,looseness=10] node[below] {$\scriptstyle y$} (C2a);
\end{tikzpicture}
\end{array}
\begin{array}{cc}
\begin{array}{c}
\begin{tikzpicture} [bend angle=45,looseness=1.2]
\node[name=s,regular polygon, regular polygon sides=3, minimum size=2cm] at (0,0) {};
\node (C1) at (s.corner 1) [vertex] {};
\node (C2) at (s.corner 2) [cvertex] {};
\node (C3) at (s.corner 3) [vertex] {};
\draw[->] (C2) -- node [gap] {$\scriptstyle c_1$} (C1);
\draw[->] (C1) -- node [gap] {$\scriptstyle c_2$} (C3);
\draw[->] (C3) -- node [gap] {$\scriptstyle c_3$} (C2);
\draw [->,bend right] (C1) to node [left] {$\scriptstyle a_1$} (C2);
\draw [->,bend right] (C2) to node [below] {$\scriptstyle a_3$} (C3);
\draw [->,bend right] (C3) to node [right] {$\scriptstyle a_2$} (C1);
\node (C2a) at ($(s.corner 2)+(-135:2pt)$) {};
\draw[<-] (C2a) edge [in=-100,out=-170,loop,looseness=10] node[below] {$\scriptstyle y$} (C2a);
\end{tikzpicture}\end{array}&
\begin{array}{l}
\end{array}}}
\end{array}
\end{array}
\]
From this, we see immediately that $\CAR\cong\mathbb{C}[[y]]$, and $\CA$ is the completion of the quiver
\[
\begin{array}{cc}
\begin{array}{c}
\begin{tikzpicture}[bend angle=20, looseness=1]
\node (a) at (-1.5,0) [vertex] {};
\node (b) at (0,0) [vertex] {};
\node (a1) at (-1.5,-0.2) {$\scriptstyle 1$};
\node (b1) at (0,-0.2) {$\scriptstyle 2$};
\draw[->,bend left] (b) to node[below] {$\scriptstyle a_2$} (a);
\draw[->,bend left] (a) to node[above] {$\scriptstyle c_2$} (b);
\end{tikzpicture}
\end{array}
\end{array}
\]
with no relations. Thus in this example both $\CAR$ and $\CA$ are infinite dimensional, which by <ref> confirms that the contraction morphism contracts a divisor to a curve.
We remark that the above example, <ref>, also appears in <cit.> and <cit.>.
§.§ Failure of Commutative Deformations
Here we give two more complicated examples. The first shows that the converse to <ref>(<ref>) is false, and the second shows that <ref> fails if we replace noncommutative deformations by commutative ones. In both examples, there is only one curve in the fibre above the closed point $\m$.
[$\Def^{\cO_C}$ does not detect divisors] Consider the group $G:=A_4$ acting on its three-dimensional irreducible representation, and set $\mathfrak{R}:=\mathbb{C}[[x,y,z]]^G$. It is well-known that in the crepant resolutions of $\Spec \mathfrak{R}$, the fibre above the origin of $\Spec \mathfrak{R}$ is one-dimensional: see <cit.> and <cit.>.
For the crepant resolution given by $h\colon G\mbox{-Hilb}\to\Spec \mathfrak{R}$, there are three curves above the origin meeting transversally in a Type $A$ configuration. Further, in this case the tilting bundle from $G$-Hilb has endomorphism ring isomorphic to the completion of the following McKay quiver with relations (see e.g. <cit.> <cit.>)
\[
\End_\mathfrak{R}(\mathfrak{R}\oplus M_1\oplus M_2\oplus M_3)\cong
\begin{array}{cc}
\begin{array}{c}
\begin{tikzpicture}
\node[rotate=60,name=s,regular polygon, regular polygon sides=3, minimum size=3cm] at (0,0) {};
\node (R) at (s.corner 2) {$\scriptstyle \mathfrak{R}$};
\node (1) at (s.corner 1) {$\scriptstyle M_{1}$};
\node (2) at (s.corner 3) {$\scriptstyle M_{3}$};
\node (M) at (s.center) {$\scriptstyle M_{2}$};
\node (Ma) at ($(s.center)+(-155:5pt)$) {};
\node (Mb) at ($(s.center)+(-25:5pt)$) {};
\draw[->] (Ma) edge [in=180,out=-110,loop,looseness=9] node[left] {$\scriptstyle u$} (Ma);
\draw[->] (Mb) edge [in=0,out=-70,loop,looseness=9] node[right] {$\scriptstyle v$} (Mb);
\draw[->] (R) edge [in=-100,out=100,looseness=1] node[left] {$\scriptstyle a$} (M);
\draw[->] (M) edge [in=80,out=-80,looseness=1] node[right] {$\scriptstyle A$} (R);
\draw[->] (1) edge [in=160,out=-40,looseness=1] node[below] {$\scriptstyle b$} (M);
\draw[->] (M) edge [in=-20,out=140,looseness=1] node[above] {$\scriptstyle B$} (1);
\draw[->] (2) edge [in=40,out=-160,looseness=1] node[above] {$\scriptstyle c$} (M);
\draw[->] (M) edge [in=-140,out=20,looseness=1] node[below] {$\scriptstyle C$} (2);
\end{tikzpicture}
\end{array}
\begin{array}{ll}
uA=vA & au=av\\
uB=\rho vB & bu=\rho bv\\
uC=\rho^2vC & cu=\rho^2cv\\
\end{array}\\ \\
\begin{array}{l}
u^2=Aa+\rho Bb+\rho^2Cc\\
v^2=Aa+\rho^2 Bb+\rho Cc
\end{array}
\end{array}}
\end{array}
\]
where $\rho$ is a cube root of unity. By <cit.>, since there are two loops on the middle vertex we see that the middle curve is a $(-3,1)$-curve, and since there are no loops on the outer vertices, the outer curves are $(-1,-1)$-curves. Also, by inspection
\begin{equation}
\End_\mathfrak{R}(M_2)/[\mathfrak{R}\oplus M_1\oplus M_3]\cong\frac{\mathbb{C}\langle\langle u,v\rangle\rangle}{\mathrm{cl}(u^2,v^2)},
\label{zigzag algebra}
\end{equation}
where $\mathrm{cl}(u^2,v^2)$ denotes the closure of the two-sided ideal $(u^2,v^2)$. Evidently, the above factor is infinite dimensional. Since there is a surjective map
\[
\End_\mathfrak{R}(M_2)/[\mathfrak{R}]\twoheadrightarrow \End_\mathfrak{R}(M_2)/[\mathfrak{R}\oplus M_1\oplus M_3]
\]
it follows that $\End_\mathfrak{R}(M_2)/[\mathfrak{R}]$ must also be infinite dimensional.
Now, contracting both the outer $(-1,-1)$-curves in $G$-Hilb we obtain a scheme $\mathfrak{U}$ and a factorization
\[
\begin{tikzpicture}
\node (A) at (0,0) {$G$-Hilb};
\node (B2) at (1.5,-0.75) {$\mathfrak{U}$};
\node (C) at (3,0) {$\Spec \mathfrak{R}$};
\draw[->] (A) -- node[below] {} (B2);
\draw[->] (B2) -- node[below] {$\scriptstyle f$} (C);
\draw[->] (A) -- node[above] {$\scriptstyle h$}(C);
\end{tikzpicture}
\]
The example we consider is $f\colon \mathfrak{U}\to \Spec \mathfrak{R}$. By construction, there is only one curve above the origin. As in <cit.>
\[
\Db(\coh \mathfrak{U})\cong\Db(\mod \End_\mathfrak{R}(\mathfrak{R}\oplus M_2)).
\]
Set $\AB:=\End_\mathfrak{R}(\mathfrak{R}\oplus M_2)$, then the quiver for $\AB$ is obtained from the above McKay quiver by composing two-cycles that pass through $M_1$ and $M_3$. In particular, in the quiver for $\AB$ there is no loop at the vertex corresponding to $\mathfrak{R}$, so $\AB_{\fib}=\mathbb{C}$. In particular, by <ref>, $\Def^{\cO_C}$ is representable.
On the other hand $\CA=\End_\mathfrak{R}(M_2)/[\mathfrak{R}]$, and we have already observed that this is infinite dimensional, so by <ref> $f$ contracts a divisor to a curve. It is also possible to observe this divisorial contraction using the explicit calculations of open covers in <cit.>.
[$\cDef^{\cJ}$ does not detect divisors] Consider again $\mathfrak{R}:=\mathbb{C}[[x,y,z]]^G$ where $G$ is the alternating group above in <ref>. Now, contracting instead the middle curve in $G$-Hilb (instead of the outer curves we contracted above) gives a factorization
\[
\begin{tikzpicture}
\node (A) at (0,0) {$G$-Hilb};
\node (B2) at (1.5,-0.75) {$W$};
\node (C) at (3,0) {$\Spec \mathfrak{R}$};
\draw[->] (A) -- node[below] {$\scriptstyle g$} (B2);
\draw[->] (B2) -- node[below] {$\scriptstyle $} (C);
\draw[->] (A) -- node[above] {$\scriptstyle h$}(C);
\end{tikzpicture}
\]
The original middle curve contracts to a closed point $\m$ in $W$, so picking an affine open $\Spec T$ in $W$ containing $\m$, and passing to the formal fibre of $g$ over this point, we obtain a morphism
\[
g\colon \mathfrak{W}\to \Spec \mathfrak{T}.
\]
To this contraction, we associate the contraction algebra $\CA$ using the procedure in <ref>. By <cit.>, it follows from the uniqueness of prorepresenting object that
\[
\CA\cong \End_{\mathfrak{R}}(M_2)/[\mathfrak{R}\oplus M_1\oplus M_3],
\]
where we have recycled notation from <ref>. Hence by (<ref>) we see that
\[
\CA\cong \frac{\mathbb{C}\langle\langle u,v\rangle\rangle}{\mathrm{cl}(u^2,v^2)}.
\]
We have already observed that this is infinite dimensional, so $g$ contracts a divisor by <ref>. Alternatively, we can see that $g$ contracts a divisor by using the explicit open cover as in <cit.>.
On the other hand, by general theory (see e.g. <cit.>) $\cDef^\cJ$ is prorepresented by the abelianization of $\CA$, which in this case is simply
\[
\CA^{\ab}\cong \frac{\mathbb{C}[[u,v]]}{(u^2,v^2)}.
\]
By inspection, this is finite dimensional. This shows that the commutative deformation functor $\cDef^\cJ$ is representable, even although a divisor is contracted to a curve.
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PAMELA's measurements of geomagnetic cutoff variations during SEP events
Bruno et al.
$^{1}$ Department of Physics, University of Bari “Aldo Moro”, I-70126 Bari, Italy.
$^{2}$ Department of Physics and Astronomy, University of Florence, I-50019 Sesto Fiorentino, Florence, Italy.
$^{3}$ INFN, Sezione di Florence, I-50019 Sesto Fiorentino, Florence, Italy.
$^{4}$ Department of Physics, University of Naples “Federico II”, I-80126 Naples, Italy.
$^{5}$ INFN, Sezione di Naples, I-80126 Naples, Italy.
$^{6}$ Lebedev Physical Institute, RU-119991 Moscow, Russia.
$^{7}$ INFN, Sezione di Bari, I-70126 Bari, Italy.
$^{8}$ INFN, Sezione di Trieste, I-34149 Trieste, Italy.
$^{9}$ Ioffe Physical Technical Institute, RU-194021 St. Petersburg, Russia.
$^{10}$ Space Science Center, University of New Hampshire, Durham, NH, USA.
$^{11}$ KTH, Department of Physics, and the Oskar Klein Centre for Cosmoparticle Physics, AlbaNova University Centre, SE-10691 Stockholm, Sweden.
$^{12}$ INFN, Sezione di Rome “Tor Vergata”, I-00133 Rome, Italy.
$^{13}$ RIKEN, Advanced Science Institute, Wako-shi, Saitama, Japan.
$^{14}$ IFAC, I-50019 Sesto Fiorentino, Florence, Italy.
$^{15}$ Heliophysics Division, NASA Goddard Space Flight Ctr, Greenbelt, MD, USA.
$^{16}$ National Research Nuclear University MEPhI, RU-115409 Moscow, Russia.
$^{17}$ Department of Physics, University of Rome “Tor Vergata”, I-00133 Rome, Italy.
$^{18}$ Agenzia Spaziale Italiana (ASI) Science Data Center,
I-00133 Rome, Italy.
$^{19}$ Department of Physics, University of Trieste, I-34147 Trieste, Italy.
$^{20}$ INFN, Laboratori Nazionali di Frascati,
I-00044 Frascati, Italy.
$^{21}$ Department of Physics, Universität Siegen, D-57068 Siegen, Germany.
$^{22}$ Indian Centre for Space Physics, 43 Chalantika,
Kolkata 700084, West Bengal, India.
$^{23}$ Previously at INFN, Sezione di Trieste, I-34149 Trieste, Italy.
$^{24}$ Electrical and Computer Engineering, New Mexico State University, Las Cruces, NM, USA.
*Corresponding author. E-mail address: alessandro.bruno@ba.infn.it.
Data from the PAMELA satellite experiment were used to measure the geomagnetic cutoff for high-energy ($\gtrsim$ 80 MeV) protons during the solar particle events on 2006 December 13 and 14. The variations of the cutoff latitude as a function of rigidity were studied on relatively short timescales, corresponding to single spacecraft orbits (about 94 minutes). Estimated cutoff values were cross-checked with those obtained by means of a trajectory tracing approach based on dynamical empirical modeling of the Earth's magnetosphere. We find significant variations in the cutoff latitude, with a maximum suppression of about 6 deg for $\sim$80 MeV protons during the main phase of the storm. The observed reduction in the geomagnetic shielding and its temporal evolution were compared with the changes in the magnetosphere configuration, investigating the role of IMF, solar wind and geomagnetic (Kp, Dst and Sym-H indexes) variables and their correlation with PAMELA cutoff results.
§ INTRODUCTION
Solar Energetic Particle (SEP) events are major space weather phenomena associated with explosive processes occurring in the solar atmosphere, such solar flares and Coronal Mass Ejections (CMEs). SEPs can produce hazardous effects to manned and robotic flight missions in the near-Earth space environment, and
influence the atmospheric chemistry and dynamics. Large SEP events can strongly perturb the Earth's magnetic field, inducing geomagnetic storms and modifying the Cosmic-Ray (CR) access to the inner magnetosphere. The consequent reduction in the geomagnetic shielding can significantly increase the potential radiation exposure compared with geomagnetically quiet times.
Estimates of geomagnetic cutoffs have been provided by satellite observations and theoretical calculations <cit.> mainly based on tracing particles through models of the Earth's magnetic field <cit.>.
In this work we present PAMELA's measurements of the variability of the geomagnetic cutoff during the SEP events on 2006 December 13 and 14,
with the focus on the strong magnetic storm on December 14 and 15.
§ DATA ANALYSIS
§.§ The PAMELA experiment
PAMELA is a space-based experiment designed for a precise measurement of the charged
cosmic radiation
in the kinetic energy range from some tens of MeV up to several hundreds of GeV <cit.>.
In particular, PAMELA is providing accurate measurements of SEPs in a wide energy interval <cit.>, bridging the low energy data by
other spacecrafts
and the GLE data by the worldwide network of neutron monitors; in addition, the detector is sensitive to the particle composition and is able to reconstruct flux angular distributions <cit.>, enabling a
more complete view of SEP events.
The Resurs-DK1 satellite, which hosts the apparatus, was launched into a semi-polar (70 deg inclination) and elliptical (350$\div$610 km altitude) orbit on 2006 June 15. The spacecraft is 3-axis stabilized; its orientation is calculated by an onboard processor with an accuracy better than 1 deg. Particle directions are measured with a high angular resolution ($<$ 2 deg). Details about apparatus performance, proton selection, detector efficiencies and experimental uncertainties can be found elsewhere (e.g. <cit.>).
The selected data set
includes protons acquired by PAMELA between 2006 December 12 and 18.
§.§ Geomagnetic Field Models
The analysis described in this work is based on the IGRF-11 <cit.> and the TS05 <cit.> models for the description of the internal and external geomagnetic field, respectively.
The TS05 model is a high resolution dynamical model of the storm-time geomagnetic field,
based on recent satellite measurements; consistent with the data-set coverage, it is valid
for $X_{GSM}$ $>$ -15 Earth's radii (Re). For comparison purposes, the T96 model <cit.> (valid up to 40 Re) was used as well.
Solar Wind (SW) and Interplanetary Magnetic Field (IMF) parameters were obtained from the high resolution (5-min) Omniweb database <cit.>.
§.§ Coordinate Systems
Data were analyzed in terms of Altitude Adjusted Corrected GeoMagnetic (AACGM) coordinates, developed to provide a more realistic description of high latitude regions by accounting for the multipolar geomagnetic field. They are defined such that all points along a magnetic field line have the same geomagnetic latitude and longitude, so that they are closely related to invariant magnetic coordinates <cit.>. The AACGM reference frame coincides with the standard Corrected GeoMagnetic (CGM) coordinate system <cit.> at the Earth's surface. Unlike other commonly used variables such as the invariant latitude,
the computation of such coordinates at low Earth orbits is not significantly affected by the modeling of external geomagnetic sources.
§.§ Evaluation of geomagnetic cutoff latitudes
The lowest magnetic latitude to which a CR particle can penetrate the Earth's magnetic field is known as its cutoff latitude and is a function of the particle momentum per unit charge, which is referred to as its rigidity. Alternatively one may consider a cutoff rigidity corresponding to a given location in space,
i.e. the minimum rigidity needed to access to that location.
Some complications arise from the presence of the Earth's solid body (together with its atmosphere):
both “allowed” and “forbidden” bands of CR particle access are present in the so-called “penumbra” region <cit.>.
The numerical algorithm developed to extract cutoff latitudes from the PAMELA data is similar to one used by <cit.> and <cit.>.
For each rigidity bin, a mean flux was obtained by averaging fluxes above 65 degrees latitude, and the cutoff latitude was evaluated as the latitude where the flux intensity is equal to the half of the average value.
Alternatively, cutoff latitudes were estimated with back-tracing techniques <cit.>. Using the spacecraft ephemeris data,
and the particle rigidity and direction provided by the PAMELA tracking system, trajectories of all detected protons were reconstructed by means of a tracing program based on numerical integration methods <cit.>, and implementing the afore-mentioned geomagnetic field models.
Trajectories were back propagated from the measurement location
until they escaped the model magnetosphere boundaries (Solar or Galactic CRs) or they reached an altitude[Corresponding to the mean production altitude for albedo protons.] of 40 km (re-entrant albedo CRs).
At a given rigidity, the cutoff latitude was evaluated as the latitude where an equal percentage of interplanetary and albedo CRs was registered.
The calculation was performed for 13 rigidity logarithmic bins, covering the interval 0.39$\div$3.29 GV. Accounting for the limited statistics at highest rigidities, final cutoff values were derived by
fitting averaged
PAMELA observations over single orbital periods ($\sim$94 min).
Time profiles of the IMF (Bx, By and Bz components in the GSM frame) and solar wind (dynamic pressure, velocity and density) parameters between 2006 December 12$\div$18 <cit.>.
§ THE 2006 DECEMBER 13 AND 14 EVENTS
On 2006 December 13 at 02:14 UT, an X3.4/4B solar flare occurred in the active region NOAA 10930 (S06W23; NOAA-STP 2006). This event also produced a full-halo CME with the sky plane projected speed of 1774 km s$^{-1}$.
The forward shock of the CME reached Earth at about 14:10 UT on December 14, causing a Forbush decrease of Galactic CR intensities that lasted for several days.
Such large events are untypical of the intervals of low solar activity. The flare X1.5 (S06W46) at 21:07 UT on December 14 gave start to a new growth of particle intensity as recorded by PAMELA and other satellites. The maximum energy of protons was below 1 GeV, and therefore no ground level enhancement (GLE) was recorded. The corresponding CME had a velocity of 1042 km s$^{-1}$.
PAMELA's measurements of the 2006 December SEP fluxes can be found in publications <cit.>.
Figure <ref> reports the variations in the
IMF (Bx, By and Bz components in the GSM frame) and
SW (dynamic pressure, velocity and density) variables
between 2006 December 12$\div$18.
The large increase in the SW velocity associated with the leading edge of the CME caused a sudden commencement of a geomagnetic storm.
The initial phase of the storm, lasting up to about 23:00 UT, was characterized by intense fluctuations in the SW density and in all IMF components.
At a later stage, the IMF Bz component became negative, the SW density decreased, and the main phase of the
storm started, reaching a maximum between 02:00$\div$08:00 UT on December 15. Another interplanetary shock associated with a different CME was observed on December 16.
Time profile of the geomagnetic cutoff latitudes measured by PAMELA, for different rigidity bins.
Comparison between measured (black) and modeled (blue - T96 model; red - TS05 model) cutoff variations in the lowest rigidity interval: 0.39-0.46 GV.
Geomagnetic cutoff latitude variations measured by PAMELA in the rigidity interval 0.39-0.46 GV (top panel), compared with the time profiles of Kp, Dst and Sym-H indexes.
§ RESULTS
Figure <ref> shows the geomagnetic cutoff latitudes measured by PAMELA for different rigidity bins (color code). Each point denotes the cutoff latitude value averaged over a single spacecraft orbit; the error bars include the statistical uncertainties of the measurement. Data were missed from 10:00 UT on December 13 until 09:14 UT on December 14 because of an onboard system reset of the satellite.
The evolution of the magnetic storm of December 14 and 15 followed the typical scenario in which the
cutoff latitudes move equatorward as a consequence of a CME impact on the magnetosphere with an associated transition to southward Bz.
The registered cutoff variation decreases with increasing rigidity, with a maximum suppression ranging from about 6 deg in the lowest rigidity bin ($\sim$80 MeV energy) to about 2 deg in the highest rigidity bin ($\sim$3 GeV energy).
Figure <ref> reports the comparison between measured and modeled cutoff latitudes, for the lowest rigidity bin 0.39$\div$0.46 GV.
While the T96 model appears to underestimate (up to 4%) the observations, a much better agreement can be noted between PAMELA and TS05 results. However, the TS05 cutoff latitudes overestimate (up to 2%) the PAMELA ones during the storm main phase.
Finally, Figure <ref> demonstrates the utility of the three indices used to infer cutoff latitude: the magnetic activity index (Kp), the disturbance storm time index (Dst) and the Sym-H index[Sym-H represents the longitudinally symmetric part of the northward magnetic field variations.], measured using ground-based magnetometers, at 3-hour, 1-hour, and 1-min resolutions, respectively. In general, the shapes of the time variations in the cutoff measurements are well correlated with corresponding
indexes changes (corresponding correlation coefficients are 0.8, 0.78 and 0.78, respectively).
A better agreement is observed for Kp during the initial phase of the storm, while the Dst and the Sym-H indexes
show an improved correlation
during the main and the recovery phases.
§ SUMMARY AND CONCLUSIONS
In this study we have exploited the data of the PAMELA satellite experiment to perform a measurement of the geomagnetic cutoff variations during the long lasting SEP events of 2006 December 13 and 14. A significant reduction in the geomagnetic shielding was observed during the consequent strong magnetic storm on December 14 and 15, with a maximum cutoff latitude suppression of about 6 deg for $\sim$80 MeV protons. Results were compared with those obtained with back-tracing techniques. The observed cutoff variations are well correlated with the time profiles of the geomagnetic indexes (Kp, Dst and Sym-H).
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Spectral-Line Survey toward OMC 2-FIR 4
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Deep Neural Networks (DNN) have achieved state-of-the-art results in a wide range of tasks,
with the best results obtained with large training sets and large models. In the past, GPUs enabled these
breakthroughs because of their greater computational speed. In the future, faster computation
at both training and test time is likely to be crucial for further progress and for
consumer applications on low-power devices. As a result, there is much interest in research and development
of dedicated hardware for Deep Learning (DL).
Binary weights, i.e., weights which are constrained to only two possible values (e.g. -1 or 1),
would bring great benefits to specialized DL hardware by replacing many multiply-accumulate operations by
simple accumulations, as multipliers are the most space and power-hungry components of the digital implementation
of neural networks.
We introduce BinaryConnect,
a method which consists in training a DNN with binary weights during the forward and backward propagations,
while retaining precision of the stored weights in which gradients are accumulated.
Like other dropout schemes, we show that BinaryConnect acts as regularizer and
we obtain near state-of-the-art results with BinaryConnect on the permutation-invariant MNIST, CIFAR-10 and SVHN.
§ INTRODUCTION
Deep Neural Networks (DNN) have substantially pushed the state-of-the-art in a wide range of tasks,
especially in speech recognition <cit.>
and computer vision, notably object recognition from images <cit.>.
More recently, deep learning is making important strides in natural language processing, especially
statistical machine translation <cit.>.
Interestingly, one of the key factors that enabled this major progress has been the advent of
Graphics Processing Units (GPUs), with speed-ups on the order of 10 to 30-fold, starting with <cit.>,
and similar improvements with distributed training <cit.>.
Indeed, the ability to train larger models on more data has enabled the kind of breakthroughs
observed in the last few years. Today, researchers and developers designing new deep learning algorithms
and applications often find themselves limited by computational capability. This along, with the drive to put deep learning
systems on low-power devices (unlike GPUs) is greatly increasing the interest in research and development of
specialized hardware for deep networks <cit.>.
Most of the computation performed during training and application of deep networks regards the multiplication
of a real-valued weight by a real-valued activation (in the recognition or forward propagation phase
of the back-propagation algorithm) or gradient (in the backward propagation phase of the
back-propagation algorithm). This paper proposes an approach called BinaryConnect to eliminate the need for these multiplications
by forcing the weights used in these forward and backward propagations to be binary, i.e. constrained
to only two values (not necessarily 0 and 1). We show that state-of-the-art results can be achieved
with BinaryConnect on the permutation-invariant MNIST, CIFAR-10 and SVHN.
What makes this workable are two ingredients:
* Sufficient precision is necessary to accumulate and
average a large number of stochastic gradients, but noisy weights (and we can view discretization
into a small number of values as a form of noise, especially if we make this discretization
stochastic) are quite compatible with Stochastic Gradient Descent (SGD), the main type
of optimization algorithm for deep learning.
SGD explores the space of parameters by making small and noisy steps and that noise is
averaged out by the stochastic gradient contributions accumulated in each weight.
Therefore, it is important to keep sufficient resolution for these accumulators, which at first sight
suggests that high precision is absolutely required. <cit.> and <cit.>
show that randomized or stochastic rounding can be used to provide unbiased discretization.
<cit.> have shown that SGD requires weights with a precision of at least 6 to 8 bits
and <cit.> successfully train DNNs with 12 bits dynamic fixed-point
Besides, the estimated precision of the brain synapses varies between 6 and 12 bits <cit.>.
* Noisy weights actually provide a form of regularization which can help to
generalize better, as previously shown with variational weight noise <cit.>,
Dropout <cit.> and DropConnect <cit.>, which add noise to the
activations or to the weights. For instance, DropConnect <cit.>, which is closest
to BinaryConnect, is a very efficient regularizer that randomly substitutes half of the weights with zeros during propagations.
What these previous works show is that only the expected value
of the weight needs to have high precision, and that noise can actually be beneficial.
The main contributions of this article are the following.
* We introduce BinaryConnect,
a method which consists in training a DNN with binary weights during the forward and backward propagations (Section <ref>).
* We show that BinaryConnect is a regularizer
and we obtain near state-of-the-art results on the permutation-invariant MNIST, CIFAR-10 and SVHN (Section <ref>).
* We make the code for BinaryConnect available
§ BINARYCONNECT
In this section we give a more detailed view of BinaryConnect, considering which two values to choose, how to
discretize, how to train and how to perform inference.
§.§ $+1$ or $-1$
Applying a DNN mainly consists in convolutions and matrix multiplications.
The key arithmetic operation of DL is thus the multiply-accumulate operation.
Artificial neurons are basically multiply-accumulators computing weighted sums of their inputs.
BinaryConnect constraints the weights to either $+1$ or $-1$ during propagations.
As a result, many multiply-accumulate operations are replaced by simple additions (and subtractions).
This is a huge gain, as fixed-point adders are much less expensive both in terms of
area and energy than fixed-point multiply-accumulators <cit.>.
§.§ Deterministic vs stochastic binarization
The binarization operation transforms the real-valued weights into the two possible values.
A very straightforward binarization operation would be based on the sign function:
\begin{equation}
w_b = \left\{ \begin{array}{ll}
+1 & \mbox{if $w \geq 0$},\\
-1 & \mbox{otherwise}.\end{array} \right.
\end{equation}
Where $w_b$ is the binarized weight and $w$ the real-valued weight. Although this is
a deterministic operation, averaging this discretization over the many input weights
of a hidden unit could compensate for the loss of information. An alternative that
allows a finer and more correct averaging process to take place is to binarize
\begin{align}
\label{eq:sampled-wb}
w_b = \left\{ \begin{array}{ll}
+1 & \mbox{with probability $p = \sigma(w)$},\\
-1 & \mbox{with probability $1-p$}.\end{array} \right.
\end{align}
where $\sigma$ is the “hard sigmoid” function:
\begin{equation}
\sigma(x) = {\rm clip}(\frac{x+1}{2},0,1) = \max(0,\min(1,\frac{x+1}{2}))
\end{equation}
We use such a hard sigmoid rather than the soft version
because it is far less computationally expensive (both in software and
specialized hardware implementations) and yielded excellent results in our experiments.
It is similar to the “hard tanh” non-linearity introduced by <cit.>.
It is also piece-wise linear and corresponds to a bounded form of the rectifier <cit.>.
§.§ Propagations vs updates
Let us consider the different steps of back-propagation with SGD udpates and whether it makes
sense, or not, to discretize the weights, at each of these steps.
* Given the DNN input, compute the unit activations layer by layer, leading to the top layer which is the output of the DNN, given its input. This step is referred as the forward propagation.
* Given the DNN target, compute the training objective's gradient w.r.t. each layer's activations,
starting from the top layer and going down layer by layer until the first hidden layer.
This step is referred to as the backward propagation or backward phase of back-propagation.
* Compute the gradient w.r.t. each layer's parameters
and then update the parameters using their computed gradients and their previous values.
This step is referred to as the parameter update.
a minibatch of (inputs, targets),
previous parameters $w_{t-1}$ (weights) and $b_{t-1}$ (biases),
and learning rate $\eta$.
updated parameters $w_t$ and $b_t$.
1. Forward propagation:
$w_b \leftarrow {\rm binarize}(w_{t-1})$
For $k=1$ to $L$, compute $a_k$ knowing $a_{k-1}$, $w_b$ and $b_{t-1}$
2. Backward propagation:
Initialize output layer's activations gradient $\frac{\partial C}{\partial a_L}$
For $k=L$ to $2$, compute $\frac{\partial C}{\partial a_{k-1}}$
knowing $\frac{\partial C}{\partial a_k}$ and $w_b$
3. Parameter update:
Compute $\frac{\partial C}{\partial w_b}$ and $\frac{\partial C}{db_{t-1}}$
knowing $\frac{\partial C}{\partial a_k}$ and $a_{k-1}$
$w_t \leftarrow {\rm clip}(w_{t-1} - \eta \frac{\partial C}{\partial w_b})$
$b_t \leftarrow b_{t-1} - \eta \frac{\partial C}{\partial b_{t-1}}$
SGD training with BinaryConnect. $C$ is the cost function for minibatch and the functions binarize($w$) and clip($w$) specify how to binarize and clip weights. $L$ is the number of layers.
A key point to understand with BinaryConnect is that we only binarize the weights during the forward and backward propagations
(steps 1 and 2) but not during the parameter update (step 3), as illustrated in Algorithm <ref>.
Keeping good precision weights during the updates is necessary for SGD to work at all.
These parameter changes are tiny by virtue of being obtained
by gradient descent, i.e., SGD performs a large number of almost infinitesimal changes in the direction that most
improves the training objective (plus noise).
One way to picture all this is to hypothesize that what matters most at the end of training is the sign of
the weights, $w^*$, but that in order to figure it out, we perform a lot of small changes to a continuous-valued
quantity $w$, and only at the end consider its sign:
\begin{equation}
w^* = {\rm sign}(\sum_t g_t)
\end{equation}
where $g_t$ is a noisy estimator of $\frac{\partial C(f(x_t,w_{t-1},b_{t-1}),y_t)}{\partial w_{t-1}}$,
where $C(f(x_t,w_{t-1},b_{t-1}),y_t)$ is the value of the objective function on (input,target) example $(x_t,y_t)$,
when $w_{t-1}$ are the previous weights and $w^*$ is its final discretized value of the weights.
Another way to conceive of this discretization is as a form of corruption, and hence as a regularizer, and our
empirical results confirm this hypothesis. In addition, we can make the discretization errors on different weights
approximately cancel each other while keeping a lot of precision by randomizing the discretization
appropriately. We propose a form of randomized discretization that preserves the expected
value of the discretized weight.
Hence, at training time, BinaryConnect randomly picks one of two values for each weight, for each
minibatch, for both the forward and backward propagation phases of backprop. However, the SGD update
is accumulated in a real-valued variable storing the parameter.
An interesting analogy to understand BinaryConnect is the DropConnect algorithm <cit.>.
Just like BinaryConnect, DropConnect only injects noise to the weights during the propagations. Whereas
DropConnect's noise is added Bernouilli noise, BinaryConnect's noise is a binary sampling process. In both cases
the corrupted value has as expected value the clean original value.
§.§ Clipping
Since the binarization operation is not influenced by variations of the real-valued weights $w$
when its magnitude is beyond the binary values $\pm 1$, and since it is a common practice
to bound weights (usually the weight vector) in order to regularize them, we have chosen to
clip the real-valued weights within the $[-1,1]$ interval right after the weight updates,
as per Algorithm <ref>.
The real-valued weights would otherwise grow very large without any impact on the binary weights.
§.§ A few more tricks
Optimization No learning rate scaling Learning rate scaling
SGD 11.45%
Nesterov momentum 15.65% 11.30%
ADAM 12.81% 10.47%
Test error rates of a (small) CNN trained on CIFAR-10
depending on optimization method and on whether the learning rate is scaled
with the weights initialization coefficients from <cit.>.
We use Batch Normalization (BN) <cit.> in all of our experiments,
not only because it accelerates the training by reducing internal covariate shift,
but also because it reduces the overall impact of the weights scale.
Moreover, we use the ADAM learning rule <cit.> in all of our CNN experiments.
Last but not least, we scale the weights learning rates
respectively with the weights initialization coefficients from <cit.> when optimizing with ADAM,
and with the squares of those coefficients when optimizing with SGD or Nesterov momentum <cit.>.
Table <ref> illustrates the effectiveness of those tricks.
§.§ Test-Time Inference
Up to now we have introduced different ways of training a DNN with on-the-fly weight binarization.
What are reasonable ways of using such a trained network, i.e., performing test-time inference on new examples?
We have considered three reasonable alternatives:
* Use the resulting binary weights $w_b$ (this makes most sense with the deterministic form of BinaryConnect).
* Use the real-valued weights $w$, i.e., the binarization only helps to achieve faster training but
not faster test-time performance.
* In the stochastic case, many different networks can be sampled by sampling a $w_b$ for each weight
according to Eq. <ref>. The ensemble output of these networks can then be obtained by averaging
the outputs from individual networks.
We use the first method with the deterministic form of BinaryConnect.
As for the stochastic form of BinaryConnect,
we focused on the training advantage and used the second method in the experiments, i.e.,
test-time inference using the real-valued weights. This follows the practice of Dropout methods, where
at test-time the “noise” is removed.
Method MNIST CIFAR-10 SVHN
No regularizer 1.30 $\pm$ 0.04% 10.64% 2.44%
BinaryConnect (det.) 1.29 $\pm$ 0.08% 9.90% 2.30%
BinaryConnect (stoch.) 1.18 $\pm$ 0.04% 8.27% 2.15%
50% Dropout 1.01 $\pm$ 0.04%
Maxout Networks <cit.> 0.94% 11.68% 2.47%
Deep L2-SVM <cit.> 0.87%
Network in Network <cit.> 10.41% 2.35%
DropConnect <cit.> 1.94%
Deeply-Supervised Nets <cit.> 9.78% 1.92%
Test error rates of DNNs trained on the MNIST (no convolution and no unsupervised pretraining),
CIFAR-10 (no data augmentation) and SVHN, depending on the method.
We see that in spite of using only a single bit per weight during propagation, performance
is not worse than ordinary (no regularizer) DNNs, it is actually better, especially with the
stochastic version, suggesting that BinaryConnect acts as a regularizer.
Features of the first layer of an MLP trained on MNIST depending on the regularizer.
From left to right: no regularizer, deterministic BinaryConnect, stochastic BinaryConnect and Dropout.
Histogram of the weights of the first layer of an MLP trained on MNIST depending on the regularizer.
In both cases, it seems that the weights are trying to become deterministic to reduce the training error.
It also seems that some of the weights of deterministic BinaryConnect are stuck around 0, hesitating between $-1$ and $1$.
Training curves of a CNN on CIFAR-10 depending on the regularizer.
The dotted lines represent the training costs (square hinge losses)
and the continuous lines the corresponding validation error rates.
Both versions of BinaryConnect significantly augment the training cost, slow down the training
and lower the validation error rate, which is what we would expect from a Dropout scheme.
§ BENCHMARK RESULTS
In this section,
we show that BinaryConnect acts as regularizer and
we obtain near state-of-the-art results with BinaryConnect on the permutation-invariant MNIST, CIFAR-10 and SVHN.
§.§ Permutation-invariant MNIST
MNIST is a benchmark image classification dataset <cit.>.
It consists in a training set of 60000 and a test set of 10000 28 $\times$ 28 gray-scale images representing digits ranging from 0 to 9.
Permutation-invariance means that the model must be unaware of the image (2-D) structure of the data (in other words, CNNs are forbidden).
Besides, we do not use any data-augmentation, preprocessing or unsupervised pretraining.
The MLP we train on MNIST consists in 3 hidden layers of 1024 Rectifier Linear Units (ReLU)
and a L2-SVM output layer (L2-SVM has been shown to perform
better than Softmax on several classification benchmarks <cit.>).
The square hinge loss is minimized with SGD without momentum.
We use an exponentially decaying learning rate.
We use Batch Normalization with a minibatch of size 200 to speed up the training.
As typically done, we use the last 10000 samples of the training set as a validation set for early stopping and model selection.
We report the test error rate associated with the best validation error rate after 1000 epochs
(we do not retrain on the validation set).
We repeat each experiment 6 times with different initializations.
The results are in Table <ref>.
They suggest that the stochastic version of BinaryConnect can be considered a regularizer,
although a slightly less powerful one than Dropout, in this context.
§.§ CIFAR-10
CIFAR-10 is a benchmark image classification dataset.
It consists in a training set of 50000 and a test set of 10000 32 $\times$ 32 color images representing
airplanes, automobiles, birds, cats, deers, dogs, frogs, horses, ships and trucks.
We preprocess the data using global contrast normalization and ZCA whitening.
We do not use any data-augmentation (which can really be a game changer for this dataset <cit.>).
The architecture of our CNN is:
\begin{equation}
(2 \times 128C3)-MP2-(2 \times 256C3)-MP2-(2 \times 512C3)-MP2-(2 \times 1024FC)-10SVM
\end{equation}
Where $C3$ is a $3\times3$ ReLU convolution layer, $MP2$ is a $2\times2$ max-pooling layer,
$FC$ a fully connected layer, and SVM a L2-SVM output layer.
This architecture is greatly inspired from VGG <cit.>.
The square hinge loss is minimized with ADAM.
We use an exponentially decaying learning rate.
We use Batch Normalization with a minibatch of size 50 to speed up the training.
We use the last 5000 samples of the training set as a validation set.
We report the test error rate associated with the best validation error rate after 500 training epochs
(we do not retrain on the validation set).
The results are in Table <ref> and Figure <ref>.
§.§ SVHN
SVHN is a benchmark image classification dataset.
It consists in a training set of 604K and a test set of 26K 32 $\times$ 32 color images representing
digits ranging from 0 to 9.
We follow the same procedure that we used for CIFAR-10, with a few notable exceptions:
we use half the number of hidden units and we train for 200 epochs instead of 500 (because SVHN is quite a big dataset).
The results are in Table <ref>.
§ RELATED WORKS
Training DNNs with binary weights has been the subject of very recent works
Even though we share the same objective, our approaches are quite different.
<cit.> do not train their DNN with Backpropagation (BP)
but with a variant called Expectation Backpropagation (EBP).
EBP is based on Expectation Propagation (EP) <cit.>,
which is a variational Bayes method used to do inference in probabilistic graphical models.
Let us compare their method to ours:
* It optimizes the weights posterior distribution (which is not binary).
In this regard, our method is quite similar as we keep a real-valued version of the weights.
* It binarizes both the neurons outputs and weights,
which is more hardware friendly than just binarizing the weights.
* It yields a good classification accuracy for fully connected networks (on MNIST)
but not (yet) for ConvNets.
<cit.> retrain neural networks
with ternary weights during forward and backward propagations, i.e.:
* They train a neural network with high-precision,
* After training, they ternarize the weights to three possible values $-H$, $0$ and $+H$
and adjust $H$ to minimize the output error,
* And eventually, they retrain with ternary weights during propagations and high-precision weights during updates.
By comparison, we train all the way with binary weights during propagations, i.e.,
our training procedure could be implemented with efficient specialized hardware
avoiding the forward and backward propagations multiplications,
which amounts to about $2/3$ of the multiplications (cf. Algorithm <ref>).
§ CONCLUSION AND FUTURE WORKS
We have introduced a novel binarization scheme for weights during forward and backward propagations
called BinaryConnect.
We have shown that it is possible to train DNNs with BinaryConnect
on the permutation invariant MNIST, CIFAR-10 and SVHN datasets
and achieve nearly state-of-the-art results.
The impact of such a method on specialized hardware implementations of deep
networks could be major, by removing the need for about 2/3 of the multiplications,
and thus potentially allowing to speed-up by a factor of 3 at training time.
With the deterministic version of BinaryConnect the impact at test time could
be even more important, getting rid of the multiplications altogether
and reducing by a factor of at least 16 (from 16 bits single-float
precision to single bit precision) the memory requirement of deep networks,
which has an impact on the memory to computation bandwidth and on the size of
the models that can be run.
Future works should extend those results to other models and datasets,
and explore getting rid of the multiplications altogether during training, by
removing their need from the weight update computation.
§ ACKNOWLEDGMENTS
We thank the reviewers for their many constructive comments.
We also thank Roland Memisevic for helpful discussions.
We thank the developers of Theano <cit.>,
a Python library which allowed us to easily develop a fast and optimized code for GPU.
We also thank the developers of Pylearn2 <cit.>
and Lasagne <cit.>,
two Deep Learning libraries built on the top of Theano.
We are also grateful for funding from NSERC, the Canada Research Chairs, Compute Canada, Nuance Foundation, and CIFAR.
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1511.00469
|
Department of Physics, University of Oslo, N-0316 Oslo, Norway
Department of Physics, University of Oslo, N-0316 Oslo, Norway
Department of Physics, University of Oslo, N-0316 Oslo, Norway
Department of Physics, University of Oslo, N-0316 Oslo, Norway
Department of Physics, University of Oslo, N-0316 Oslo, Norway
Department of Physics, University of Oslo, N-0316 Oslo, Norway
Department of Physics, University of Oslo, N-0316 Oslo, Norway
We have analyzed primary $\gamma$-ray spectra of the odd-odd $^{238}$Np nucleus extracted
from $^{237}$Np($d,p\gamma$)$^{238}$Np coincidence data measured at the Oslo Cyclotron Laboratory.
The primary $\gamma$ spectra cover an excitation-energy region of $0 \leq E_i \leq 5.4$ MeV,
and allowed us to perform a detailed study of the $\gamma$-ray strength as function of
excitation energy. Hence, we could test the validity of the generalized Brink-Axel hypothesis,
which, in its strictest form, claims no excitation-energy dependence on the $\gamma$ strength.
In this work, using the available high-quality $^{238}$Np data,
we show that the $\gamma$-ray strength function is to a very large extent independent on
the initial and final states.
Thus, for the first time, the generalized Brink-Axel hypothesis
has been experimentally verified for $\gamma$ transitions between states in the quasi-continuum region,
not only for specific collective resonances, but also for the full strength below the neutron
separation energy. Based on our findings, the necessary criteria for the generalized
Brink-Axel hypothesis to be fulfilled are outlined.
24.30.Gd, 21.10.Ma, 25.40.Hs
Sixty years ago, David M. Brink proposed in his PhD thesis <cit.> that
the photoabsorption cross section of the giant electric dipole resonance (GDR)
is independent of the detailed structure of the initial state.
In his thesis, he expressed his hypothesis as follows: "If it
were possible to perform the photo effect on an excited state, the cross section
for absorption of a photon of energy $E$ would still
have an energy dependence given by (15)", where equation (15) refers
to a Lorentzian shape of the photoabsorption cross section.
Brink's original idea, the Brink hypothesis, was first
intended for the photoabsorption process on the GDR, but
has been further generalized, applying the principle of detailed balance,
to include absorption and emission of $\gamma$ rays between resonant states <cit.>.
In addition to assuming independence of excitation energy,
there is no explicit dependence of initial and final spins except the obvious
dipole selection rules, implying that all levels exhibit the same dipole strength
regardless of their initial spin quantum number.
We will refer to this as the generalized Brink-Axel (gBA) hypothesis.
A review of the history of the hypothesis was given
by Brink in Ref. <cit.>.
The gBA hypothesis has implications for almost
any situation where nuclei are brought to an excited state above $\approx 2\Delta$,
where $\Delta \approx 1$ MeV is the pair-gap parameter.
Here, the nucleus will typically de-excite via $\gamma$-ray emission
and/or by emission of particles. In this context, it is usual to translate the
$\gamma$-ray cross section $\sigma(E_{\gamma})$ into $\gamma$-ray strength function ($\gamma$SF)
by $f(E_{\gamma})=(3\pi^2\hbar^2c^2)^{-1}\sigma(E_{\gamma})/E_{\gamma}$.
To describe and model the electric dipole part of the $\gamma$-decay channel, the gBA hypothesis is
frequently used, applying in particular the assumption of spin independence <cit.>.
For example, a rather standard approach to calculating $E1$ strength is to apply some variant of the
quasi-particle random-phase approximation (QRPA) to obtain $B(E1)$ values as function of excitation energy,
and assuming that this $E1$ distribution corresponds to the one in the quasi-continuum;
see, e.g., Ref. <cit.> and references therein.
Also for $M1$ transitions the gBA hypothesis has been utilized, see e.g. Ref. <cit.>.
Furthermore, the hypothesis is also often applied to $\beta$-decay and electron capture
for calculating Gamow-Teller and Fermi transition strengths,
see e.g. Ref. <cit.> and references therein. The main reason for its wide
range of applications is the drastic simplification of the considered problem, and
in some cases it is a key necessity to be able to perform the desired calculation <cit.>.
Hence, the question of whether the hypothesis is valid or not, and under which circumstances,
is of utmost importance for multiple reasons: its fundamental impact on nuclear structure
and dynamics, and its pivotal role for the description of $\gamma$ and $\beta$ decay
for applied nuclear physics, such as input for ($n,\gamma$) cross-section calculations
relevant to the $r$-process nucleosynthesis in extreme astrophysical
environments <cit.> and next-generation nuclear power plants <cit.>.
However, it is not at all obvious neither from
experiment nor theory that the gBA hypothesis is valid.
From an experimental point of view, there are two main reasons for this;
the hypothesis has primarily been tested at very high excitation energies or with
only a few states included.
In the first case, compilations show that the width of the GDR varies
with temperature and spin, in contradiction to the gBA hypothesis <cit.>. However,
the hypothesis was not originally considered for building the GDR on such highly excited states.
Obviously, thermal fluctuations will affect the width of the GDR,
but the GDR energy centroid stays rather fixed.
Other test cases suffer from large Porter-Thomas (PT) fluctuations <cit.>, since
the $\gamma$SF could not be averaged over a sufficient amount of levels <cit.>.
In particular, this is the case for lighter nuclei or if levels close to the ground state are considered.
In general, experimental data supporting the gBA hypothesis are rather scarce.
For example, $(n,\gamma)$ reactions give
$\gamma$SFs consistent with the gBA hypothesis, but in a limited
$\gamma$-ray energy range <cit.>.
Furthermore, data on the $^{89}$Y$(p,\gamma)^{90}$Zr
reaction point towards deviations from the gBA hypothesis <cit.>.
There have also been various theoretical attempts to test the gBA hypothesis and modifications or
even violations are found <cit.>.
For some theoretical applications, the assumption
of the gBA hypothesis is successfully applied <cit.>.
We may learn from these experimental and theoretical attempts that
the structure and dynamics of the system represents important constraints.
In this Letter, we address the gBA hypothesis
from an experimental point of view, and we provide the needed criteria for the hypothesis
to be valid for $\gamma$ decay below the neutron threshold by a detailed analysis of the
$^{238}$Np $\gamma$SF.
The $^{238}$Np nucleus is probably the ultimate case to test the gBA hypothesis,
as it is an odd-odd system with extremely high level density.
Already a few hundred keV above the ground state, we find a level density of $\approx 200$ MeV$^{-1}$,
which increases to $\approx 43\cdot 10^{6}$ MeV$^{-1}$ at the neutron separation energy of $S_n=5.488$ MeV.
In a previous study <cit.>, the level density and $\gamma$SF were
extracted from the distributions of primary $\gamma$-rays measured in the
$^{237}$Np$(d, p \gamma)^{238}$Np reaction. This very rich data set represents ideal
conditions for testing the gBA hypothesis where
the PT fluctuations are negligible due to the high level density.
In the following, we utilize the primary matrix of initial excitation energy $E_i$ versus $\gamma$-ray
energy <cit.>.
(Color online) Primary $\gamma$-ray matrix of $^{238}$Np <cit.>.
(Color online) Primary $\gamma$-ray spectra (data points) at various initial excitation energies
compared to the product $\rho(E_i-E_{\gamma}){\cal{T}}(E_{\gamma})$ (blue curve).
(Color online) The procedure to extract $\gamma$SFs as function of
initial $E_i$ (left) and final $E_f$ (right) excitation energies.
The blue-shaded region (middle) illustrates the exponentially increasing
level density as function of excitation energy. The two $\gamma$SFs
are limited to $E_{\gamma} < E_i$ and
$E_{\gamma} < S_n - E_f$, respectively, where $S_n$ is the neutron separation energy.
(Color online) Comparison between the three $\gamma$SFs obtained by
Eqs. (<ref>), (<ref>) and (<ref>).
Figure <ref> shows the primary $U(E_{\gamma},E_i)$ $\gamma$ spectra
(unfolded with the detector response functions)
as function of initial excitation energy $E_i$.
We now normalize $U$ to obtain the probability that the nucleus emits
a $\gamma$-ray with energy $E_{\gamma}$ at excitation energy $E_i$ by
$P(E_{\gamma},E_i )= U(E_{\gamma},E_i)/\sum_{E_{\gamma}}U(E_{\gamma},E_i)$.
The probability is assumed to be factorized into:
\begin{equation}
P(E_{\gamma},E_i ) \propto \rho(E_i-E_{\gamma}){\cal{T}}(E_{\gamma}) .\
\label{eq:rhoT}
\end{equation}
According to Fermi's golden rule <cit.>, the decay probability $P$ is
proportional to the level density at the final energy $\rho(E_i-E_{\gamma})$.
The decay probability is also proportional to the squared transition matrix element
$|\langle f| \hat{T}|i \rangle|^2$ between initial $|i\rangle$ and final $\langle f|$ states,
which is represented by
the $\gamma$-ray transmission coefficient ${\cal{T}}$ when averaged over many transitions
with the same transition energy $E_\gamma$. For now, let us assume
that the transmission coefficient depends only on $E_{\gamma}$, in accordance with the
gBA hypothesis.
The factorization given in Eq. (<ref>) allows us
to simultaneously extract the functions $\rho$
and ${\cal{T}}$ from the two-dimensional probability landscape $P$.
The technique used, the Oslo method <cit.>, requires no
models for these functions.
In the present case <cit.>,
we have fitted each vector element of the two functions to the following region of the $P$ matrix:
$3.0 \leq E_i \leq 5.4$ MeV and $E_{\gamma}> 0.84$ MeV.
The justification
of this standard procedure has been experimentally tested for many nuclei by the Oslo group <cit.>
and a survey of possible errors for the Oslo method was presented in Ref. <cit.>.
The applicability of Eq. (<ref>) and the quality of the least $\chi ^2$ fitting
procedure are demonstrated in Fig. <ref>.
The agreement is very satisfactory with a $\chi ^2_{\rm reduced} = 0.81$, and indicates
that the determination of the level density $\rho$
and the transmission coefficient ${\cal{T}}$ works well. In the following we assume that $\rho$ and ${\cal{T}}$ are
normalized according to the procedure in Ref. <cit.>. Thus, we introduce
a normalization factor $N$ in Eq. (<ref>), which only depends on the initial excitation energy,
and rewrite:
\begin{equation}
N(E_i)P(E_{\gamma},E_i ) = \rho(E_i-E_{\gamma}){\cal{T}}(E_{\gamma}) ,\
\label{eq:rhoTnorm}
\end{equation}
which determines the normalization factor by
\begin{equation}
N(E_i)=\frac{\int_0^{E_i} {\cal T}(E_{\gamma}) \rho(E_i-E_{\gamma})\, {\mathrm{d}} E_{\gamma }
}{\int_0^{E_i} \, P(E_{\gamma},E_i)\, {\mathrm{d}} E_{\gamma}}.
\label{eq:nei}
\end{equation}
It is an open question
if the transmission coefficient ${\cal{T}}$ actually changes with excitation energy,
as this procedure gives an average ${\cal{T}}$ for a rather wide range of initial
excitation energies in the standard Oslo method.
Hence, it is possible that variations of ${\cal{T}}$ as function
of initial (or final) excitation energy might be masked.
In the following, we will investigate whether ${\cal{T}}$ depends on initial
and final excitation energies in order to test the gBA hypothesis. For this we
collect $\gamma$ transitions from certain initial states or $\gamma$ transitions into
certain final states as illustrated in Fig. <ref>.
The idea is that, as the level density $\rho$ is known, the
$\gamma$ transmission coefficient can be studied in detail
per excitation energy bin simply by $NP/\rho$ from Eq. (<ref>).
(Color online) The $\gamma$SFs as function of initial excitation energies (data points),
see Eq. (<ref>). The blue curve
is obtained by the standard Oslo method, see Eq. (<ref>). The excitation energy bins are 121 keV broad.
(Color online) The $\gamma$SFs as function of final excitation energies (data points),
see Eq. (<ref>). See text of Fig. <ref>.
More specifically, we get for initial states:
\begin{equation}
{\cal T} (E_{\gamma}, E_i ) =
\frac{N(E_i)P(E_{\gamma},E_i)}{\rho(E_i - E_{\gamma})},
\label{eq:Ei}
\end{equation}
or alternatively for final states:
\begin{equation}
{\cal T} (E_{\gamma},E_f) =
\frac{N(E_f + E_{\gamma})P(E_{\gamma},E_f + E_{\gamma})}{\rho(E_f)},
\label{eq:Ef}
\end{equation}
where $E_f=E_i-E_{\gamma}$.
One should note that
the normalization factor $N$ is calculated from the assumption that
both ${\cal T} (E_{\gamma},E_i)$ and ${\cal T} (E_{\gamma},E_f)$
fluctuate on the average around the excitation-independent ${\cal T} ( E_{\gamma})$,
see Eq. (<ref>).
We now translate the $\gamma$ transmission coefficient into $\gamma$SF by <cit.>
\begin{equation}
f(E_{\gamma}) = \frac{1}{2\pi}\frac{{\cal T}(E_{\gamma})}{E_{\gamma}^{2L +1}},
\label{eq:gSF}
\end{equation}
where we assume that dipole radiation ($L=1 $) dominates the $\gamma$ decay in the quasi-continuum.
This is motivated by known discrete $\gamma$-ray
transitions from neutron capture <cit.>
and angular distributions of primary $\gamma$-rays measured at
high excitation energies <cit.>.
To check that the normalization function $N(E_i)$ is reasonable, we compare
the three $\gamma$SFs obtained from Eqs. (<ref>), (<ref>) and (<ref>),
where $f(E_{\gamma},E_i )$ and $f(E_{\gamma},E_f)$ are averaged over
initial and final excitation energies by
\begin{eqnarray}
f_i (E_{\gamma}) &=& \frac{1}{S_n - E_{\gamma}}\int_{0}^{S_n-E_{\gamma}} f (E_{\gamma},E_i )\, \mathrm{d}E_i,\\
f_f (E_{\gamma}) &=& \frac{1}{S_n - E_{\gamma}}\int_{E_{\gamma}}^{S_n} f (E_{\gamma},E_f)\, \mathrm{d}E_f,
\label{eq:3gsf}
\end{eqnarray}
respectively. Figure <ref> demonstrates that the three extraction methods
give $f(E_{\gamma}) = f_i(E_{\gamma})= f_f(E_{\gamma})$ within the experimental errors.
This supports the normalization function $N$ used in Eqs. (<ref>) and (<ref>).
We are now ready to compare our data with the gBA hypothesis. Figure <ref>
shows the initial excitation energy dependent $f(E_{\gamma}, E_i)$ compared to $f(E_{\gamma}$) obtained
with the standard Oslo method (blue curve). The excitation-energy bins are $\Delta E_i = 121$ keV wide,
and only every fourth gate is shown. The overall agreement is excellent, and
the same holds also for all the bins not shown. It is clear that each of these $\gamma$SFs are
built on a specific initial excitation-energy gate, but with no specific final state, as illustrated
in Fig. <ref>. However, for a given
$E_{\gamma}$ and $E_i$, the final excitation energy is determined. Since all
data points coincide with $f(E_{\gamma})$,
this also indicates independence of the final state.
Any potential dependence of the final state is best analyzed by $f(E_{\gamma},E_f )$ as given
by Eq. (<ref>) and shown in Fig. <ref>.
Again, we find an excellent agreement between the various $\gamma$SFs with $\gamma$ transitions
into specific final excitation-energy bins. However, there are discrepancies for $E_{\gamma} < 1$ MeV, which feed
final states below $\approx 1$ MeV. At these energies, $f(E_{\gamma},E_f)$ shows an increase compared to the average
$f(E_{\gamma})$. These $\gamma$ transitions could possibly be due to vibrational modes built on
the ground state, and, if this be true, not part of a general $\gamma$SF extracted
in the quasi-continuum with the standard Oslo method. Vibrational levels are
strongly populated in inelastic scattering, such as the reactions $^{237,239}$Np($d,d^{\prime}$)$^{237,239}$Np
performed by Thompson et al. <cit.>. In that work, levels built on vibration modes were seen
for excitation energies in the $\approx 0.9$-MeV and $\approx 1.6$-MeV regions.
A similar population of vibrational states
has been observed in the $^{238}$U($^{16}$O,$^{16}$O$^{\prime}$)$^{238}$U and $^{238}$U($\alpha$,$\alpha^{\prime}$)$^{238}$U
reactions <cit.>. By means of $\alpha\gamma$-coincidences, a concentration
of $E_{\gamma}\approx 1$ MeV transitions depopulating $\beta$-, $\gamma$- and octupole vibrational bands
has been seen.
Thus, the enhanced $\gamma$SF found in our data at low excitation energies with $E_{\gamma} \approx 1$ MeV
is likely due to deexcitation of vibrational structures, which do not show up in the
high level density quasi-continuum.
The excellent agreement between excitation energy dependent
and independent $\gamma$SFs indicate that PT
fluctuations are small compared to the experimental errors for the system studied.
For the $\chi_{\nu}^2$ distribution, which governs the PT fluctuations,
the relative fluctuations are given by $r=\sqrt{(2/\nu)}$ where
$\nu$ is the number of degrees of freedom <cit.>. Typically, we have at 2.0 and 4.0
MeV of excitation energies, $\approx 1200$ and $\approx 120000$ levels within the 121-keV excitation energy bins.
Taking the number of levels as the number of degrees of freedom,
we obtain $r = 4.1$ % and 0.4 %, respectively, which
should be compared with the data error bars of Figs. <ref> and <ref> of
typically 10%. Thus, in the $^{238}$Np case, the PT
fluctuations are smaller than the statistical errors and
not significant. However, for systems with less than $\approx 200$ levels
per bin the PT fluctuations will exceed the experimental statistical error of 10%.
This gives guidance for the necessary statistics and the required level density
for the gBA hypothesis to be fulfilled.
In conclusion, we have studied the $\gamma$-ray strength function between
well defined excitation energy bins in $^{238}$Np.
For the first time, the generalized Brink-Axel hypothesis has been
verified in the nuclear quasi-continuum. The
discrepancies seen in the low excitation energy region are probably caused by
decay of vibrational states built on the ground state. These excitation modes
are not part of the $\gamma$-ray strength function of the
quasi-continuum. The validity of the generalized Brink-Axel hypothesis
requires that Porter-Thomas fluctuations are low by averaging over a sufficient amount of levels
compared to the experimental errors.
The authors wish to thank J.C. Müller, E.A. Olsen, A. Semchenkov and J. Wikne
at the Oslo Cyclotron Laboratory for providing excellent experimental conditions.
A.C.L. acknowledges financial support through the ERC-STG-2014 under grant agreement no. 637686.
S.S. and G.M.T. acknowledge financial support by the NFR under project grants no. 210007 and 222287, respectively.
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1511.00315
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The study of the birational properties of algebraic $k$-tori began in the
sixties and seventies with work of Voskresenkii, Endo, Miyata, Colliot-Thélène and Sansuc. There was particular interest in determining the rationality of a given algebraic $k$-tori. As rationality problems for algebraic varieties are in general difficult, it is natural to consider relaxed notions such as stable rationality, or even retract rationality. Work of the above authors
and later Saltman in the eighties determined
necessary and sufficient conditions to determine when an algebraic torus is
stably rational, respectively retract rational in terms of the integral
representations of its associated character lattice. An interesting question
is to ask whether a stably rational algebraic $k$-torus
is always rational. In the general case, there exist examples of non-rational stably rational $k$-varieties. Algebraic $k$-tori of dimension $r$ are classified up to isomorphism by conjugacy classes of finite subgroups of $\GL_r(\Z)$.
This makes it natural to examine the rationality problem for algebraic $k$
tori of small dimensions.
In 1967, Voskresenskii <cit.> proved that all algebraic tori of dimension 2 are rational. In 1990, Kunyavskii <cit.> determined which algebraic tori of dimension 3 were rational. In 2012, Hoshi and Yamasaki <cit.> determined which algebraic tori
of dimensions 4 and 5 were stably (respectively retract) rational with the aid of GAP. They did not address the rationality question in dimensions 4 and 5.
In this paper, we show that all stably rational algebraic $k$-tori of dimension 4 are rational, with the possible exception of 10 undetermined cases.
Hoshi and Yamasaki found 7 retract rational but not stably rational dimension 4 algebraic $k$-tori. We give a non-computational proof of these results.
§ INTRODUCTION
Rationality problems in algebraic geometry are central but notoriously difficult questions, making it natural to consider relaxed notions.
Let $k$ be a field and let $X$ be a $k$-variety.
$X$ is $k$-rational if it is birationally isomorphic to projective $n$ space, $\PP^n_k$; $X$ is stably $k$-rational if $X\times_k \A^r$ is rational for some $r\ge 0$.
$X$ is retract $k$-rational if there exist rational maps
$f:X\brokrarr \A^n$ and $g:\A^n\brokrarr X$ such that $g\circ f=\id_X$.
$X$ is $k$-unirational if there exists a dominant rational map
$\PP^m_k\brokrarr X$, for some $m$.
Note that rationality implies stably rationality implies retract rationality
implies unirationality.
It is not too hard to find non-stably rational retract rational $k$-varieties
- we will discuss such examples later. It is much more difficult to find non-rational stably rational $k$-varieties - although examples were found by
Beauville, Colliot-Thélène, Sansuc and Swinnerton-Dyer <cit.>
in 1985.
In this paper, we examine the rationality problem for algebraic $k$-tori of dimension 4 where $k$ is a field of characteristic 0. For algebraic $k$-tori, the question of whether
there exist non-rational stably rational algebraic $k$-tori remains open to the
best of the author's knowledge.
For any fixed quasi-projective $k$-variety $X$, a $k$-form of $X$
is another $k$-variety $Y$ which is isomorphic to $X$ after extending to the
separable closure $k_s$ of $k$.
Isomorphism classes of $k$-forms of $X$ are in bijection with elements
of non-abelian Galois cohomology set $H^1(\G_k,\Aut(X))$ where
$\G_k=\Gal(k_s/k)$ is the absolute Galois group.
An algebraic $k$-torus $T$ of dimension $r$ is a $k$-form of a split $k$-torus so that
$$T\times_k k_s\cong \GG^r_{m,k_s}$$
Algebraic $k$-tori of dimension $r$ are determined up to isomorphism
by continuous representations of $\G_k$ into $\GL_r(\Z)$.
Any algebraic $k$-tori of dimension $r$ is split by a finite Galois extension $L$ over $k$.
That is, there exists a finite Galois extension $L/k$ such that
$$T\times_k L\cong \GG^r_{m,L}$$
Algebraic $k$-tori of dimension $r$ which are split by a finite Galois extension $L/k$ are in bijection with conjugacy classes of finite subgroups of $\GL_r(\Z)$.
More precisely, a finite subgroup $G$ of $\GL_r(\Z)$ up to conjugacy determines a $G$-lattice $M_G$ up to isomorphism.
Then $T_G=\Spec(L[M_G]^G)$ is an algebraic $k$-torus of dimension $r$ split by $L$.
Conversely, the character lattice of an algebraic $k$-torus of dimension $r$ split by $L$ is a $G$-lattice of rank $r$ and so determines a conjugacy class of finite subgroups of $\GL_r(\Z)$.
Given a finite subgroup $G$ of $\GL_r(\Z)$ and a Galois $G$-extension $L/k$,
the function field
$T_G=\Spec(L[M_G]^G)$ is the quotient field $K=L(M_G)^G$ of $L[M_G]^G$.
This concrete description allows us to take advantage of the fact that each of the rationality concepts defined above for a $k$-variety $X$ can be rephrased
in terms of its function field $K=k(X)$.
$X$ is $k$-rational if and only if $K/k$ is a rational extension, i.e.
there exist $z_1,\dots, z_n\in K$ which are algebraically independent over $k$
and such that $K=k(z_1,\dots,z_n)$.
$X$ is stably $k$-rational if and only if there exists a field $L$ containing $K$ which is rational over both $k$ and $K$.
$X$ is retract $k$-rational if and only if $K$ contains a $k$-algebra $R$ such that $K$ is the quotient field of $R$ and the
identity map $1_R$ factors through the localisation of a polynomial ring over $k$.
$X$ is $k$-unirational if and only if $k\subseteq K \subseteq k(x_1,\dots,x_m)$ for some $m$.
Voskresenskii, Endo, Miyata, Colliot-Thélène, Sansuc, Saltman studied the rationality problem for algebraic $k$-tori from the sixties to the
eighties <cit.>. They determined conditions under which a algebraic $k$-torus is stably
rational, and retract rational respectively. The conditions were phrased in terms of the character lattice of the algebraic $k$-torus as a lattice over its
splitting group.
For algebraic tori of small dimensions, the question of retract/stable rationality has been addressed. Since algebraic $k$-tori of dimension $r$ split by a Galois extension $L/k$ are determined up to isomorphism by conjugacy classes of finite subgroups of $\GL_r(\Z)$, this is a finite set by a theorem of Jordan.
For small values of $r$, the classification of conjugacy classes of finite
subgroups is known but the number of such conjugacy classes grows rapidly
with $r$.
For $r=2$, there are 13 conjugacy classes of finite subgroups of $\GL_2(\Z)$ and hence 13 possible isomorphism classes of algebraic $k$-tori.
Voskresenskii <cit.> showed in 1967 that all 2-dimensional algebraic $k$-tori are
For $r=3$, there are 73 conjugacy classes of finite subgroups of $\GL_3(\Z)$,
a classification due to Tahara <cit.> in 1971.
In 1990, Kunyavskii <cit.> classified algebraic $k$-tori of dimension 3 up to birational equivalence. He found that all but 15 were rational. The remaining 15 were
not even retract rational.
There are 710 conjugacy classes of finite subgroups of $\GL_4(\Z)$. Dade <cit.>
found the maximal conjugacy classes of finite subgroups of $\GL_4(\Z)$ in 1965
without the use of a computer by analyzing the quadratic forms stabilized by the subgroups. In 1971, Bülow, Neubüser <cit.> determined all the conjugacy classes of finite subgroups of $\GL_4(\Z)$ using Dade's classification
and computer techniques. In 1978, together with H. Brown, H. Wondratschek, and H. Zassenhaus <cit.>, they wrote a book
on Crystallographic Groups of Four-Dimensional Space.
This classification determined the library of crystallographic groups of
dimensions 2,3,4 which is programmed into GAP <cit.>.
In 2012, Hoshi and Yamasaki <cit.> determined which algebraic tori of dimensions 4 and 5 are stably (respectively retract) rational using GAP.
For dimension 4, they found that 487 algebraic tori are stably rational, 7 are
retract but not stably rational and the remaining 216 are not retract rational.
They did not address the issue of rationality.
In this paper, I show that all but possibly 10 of the stably rational algebraic tori
of dimension 4 are rational.
An important source of examples of algebraic tori are the norm one tori.
Given a separable field extension $K/k$ of degree $n$ and $L/k$ the Galois closure of $K/k$, the norm one torus $R^{(1)}_{K/k}(\Gm)$ is the kernel of the
norm map $R_{K/k}(\Gm)\to \Gm$ where $R_{K/k}$ is the Weil restriction <cit.>.
Note $R_{K/k}(\Gm)$ is split by $L/k$. Let $G=\Gal(L/k)$ and $H=\Gal(L/K)$.
The character lattice of $R^{(1)}_{K/k}(\Gm)$ is given by the $G$-lattice $J_{G/H}$, the dual of the kernel of the augmentation ideal of the permutation
lattice $\Z[G/H]$.
The rationality problem for norm one tori was studied by many authors
The stable and retract rationality of norm one tori corresponding to Galois
field extensions is well understood by work of Endo, Miyata, Saltman, Colliot-
Thélène, Sansuc.
Let $K/k$ be a finite Galois extension with Galois
group $G$. Let $R^{(1)}_{K/k}(\Gm)$ be the corresponding norm 1 torus.
* $R^{(1)}_{K/k}(\Gm)$ is retract rational
if and only if the Sylow subgroups of $G$ are cyclic. <cit.>
* $R^{(1)}_{K/k}(\Gm)$ is stably rational if and only if the
Sylow subgroups of $G$ are cyclic and $\hat{H}^0(G,\Z)=\hat{H}^4(G,\Z)$
where $\hat{H}$ indicates Tate cohomology. <cit.>
Note that the conditions above are equivalent to the following restrictions on the structure of the group $G$:
$$G=C_m\mbox{ or } G=C_n\times\langle \sigma,\tau: \sigma^k=\tau^{2^d}=1,
\tau\sigma\tau^{-1}=\sigma^{-1}\rangle$$
where $d\ge 1,k\ge 3, n,k$ odd and $\gcd(n,k)=1$.
These theorems were proven in the 1970s. Some cases of non-Galois separable
extensions were dealt with much more recently.
Let $K/k$ be a finite non-Galois separable field extension
and let $L/k$ be the Galois closure of $K/k$. Let $H\le G$ be the Galois groups
of $L/K$ and $L/k$ respectively.
* If $G$
is nilpotent, then the norm one torus $R^{(1)}_{K/k}(\Gm)$ is not retract rational.
* If all the Sylow subgroups of $G$ are cyclic,
then the norm one torus $R^{(1)}_{K/k}(\Gm)$ is retract $k$-rational and, we
have that
$R^{(1)}_{K/k}(\Gm)$ is stably $k$-rational if and only if
either $G=D_{2n}$ where $n$ is odd or $G=C_m\times D_{2n}$ where $n$ is odd and
$\gcd(m,n)=1$ and $H\le D_{2n}$ is a cyclic group of order 2.
Let $K/k$ be a finite non-Galois separable field extension and $L/k$ be the Galois closure of $K/k$ such that the Galois group of $L/k$ is $S_n$ and that of $L/K$ is $S_{n-1}$.
* $R^{(1)}_{K/k}(\Gm)$ is retract $k$-rational if and only if $n$ is prime.
* $R^{(1)}_{K/k}(\Gm)$ is (stably) $k$-rational if and only if $n=3$.
<cit.> Let $K/k$ be a non-Galois separable field extension of degree
$n$ and let $L/k$ be the Galois closure of $K/k$ such that the Galois
group of $L/k$ is $A_n, n\ge 4$ and that of $L/K$ is $A_{n-1}$.
* $R^{(1)}_{K/k}(\Gm)$ is retract $k$-rational if and only if $n$ is prime.
* For some positive integer $t$, $[R^{(1)}_{K/k}(\Gm)]^{(t)}$ is stably $k$-rational
if and only if $n=5$.
S. Endo asked in <cit.> about the stable rationality of $R^{(1)}_{K/k}(\Gm)$
in the $A_5$ case. Hoshi and Yamasaki <cit.> used GAP to show that in fact
$R^{(1)}_{K/k}(\Gm)$ is stably rational in this case.
In this paper, we present a non-computational proof of that fact.
The norm one torus $R^{(1)}_{K/k}(\Gm)$ corresponding to $A_5$ has
character lattice $J_{A_5/A_4}$.
It is one of the ten algebraic $k$-tori of dimension 4 which is stably
rational but whose rationality is unknown.
Another of these exceptional stably rational algebraic $k$-torus of dimension
4 whose rationality is unknown is intimately
related to this norm one torus. In fact its splitting group is $A_5\times C_2$
and its character lattice restricts to $J_{A_5/A_4}$ on $A_5$.
The character lattices of the remaining 8 stably rational algebraic $k$-tori whose rationality is unknown are also intimately related.
See Section 5 for more details.
This paper is organised as follows.
In Section 2, we will recall some facts about algebraic $k$-tori and
conditions for stable and retract rationality.
In Section 3, we will determine some families of hereditarily rational algebraic $k$-tori. We call an rational algebraic $k$-torus corresponding to a finite subgroup
$G$ of $\GL_r(\Z)$ hereditarily rational if an algebraic $k$-tori corresponding to any subgroup of $G$ is rational. We do not know whether all rational algebraic $k$-tori are hereditarily rational but we show this to be the case for
a number of natural families of algebraic tori.
In Section 4, we show that 8 of the 10 maximal conjugacy classes of
$\GL(4,\Z)$ which correspond to stably rational tori from Hoshi and Yamasaki's list have corresponding algebraic tori which are hereditarily rational.
We then use GAP to show that the
set of rational $k$-tori obtained from subgroups of these groups is of size 477
and matches the set of stably rational $k$-tori of dimension 4 obtained
by Hoshi and Yamasaki with 10 undetermined exceptions. Note that the use of GAP
is very minimal: it is limited to finding conjugacy classes of finite subgroups of $\GL_4(\Z)$ which are subgroups of the groups corresponding to the 8 hereditarily rational algebraic $k$-tori mentioned above.
In Sections 5 and 6, we give non-computational proofs that the remaining two maximal
algebraic $k$-tori of dimension 4 are stably rational, recovering the results of Hoshi and Yamasaki.
We also show that 7 algebraic $k$-tori of dimension 4 are retract but not stably rational in a non-computational way, also recovering the results of Hoshi
and Yamasaki.
We remark that some rationality results for stably rational algebraic tori of dimension 5 have been found by Armin Jamshidpey in a thesis which is soon to be defended.
§ PRELIMINARIES
We begin with some remarks on notation used in this paper.
Throughout, $k$ will be a field of characteristic zero. We will see that this
hypothesis is necessary to apply the criteria for stable and retract rationality.
For finite groups, we will denote by $C_n$
the cyclic group of order $n$; by $D_{2n}$ the dihedral group of order $2n$;
by $S_n$ the symmetric group on $n$ letters and by $A_n$ the alternating group
on $n$ letters.
We will discuss below root system terminology, algebraic torus - lattice correspondence, lattice terminology and birational properties of algebraic tori.
§.§ Root Systems
For more information on root systems, see, for example,
Humphreys <cit.>.
Here we will review the notation that we will use on root systems.
Note that we use root systems on vector spaces over $\Q$, instead of $\R$,
but as our root systems are all crystallographic, this does not affect the theory.
Let $\Phi$ be a root system on a finite dimensional $\Q$ vector space $V$
equipped with a fixed symmetric bilinear form $(\cdot,\cdot)$.
Then, by definition, $V=\Q\Phi$ and
the reflection $s_{\alpha}:V\to V$ in the root $\alpha\in \Phi$,
is given by
$$s_{\alpha}(x)=x-\langle x,\alpha\rangle \alpha$$
$$\langle x,y\rangle=2\frac{(x,y)}{(y,y)},x,y\in V$$
The Weyl group of $\Phi$, or the group generated by these reflections, will be denoted by $W(\Phi)$.
Recall that for a basis $\Delta$ for the root system $\Phi$, we have
$$W(\Phi)=\langle s_{\alpha}:\alpha\in \Phi\rangle =\langle s_{\alpha}:\alpha\in \Delta\rangle$$
so that the Weyl group can be generated by just the simple reflections, those with respect to roots in $\Delta$.
Recall that $\Aut(\Phi)$, the group of automorphisms of the root system $\Phi$,
$$\Aut(\Phi)=\{g\in \GL(\Q\Phi): \langle gv,gw\rangle =\langle v,w\rangle, v,w
\in \Q\Phi\}$$
The Weyl group $W(\Phi)$ is a subgroup of $\Aut(\Phi)$.
We will denote by $\Z\Phi$, the root lattice of $\Phi$. $\Z \Phi$ is the $\Z$-span of $\Phi$
and has $\Z$-basis $\Delta$, where $\Delta$ is a basis of $\Phi$.
Its weight lattice is
$$\Lambda(\Phi)=\{x\in \Q\Phi: \langle x,\alpha\rangle\in \Z\mbox{ for all }\alpha\in \Phi\}.$$
$\Z \Phi\subseteq \Lambda(\Phi)$ are lattices of the same rank
and both are stabilised by the finite subgroup $\Aut(\Phi)$ and so by its
subgroup $W(\Phi)$.
A root system is irreducible if $\Phi$ cannot be partitioned into two proper
orthogonal subsets.
Root systems can be decomposed into a disjoint union of irreducible root systems. There is a classification of irreducible root systems.
There are 4 infinite families $A_n,B_n,C_n,D_n$ and exceptionals $E_6,E_7,E_8,
For our applications, we will mainly be referring to the irreducible root systems of type
$A_n$, $B_n$, $F_4$ and $G_2$. We will refer to Humphreys <cit.> for explicit constructions of the irreducible root systems, their bases,
Weyl groups, weight lattices and automorphism groups.
We make some particular remarks about automorphism groups of root systems that will be used subsequently.
The automorphism group of $A_n$ is $W(A_n)\times C_2=S_{n+1}\times C_2$.
As an $W(A_n)=S_{n+1}$ representation, the root lattice of $A_n$ is $I_{X_{n+1}}$, the augmentation ideal of the
$S_{n+1}$-permutation lattice corresponding to the natural transitive $S_{n+1}$-set $X_{n+1}$ with stabilizer subgroup $S_n$. As an $\Aut(A_n)$-representation, the root lattice of $A_n$ is
$I_{X_{n+1}}\otimes I_{Y_2}$ where $X_{n+1}$ is the $\Aut(A_n)$-set on which $C_2$ acts trivially and $S_{n+1}$ acts as above; and $Y_2$ is the $\Aut(A_n)$-set of size 2 on which $C_2$ acts transitively and $S_{n+1}$ acts trivially.
Note that $\Aut(A_2)=W(G_2)$ and the root lattice of $G_2$ and $A_2$ coincide.
This implies that the root lattice of $G_2$ is $I_{X_3}\otimes I_{Y_2}$ as
an $\Aut(A_2)=S_3\times C_2$-lattice.
§.§ Lattice-Tori Correspondence
We will discuss in more detail the correspondence between isomorphism
classes of algebraic $k$-tori of dimension $r$ and conjugacy classes
of finite subgroups of $\GL_r(\Z)$ as well as criteria
for determining whether a given algebraic $k$-torus is retract or stably
rational. <cit.>
An algebraic $k$-torus of dimension $r$ is a $k$-form of a split $k$-torus
$\Gmr$. As discussed in the introduction, this implies that
isomorphism classes of algebraic $k$-tori are in bijection
with elements of the non-abelian Galois cohomology
set $H^1(\cG_k,\Aut(\Gmr))$ where $\cG_k=\Gal(k_s/k)$ is the absolute Galois
group of $k$. Since $\cG_k$ acts trivially on $\Aut(\Gmr)=\GL_r(\Z)$,
elements of $H^1(\cG_k,\Aut(\Gmr))$ are actually continuous homomorphisms
of the compact profinite group $\cG_k$ into the discrete group $\GL_r(\Z)$
and as such have finite image.
This shows that algebraic $k$-tori of dimension $r$ are determined up to isomorphism
by continuous representations of $\cG_k$ into $\GL_r(\Z)$
or equivalently by lattices of rank $r$ with a continuous action of $\cG_k$ up to isomorphism.
Given such a continuous representation $\rho:\cG_k\to \GL_r(\Z)$,
and the associated $\cG_k$ lattice $M$ of rank $r$,
$\Spec(k_s[M]^{\cG_k})$ is an algebraic $k$-torus.
Conversely, an algebraic $k$-torus $T$ determines its character
lattice $\hat{T}=\Hom(T,\Gm)$ which is a lattice equipped with a continuous action of $\cG_k$.
Every algebraic $k$-torus is split by a finite Galois extension.
In fact, if the algebraic $k$-torus is determined by a continuous representation
$\rho:\cG_k\to \GL_r(\Z)$ with associated $\cG_k$ lattice $M$ of rank $r$,
then for $N=\ker(\rho)$, $L=k_s^N$ is a finite Galois extension of $k=k_s^{\cG_k}$
with finite Galois group $G\cong \cG_k/N\cong \rho(\cG_k)$.
Note that $L[M]^G=(k_s[M]^N)^{\cG_k/N}=k_s[M]^{\cG_k}$.
Isomorphism classes of algebraic tori of rank $r$ split by a finite Galois extension $L/k$
with Galois group $G$ are in bijection with isomorphism classes of $G$-lattices of rank $r$. Here the $G$-lattice $M$ determines the algebraic torus
split by $L/k$ as $\Spec(L[M]^G)$ and conversely the algebraic torus $T$ with
splitting group $G$ determines its character lattice which is a $G$-lattice.
Criteria for determining the stable (respectively retract)
rationality of an algebraic $k$-torus split by a finite Galois extension $L/k$
with Galois group $G$ are phrased in terms of the integral representation theory of the character lattice as a $G$-lattice.
To describe these criteria, we need some definitions about $G$-lattices.
We introduce some notation:
For a finite Galois extension $L/k$ with Galois group $G$, we will denote
by $C(L/k)$, the category of algebraic $k$-tori split by $L$
and by $C(G)$, the dual category of $G$-lattices (torsion-free $G$ modules)
of finite rank.
§.§ Lattice terminology
For more details, see for example, Lorenz <cit.>.
Let $G$ be a finite group. Note that for a $G$-lattice
$M$ and a subgroup $H$ of $G$, $\hat{H}^k(H,M)$ refers to the
$k$th Tate cohomology group of $M$ as an $H$ module, where $k\in \Z$.
* For a $G$-lattice the dual lattice $M^*=\Hom(M,\Z)$ is a $G$-lattice
with $(g\cdot f)(m)=g\cdot f(g^{-1}\cdot m)$ for $f\in M^*$, $g\in G$, $m\in M$.
* A permutation $G$-lattice is a $G$-lattice with a finite $\Z$
basis which is permuted by the action of $G$. All transitive
permutation lattices with stabilizer subgroup $H$ are isomorphic to
$\Z[G/H]$ where $G/H$ is the set of left cosets of $H$ in $G$.
Any permutation $G$-lattice is the direct sum of a finite number
of transitive $G$ permutation lattices.
* A sign permutation $G$-lattice is a $G$-lattice with
$\Z$-basis which is permuted by the action of $G$ up to sign.
* Given an $H$-lattice $N$ where $H\le G$, $\Ind^G_H(M)=\Z G\otimes_{\Z H}M$
is the $G$-lattice induced from the $H$ lattice $M$.
Note that $\Ind^G_H(\Z)=\Z[G/H]$.
* An invertible or permutation projective $G$-lattice $M$
is a $G$-lattice which is a direct summand of a $G$ permutation lattice.
That is, there exists a $G$-lattice $M'$ such that $M\oplus M'=P$
for some $G$ permutation lattice $P$.
* A $G$-lattice $M$ is quasipermutation if there exists
a short exact sequence $0\to M\to P\to Q\to 0$ of $G$-lattices
with $P,Q$ $G$-permutation lattices.
The following are some natural $G$-lattices associated with each $G$-permutation lattice:
For a finite $G$-set $X$, let the associated permutation $G$-lattice be denoted as $\Z[X]$. A natural $G$ sublattice is the augmentation ideal $I_X$
given by the kernel of the $G$-equivariant homomorphism
$\epsilon_X:\Z[X]\to \Z$, sending $x\to 1$ for all $x\in X$.
But then
$$0\to I_X\to \Z[X]\to \Z\to 0$$
is a short exact sequence of $G$-lattices.
Let $J_X=(I_X)^*$ be its $\Z$ dual.
Then $J_X$ satisfies the short exact sequence of $G$-lattices
$$0\to \Z\to \Z[X]\to J_X\to 0$$
Facts and Theorems about Tate Cohomology:
Let $G$ be a finite group and $M$ be a $G$-lattice. Then
* Let $N_G:M\to M^G$ be the norm map $N_G(m)=\sum_{g\in G}g\cdot m$.
\begin{cases}\hat{H}^k(G,M)=H^k(G,M)&\mbox{ if }k\ge 1\\
\hat{H}^0(G,M)=\ker_M(N_G)/I_G(M)& \\
\hat{H}^{-1}(G,M)=M^G/\ker_M(N_G)&\\
\hat{H}^{-i-1}(G,M)=H_i(G,M)&\mbox{ if }i\ge 1
\end{cases}
* (Duality)
$\hat{H}^{k}(G,M)\cong \hat{H}^{-k}(G,M^*)$ where $M^*$ is the $\Z$ dual lattice of $M$.
* (Shapiro's Lemma): For an $H$-lattice $N$, where $H\le G$, we have
$$\hat{H}^k(G,\Ind^G_H(N))\cong \hat{H}^k(H,N)$$
where $\Ind^G_H(M)=\Z G\otimes_{\Z H}M$ is the $G$-lattice induced from the
$H$-lattice $M$.
* A $G$-lattice $M$ is flasque if $\hat{H}^{-1}(H,M)=0$ for all subgroups $H\le G$.
* A $G$-lattice $M$ is coflasque if $\hat{H}^1(H,M)=0$ for all subgroups
$H\le G$.
Following Voskresenskii, we denote by $S(G)$ the class of all permutation $G$-lattices, $D(G)$ the
class of all invertible $G$-lattices, $\hat{H}^{-1}(G)$ the class of all
flasque $G$-lattices, $\hat{H}^1(G)$ the class of all coflasque $G$-lattices
and $\tilde{H}(G)=H^1(G)\cap H^{-1}(G)$.
$$S(G)\subset D(G)\subset \tilde{H}(G)\subset H^i(G)\subset C(G)$$
where each inclusion is proper.
Most inclusions are clear: permutation lattices
are flasque and coflasque due to Shapiro's Lemma, so invertible lattices
must also be flasque and coflasque as the direct summands of permutation lattices.
Note also permutation lattices are self-dual and that the dual of a flasque $G$-lattice is coflasque (and vice-versa).
The following definitions and results are due to Voskresenskii, Endo-Miyata, Colliot-Thélène and Sansuc
We say that two $G$-lattices $M,N$ are stably isomorphic and write
$[M]=[N]$ if and only if there exist permutation lattices $P,Q$ such that
$M\oplus P\cong N\oplus Q$.
(See <cit.>)
A $G$-lattice $M$ has a flasque resolution. That is,
there exists a short exact sequence of $G$-lattices
$$0\to M\to P\to F\to 0$$
such that $P$ is $G$-permutation and $F$ is $G$-flasque.
Given 2 flasque resolutions of a $G$-lattice $M$
$$0\to M\to P_i\to F_i\to 0, i=1,2$$
we have that $[F_1]=[F_2]$.
We may then define the flasque class of the $G$-lattice $M$
to be $\rho_G(M)=[F]$ where $0\to M\to P\to F\to 0$ is a flasque resolution
and $[F]$ is the stable isomorphism class of $F$.
The proof that every $G$-lattice has a flasque resolution (see eg. <cit.>) is straightforward and constructive
but relies strongly on knowledge of the conjugacy classes of subgroups of $G$
and the restrictions of the $G$-lattice $M$ to these subgroups.
Hoshi and Yamasaki <cit.> gave algorithms to construct these flasque
resolutions for $G$-lattices of ranks up to 5 for which all of this information is known.
We say that $G$-lattices $M$ and $N$ are flasque equivalent if
there exist short exact sequences of $G$-lattices such that
$$0\to M\to E\to P\to 0, 0\to N\to E\to Q\to 0$$
Note that $M\sim_G 0$ if and only if $M$ is a quasipermutation $G$-lattice.
Note that $M\sim_G N$ if and only if $\rho_G(M)=\rho_G(N)$. (See <cit.>).
§.§ Birational properties of algebraic $k$-tori
We will give a brief summary of how the concepts in the
last section can be used to discuss birational properties of algebraic $k$-tori.
A more in depth discussion can be found in <cit.>.
Given an algebraic $k$-torus $T\in C(L/k)$ where $L/k$ is a finite Galois
extension with group $G$, we may find by resolution of singularities (due
to Hironaka in characteristic 0)
a smooth projective $k$-variety $X$ which contains $T$ as an open subset.
[Note, this is the reason for our characteristic zero assumption.]
$X$ is called a projective model of $T$. Then $X$ and $T$ are birationally
isomorphic. Then $X_L=L\otimes_k X$ is rational since $X_L$ and $T_L$ are
birationally equivalent. Then the Picard group of $X_L$, $\Pic(X_L)$, is a $G$-lattice. Given two such projective models $X,Y$ of $T$, $\Pic(X_L)$
is stably $G$ isomorphic to $\Pic(Y_L)$. So the class of $[\Pic(X_L)]$
is a birational invariant.
For any intermediate field extension $k\subset K\subset L$ with $H=\Gal(L/K)$,
$\hat{H}^{\pm 1}(H,\Pic(X_L))$ are also birational invariants of $T$
and $\hat{H}^{-1}(H,\Pic(X_L))=0$.
The inclusion $T_L$ into $X_L$ induces
a short exact sequence of $G$-lattices
$$0\to \hat{T}\to \hat{S}\to \Pic(X_L)\to 0$$
where $\hat{S}$ is generated by the components of the closed subset $X_L-T_L$.
$\hat{S}$ is a $G$-permutation lattice and as observed above, $\Pic(X_L)$
is a flasque $G$-lattice.
This gives a geometric construction of a flasque resolution of the
$G$-lattice $\hat{T}$.
If $T$ is $k$-rational, so is $X$ and then $[\Pic X_L]=0$.
In fact if $T$ is stably $k$-rational then so is $X$ and so
$X\times_k \Gmr$ is rational for some $r\ge 0$ but $\Pic(X_L\times_k \GG^r_{m,L})=\Pic(X_L)$ and so we again have $[\Pic(X_L)]=0$.
This gives the basic setup behind the following results
connecting birational properties of algebraic $k$-tori to properties
about the integral representation of their character lattices over a splitting group.
(Endo-Miyata, Voskresenskii, Saltman) Let $L/k$ be a finite
Galois extension with Galois group $G$ and let $T,T'\in C(L/k)$
with character lattices $\hat{T}, \hat{T'}$ in $C(G)$.
* (<cit.>) $T$ is stably $k$-rational if and only if
$\hat{T}\sim_G 0$ (equivalently $\hat{T}$ is a quasipermutation $G$-lattice).
* (<cit.>) $T$ and $T'$ are stably birational $k$ equivalent
if and only if $\hat{T}\sim_G \hat{T'}$.
* (<cit.>) $T$ is retract $k$-rational if and only
if $\rho_G(\hat{T})$ is invertible.
A torus $T\in C(L/k)$ with permutation character lattice
$\oplus_{i=1}^k\Z[G/H_i]$ for subgroups $H_i, i=1,\dots,k$
corresponds to an algebraic $k$-torus
$T=\prod_{i=1}^kR_{K_i/k}(\Gm)$ where $K_i=L^{H_i}$ and
$R_{K/k}$ refers to Weil restriction of scalars.
Such tori are called quasisplit.
<cit.>. See also <cit.>.
Let $L/k$ be a Galois extension with Galois group $G$.
* A torus $T\in C(L/k)$ is rational if $\hat{T}\in C(G)$ is permutation.
* If there exists an exact sequence $0\to S\to T\to T'\to 0$ of tori in $C(L/k)$
where $S$ is quasisplit, then $T$ is birationally equivalent to $T'\times_k S$.
Note that the last theorem is equivalent to the following in lattice theoretic terms:
Let $L/k$ be a Galois extension of fields with $G=\Gal(L/k)$.
* $L(P)^G$ is rational over $L^G$ if $P$ is a permutation $G$-lattice.
* If $0\to M\to N\to P\to 0$ is an exact sequence of $G$-lattices with
$P$ $G$-permutation, then $L(N)\cong L(M)(P)$ as $G$ fields and so
$L(N)^G\cong L(M)(P)^G$ is rational over $L(M)^G$.
§ FAMILIES OF RATIONAL ALGEBRAIC $K$-TORI
In this section, we will gather families of rational algebraic $k$-tori from
results in the literature.
Let $M$ be a $G$-lattice for a finite group $G$. If all algebraic tori with character lattice $\Res_H^G(M)$ and splitting group $H$ are rational for any subgroup $H$ of $G$, we call $M$ hereditarily rational. Note that a $G$-lattice $M$ is
hereditarily rational if and only if for any subgroup $H$ of $G$
and for any Galois extension $K/k$ with Galois group $H$,
the function field $K(M)^H$ is rational over $K^H$.
Let $T$ be an algebraic $k$-torus of dimension $r$, $h_T:\cG_k\to \GL_r(\Z)$
the associated continuous representation and $W_T=h_T(\cG_k)\le \GL_r(\Z)$.
We call $T$ hereditarily rational if the associated $W_T$-lattice $\hat{T}$
is hereditarily rational.
In the definition of hereditarily rational algebraic $k$-torus, rationality is completely determined by the structure of its character lattice as a $\cG_k$ representation.
Note that an algebraic $k$-torus $T$ is determined uniquely
by the $G$-lattice $\hat{T}$ and the Galois extension $L/k$ with Galois group
$G$. Every $G$ module $\hat{T}$ determines many algebraic $k$-tori
corresponding to different Galois extensions $L/k$ with Galois group $G$
and these tori may be non-isomorphic over $k$.
Stable rationality and retract rationality of an algebraic $k$-torus only
depend on the $G$-lattice $\hat{T}$. However it is not clear whether this
is true for rationality, although the author does not know of a counterexample.
A reason for caution is given by the example given in [p. 54]<cit.> of 2 norm one
tori $R^{(1)}_{L_i/\Q}(\Gm)$, $i=1,2$ corresponding to distinct biquadratic
extensions which are not even stably birationally equivalent over $\Q$.
We will now produce families of examples of hereditarily rational algebraic $k$-tori. These are gathered from known examples in the literature, which will be cited accordingly.
Let $T$ be an algebraic $k$-torus with Galois splitting field $K/k$,
$G=\Gal(K/k)$ and character lattice $M$. Then $T$ is hereditarily rational in the following cases:
* <cit.> $M$ is a permutation $G$-lattice. ($T$ is then a quasi-split torus).
* <cit.> $M=I_X$ is an augmentation ideal for a $G$ set $X$.
* <cit.> $M$ is a sign-permutation $G$-lattice.
* <cit.> $T_M$ is $G$-equivariantly linearisable.
* <cit.> $M=I_X\otimes I_Y$ is a tensor product of augmentation ideals
for finite $G$-sets $X,Y$ with $\gcd(|X|,|Y|)=1$.
* <cit.> $M$ is the root lattice for the $G_2$ root system for the group $G=W(G_2)$.
* Any permutation $G$-lattice $P$ is hereditarily rational since
$K(P)^G$ is rational over $k$ for any Galois $G$ extension $K/k$ <cit.> and permutation
lattices are preserved under restriction.
Since quasi-split tori are precisely those with permutation character
lattices, these tori are hereditarily rational.
More explicitly, if $P=\oplus_{i=1}^r\Z[G/H_i]$ is a permutation $G$-lattice
where $H_1,\dots, H_r$ are subgroups of $G$, and $L/k$ is a Galois extension
with Galois group $G$, the corresponding quasi-split torus
$\prod_{i=1}^rR_{K_i/k}(\Gm)\in C(L/k)$ is hereditarily rational.
* Let $P=\Z X$ be a permutation $G$-lattice corresponding to any $G$ set $X$.
Then $\epsilon_X:\Z X\to \Z, x\to 1$ is a $G$ invariant surjection.
The augmentation ideal $I_X=\ker(\epsilon_X)$ is an hereditarily rational $G$
lattice. Suppose $\Z X=\oplus_{i=1}^s \Z[G/H_i]$.
Given a Galois $G$ extension $K/k$, the exact sequence
$0\to I_X\to \Z X\to \Z\to 0$
corresponds to the exact sequence of $k$-tori
$$1\to \GG_{m,k}\to R_{F_1/k}(\Gm)\times \cdots \times R_{F_s/k}(\Gm)\to T\to 1$$
where $K_i/k$, $i=1,\dots,s$ are intermediate field extensions of $L/k$
such that $L/K_i$ is Galois with group $H_i$ for each $i$.
An algebraic $k$-torus with character lattice $I_X$ is given by
Since each $R_{F_i/k}(\Gm)$ can be identified with an open set of
$R_{F_i/k}(\A^1)=\A^{n_i}_k$, $T=\prod_{i=1}^sR_{F_i/k}(\Gm)/\Gm$ admits an open embedding into
projective space and hence is rational.
Note that for any subgroup $H$ of $G$, $(I_X)_H$ is also such an augmentation
ideal and so any algebraic torus with this character lattice would be
rational by the same argument.
* A geometric proof that a torus with sign permutation character lattice is rational is given in
<cit.>. Note here that an orthogonal integral representation corresponds to a sign permutation lattice in our language. The restriction of a sign permutation lattice to any subgroup is again a sign permutation lattice, so that tori with sign permutation character lattices are hereditarily rational.
* $T_M$ is $G$-equivariantly linearisable if for any Galois $G$-extension
$K/k$, we have that $K(M)$
is $G$-equivariantly birational to $K(V)$ where $G$ acts $k$ linearly on $V$.
Then as $G$-fields, $K(M)\cong K(V)$. Since $G$ acts faithfully on $K$ and linearly on $V$, $K(M)^G\cong K(V)^G$ is rational over $K^G$.
If $K(M)$ is $G$-equivariantly birationally linearisable, then $K(\Res^G_H(M))$ is $H$ equivariantly birationally linearisable for any subgroup $H$ of $G$.
So a torus with Galois splitting field $K$ and character lattice $M$ is hereditarily rational.
Note that this gives an alternative proof that an algebraic torus with a (sign) permutation or augmentation ideal character lattice is hereditarily rational.
For algebraic tori with permutation character lattices, this is clear.
For algebraic tori with sign permutation character lattices,
$K(X_1,\dots,X_n)=K(Y_1,\dots,Y_n)$ where $Y_i=\frac{1-X_i}{1+X_i}$ for all $i=1,\dots,n$. Then $\sigma(Y_i)=\pm Y_j$ for all $i=1,\dots, n$ and so the action of $G$ on $V=\sum_{i=1}^nkY_i$ is linear.
* A result of Klyachko <cit.> with a simpler proof given by Florence <cit.> shows that for any
finite $\cG_k$ sets $X$ and $Y$ which are relatively prime, an algebraic $k$-torus with character lattice $I_X\otimes I_Y$ is rational.
This shows that for any finite group $G$, and any 2 relatively prime $G$ sets
$X,Y$, the $G$-lattice $I_X\otimes I_Y$ is hereditarily rational.
Note that any $G$ set $Y$ of size 2 gives a rank 1 sign lattice $I_Y$.
Then for any other $G$ set $Y$ of odd order, $I_X\otimes I_Y$ is hereditarily
rational. In particular, this recovers the result for the root lattice of $G_2$
since it may be expressed as a tensor product
$I_{X_3}\otimes I_{Y_2}$ as described earlier. More generally, it shows that a algebraic $k$-torus
with character lattice $(\Z \bA_{2k},\Aut(\bA_{2k}))$ <cit.> is hereditarily rational.
* Note that the root lattice for the root system $\bG_2$ can also be described as the action of the automorphism group of the root system $\bA_2$ on the
root lattice of $\bA_2$.
The root lattice for $G_2$ restricted to $S_3$ is $I_{S_3/S_2}$ and $C_2$ acts as $-1$ on the lattice. For an algebraic $k$-torus $T$ with character lattice $G_2$,
the group $W(G_2)$ acts regularly on $\GG_m^2$. Voskresenskii shows that the action
of $W(G_2)$ on $\GG_m^2$ can be extended to a birational action on $(\PP^1)^2$
and so $T$ is an open part of the $k$-variety $((\PP^1)^2\otimes_k k_s)/W(G_2)$
which is a $k$-form of $\PP^1\times \PP^1$. Since any $k$-form of $\PP^1\times \PP^1$
is rational if it has a point, $T$ is rational.
Let $M=\oplus_{i=1}^rM_i$ be the direct sum of $G$-lattices
$M_i$, $i=1,\dots,k$. If $T_i,i=1,\dots,r$ are algebraic $k$-tori with character lattices $M_i$, then $T=T_1\times_k \cdots \times_k T_r$ is the algebraic $k$-torus with character lattice $\oplus_{i=1}^rM_i$. If $K/k$ is a $G$-Galois extension, $K(M)$ is the composite of the $G$-fields $K(M_i)$ and so $K(M)^G$ is the composite of the fields $K(M_i)^G$. So if each $T_i$ is (hereditarily) rational, so is $T$.
So the direct sum of hereditarily rational $G$-lattices is hereditarily rational.
Let $M$ be a $G$-lattice of rank $r$.
Let $P_n=\oplus_{i=1}^n\Z\e_i$ be a permutation lattice for the wreath product $G^n\rtimes S_n$ on which $G^n$ acts trivially and $S_n$ acts by permuting the basis. Then $M\otimes P_n$ is the natural $G^n\rtimes S_n$ lattice of rank $rn$ for the wreath product. Let $M_i=M\otimes \Z\e_i$.
Then $M\otimes P_n=\oplus_{i=1}^nM_i$. Let $\pi_i:G^n\to G$ be the $i$th projection. The action of $G^n\rtimes S_n$ on $M\otimes P_n$ is then given by
$$g\cdot (m\otimes \e_i)=(\pi_i(g)\cdot m)\otimes \e_i, g\in G^n$$
$$\sigma\cdot (m\otimes \e_i)=m\otimes \e_{\sigma(i)}, \sigma\in S_n$$
Suppose $T$ is an algebraic $F$-torus with character lattice $M$ and Galois
splitting field $K/F$ with Galois group $G$.
If $F/k$ is a separable extension of degree $n$, then $R_{F/k}(T)$ is an algebraic $k$-torus with character lattice $M\otimes P_n$.
Let $M$ be a $G$-lattice of rank $r$.
Using the notation from the definition,
let $M\otimes P_n$ be the natural $G^n\rtimes S_n$ lattice of rank $rn$.
Let $T^{n,\wr}_M$
be an algebraic torus with character lattice $M\otimes P_n$.
Then if any algebraic torus with character lattice $M$ is (stably) rational, so is $T^{n,\wr}_M$.
If any algebraic torus with character lattice $M$ is linearisable, so is $T^{n,\wr}_M$.
If any algebraic torus with character lattice $M$ is hereditarily rational, so is $T^{2,\wr}_{M}$.
Let $L/k$ be a Galois $G^n\rtimes S_n$-extension.
For preciseness, we will write $G^n=G_1\times \cdots \times G_n$ where $G_i=G$
and $M\otimes P_n=\oplus_{i=1}^nM_i$ where $M_i=M$. Here $G_i$ acts on $M_i$ as $G$ acts on $M$ and $G_j$ acts trivially on $M_i,i\ne j$
so that $M\otimes P_n$ restricted to $G^n$ is $M^n$.
Now $L(M^n)$ is a composite of $G^n=\prod_{i=1}^nG_i$ fields $L(M_i)$.
Since any algebraic torus with character lattice $M$ is rational,
we have that $L(M_i)^{G^n}=L^{\prod_{j\ne i}G_j}(M_i)^{G_i}=L^{G^n}(f_1^i,\dots,f_r^i)$, for all
Then $L(M)^{G^n}=\prod_{i=1}^n(L^{\prod_{j\ne i}G_j}(M_i))^G_i=L^{G^n}(f_1^1,\dots,f_r^1,\dots,f_1^n,\dots,f_r^n)$
and $L(M\otimes P_n)^{G^n\rtimes S_n}=(L(M^n)^{G^n})^{S_n}$.
Since $S_n$ permutes the $n$ copies of $M$ and correspondingly permutes the
fields $L^{\prod_{j\ne i}G_j}$, it permutes the corresponding invariants, and so $L(M\otimes P_n)^{G^n\rtimes S_n}$ is rational over $L^{G^n\rtimes S_n}$.
If an algebraic torus with character lattice $M$ is $G$-linearisable,
then $L(M)\cong L(V)$ as $G$ fields where $G$ acts linearly on $V$.
Then $L(M\otimes P_n)\cong L(V\otimes P_n)$
as $G^n\rtimes S_n$ fields.
If an algebraic torus with character lattice $M$ is stably rational then
$M$ is quasi-permutation and satisfies an exact sequence
$$0\to M\to Q_1\to Q_2\to 0$$
where $Q_1,Q_2$ are permutation $G$-lattices.
$$0\to M\otimes P_n\to Q_1\otimes P_n\to Q_2\otimes P_n\to 0$$
is an exact sequence of $G^n\rtimes S_n$ lattices with $Q_i\otimes P_n$
$G^n\rtimes S_n$-permutation for $i=1,2$.
Let $n=2$ and let $H$ be any subgroup of $G^2\rtimes S_2$.
Then either $H\le G^2$ or $H/H\cap G^2\cong C_2$.
Let $N=H\cap G^2$. Then $L(M_1\oplus M_2)=L(M_1)L(M_2)$ is a composite of $N$ fields.
Let $\pi_i$ be the restriction of the
$i$th projection map $G^2\to G$ to $N$. Then $L(M_i)^N=L^{\ker(\pi_i)}(M_i)^{\pi_i(N)}$. Since $\pi_i(N)\le G$, we see by assumption that $L(M_i)^N$ is rational over $L^N$, $i=1,2$ and so $L(M_1\oplus M_2)^N=L(M_1)^NL(M_2)^N$
is rational over $L^N$ as the composite of rational extensions.
So we need only consider the case when $H/N\cong C_2$. Let $h\in H\setminus N$. Then $h(M_1)=M_2$ and $h(L)=L$ shows that $L(M_2)=h(L(M_1))$.
We have that $L(M_1\oplus M_2)^N=L(M_1)^N(h(L(M_1))^N=L(M_1)^Nh(L(M_1)^N)$
since $N$ is normal in $H$. Since $L(M_1)^N=L^N(f_1,\dots,f_r)$,
$h(L(M_1)^N)=L^N(hf_1,\dots,hf_r)$ and so
$L(M_1\oplus M_2)^N=L^N(f_1,\dots,f_r,hf_1,\dots,hf_r)$. Since $h^2\in N$,
$h(hf_i)=f_i$ for all $i=1,\dots,r$. Then $H/N$ acts by permutations on
$L(M_1\oplus M_2)^N$ and so $L(M\otimes P_2)^H=L^N(f_1,\dots,f_r,hf_1,\dots,hf_r)^{H/N}$ is rational over $L^H$ as required.
If all algebraic tori with character lattice $M$ are hereditarily rational, an algebraic torus, $T^{n,\wr}_M$ with character lattice $M\otimes P_n$ is rational and hence stably rational.
This means that an algebraic torus with character lattice $M\otimes P_n$ corresponding to a subgroup of $G^n\rtimes S_n$ must also be stably rational.
However it is not clear whether such an algebraic torus must be hereditarily rational if $n\ge 3$.
However for the case of 4 dimensional algebraic tori, the above result will be sufficient.
Let $M$ be an hereditarily rational faithful $G$-lattice
and let $L$ be a $G$-lattice such that there exists an exact sequence
$$0\to M\to L\to P\to 0$$
for some permutation $G$-lattice $P$.
Then an algebraic $k$-torus with character lattice $(L,G)$ is hereditarily rational.
More precisely, if $T$ and $T'$ are algebraic $k$-tori fitting into an exact
$$1\to S\to T\to T'\to 1$$
where $S$ is a quasi-split $k$-torus, then $T$ is birationally equivalent
to $T'\times_k S$. If $T'$ is (hereditarily) $k$-rational, so is $T$.
Let $K/k$ be a Galois extension with Galois group $G$. Then
$K(L)\cong K(M)(P)$ as $G$-fields. Then $K(L)^G\cong K(M)(P)^G$ is rational over
$K(M)^G$ which is in turn rational over $K^G$ <cit.>.
The same result would hold for any subgroup $H$ and the $H$ lattice $L_H$.
§ HEREDITARILY RATIONAL $K$-TORI IN DIMENSIONS 2,3,4
We now have enough information to show that all but 10 of the stably rational
algebraic $k$-tori of dimension 4 are hereditarily rational.
We will first quickly illustrate our approach for the rank 2 and 3 cases
due to Voskresenskii and Kunyavskii respectively.
Note in order to do this, we will need to identify certain $G$-lattices
corresponding to conjugacy classes of finite subgroups of $\GL_r(\Z)$.
We remark that for a $G$-lattice $M$, we could use the character to
identify the corresponding $\Q G$ module $\Q M$, but that
is not sufficient to identify its isomorphism type as a $G$-lattice.
We identify $G$-lattices up to isomorphism using explicit isomorphisms.
Given a finite subgroup $G\in \GL_r(\Z)$, we will denote by $M_G$ the associated $G$-lattice of rank $r$.
For the standard basis $\e_1,\dots,\e_r$ of $M_G$, and given $g=(a_{ij})_{i,j=1}^r\in \GL_r(\Z)$, the action of $G$ on $M_G$ will be given on rows
to agree with the notation in GAP and in Hoshi and Yamasaki's paper.
That is, $g\cdot \e_i=\sum_{j=1}^ra_{ij}\e_j$ for all $i=1,\dots,r$.
We will frequently wish to show that a $G$-lattice $L$ is isomorphic to $I_X$ or its dual $J_X$ for some transitive $G$-set $X$ with stabilizer subgroup $H$.
Note that if there exists $\y\in L$ such that $\Z G\cdot \y=L$, $\sum_{g\in G}g\y=\0$ and $\rk(L)=[G:G_{\y}]-1$ then
we see that for $H=G_{\y}$,
$\pi:\Z[G/H]\to L$ is a surjective $G$-homomorphism with
$\ker(\pi)=\Z(\sum_{gH\in G/H}gH)$ so that $L\cong J_{G/H}$.
If instead we show that $L^*\cong J_{G/H}$ in this way, then, we have
$L\cong I_{G/H}$.
A rank 1 $G$-lattice is determined by a group homomorphism $\chi:G\to \GL_1(\Z)=\{\pm 1\}$. If $\chi$ is trivial, we will write the lattice as $\Z$.
If $\chi:G\to \{\pm 1\}$ is non-trivial, it is completely determined by its kernel which is necessarily a normal subgroup $N$ of index 2 in $G$.
We will then write the rank 1 $G$ lattice corresponding to $\chi$ as $\Z^{-}_{N}$ where $N=\ker(\chi)$.
(Recognising augmentation ideals and their duals).
Let $X_{n+1}=\{x_i:i=1,\dots,n+1\}$ be the standard $S_{n+1}$ set on which $S_{n+1}$ acts transitively with stabilizer subgroup $S_n$.
Let $\pi:\Z[X_{n+1}]\to J_{X_{n+1}}$ be the natural surjection.
Note that $\ker(\pi)=\Z(\sum_{i=1}^{n+1}x_i)$.
Then $\{\overline{x_i}:i=1,\dots,n\}$ forms a $Z$-basis of $J_{X_{n+1}}$
where $\overline{x_i}=\pi(x_i)$.
With respect to this basis,
$\sigma(\overline{x_i}))=\overline{x_{\sigma(i)}}, \sigma(i)\ne n+1$
and $\sigma(\overline{x_i})=-\sum_{i=1}^n\overline{x_{\sigma(i)}}, \sigma(i)=n+1$.
We will write $\rho_n:S_{n+1}\to \GL_n(\Z)$ for the representation of $S_{n+1}$ associated with $J_{X_{n+1}}$ with respect to the
basis $\overline{X_{n+1}}$ determined on rows.
We will write $\rho_n^*:S_{n+1}\to \GL_n(\Z)$ for the dual of this representation of $S_{n+1}$ associated with $I_{X_{n+1}}\cong J_{X_{n+1}}^*$.
Note that the matrices in the image of $\rho_n$ (resp. $\rho_n$) are either
permutation matrices or permutation matrices with one row (resp. column)
replaced by $[-1,\dots,-1]$. It is then easy to determine which
permutation determined them.
We will extend the action of $S_{n+1}$ on $X_{n+1}$ to an action of
$S_{n+1}\times C_2$ on $X_{n+1}$ by inflation.
That is, $S_{n+1}$ acts on $X_{n+1}$ as before and $C_2$ acts trivially on
We will denote by $\rho_n^-:S_{n+1}\times C_2\to \GL_n(\Z)$ the
representation associated to the $S_{n+1}\times C_2$-lattice
$J_{X_{n+1}}\otimes \Z^-_{S_{n+1}}$ with respect to the $\Z$-basis
$\overline{x_i}\otimes 1$.
Then if $C_2=\langle \gamma\rangle$, note that $\rho_n^-(\sigma,1)=\rho_n(\sigma)$
and $\rho_n(\sigma,\gamma)=-\rho_n(\sigma)$ for all $\sigma\in S_{n+1}$.
The matrices in the image of $\rho_n^-$ are then also easy
to recognise.
We will denote by
$(\rho_n^-)^*:S_{n+1}\times C_2\to \GL_n(\Z)$ the representation
corresponding to the dual lattice
$I_{X_{n+1}}\otimes \Z_-^{S_{n+1}}=J_{Y_{n+1}}\otimes \Z_-^{S_{n+1}}$.
Note that the root lattice $\Z A_n$ as a $W(A_n)=S_{n+1}$-lattice is $(I_{X_{n+1}})$
so that the weight lattice $\Lambda(A_n)$ is
$J_{X_{n+1}}$ as a $W(A_n)=S_{n+1}$ lattice.
Note that $\Z A_n$ as an $\Aut(A_n)=S_{n+1}\times C_2$ lattice is
$I_{X_{n+1}}\otimes \Z^-_{S_{n+1}}$
so that $\Lambda(A_n)$ is its dual
$J_{X_{n+1}}\otimes \Z^-_{S_{n+1}}$
as an $\Aut(A_n)$-lattice.
With respect to the standard basis $\e_1,\dots,\e_n$, the Weyl group
of $B_n$, has reflections $\tau_i=s_{\e_i}$, $i=1,\dots,n$ and
$\sigma_{ij}=s_{\e_i-\e_j}$. On the root lattice $\Z B_n$, with $\Z$-basis $\e_1,\dots,\e_n$, the $\tau_i$ fix $\e_j,j\ne i$ and
and $\sigma_{ij}$ acts by swapping $\e_i$ and $\e_j$ and fixing the
other basis elements.
So $W(B_n)=(C_2)^n\rtimes S_n$
where $C_2^n=\langle \tau_i:i=1,\dots,n\rangle$ and $S_n=\langle \sigma_{ij}:i\ne j\rangle$.
We denote
$\eta_n:W(B_n)\to \GL_n(\Z)$
by the representation of $W(B_n)$ corresponding to its lattice $\Z B_n$
with respect to the standard basis $\e_1,\dots,\e_n$.
Note that the images of $\eta_n$ are sign permutation matrices.
We will write elements of $W(B_n)$ as $\tau \sigma$ where $\tau\in C_2^n$
is a product of $\tau_i$ and $\sigma\in S_n$.
Let $n$ be an odd integer
and let $D_{2n}$ be the dihedral group
of size $2n$ given by the presentation
$$D_{2n}=\langle \sigma,\tau: \sigma^n=1=\tau^2, \tau\sigma\tau^{-1}=\sigma^{-1}\rangle$$
Via the injective group homomorphism $\varphi:D_{2n}\to S_n$
given on generators by $\sigma\mapsto (1,2,\dots,n)$ and $\tau\mapsto \prod_{i=1}^{\frac{n-1}{2}}(i,n-i)$, $D_{2n}$ acts by restriction on the
$S_n$-set $X_n$.
$$J_{X_n}\cong I_{X_n}\otimes \Z^{-}_{C_n}$$
as $D_{2n}$-lattices.
Considering $X_n$ as a $D_{2n}\times C_2$-set by inflation,
$$J_{X_n}\otimes \Z^{-}_{D_{2n}}\cong I_{D_{2n}\times C_2/C_2\times C_2}\otimes \Z^{-}_{D_{2n}}$$
as $D_{2n}\times C_2$-lattices.
Restricting the standard $S_n$-set $X_n$ to $D_{2n}$ via $\varphi$,
we see that $\sigma(x_i)=x_{i+1\bmod n}$ and $\tau(x_i)=x_{n-i}$
for all $i=1,\dots,n$.
In particular, $D_{2n}\cdot x_{\frac{n+1}{2}}=X_n$ and the stabilizer subgroup of $x_{\frac{n+1}{2}}$ is $\langle \tau\rangle\cong C_2$.
So $X_n\cong D_{2n}/C_2$ as a $D_{2n}$-set.
Note that all elements of order 2 are conjugate in $D_{2n}$
so that there is a unique conjugacy class of subgroups isomorphic to $C_2$ in
This implies that $J_{D_{2n}/C_2}$ and $I_{D_{2n}/C_2}$ are well-defined,
since $\Z[G/H]\cong \Z[G/gHg^{-1}]$.
By the remark, it will suffice to find an element $z$ of
$I_{X_n}\otimes \Z^-_{C_n}$ such that the distinct elements of the orbit of $z$ under $D_{2n}$
form a $\Z$-basis for $I_{X_n}\otimes \Z^{-}_{C_n}$ and the stabilizer
subgroup of $z$ is a cyclic subgroup of order 2.
Take $z=(x_1-x_n)\otimes 1\in I_{X_n}\otimes \Z^-_{C_n}$.
We will show that $(D_{2n})_{z}=\langle \tau\rangle$, $\sum_{g\in D_{2n}}\cdot z=\0$ and $\Z D_{2n}\cdot z$ has rank $n-1$. By our previous observations, this will imply that
$$I_{X_n}\otimes \Z^-_{C_n}=\Z D_{2n}\cdot z\cong J_{X_n}$$
as $D_{2n}$-lattices.
Note that $\tau(z)=z$.
$$\sigma^i((x_1-x_n)\otimes 1)=(x_{1+i}-x_{n+i\bmod n})\otimes 1,$$
it is clear that the stabilizer subgroup of $z$ in $D_{2n}$ is $\langle \tau\rangle$.
It is also clear that the orbit sums to zero.
We need only check that $\Z D_{2n}\cdot z=I_{X_n}\otimes \Z^-_{C_n}$.
Let $\sum_{i=1}^nb_i(x_i\otimes 1)\in I_{X_n}\otimes \Z^-_{C_n}$.
Then $\sum_{i=1}^nb_i=0$. We need to show that we can solve
$$\sum_{i=1}^{n-1}a_i(x_i-x_{n+i\bmod n})\otimes 1
=\sum_{i=1}^{n}b_i(x_i\otimes 1)$$
for some unique $a_i$ if $\sum_{i=1}^{n}b_i=0$.
We obtain the following equations:
$$a_i-a_{i+1}=b_i, i=1,\dots,n-2; a_{n-1}=b_{n-1}, -a_1=b_n$$
One can easily see that these equations correspond to a matrix system
of the form $C\aa=\bb$ where
\dots&\dots&\dots&\dots&\dots\\0&\dots&0&1&-1\\0&0&\dots&\dots&1\end{array}\right]$$
is an $n\times n$ matrix.
Since the rows of this matrix add to 0, and the last $n-1$ rows form a triangular
system with ones on the diagonal, it is clear that one could solve this
system uniquely for $\bb=[b_1,\dots,b_{n}]^T\in \Z^n$ where $\sum_{i=1}^{n}b_i=0$.
So we have proved that $J_{X_n}\cong I_{X_n}\otimes \Z^-_{C_n}$.
For the last statement,
we will write
$$D_{2n}\times C_2=\langle \sigma,\tau,\gamma: \sigma^n=\tau^2=\gamma^2=1,
\tau\sigma\tau^{-1}=\sigma^{-1},\gamma\tau=\tau\gamma, \gamma\sigma=
\sigma\gamma\rangle$$
Note that $X_n$ as an $(D_{2n}\times C_2)$-set is isomorphic to
$(D_{2n}\times C_2)/(C_2\times C_2)$.
Inflation preserves exactness, so in particular isomorphisms and commutes
with tensor products.
So inflating the $D_{2n}$-lattice isomorphism
$$J_{X_n}\cong I_{X_n}\otimes \Z^-_{C_n}$$
we obtain a $D_{2n}\times C_2$-lattice isomorphism
\begin{equation}
J_{X_n}\cong I_{X_n}\otimes \Z^-_{C_n\times C_2}\label{eq:jinfl}
\end{equation}
Note that
$$\Z^{-}_{C_n\times C_2}\otimes \Z^{-}_{D_{2n}}\cong \Z^{-}_{\langle \sigma,\tau\gamma\rangle}$$
Note also that
$\langle \sigma,\tau\gamma\rangle\cong D_{2n}$
and $\langle \sigma,\tau\gamma,\gamma\rangle=\langle \sigma,\tau,\gamma\rangle=D_{2n}\times C_2$, $\langle \tau\gamma,\gamma\rangle=\langle \tau,\gamma\rangle=C_2\times C_2$.
So we see that after tensoring (<ref>) by $\Z^-_{D_{2n}}$ we obtain
$$J_{X_n}\otimes \Z^-_{D_{2n}}\cong
I_{X_n}\otimes \Z^{-}_{\langle \sigma,\tau\gamma\rangle}\cong I_{X_n}\otimes \Z^-_{D_{2n}}$$
Note that although $H=\langle \sigma,\tau\gamma\rangle$ is a different non-conjugate subgroup isomorphic to $D_{2n}$, $I_{X_n}=I_{\langle \sigma,\tau\gamma,\gamma\rangle}/{\langle \tau\gamma,\gamma\rangle}$.
This technical result has some very interesting consequences.
Note that $\Lambda (A_2)\cong J_{X_3}$ as an $S_3$-lattice. The symmetric group $S_3$ is also
the dihedral group $D_6$. We then see that $\Lambda (A_2)\cong J_{X_2}\cong
I_{X_2}\otimes \Z^-_{C_3}=I_{X_2}\otimes I_{Y_2}$
where $Y_2$ is the $S_3$ set of size 2 permuted by $S_3/C_3$ and fixed by $C_3$.
Using Proposition <ref>, we easily recover the fact that an algebraic torus
with character lattice $(\Lambda (A_2),S_3)$ is hereditarily rational.
Note also that we can recover the fact that an algebraic torus with character lattice $(\Z G_2,W(G_2))$
or equivalently $(\Z A_2,\Aut(A_2))$ is hereditarily rational.
More generally, it shows that an algebraic torus with character lattice
$J_{X_n}\otimes \Z^-_{D_{2n}}$, $n$ odd, is hereditarily rational.
Given a finite subgroup $G$ of $\GL_r(\Z)$ up to conjugacy, the lattice determined by $G$, $M_G$ is determined by the action of $G$ by multiplying elements
of $\Z^r$ (considered as rows) by elements of $G$ on the right.
So for the standard basis $\e_1,\dots, \e_r$ of $\Z^r$,
$\e_i\cdot g=\sum_{j=1}^ra_{ij}\e_j$
where $g=[a_{ij}]_{i,j=1}^r\in \GL_r(\Z)$.
There is a library of conjugacy class representatives of finite
subgroups of $\GL_r(\Z)$ for $r=2,3,4$ in GAP.
The maximal finite subgroups of $\GL_r(\Z)$ for $r=2,3,4$ are encoded in
GAP as DadeGroup(r,k), in honour of Dade who determined the maximal
finite subgroups of $\GL_4(\Z)$ without the use of a computer.
I will use the GAP labelling to refer to conjugacy class representatives.
I will identify $M_G$ for each maximal finite subgroup of $\GL_r(\Z)$, $r=2,3,4$. These results are probably folklore (at least for $r=2,3$) but are not phrased in these terms in the literature.
Note, in GAP, the command
where $r=2,3,4$ and $k$ is in the
correct range for the rank gives
The command
gives the group generators of the
group M.
For better legibility, one could use the command
The use of GAP to list generators is a convenience. This
information could also be found in <cit.>.
§.§ Algebraic $k$-tori of dimension 2
For rank 2, there are 13 conjugacy classes of finite subgroups.
There are 2 maximal such classes.
The maximal finite subgroups of $\GL_2(\Z)$ up to conjugacy
are $G_i$=, $i=1,2$.
* $M_{G_1}=(\Z B_2,W(B_2))=(\Z B_2,\Aut(B_2)$ and
* $M_{G_2}=(\Z A_2,\Aut(A_2)=(\Z G_2,W(G_2))$.
The corresponding algebraic $k$-tori are hereditarily rational.
The generators of $G_1=$ DadeGroup(2,1) with GAP ID [2,3,2,1] given by
are the images of
under the faithful representation $\eta_2:W(B_2)\to \GL(2,\Z)$ determined by
$\Z B_2$ with respect to the standard basis determined earlier.
Since $\langle \tau_2,\tau_2(12),\tau_1\tau_2\rangle =\langle \tau_1,\tau_2,(12)\rangle=W(B_2)$, we may claim that $M_{G_1}=(\Z B_2,W(B_2))$ as required.
Since this is a sign permutation lattice, the corresponding torus is
hereditarily rational.
The generators of $G_2=$ DadeGroup(2,2) with GAP code [2,4,4,1] given by
\left[\begin{array}{rr}0&-1\\1&-1\end{array}\right]$$
are the images of
under the faithful representation $(\rho_2^-)^*:S_3\times C_2\to \GL_2(\Z)$
determined by the $S_3\times C_2$-lattice
$I_{X_3}\otimes \Z^-_{S_3}$ described earlier.
We have already noticed that this coincides with $(\Z A_2,\Aut(A_2))$.
We need only check that $\langle ((12),1),((132),1),(1,\gamma)\rangle=S_3\times C_2$, which is clear.
So for rank 2, the lattices corresponding to maximal finite subgroups of $\GL_2(\Z)$ are $(\Z G_2,W(G_2))=(\Z A_2,\Aut(A_2))$ and $(\Z B_2,W(B_2))$. As explained above,
these are hereditarily rational and so the corresponding tori and those
corresponding to their subgroups are rational. This is effectively
a rephrasing of how Voskresenskii proves that all algebraic k-tori of dimension 2 are rational.
§.§ Algebraic $k$-tori of dimension 3
For rank 3, the 73 conjugacy classes of finite subgroups of $\GL_3(\Z)$
were determined by
Tahara <cit.> There are 4 maximal such classes.
Kunyavskii <cit.> classified the algebraic $k$-tori of dimension 3 up to birational
equivalence. For each of the tori corresponding to maximal subgroups, he
constructed a nonsingular projective toric model of the algebraic $k$
tori and used the geometric construction of the flasque resolution of the
character lattice of each to understand birational properties of the
algebraic $k$-tori corresponding to maximal subgroups.
See Kunyavskii <cit.> and the description of his work in Voskresenskii <cit.>.
Let $G_k$= for $k=1,\dots,4$.
Then the corresponding lattices are:
* $(M_{G_1},G_1)=(\Z A_2\oplus \Z A_1,\Aut(A_2)\times \Aut(A_1))$.
* $(M_{G_2},G_2)=(\Z B_3,W(B_3))=(\Z B_3,\Aut(B_3))$.
* $(M_{G_3},G_3)=(\Lambda(A_3),\Aut(A_3))$.
* $(M_{G_4},G_4)=(\Z A_3,\Aut(A_3))$.
We claim DadeGroup(3,1) with GAP ID [3,6,7,1] corresponds to
$(\Z A_2\oplus \Z A_1,\Aut(A_2\times A_1))$.
$\Aut(A_2)$ acts as above on $\Z A_2$ and trivially on $\Z A_1$
and $\Aut(A_1)$ acts as $-1$ on $\Z A_1$ and trivially on $\Z A_2$.
The generators given by GAP are
$$ A=(\rho_2^-)^*((12),1), B=(\rho_2^-)^*((132),1)$$
One may replace these generators by
Then by the above argument, we have already seen that $\langle A,B,-I_2\rangle$
determines $(\Z A_2,\Aut(A_2))$ and so we clearly
have that the full group determines $(\Z A_2\oplus \Z A_1,\Aut(A_2)\times \Aut(A_1))$.
We claim that DadeGroup(3,2) with GAP ID [3,7,5,1] corresponds to $(\Z B_3,W(B_3))$.
It is clear from the matrix generators that this corresponds to a sign
permutation lattice.
Recall the faithful representation $\eta_3:W(B_3)\to \GL_3(\Z)$ determined by $\Z B_3$
with respect to the standard basis.
The generators given by GAP are the images under $\eta_3$ of the following elements of $W(B_3)$:
so it is not hard to see that the generators
can be replaced by
$$\langle \tau_1,\tau_2,\tau_3,(12),(132)\rangle\cong C_2^3\rtimes S_3$$
DadeGroup(3,3) with GAP ID [3,7,5,2] corresponds to $(\Lambda(A_3),\Aut(A_3))$
where $G=\Aut(A_3)=S_4\times C_2$.
The generating set given by GAP are the images under $\rho^-_3:S_4\times C_2\to \GL(3,\Z)$ of the elements:
where $\rho_3^-:S_4\times C_2\to \GL_3(\Z)$
is the representation determined by $J_{X_4}\otimes \Z^-_{S_4}$
with respect to a natural basis described earlier.
It suffices to check that $\langle (3,4),(1,3,2),(1,3)(2,4),(1,2)(3,4)\rangle=S_4$ which is straightforward.
Note that $\Aut(A_3)$ acts on $\Lambda(A_3)$ as
$J_{X_4}\otimes \Z^-_{S_4}$, where $X_4$ is the natural $S_4\times C_2$-set.
DadeGroup(3,4) with GAP code [3,7,5,3] corresponds to $(\Z A_3,\Aut(A_3))$.
The matrix generators of this group given by GAP are the transposes of
those for DadeGroup(3,4). So the corresponding lattice is accordingly the
$\Z$ dual $S_4\times C_2$-lattice $I_{X_4}\otimes \Z^-_{S_4}$ which indeed is the
representation of $\Aut(A_3)$ on $\Z A_3$.
Note that the last 3 Dade groups for dimension 3 are all $\Z$-forms
of the root lattice of $A_3$. [Since $\SL_4/C_2\cong \SO_3$, the character lattice
of $\SL_4/C_2$ would be a $\Z$-form of the character lattice of $\SL_4/C_4$
which is $\Z A_3$.]
Although this is not how Kunyavskii determined the rational algebraic $k$
tori of dimension 3, the following argument is more or less equivalent.
He did not need to determine which ones were maximal rational as the numbers
were relatively small.
(Rational algebraic tori of dimension 3)
The maximal finite subgroups of $\GL(3,\Z)$ corresponding to hereditarily rational tori of dimension 3 have the following lattices:
* $(\Z A_2\oplus \Z A_1,\Aut(A_2)\times \Aut(A_1))$
* $(\Z B_3,W(B_3))=(\Z B_3,\Aut(B_3))$.
* $(\Z A_3,W(A_3))$
* $(L,W(B_2))$ where $L$ fits into a short exact sequence of
$W(B_2)$ lattices
$$0\to \Z B_2\to L\to \Z\to 0$$
There are 58 conjugacy classes of finite subgroups of $\GL(3,\Z)$ which are conjugate to a subgroup of one of the above 4 groups. These correspond to the list of 58 rational algebraic tori of dimension 3.
We note that the lattice $(\Z A_2\oplus \Z A_1,\Aut(A_2)\times \Aut(A_1))$
is hereditarily rational as it is a direct sum of hereditarily rational
lattices. This corresponds to DadeGroup(3,1) with GAP code [3,6,7,1].
The lattice $(\Z B_3,W(B_3))$ is hereditarily rational as it is a sign permutation lattice. It corresponds to DadeGroup(3,2) with GAP code [3,7,5,1].
The lattice $(\Z A_3,W(A_3))=(I_{X_4},S_4)$ is hereditarily rational.
From our identification of DadeGroup(3,4) with lattice $(\Z A_3,\Aut(A_3))$ we see that this should correspond to a maximal subgroup.
The subgroup with GAP code [3,7,4,3]
has generators
$$\rho_3^*((3,4)), \rho_3^*((1,3,2)), \rho_3^*((13)(24)), \rho_3^*((12)(34))$$
where $\rho_3^*:S_4\to \GL_3(\Z)$ is the dual representation of $\rho_3$
and hence corresponds to the lattice $I_{X_4}$.
The group with GAP code [3,4,5,2] is abstractly isomorphic to $D_8$.
The following are a set of generators given by GAP:
We note that the sublattice spanned by $\{\e_1-\e_2,\e_3\}$ is stable under the
action of the group. We then recompute the action of the
generators on the $\Z$-basis $\{\e_1-\e_2,\e_3,\e_2\}$ or equivalently
conjugate the generators by the change of basis matrix.
Then the conjugate generators are
With respect to this new basis, it is clear that
the corresponding lattice $L$ fits into the short exact sequence of $W(B_2)\cong D_8$
$$0\to \Z B_2\to L\to \Z\to 0$$
where we recall that $W(B_2)=C_2^2\rtimes C_2\cong D_8$.
Then we see that this lattice is also hereditarily rational.
We can then check using GAP that the union of conjugacy classes of subgroups of
the groups with the above GAP IDs corresponds to the complete list of 58
rational algebraic $k$-tori given by Kunyavskii. We will give more
details of our minimal GAP calculations after the dimension 4 case.
§.§ Algebraic $k$-tori of dimension 4
We now examine the dimension 4 case.
The classification of maximal finite subgroups of $\GL_4(\Z)$ up to conjugacy
is due to Dade. There are 9 maximal finite subgroups.
There are 710 conjugacy classes of finite subgroups of $\GL_4(\Z)$.
Let $G_k=$ for $k=1,\dots,9$.
Then the corresponding lattices are:
* $(M_{G_1},G_1)=(\Z B_2\oplus \Z A_2,\Aut(B_2)\times \Aut(A_2))$.
* $(M_{G_2},G_2)=(\Lambda A_3\oplus \Z A_1,\Aut(A_3)\times \Aut(A_1))$.
* $(M_{G_3},G_3)=(\Z A_3\oplus \Z A_1,\Aut(A_3)\times \Aut(A_1))$
* $(M_{G_4},G_4)=(L,((W(A_2)\times W(A_2))\rtimes C_2)\times C_2)$
where $L$ is the non-trivial intermediate lattice between $\Z A_2\oplus \Z A_2$
and $\Lambda(A_2)\oplus \Lambda(A_2)$.
* $(M_{G_5},G_5)=(\Z A_2\oplus \Z A_2, (\Aut(A_2)\times \Aut(A_2))\rtimes C_2)$.
* $(M_{G_6},G_6)=(\Z A_4,\Aut(A_4))$.
* $(M_{G_7},G_7)=(\Lambda(A_4),\Aut(A_4))$.
* $(M_{G_8},G_8)=(\Z B_4,W(B_4))=(\Z B_4,\Aut(B_4))$.
* $(M_{G_9},G_9)=(\Z F_4,W(F_4))=(\Z F_4,\Aut(F_4))$.
$G_1=$ DadeGroup(4,1) with GAP ID [4,20,22,1]
corresponds to $(\Z B_2\oplus \Lambda(A_2),W(B_2)\times \Aut(A_2))$
which is hereditarily rational as the direct sum of hereditarily rational
The generators of this group are given as
$$A_1=\eta_2(\tau_2),A_2=\eta_2(\tau_1 (12)),$$
$$B_1=\rho^-_3((23),1), B_2=\rho^-_3((13),1)$$
Since $B_1B_2=\rho^-_3((132),1)$ has order 3, and $A_1$ has order 2,
$\diag(A_1,I_2)=\diag(A_1,B_1B_2)^3\in G_1$. Since also $A_2^2=-I_2$, we may
replace the above set of generators by
\diag(I_2,-I_2)\}$$
Then the associated lattice is $M=M_1\oplus M_2$ where
where $M_1=\Z\e_1\oplus \Z\e_2$ and $M_2=\Z\e_3\oplus \Z\e_4$
are both $G$ invariant.
Then since
$$A_1=\eta_2(\tau_2),A_2=\eta_2(\tau_2 (12)),-I_2=\eta_2(\tau_1\tau_2\tau_3)$$
$$B_1=\rho^-_3((23),1), B_2=\rho^-_3((13),1),-I_2=\rho^-_3(1,\gamma)$$
we may see that the lattice $M_1$ is $(\Z B_2,W(B_2))$ and the
lattice $M_2$ is $(J_{X_3}\otimes \Z^-_{S_3},S_3\times C_2)$, where
$X_3$ is the natural $S_3\times C_2$-set.
$M_2$ corresponds to the natural action of $\Aut(A_2)$ on $\Lambda(A_2)$.
So $(M_{G},G)=(\Z B_2\oplus \Lambda(A_2),\Aut(B_2)\times \Aut(A_2))$.
The lattice is hereditarily rational as the direct sum of 2 hereditarily
rational lattices.
DadeGroup(4,2) has GAP ID [4,25,11,2].
The generators are
We then clearly see that the lattice decomposes as a direct sum of
$M_1=\Z\e_1$ and $M_2=\oplus_{i=2}^4\Z \e_i$.
To determine $M_2$, we
note that for the faithful representation of $S_4\times C_2$ given
by $(\rho_3^-)^*:S_4\times C_2\to \GL(3,\Z)$ corresponding to the
$S_4\times C_2$ lattice $I_{X_4}\otimes \Z^-_{S_4}$, we observe that
Since $A_2^3=-I_3$, we see that the elements
are contained in this group.
Note that $\langle A_1,-A_2,A_3,-I_3\rangle=(\rho_3^-)^*(S_4\times C_2)$
$$\langle ((12),1),(134),1),((14)(23),1),(1,\gamma)\rangle=S_4\times C_2.$$
So $H=(1,(\rho_3^-)^*(S_4\times C_2))$ is a subgroup of $G_2$.
Since $(1,A_4)\in H\le G_2$, then we also have $(-1,I_3)\in G_2$ and
so $G_2=\langle (-1,I_3)\rangle \times (\rho_3^-)^*(S_4\times C_2)$.
This shows that $M_2$ corresponds to the $S_4\times C_2$-lattice
$I_{X_4}\otimes \Z^-_{S_4}$.
This is the natural action of $\Aut(A_3)$ on $\Z A_3$.
So the lattice corresponding to DadeGroup(4,2) is $(\Z A_1\oplus \Z A_3,\Aut(A_1)\times \Aut(A_3))$.
$G_3=$DadeGroup(4,3) has GAP ID [4,25,11,4].
The generators are
We then clearly see that the lattice decomposes as a direct sum of
$M_1=\Z\e_1$ and $M_2=\oplus_{i=2}^4\Z \e_i$.
To determine $M_2$, we
note that
\rho_3^-((12)(34),1)=-B_4$$
where $\rho_3^-:S_4\times C_2\to \GL_3(\Z)$ is the representation
corresponding to $J_{X_4}\otimes \Z^-_{S_4}$
which in turn corresponds to $\Aut(A_3)$ acting on $\Lambda(A_3)$.
A similar argument to the one for $G_2$ shows that
$G_3=\langle (-1,I_3)\rangle \times \rho_3^-(S_4\times C_2)$
and the group determines the lattice
$(\Z A_1\oplus \Lambda(A_3),\Aut(A_1)\times \Aut(A_3))$.
$G_4$=DadeGroup(4,4) with GAP ID [4,29,9,1]
is abstractly isomorphic to $((S_3\times S_3)\rtimes C_2)\times C_2$
and has generators given by
$$\left[\begin{array}{rrrr}1&-1&0&0\\0&-1&0&0\\0&0&1&-1\\0&0&0&-1\end{array}\right], \left[\begin{array}{rrrr}1&-1&0&0\\0&0&1&-1\\0&-1&0&0\\0&0&0&-1\end{array}\right],\left[\begin{array}{rrrr}1&-1&0&0\\1&0&0&0\\0&0&1&-1\\0&0&1&0\end{array}\right],\left[\begin{array}{rrrr}0&0&0&1\\0&0&-1&1\\0&-1&0&1\\1&-1&-1&1\end{array}\right]$$
We claim that the lattice determined by this group is
the proper intermediate lattice $L$ between $\Z A_2\oplus \Z A_2$
and $\Lambda(A_2)\oplus\Lambda(A_2)$ and the group action is that
induced by the index 2 subgroup of the automorphism group of the
root system $A_2\times A_2$ given by
$(W(A_2)\times W(A_2))\rtimes C_2)\times C_2$.
Let $\omega_1,\omega_2$ be the fundamental dominant weights of $A_2$
with respect to a basis $\alpha_1,\alpha_2$ of the $A_2$ root system.
We will write the basis of $\Lambda(A_2)\oplus \Lambda(A_2)$
as $\{\omega_1,\omega_2,\omega_1',\omega_2'\}$
and that of $\Z A_2\oplus \Z A_2$ as
Then the claimed sublattice $L$ of $\Lambda(A_2)\oplus \Lambda(A_2)$
satisfies $L=\langle \omega_1+\omega_1',\Z A_2\oplus \Z A_2\rangle$
Let $s_1=s_{\alpha_1},s_1'=s_{\alpha_1'},s_2=s_{\alpha_2},s_2'=s_{\alpha_2'}$
be the generators of $W(A_2)\times W(A_2)$. Let $\tau$ be the element of
order 2 which swaps the 2 copies of $\Lambda(A_2)$.
Then our group is $(\langle s_1,s_2,s_1',s_2'\rangle \rtimes \tau)\times \langle -\id \rangle$.
We claim that
is a $\Z$-basis of our lattice $L$.
We also recall that $\alpha_1=2\omega_1-\omega_2$ and $\alpha_2=-\omega_1+2\omega_2$. The same results hold for the prime copies.
$\langle s_1,s_2\rangle$ acts trivially on $\oplus_{i=1}^2\omega_i'$
and similarly for the prime copies.
So $s_2s_1(\omega_1+\omega_1')=\omega_1-\alpha_1-\alpha_2+\omega_1'=-\omega_2+\omega_1'$
and $s_2's_1'(\alpha_1')=-\alpha_1'-\alpha_2'$.
Note that $s_2s_1(\omega_1)=\omega_1+s_2s_1(\alpha_1)$.
From these calculations, one can show that $\beta$ is a basis of $L$.
They also allow us to find the matrices of the generators
$s_1,s_2s_1,s_1',s_2's_1',\tau,-\id$ on the basis $\beta$.
[We omit the details but note that the worst calculation is $s_1(\omega_1+\omega_1')$ made above.]
We obtain:
$$s_1=\left[\begin{array}{rrrr}-1&-1&1&0\\0&1&0&0\\0&0&1&0\\0&0&1&0\end{array}\right], s_1'=\left[\begin{array}{rrrr}1&0&-1&0\\0&1&-1&0\\0&0&-1&0\\0&0&1&1\end{array}\right],s_2s_1=\left[\begin{array}{rrrr}0&1&0&0\\-1&-1&1&-1\\0&0&1&0\\0&0&0&1\end{array}\right]$$
$$s_2's_1'=\left[\begin{array}{rrrr}1&0&0&1\\0&1&0&1\\0&0&0&1\\0&0&-1&-1\end{array}\right], \tau=\left[\begin{array}{rrrr}1&0&0&0\\1&0&0&1\\2&1&-1&1\\-1&1&0&0\end{array}\right], -I_4$$
The group generated by these generators is conjugate to the
group DadeGroup(4,4).
This is determined by GAP by checking that the
CrystCatZClass of the two matrix groups given by generators are in fact the
same. See the remark below on GAP calculations.
[In fact, the index 2 subgroup generated by the generators
$\langle s_1,s_2s_1,s_1',s_2's_1',-\id\rangle$
is precisely equal to the group with GAP ID [4,22,11,1].
which is conjugate to a subgroup of the group DadeGroup(4,4).]
So the lattice
determined by DadeGroup(4,4) is indeed the intermediate lattice
$L$ between $\Z A_2\oplus \Z A_2$ and $\Lambda(A_2)\oplus \Lambda(A_2)$
as a $(W(A_2)\times W(A_2))\rtimes C_2 \times C_2$ lattice.
For G=DadeGroup(4,5) with GAP ID [4,30,13,1], we claim that the
corresponding lattice is $(\Z A_2\oplus \Z A_2,\Aut(A_2\times A_2))$
which is hereditarily rational as a wreath product of 2 hereditarily
rational lattices.
$\Aut(A_2\times A_2)=(\Aut(A_2)\times \Aut(A_2))\rtimes S_2$ acts naturally
on the root lattice for $A_2\times A_2$ where $\Aut(A_2)\times \Aut(A_2)$
acts diagonally on $\Z A_2\oplus A_2$ and the subgroup $S_2$ permutes the
two copies of $A_2$.
The generators given by GAP are
\left[\begin{array}{rr}0&I_2\\P&0\end{array}\right],
\left[\begin{array}{rr}I_2&0\\0&Y\end{array}\right],
\left[\begin{array}{rr}Z&0\\0&I_2\end{array}\right]$$
Since $Y,Z$ have order 6 and $Y^3=Z^3=-I_2$,
we may replace these generators by
\left[\begin{array}{rr}0&I_2\\I_2&0\end{array}\right],
\left[\begin{array}{rr}I_2&0\\0&Y^2\end{array}\right],
\left[\begin{array}{rr}Z^2&0\\0&I_2\end{array}\right],
X_5=\diag(I_2,-I_2), X_6=\diag(-I_2,I_2).$$
$$Y^2=\left[\begin{array}{rr}-1&1\\-1&0\end{array}\right]=(\rho_2^-)^*((123))\qquad Z^2=\left[\begin{array}{rr}0&-1\\1&-1\end{array}\right]=(\rho_2^-)^*((132)), P=(\rho_2)^*((12))$$
We see that the lattice defined on $\Z\e_1\oplus \Z\e_2$ defined by $\langle P,Y^2\rangle$ is the $S_3$-lattice $I_{X_3}$.
We also see that the lattice defined on $\Z\e_3\oplus \Z\e_4$
defined by $\langle P, Z^2\rangle$ is $I_{X_3}$.
This shows that the lattices defined by both
$\langle P,Y^2,-I_2\rangle$ and $\langle P,Z^2,-I_2\rangle$ are
isomorphic to the lattice $(I_{X_3}\otimes \Z^-_{S_3},S_3\times C_2)$
or equivalently $(\Z A_2,\Aut(A_2))$.
So the lattice restricted to
$$\langle \diag(I_2,P),\diag(I_2,Y^2),\diag(I_2,-I_2),\diag(P,I_2),\diag(Z^2,I_2),\diag(-I_2,I_2)\rangle$$
is $(\Z A_2\oplus \Z A_2,\Aut(A_2)\times \Aut(A_2))$.
swaps the 2 copies of $\Z A_2$, we see that the full lattice structure is
given by
$$(\Z A_2\oplus \Z A_2,(\Aut(A_2)\times \Aut(A_2))\rtimes C_2)$$
as required.
DadeGroup(4,6) has GAP ID [4,31,7,1].
We will show that it determines the lattice $(\Z A_4, \Aut(A_4))$
which is hereditarily rational since it is the tensor product of
2 augmentation ideals of relatively prime ranks.
This lattice is $(I_{X_5}\otimes \Z^-_{S_5},S_5\times C_2)$.
Recalling our representation $((\rho^-_4)^*:S_5\times C_2\to \GL_4(\Z)$
determined by $(I_{X_5}\otimes \Z^-_{S_5})$,
we see that the generators of DadeGroup(4,6) given by GAP
$$(\rho^-_4)^*(((15)(234),\gamma),(\rho^-_4)^*((14532)),\gamma), (\rho_4)^*((1423),\gamma)$$
Since $(14532)$ has odd order, we see that $((14523),\gamma)^5=(\id,\gamma)$,
and we know that $S_5$ is generated by any 5 cycle and any transposition,
so it suffices to note that $[(15)(234)]^3=(15)$, in order to conclude that
the preimages generate $S_5\times C_2$.
So, as required, DadeGroup(4,6) determines the
lattice $(I_{X_5}\otimes \Z^-_{S_5},S_5\times C_2)=(\Z A_4,\Aut(A_4))$.
DadeGroup(4,7) has GAP ID [4,31,7,2].
We will show that it determines the lattice $(\Lambda(A_4), \Aut(A_4))$.
This lattice is $(J_{X_5}\otimes \Z^-_{S_5},S_5\times C_2)$.
In terms of our representation $\rho^-_4:S_5\times C_2\to \GL_4(\Z)$
determined by $(J_{X_5}\otimes \Z^-_{S_5})$,
we see that the generators of DadeGroup(4,7) given by GAP
$$\rho^-_4((132)(45),\gamma),\rho^-_4((15234),\gamma), \rho^-_4((1324),\gamma)$$
Since $(15234)$ has odd order, we see that $((15234),\gamma)^5=(\id,\gamma)$,
and we know that $S_5$ is generated by any 5 cycle and any transposition,
so it suffices to note that $[(132)(45]^3=(45)$, in order to conclude that
the preimages generate $S_5\times C_2$.
So, as required, DadeGroup(4,7) determines the lattice
$(J_{X_5}\otimes \Z^-_{S_5},S_5\times C_2)=(\Lambda(A_4),\Aut(A_4))$.
DadeGroup(4,8) has GAP ID [4,32,21,1].
We claim that it determines the lattice
$(\Z B_4,W(B_4))$.
In terms of our representation $\eta_4:W(B_4)\to \GL_4(\Z)$, the generators
given by GAP are
$$\eta_4(\tau_2\tau_4(24)), \eta_4(\tau_1\tau_3\tau_4(234)),
\eta_4(\tau_1\tau_2),
\eta_4(\tau_3\tau_4(12)(34)),\eta_4(\tau_1\tau_2(13)(24)),
\eta_4(\tau_1\tau_4(14)(23))$$
We need only check that the subgroup $H$ of $W(B_4)$ generated by the
preimages of the generators under $\eta_4$ is $W(B_4)$.
(Note that GAP gives a structure description for the group as the wreath
product of $C_2$ by $S_4$ so this is just a check).
Recall that $(\tau\sigma)^2=\tau\tau^{\sigma}\sigma^2$ and $\tau_i^{\tau\sigma}=\tau_{\sigma(i)}$ for any $\tau\in C_2^4$ and $\sigma\in S_4$.
Noting that $(\tau_1\tau_2)$ is in $H$ and conjugating this element by
each of the other above generators, one can see that
$\tau_i\tau_j$ is in the group for all $1\le i\ne j\le 4$.
Looking again at the generators, we see that $(24),(12)(34),(13)(24),(14)(23)$
are also in the group.
But then since also $[\tau_1\tau_3\tau_4(234)]^2=\tau_2\tau_3(243)$
is in $H$, we have that $(243)$ is in $H$ too.
We now have $\langle (24),(12)(34),(13)(24),(14)(23),(243)\rangle=S_4$
and $\langle \tau_i\tau_j: 1\le i\ne j\le 4\rangle$ as subgroups.
But looking at the original generator $\tau_1\tau_3\tau_4(234)$, we see that
$\tau_1$ and hence all its conjugates under $S_4$ are in the group.
This shows that the group generated by its preimages contains
$\langle \tau_i:i=1,\dots,4\rangle$ and $S_4$ and then must be $W(B_4)$.
Note that $(\Z B_4,W(B_4))$ is a hereditarily rational lattice as it is
sign permutation.
DadeGroup(4,9) has GAP ID [4,33,16,1].
Its generators are
$$X_1=\left[\begin{array}{rrrr}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{array}\right], X_2=\left[\begin{array}{rrrr}1&0&0&0\\0&0&1&0\\0&1&0&0\\1&1&1&-1\end{array}\right],
$$X_4=\left[\begin{array}{rrrr}-1&0&0&1\\0&0&-1&1\\-1&-1&-1&1\\-1&-1&-1&2\end{array}\right], X_5=\left[\begin{array}{rrrr}0&0&-1&0\\1&1&1&-2\\-1&0&0&0\\0&0&0&-1\end{array}\right],$$
We wish to show that the associated lattice is $(\Z F_4,W(F_4))$.
We recall a standard basis of the root system $F_4$ is given by
The roots in the root system $F_4$ are of the
$$\pm \e_i\pm \e_j,1\le i<j\le 4; \pm \e_i, 1\le i\le 4; \frac{\pm \e_1\pm \e_2\pm \e_3\pm \e_4}{2}$$
A $\Z$-basis for $\Z F_4$ is given by
For each of the simple roots in $\Delta$
we may compute the matrices of $s_{\alpha_i}$ with respect to the basis $\beta$
$$s_{\alpha}(x)=x-2\frac{x\cdot \alpha}{\alpha\cdot \alpha}\alpha$$
is the simple reflection corresponding to $\alpha$.
We find that
$$s_{\alpha_1}=\left[\begin{array}{rrrr}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{array}\right], s_{\alpha_2}=\left[\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\-1&-1&-1&2\\
$$s_{\alpha_3}=\left[\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\0&0&1&0\\1&1&1&-1\end{array}\right], s_{\alpha_4}=\left[\begin{array}{rrrr}0&0&0&1\\1&1&0&-1\\1&0&1&-1\\
We may show that $s_{\alpha_1},\dots,s_{\alpha_4}$ are all contained in DadeGroup(4,9). That is, we may express them as products of the generators given by GAP. Explicitly, $s_{\alpha_1}=X_1$, $s_{\alpha_2}=X_5X_3X_4^{-1}X_1$,
$s_{\alpha_3}=X_2X_1$, $s_{\alpha_4}=X_6^{-1}X_4^{-1}X_3^{-1}X_2X_1$.
Since DadeGroup(4,9) has order $1152=|W(F_4)|$ we see that they coincide.
(Stably Rational Tori of Dimension 4 (Hoshi Yamasaki))
The 487 finite subgroups of $\GL(4,\Z)$ which correspond to stably rational tori from Hoshi and Yamasaki's list
are conjugate to a subgroup of one of the groups with
GAP IDs on the following list:
* (Irreducible maximal finite subgroups)
* (Irreducible non-maximal finite subgroup)
* (Decomposable finite subgroups)
[4,20,22,1], [4,25,9,2],[4,25,11,1]
* (Reducible finite subgroups)
That is, these are the maximal conjugacy classes of subgroups corresponding to
stably rational tori.
In the list above, the maximal indecomposable subgroups corresponding to stably rational tori are listed in their
paper in the proof of the result for dimension 4. The maximal decomposable subgroups are not listed in their paper but could be
easily derived from their results or directly from GAP.
The 7 finite subgroups of $\GL(4,\Z)$ which correspond to retract rational
tori which are not stably rational from Hoshi and Yamasaki's list are
* (6 Subgroups of DadeGroup(4,7) of GAP ID [4,31,7,2]):
* (1 exceptional subgroup)
(Hereditarily rational tori of dimension 4)
Among the 10 finite conjugacy classes of subgroups of $\GL(4,\Z)$ which are maximal among those corresponding to stably rational tori of dimension 4, the following 8 correspond to
hereditarily rational algebraic tori. We list the GAP ID of each group $G$ together with the
lattice structure of $M_G$.
* Dade Groups which correspond to hereditarily rational tori:
* $(\Z B_2\oplus \Z A_2,\Aut(B_2)\times \Aut(A_2))$
* $(\Z A_2\oplus \Z A_2,(\Aut(A_2)\times \Aut(A_2))\rtimes C_2)$.
* $(\Z A_4,\Aut(A_4))$.
* $(\Z B_4,W(B_4))$.
* Direct Products $G=H\times C_2$ of a finite matrix group H of rank 3 and
a finite matrix group $C_2$ of rank 1 where $H$ corresponds to a maximal hereditarily rational torus of rank 3.
In this case the lattice is $M_G=\inf_H^G(M_H)\oplus \Z^-_H$
* $(\inf_{S_4}^{S_4\times C_2}\Z A_3 \oplus \Z^-_{S_3},W(A_3)\times C_2)$.
* $(\inf_{D_8}^{D_8\times C_2}L\oplus \Z^-_{D_8},D_8\times C_2)$.
* Groups whose corresponding lattice is indecomposable but reducible with a rank 3 invariant maximal hereditarily rational sublattice and a fixed rank 1 quotient lattice.
* $(L_1,W(B_3))$ where $L_1$ is a non-split extension of $W(B_3)$-lattices
$$0\to \Z B_3\to L_1\to \Z\to 0$$
* $(L_2,W(A_3))$ where $L_2$ is a non-split extension of
$$0\to \Z A_3\to L_2\to \Z\to 0$$
The union of the conjugacy classes of these 8 finite subgroups of $\GL(4,\Z)$
produces 477 of the 487 stably rational algebraic $k$-tori
determined by Hoshi and Yamasaki.
We have already determined the Dade groups which correspond to hereditarily
rational tori.
We first determine the GAP IDs of the groups corresponding to a direct
sum of a maximal hereditarily rational lattice of rank 3 and a sign lattice.
We claim that the group with GAP ID [4,25,9,2] has lattice given by
$$(\Z^-_{W(A_3)}\oplus \inf^{W(A_3)\times C_2}_{W(A_3)}\Z A_3,W(A_3)\times C_2)$$
The generators given by GAP are
where $A_i\in \GL_3(\Z)$ are given
$$A_1=(\rho_3)^*((34)), A_2=(\rho_3)^*((124)), A_3=(\rho_3)^*((14)(23)),
where $(\rho_3)^*:S_4\to \GL_3(\Z)$ is the representation associated to the root lattice $(I_{X_4},S_4)=(\Z A_3,W(A_3))$ described earlier.
Since $(124)$ is odd, so is $A_2$ and so the generator $(-1,A_2)$
can be replaced by $(1,A_2)$ and $(-1,I_3)$.
Then it is clear that the group is a direct product of $\langle (-1,I_3)\rangle$ and $\langle (1,A_i):i=1,\dots,4\rangle$.
It suffices to show that $\langle (34),(124),(14)(23),(12)(34)\rangle =S_4$,
which is easily checked.
So $\langle (1,A_i\rangle=(\rho_3)^*(S_4)$
which implies that the corresponding lattice is the $C_2\times S_4$-lattice inflated from the $S_4$-lattice $I_{X_4}$ as required.
The group with GAP ID [4,13,6,4] has
Since we may replace the generators by $(I_3,-1),(A,1),(B,1)$,
the group is a direct product $\langle (I_3,-1)\rangle\times \langle (A,1),(B,1)\rangle$ and so we may just determine the group $\langle A,B\rangle$.
We note that $\Z(\e_1-\e_3)\oplus \Z\e_2$ is stable under $\langle A,B\rangle$.
Computing the matrices with respect to the new basis
$\{\e_1-\e_3,\e_2,\e_3\}$ (or equivalently conjugating by the change of basis matrix)
we obtain
It is then clear that the $W(B_2)$ lattice determined by $\langle A',B'\rangle$
$$0\to \Z B_2\to L\to \Z\to 0$$
Then our lattice is $(\inf_{W(B_2)}^{W(B_2)\times C_2}L\oplus \Z^-_{W(B_2)},W(B_2)\times C_2)$.
We next look at the groups which correspond to a reducible
lattice with a 3 dimensional invariant sublattice:
For the group with GAP ID [4,25,7,5], the generators are
Note that this determines a lattice $M$ which contains a sublattice
with basis $\{\e_2,\e_3,\e_4\}$ which is stable under the action of the
group. Note also that $M/(\oplus_{i=2}^4\Z\e_i)\cong \Z$.
The action on $\oplus_{i=2}^4\Z\e_i$ is determined by
the group generated by
\eta_3(\tau_1\tau_2), \eta_3(\tau_2\tau_3)$$
Since $(\tau_1(123))^2=\tau_1\tau_2(132)$, $(132)$ is in the preimage,
and then so is $\tau_1$. Then one can easily
show that $\tau_2,\tau_3$ and $(23)$ are in the preimage too.
This shows that the group determines a $W(B_3)$-lattice $M$
which satisfies
$$0\to \Z B_3\to M\to \Z\to 0$$
So the corresponding algebraic torus is hereditarily rational.
For the group with GAP ID [4,24,3,4], the generators are
Note that this determines a lattice $M_4$ which contains a sublattice
$M_3=\oplus_{i=2}^4\Z\e_i$ which is stable under the action of the
group. Note also that $M_4/M_3\cong \Z$.
The action restricted to the sublattice $M_3$ is determined by
the group generated by
for the representation $\rho_3^*:S_4\to \GL(3,\Z)$
associated to the $S_4$-lattice $I_{X_4}$. It is easily checked that the restriction of the group action on $M_3$ is a faithful action.
Since the group generated by preimages is $\langle (14),(134),(13)(24),(14)(23)\rangle$, we see that it contains the normal subgroup $\langle (13)(24),(14)(23)\rangle$ of $S_4$. Then clearly $\langle (13)(24),(14)(23),(143)\rangle =A_4$
and $\langle (13)(24),(14)(23),(143),(14)\rangle =S_4$.
So the lattice determined by this group satisfies a short exact sequence of
$W(A_3)=S_4$ lattices given by
$$0\to \Z A_3\to M_3\to \Z\to 0$$
This again shows that the associated group is hereditarily rational.
We then use GAP to take the union of the conjugacy classes of subgroups
corresponding to these 10 hereditarily rational lattices. (See below for our simple use of GAP to obtain this information.)
We find that we obtain 477 hereditarily rational tori, all but 10 of the
stably rational tori obtained by
Hoshi and Yamasaki.
Of these 10, there are 2 maximal groups having GAP IDs [4,25,8,5] and [4,31,6,2]. In the next proposition we will describe the lattice structure of these
two groups and list the 10 exceptional subgroups.
In a subsequent section, we will give non-computational
proofs that the tori corresponding to these two groups are stably rational.
Let $H_i$, $i=1,2$ be finite matrix groups of rank $r_i$, $i=1,2$ where each is a maximal subgroup corresponding to a hereditarily rational torus of the appropriate rank. Then $H_1\times H_2$ is a finite matrix group corresponding to a hereditarily rational torus of rank $r+s$ but it may not be maximal among the finite matrix groups of rank $r+s$ whose corresponding algebraic torus is stably rational.
The following are the GAP IDs and lattices
corresponding to the two finite subgroups of $\GL(4,\Z)$ which are maximal among those corresponding to stably rational tori of dimension 4 but are not known to be hereditarily rational:
* $(J_{X_5}\otimes \Z_{A_5}^-,A_5\times C_2)$
* The corresponding lattice $L$ is a non-split extension
of $W(B_3)$-lattices
$$0\to \Z B_3\to L\to \Z^-_{C_2^3\rtimes A_3}\to 0$$
There are 10 conjugacy classes of subgroups of these 2 groups which are not
conjugate to a subgroup of one of the 8 groups from Theorem <ref>.
The rationality of the corresponding algebraic tori is hence unknown.
The full list is
* Subgroups of
* Subgroups of .
Note that the lattice corresponding to
is $(J_{X_5},A_5)$ which corresponds to a norm one torus.
From the proof of the last theorem, we see that the tori corresponding to all
of the maximal groups corresponding to stably rational tori except for those with GAP IDs [4,25,8,5] and [4,31,6,2] are hereditarily rational.
For the group with GAP ID [4,25,8,5], the generators are
Note that these are in fact the same generators as in the [4,25,7,5]
case except for the first one.
Note that this determines a lattice $M$ which contains a sublattice $M_0$
with basis $\{\e_2,\e_3,\e_4\}$ which is stable under the action of the
group. Note also that $M_1=M/M_0$ is a rank 1 lattice with
non-trivial action.
The action on $M_0=\oplus_{i=2}^4\Z\e_i$ is determined by
the group generated by
\eta_3(\tau_1\tau_2), \eta_3(\tau_2\tau_3)$$
Since $(\tau_1(123))^2=\tau_1\tau_2(132)$, then $(132)$ is in the preimage
and hence so is $\tau_1$.
Then one can easily
show that $\tau_2,\tau_3$ and $(23)$ are too. Then it is clear that the group
acts on the lattice $M_0$ as $(\Z B_3,W(B_3))$.
It is easy to check that the restriction of the group action to $M_0$
is faithful, so that we may identify the group elements with the elements of
Note that $N=\langle \tau_1(123),\tau_1\tau_2,\tau_2\tau_3\rangle$ acts trivially on $M_1=M/M_0$. Since $N$ contains $(\tau_1(123))^2= \tau_1,\tau_2,\tau_3$, it can be shown to contain $\langle \tau_1,\tau_2,\tau_3,(123)\rangle=C_2^3\rtimes A_3$.
This shows that the group $G$ determines a lattice $M$
which satisfies
$$0\to \Z B_3\to M\to \Z^-_{C_2^3\rtimes A_3}\to 0$$
Note also that $W(B_3)=C_2^3\rtimes S_3\cong C_2\times S_4$ and $N=C_2^3\rtimes A_3\cong C_2\times A_4$.
We will present a non-computational proof that the lattice corresponding to
[4,25,8,5] is quasi-permutation.
The lattice determined by [4,31,6,2] is $
(J_{X_5}\otimes \Z_{A_5}^-,A_5\times C_2)$.
This is because the generators given by GAP are
Since $(15243)$ has odd order, we see that $(1,\gamma)\in C_2$ is in the preimage.
Then it is not hard to see that $\langle (13)(24),(12)(34),(132)\rangle=A_4$
and so $\langle (15243),(132),(12)(34),(13)(24)\rangle=A_5$.
The lattice determined by [4,31,3,2] is $(J_{X_5},A_5)$.
This is because the generators given by GAP are
Again, it's easy to see that $\langle (13)(25),(12)(35)\rangle=C_2\times C_2$
and $\langle (13)(25),(12)(35),(123)\rangle=A_4$
and finally $\langle (13)(25),(12)(35),(123)\rangle=A_5$.
We will show in the next section that the
lattices $(J_{A_5\times C_2/A_4\times C_2}\otimes \Z_{A_5}^-,A_5\times C_2)$
and $(J_{A_5/A_4},A_5)$ are quasi-permutation so that the corresponding tori are stably
rational. We are not able to show that these tori are rational.
Note that that determined by the $A_5$-lattice $J_{A_5/A_4}$ is a norm one torus and we
will see that the other torus is closely related.
To explain why there are only 2 missing subgroups of [4,31,6,2] not known to correspond to stably rational tori, note that the
restriction of $(J_{X_5}\otimes \Z_{A_5}^-,A_5\times C_2)$ to the maximal subgroup $D_5\times C_2$
is $(J_{X_5}\otimes \Z_{D_{10}}^-,D_{10}\times C_2)$
which corresponds to a hereditarily rational torus.
Note that all subgroups of $A_5\times C_2$ except $A_5$ are subgroups of $D_{10}\times C_2$.
We explain our very basic use of GAP.
We mainly used the generating sets (which could have been found in <cit.>) and as a calculation tool to check our hypotheses.
All the calculations of the lattices corresponding to the groups could be done by hand as explained above directly from the
generating sets, with only 2 exceptions. In the case of the lattice for the Weyl group of $F_4$, we used GAP to find the simple reflections in the generators. In the case of DadeGroup(4,4) we used GAP to check that our proposed group
was conjugate to DadeGroup(4,4). We hope to find simpler proofs in those 2 cases. Note that they do not come into play in checking for rational tori.
To check in the dimension 3 and 4 cases that the conjugacy classes of subgroups of the groups corresponding to our hereditarily rational algebraic tori are
rational give all (respectively all but 10) stably rational algebraic tori
we mainly use the following function:
This function returns the conjugacy classes of subgroups of the
group MatGroupZClass(r,m,n,k) given as a list of GAP IDs.
It depends on the GAP script written by Hoshi and Yamasaki
crystcat.gap. This script which is available on the
second author's website,
determines the GAP ID of a finite
subgroup $G$ of $\GL_n(\Z)$ where $n=2,3,4$ by the function
It uses the data of the
book <cit.> to find the crystal class, Q class and Z class of a finite
subgroup of $\GL_n(\Z)$.
It is invoked using
With that tool, for the proposed maximal hereditarily rational subgroups, one can find the
union of all the conjugacy classes of subgroups in terms of their GAP IDs.
One can also find the union of all conjugacy classes of subgroups of the
Dade Groups. By taking the difference of these two sets, we find the
list of all GAP IDs correponding to non-rational tori. One can then
check them against the lists in Hoshi and Yamasaki <cit.> which I do not
reproduce here.
§ STABLE RATIONALITY OF EXCEPTIONAL TORI
In this section, we will show that algebraic $k$-tori whose
character lattices are given by $(J_{X_5},A_5)$
or $(J_{X_5}\otimes \Z^-_{A_5},A_5\times C_2)$ are stably rational
recovering results of Hoshi and Yamasaki in a non-computational way.
We will also give new non-computational proofs showing that the 7 algebraic
$k$-tori of dimension 4 which are retract but not stably rational.
Note that, for a prime $p$, Beneish <cit.> proved that the $S_p$-lattice $J_{X_p}$ is flasque equivalent to
as $S_p$-lattices where $N_{S_p}(C_p)=C_p\rtimes C_{p-1}$ is the normaliser
of the cyclic $p$ Sylow subgroup of $S_p$.
[Recall however, that it is known that the $N_{S_p}(C_p)$-lattice $J_{X_p}$ is not
$C_p\rtimes C_{p-1}$-quasi permutation for primes $p\ge 5$.]
We intend to prove a similar result for $A_5$ and $N_{A_5}(C_5)=C_5\rtimes C_2=D_{10}$. The arguments are similar at the start but diverge at a critical point.
This result and the fact that $J_{X_5}$ is $D_{10}$-quasi-permutation,
will allow us to show that $J_{X_5}$ is also $A_5$-quasi-permutation.
Note that this is equivalent to the result that for a separable extension $K/k$ of degree 5
with Galois closure $L/k$ such that $\Gal(L/k)=A_5$ and $\Gal(L/K)=A_4$,
the norm one torus $R^{(1)}_{K/k}(\Gm)$ is stably rational.
We intend also to show that $J_{X_5}\otimes \Z^-_{A_5}$ is $A_5\times C_2$-quasi-permutation (and the corresponding torus stably rational)
using the result for the corresponding norm one torus and a useful Lemma
due to Florence (see below).
The following lemma was observed by Bessenrodt-Lebruyn <cit.>.
For the transitive $S_n$-set $X_n$ with stabilizer subgroup $S_{n-1}$,
$$I_{X_n}\otimes \Z[X_n]\cong \Z[S_n/S_{n-2}]$$
Let $\{\e_i:i=1,\dots,n\}$ be the $\Z$-basis of the $S_n$-set $X_n$ permuted by $S_n$
via $\sigma(\e_i)=\e_{\sigma(i)}$.
It suffices to show that $I_{X_n}\otimes \Z[X_n]$
has $\Z$-basis
$$\{(\e_i-\e_j)\otimes \e_i: i\ne j\}$$
as then this basis is clearly transitively permuted by the action of $S_n$
with stabilizer subgroup $S_{n-2}$.
Since a $\Z$-basis of $I_{X_n}\otimes \Z[X_n]$ is
given by
$$\{(\e_i-\e_{i+1})\otimes \e_j: 1\le i\le n-1, 1\le j\le n\}$$
we need only show the $\Z$-span of each set contains the other.
$$(\e_i-\e_j)\otimes \e_j=\sum_{k=i}^{j-1}(\e_k-\e_{k+1})\otimes \e_j, i<j$$
$$(\e_i-\e_j)\otimes \e_j=-\sum_{k=j}^{i-1}(\e_k-\e_{k+1})\otimes \e_j, i>j.$$
$$(\e_i-\e_{i+1})\otimes \e_j=(\e_j-\e_{i+1})\otimes \e_j-(\e_j-\e_i)\otimes \e_j.$$
The following lemma was proved by Bessenrodt and Lebruyn but unpublished.
It was proved in Beneish <cit.>. Here is a simpler proof.
For $p$ prime,
let $B_p$ be the $S_p$-lattice
$$B_p=J_{X_p}\otimes I_{X_p}$$
$$B_p\oplus \Z[X_p]\cong \Z[S_p/S_{p-2}]\oplus \Z$$
So $B_p$ is $S_p$-stably permutation.
Tensoring the exact sequence
$$0\to I_{X_p}\to \Z [X_p]\to \Z\to 0 \qquad (*)$$
by $J_{X_p}=(I_{X_p})^*$, and noting that
$$(I_{X_p})^*\otimes (\Z [S_p/S_{p-1}])^*\cong (I_{X_p}\otimes \Z [S_p/S_{p-1}])^*\cong \Z [S_p/S_{p-2}]$$
as well as the fact that permutation lattices are self-dual,
we see that
\xymatrix{
&\Z \ar[r]^{=}\ar@{>->}[d] &\Z\ar@{>->}[d] \\
B_p \ar@{>->}[r]\ar[d]^{=} & \mbox{pull-back} \ar@{->>}[r]\ar@{->>}[d]
B_p \ar@{>->}[r]& \Z[S_p/S_{p-2}] \ar@{->>}[r] &I_{X_p}^*
Since extensions of permutation lattices by permutation lattices
are always split,
we have the exact sequence
$$0\to B_p\to \Z\oplus \Z[S_p/S_{p-2}]\to \Z[X_p]\to 0$$
To prove the result, we need only show that
$B_p$ is invertible since extensions of permutation lattices
by invertible lattices are split.
To show that $B_p$ is $S_p$-invertible,
it suffices to show that $B_p$ is $Q$-invertible for each Sylow $q$-subgroup $Q$ of $S_p$.
Let $Q$ be a Sylow $q$-subgroup of $S_p$ where $q\ne p$.
Then $Q$ must fix some $\e_i\in \Z[X_p]$.
Then $I_{X_p}\vert_Q\oplus \Z=\Z[X_p]\vert_Q$.
In fact $I_{X_p}\vert_Q$ is then $Q$-permutation with
$\Z$-basis $\{\e_i-\e_j: j\ne i\}$.
Dualising we get
$(I_{X_p})^{*}\vert_Q\oplus \Z=\Z[X_p]\vert_Q$
and tensoring with $I_{X_p}\vert_Q$
we obtain
$$(B_p)\vert_Q\oplus I_{X_p}\vert_Q\cong \Z[S_p/S_{p-2}]\vert_Q$$
So $B_p\vert_Q$ is $Q$-stably permutation and hence $Q$-invertible.
It suffices to show that $B_p\vert P$ is $P$-invertible
where $P$ is a Sylow $p$-subgroup of $S_p$. Note that $P\cong C_p$
and $\Z[X_p]_{C_p}\cong \Z[C_p]$.
$$0\to (B_p)_P\to (\Z\oplus \Z[S_p/S_{p-2}])_P\to \Z[P]\to 0$$
splits since $\Z[P]$ is free.
So $B_p$ is $S_p$-invertible and then
the sequence
$$0\to B_p\to \Z\oplus \Z[S_p/S_{p-2}]\to \Z[X_p]\to 0$$
splits to give us the result.
For a group $G$, a $G$ module $M$ is cohomologically trivial if $\hat{H}^k(H,M)=0$ for all $k$
and for all subgroups $H$ of $G$. Projective $G$ modules are cohomologically
trivial. A result in Brown <cit.> shows that faithful cohomologically trivial $G$-lattices (torsion-free $G$ modules) are $G$-projective.
$\Fp I_{S_p/N_p}:=\Fp\otimes_{\Z}I_{S_p/N_p}$
is $S_p$-cohomologically trivial where $N_p=N_{S_p}(C_p)$ is the normaliser of a cyclic Sylow $p$-subgroup $C_p$ of $S_p$.
Note that tensoring the augmentation sequence for $S_p/N_p$
by $\Fp$ is exact.
Since $p$ does not divide $[S_p:N_p]$,
the $\Fp S_p$-exact sequence
$$0\to \Fp I_{S_p/N_p}\to \Fp[S_p/N_p]\to \Fp\to 0$$
and so
$$\Fp[S_p/N_p]\cong \Fp I_{S_p/N_p}\oplus \Fp$$
It suffices to check whether the restrictions to Sylow subgroups
are cohomologically trivial.
For any Sylow $q$-subgroup for $q\ne p$, representations of $\Fp Q$
are completely reducible and so all are projective and hence
cohomologically trivial.
So it suffices to check whether $\Fp I_{S_p/N_p}\vert P$ is cohomologically trivial for a cyclic Sylow $p$-subgroup $P=C_p$.
By Mackey's Theorem,
$$\Res^{S_p}_P\Ind^{S_p}_{N_p}\Fp=\oplus_{x\in P\backslash S_p/N_p}\Fp[P/P\cap N_p^x]$$
Since $P\cong C_p$, $P\cap N_p^x$ is either $P$ or $\{1\}$.
We claim that the unique double coset with $P\cap N_p^x=P$ is $PxN_p=N_p$.
Suppose $P\cap N_p^x=P$. Then $P\le N_p^x$ and so $P^{x^{-1}}\le N_p$.
But $P$ is the unique $p$-Sylow subgroup of $N_p$ and so $P^{x^{-1}}=P$
which means $x^{-1}\in N_p$. Then we have $x\in N_p$ and $PxN_p=N_p$.
So for all non-trivial double cosets $PxN_p\ne N_p$, we have
$P\cap N_p^x=1$.
This means that
$$\Res^{S_p}_P\Ind^{S_p}_{N_p}\Fp=\Fp\oplus (\Fp P)^k$$
for some $k$.
Since $\Fp[S_p]$ satisfies Krull Schmidt, we have that
$$\Res^{S_p}_P\Fp I_{S_p/N_p}\oplus \Fp\cong \Fp \oplus (\Fp P)^k$$
implies that
$\Res^{S_p}_P\Fp I_{S_p/N_p}\cong (\Fp P)^k$ is free
and so cohomologically trivial.
A projective $A_5$-lattice is
$A_5$-stably permutation.
By Endo and Miyata <cit.>, the projective class group of the group ring of $A_5$, $\Z [A_5]$, is $\Cl(\Z [A_5])=0$. But the projective class group is the
kernel of the rank homomorphism $K_0(\Z [A_5])\to \Z$ which sends any finitely
generated projective $A_5$ lattice to its rank.
Since this class group is zero, it shows that any projective $A_5$-lattice
has the same class in $K_0(\Z A_5)$ as a free $A_5$-lattice of the same rank.
But then by <cit.>, we see that a projective $A_5$-lattice is $A_5$-stably free.
This shows that all cohomologically trivial faithful $A_5$-lattices are stably free and hence
stably permutation.
Projective $G$-lattices are stably permutation for any finite group $G$
with splitting field $\Q$ (e.g. $S_n$). <cit.>, see also <cit.>. This was used in <cit.>.
Note that the splitting field for $A_5$ is $\Q(\sqrt{5})$.
An $\Fp G$-module $M$ is projective if and only if
$\Res^G_PM$ is projective for a Sylow $p$-subgroup $P$ of $G$.
In particular, if $G$ is a transitive subgroup of $S_p$, then for the $G$-set $X_p\cong G/G\cap S_{p-1}$, we have that $\Fp[X_p]$ is projective as an $\Fp G$-module.
The natural surjection
$\pi:\Ind^G_P\Res^G_PM\to M$
has a section
$$s:M\to \Ind^G_P\Res^G_PM, m\to \frac{1}{[G:P]}\sum_{gP\in G/P}g\otimes g^{-1}m$$
since $[G:P]$ is invertible in $\Fp$.
Then $M$ is an $\Fp G$ direct summand of $\Ind^G_P\Res^G_PM$.
If $\Res^G_PM$ is projective, so is $\Ind^G_P\Res^G_PM$ and hence so
is $M$ by the previous remark.
The converse is clear.
Since $G$ is a transitive subgroup of $S_p$, it has a cyclic $p$-Sylow subgroup
$C_p$. Then $\Fp[X_p]$ restricted to the cyclic $p$-Sylow subgroup $C_p$ is isomorphic to the
free $\Fp[C_p]$-module $\Fp[C_p]$.
If $0\to M\to P\to L\to 0$ and $0\to M'\to Q\to L\to 0$
are 2 short exact sequences of $G$ modules with $P,Q$ $G$-permutation,
then $M\sim M'$ where $\sim$ denotes flasque equivalence.
The pullback diagram gives two exact
sequences $0\to M\to E\to Q\to 0$ and $0\to M'\to E\to P\to 0$.
Note that the pullback module $E$ is a $G$-lattice (i.e. is $\Z$-torsion
free). So $M\sim M'$ as required.
Suppose $G$ is a transitive subgroup of $S_p$
for which all $G$-projective lattices are $G$-stably permutation.
Let $N=N_G(C_p)$ be the normaliser subgroup of a (cyclic) Sylow $p$ subgroup $C_p$.
Let $M$ be a $G$-lattice such that
there exists a short exact sequence of $G$ modules
$$0\to M\to P\to X\to 0$$
* $P$ is $G$-permutation.
* $X$ $p$-torsion
* $X\otimes \Fp I[G/N]$ is cohomologically trivial
$$M\sim \Ind^G_N\Res^G_N(M)$$
where $\sim$ denotes flasque equivalence.
Note that the hypothesis of transitivity implies that a Sylow $p$ subgroup
$C_p$ of $G$ is cyclic of order $p$.
So $N=N_G(C_p)\le N_{S_p}(C_p)=C_p\rtimes C_{p-1}$
and so $N=N_G(C_p)=C_p\rtimes (C_{p-1}\cap G)$.
Applying $\Ind^G_N\Res^G_N$,
we get
$$0\to \Ind^G_N\Res^G_NM\to \Ind^G_N\Res^G_NP\to \Ind^G_N\Res^G_NX\to 0$$
Since $X$ is $p$-torsion,
$\Ind^G_N\Res^G_NX= \Fp[G/N]\otimes_{\Fp}X$.
We have already noted that $\Fp[G/N]=\Fp\oplus \Fp I_{G/N}$.
\begin{equation}
\Ind^G_N\Res^G_NX=X\oplus (\Fp I_{G/N}\otimes X)\label{eq:indresx}
\end{equation}
By hypothesis, $\Fp I_{G/N}\otimes X$ is cohomologically trivial.
$$0\to K\to F\to \Fp I_{G/N}\otimes X\to 0$$
be an exact sequence of $G$-modules with $F$ a free $G$-module.
Then $K$ is also cohomologically trivial and $\Z$-free.
By <cit.>, this implies that $K$ is $G$-projective and hence $G$-stably permutation
by hypothesis.
Adding this sequence to the original and recalling (<ref>), we obtain
$$0\to M\oplus K\to P\oplus F\to \Ind^G_N\Res^G_NX\to 0$$
But then by Lemma <ref>, we see that $M\oplus K\sim \Ind^G_N\Res^G_NM$.
Since $K$ is $G$-stably permutation, we see that $M\sim M\oplus K$
and so $M\sim \Ind^G_N\Res^G_NM$.
For an odd prime $p$, the transitive $S_p$-set $X_p$,
and $N=N_{S_p}(C_p)$, we have that
$$J_{X_p}\sim \Ind^{S_p}_{N}J_{N/C_{p-1}}$$
as $S_p$-lattices.
Also, for the transitive $A_5$-set $X_5$,
we have that
$$J_{X_5}\sim \Ind^{A_5}_{D_{10}}J_{D_{10}/C_2}$$
as $A_5$-lattices. It follows that $J_{X_5}$ is $A_5$-quasi-permutation.
We will apply the previous proposition. Note that $G=S_p$, for an odd prime $p$ and $G=A_5$
satisfy the hypothesis that all projective $G$-lattices are $G$-stably permutation.
Note also that $A_5$ is a transitive subgroup of $S_5$ such that the normaliser of a cyclic
subgroup of order 5 is $D_{10}$, the dihedral group of order 10.
We need to construct an appropriate $G$-exact sequence to apply the proposition.
For any $G$ set $Y$ of size $n$,
there is an inclusion of $G$-lattices
$\alpha: I_{Y}\oplus \Z\to \Z[Y]$
where $\alpha\vert_{I_{Y}}$ is the inclusion and for $n=|Y|$ and $\{\e_i:i=1,\dots,n\}$ a $\Z$-basis of $\Z[Y]$,
$\alpha:\Z\to \Z[Y], 1\to \sum_{i=1}^n\e_i$.
Since $\{\e_1-\e_2,\dots,\e_{n-1}-\e_n,\e_n\}$
is a basis for $\Z[Y]$
is a basis for $I_Y\oplus \Z$,
it is easily checked that
$$0\to I_Y\oplus \Z\to \Z[Y]\to \Z/n\Z\to 0$$
is a short exact sequence of $G$-lattices
with $\Z/n\Z$ having trivial action.
Letting $Y=X_p$, we set $B_p=J_{X_p}\otimes I_{X_p}=(I_{X_p})^*\otimes I_{X_p}$
where $X_p$ is a transitive $G$-set of size $p$. Then tensoring by $J_{X_p}$,
we obtain
$$0\to B_p\oplus J_{X_p}\to \Z[X_p]\otimes J_{X_p}\to \Fp J_{X_p}\to 0$$
We need to show that $\Fp J_{X_p}\otimes \Fp I_{G/N}$ is $G$-cohomologically trivial.
Tensoring the following $G$-exact sequence by $\Fp I_{G/N}$:
$$0\to \Fp\to \Fp[X_p]\to \Fp J_{X_p}\to 0$$
we obtain
$$0\to \F_pI_{G/N}\to \Fp[X_p]\otimes I_{G/N}\to \Fp J_{X_p}\otimes I_{G/N}\to 0$$
Now $\Fp[X_p]$ is $\Fp[G]$-projective by Lemma <ref>. So $\Fp[Y]\otimes I_{G/N}$ is also
$\Fp G$-projective and so cohomologically trivial as an $\Fp G$ module. We have
already seen that $\F_pI_{G/N}$ is cohomologically trivial as an
$\Fp G$-module. This shows that $(\Fp J_{X_p})\otimes (\Fp I_{G/N})$
is cohomologically trivial as an $\Fp G$-module and hence also as a $G$-module as $\Fp G$ is cohomologically trivial as a $G$-module.
Note that we have constructed an exact sequence of the required
form for $B_p\oplus J_{X_p}$. But since $B_p$ is $G$-stably
permutation, the fact that $B_p\oplus J_{X_p}\sim \Ind^G_N\Res^G_N(B_Y\oplus J_y)
=\Ind^G_N\Res^G_N(B_p)\oplus \Ind^G_N\Res^G_N(J_{X_p})$ shows that $J_{X_p}\sim
\Ind^G_N\Res^G_N(J_{X_p})$ since $\Ind^G_N\Res^G_N$ preserve stably permutation
For $G=S_p$, $p$ an odd prime, and $N=N_{S_p}(C_p)=C_p\rtimes C_{p-1}$, we see that
$\Res^G_N(J_{X_p})=J_{N/C_{p-1}}$ and
so $J_{X_p}\sim \Ind^{S_p}_{N}J_{N/C_{p-1}}$ as $S_p$-lattices.
For $G=A_5, G\cap S_4=A_4$, $N=N_G(C_5)=D_{10}$ and $N\cap S_4=C_2$,
$$J_{X_5}=J_{A_5/A_4}\sim \Ind^{A_5}_{D_{10}}(J_{D_{10}/C_2})$$
as $A_5$-lattices as required.
For an odd prime $p$ and the transitive $S_p$ set $X_p$ of size $p$,
a flasque resolution of $J_{X_p}$ is given
$$0\to J_{X_p}\to \Z[S_p/S_{p-2}]\to J_{X_p}^{\otimes 2}\to 0$$
In fact, the $S_p$-lattice $J_{X_p}^{\otimes 2}$ is invertible.
This is well-known and was proven in <cit.> using somewhat different
As we require this result, we give a quick self-contained proof.
Tensoring the $S_p$-exact sequence
$$0\to \Z\to \Z[X_p]\to J_{X_p}\to 0$$
by $J_{X_p}$ and noting that $J_{X_p}\otimes \Z[X_p]\cong (I_{X_p}\otimes \Z[X_p])^*\cong \Z[S_p/S_{p-2}]$ by *,
we obtain the $S_p$-exact sequence of the statement.
It suffices to show that $J_{X_p}^{\otimes 2}$ is invertible when restricted to Sylow $q$-subgroups of $S_p$. Note that any Sylow $q$ subgroup $Q$ of $S_p$
for $q\ne p$, must fix $\e_i$ for some $i=1,\dots,p$, where
$\Z[X_p]$ has $\Z$-basis $\e_i,i=1,\dots,p$. Then $\Res^{S_p}_Q(I_{X_p})$
permutation $Q$-lattice with $\Z$-basis
$\e_i-\e_j, j\ne i$. So its dual $J_{X_p}$ and $J_{X_p}^{\otimes 2}$
must also be $Q$-permutation lattices.
As for the Sylow $p$-subgroup $C_p$, a result of Endo-Miyata shows that
for cyclic $p$-groups, every flasque lattice is invertible.
Since $(J_{X_p}^{\otimes 2})^*=I_{X_p}^{\otimes 2}$, we need only
check that $\Res^{S_p}_{C_p}I_{X_p}^{\otimes 2}=I_{C_p}^{\otimes 2}$ is coflasque.
Dualising the exact sequence of the statement and restricting to $C_p$, we obtain the $C_p$-exact sequence
$$0\to (I_{C_p})^{\otimes 2}\to \Z[C_p]^{p-1}\to I_{C_p}\to 0$$
But then since $(I_{C_p})^{C_p}=0$, and $H^1(C_p,\Z[C_p]^{p-1})=0$,
we see that $H^1(C_p,(I_{C_p})^{\otimes 2})=0$ as required.
Recall the following useful lemma from Florence <cit.>. The original was
stated for lattices for a profinite group. The proof for $G$-lattices
follows immediately.
Let $A_i,B_i,C_i, i=1,2$ be $G$-lattices fitting into
two exact sequences
$$0\to A_i\stackrel{j_i}{\to}B_i\stackrel{\pi_i}{\to}C_i\to 0$$
Assume we are given ax $G$ module map $s_i:C_i\to B_i$, and $d_1,d_2$ two coprime integers, such that $\pi\circ s_i=d_i\id$, $i=1,2$. Let $A_3=A_1\otimes A_2$,
$$B_3=(B_1\otimes B_2)\oplus (C_1\otimes C_2),\qquad
C_3=(C_1\otimes B_2) \oplus (B_1\otimes C_2).$$
Then there is an exact sequence
$$0\to A_3\stackrel{j_3}{\to} B_3\stackrel{\pi_3}{\to} C_3\to 0$$
together with a $G$ module map $s_3:C_3\to B_3$ such that
$\pi_3\circ s_3=d_1d_2\id$.
Observe that this lemma is very handy for showing that
in certain circumstances, the tensor product of two quasi-permutation lattices
is again quasi-permutation. Indeed, with the hypotheses of the Lemma
and the additional assumption that $B_i,C_i,i=1,2$ are all permutation lattices,
then $B_3,C_3$ are also permutation, since the tensor product of permutation
lattices is permutation and the direct sum of permutation lattices is
permutation. Indeed, in Theorem 2.2 of the same paper, he
shows that the tensor product of augmentation ideals of $G$ sets of
pairwise relatively prime order is quasipermutation as a consequence of this
Lemma. He then goes on to give a simple proof of Klyachko's result that
a $k$-torus with character lattice isomorphic to the tensor product
of two augmentation ideals for $G$ sets of relatively
prime order is rational.
We will apply this Lemma to prove that the $A_5\times C_2$ lattice
$J_{X_5}\otimes \Z^-_{A_5}$ is quasi-permutation.
$$0\to A\to B\stackrel{\pi}{\to} C\to 0$$
is an exact sequence of $G$-lattices and $s:C\to B$ is a $G$-equivariant map
such that $\pi\circ s=n\id$.
Let $D$ be another $G$-lattice.
Then for the $G$-exact sequence
$$0\to (D\otimes A)\to (D\otimes B)\stackrel{\hat{\pi}}{\to} (D\otimes C)\to 0$$
where $\hat{\pi}=(\id_D\otimes \pi)$,
there exists a $G$-equivariant map
$$\hat{s}=(\id_D\otimes s): (D\otimes C)\to
(D\otimes B)$$
such that $\hat{\pi}\circ \hat{s}=n\id$.
In particular, this remark applies to the natural exact sequence for the $G$-lattice $J_{X_n}$ where $X_n$ is a $G$-set of size $n$.
Then for the $G$-exact sequence
$$0\to \Z\to \Z[X_n]\stackrel{\pi}{\to} J_{X_n}\to 0$$
there is a natural $G$-equivariant map $s:J_{X_n}\to \Z[X_n]$
given by $s(\pi(x))=nx-\sum_{y\in X}y$ which satisfies $\pi\circ s=n\id$.
Tensoring this sequence with $J_{X_n}$,
we obtain a $G$-exact sequence
$$0\to J_{X_n}\to J_{X_n}\otimes \Z[X_n]\stackrel{\hat{\pi}}\to J_{X_n}^{\otimes 2}\to 0$$
For this sequence, there exists a $G$-equivariant map $\hat{s}:J_{X_n}^{\otimes 2}\to J_{X_n}\otimes \Z[X_n]$ such that $\hat{\pi}\circ \hat{s}=n\id$.
Recall that:
<cit.> Let $K/k$ be a separable extension of prime degree $p$, $p\ge 5$ with Galois closure $L/k$ having Galois group $\Gal(L/k)=C_p\rtimes C_{p-1}$
and $H=\Gal(L/K)=C_{p-1}$. Then $R^{(1)}_{K/k}(\Gm)$ is not a stably
rational variety.
Note that this is equivalent to $J_{X_p}$ is not stably permutation as an $C_p\rtimes C_{p-1}$ lattice if $p\ge 5$ is prime. This applies in particular to the $F_{20}=C_5\rtimes C_4$-lattice $J_{X_5}$.
For a prime $p\ge 5$, the $S_p\times C_2$-lattice $J_{X_p}\otimes \Z^-_{S_p}$ is $S_p\times C_2$-quasi-invertible but not $S_p\times C_2$-quasipermutation.
The $A_5\times C_2$-lattice $ J_{X_5}\otimes \Z^-_{A_5}$ is
$A_5\times C_2$-quasipermutation.
Let $p\ge 5$ be a prime.
A flasque resolution for $J_{X_p}$ for the transitive $S_p$-set $X_p$ can be given by
$$0\to J_{X_p}\to J_{X_p}\otimes \Z[X_p]\stackrel{\pi_1}{\to} J_{X_p}^{\otimes 2}\to 0$$
As we have seen, $J_{X_p}\otimes \Z[X_p]$ is permutation as the dual of
$I_{X_p}\otimes \Z[X_p]$. We have shown that $J_{X_p}^{\otimes 2}$ is $S_p$-quinvertible.
We see from the remark that there exists an $S_p$-equivariant map
$$s_1:J_{X_p}^{\otimes 2}\oplus P\to \Z[X_p]\otimes J_{X_p}\oplus P$$
such that $\pi_1\circ s_1=p\id$.
Inflating this sequence from $S_p$ to $S_p\times C_2$ gives us the
same statements for the $S_p\times C_2$-set $X_p$.
That is, the above sequence can be considered also a flasque resolution for
the $S_p\times C_2$-lattice $J_{X_p}$ with all maps considered above
$S_p\times C_2$-equivariant.
We remark that the sign lattice $\Z^{-}_{S_p}$ for $S_p\times C_2$ is in
fact the augmentation ideal $I_{Y_2}$ for $Y_2=(S_p\times C_2)/S_p$
and so its augmentation sequence
$$0\to I_{Y_2}\to \Z[Y_2]\stackrel{\pi_2}{\to}\Z\to 0$$
admits an $S_p\times C_2$-equivariant map $s_2:\Z\to \Z[Y_2]$ such that
$\pi_2\circ s_2=2\id$.
We may apply Florence's Lemma to the 2 exact sequences above to obtain an
$S_p\times C_2$-exact sequence:
\begin{equation} \label{eq:signedweightseq}
0\to J_{X_p}\otimes I_{Y_2}\to M\to N\to 0
\end{equation}
$$M=J_{X_p}\otimes \Z[X_p]\otimes \Z[Y_2]\oplus J_{X_p}^{\otimes 2}\otimes \Z$$
$$N=J_{X_p}\otimes \Z[X_p]\otimes \Z\oplus J_{X_p}^{\otimes 2}\otimes \Z[Y_2]$$
Since $J_{X_p}\otimes \Z[X_p]$ is permutation and $J_{X_p}^{\otimes 2}$ is
invertible as $S_p\otimes C_2$-lattices, we see that the same holds for $M$ and $N$.
We may then find an appropriate $S_p\times C_2$-lattice $L$ such that $M\oplus L\cong P$ is permutation. Note that $L$ is also invertible.
Then the $S_p\times C_2$-exact sequence
$$0\to J_{X_p}\otimes I_{Y_2}\to P\to N\oplus L\to 0$$
shows that the $S_p\times C_2$-lattice $J_{X_p}\otimes \Z^-_{S_p}=J_{X_p}\otimes I_{Y_2}$ is quasi-invertible as required.
Note though, that the $S_p\times C_2$-lattice $J_{X_p}\otimes \Z^-_{S_p}$ is not quasi-permutation as its restriction to $C_p\rtimes C_{p-1}$ is the lattice
$J_{X_p}$ which is not quasi-permutation.
Now setting $p=5$, we restrict the above sequence (<ref>) to $A_5\times C_2$.
We have an $A_5\times C_2$-exact sequence
$$0\to J_{X_5}\otimes I_{Y_2}\to M\to N\to 0$$
$$M=J_{X_5}\otimes \Z[X_5]\otimes \Z[Y_2]\oplus J_{X_5}^{\otimes 2}\otimes \Z$$
$$N=J_{X_5}\otimes \Z[X_5]\otimes \Z\oplus J_{X_5}^{\otimes 2}\otimes \Z[Y_2]$$
Since the $A_5$-lattice $J_{X_5}$ is quasi-permutation, its inflation, the $A_5\times C_2$-lattice $J_{X_5}$ is also quasi-permutation.
But then the its flasque lattice, the $A_5\times C_2$-lattice $J_{X_5}^{\otimes 2}$, is $A_5\times C_2$-stably permutation. This shows that as $A_5\times C_2$-lattices, $M$
and $N$ are stably permutation.
We may then find an appropriate $A_5\times C_2$-permutation lattice $Q$ such that $M\oplus Q\cong P$ is permutation.
Then the $A_5\times C_2$-exact sequence
$$0\to J_{X_5}\otimes I_{Y_2}\to P\to N\oplus Q\to O$$
shows that the $A_5\times C_2$-lattice $J_{X_5}\otimes \Z^-_{S_5}=J_{X_5}\otimes I_{Y_2}$ is quasi-permutation as required.
$\Z[F_{20}/D_{10}]\otimes J_{F_{20}/C_4}^{\otimes 2}$ is
$F_{20}$-stably permutation.
$\Z[S_5/A_5]\otimes J_{S_5/S_4}^{\otimes 2}$ is $S_5$-stably permutation.
For any $G$-lattice $M$ and a subgroup $H$ of $G$,
$$\Z[G/H]\otimes M\cong \Ind^G_H\Res^G_H(M)$$
Since we have seen that
$$0\to J_{X_5}\to J_{X_5}\otimes \Z[X_5]\to J_{X_5}^{\otimes 2}\to 0$$
is a flasque resolution for the $S_5$-lattice $J_{X_5}$,
it is also a flasque resolution for its restrictions to any subgroup.
Note that $J_{X_5}\cong J_{S_5/S_4}$ as an $S_5$-lattice,
$J_{X_5}\cong J_{A_5/A_4}$ as an $A_5$-lattice,
$J_{X_5}\cong J_{F_{20}/C_4}$ as an $F_{20}$-lattice and
$J_{X_5}\cong J_{D_{10}/C_2}$ as an $D_{10}$-lattice.
Since the $D_{10}$ lattice $J_{X_5}\cong J_{D_{10}/C_2}$ is
quasi-permutation, $J_{X_5}^{\otimes 2}$ must be stably permutation as $D_{10}$-lattice.
$$\Z[F_{20}/D_{10}]\otimes J_{X_5}^{\otimes 2}\cong \Ind^{F_{20}}_{D_{10}}
\Res^{F_{20}}_{D_{10}}J_{X_5}$$
shows that $\Z[F_{20}/D_{10}]\otimes J_{X_5}^{\otimes 2}$
is $F_{20}$-stably permutation.
Similarly, since the
$A_{5}$-lattice $J_{X_5}\cong J_{A_{5}/A_4}$ is
quasi-permutation, $J_{X_5}^{\otimes 2}$ must be stably permutation as
an $A_{5}$-lattice.
$$\Z[S_5/A_5]\otimes J_{X_5}^{\otimes 2}\cong \Ind^{S_5}_{A_5}
\Res^{S_5}_{A_5}J_{X_5}$$
shows that $\Z[S_5/A_5]\otimes J_{X_5}^{\otimes 2}$
is $S_5$-stably permutation.
For the $S_5$-lattices $J_{X_5}$ and
$J_{X_5}\otimes \Z^-_{A_5}$, we have
$$\rho_{S_5}(J_{X_5}\otimes \Z^-_{A_5})=-\rho_{S_5}(J_{X_5})\ne 0$$
The GAP IDs of the corresponding groups are $[4,31,4,2]$ and $[4,31,5,2]$.
For the $F_{20}$-lattices $J_{X_5}$ and $J_{X_5}\otimes \Z^-_{D_{10}}$,
$$\rho_{F_{20}}(J_{X_5}\otimes \Z^-_{D_{10}})=-\rho_{F_{20}}(J_{X_5})\ne 0$$
The GAP IDs of the corresponding groups are $[4,31,1,3]$ and $[4,31,1,4]$.
From the remark, we have observed that the $S_5$-flasque resolution
for $J_{X_5}$ given by
$$0\to J_{X_5}\to J_{X_5}\otimes \Z[X_5]\stackrel{\pi_1}{\to} J_X^{\otimes 2}\to 0$$
admits an equivariant map $s_1:J_{X_5}^{\otimes 2}\to J_{X_5}\otimes \Z[X_5]$
such that $\pi_1\circ s_1=5\id$.
For the second sequence, we will take instead the $S_5$-set $S_5/A_5$,
and the quasi-permutation resolution for $I_{S_5/A_5}=\Z^-_{A_5}$
given by
$$0\to I_{S_5/A_5}\to \Z[S_5/A_5]\stackrel{\pi_2}{\to} \Z\to 0$$
which admits an equivariant map $s_2:\Z\to \Z[S_5/A_5]$ such that
$\pi_2\circ s_2=2\id$.
So the 2 sequences satisfy the hypotheses for Florence's Lemma
and we obtain an $S_5$-exact sequence
$$0\to J_{X_5}\otimes I_{S_5/A_5}\to M\to Q\to 0$$
$$M=J_{X_5}\otimes \Z[X_5]\otimes \Z[S_5/A_5]\oplus J_{X_5}^{\otimes 2}\otimes \Z$$
$$Q=J_{X_5}\otimes \Z[X_5]\otimes \Z\oplus J_{X_5}^{\otimes 2}\otimes \Z[S_5/A_5]$$
Since $J_{X_5}\otimes \Z[X_5]$ is $S_5$-permutation and
$J_{X_5}^{\otimes 2}\otimes \Z[S_5/A_5]$ is $S_5$-permutation by the Lemma,
we see that $Q$ is permutation. Since also $J_{X_5}^{\otimes 2}\otimes \Z\cong J_{X_5}^{\otimes 2}$ is $S_5$-invertible,
we see that $M$ is also $S_5$-invertible. In fact if $L$ is an $S_5$-lattice
such that $J_{X_5}^{\otimes 2}\oplus L$ is $S_5$-permutation, we may
adjust this $S_5$-exact sequence to
$$0\to J_{X_5}\otimes I_{S_5/A_5}\to M\oplus L\to Q\oplus L\to 0$$
Then $Q\oplus L$ is $S_5$-invertible, $M\oplus L$ is $S_5$-permutation,
and the $S_5$-flasque class of $J_{X_5}\otimes I_{S_5/A_5}$ is $[Q\oplus L]=[L]$
since $Q$ is permutation. But the $S_5$-flasque class of $J_{X_5}$
is $[J_{X_5}^{\otimes 2}]$ and since $J_{X_5}^{\otimes 2}\oplus L$ is permutation, we see that for the $S_5$-lattices $J_{X_5}\otimes \Z^-_{A_5}$
$$\rho_{S_5}(J_{X_5}\otimes \Z^-_{A_5})=-\rho_{S_5}(J_{X_5})$$
as required. We also know that $\rho_{S_5}(J_{X_5})\ne 0$ as the lattice $J_{X_5}$
restricted to $F_{20}$ is not quasi-permutation and so neither is $J_{X_5}$
as an $S_5$-lattice.
The above argument restricted to the subgroup $F_{20}$ gives the other statements, where we note that $S_5$-set $S_5/A_5$ restricted to $F_{20}$ is
The GAP ID identifications will be discussed below.
Note that the Corollary was obtained computationally in <cit.>.
By results of Hoshi and Yamasaki in <cit.>, there are 7 conjugacy classes of finite subgroups of $\GL_4(\Z)$ which correspond to
retract but not stably rational algebraic tori. It turns out that the DadeGroup(4,7) with GAP ID [4,31,7,2] and corresponding lattice
$(\Lambda(A_4),\Aut(A_4))$ is one such example. All but one of the other such
examples can be seen to be subgroups (not just conjugate subgroups) of this group. Their lattices correspond to restrictions of $\Lambda(A_4)$ to the appropriate subgroup. The justifications are
similar to identifying the lattice corresponding to DadeGroup(4,7) since they correspond to subgroups. For this reason, they are omitted.
The following subgroups of with GAP ID
and lattice
$$(\Lambda(A_4),\Aut(A_4))=(J_{X_5}\otimes \Z^-_{S_5},S_5\times C_2)$$
have GAP IDs and lattices given by:
* $(J_{X_5},F_{20})$.
* $(J_{X_5}\otimes \Z^-_{D_{10}},F_{20})$
* $(J_{X_5}\otimes \Z^-_{F_{20}},F_{20}\times C_2)$.
* $(J_{X_5},S_5)$.
* $(J_{X_5}\otimes \Z^-_{A_5},S_5)$.
All of these correspond to retract rational tori which are not stably rational.
Note that these observations for [4,31,1,3] and [4,31,4,2] were made in Hoshi
Yamasaki. It was shown above that
$(J_{X_p}\otimes \Z^-_{S_p},S_p\times C_2)$ is quasi-invertible for $p\ge 5$ prime (or equivalently the corresponding norm one torus is retract
rational). Restrictions of quasi-invertible
lattices are quasi-invertible. So all the lattices on the list above are
quasi-invertible with corresponding tori retract rational.
By the result above, we know that
$(J_{X_5},F_{20})$ is not quasi-permutation. So $\rho(J_{X_5},F_{20})\ne 0$. Since every lattice in the list above
contains either $(J_{X_5},F_{20})$ or $(J_{X_5}\otimes \Z^-_{D_{10}},F_{20})$,
we see that the result will follow since
$$\rho_{F_{20}}(J_{X_5}\otimes \Z^-_{D_{10}})=-\rho_{F_{20}}(J_{X_5})\ne 0$$
We will now address the remaining case of a retract but not stably rational
algebraic $k$-torus of dimension 4.
For any cyclic group $C_n$ and given a primitive
$n$th root of unity $\omega_n$, there is a natural
$C_n$ lattice given by $\Z[\omega_n]$, which is the ring of integers of the
field $\Q(\omega_n)$. The generator of $C_n$, $\sigma_n$ then acts
as multiplication by $\omega_n$ on $\Z[\omega_n]$.
As a ring, $\Z[\omega_n]\cong \Z[X]/(\Phi_n(X))$ where $\Phi_n(X)$ is the
$n$th cyclotomic polynomial of degree $\varphi(n)$, the Euler $\phi$ function of $n$.
With respect to the basis $1,\omega_n,\dots,\omega_n^{\varphi(n)-1}$,
the matrix of $\sigma_n$ acting on $\Z[\omega_n]$ is the
companion matrix of $\Phi_n(X)$.
The lattice corresponding to the group with GAP ID is quasi-invertible
but not quasi-permutation and hence the associated algebraic
torus is retract but not stably rational.
The generators of the group with GAP ID
[4,33,2,1] given by GAP
It is easy to check that $A$ has order 8 and $B$ has order 12.
One could then replace the generators by $A$ and $C=B^4$
and check that $ACA^{-1}=C^{-1}$. This implies that for the semi-direct product
$$G=C_3\rtimes C_8=\langle \sigma,\tau: \sigma^3=1,\tau^8=1,\tau\sigma\tau^{-1}=\sigma^{-1}\rangle$$
$\rho:G\to \GL(4,\Z)$ with $\rho(\sigma)=C$ and $\rho(\tau)=A$
gives a faithful representation of the group.
Note that $G$ has centre
$\Z(G)=\langle \tau^2\rangle\cong C_4$ and $H_3=\langle \sigma\rangle\cong C_3$
is normal.
The group $H_8=\langle \tau\rangle\cong C_8$ is a Sylow 2-subgroup with
3 distinct conjugates $H_8=\langle \tau\rangle, (H_8)^{\sigma}=\langle \tau\sigma\rangle,(H_8)^{\sigma^{2}}=\langle \tau\sigma^2\rangle$ which pairwise intersect
in $\langle \tau^2\rangle$.
There are then $3\varphi(8)=12$ elements of order 8.
Since $\tau^2$ is central, $\tau^2\sigma$ is an element of order 12.
As the group has order 24, the elements of
$H_{12}=\langle \tau^2\sigma\rangle\cong C_{12}$ account for the remaining 12 elements of the group.
This shows that the only proper subgroups of $G$ are cyclic.
There is exactly one subgroup $H_d$ of $G$ of order $d|12$ and each is a cyclic subgroup of $H_{12}\cong C_{12}$, and is normal in $G$.
The only non-normal subgroups are the 3 conjugates of $H_8$ which
are cyclic of order 8.
Let $L$ be the rank 4 lattice with the action of $G$ induced by $\rho$.
We will show that $L_{H_3}\cong (I_{C_3})^2$ and $L_{H_8}\cong \Z[\omega_8]$.
For the restriction to $H_3=\langle \sigma\rangle$,
since the minimal polynomial of $C=\rho(\sigma)$ is $x^2+x+1$,
and $C\in \GL(4,\Z)$, it must have invariant factors $x^2+x+1,x^2+x+1$.
This is sufficient to show that $\Q L\cong \Q[\omega_3]^2\cong \Q I_{C_3}^2$ as $\Q H_3$-modules. We need to check that $L\cong \Z[\omega_3]^2\cong I_{C_3}^2$ as $H_3\cong C_3$-lattices.
In fact, one can show that the matrix of $\rho(\sigma)=C$ with respect to
the $\Z$-basis
of $L$ is $\diag(C_{x^2+x+1},C_{x^2+x+1})$
is the companion matrix of $x^2+x+1$. We may then conclude that $L_{H_3}\cong (I_{C_3})^2$.
For the restriction to $H_8$, since $A=\rho(\tau)$ is a matrix of order 8 in $\GL_4(\Z)$, it must have minimal polynomial $x^4+1$.
Again this is sufficient to conclude that the $\Q H_8$-module $\Q L$ is congruent to $\Q[\omega_8]$.
In fact, one can show that the matrix of $A$ with respect to the
of $L$ is the companion matrix
$C_{x^4+1}$ of $x^4+1$ given by
So $L$ restricted to $H_8=\langle \tau\rangle\cong C_8$ must be isomorphic to $\Z[\omega_8]$. We see that this lattice is a sign-permutation lattice
isomorphic to $\Ind^{C_8}_{C_2}\Z_-$.
Note that $L$ restricted to $H_3$ is quasi-permutation as it is isomorphic to
$I_{C_3}^2$, the direct sum of 2 quasi-permutation lattices
and $L_{H_8}$ is quasi-permutation as it is sign-permutation.
This implies that $L$ is quasi-invertible, since it is quasi-permutation on restriction to its Sylow subgroups.
Now, we need to construct a flasque resolution of $L$.
Note that $L\cong L^*$ as a $G$-lattice. This means we can construct a coflasque resolution of $L$
and dualise.
It turns out that this is easy. Any non-trivial subgroup $H$ of $G$ contains
$H_2=\langle \tau^4\rangle \cong C_2$ or $H_3=\langle \sigma\rangle\cong C_3$.
Since $\rho(\tau^4)=-I_4$ shows that $L^{\langle \tau^4\rangle}=0$
and $L^{H_3}=(IC_3\oplus IC_3)^{C_3}=0$, we see that for any non-trivial subgroup
$H$ of $G$, we have $L^H=0$.
So to find a coflasque resolution of $L$, we need only find
a permutation lattice which surjects onto $L$.
Since $L$ is an irreducible $G$-lattice,
there is a surjection $\pi:\Z G\to L$. If $K=\ker(\pi)$,
$$0\to K\to \Z G\to L\to 0$$
is a coflasque resolution and its dual is
a flasque resolution of $L\cong L^*$.
Let $F=K^*$.
Then we have the flasque resolution
$$0\to L\to \Z G\to F\to 0$$
Since $\Z G$ is free as a $G$-lattice, we have $\hat{H}^0(H,F)\cong \hat{H}^1(H,L)$ for any subgroup $H$ of
Now, $\hat{H}^0(H_3,F)\cong\hat{H}^1(H_3,L)\cong \hat{H}^1(C_3,IC_3)^2\cong (\Z/3\Z)^2$.
Since $L_{H_8}\cong \Ind^{C_8}_{C_2}\Z_-$ is a sign-permutation lattice
we see that
\hat{H}^1(C_2,\Z_-)=\Z/2\Z.$$
Suppose $F\oplus P\cong Q$ for some permutation $G$-lattices $P$
and $Q$.
Setting $H_{24}=G$, representatives of the conjugacy classes of subgroups of $G$ can be given by $\{H_d, d|24\}$ as described earlier.
We may then write
$P=\oplus_{d|24}a_d\Z[G/H_d]$, $Q=\oplus_{d|24}b_d\Z[G/H_d]$ for some
$a_d,b_d\in \Z$. We will obtain a contradiction by applying $\hat{H}^0$
to the equation $F\oplus P\cong Q$ for the group $G$ and the subgroup $H_8$.
We observe that
$\hat{H}^0(G,F)\cong \hat{H}^1(G,L)=0$.
Indeed, since $L\cong L^*$ as $G$-lattices, $\hat{H}^{1}(H,L)\cong
\hat{H}^{-1}(H,L^*)=\ker_L(N_H)/I_H(L)$ for any subgroup $H$ of $G$.
Since $H_3\cong C_3$ is a normal subgroup and is generated by $\sigma$
with image having minimal polynomial $1+x+x^2$, it is clear that
$\ker_L(N_{H_3})=L$. As $H_3$ is normal in $G$, $N_G=\sum_{g\in G/{H_3}}gN_{H_3}$, and
so $\ker_L(N_G)=L$. Similarly, since $H_8\cong C_8$ is generated by $\tau$
with image having minimal polynomial $x^4+1$, we see that $N_{H_8}=\sum_{i=0}^7\tau^i
=\sum_{i=0}^3\tau^i(\tau^4+1)$ has $\ker_L(N_{H_8})=L$. But we have seen that $\hat{H}^{-1}(H_8,L)\cong \hat{H}^1(H_8,L)\cong \Z/2\Z$ and $\hat{H}^{-1}(H_3,L)\cong \hat{H}^1(H_3,L)\cong (\Z/3\Z)$. Since $\hat{H}^{-1}(H,L)=\ker_L(N_H)/I_H(L)$, we see that
$3L\subseteq I_{H_3}(L)\subseteq I_G(L)$ and $2L\subseteq I_{H_8}(L)\subseteq I_G(L)$. Then we see that $I_G(L)=L$ and so $\hat{H}^1(G,L)=0$.
Applying $\hat{H}^0(G,\cdot)$ to $F\oplus P\cong Q$, and observing that $\hat{H}^0(G,\Z[G/H_i])\cong \Z/|H_i|\Z$,
we see that we have
\begin{equation}\label{eq:contradiction}
\sum_{d|24}x_d\Z/d\Z=0
\end{equation}
where $x_d=a_d-b_d\in \Z$.
Tensoring (<ref>) by $\Z/8\Z$ and equating coefficients of $\Z/2^i\Z$ for $i=1,2,3$, we obtain
$$x_2+x_6=0, x_4+x_{12}=0,x_8+x_{24}=0$$
Now restrict the isomorphism $F\oplus P\cong Q$
to $H_8$.
Note that $\hat{H}^0(H_8,F)=\Z/2\Z$.
By Mackey's Theorem,
$$\Res^G_{H_8}\Ind^G_{H_d}\Z=\oplus_{H_8xH_d\in H_8\setminus G/H_d}\Z[H_8/H_8\cap H_d^x]$$
For all $d\ne 8$, $H_d$ is a normal subgroup,
and so $H_8xH_d=H_8H_dx$ is a coset of $H_8H_d$ in $G$. Then
$$\Res^G_{H_8}\Z[G/H_d]\cong \Z[H_8/H_8\cap H_d]^{[G:H_8H_d]}, d\ne 8$$
In particular, if $d=1,2,4$, then $H_d\le H_8$, $H_8\cap H_d=H_d$ and $H_8H_d=H_8$ and
$$\Res^G_{H_8}\Z[G/H_d]\cong \Z[H_8/H_d]^3, d=1,2,4$$
If $d=3(2^k), k=0,1,2,3$, then $H_d\cap H_8=H_{2^k}$, $H_8H_d=G$ and so
$$\Res^G_{H_8}\Z[G/H_d]\cong \Z[H_8/H_{2^k}], d=2^k3, k=0,1,2,3$$
For $H_8$, we see that $G=H_81H_8\cup H_8\sigma H_8=H_8\cup H_8\sigma H_8$
and $H_8\cap H_8^{\sigma}=H_4$.
$$\Res^G_{H_8}\Z[G/H_8]\cong\Z\oplus \Z[H_8/H_4]$$
Taking $\hat{H}^0(H_8,\cdot)$ of $F\oplus P\cong Q$ and comparing coefficients
of $\Z/2^i\Z$, $i=1,2,3$, we obtain
$$3x_2+x_6=1, 3x_4+x_8+x_{12}=0, x_{8}+x_{24}=0$$
Together with our previous set of equations, we obtain a contradiction
since $x_2+x_6=0$ and $3x_2+x_6=1$
implies $2x_2=1$ but $x_2\in \Z$.
The contradiction implies that the flasque lattice $F$ cannot be $G$-stably
permutation and so our lattice $L$ cannot be $G$-quasi-permutation.
§ STABLE RATIONALITY OF THE TORUS CORRESPONDING TO [4,25,8,5]
In this section we present a non-computational proof that the torus
corresponding to [4,25,8,5] is stably rational. In an earlier version, we
erroneously concluded that we could prove that this algebraic torus was
rational. It is not clear to this author at this point whether this torus
is rational or not. However, this example may shed some further light on the question of whether or not stably rational tori are all rational. We provide a lot of information about this case for this reason.
Let $G$= MatGroupZClass(4,25,8,5).
This group is generated by the following elements
Let $M_4=\oplus_{i=1}^4\Z\e_i$. $G$ stabilises the sublattice $M_3=\oplus_{i=2}^4\Z\e_i$.
In fact, $M_3\cong \Z B_3$ as a faithful $G\cong W(B_3)$-lattice. Under the restriction to $M_3$, $g_1$ acts as $\tau_1\tau_2\tau_3(23)$,
$g_2$ acts as $\tau_1(123)$, $g_3$ acts as $\tau_1\tau_2$ and $g_4$ acts as $\tau_2\tau_3$.
Then $(g_2)^3$ acts as $\tau_1\tau_2\tau_3$, so that it may be seen that the restriction generates the same group as $\langle \tau_1,\tau_2,\tau_3,(23),(123)\rangle=W(B_3)\cong C_2\times S_4$.
It can then be checked that $G$ acts on $M_4/M_3\cong \Z_N^{-}$ where $N=\langle g_2,g_3,g_4\rangle=\langle g_2^3\rangle \times \langle g_3,g_4\rangle \rtimes \langle g_2^2\rangle\cong C_2\times A_4$.
Since $G$ acts faithfully on $M_3$, the restriction map
$\GL(M_4)\to \GL(M_3)$ restricts to an isomorphism on $G$.
Then as in its restriction to $M_3$, we see that $\langle g_2^3\rangle\cong C_2$ is the centre of $G$, $\langle g_2^2\rangle$ normalizes the subgroup
$\langle g_3,g_4\rangle \cong C_2\times C_2$ so that $\langle g_2^3,g_3,g_4\rangle\cong A_4$ and $\langle g_2,g_3,g_4\rangle =\langle g_2^3\rangle\times
\langle g_2^2,g_3,g_4\rangle\cong C_2\times A_4$.
Since $N\cdot \e_1$ contains
$$\langle g_2\rangle \cdot\e_1=\{\e_1,\e_1+\e_3,
\e_1+\e_3+\e_4,\e_1-\e_2+\e_3+\e_4,\e_1-\e_2+\e_4,\e_1-\e_2\},$$
we see that $|N\cdot \e_1|\ge 6$. Since $g_2^2(\e_1)=g_4(\e_1)$, and $g_4$ has order 2, then $C_3\cong \langle g_4g_2^2\rangle \le N_{\e_1}$. So $3| |
N_{\e_1}|$. But then $|N_{\e_1}|=|N|/|N\cdot \e_1|\le 4$ shows that
$N_{\e_1}=\langle g_4g_2^2\rangle \cong C_3$ and so $|N\cdot \e_1|=8$.
We need only 2 other orbit elements in addition to the above and they
can be seen to be $g_3\e_1=\e_1-\e_2+\e_3$ and $g_2g_3\e_1=\e_1+\e_4$.
The orbit $N\cdot \e_1$ determines the orbit $N\cdot t_1$.
Clearly $\frac{1}{N_{t_1}}\sum_{n\in N}nt_1\in K(M_4)^N$.
From the computation of the $N$ orbit of $\e_1$, we see that
$$z=\frac{1}{N_{t_1}}\sum_{n\in N}nt_1=
Using the generators of $N$, it is not difficult to double check that the
element $z=t_1(1+t_2^{-1})(1+t_3)(1+t_4)$ is indeed fixed by $N$.
We will write $\beta=(1+t_2^{-1})(1+t_3)(1+t_4)\in K(M_3)$ so that $z=t_1\beta$.
Then $K(M_4)=K(M_3)(t_1)=K(M_3)(z)$. Since $z$ is fixed by $N$, we have
where $\sigma=g_1N$ and $L=K(M_3)^N$
Now $\sigma(z)=z^{-1}\beta\sigma(\beta)$.
It is well known that $L(z)^\sigma$ is rational over $L^\sigma=K(M_3)^G$
if and only if $\beta\sigma(\beta)$ is a norm of the extension $L^N/L^G$
or equivalently if there exists $\alpha\in L^N$ such that $\beta\sigma(\beta)=
\alpha\sigma(\alpha)$ <cit.>. We will rephrase this condition for our particular situation.
Let $G$ act on a field $L$ and let $M$ be a $G$ lattice.
Then $G$ acts quasi-monomially on the group ring $L[M]$ and hence its quotient field $L(M)$
if its action extends the action on the units $L[M]^*=L^*\oplus M$
from its given action on $L^*$ and $L[M]^*/L^*\cong M$
so that
$$1\to L^*\to L[M]^*\to M\to 0$$
is a short exact sequence of $G$ modules.
Let $L$ be a field on which $G$ acts faithfully.
Let $N$ be a normal subgroup of $G$ of index 2.
Let $G$ act on $L(\Z_N^-)=L(t)$ by a quasi monomial action, extending its action on $L$. Then $L(\Z_N^-)^G$ is rational over $L^G$ if and only if
the action of $G$ on $L(\Z_N^-)$ can be extended to a quasi-monomial
action on $L(\Z[G/N])$ if and only if $b=gN(t)t\in N_{L^N/L^G}(L^N)^{\times})$.
Suppose the quasi-monomial action of $G$ on $L(\Z_N^-)$ can be extended to a
quasi-monomial action on $L(\Z[G/N])$. We write $L(\Z[G/N])=L(X,Y)$ where
$X,Y$ are the multiplicative generators of $\Z[G/N]$ in $L[\Z[G/N]]^{\times}$ corresponding to a permutation basis. Then if $\sigma=gN$ generates $G/N\cong C_2$, $\sigma(X)=\alpha Y$ for some $\alpha\in L^{\times}$.
Since $\Z[G/N]$ is fixed by $N$, we see that $\alpha\in (L^N)^{\times}$.
Then note that $\sigma^2=1$ implies that $\sigma(\alpha)\sigma(Y)=X$ and so
$\sigma(Y)=\frac{1}{\sigma(\alpha)}X$. We assumed that the quasi-monomial action of $G$ on $L(\Z_N^{-})=L(t)$ can be extended to that on $L(\Z[G/N])=L(X,Y)$.
Then $t$ would be mapped to $X/Y$ and so $\sigma(t)t=
\frac{\sigma(X)X}{\sigma(Y)Y}=\sigma(\alpha)\alpha$ as required.
Then $L(\Z_N^-)^G\subset L(\Z[G/N])^G$. The latter field is rational over $L^G$ and so $L(\Z_N^-)^G$ is unirational over $L^G$. Since $L(\Z_N^-)^G$ is of transcendence degree 1 over $L^G$, this means it is rational over $L^G$.
Conversely, if there exists $\alpha\in (L^N)^{\times}$ such that $\sigma(t)t=
\sigma(\alpha)\alpha$, we may extend the quasi-monomial action of $G$ on $L(\Z_N^-)=L(t)$ to one on $L(\Z[G/N])=L(X,Y)$ by the same rule.
That is, if $\sigma$ is a generator of $G/N$, then set $\sigma(X)=\alpha Y$.
This implies that $\sigma(Y)=\frac{1}{\sigma(\alpha)}X$ and the
inclusion mapping $t\to X/Y$ can be seen to be $G$ invariant.
Note that $N$ fixes $t,X,Y$ and so it is only necessary to determine the action of $G/N$.
We could apply the lemma to our situation. Here $L=K(M_3)$ and
$G$ acts quasi-monomially on $K(M_4)=K(M_3)(\Z_N^-)=K(M_3)(z)$.
Then $K(M_4)^G=(K(M_3)^N(z))^{G/N}$ would be rational over $K(M_3)^G$
if and only if $\sigma(z)z=\prod_{i=2}^4(1+t_i)(1+t_i^{-1})$ can be written as a norm from the quadratic extension $K(M_3)^N/K(M_3)^G$.
Unfortunately, as we do not have an explicit description of the field $K(M_3)^N$, it is not obvious to the author how to solve such a norm equation.
If this were the case, then $K(M_4)^N$ would be rational over $K(M_3)^G$ since
we already know that $K(M_3)^G$ is rational over $K^G$ as $M_3$ is a sign permutation lattice. However, if it is not the case, which may well be true,
this does not solve the rationality problem for $K(M_4)^G$.
Finding a subfield $E$ such that $K(M_4)^G/E$ is non-rational but $E/K^G$ is rational does not rule out the rationality of $K(M_4)^G/K^G$.
We can however give an explicit proof of the stably rationality of $K(M_4)^G$ over $K^G$.
We first construct a coflasque resolution of $M_4^*$.
Let $\e_i^*$,$i=1,\dots,4$ be the dual basis of the standard basis $\e_1,\dots \e_4$ and recall that the matrix of the
action of each group element on a dual lattice with respect to the dual basis is the inverse transpose of its matrix with respect to the original basis.
We need to determine a permutation $G$-lattice which surjects onto $M_4^*$.
Note that $G\cdot \e_1^*=\{\pm \e_1^*\}$. It is clear that $N=\langle g_2,g_3,g_4\rangle\subset G_{\e_1^*}$.
We have seen that $N\cong C_2\times A_4$, so $N$ is of index 2 in $G$ and so
Note also that $\langle g_2\rangle$ has orbit $\{\e_2^*,-\e_4^*,-\e_3^*,-\e_1^*-\e_2^*,-\e_1^*+\e_4^*,-\e_1^*-\e_3^*\}$.
Since $g_1\e_2^*=-\e_2^*$, it is clear that
$$\gamma=\{\pm \e_2^*,\pm \e_3^*,\pm \e_4^*, \pm (\e_1^*+\e_2^*), \pm (\e_1^*-\e_3^*), \pm (\e_1^*-\e_4^*)\}$$
is contained in $G\cdot \e_2^*$.
Since $g_4\e_2^*=\e_2^*$, $g_3(\e_2^*)=g_2^3(\e_3^*)=-\e_1^*-\e_2^*$, and
$g_3^2=g_4^2=1$, we see that $G_{\e_2^*}$ contains $\langle g_3g_2^3,g_4\rangle\cong C_2\times C_2$.
Then since $|G\cdot \e_2^*|\ge 12$ implies $|G_{\e_2^*}|\le 4$,
the orbit calculation shows that we must have $G_{\e_2^*}=\langle g_3g_2^3,g_4\rangle$ and $G\cdot \e_2^*=\gamma$.
Since $\Z G\cdot\e_1^*+\Z G\cdot\e_2^*$ contains $\{\e_1^*,\e_2^*,\e_3^*,\e_4^*\}$, we see that the map of $G$-lattices
$\pi:\Z[G/G_{\e_1^*}]\oplus \Z[G/G_{\e_2^*}]\to M_4^*$
sending $gG_{\e_1*}\to g\cdot \e_1^*$ and $gG_{\e_2^*}\to g\cdot \e_2^*$
is surjective.
Note for a finite group $G$ acting on a set $X$, a permutation $\Z$-basis of a transitive permutation $G$-lattice
$\Z[G/G_x]$ for some $x\in X$, is in bijection with the elements of the orbit $G\cdot x$.
We will write $p(g\cdot x)$ in place of $gG_x$.
Then this basis of $\Z[G/G_{\e_1^*}]\oplus \Z[G/G_{\e_2^*}]$ will be written as
$$\{p(x): x\in G\cdot \e_1^*\}\cup \{p(y): y\in G\cdot \e_2^*\}$$
The map $\pi$ then sends $p(x)\to x$ for all $x\in G\cdot \e_1^*\cup G\cdot \e_2^*$.
As $x\in G\cdot \e_i^*, i=1,2$ if and only if $-x\in G\cdot \e_i^*$,
it is then easy to see that $p(x)+p(-x)$ is in $\ker(\pi)$ for all
$x\in G\cdot \e_1^*\cup G\cdot \e_2^*$ and that the set
$$\{p(x)+p(-x): x\in G\cdot \e_1^*\cup G\cdot \e_2^*\}$$
is $G$-stable.
So we see that $\ker(\pi)$ has a permutation $G$-sublattice
$$Q=\Z[G/G_{p(\e_1^*)+p(-\e_1^*)}]\oplus \Z[G/G_{p(\e_2^*)+p(-\e_2^*)}]\cong \Z\oplus \Z[G/D_8]$$
where we note that $G/G_{p(\e_2^*)+p(-\e_2^*)}\cong \langle g_1,g_3g_2^3,g_4\rangle\cong D_8$ since $g_1(g_3g_2^3)g_1^{-1}=g_4$.
From the calculation of the $G$-orbits $G\cdot \e_1^*$ and $G\cdot \e_2^*$
above, we see that a permutation basis of $Q$ is
given by
$$\beta=\{p(x)+p(-x): x\in \{\e_i^*:i=1,\dots,4\}\cup \{\e_1^*+\e_2^*,\e_1^*-\e_3^*,\e_1^*-\e_4^*\}\}$$
A straightforward calculation
shows that
we may extend the permutation basis
for $Q$ to
a $\Z$-basis for $C=\ker(\pi)$ by adding the vectors
As we have seen, $G$ acts by permutations on the sublattice $Q\cong \Z\oplus \Z[G/D_8]$ of $C$. We will be interested in the action of $G$ on the remaining basis vectors and hence on $C/Q$.
As our calculations will involve the restrictions to the Sylow subgroups of $G$, we will replace the generators with
$\{g_1,g_2^3,g_2^3g_3,g_4,g_2^4\}$. We note that a Sylow 3-subgroup of $G$ is $\langle g_2^4\rangle\cong C_3$ and a Sylow 2-subgroup of $G$ is $\langle g_2^3,g_1,g_2^3g_3,g_4\rangle
\cong C_2\times D_8$.
Note that $\langle g_2^3\rangle\cong C_2$ is central and $g_1$ conjugates $g_2^3g_3$ to $g_4$.
We will write the basis vectors for $Q$ as $s_i=p(\e_i^*)+p(-\e_i^*)$, $i=1,\dots,4$,
We will compute the action of the generators of $G$ on $C$:
$g_1$ swaps $\e_1^*$ and $-\e_1^*$, $\e_2^*$ and $-\e_2^*$,
$\e_3^*$ and $-\e_4^*$,
This shows that
$g_1$ fixes $s_1,s_2,s_{12}$ and swaps $s_3$ and $s_4$, $s_{13}$ and $s_{14}$.
$g_2^3$ fixes $\e_1^*$,
and swaps $\e_2^*$ and $-(\e_1^*+\e_2^*)$,
$\e_3^*$ and $\e_1^*-\e_3^*$,
$\e_4^*$ and $\e_1^*-\e_4^*$.
This shows that
$g_2^3$ fixes $s_1$, and swaps $s_2$ and $s_{12}$; $s_3$ and $s_{13}$; $s_4$ and $s_{14}$.
and $g_2^3$ fixes $r_{13}$ and $r_{14}$.
$g_2^3g_3$ fixes $\e_1^*,\e_2^*,\e_3^*$, and swaps $\e_4^*$ with $\e_1^*-\e_4^*$.
$g_2^3g_3$ fixes $s_1,s_2,s_3,s_{12},s_{13}$ and swaps $s_4$ and $s_{14}$.
This shows also that $g_2^3g_3$ fixes $r_{12},r_{13},r_{14}$.
$g_4$ fixes $\e_1^*,\e_2^*,\e_4^*$, and swaps $\e_3^*$ with $\e_1^*-\e_3^*$.
$g_4$ fixes $s_1,s_2,s_4,s_{12},s_{14}$ and swaps $s_3$ and $s_{13}$.
This shows also that $g_4$ fixes $r_{12},r_{13},r_{14}$.
$g_2^4$ fixes $\e_1^*$ and sends $\e_2^*$ to $-\e_3^*$, $\e_3^*$ to $\e_1^*-\e_4^*$, $\e_4^*$ to $\e_1^*+\e_2^*$.
Then $g_2^4$ fixes $s_1$ and sends $s_2$ to $s_3$, $s_3$ to $s_{14}$, $s_4$ to $s_{12}$, $s_{12}$ to $s_{13}$, $s_{13}$ to $s_4$ and $s_{14}$ to $s_2$.
Then $g_2^4(r_{12})
=r_{13}-s_3$, $g_2^4(r_{13})
=r_{14}$, $g_2^4(r_{14})%=p(-\e_2^*)-p(\e_1^*)+p(\e_1^*+\e_2^*)
From this calculation, it is clear that $G$ acts on $C/Q$ as a sign permutation lattice. In fact, $\langle g_2^3,g_2^3g_3,g_4\rangle\cong C_2^3$ fix $C/Q$
pointwise, $g_1(r_{12}+Q)=-r_{12}+Q$, $g_1(r_{13}+Q)=-r_{14}+Q$ and
$g_1(r_{14}+Q)=-r_{13}+Q$, $g_2^4$ acts as $r_{12}+Q\to r_{13}+Q\to r_{14}+Q\to
We will show that $C$ is invertible which in turn implies that it is coflasque and flasque.
According to <cit.>, to check that $C$ is invertible it suffices to check that $\F_pC$ is $\F_p\Syl_p$ permutation for each
Sylow $p$-subgroup $\Syl_p$ of $G$ and for any Sylow $2$ subgroup $\Syl_2$, additionally
we must check that $\dim_{\Q}(\Q C)^{\Syl_2}=\dim_{\F_2}(\F_2 C)^{\Syl_2}$.
For $p=3$, we note that a Sylow $p$-subgroup $\langle g_2^4\rangle\cong C_3$
acts by permutations on $C/Q$ and so $0\to Q\to C\to C/Q\to 0$ splits
as a $C_3=\langle g_2^4\rangle$ lattice. Then $C$ restricted to $C_3$
is isomorphic to $Q\oplus C/Q$ restricted to $C_3$ and so is permutation.
In fact $C_{C_3}\cong \Z C_3^3\oplus \Z$. Then it is clear that $\F_3 C$
is also $\F_3 C_3$ permutation.
For $p=2$, from the earlier description of the action of $G$ on $C$, one can check that the following is a permutation basis for $\F_2C$ as an $\F_2\Syl_2$
module where $\Syl_2=\langle g_1,g_2^3,g_2^3g_3,g_4\rangle$:
A calculation also shows that the orbits of this permutation basis under $\Syl_2$
are $\{s_2,s_{12}\}$, $\{s_3,s_4,s_{13},s_{14}\}$, $\{r_{12}+s_1+s_2,r_{12}+s_{12}\}$ and $\{r_{14},r_{13}+s_{13}+s_1+s_3\}$
and so $\dim_{\F_2}(\F_2C)^{\Syl_2}=4$.
On the other hand, from the exact sequence $0\to Q\to C\to C/Q\to 0$,
we see that
$$0\to Q^{\Syl_2}\to C^{\Syl_2}\to (C/Q)^{\Syl_2}\to 0$$
since $Q$ is permutation and so is coflasque.
Since the permutation lattice $Q$ has orbits
$\{s_1\}\cup \{s_2,s_{12}\}\cup \{s_3,s_4,s_{13},s_{14}\}$ under $\Syl_2$, we see that
$\rk(Q^{\Syl_2})=3$. Since $C/Q$ is sign permutation and is fixed by $\langle g_2^3,g_2^3g_3,g_4\rangle$, then $(C/Q)^{\Syl_2}=(C/Q)^{\langle g_1\rangle}=\Z (r_{13}-r_{14})$.
$\dim_{\Q}(\Q C)^{\Syl_2}=\rk(C^{\Syl_2})=\rk(Q^{\Syl_2})+\rk((C/Q)^{\Syl_2})
=3+1=4$. Since this agrees with $\dim_{\F_2}(\F_2C)^{\Syl_2}$ and $\F_pC$
is $\F_p\Syl_p$-permutation for all Sylow $p$-subgroups $\Syl_p$ of $G$, then
$C$ is invertible as required.
[Note that although
$$0\to \F_2Q\to \F_2C\to \F_2(C/Q)\to 0$$
is an exact sequence of $\F_2\Syl_2$ modules with 2 end modules permutation,
this sequence does not split. This can already be seen when restricting the exact sequence above to the subgroup $\langle g_1\rangle$.
This means that one cannot use the same argument as in the $p=3$ where the
sequence splits integrally.]
So we now have a coflasque resolution for $M_4^*$ for which the coflasque lattice is invertible. This gives us a flasque resolution for $M_4$ for which the
flasque lattice $C^*$ is invertible. We need to show that $C$ is stably permutation, which would then imply the same for $C^*$.
Since $C/Q$ is sign-permutation, it is quasi-permutation and self-dual, so that
there must exist a sequence $0\to P_0\to P_1\to C/Q\to 0$ where $P_0,P_1$ are
permutation. Explicitly, $C/Q\cong \Ind_{\Syl_2}^G\Z_{C_2^3}^-$,
Indeed, for any transitive sign permutation lattice $S$ for a group $G$
with $v$ an element of the sign permutation basis for $S$, if $G_v$ is the stabilizer subgroup of $v$ and $G_v^{\pm}=\{g\in G:gv=\pm v\}$,
we may see that $S\cong \Ind_{G_v^{\pm}}^G\Z_{G_v}^-$, via the map
$gv\mapsto g\otimes z, g\in G$ where $z$ is a basis of the rank 1 sign lattice $\Z_{G_v}^-$ of
$G_v^{\pm}$ with kernel $G_v$.
In our case $C/Q$ is a transitive sign permutation lattice with sign permutation basis $\{\overline{r_{12}},\overline{r_{13}},\overline{r_{14}}\}$.
It is easy to see that $C_2^3$ fixes $\overline{r_{12}}$ pointwise and that the
orbit of $\overline{r_{12}}$ under $G$ is
So the stabilizer subgroup of $\overline{r_{12}}$ is $C_2^3$.
It is similarly easy to check that $G_{\overline{r_{12}}}^{\pm}$ is $\Syl_2$.
Then the exact sequence of $\Syl_2$ lattices
$$0\to Z\to \Z[\Syl_2/C_2^3]\to \Z_{C_2^3}^-\to 0$$
induces an exact sequence of $G$ lattices
$$0\to \Z[G/\Syl_2]\to \Z[G/C_2^3]\to \Ind_{\Syl_2}^G\Z_{C_2^3}^-\to 0$$
Since the last term of this exact sequence is isomorphic to $C/Q$,
we can produce a pullback diagram:
\xymatrix{
&\Z[G/\Syl_2] \ar[r]^{=}\ar@{>->}[d] &\Z[G/\Syl_2]\ar@{>->}[d] \\
Q \ar@{>->}[r]\ar[d]^{=} & \mbox{pull-back} \ar@{->>}[r]\ar@{->>}[d]
Q \ar@{>->}[r]& C \ar@{->>}[r] &\Ind_{\Syl_2}^G\Z_{C_2^3}^-
Since $Q$ and $\Z[G/C_2^3]$ are permutation, the middle row splits.
Also, since we have shown that $C$ is invertible and $\Z[G/\Syl_2]$ is permutation, we also have $\Ext^1_G(C,\Z[G/\Syl_2])=0$ since $C$ is the direct summand of a permutation lattice. Thus, also the middle column splits.
We then see that
$$C\oplus \Z[G/\Syl_2]\cong Q\oplus \Z[G/C_2^3]=\Z\oplus \Z[G/D_8]\oplus \Z[G/C_2^3]$$
is stably permutation as required.
This shows that $M_4$ is $G$ quasi-permutation as it then satisfies a $G$ exact sequence:
$$0\to M_4\to \Z[G/N]\oplus \Z[G/C_2^2]\oplus \Z[G/\Syl_2]\to
\Z\oplus \Z[G/D_8]\oplus \Z[G/C_2^3]\to 0$$
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The study of the birational properties of algebraic $k$ tori began in the
sixties and seventies with work of Voskresenkii, Endo, Miyata, Colliot-Thélène and Sansuc. There was particular interest in determining the rationality of a given algebraic $k$ tori. As rationality problems for algebraic varieties are in general difficult, it is natural to consider relaxed notions such as stable rationality, or even retract rationality. Work of the above authors
and later Saltman in the eighties determined
necessary and sufficient conditions to determine when an algebraic torus is
stably rational, respectively retract rational in terms of the integral
representations of its associated character lattice. An interesting question
is to ask whether a stably rational algebraic $k$ torus
is always rational. In the general case, there exist examples of non-rational stably rational $k$ varieties. Algebraic $k$ tori of dimension $r$ are classified up to isomorphism by conjugacy classes of finite subgroups of $\GL_r(\Z)$.
This makes it natural to examine the rationality problem for algebraic $k$
tori of small dimensions.
In 1967, Voskresenskii <cit.> proved that all algebraic tori of dimension 2 are rational. In 1990, Kunyavskii <cit.> determined which algebraic tori of dimension 3 were rational. In 2012, Hoshi and Yamasaki <cit.> determined which algebraic tori
of dimensions 4 and 5 were stably (respectively retract) rational with the aid of GAP. They did not address the rationality question in dimensions 4 and 5.
In this paper, we show that all stably rational algebraic $k$ tori of dimension 4 are rational, with the possible exception of 2 undetermined cases.
Hoshi and Yamasaki found 7 retract rational but not stably rational dimension 4 algebraic $k$ tori. We give a non-computational proof of these results.
§ INTRODUCTION
Rationality problems in algebraic geometry are central but notoriously difficult questions, making it natural to consider relaxed notions.
A $k$ variety is $k$-rational if it is birationally isomorphic to projective $n$ space, $\PP^n_k$;
or equivalently if its function field is a purely transcendental extension of $k$. A $k$ variety $X$ is $k$-stably rational if $X\times_k \A^r$ is rational for some $r\ge 0$. A $k$ variety is $k$-retract rational if there exist rational maps
$f:X\brokrarr \A^n$ and $g:\A^n\brokrarr X$ such that $g\circ f=\id_X$.
A $k$ variety $X$ is $k$-unirational if there exists a dominant rational map
$\PP^m_k\brokrarr X$, for some $m$.
Note that rationality implies stably rationality implies retract rationality
implies unirationality.
It is not too hard to find non-stably rational retract rational $k$ varieties
- we will discuss such examples later. It is much more difficult to find non-rational stably rational $k$ varieties - although examples were found by
Beauville, Colliot-Thélène, Sansuc and Swinnerton-Dyer <cit.>
in 1985.
In this paper, we examine the rationality problem for algebraic $k$ tori of dimension 4 where $k$ is a field. For algebraic $k$ tori, the question of whether
there exist non-rational stably rational algebraic $k$ tori remains open to the
best of the author's knowledge.
For any fixed quasi-projective $k$-variety $X$, a $k$ form of $X$
is another $k$ variety $Y$ which is isomorphic to $X$ after extending to the
separable closure $k_s$ of $k$.
Isomorphism classes of $k$ forms of $X$ are in bijection with elements
of non-abelian Galois cohomology set $H^1(\G_k,\Aut(X))$ where
$\G_k=\Gal(k_s/k)$ is the absolute Galois group.
An algebraic $k$ torus $T$ of dimension $r$ is a $k$ form of a split $k$ torus so that
$$T\times_k k_s\cong {\Gm}^r_{k_s}$$
Algebraic $k$ tori of dimension $r$ are determined up to isomorphism
by continuous representations of $\G_k$ into $\GL_r(\Z)$.
Any algebraic $k$ tori of dimension $r$ is split by a finite Galois extension $L$ over $k$.
That is, there exists a finite Galois extension $L/k$ such that
$$T\times_k L\cong {\Gm}^r_L$$
Algebraic $k$ tori of dimension $r$ which are split by a finite Galois extension $L/k$ are in bijection with conjugacy classes of finite subgroups of $\GL_r(\Z)$.
More precisely, a finite subgroup $G$ of $\GL_r(\Z)$ up to conjugacy determines a $\Z G$ lattice $M_G$ up to isomorphism.
Then $T_G=\Spec(L[M_G]^G)$ is an algebraic $k$ torus of dimension $r$ split by $L$.
Conversely, the character lattice of an algebraic $k$ torus of dimension $r$ split by $L$ is a $G$ lattice of rank $r$ and so determines a conjugacy class of finite subgroups of $\GL_r(\Z)$.
Given a finite subgroup $G$ of $\GL_r(\Z)$ and a Galois $G$ extension $L/k$,
one can phrase the rationality problems for the corresponding algebraic $k$-torus
$T_G=\Spec(L[M_G]^G)$ in terms of its function field $K=L(M_G)^G$.
$T_G$ is $k$-rational if and only if $K$ is a rational or purely transcendental extension of $k$, i.e. $K$ is isomorphic to $k(x_1,\dots,x_n)$, the quotient field of the polynomial ring in $n$ variables over $k$ where $n$ is the dimension of $T_G$. $T_G$ is $k$-stably rational if and only if there exist
algebraically independent variables $y_1,\dots,y_m$ such that $K(y_1,\dots,y_m)$ is $k$-rational.
For an infinite field $k$, $T_G$ is $k$-retract rational if and only if $K$ contains a $k$-algebra $R$ such that $K$ is the quotient field of $R$ and the
identity map $1_R$ factors through the localisation of a polynomial ring over $k$.
$T_G$ is unirational if and only if $k\subseteq K \subseteq k(x_1,\dots,x_m)$ for some $m$.
Voskresenskii, Endo, Miyata, Colliot-Thélène, Sansuc, Saltman studied the rationality problem for algebraic $k$ tori from the sixties to the
eighties <cit.>. They determined conditions under which a algebraic $k$ torus is stably
rational, and retract rational respectively. The conditions were phrased in terms of the character lattice of the algebraic $k$ torus as a lattice over its
splitting group.
For algebraic tori of small dimensions, the question of retract/stable rationality has been addressed. Since algebraic $k$ tori of dimension r split by a Galois extension $L/k$ are determined up to isomorphism by conjugacy classes of finite subgroups of $\GL_r(\Z)$, this is a finite set by a theorem of Jordan.
For small values of $r$, the classification of conjugacy classes of finite
subgroups is known but the number of such conjugacy classes grows rapidly
with $r$.
For $r=2$, there are 13 conjugacy classes of finite subgroups of $\GL_2(\Z)$ and hence 13 possible isomorphism classes of algebraic $k$ tori.
Voskresenskii <cit.> showed in 1967 that all 2-dimensional algebraic $k$ tori are
For $r=3$, there are 73 conjugacy classes of finite subgroups of $\GL_3(\Z)$,
a classification due to Tahara <cit.> in 1971.
In 1990, Kunyavskii <cit.> classified algebraic $k$-tori of dimension 3 up to birational equivalence. He found that all but 15 were rational. The remaining 15 were
not even retract rational.
There are 710 conjugacy classes of finite subgroups of $\GL_4(\Z)$. Dade <cit.>
found the maximal conjugacy classes of finite subgroups of $\GL_4(\Z)$ in 1965
without the use of a computer by analyzing the quadratic forms stabilized by the subgroups. In 1971, Bülow, Neubüser <cit.> determined all the conjugacy classes of finite subgroups of $\GL_4(\Z)$ using Dade's classification
and computer techniques. In 1978, together with H. Brown, H. Wondratschek, and H. Zassenhaus <cit.>, they wrote a book
on Crystallographic Groups of Four-Dimensional Space.
This classification determined the library of crystallographic groups of
dimensions 2,3,4 which is programmed into GAP <cit.>.
In 2012, Hoshi and Yamasaki <cit.> determined which algebraic tori of dimensions 4 and 5 are stably (respectively retract) rational using GAP.
For dimension 4, they found that 487 algebraic tori are stably rational, 7 are
retract but not stably rational and the remaining 216 are not retract rational.
They did not address the issue of rationality.
In this paper, I show that all but 2 of the stably rational algebraic tori
of dimension 4 are rational.
An important source of examples of algebraic tori are the norm one tori.
Given a separable field extension $K/k$ of degree $n$ and $L/k$ the Galois closure of $K/k$, the norm one torus $R^{(1)}_{K/k}(\Gm)$ is the kernel of the
norm map $R_{K/k}(\Gm)\to \Gm$ where $R_{K/k}$ is the Weil restriction <cit.>.
Note $R_{K/k}(\Gm)$ is split by $L/k$. Let $G=\Gal(L/k)$ and $H=\Gal(L/K)$.
The character lattice of $R^{(1)}_{K/k}(\Gm)$ is given by the $G$ lattice $J_{G/H}$, the dual of the kernel of the augmentation ideal of the permutation
lattice $\Z[G/H]$.
The rationality problem for norm one tori was studied by many authors
The stable and retract rationality of norm one tori corresponding to Galois
field extensions is well understood by work of Endo, Miyata, Saltman, Colliot-
Thélène, Sansuc.
Let $K/k$ be a finite Galois extension with Galois
group $G$. Let $R^{(1)}_{K/k}(\Gm)$ be the corresponding norm 1 torus.
* $R^{(1)}_{K/k}(\Gm)$ is retract rational
if and only if the Sylow subgroups of $G$ are cyclic. <cit.>
* $R^{(1)}_{K/k}(\Gm)$ is stably rational if and only if the
Sylow subgroups of $G$ are cyclic and $\hat{H}^0(G,\Z)=\hat{H}^4(G,\Z)$
where $\hat{H}$ indicates Tate cohomology. <cit.>
Note that the conditions above are equivalent to the following restrictions on the structure of the group $G$:
$$G=C_m\mbox{ or } G=C_n\times\langle \sigma,\tau: \sigma^k=\tau^{2^d}=1,
\tau\sigma\tau^{-1}=\sigma^{-1}\rangle$$
where $d\ge 1,k\ge 3, n,k$ odd and $\gcd(n,k)=1$.
These theorems were proven in the 1970s. Some cases of non-Galois separable
extensions were dealt with much more recently.
Let $K/k$ be a finite non-Galois separable field extension
and let $L/k$ be the Galois closure of $K/k$. Let $H\le G$ be the Galois groups
of $L/K$ and $L/k$ respectively.
* If $G$
is nilpotent, then the norm one torus $R^{(1)}_{K/k}(\Gm)$ is not retract rational.
* If all the Sylow subgroups of $G$ are cyclic,
then the norm one torus $R^{(1)}_{K/k}(\Gm)$ is retract $k$-rational and, we
have that
$R^{(1)}_{K/k}(\Gm)$ is stably $k$-rational if and only if
either $G=D_{2n}$ where $n$ is odd or $G=C_m\times D_{2n}$ where $n$ is odd and
$\gcd(m,n)=1$ and $H\le D_{2n}$ is a cyclic group of order 2.
Let $K/k$ be a finite non-Galois separable field extension and $L/k$ be the Galois closure of $K/k$ such that the Galois group of $L/k$ is $S_n$ and that of $L/K$ is $S_{n-1}$.
* $R^{(1)}_{K/k}(\Gm)$ is retract $k$-rational if and only if $n$ is prime.
* $R^{(1)}_{K/k}(\Gm)$ is (stably) $k$-rational if and only if $n=3$.
<cit.> Let $K/k$ be a non-Galois separable field extension of degree
$n$ and let $L/k$ be the Galois closure of $K/k$ such that the Galois
group of $L/k$ is $A_n, n\ge 4$ and that of $L/K$ is $A_{n-1}$.
* $R^{(1)}_{K/k}(\Gm)$ is retract $k$-rational if and only if $n$ is prime.
* For some positive integer $t$, $[R^{(1)}_{K/k}(\Gm)]^{(t)}$ is stably $k$-rational
if and only if $n=5$.
S. Endo asked in <cit.> about the stable rationality of $R^{(1)}_{K/k}(\Gm)$
in the $A_5$ case. Hoshi and Yamasaki <cit.> used GAP to show that in fact
$R^{(1)}_{K/k}(\Gm)$ is stably rational in this case.
In this paper, we present a non-computational proof of that fact.
The norm one torus $R^{(1)}_{K/k}(\Gm)$ corresponding to $A_5$ has
character lattice $J_{A_5/A_4}$.
It is one of the two algebraic $k$ tori of dimension 4 which I can prove to be stably
rational but cannot determine whether or not it is rational.
The other such algebraic $k$ torus of dimension 4 is intimately
related to this norm one torus. In fact its splitting group is $A_5\times C_2$
and its character lattice restricts to $J_{A_5/A_4}$ on $A_5$.
See Section 5 for more details.
This paper is organised as follows.
In Section 2, we will recall some facts about algebraic $k$ tori and
conditions for stable and retract rationality.
In Section 3, we will determine some families of hereditarily rational algebraic $k$ tori. We call an rational algebraic $k$ torus corresponding to a finite subgroup
$G$ of $\GL_r(\Z)$ hereditarily rational if an algebraic $k$ tori corresponding to any subgroup of $G$ is rational. We do not know whether all rational algebraic $k$ tori are hereditarily rational but we show this to be the case for
a number of natural families of algebraic tori.
In Section 4, we show that in dimension 4, we identify 11 algebraic $k$ tori
which are maximally hereditarily $k$-rational. We then use GAP to show that the
set of rational $k$ tori obtained in this way is of size 485
and matches the set of stably rational $k$ tori of dimension 4 obtained
by Hoshi and Yamasaki with 2 undetermined exceptions. Note that the use of GAP
is very minimal: it is limited to finding conjugacy classes of finite subgroups of $\GL_4(\Z)$ which are subgroups of the groups corresponding to the 11 algebraic $k$ tori mentioned above.
In Section 5, we give a non-computational proof that the remaining two
algebraic $k$ tori of dimension 4 are stably rational.
We also show that 7 algebraic $k$ tori of dimension 4 are retract but not stably rational in a non-computational way, recovering the results of Hoshi
and Yamasaki. In an upcoming paper, we will show that the remaining 216 algebraic $k$ tori of dimension 4 are not retract rational in a non-computational
way using only knowledge of the lattice of conjugacy classes of finite
subgroups of $\GL_4(\Z)$.
§ PRELIMINARIES
We begin with some remarks on notation used in this paper.
Throughout, $k$ will be a field of characteristic zero. We will see that this
hypothesis is necessary to apply the criteria for stable and retract rationality.
For finite groups, we will denote by $C_n$
the cyclic group of order $n$; by $D_{2n}$ the dihedral group of order $2n$;
by $S_n$ the symmetric group on $n$ letters and by $A_n$ the alternating group
on $n$ letters.
We will discuss below root system terminology, algebraic torus - lattice correspondence, lattice terminology and birational properties of algebraic tori.
§.§ Root Systems
For a root system $\Phi$, with basis $\Delta$,
we will denote by $W(\Phi)$, its Weyl group.
$$W(\Phi)=\langle s_{\alpha}: \alpha\in \Delta\rangle$$
where $s_{\alpha}$ is the reflection in $\alpha\in \Phi$.
$$s_{\alpha}(x)=x-\langle x,\alpha\rangle \alpha$$
and $\Aut(\Phi)$ the group of automorphisms of the root system. We will denote by $\Z\Phi$
its root lattice. $\Z \Phi$ is the $\Z$-span of $\Phi$
and has $\Z$-basis $\Delta$. Its weight lattice is
$$\Lambda(\Phi)=\{x\in \Q\Phi: \langle x,\alpha\rangle\in \Z\mbox{ for all }\alpha\in \Phi\}.$$
$\Z \Phi\subseteq \Lambda(\Phi)$ are lattices of the same rank
and both are stabilised by the finite subgroup $\Aut(\Phi)$ and so by its
subgroup $W(\Phi)$.
We will mainly be referring to the irreducible root systems of type
$A_n$, $B_n$, $F_4$ and $G_2$.
For more information on root systems, see, for example,
Humphreys <cit.>.
§.§ Lattice-Tori Correspondence
We will discuss in more detail the correspondence between isomorphism
classes of algebraic $k$ tori of dimension $r$ and conjugacy classes
of finite subgroups of $\GL_r(\Z)$ as well as criteria
for determining whether a given algebraic $k$ torus is retract or stably
rational. <cit.>
An algebraic $k$ torus of dimension $r$ is a $k$ form of a split $k$ torus
$\Gm^r$. As discussed in the introduction, this implies that
isomorphism classes of algebraic $k$ tori are in bijection
with elements of the non-abelian Galois cohomology
set $H^1(\cG_k,\Aut(\Gm^r))$ where $\cG_k=\Gal(k_s/k)$ is the absolute Galois
group of $k$. Since $\cG_k$ acts trivially on $\Aut(\Gm^r)=\GL_r(\Z)$,
elements of $H^1(\cG_k,\Aut(\Gm^r))$ are actually continuous homomorphisms
of the compact profinite group $\cG_k$ into the discrete group $\GL_r(\Z)$
and as such have finite image.
This shows that algebraic $k$ tori of dimension $r$ are determined up to isomorphism
by continuous representations of $\cG_k$ into $\GL_r(\Z)$
or equivalently by lattices of rank $r$ with a continuous action of $\cG_k$ up to isomorphism.
Given such a continuous representation $\rho:\cG_k\to \GL_r(\Z)$,
and the associated $\cG_k$ lattice $M$ of rank $r$,
$\Spec(k_s[M]^{\cG_k})$ is an algebraic $k$ torus.
Conversely, an algebraic $k$ torus $T$ determines its character
lattice $\hat{T}=\Hom(T,\Gm)$ which is a lattice equipped with a continuous action of $\cG_k$.
Every algebraic $k$ torus is split by a finite Galois extension.
In fact, if the algebraic $k$ torus is determined by a continuous representation
$\rho:\cG_k\to \GL_r(\Z)$ with associated $\cG_k$ lattice $M$ of rank $r$,
then for $N=\ker(\rho)$, $L=k_s^N$ is a finite Galois extension of $k=k_s^{\cG_k}$
with finite Galois group $G\cong \cG_k/N\cong \rho(\cG_k)$.
Note that $L[M]^G=(k_s[M]^N)^{\cG_k/N}=k_s[M]^{\cG_k}$.
In fact, algebraic tori of rank $r$ split by a finite Galois extension $L/k$
with Galois group $G$ up to isomorphism are in bijection with $G$ lattices of rank $r$ up to
isomorphism. Here the $G$ lattice $M$ determines the algebraic torus
split by $L/k$ as $\Spec(L[M]^G)$ and conversely the algebraic torus $T$ with
splitting group $G$ determines its character lattice which is a $G$ lattice.
Criteria for determining the stable (respectively retract)
rationality of an algebraic $k$ torus split by a finite Galois extension $L/k$
with Galois group $G$ are phrased in terms of the integral representation theory of the character lattice as a $G$ lattice.
To describe these criteria, we need some definitions about $G$ lattices.
We introduce some notation:
For a finite Galois extension $L/k$ with Galois group $G$, we will denote
by $C(L/k)$, the category of algebraic $k$ tori split by $L$
and by $C(G)$, the dual category of $G$ lattices (torsion-free $G$ modules)
of finite rank.
§.§ Lattice terminology
For more details, see for example, Lorenz <cit.>.
Let $G$ be a finite group. Note that for a $G$ lattice
$M$ and a subgroup $H$ of $G$, $\hat{H}^k(H,M)$ refers to the
$k$th Tate cohomology group of $M$ as an $H$ module, where $k\in \Z$.
* For a $G$ lattice the dual lattice $M^*=\Hom(M,\Z)$ is a $G$ lattice
with $(g\cdot f)(m)=g\cdot f(g^{-1}\cdot m)$ for $f\in M^*$, $g\in G$, $m\in M$.
* A (transitive) permutation $G$ lattice is a $G$ lattice with $\Z$
basis which is permuted by the action of $G$. All transitive
permutation lattices with stabilizer subgroup $H$ are isomorphic to
$\Z[G/H]$ where $G/H$ is the set of left cosets of $H$ in $G$.
Any permutation $G$ lattice is the direct sum of a finite number
of transitive $G$-permutation lattices.
* A sign permutation $G$ lattice is a $G$ lattice with
$\Z$-basis which is permuted by the action of $G$ up to sign.
* Given an $H$ lattice $N$ where $H\le G$, $\Ind^G_H(M)=\Z G\otimes_{\Z H}M$
is the $G$ lattice induced from the $H$ lattice $M$.
Note that $\Ind^G_H(\Z)=\Z[G/H]$.
* An invertible or permutation projective $G$ lattice $M$
is a $G$ lattice which is a direct summand of a $G$-permutation lattice.
That is, there exists a $G$ lattice $M'$ such that $M\oplus M'=P$
for some $G$-permutation lattice $P$.
* A $G$ lattice $M$ is flasque if $\hat{H}^{-1}(H,M)=0$ for all subgroups $H\le G$.
* A $G$ lattice $M$ is coflasque if $\hat{H}^1(H,M)=0$ for all subgroups
$H\le G$.
* A $G$ lattice $M$ is quasipermutation if there exists
a short exact sequence $0\to M\to P\to Q\to 0$ of $G$ lattices
with $P,Q$ $G$-permutation.
Facts and Theorems about Tate Cohomology:
Let $M$ be a $G$ lattice. Then
* Let $N_G:M\to M^G$ be the norm map $N_G(m)=\sum_{g\in G}g\cdot m$.
\begin{cases}\hat{H}^k(G,M)=H^k(G,M)&\mbox{ if }k\ge 1\\
\hat{H}^0(G,M)=\ker_M(N_G)/I_G(M)& \\
\hat{H}^{-1}(G,M)=M^G/\ker_M(N_G)&\\
\hat{H}^{-i-1}(G,M)=H_i(G,M)&\mbox{ if }i\ge 1
\end{cases}
* (Duality)
$\hat{H}^{k}(G,M)\cong \hat{H}^{-k}(G,M^*)$ where $M^*$ is the $\Z$ dual lattice.
* (Shapiro's Lemma): For an $H$ lattice $N$, where $H\le G$, we have
$$\hat{H}^k(G,\Ind^G_H(N))\cong \hat{H}^k(H,N)$$
where $\Ind^G_H(M)=\Z G\otimes_{\Z H}M$ is the $G$ lattice induced from the
$H$ lattice $M$.
Following Voskresenskii, we denote by $S(G)$ the class of all permutation $G$ lattices, $D(G)$ the
class of all invertible $G$ lattices, $\hat{H}^{-1}(G)$ the class of all
flasque $G$ lattices, $\hat{H}^1(G)$ the class of all coflasque $G$ lattices
and $\tilde{H}(G)=H^1(G)\cap H^{-1}(G)$.
$$S(G)\subset D(G)\subset \tilde{H}(G)\subset H^i(G)\subset C(G)$$
where each inclusion is proper.
Most inclusions are clear: $D(G)\subset \tilde{H}(G)$ follows from Shapiro's
Lemma and the duality statement above since permutation $G$ lattices
are self-dual.
Note also that the dual of a flasque $G$ lattice is coflasque.
For a $G$ set $X$, let the associated permutation $G$ lattice be denoted as $\Z[X]$. A natural $G$ sublattice is the augmentation ideal $I_X$
given by the kernel of the $G$-equivariant homomorphism
$\epsilon_X:\Z[X]\to \Z$, sending $x\to 1$ for all $x\in X$.
But then
$$0\to I_X\to \Z[X]\to \Z\to 0$$
is a short exact sequence of $G$ lattices.
Let $J_X=(I_X)^*$ be its $\Z$ dual.
Then $J_X$ satisfies the short exact sequence of $G$ lattices
$$0\to \Z\to \Z[X]\to J_X\to 0$$
The following definitions and results are due to Voskresenskii, Endo-Miyata, Colliot-Thélène and Sansuc
We say that two $G$ lattices $M,N$ are stably isomorphic and write
$[M]=[N]$ if and only if there exist permutation lattices $P,Q$ such that
$M\oplus P\cong N\oplus Q$. (In many places, such lattices are called
similar, but I find this terminology ambiguous).
(See <cit.>)
A $G$ lattice $M$ has a flasque resolution. That is,
there exists a short exact sequence
$$0\to M\to P\to F\to 0$$
such that $P$ is $G$-permutation and $F$ is $G$ flasque.
Given 2 flasque resolutions of a $G$ lattice $M$
$$0\to M\to P_i\to F_i\to 0, i=1,2$$
we have that $[F_1]=[F_2]$.
We may then define the flasque class of the $G$ lattice $M$
to be $\rho_G(M)=[F]$ where $0\to M\to P\to F\to 0$ is a flasque resolution
and $[F]$ is the stable isomorphism class of $F$.
The proof that every $G$ lattice has a flasque resolution which can be found in <cit.> for example is straightforward and constructive
but relies strongly on knowledge of the conjugacy classes of subgroups of $G$
and the restrictions of the $G$ lattice $M$ to these subgroups.
Hoshi and Yamasaki <cit.> gave algorithms to construct these flasque
resolutions for $G$ lattices of ranks up to 5 in which all of this information is known.
We say that $G$ lattices $M$ and $N$ are flasque equivalent if
there exist short exact sequences of $G$ lattices such that
$$0\to M\to E\to P\to 0, 0\to N\to E\to Q\to 0$$
Note that $M\sim 0$ if and only if $M$ is a quasipermutation $G$ lattice.
Note that $M\sim N$ if and only if $\rho(M)=\rho(N)$. (See <cit.>).
§.§ Birational properties of algebraic $k$ tori
We will give a brief summary of how the concepts in the
last section can be used to discuss birational properties of algebraic $k$ tori.
A more in depth discussion can be found in <cit.>.
Given an algebraic $k$ torus $T\in C(L/k)$ where $L/k$ is a finite Galois
extension with group $G$, we may find by resolution of singularities (due
to Hironaka in characteristic 0)
a smooth projective $k$ variety $X$ which contains $T$ as an open subset.
[Note, this is the reason for our characteristic zero assumption.]
$X$ is called a projective model of $T$. Then $X$ and $T$ are birationally
isomorphic. Then $X_L=L\otimes_k X$ is rational since $X_L$ and $T_L$ are
birationally equivalent. Then the Picard group of $X_L$, $\Pic(X_L)$, is a $G$ lattice. Given two such projective models $X,Y$ of $T$, $\Pic(X_L)$
is stably $G$ isomorphic to $\Pic(Y_L)$. So the class of $[\Pic(X_L)]$
is a birational invariant.
For any intermediate field extension $k\subset K\subset L$ with $H=\Gal(L/K)$,
$\hat{H}^{\pm 1}(H,\Pic(X_L))$ are also birational invariants of $T$
and $\hat{H}^{-1}(H,\Pic(X_L))=0$.
The inclusion $T_L$ into $X_L$ induces
a short exact sequence of $G$ lattices
$$0\to \hat{T}\to \hat{S}\to \Pic(X_L)\to 0$$
where $\hat{S}$ is generated by the components of the closed subset $X_L-T_L$.
$\hat{S}$ is a $G$-permutation lattice and as observed above, $\Pic(X_L)$
is a flasque $G$ lattice.
This gives a geometric construction of a flasque resolution of the
$G$ lattice $\hat{T}$.
If $T$ is $k$-rational, so is $X$ and then $[\Pic X_L]=0$.
In fact if $T$ is stably $k$-rational then so is $X$ and so
$X\times_k \Gm^r$ is rational for some $r\ge 0$ but $\Pic(X_L\times_k \Gm^r_L)=\Pic(X_L)$ and so we again have $[\Pic(X_L)]=0$.
This gives the basic setup behind the following results
connecting birational properties of algebraic $k$ tori to properties
about the integral representation of their character lattices over a splitting field.
(Endo-Miyata, Voskresenskii, Saltman) Let $L/k$ be a finite
Galois extension with Galois group $G$ and let $T,T'\in C(L/k)$
with character lattices $\hat{T}, \hat{T'}$ in $C(G)$.
* (<cit.>) $T$ is stably $k$-rational if and only if
$\hat{T}\sim 0$ (equivalently $\hat{T}$ is a quasipermutation $G$ lattice).
* (<cit.>) $T$ and $T'$ are stably birational $k$ equivalent
if and only if $\hat{T}\sim \hat{T'}$.
* (<cit.>) $T$ is retract $k$-rational if and only
if $\rho(\hat{T})$ is invertible.
A torus $T\in C(L/k)$ with permutation character lattice
$\oplus_{i=1}^k\Z[G/H_i]$ for subgroups $H_i, i=1,\dots,k$
corresponds to an algebraic $k$ torus
$T=\prod_{i=1}^kR_{K_i/k}(\Gm)$ where $K_i=L^{H_i}$ and
$R_{K/k}$ refers to Weil restriction of scalars.
Such tori are called quasisplit.
<cit.>. See also <cit.>.
Let $L/k$ be a Galois extension with Galois group $G$.
* A torus $T\in C(L/k)$ is rational if $\hat{T}\in C(G)$ is permutation.
* If there exists an exact sequence $0\to S\to T\to T'\to 0$ of tori in $C(L/k)$
where $S$ is quasisplit, then $T$ is birationally equivalent to $T'\times_k S$.
Note that the last theorem is equivalent to the following in lattice theoretic terms:
Let $L/k$ be a Galois extension of fields with $G=\Gal(L/k)$.
* $L(P)^G$ is rational over $L^G$ if $P$ is a permutation $G$ lattice.
* If $0\to M\to N\to P\to 0$ is an exact sequence of $G$ lattices with
$P$ $G$-permutation, then $L(N)\cong L(M)(P)$ as $G$ fields and so
$L(N)^G\cong L(M)(P)^G$ is rational over $L(M)^G$.
§ FAMILIES OF RATIONAL ALGEBRAIC $K$ TORI
Note that an algebraic $k$ torus $T$ is determined uniquely
by the $G$ lattice $\hat{T}$ and the Galois extension $L/k$ with Galois group
$G$. Every $G$ module $\hat{T}$ determines many algebraic $k$ tori
corresponding to different Galois extensions $L/k$ with Galois group $G$
and these tori may be non-isomorphic over $k$.
Stable rationality and retract rationality of an algebraic $k$ torus only
depend on the $G$ lattice $\hat{T}$. However it is not clear whether this
is true for rationality, although the author does not know of a counterexample.
A reason for caution is given by the example given in [p. 54]<cit.> of 2 norm one
tori $R^{(1)}_{L_i/\Q}(\Gm)$, $i=1,2$ corresponding to distinct biquadratic
extensions which are not even stably birationally equivalent over $\Q$.
Let $M$ be a $G$ lattice for a finite group $G$. If all algebraic tori with character lattice $\Res_H^G(M)$ and splitting group $H$ are rational for any subgroup $H$ of $G$, we call $M$ hereditarily rational. Note that a $G$ lattice $M$ is
hereditarily rational if and only if for for any subgroup $H$ of $G$
and for any Galois extension $K/k$ with Galois group $H$,
the function field $K(M)^H$ is rational over $K^H$.
Let $T$ be an algebraic $k$ torus of dimension $r$, $h_T:G_k\to \GL_r(\Z)$
the associated continuous representation and $W_T=h_T(G_k)\le \GL_r(\Z)$.
We call $T$ hereditarily rational if the associated $W_T$ lattice $\hat{T}$
is hereditarily rational.
We will now produce families of examples of hereditarily rational algebraic $k$ tori. These are gathered from known examples in the literature.
(Quasi-split tori: tori with permutation character lattices)
Any permutation $G$ lattice $P$ is hereditarily rational since
$K(P)^G$ is rational over $k$ for any Galois $G$ extension $K/k$ <cit.> and permutation
lattices are preserved under restriction.
Since quasi-split tori are precisely those with permutation character
lattices, these tori are hereditarily rational.
More explicitly, if $P=\oplus_{i=1}^k\Z[G/H_i]$ is a permutation $G$ lattice
where $H_1,\dots, H_k$ are subgroups of $G$, and $L/k$ is a Galois extension
with Galois group $G$, the corresponding quasi-split torus
$\prod_{i=1}^kR_{K_i/k}(\Gm)\in C(L/k)$ is hereditarily rational.
<cit.> (Tori with augmentation ideal character lattices)
Let $P=\Z X$ be a permutation $G$ lattice corresponding to any $G$ set $X$.
Then $\epsilon_X:\Z X\to \Z, x\to 1$ is a $G$ invariant surjection.
The augmentation ideal $I_X=\ker(\epsilon_X)$ is an hereditarily rational $G$
lattice. Suppose $\Z X=\oplus \Z[G/H_i]$.
Given a Galois $G$ extension $K/k$, the exact sequence
$0\to I_X\to \Z X\to \Z\to 0$
corresponds to the exact sequence of $k$ tori
$$1\to {\Gm}_k\to R_{F_1/k}(\Gm)\times \cdots \times R_{F_s/k}(\Gm)\to T\to 1$$
where $K_i/k$, $i=1,\dots,s$ are intermediate field extensions of $L/k$
such that $L/K_i$ is Galois with group $H_i$ for each $i$.
An algebraic $k$ torus with character lattice $I_X$ is given by
Since each $R_{F_i/k}(\Gm)$ can be identified with an open set of
$R_{F_i/k}(\A^1)=\A^{n_i}_k$, $T=\prod_{i=1}^sR_{F_i/k}(\Gm)/\Gm$ admits an open embedding into
projective space and hence is rational.
Note that for any subgroup $H$ of $G$, $(I_X)_H$ is also such an augmentation
ideal and so any algebraic torus with this character lattice would be
rational by the same argument.
(Tori with sign permutation character lattices)
A sign permutation $G$ lattice is hereditarily rational. A geometric proof is given in
<cit.>. An algebraic proof is given below in Proposition <ref>.
(Torus with character lattice $(\Z \bG_2,W(\bG_2))$) or equivalently $(\Z\bA_2,\Aut(\bA_2))$).
See Voskresenskii <cit.>.
The root lattice for $G_2$ is hereditarily rational as a $W(G_2)=S_3\times C_2$ lattice. Note that the root
lattice for $G_2$ restricted to $S_3$ is $I_{S_3/S_2}$ and $C_2$ acts as $-1$ on the lattice. For an algebraic $k$ torus $T$ with character lattice $G_2$,
the group $W(G_2)$ acts regularly on $\Gm^2$. Voskresenskii shows that the action
of $W(G_2)$ on $\Gm^2$ can be extended to a birational action on $(\PP^1)^2$
and so $T$ is an open part of the $k$ variety $((\PP^1)^2\otimes_k k_s)/W(G_2)$
which is a $k$ form of $\PP^1\times \PP^1$. Since any $k$ form of $\PP^1\times \PP^1$
is rational if it has a point, $T$ is rational.
(Tori with character lattices $I_X\otimes I_Y$ where $\gcd(|X|,|Y|)=1$)
A result of Klyachko <cit.> with a simpler proof given by Florence <cit.> shows that for any
finite $\cG_k$ sets $X$ and $Y$ which are relatively prime, an algebraic $k$ torus with character lattice $I_X\otimes I_Y$ is rational.
This shows that for any finite group $G$, and any 2 relatively prime $G$ sets
$X,Y$, the $G$ lattice $I_X\otimes I_Y$ is hereditarily rational.
Note that any $G$ set $Y$ of size 2 gives a rank 1 sign lattice $I_Y$.
Then for any other $G$ set $Y$ of odd order, $I_X\otimes I_Y$ is hereditarily
rational. In particular, this recovers the result for the root lattice of $G_2$
since it may be expressed as a tensor product $I_{S_3\times C_2/S_2\times C_2}\otimes I_{S_3\times C_2/S_3}$. More generally, it shows that a algebraic $k$ torus
with character lattice $(\Z \bA_{2k},\Aut(\bA_{2k}))$ <cit.> is hereditarily rational.
Let $M=\oplus_{i=1}^kM_i$ be the direct sum of $G$ lattices
$M_i$, $i=1,\dots,k$. If $K/k$ is $G$ Galois, then
$K(\oplus_{i=1}^kM_i)^G=\prod_{i=1}^rK(M_i)^G$ is rational over $k$.
Let $M$ be an hereditarily rational $G$ lattice of rank $r$.
Let $M^n_{wr}$ be the natural $\Z (G^n\rtimes S_n)$ lattice of rank $rn$
for the wreath product. Then $M^n_{wr}$ is hereditarily $G^n\rtimes S_n$
rational. So an algebraic $k$ torus with character lattice $(M^n_{wr},G^n\rtimes S_n)$ is hereditarily rational.
If $M$ is an hereditarily rational $G$ lattice so is
$M^n$ as a $G^n$ lattice since $K(M^n)^{G^n}$ is the composite of $n$ copies
of $K(M)^G$. Let $H$ be any subgroup of the wreath product
$G^n\rtimes S_n$. Then $N=H\cap G^n$ is a normal subgroup of $H$.
Let $K/k$ be a Galois $H$ extension. Then $K(M^n_{wr})^H=(K(M^n_{wr})^N)^{H/N}$.
Since $M^n$ is $G^n$ hereditarily rational, $K(M^n_{wr})^N$ is rational over
$K^N$. So there exist $f_1,\dots,f_{nr}$ in $K(M^n_{wr})^N$ which are
algebraically independent over $K^N$ such that
$K(M^n_{wr})^N=K^N(f_1,\dots,f_{nr})$. Since $H/N$ permutes the invariants
$f_1,\dots,f_{nr}\in K(M^n_{wr})^N$ and $K^N/k$ is a Galois $H/N$ extension, we see that
$K(M^n_{wr})^H=(K^N(f_1,\dots,f_{nr}))^{H/N}$ is rational as required.
This shows that the $W(B_n)$ lattice $\Z B_n$ for the root
system $B_n$ is hereditarily rational. $W(B_n)=C_2^n\rtimes S_n$ and the root
lattice for $B_n$ can be expressed as $(\Z_-)^n_{wr}$. That is, $S_n$ acts
by permutation and the group $C_2^n$ acts by sign changes.
Note that any sign-permutation lattice of rank n is isomorphic to the restriction of
$\Z B_n$ to a subgroup of $W(B_n)$.
Let $M$ be an hereditarily rational faithful $G$ lattice
and let $N$ be a $G$ lattice such that there exists an exact sequence
$$0\to M\to N\to P\to 0$$
for some permutation $G$ lattice $P$.
Then an algebraic $k$ torus with character lattice $(N,G)$ is hereditarily rational.
More precisely, if $T$ and $T'$ are algebraic $k$ tori fitting into an exact
$$1\to S\to T\to T'\to 1$$
where $S$ is a quasi-split $k$ torus, then $T$ is birationally equivalent
to $T'\times_k S$. If $T'$ is (hereditarily) $k$-rational, so is $T$.
Let $K/k$ be a Galois extension with Galois group $G$. Then
$K(N)\cong K(M)(P)$ as $G$ fields. Then $K(N)^G\cong K(M)(P)^G$ is rational over
$K(M)^G$ which is in turn rational over $K^G$. <cit.>
The same result would hold for any subgroup $H$ and the $H$ lattice $M_H$.
§ MAXIMAL HEREDITARILY RATIONAL $K$ TORI IN DIMENSIONS 2,3,4
We now have enough information to find the maximal hereditarily rational tori of dimension 4
(with 2 exceptions).
We will first quickly illustrate our approach for the rank 2 and 3 cases
due to Voskresenskii and Kunyavskii respectively.
Note in order to do this, we will need to identify certain $G$ lattices
corresponding to conjugacy classes of finite subgroups of $\GL_r(\Z)$.
We remark that for a $G$ lattice $M$, we could use the character to
identify the corresponding $\Q G$ module $\Q\otimes_{\Z}M$, but that
is not sufficient to identify its isomorphism type as a $G$ lattice.
We identify $G$ lattices up to isomorphism using explicit isomorphisms.
We will frequently wish to show that a $G$ lattice $L$ is isomorphic to $I_{G/H}$ or its dual $J_{G/H}$ for some subgroup $H$. We will use the following observation. If we can find an element $\x\in L$ such that
$G\x=L$, i.e. such that $\{g\x: g\in G\}$ is a $\Z$-spanning set for $L$
and such that $\sum_{g\in G}g\x=\0$ and such that the stabiliser subgroup $G_{\x}\cong H$, then $L\cong \Z[G/H]/\Z(\sum_{gH\in G/H}g\x)\cong J_{G/H}$.
If we show that $L^*\cong J_{G/H}$ in this way, then, we have
$L\cong I_{G/H}$. Note that the matrix generators of representation into the
dual lattice are given as the transposes of the original generators.
A rank 1 $G$ lattice is determined by a group homomorphism $\chi:G\to \GL_1{\Z}=\{\pm 1\}$. If $\chi$ is trivial, we will write the lattice as $\Z$.
If $\chi$ is non-trivial, we will write $\Z^{-}_{\ker(\chi)}$ since any non-trivial group representation $\chi:G\to \{\pm 1\}$ is completely determined by its kernel which is necessarily a normal subgroup of index 2 in $G$.
Corresponding to the $\Z S_{n+1}$ lattice $J_{S_{n+1}/S_n}$, we
will frequently see a certain natural representation of $S_{n+1}$.
be a $\Z$-basis of $\Z[S_{n+1}/S_n]$
on which $\sigma(\e_i)=\e_{\sigma(i)}$ for all $i=1,\dots,n+1, \sigma\in S_{n+1}$.
Let $\overline{\e_i}, i=1,\dots,n$ be the images of $\e_i$ in
$$\{\overline{\e_i}, i=1,\dots,n\}$$
forms a $\Z$-basis for $J_{S_{n+1}/S_n}$.
Note that $\sigma(\overline{\e_i})=\overline{\e_{\sigma(i)}}$
so that $\sigma(\overline{\e_i})=\overline{\e_{\sigma(i)}}$ unless $\sigma(i)=n$
and $\sigma(\overline{\e_i})=-\sum_{j=1}^n\overline{\e_j}$ otherwise.
We denote by $\rho_n:S_{n+1}\to \GL_n(\Z)$, the representation associated to
$J_{S_{n+1}/S_n}$ with respect to the $\Z$-basis $\overline{\e_1},\dots,\overline{\e_n}$.
Explicitly, $\rho_n(\sigma)=[a_{ij}(\sigma)]_{i,j}^n$
where $\sigma(\overline{\e_i})=\sum_{j=1}^na_{ij}(\sigma)\overline{\e_j}$
for all $i=1,\dots,n$. Note that the $i$th row is determined by
$\sigma(\overline{\e_i})$. This is done to match the notation of Hoshi and
Note that the matrices in the image of $\rho_n$ are either
permutation matrices or permutation matrices with one row
replaced by $[-1,\dots,-1]$. It is then easy to determine which
permutation determined them.
We will denote by $\rho_n^-:S_{n+1}\times C_2\to \GL_n(\Z)$, the
representation associated to the $S_{n+1}\times C_2$ lattice $J_{S_{n+1}\times C_2/S_n\times C_2}\otimes \Z^-_{S_{n+1}}$ with respect to the $\Z$-basis
$\overline{\e_i}\otimes 1$.
Then if $C_2=\langle \gamma\rangle$, note that $\rho_n^-(\sigma,1)=\rho_n(\sigma)$
and $\rho_n(\sigma,\gamma)=-\rho_n(\sigma)$ for all $\sigma\in S_{n+1}$.
The matrices in the image of $\rho_n^-$ are then also easy
to recognise.
We will denote by
$(\rho_n^-)^*:S_{n+1}\to \GL_n(\Z)$, and
$(\rho_n^-)^*:S_{n+1}\times C_2\to \GL_n(\Z)$ the representations
corresponding to the dual lattices $I_{S_{n+1}/S_n}$ and $I_{S_{n+1}\times C_2/S_n\times C_2}\otimes \Z_-^{S_{n+1}}$. Note that the root lattice $\Z A_n$ as a $W(\b A_n)=S_{n+1}$ lattice is $(I_{S_{n+1}/S_n})$
so that the weight lattice $\Lambda(A_n)$ is
$J_{S_{n+1}/S_n}$ as a $W(A_n)=S_{n+1}$ lattice.
Note that $\Z A_n$ as an $\Aut(A_n)=S_{n+1}\times C_2$ lattice is
$I_{S_{n+1}\times C_2/S_n\times C_2}\otimes \Z^-_{S_{n+1}}$
so that $\Lambda(A_n)$ is its dual
$J_{S_{n+1}\times C_2/S_n\times C_2}\otimes \Z^-_{S_{n+1}}$
as an $\Aut(A_n)$ lattice.
With respect to the standard basis $\e_1,\dots,\e_n$, the Weyl group
of $B_n$, has reflections $\tau_i=s_{\e_i}$, $i=1,\dots,n$ and
$\sigma_{ij}=s_{\e_i-\e_j}$. On the root lattice $\Z B_n$, with $\Z$-basis $\e_1,\dots,\e_n$, the $\tau_i$ fix $\e_j,j\ne i$ and $\tau_i(\e_i)=-\tau_i(\e_i)$.
and $\sigma_{ij}$ acts by swapping $\e_i$ and $\e_j$ and fixing the
other basis elements.
So $W(B_n)=(C_2)^n\rtimes S_n$
where $C_2^n=\langle \tau_i:i=1,\dots,n\}$ and $S_n=\langle \sigma_{ij}:i\ne j\}$.
We denote
$\eta_n:W(B_n)\to \GL_n(\Z)$
by the representation of $W(B_n)$ corresponding to its lattice $\Z B_n$
with respect to the standard basis $\e_1,\dots,\e_n$.
Note that the images of $\eta_n$ are sign permutation matrices.
We will write elements of $W(B_n)$ as $\tau \sigma$ where $\tau\in C_2^n$
is a product of $\tau_i$ and $\sigma\in S_n$.
Let $n$ be an odd integer and $D_{2n}$ be the dihedral group
of size $2n$ given by the presentation
$$D_{2n}=\langle \sigma,\tau: \sigma^n=1=\tau^2, \tau\sigma\tau^{-1}=\sigma^{-1}\}$$
$$J_{D_{2n}/C_2}\cong I_{D_{2n}/C_2}\otimes \Z^{-}_{C_n}$$
which implies that
$$I_{D_{2n}/C_2}\cong J_{D_{2n}/C_2}\otimes Z^{-}_{C_n}$$
$$J_{D_{2n}\times C_2/C_2\times C_2}\otimes \Z^{-}_{D_{2n}}\cong I_{D_{2n}\times C_2/C_2\times C_2}\otimes \Z^{-}_{D_{2n}}$$
Note that all elements of order 2 are conjugate in $D_{2n}$
so that there is a unique conjugacy class of subgroups isomorphic to $C_2$ in
This implies that $J_{D_{2n}/C_2}$ and $I_{D_{2n}/C_2}$ are well-defined,
since $\Z[G/H]\cong \Z[G/gHg^{-1}]$.
Note that given the presentation for $D_{2n}$ above, we find that
the subgroup $\langle \tau\rangle\cong C_2$ has transversal
$$\{\sigma^i: i=0,\dots,n-1\}$$
[In fact, a cyclic 2 group in $D_{2n}$ must
take the form $\langle \sigma^j\tau\rangle$ and so has the same transversal.]
Then the permutation lattice $\Z[D_{2n}/C_2]\cong \Z[D_{2n}/\langle \tau\rangle]$
has $\Z$-basis $\{\sigma^iC_2:i=0,\dots,n-1\}$.
The action of the dihedral group on this permutation lattice is then determined by
$\sigma(\sigma^iC_2)=\sigma^{i+1}C_2$ and $\tau(\sigma^iC_2)=\tau\sigma^i\tau\tau C_2=\sigma^{-i}C_2$.
The first 2 statements are equivalent by duality.
We will prove the second statement.
By the remark, it will suffice to find an element $\x$ of
$I_{D_{2n}/C_2}\otimes \Z^-_{C_n}$ such that the distinct elements of the orbit of $\x$ under $D_{2n}$
form a $\Z$-basis for $I_{D_{2n}/C_2}\otimes \Z^{-}_{C_n}$ and the stabilizer
subgroup of $\x$ is a cyclic subgroup of order 2.
Take $\x=(\sigma C_2-\sigma^{-1}C_2)\otimes 1\in I_{D_{2n}/C_2}\otimes \Z^-_{C_n}$.
We will show that $\x$ satisfies the two properties above and so
$$I_{D_{2n}/C_2}\otimes \Z^-_{C_n}=\Z D_{2n}\cdot \x\cong J_{D_{2n}/C_2}$$
Note that $\tau(\x)=\tau((\sigma C_2-\sigma^{-1}C_2)\otimes 1)=(\sigma^{-1} C_2-\sigma C_2)\otimes -1=\x$.
The orbit of $\x$ under $D_{2n}$ is then
$$\{(\sigma^{i} C_2-\sigma^{i-2}C_2)\otimes 1: i=1,\dots,n\}$$
We need to carefully prove that this set is a $\Z$-basis for
$I_{D_{2n}/C_2}\otimes \Z^-_{C_n}$ since this fails when $n$ is even
although the elements of the orbit are distinct. [This is very clearly false in the
case $n=4$ when the orbit is
$$\{(\sigma C_2-\sigma^3 C_2))\otimes 1,(\sigma^2 C_2-C_2)\otimes 1,(\sigma^3 C_2-\sigma C_2)\otimes 1\}]$$
Indeed, if $\sum_{i=0}^{n-1}b_i(\sigma^iC_2\otimes 1)\in I_{D_{2n}/C_2}\otimes \Z^-_{C_n}$
then $\sum_{i=0}^{n-1}b_i=0$. We need to show that we can solve
$$\sum_{i=1}^{n-1}x_i(\sigma^iC_2-\sigma^{i-2}C_2)\otimes 1
=\sum_{i=0}^{n-1}b_i(\sigma^iC_2\otimes 1)$$
for some unique $x_i$ if $\sum_{i=0}^{n-1}b_i=0$.
We obtain the following equations:
$$-x_2=b_0, x_{n-2}=b_{n-2}, x_i-x_{i+2}=b_i, i=1,\dots,n-3, x_{n-1}-x_1=b_{n-1}$$
Reordering these equations as
$$-x_2=b_0,x_{2i}-x_{2i+2}=b_{2i},i=1,\dots (n-3)/2;$$
$$x_{n-1}-x_1=b_{n-1}; x_{2i-1}-x_{2i+1}=b_{2i-1}, i=1,\dots, (n-3)/2; x_{n-2}=b_{n-2},$$
one can easily see that these equations correspond to a matrix system
of the form $A\x=\bb$ where
\dots&\dots&\dots&\dots&\dots\\0&\dots&0&1&-1\\0&0&\dots&\dots&1\end{array}\right]$$
is an $n\times n$ matrix.
Since the rows of this matrix add to 0, and the last n-1 rows form a triangular
system with ones on the diagonal, it is clear that one could solve this
system uniquely for $\bb=[b_0,\dots,b_{n-1}]^T\in \Z^n$ where $\sum_{i=0}^{n-1}b_i=0$.
So we have proved that $J_{D_{2n}/C_2}\cong I_{D_{2n}/C_2}\otimes \Z^-_{C_n}$
and by duality we have $I_{D_{2n}/C_2}\cong J_{D_{2n}/C_2}\otimes \Z^-_{C_n}$.
For the last statement,
we will write
$$D_{2n}\times C_2=\langle \sigma,\tau,\gamma: \sigma^n=\tau^2=\gamma^2=1,
\tau\sigma\tau^{-1}=\sigma^{-1},$$
$$ \gamma\tau=\tau\gamma, \gamma\sigma=
\sigma\gamma\rangle$$
note that $\Z[D_{2n}\times C_2/C_2\times C_2]=\Z[\langle \sigma,\tau,\gamma\rangle/\langle \tau,\gamma\rangle]$
has $\Z$-basis $\sigma^i(C_2\times C_2),i=0,\dots,n-1$.
This shows that $\gamma$ acts trivially on $\Z[D_{2n}\times C_2/C_2\times C_2]$
and $\Z[D_{2n}\times C_2/C_2\times C_2]$ restricts to $\Z[D_{2n}/C_2]$ as
a $D_{2n}$ lattice.
That is,
$$\inf_{D_{2n}}^{D_{2n}\times C_2}\Z[D_{2n}/C_2]=\Z[D_{2n}\times C_2/C_2\times C_2]$$
where $\inf_{G/N}^GL$ represents the $G$ lattice inflated from the $G/N$
lattice $L$ where $N$ is a normal subgroup of $G$.
But then from the defining exact sequences of $J_{D_{2n}/C_2}$
and $I_{D_{2n}/C_2}$ and the fact that inflation is exact, we may see that
$$\inf^{D_{2n}\times C_2}_{D_{2n}}I_{D_{2n}/C_2}=I_{D_{2n}\times C_2/C_2\times C_2},
\inf^{D_{2n}\times C_2}_{D_{2n}}J_{D_{2n}/C_2}=J_{D_{2n}\times C_2/C_2\times C_2}$$
So we have
$$J_{D_{2n}\times C_2/C_2\times C_2}\cong I_{D_{2n}\times C_2/C_2\times C_2}\otimes \Z^{-}_{C_n\times C_2}$$
since inflation commutes with tensor products.
Note that
$$\Z^{-}_{C_n\times C_2}\otimes \Z^{-}_{D_{2n}}\cong \Z^{-}_{\langle \sigma,\tau\gamma\rangle}$$
$\langle \sigma,\tau\gamma\rangle\cong D_{2n}$
and $\langle \sigma,\tau\gamma,\gamma\rangle=\langle \sigma,\tau,\gamma\rangle=D_{2n}\times C_2$. $\langle \tau\gamma,\gamma\rangle=\langle \tau,\gamma\rangle$.
So we see that after tensoring (*) by $\Z^-_{D_{2n}}$ we obtain
$$J_{D_{2n}\times C_2/C_2\times C_2}\otimes \Z^-_{D_{2n}}\cong
I_{D_{2n}\times C_2/C_2\times C_2}\otimes \Z^{-}_{\langle \sigma,\tau\gamma\rangle}$$
This technical result has some very interesting consequences.
Note that $\Lambda (A_2)\cong J_{S_3/S_2}$. The symmetric group $S_3$ is also
the dihedral group $D_6$. We then see that $\Lambda (A_2)\cong J_{D_6/C_2}\cong
I_{D_6/C_2}\otimes \Z^-_{C_3}=I_{D_6/C_2}\otimes I_{D_6/C_3}$.
Using Proposition <ref>, we easily recover the fact that an algebraic torus
with character lattice $(\Lambda (A_2),S_3)$ is hereditarily rational.
Note also that we can recover the fact that an algebraic torus with character lattice $(G_2,W(G_2))$
or equivalently $(\Z A_2,\Aut(A_2))$ is hereditarily rational.
More generally, it shows that an algebraic torus with character lattice
$J_{D_{2n}\times C_2/C_2\times C_2}\otimes \Z^-_{D_{2n}}$, $n$ odd, is hereditarily rational.
Given a finite subgroup $G$ of $\GL_r(\Z)$ up to conjugacy, the lattice determined by $G$, $M_G$ is determined by the action of $G$ by multiplying elements
of $\Z^r$ (considered as rows) by elements of $G$ on the right.
So for the standard basis $\e_1,\dots, \e_r$ of $\Z^r$,
$\e_i\cdot g=\sum_{j=1}^ra_{ij}\e_j$
where $g=[a_{ij}]_{i,j=1}^r\in \GL_r(\Z)$.
There is a library of conjugacy class representatives of finite
subgroups of $\GL_r(\Z)$ for $r=2,3,4$ in GAP.
The maximal finite subgroups of $\GL_r(\Z)$ for $r=2,3,4$ are encoded in
GAP as DadeGroup(r,k), in honour of Dade who determined the maximal
finite subgroups of $\GL_4(\Z)$ without the use of a computer.
I will use the GAP labelling to refer to conjugacy class representatives.
I will identify $M_G$ for each maximal finite subgroup of $\GL_r(\Z)$, $r=2,3,4$. These results are probably folklore (at least for $r=2,3$) but are not phrased in these terms in the literature.
Note, in GAP, the command
where $r=2,3,4$ and $k$ is in the
correct range for the rank gives
The command
gives the group generators of the
group M.
For better legibility, one could use the command
The use of GAP to list generators is a convenience. This
information could also be found in <cit.>.
§.§ Algebraic $k$ tori of dimension 2
For rank 2, there are 13 conjugacy classes of finite subgroups.
There are 2 maximal such classes.
The maximal finite subgroups of $\GL_2(\Z)$ up to conjugacy
are $G_i$=DadeGroup(2,i)$, i=1,2$.
* $M_{G_1}=(\Z B_2,W(B_2))=(\Z B_2,\Aut(B_2)$ and
* $M_{G_2}=(\Z A_2,\Aut(A_2)=(\Z G_2,W(G_2))$.
The corresponding algebraic $k$ tori are hereditarily rational.
The generators of $G_1=$DadeGroup(2,1) with GAP ID [2,3,2,1]
where $\eta_2$ is the faithful representation of $W(B_2)$ determined by
$\Z B_2$ with respect to the standard basis determined earlier.
Since $\langle \tau_2,\tau_2(12)\rangle =\langle \tau_2,(12)\rangle=W(B_2)$, we may claim that $M_{G_1}=(\Z B_2,W(B_2))$ as required.
Since this is a hereditarily rational lattice by Example <ref>, we are done.
The group $G_2=$DadeGroup(2,2) has GAP ID [2,4,4,1]
$$(\rho_2^-)^*((12),1),(\rho_2^-)^*((132),1)=\rho_2((123))^T, (\rho_2^-)^*(1,\gamma)=-I_2$$
where $\rho_2^-:S_3\times C_2\to \GL_2(\Z)$ is the natural representation of
$I_{S_3\times C_2/S_2\times C_2}\otimes \Z^-_{S_3}$ defined earlier.
We have already noticed that this coincides with $(\Z A_2,\Aut(A_2))$.
We need only check that $\langle ((12),1),((132),1),(1,\gamma)\rangle=S_3\times C_2$, which is clear.
So for rank 2, the lattices corresponding to maximal finite subgroups of $\GL_2(\Z)$ are $(G_2,W(G_2))=(\Z A_2,\Aut(A_2))$ and $(\Z B_2,W(B_2))$. As explained above,
these are hereditarily rational and so the corresponding tori and those
corresponding to their subgroups are rational. This is effectively
a rephrasing of how Voskresenskii proves that all algebraic k-tori of dimension 2 are rational.
§.§ Algebraic $k$ tori of dimension 3
For rank 3, the 73 conjugacy classes of finite subgroups of $\GL_3(\Z)$
were determined by
Tahara <cit.> There are 4 maximal such classes.
Kunyavskii <cit.> classified the algebraic $k$ tori of dimension 3 up to birational
equivalence. For each of the tori corresponding to maximal subgroups, he
constructed a nonsingular projective toric model of the algebraic $k$
tori and used the geometric construction of the flasque resolution of the
character lattice of each to understand birational properties of the
algebraic $k$ tori corresponding to maximal subgroups.
See Kunyavskii <cit.> and the description of his work in Voskresenskii <cit.>.
Let $G_k$ be the DadeGroup(3,k) for $k=1,\dots,4$.
Then the corresponding lattices are:
* $(M_{G_1},G_1)=(\Z A_2\oplus \Z A_1,\Aut(A_2)\times \Aut(A_1))$.
* $(M_{G_2},G_2)=(\Z B_3,W(B_3))=(\Z B_3,\Aut(B_3))$.
* $(M_{G_3},G_3)=(\Lambda(A_3),\Aut(A_3))$.
* $(M_{G_4},G_4)=(\Z A_3,\Aut(A_3))$.
We claim DadeGroup(3,1) with GAP ID [3,6,7,1] corresponds to
$(\Z A_2\oplus \Z A_1,\Aut(A_2\times A_1))$.
$\Aut(A_2)$ acts as above on $\Z A_2$ and trivially on $\Z A_1$
and $\Aut(A_1)$ acts as $-1$ on $\Z A_1$ and trivially on $\Z A_2$.
The generators given by GAP are
$$ A=(\rho_2^-)^*((12),1), B=(\rho_2^-)^*((132),1)$$
One may replace these generators by
Then by the above argument, we have already seen that $\langle A,B,-I_2\rangle$
determines $(\Z A_2,\Aut(A_2))$ and so we clearly
have that the full group determines $(\Z A_2\oplus \Z A_1,\Aut(A_2)\times \Aut(A_1))$.
DadeGroup(3,2) with GAP ID [3,7,5,1] corresponds to $(\Z B_3,W(B_3))$.
It is clear from the matrix generators that this corresponds to a sign
permutation lattice.
Recall the faithful representation $\eta_3:W(B_3)\to \GL_3(\Z)$ determined by $\Z B_3$
with respect to the standard basis.
The generators given by GAP are
so it is not hard to see that the generators
can be replaced by
$$\langle \tau_1,\tau_2,\tau_3,(12),(132)\rangle\cong C_2^3\rtimes S_3$$
DadeGroup(3,3) with GAP ID [3,7,5,2] corresponds to $(\Lambda(A_3),\Aut(A_3))$
where $G=\Aut(A_3)=S_4\times C_2$.
The generating set given by GAP is:
\rho^-_3((13)(24),1),\rho^-_3((12)(34),1)$$
where $\rho_3^-:S_4\times C_2\to \GL_3(\Z)$
is the representation determined by $J_{S_4\times C_2/S_3\times C_2}\otimes \Z^-_{S_4}$
with respect to a natural basis.
It suffices to check that $\langle (3,4),(1,3,2),(1,3)(2,4),(1,2)(3,4)\rangle=S_4$ which is straightforward.
Note that $\Aut(A_3)$ acts on $\Lambda(A_3)$ as
$J_{S_4\times C_2/S_3\times C_2}\otimes \Z^-_{S_4}$.
DadeGroup(3,4) with GAP ID [3,7,5,3] corresponds to $(\Z A_3,\Aut(A_3))$.
The matrix generators of this group given by GAP are the transposes of
those for Dade(3,4). So the corresponding lattice is accordingly the
$\Z$ dual $I_{S_4\times C_2/S_3\times C_2}\otimes \Z^-_{S_4}$ which indeed is the
representation of $\Aut(A_3)$ on $\Z A_3$.
Note that the last 3 Dade groups for dimension 3 are all $\Z$ forms
of the root lattice of $A_3$. [Since $SL_4/C_2\cong SO_3$, the character lattice
of $SL_4/C_2$ would be a $\Z$ form of the character lattice of $\SL_4/C_4$
which is $\Z A_3$.]
Although this is not how Kunyavskii determined the rational algebraic $k$
tori of dimension 3, the following argument is more or less equivalent.
He did not need to determine which ones were maximal rational as the numbers
were relatively small.
(Maximal hereditarily rational tori of dimension 3)
The maximal hereditarily rational tori of dimension 3
correspond to the following lattices:
* $(\Z A_2\oplus \Z A_1,\Aut(A_2)\times \Aut(A_1))$
* $(\Z B_3,W(B_3))=(\Z B_3,\Aut(B_3))$.
* $(\Z A_3,W(A_3))$
* $(L,W(B_2))$ where $L$ fits into a short exact sequence of
$W(B_2)$ lattices
$$0\to \Z B_2\to L\to \Z\to 0$$
We note that the lattice $(\Z A_2\oplus \Z A_1,\Aut(A_2)\times \Aut(A_1))$
is hereditarily rational as it is a direct sum of hereditarily rational
lattices. This corresponds to DadeGroup(4,1) with GAP ID [3,6,7,1].
The lattice $(\Z B_3,W(B_3))$ is hereditarily rational as it is a sign permutation lattice. It corresponds to DadeGroup(4,2) with GAP ID [3,7,5,1].
The lattice $(\Z A_3,W(A_3))=(I_{S_4/S_3},S_4)$ is hereditarily rational.
From our identification of DadeGroup(3,4) with lattice $(\Z A_3,\Aut(A_3))$ we see that this should correspond to a maximal subgroup.
The subgroup with GAP ID [3,7,5,3]
has generators
$$\rho_3^*((3,4)), \rho_3^*((1,3,2)), \rho_3^*((13)(24)), \rho_3^*((12)(34))$$
where $\rho_3^*:S_4\to \GL_3(\Z)$ is the dual representation of $\rho_3$
and hence corresponds to the lattice $I_{S_4/S_3}$.
The group with GAP ID [3,4,5,2] is abstractly isomorphic to $D_8$
and has generating set given by GAP
We note that the sublattice spanned by $\{\e_1-\e_2,\e_3\}$ is stable under the
action of the group. We then recompute the action of the
generators on the $\Z$-basis $\{\e_1-\e_2,\e_3,\e_2\}$ or equivalently
conjugate the generators by the change of basis matrix.
Then the conjugate generators are
With respect to this new basis, it is clear that
the corresponding lattice $L$ fits into the short exact sequence of $W(B_2)\cong D_8$
$$0\to \Z B_2\to L\to \Z\to 0$$
where we recall that $W(B_2)=C_2^2\rtimes C_2\cong D_8$.
Then we see that this lattice is also hereditarily rational.
We can then check using GAP that the union of conjugacy classes of subgroups of
the groups with the above GAP IDs give the complete list of 58
rational algebraic $k$ tori given by Kunyavskii. We will give more
details of our minimal GAP calculations after the dimension 4 case.
§.§ Algebraic $k$ tori of dimension 4
We now examine the dimension 4 case.
The classification of maximal finite subgroups of $\GL_4(\Z)$ up to conjugacy
is due to Dade. There are 9 maximal finite subgroups.
There are 710 conjugacy classes of finite subgroups of $\GL_4(\Z)$.
Let $G_k$ be the DadeGroup(4,k) for $k=1,\dots,9$.
Then the corresponding lattices are:
* $(M_{G_1},G_1)=(\Z B_2\oplus \Z A_2,\Aut(B_2)\times \Aut(A_2))$.
* $(M_{G_2},G_2)=(\Lambda A_3\oplus \Z A_1,\Aut(A_3)\times \Aut(A_1))$.
* $(M_{G_3},G_3)=(\Z A_3\oplus \Z A_1,\Aut(A_3)\times \Aut(A_1))$
* $(M_{G_4},G_4)=(L,((W(A_2)\times W(A_2))\rtimes C_2)\times C_2)$.
where $L$ is the non-trivial intermediate lattice between $\Z A_2\oplus \Z A_2$
and $\Lambda(A_2)\oplus \Lambda(A_2)$.
* $(M_{G_5},G_5)=(\Z A_2\oplus \Z A_2, (\Aut(A_2)\times \Aut(A_2))\rtimes C_2)$.
* $(M_{G_6},G_6)=(\Z A_4,\Aut(A_4))$.
* $(M_{G_7},G_7)=(\Lambda(A_4),\Aut(A_4))$.
* $(M_{G_8},G_8)=(\Z B_4,W(B_4))=(\Z B_4,\Aut(B_4))$.
* $(M_{G_9},G_9)=(\Z F_4,W(F_4))=(\Z F_4,\Aut(F_4))$.
$G=$ DadeGroup(4,1) with GAP ID [4,20,22,1]
corresponds to $(\Z B_2\oplus \Lambda(A_2),W(B_2)\times \Aut(A_2))$
which is hereditarily rational as the direct sum of hereditarily rational
The generators of this group are given as
$$A_1=\eta_2(\tau_2),A_2=\eta_2(\tau_2 (12)),$$
$$B_1=\rho^-_3((23),1), B_2=\rho^-_3((132),1)$$
Since $A_2^2=-I_2$, we may replace this
set of generators by
\diag(I_2,-I_2)\}$$
Then the associated lattice is $M=M_1\oplus M_2$ where
where $M_1=\Z\e_1\oplus \Z\e_2$ and $M_2=\Z\e_3\oplus \Z\e_4$
are both $G$ invariant.
Then since
$$A_1=\eta_2(\tau_2),A_2=\eta_2(\tau_2 (12)),-I_2=\eta_2(\tau_1\tau_2\tau_3)$$
$$B_1=\rho^-_3((23),1), B_2=\rho^-_3((132),1),-I_2=\rho^-_3(1,\gamma)$$
we may see that the lattice $M_1$ is $(\Z B_2,W(B_2))$ and the
lattice $M_2$ is $(J_{S_3\times C_2/S_2\times C_2}\otimes \Z^-_{S_3},S_3\times C_2)$.
$M_2$ corresponds to the natural action of $\Aut(A_2)$ on $\Lambda(A_2)$.
So $(M_{G},G)=(\Z B_2\oplus \Lambda(A_2),\Aut(B_2)\times \Aut(A_2))$.
The lattice is hereditarily rational as the direct sum of 2 hereditarily
rational lattices.
DadeGroup(4,2) has GAP ID [4,25,11,2].
The generators are
We then clearly see that the lattice decomposes as a direct sum of
$M_1=\Z\e_1$ and $M_2=\oplus_{i=2}^4\Z \e_i$.
To determine $M_2$, we
note that $A_2^3=-I_3$. We may then replace the above generators
\rho_3^-((12)(34),1)=-A_4^T, \rho_3(\id,\gamma)=-I_3$,
we can see that the lattice $M_2$ corresponds to the
dual lattice of $J_{S_4\times C_2/S_3\times C_2}\otimes \Z^-_{S_4}$
and so to $I_{S_4\times C_2/S_3\times C_2}\otimes \Z^-_{S_4}$.
This is the natural action of $\Aut(A_3)$ on $\Z A_3$.
So the lattice corresponding to DadeGroup(4,2) is $(\Z A_1\oplus \Z A_3,\Aut(A_1)\times \Aut(A_3))$.
DadeGroup(4,3) has GAP ID [4,25,11,4].
The generators are
We then clearly see that the lattice decomposes as a direct sum of
$M_1=\Z\e_1$ and $M_2=\oplus_{i=2}^4\Z \e_i$.
To determine $M_2$, we
note that
\rho_3^-((12)(34),\gamma)=B_4$$
where $\rho_3^-:S_4\times C_2\to \GL_3(\Z)$ is the representation
corresponding to $J_{S_4\times C_2/S_3\times C_2}\otimes \Z^-_{S_4}$
which in turn corresponds to $\Aut(A_3)$ acting on $\Lambda(A_3)$.
As it is not hard to show that that the preimages generate
$S_4\times C_2$, we have found that $M_2$ is $(\Lambda(A_3),\Aut(A_3))$.
So the group determines the lattice $(\Z A_1\oplus \Lambda(A_3),\Aut(A_1)\times \Aut(A_3))$.
DadeGroup(4,4) with GAP ID [4,29,9,1]
is abstractly isomorphic to $((S_3\times S_3)\rtimes C_2)\times C_2$
and has generators given by
$$\left[\begin{array}{rrrr}1&-1&0&0\\0&-1&0&0\\0&0&1&-1\\0&0&0&-1\end{array}\right], \left[\begin{array}{rrrr}1&-1&0&0\\0&0&1&-1\\0&-1&0&0\\0&0&1&-1\end{array}\right],\left[\begin{array}{rrrr}1&-1&0&0\\1&0&0&0\\0&0&1&-1\\0&0&1&0\end{array}\right],\left[\begin{array}{rrrr}0&0&0&1\\0&0&-1&1\\0&-1&0&1\\1&-1&-1&1\end{array}\right]$$
We claim that the lattice determined by this group is
the proper intermediate lattice $L$ between $\Z A_2\oplus \Z A_2$
and $\Lambda(A_2)\oplus\Lambda(A_2)$.
Let $\omega_1,\omega_2$ be the fundamental dominant weights of $A_2$
with respect to a basis $\alpha_1,\alpha_2$ of the $A_2$ root system.
We will write the basis of $\Lambda(A_2)\oplus \Lambda(A_2)$
as $\{\omega_1,\omega_2,\omega_1',\omega_2'\}$
and that of $\Z A_2\oplus \Z A_2$ as
Then the claimed sublattice $L$ of $\Lambda(A_2)\oplus \Lambda(A_2)$
satisfies $L=\langle \omega_1+\omega_1',\Z A_2\oplus \Z A_2\rangle$
If $s_1=s_{\alpha_1},s_1'=s_{\alpha_1'},s_2=s_{\alpha_2},s_2'=s_{\alpha_2'}$.
then computing the action of generators
$s_1,s_2s_1,s_1',s_2's_1'$ of $(S_3\times S_3)$, $\tau$ the generator that flips the 2 copies of
$S_3$ and $-I_4$ on a $\Z$-basis
then the matrices we obtain for the above generators in terms of this basis
$$s_1=\left[\begin{array}{rrrr}-1&-1&1&0\\0&1&0&0\\0&0&1&0\\0&0&1&0\end{array}\right], s_1'=\left[\begin{array}{rrrr}1&0&-1&0\\0&1&-1&0\\0&0&-1&0\\0&0&1&1\end{array}\right],s_2s_1=\left[\begin{array}{rrrr}0&1&0&0\\-1&-1&1&-1\\0&0&1&0\\0&0&0&1\end{array}\right]$$
$$s_2's_1'=\left[\begin{array}{rrrr}1&0&0&1\\0&1&0&1\\0&0&0&1\\0&0&-1&-1\end{array}\right], \tau=\left[\begin{array}{rrrr}1&0&0&0\\1&0&0&1\\2&1&-1&1\\-1&1&0&0\end{array}\right], -I_4$$
The group generated by these generators is conjugate to the
group DadeGroup(4,5). This is determined by GAP by checking that the
CrystCatZClass of the two matrix groups given by generators are in fact the
same. See the remark below on GAP calculations. So the lattice
determined by DadeGroup(4,5) is indeed the intermediate lattice
$L$ between $\Z A_2\oplus \Z A_2$ and $\Lambda(A_2)\oplus \Lambda(A_2)$
as a $(W(A_2)\times W(A_2))\rtimes C_2 \times C_2$ lattice.
For G=DadeGroup(4,5) with GAP ID [4,30,13,1], we claim that the
corresponding lattice is $(\Z A_2\oplus \Z A_2,\Aut(A_2\times A_2))$
which is hereditarily rational as a wreath product of hereditarily
rational lattices.
$\Aut(A_2\times A_2)=(\Aut(A_2)\times \Aut(A_2))\rtimes S_2$ acts naturally
on the root lattice for $A_2\times A_2$ where $\Aut(A_2)\times \Aut(A_2)$
acts diagonally on $\Z A_2\oplus A_2$ and the subgroup $S_2$ permutes the
two copies of $A_2$.
The generators given by GAP are
\left[\begin{array}{rr}0&I_2\\P&0\end{array}\right],
\left[\begin{array}{rr}I_2&0\\0&Y\end{array}\right],
\left[\begin{array}{rr}Z&0\\0&I_2\end{array}\right]\}$$
Since $Y,Z$ have order 6 and $Y^3=Z^3=-I_2$,
we may replace these generators by
\left[\begin{array}{rr}0&I_2\\I_2&0\end{array}\right],
\left[\begin{array}{rr}I_2&0\\0&Y^2\end{array}\right],
\left[\begin{array}{rr}Z^2&0\\0&I_2\end{array}\right],
X_5=\diag(I_2,-I_2), X_6=\diag(-I_2,I_2).$$
$$Y^2=\left[\begin{array}{rr}-1&1\\-1&0\end{array}\right]\qquad Z^2=\left[\begin{array}{rr}0&-1\\1&-1\end{array}\right],$$
We see that the dual lattice defined on $\Z\e_1\oplus \Z\e_2$ defined by $\langle P,Y^2\rangle$ is $J_{S_3/S_2}=\Z S_3\cdot \e_1$.
We also see that the dual lattice defined on $\Z\e_3\oplus \Z\e_4$
defined by $\langle P, Z^2\rangle$ is $J_{S_3/S_2}=\Z S_3\cdot \e_3$.
This shows that the lattices defined by both
$\langle P,Y^2,-I_2\rangle$ and $\langle P,Z^2,-I_2\rangle$ are
isomorphic to $(I_{S_3\times C_2/S_2\times C_2}\otimes \Z^-_{S_3},S_3\times C_2)$
or equivalently $(\Z A_2,\Aut(A_2))$.
So the lattice restricted to
$$\langle \diag(I_2,P),\diag(I_2,Y^2),\diag(I_2,-I_2),\diag(P,I_2),\diag(Z^2,I_2),\diag(-I_2,I_2)\rangle$$
is $(\Z A_2\oplus \Z A_2,\Aut(A_2)\times \Aut(A_2))$.
swaps the 2 copies of $\Z A_2$, we see that the full lattice structure is
given by
$$(\Z A_2\oplus \Z A_2,(\Aut(A_2)\times \Aut(A_2))\rtimes C_2)$$
as required.
DadeGroup(4,6) has GAP ID [4,31,7,1].
We will show that it determines the lattice $(\Z A_4, \Aut(A_4))$
which is hereditarily rational since it is the tensor product of
2 augmentation ideals of relatively prime ranks.
This lattice is $(I_{S_5\times C_2/S_4\times C_2}\otimes \Z^-_{S_5},S_5\times C_2)$
Recalling our representation $\rho^-_4:S_5\times C_2\to \GL_4(\Z)$
determined by $(J_{S_5\times C_2/S_4\times C_2}\otimes \Z^-_{S_5})$,
we see that the generators of DadeGroup(4,6) given by GAP
$$\rho_4((15)(243),\gamma)^T,\rho_4((12354),\gamma)^T, \rho_4((1324),\gamma)^T$$
Since (12354) has odd order, we see that $((12354),\gamma)^5=(\id,\gamma)$,
and we know that $S_5$ is generated by any 5 cycle and any transposition,
so it suffices to note that $[(15)(243)]^3=(15)$, in order to conclude that
the preimages generate $S_5\times C_2$.
So, as required, DadeGroup(4,6) determines the dual lattice to $J_{S_5\times C_2/S_4\times C_2}\otimes \Z^-_{S_5},S_5\times C_2$ which is $I_{S_5\times C_2/S_4\times C_2}\otimes \Z^-_{S_5}$.
DadeGroup(4,7) has GAP ID [4,31,7,2].
We will show that it determines the lattice $(\Lambda(A_4), \Aut(A_4))$.
This lattice is $(J_{S_5\times C_2/S_4\times C_2}\otimes \Z^-_{S_5},S_5\times C_2)$
In terms of our representation $\rho^-_4:S_5\times C_2\to \GL_4(\Z)$
determined by $(J_{S_5\times C_2/S_4\times C_2}\otimes \Z^-_{S_5})$,
we see that the generators of DadeGroup(4,6) given by GAP
$$\rho^-_4((132)(45),\gamma),\rho^-_4((15234),\gamma)^T, \rho^-_4((1324),\gamma)^T$$
Since (15234) has odd order, we see that $((15234),\gamma)^5=(\id,\gamma)$,
and we know that $S_5$ is generated by any 5 cycle and any transposition,
so it suffices to note that $[(132)(45]^3=(45)$, in order to conclude that
the preimages generate $S_5\times C_2$.
So, as required, DadeGroup(4,7) determines the lattice $J_{S_5\times C_2/S_4\times C_2}\otimes \Z^-_{S_5},S_5\times C_2$.
DadeGroup(4,8) has GAP ID [4,32,21,1].
We claim that it determines the lattice
$(\Z B_4,W(B_4))$.
In terms of our representation $\eta_4:W(B_4)\to \GL_4(\Z)$, the generators
given by GAP are
$$\eta_4(\tau_2\tau_4(24)), \eta_4(\tau_1\tau_3\tau_4(234)), \eta_4(\tau_1\tau_2),
\eta_4(\tau_3\tau_4(12)(34)),\eta_4(\tau_1\tau_2(13)(24)),\eta_4(\tau_1\tau_4(14)(23))$$
We need only check that the group generated by the preimages is $W(B_4)$.
(Note that GAP gives a structure description for the group as the wreath
product of $C_2$ by $S_4$ so this is just a check).
Since $(234)$ is odd, the group contains $\tau_1\tau_3\tau_4$ and $\tau_1\tau_2$
Recalling that $(\tau\sigma)^2=\tau\tau^{\sigma}\sigma^2$ and $\tau_i^{\sigma}=\tau_{\sigma(i)}$, we see that
$[\tau_1\tau_2(13)(24)]^2=\tau_1\tau_2\tau_3\tau_4$. But then we have
$\tau_2=(\tau_1\tau_2\tau_3\tau_4)(\tau_1\tau_3\tau_4)$ and so $\tau_1$ and $\tau_3\tau_4$. But then we have $(12)(34)$ and $(13)(24)$ and so $(14)(23)$.
But then we also have $\tau_1\tau_4$ and so $\tau_1,\tau_2,\tau_3,\tau_4$.
Since our group now contains the normal subgroup $\langle (12)(34),(13)(24)\rangle$ and a cyclic 3 subgroup $(234)$ of $A_4$, we have $A_4$.
Since it also contains a transposition $(24)$, it contains $S_4$.
So it contains $C_2^4\rtimes S_4$ as required.
Note that $(\Z B_4,W(B_4))$ is a hereditarily rational lattice as it is
sign permutation.
DadeGroup(4,9) has GAP ID [4,33,16,1].
Its generators are
$$X_1=\left[\begin{array}{rrrr}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{array}\right], X_2=\left[\begin{array}{rrrr}1&0&0&0\\0&0&1&0\\0&1&0&0\\1&1&1&-1\end{array}\right],
$$X_4=\left[\begin{array}{rrrr}-1&0&0&1\\0&0&-1&1\\-1&-1&-1&1\\-1&-1&-1&2\end{array}\right], X_5=\left[\begin{array}{rrrr}0&0&-1&0\\1&1&1&-2\\-1&0&0&0\\0&0&0&-1\end{array}\right],$$
We wish to show that the associated lattice is $(\Z F_4,W(F_4))$.
We recall a standard basis of the root system $F_4$ is given by
The roots in the root system $F_4$ are of the
$$\pm \e_i\pm \e_j,1\le i<j\le 4; \pm \e_i, 1\le i\le 4; \frac{\pm \e_1\pm \e_2\pm \e_3\pm \e_4}{2}$$
A $\Z$-basis for $\Z F_4$ is given by
For each of the simple roots in $\Delta$
we may compute the matrices of $s_{\alpha_i}$ with respect to the basis $\beta$
$$s_{\alpha}(x)=x-2\frac{x\cdot \alpha}{\alpha\cdot \alpha}\alpha$$
is the simple reflection corresponding to $\alpha$.
We find that
$$s_{\alpha_1}=\left[\begin{array}{rrrr}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{array}\right], s_{\alpha_2}=\left[\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\-1&-1&-1&2\\
$$s_{\alpha_3}=\left[\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\0&0&1&0\\1&1&1&-1\end{array}\right], s_{\alpha_4}=\left[\begin{array}{rrrr}0&0&0&1\\1&1&0&-1\\1&0&1&-1\\
We may show that $s_{\alpha_1},\dots,s_{\alpha_4}$ are all contained in DadeGroup(4,9). That is, we may express them as products of the generators given by GAP. Explicitly, $s_{\alpha_1}=X_1$, $s_{\alpha_2}=X_5X_3X_4^{-1}X_1$,
$s_{\alpha_3}=X_2X_1$, $s_{\alpha_4}=X_6^{-1}X_4^{-1}X_3^{-1}X_2X_1$.
Since DadeGroup(4,9) has order $1152=|W(F_4)|$ we see that they coincide.
(Maximal hereditarily rational tori of dimension 4)
The maximal hereditarily rational tori of dimension 4
correspond to the following 3 groups of lattices
* Hereditarily rational lattices corresponding to Dade Groups:
* $(\Z B_2\oplus \Z A_2,\Aut(B_2)\times \Aut(A_2))$
* $(\Z A_2\oplus \Z A_2,(\Aut(A_2)\times \Aut(A_2))\rtimes C_2)$.
* $(\Z A_4,\Aut(A_4))$.
* $(\Z B_4,W(B_4))$.
* Hereditarily rational lattices which decompose into a 3 dimension lattice
and a 1 dimension lattice and hence correspond to the maximal hereditarily rational lattices of dimension 3:
For each of the 4 maximal hereditarily rational lattices $M_i$ with group $G_i$, $i=1,\dots,4$, the corresponding 4 dimensional maximal hereditarily rational lattice is
$$(\inf_{G_i}^{G_i\times C_2}M_i\oplus \Z^-_{G_i},G_i\times C_2)$$
* Reducible lattices with a rank 3 invariant sublattice:
* $(L_1,W(B_3))$ where $L_1$ fits in a short exact sequence
of $W(B_3)$ lattices
$$0\to \Z B_3\to L_1\to \Z\to 0$$
* $(L_2,W(B_3))$ where $L_2$ fits in a short exact sequence of
$W(B_3)$ lattices
$$0\to \Z B_3\to L_2\to \Z^-_{C_2^3\rtimes A_3}\to 0$$
* $(L_3,W(A_3))$ where $L_3$ fits in a short exact sequence of
$W(A_3)$ lattices
$$0\to \Z A_3\to L_3\to \Z\to 0$$
The union of the conjugacy classes of finite groups corresponding to these
lattices produces 485 of the 487 stably rational algebraic $k$ tori
determined by Hoshi and Yamasaki. The remaining two lattices corresponding
to stably rational algebraic $k$ tori
of dimension 4 are
(J_{A_5\times C_2/A_4\otimes C_2},\Z^-_{A_5},A_5\times C_2)$$
Note that these correspond to the weight lattice for the $A_4$
root system restricted to $A_5\le W(A_4)$ and restricted to
$A_5\times C_2\le \Aut(A_4)$ respectively.
We have already determined the Dade groups which correspond to hereditarily
rational lattices.
We first determine the GAP IDs of the groups corresponding to a direct
sum of a maximal hereditarily rational lattice of rank 3 and a sign lattice.
We claim that the group with GAP ID [4,25,9,2] has lattice given by
$\Z^-_{W(A_2)}\oplus \inf_{W(A_2)}^{C_2\times W(A_2)}\Z A_3,C_2\times W(A_2))$.
The generators given by GAP are
Since $(142)$ is odd, it is clear that one could replace $\diag(-1,\rho_3((142))^T)$ with $\diag(-1,I_3)$ and $\diag(1,\rho_3((142))$.
It is clear that $\langle (14)(23),(12)(34)\rangle$ generates $V_4$,
$\langle V_4,(142)\rangle=A_4$ and $\langle A_4,(34)\rangle =S_4$.
So the 3 dimensional lattice is $I_{S_4/S_3}$ as required and the
result follows.
We claim that the group with GAP ID [4,25,11,1] has corresponding lattice given by
$$(\Z^-_{W(B_3)}\oplus \inf_{W(B_3)}^{W(B_3)\times C_2}\Z B_3,C_2\times W(B_3))$$
In terms of our representation $\eta_3:W(B_3)\to \GL_3(\Z)$, the generators given by GAP are
$$\diag(1,\eta_3(\tau_2\tau_3(23))),\diag(1,\eta_3(\tau_3 (123))),\diag(1,\eta_3(\tau_1\tau_3)),\diag(-1,\eta_3(\tau_1)),\diag(-1,\eta_3(\tau_1\tau_2\tau_3)))$$
To determine the group generated by the preimages, note that it contains $(1,\tau_3)$ as $(123)$ is odd. So it also contains $(1,\tau_1)$ and hence $(-1,\id)$.
This implies it contains $(1,\tau_2)$. But then it contains $(1,(23))$
and $(1,(123))$. This means it contains all of $C_2\times W(B_3)$
where $W(B_3)=C_2^3\rtimes S_3$.
The group with GAP ID [4,13,6,4] has
generators are
We may observe that $A=\rho_3((14)(23))^T$ and $B=\rho_3((1234))^T$
so that it is clear the group $\langle A,B\rangle=D_8\cong W(B_2)$.
We note that $\Z(\e_1-\e_3)\oplus \Z\e_2$ is stable under $\langle A,B\rangle$.
Computing the matrices with respect to the new basis
$\{\e_1-\e_3,\e_2,\e_3\}$ (or equivalently conjugating by the change of basis matrix)
we obtain
So we may consider the group to be generated by the
conjugate group generated by
It is then clear that the $W(B_2)$ lattice determined by $\langle A',B'\rangle$
$$0\to \Z B_2\to L\to \Z\to 0$$
Then our lattice is $(\inf_{W(B_2)}^{W(B_2)\times C_2}L\oplus \Z^-_{W(B_2)}$.
For the group with GAP ID [4,15,12,1], the generators
It is then not difficult to see that the lattice determined
by this group is
$(\Z A_1\oplus \Z A_1\oplus \Z A_2,\Aut(A_1)\times \Aut(A_1) \times \Aut(A_2))$where each of the factor groups acts non-trivially only on the corresponding
lattice factor.
We next look at the groups which correspond to a reducible
lattice with a 3 dimensional invariant sublattice:
For the group with GAP ID [4,25,7,5], the generators are
Note that this determines a lattice $M$ which contains a sublattice
with basis $\{\e_2,\e_3,\e_4\}$ which is stable under the action of the
group. Note also that $M/(\oplus_{i=2}^4\Z\e_i)\cong \Z$.
The action on $\oplus_{i=2}^4\Z\e_i$ is determined by
the group generated by
\eta_3(\tau_1\tau_2), \eta_3(\tau_2\tau_3)$$
Since $(123)$ is odd, $\tau_1$ is in the preimage, from which one can easily
show that $\tau_2,\tau_3$ are too. Then $\langle (23),(123)\rangle= S_3$
is also in the preimage.
This shows that the group determines a lattice $M$
which satisfies
$$0\to \Z B_3\to M\to \Z\to 0$$
For the group with GAP ID [4,25,8,5], the generators are
Note that these are in fact the same generators as in the [4,25,7,5]
case except for the first one.
Note that this determines a lattice $M$ which contains a sublattice $M_0$
with basis $\{\e_2,\e_3,\e_4\}$ which is stable under the action of the
group. Note also that $M_1=M/M_0$ is a rank 1 lattice with
non-trivial action.
The action on $M_0=\oplus_{i=2}^4\Z\e_i$ is determined by
the group generated by
\eta_3(\tau_1\tau_2), \eta_3(\tau_2\tau_3)$$
Since $(123)$ is odd, $\tau_1$ is in the preimage, from which one can easily
show that $\tau_2,\tau_3$ are too. Then $\langle (23),(123)\rangle= S_3$
is also in the preimage. So the group is isomorphic to $W(B_3)$.
Note that $N=\langle \tau_1(123),\tau_1\tau_2,\tau_2\tau_3\rangle$ acts trivially on $M_1=M/M_0$. Since $N$ contains $\langle \tau_1,\tau_2,\tau_3\rangle$, it
is $C_2^3\rtimes A_3$ where $A_3=\langle (123)\rangle$.
This shows that the group $G$ determines a lattice $M$
which satisfies
$$0\to \Z B_3\to M\to \Z^-_{C_2^3\rtimes A_3}\to 0$$
Let $M_0=\Z B_3$.
Restricting to the normal subgroup of index 2, $N=C_2^3\rtimes A_3$,
and considering a Galois extension $L/k$ with group $G$,
we see that $L(M)^G= ((L(M))^N)^{G/N}=(L(M_0)^N(M_1))^{G/N}$.
Let $K=L(M_0)^N$. Then $L(M)^G=K(M_1)^{G/N}=K(t)^{G/N}$.
It suffices to show that $L(M)^G$ is rational over $L(M_0)^G$ since the
latter is rational over $k$. The generator $\sigma$ of $G/N\cong C_2$ acts as $\sigma(t)=t^{-1}$ since it is determined by $D_1$.
Note that $K^{\langle \sigma\rangle}=L(M_0)^G$.
Set $E=L(M_0)^G$.
Now $[K:E]=[K:K^{\langle \sigma\rangle}]\le 2$ but $\sigma$ does not fix $K$.
So $K/E$ is a quadratic extension. All our fields are
of characteristic zero, so we may assume that $K=E(\alpha)$
where $\sigma(\alpha)=-\alpha$.
So we have to show that $K(t)^{\langle \sigma \rangle}$ is rational over
$K^{\langle \sigma\rangle}=E$. Setting $z=\frac{1-t}{1+t}$, we see that
Then $K(t)^{\langle \sigma\rangle}=(K^{\langle \sigma \rangle}(\alpha)(t))^{\langle \sigma \rangle}
=E(\alpha)(z)^{\langle \sigma\rangle}=E(\alpha)(\alpha z)^{\langle \sigma\rangle} =E(\alpha z)$ is rational over $E$. This shows that $L(M)^G=K(t)^{\langle \sigma\rangle}$ is rational over
$L(M_0)^G=E$. But then the $k$ torus corresponding to the group [4,25,8,5] is
rational. For any subgroup $H$ of this group,
the corresponding lattice would have a decomposition as
$$0\to (M_0)_H\to M_H\to (M_1)_H\to 0$$
Then we see that a similar argument would
work to show that the algebraic torus corresponding to the subgroup is also
rational. Then the lattice correponding to the group [4,25,8,5] is hereditarily
For the group with GAP ID [4,24,3,4], the generators are
Note that this determines a lattice $M_3$ which contains a sublattice
$\oplus_{i=2}^4\Z\e_i$ which is stable under the action of the
group. Note also that $M_2/(\oplus_{i=2}^4\Z\e_i)\cong \Z$
The action on $\oplus_{i=2}^4\Z\e_i$ is determined by
the group generated by
Since the group generated by preimages is $\langle (14),(143),(13)(24),(14)(23)\rangle$, we see that it contains the normal subgroup $\langle (13)(24),(14)(23)\rangle$ of $S_4$. Then clearly $\langle (13)(24),(14)(23),(143)\rangle =A_4$
and $\langle (13)(24),(14)(23),(143),(14)\rangle =S_4$.
So the lattice determined by this group satisfies a short exact sequence of
$W(A_3)=S_4$ lattices given by
$$0\to \Z A_3\to M_3\to \Z\to 0$$
We then use GAP to take the union of the conjugacy classes of subgroups
corresponding to these 11 hereditarily rational lattices.
We find that we obtain all but 2 of the stably rational tori obtained by
Hoshi and Yamasaki. The 2 exceptions are [4,31,6,2] and [4,31,3,2].
The lattice determined by [4,31,6,2] is $
(J_{A_5\times C_2/A_4\times C_2}\otimes \Z_{A_5}^-,A_5\times C_2)$.
This is because the generators given by GAP are
Since $(15243)$ has odd order, we see that $(1,\gamma)\in C_2$ is in the preimage.
Then it is not hard to see that $\langle (13)(24),(12)(34),(132)\rangle=A_4$
and so $\langle (15243),(132),(12)(34),(13)(24)\rangle=A_5$.
The lattice determined by [4,31,3,2] is $(J_{A_5/A_4},A_5)$.
This is because the generators given by GAP are
Again, it's easy to see that $\langle (13)(25),(12)(35)\rangle=C_2\times C_2$
and $\langle (13)(25),(12)(35),(123)\rangle=A_4$
and finally $\langle (13)(25),(12)(35),(123)\rangle=A_5$.
We will show in the next section that the
lattices $(J_{A_5\times C_2/A_4\times C_2}\otimes \Z_{A_5}^-,A_5\times C_2)$
and $(J_{A_5/A_4},A_5)$ are quasi-permutation so that the corresponding tori are stably
rational. We are not able to show that these tori are rational.
Note that that determined by $J_{A_5/A_4}$ is a norm one torus and we
will see that the other tori is closely related.
To explain why there are only 2 missing groups, note that the
restriction of $(J_{A_5\times C_2/A_4\times C_2}\otimes \Z_{A_5}^-,A_5\times C_2)$ to the maximal subgroup $D_5\times C_2$
is $(J_{D_{10}\times C_2/C_2\times C_2}\otimes \Z_{D_{10}}^-,D_{10}\times C_2)$
which we showed to be hereditarily rational.
Note that all subgroups of $A_5\times C_2$ except $A_5$ are subgroups of $D_{10}\times C_2$.
We explain our very basic use of GAP.
We mainly used the generating sets (which could have been found in <cit.>) and as a calculation tool to check our hypotheses.
All the calculations of the lattices corresponding to the groups could be done by hand as explained above directly from the
generating sets, with only 2 exceptions. In the case of the lattice for the Weyl group of $F_4$, we used GAP to find the simple reflections in the generators. In the case of DadeGroup(4,4) we used GAP to check that our proposed group
was conjugate to DadeGroup(4,4). We hope to find simpler proofs in those 2 cases. Note that they do not come into play in checking for rational tori.
To check in the dimension 3 and 4 cases that the conjugacy classes of subgroups of the groups corresponding to our hereditarily rational algebraic tori are
rational give all (respectively all but 2) stably rational algebraic tori
we mainly use the following function:
This function returns the conjugacy classes of subgroups of the
group MatGroupZClass(r,m,n,k) given as a list of GAP IDs.
It depends on the GAP script written by Hoshi and Yamasaki
crystcat.gap. This script which is available on the
second author's website,
determines the GAP ID of a finite
subgroup $G$ of $\GL_n(\Z)$ where $n=2,3,4$ by the function
. It uses the data of the
book <cit.> to find the crystal class, Q class and Z class of a finite
subgroup of $\GL_n(\Z)$.
It is invoked using
With that tool, for the proposed maximal hereditarily rational subgroups, one can find the
union of all the conjugacy classes of subgroups in terms of their GAP IDs.
One can also find the union of all conjugacy classes of subgroups of the
Dade Groups. By taking the difference of these two sets, we find the
list of all GAP IDs correponding to non-rational tori. One can then
check them against the lists in Hoshi and Yamasaki <cit.> which I do not
reproduce here.
§ STABLE RATIONALITY OF TWO EXCEPTIONAL TORI
In this section, we will show that algebraic $k$ tori whose
character lattices are given by $(J_{A_5/A_4},A_5)$
or $(J_{A_5\times C_2/A_4\times C_2}\otimes \Z^-_{A_5},A_5\times C_2)$ are stably rational
recovering results of Hoshi and Yamasaki in a non-computational way.
We will also give new non-computational proofs showing that the 7 algebraic
$k$ tori of dimension 4 which are retract but not stably rational.
Note that, for a prime $p$, Beneish proved that $J_{S_p/S_{p-1}}$ is flasque equivalent to
as $S_p$ lattices where $N_{S_p}(C_p)=C_p\rtimes C_{p-1}$ is the normaliser
of the cyclic $p$ Sylow subgroup of $S_p$.
We intend to prove a similar result for $A_5$ and $N_{A_5}(C_5)=C_5\rtimes C_2=D_{10}$. The arguments are similar at the start but diverge at a critical point.
This result and the fact that $J_{D_{10}/C_2}$ is quasi-permutation,
will allow us to show that $J_{A_5/A_4}$ is also quasi-permutation.
Note that this is equivalent that for a separable extension $K/k$ of degree 5
with Galois closure $L/k$ such that $\Gal(L/k)=A_5$ and $\Gal(L/K)=A_4$,
the norm one torus $R^{(1)}_{K/k}(\Gm)$ is stably rational.
We intend also to show that $J_{A_5\times C_2/A_4\times C_2}\otimes \Z^-_{A_5}$ is $A_5\times C_2$ quasi-permutation (and the corresponding torus stably rational)
using the result for the corresponding norm one torus and a useful Lemma
due to Florence (see below).
Let $\Z A_{n-1}$ be the $A_{n-1}$ root lattice.
Let $\e_i, i=1,\dots,n$ be a $\Z$-basis permuted by $S_n$
for the permutation lattice
$$\Z A_{n-1}=\ker(\epsilon_n:\Z[S_n/S_{n-1}]\to \Z, \e_i\to 0)=I_{S_n/S_{n-1}}$$
The following lemma was observed by Bessenrodt-Lebruyn <cit.>.
$\Z A_{n-1}\otimes \Z[S_n/S_{n-1}]\cong \Z[S_n/S_{n-2}]$.
It suffices to show that $\Z A_{n-1}\otimes \Z[S_n/S_{n-1}]$
has $\Z$-basis
$$\{(\e_i-\e_j)\otimes \e_i: i\ne j\}$$
as then this basis is clearly transitively permuted by the action of $S_n$
with stabilizer subgroup $S_{n-2}$.
Since a $\Z$-basis of $\Z A_{n-1}\otimes \Z[S_n/S_{n-1}]$ is
given by
$$\{(\e_i-\e_{i+1})\otimes \e_j: 1\le i\le n-1, 1\le j\le n\}$$
we need only show the $\Z$-span of each set contains the other.
$$(\e_i-\e_j)\otimes \e_j=\sum_{k=i}^{j-1}(\e_k-\e_{k+1})\otimes \e_j, i<j$$
$$(\e_i-\e_j)\otimes \e_j=-\sum_{k=j}^{i-1}(\e_k-\e_{k+1})\otimes \e_j, i>j.$$
$$\e_i-\e_{i+1}\otimes \e_j=(\e_j-\e_{i+1})\otimes \e_j-(\e_j-\e_i)\otimes \e_j.$$
The following lemma was proved by Bessenrodt and Lebruyn but unpublished.
It was proved in Beneish <cit.>. Here is a simpler proof.
For $p$ prime,
let $B_p$ be the $W(A_{p-1})=S_p$ lattice
$$B_p=(\Z A_{p-1})^*\otimes \Z A_{p-1}$$
$$B_p\oplus \Z[S_p/S_{p-1}]\cong \Z[S_p/S_{p-2}]\oplus \Z$$
So $B_p$ is stably permutation.
Tensoring the exact sequence
$$0\to \Z A_{p-1}\to \Z [S_p/S_{p-1}]\to \Z\to 0 \qquad (*)$$
by $(\Z A_{p-1})^*$ and noting that
$$(\Z A_{p-1})^*\otimes \Z [S_p/S_{p-1}])^*\cong (\Z A_{p-1}\otimes \Z [S_p/S_{p-1}])^*\cong \Z [S_p/S_{p-2}]$$
since permutation lattices are self-dual,
we see that
\xymatrix{
&\Z \ar[r]^{=}\ar@{>->}[d] &\Z\ar@{>->}[d] \\
B_p \ar@{>->}[r]\ar[d]^{=} & \mbox{pull-back} \ar[r]\ar@{->>}[d] &\ar[d]\Z[S_p/S_{p-1}]\\
B_p \ar@{>->}[r]& \Z[S_p/S_{p-2}] \ar[r] &\Z[A_{p-1}]^*
Since extensions of permutation lattices by permutation lattices
are always split,
we have the exact sequence
$$0\to B_p\to \Z\oplus \Z[S_p/S_{p-2}]\to \Z[S_p/S_{p-1}]\to 0$$
To prove the result, we need only show that
$B_p$ is invertible since extensions of permutation lattices
by invertible lattices are split.
To show that $B_p$ is $S_p$ invertible,
it suffices to show that $B_p$ is $Q$ invertible for each Sylow $q$
subgroup $Q$ of $S_p$.
Let $Q$ be a Sylow $q$ subgroup of $S_p$ where $q\ne p$.
Then $Q$ must fix some $\e_i\in \Z[S_p/S_{p-1}]$.
Then $\Z A_{p-1}\vert_Q\oplus \Z=\Z[S_p/S_{p-1}]\vert_Q$.
In fact $\Z A_{p-1}\vert_Q$ is then $Q$ permutation with
$\Z$-basis $\{\e_i-\e_j: j\ne i\}$.
Dualising we get
$(\Z A_{p-1})^{*}\vert_Q\oplus \Z=\Z[S_p/S_{p-1}]\vert_Q$
and tensoring with $\Z A_{p-1}\vert_Q$
we obtain
$$(B_p)\vert_Q\oplus \Z A_{p-1}\vert_Q\cong \Z[S_p/S_{p-2}]\vert_Q$$
So $B_p\vert_Q$ is $Q$ stably permutation and hence $Q$
It suffices to show that $B_p\vert P$ is $P$ invertible
where $P$ is a Sylow $p$ subgroup of $S_p$. Note that $P\cong C_p$
and $\Z[S_p/S_{p-1}]_{C_p}\cong \Z[C_p]$.
$$0\to (B_p)_P\to (\Z\oplus \Z[S_p/S_{p-2}])_P\to \Z[P]\to 0$$
splits since $\Z[P]$ is free.
So $B_p$ is $S_p$ invertible and then
the sequence
$$0\to B_p\to \Z\oplus \Z[S_p/S_{p-2}]\to \Z[S_p/S_{p-1}]\to 0$$
splits to give us the result.
We will write $I_{G/H}$ for the augmentation ideal
$$I_{G/H}=\ker(\Z[G/H]\to \Z)$$
where $H$ is a subgroup of the finite group $G$.
Let $N=N_{S_p}(C_p)$ be the normaliser of a cyclic Sylow $p$ subgroup $C_p$ of $S_p$.
For a group $G$, a $G$ module $M$ is cohomologically trivial if $\hat{H}^k(H,M)=0$ for all $k$
and for all subgroups $H$ of $G$. Projective $G$ modules are cohomologically
trivial. A result in Brown <cit.> shows that faithful cohomologically trivial $G$ lattices (torsion-free $G$ modules are $G$ projective.
$\Fp I_{S_p/N}:=\Fp\otimes_{\Z}I_{S_p/N}$
is cohomologically trivial where $N$ is the normaliser of a cyclic Sylow $p$
subgroup $C_p$ of $S_p$.
Note that tensoring the augmentation sequence for $S_p/N$
by $\Fp$ is exact.
Since $p$ does not divide $[S_p:N]$,
the $\Fp S_p$ sequence
$$0\to \Fp I_{S_p/N}\to \Fp[S_p/N]\to \Fp\to 0$$
and so
$$\Fp[S_p/N]\cong \Fp I_{S_p/N}\oplus \Fp$$
It suffices to check whether the restrictions to Sylow subgroups
are cohomologically trivial.
For any Sylow $q$ subgroup for $q\ne p$, representations of $\Fp Q$
are completely reducible and so all are projective and hence
cohomologically trivial.
So it suffices to check whether $\Fp I_{S_p/N}\vert P$ is cohomologically trivial for $P=C_p$ a cyclic Sylow $p$-subgroup.
By Mackey's Theorem,
$$\Res^{S_p}_P\Ind^{S_p}_N\Fp=\oplus_{x\in P\backslash S_p/N}\Fp[P/P\cap N^x]$$
Since $P\cong C_p$, $P\cap N^x$ is either $P$ or $\{1\}$.
We claim that the unique double coset with $P\cap N^x=P$ is $PxN=N$.
Suppose $P\cap N^x=P$. Then $P\le N^x$ and so $P^{x^{-1}}\le N$.
But $P$ is the unique $p$-Sylow subgroup of $N$ and so $P^{x^{-1}}=P$
which means $x^{-1}\in N$ and so $x\in N$ and $PxN=N$.
So for all non-trivial double cosets $PxN\ne N$, we have
$P\cap N^x=1$.
This means that
$$\Res^{S_p}_P\Ind^{S_p}_N\Fp=\Fp\oplus (\Fp P)^k$$
for some $k$.
Since $\Fp[S_p]$ satisfies Krull Schmidt, we have that
$$\Res^{S_p}_PI_p\oplus \Fp\cong \Fp \oplus (\Fp P)^k$$
implies that
$I_p\vert_{P}\cong (\Fp P)^k$ is free
and so cohomologically trivial.
A projective $A_5$ lattice is
stably permutation.
By Endo and Miyata <cit.>, the projective class group of $\Z A_5$ is $\Cl(\Z A_5)=0$. But the projective class group is the
kernel of the rank homomorphism $K_0(\Z A_5)\to \Z$ which sends any finitely
generated projective $A_5$ lattice to its rank.
If this class group is zero, it shows that any projective $A_5$ lattice $P$
has the same class in $K_0(\Z A_5)$ as $(\Z A_5)^r$ where $r$ is its rank.
But then by <cit.>, we see that $P\oplus \Z A_5^s\cong (\Z A_5)^{r+s}$
and so $P$ is stably free.
This shows that all cohomologically trivial faithful $A_5$ lattices are stably free and hence
stably permutation.
Projective $G$ lattices are stably permutation for any finite group $G$
with splitting field $\Q$ (e.g. $S_n$). <cit.>, see also <cit.>. This was used in <cit.>.
Note that the splitting field for $A_5$ is $\Q(\sqrt{5})$.
An $\Fp G$ module $M$ is projective if and only if
$\Res^G_PM$ is projective for a Sylow $p$-subgroup $P$ of $G$.
In particular, $\Fp[G/G\cap S_{p-1}]$ is projective as an $\Fp G$
module if $G$ is a transitive subgroup of $S_p$.
Since $d=[G:P]$ is invertible in $\Fp$,
the natural surjection
$\pi:\Ind^G_P\Res^G_PM\to M$
has a section
$$s:M\to \Ind^G_P\Res^G_PM, m\to \frac{1}{[G:P]}\sum_{gP\in G/P}g\otimes g^{-1}m$$
and so $M$ is a direct summand of $\Ind^G_P\Res^G_PM$.
If $\Res^G_PM$ is projective, so is $\Ind^G_P\Res^G_PM$ and hence so
is $M$ by the previous remark.
The converse is clear.
Since $G$ is a transitive subgroup of $S_p$, it has a cyclic $p$ Sylow subgroup
$C_p$ and $\Fp[G/G\cap S_{p-1}]$ restricted to the cyclic $p$ Sylow subgroup
is $\Fp[C_p]$ which is free and hence projective.
If $0\to M\to P\to L\to 0$ and $0\to M'\to Q\to L\to 0$
are 2 short exact sequences of $G$ modules with $P,Q$ $G$-permutation,
then $M\sim M'$ where $\sim$ denotes flasque equivalence.
:The pullback diagram gives two exact
sequences $0\to M\to E\to Q\to 0$ and $0\to M'\to E\to P\to 0$.
Note that the pullback module $E$ is a $G$ lattice (i.e. is $\Z$ torsion
Suppose $G$ is a transitive subgroup of $S_p$
for which all $G$ projective lattices are stably permutation.
Let $M$ be a $G$ lattice such that
there exists a short exact sequence of $G$ modules
$$0\to M\to P\to X\to 0$$
such that
* $P$ is $G$-permutation.
* $X$ $p$ torsion
* $X\otimes \Fp I[G/N]$ is cohomologically trivial
$$M\sim \Ind^G_N\Res^G_N(M)$$
and also
$$M\otimes Q\sim \Ind^G_N\Res^G_N(M\otimes Q)$$
where $N=N_G(C_p)$ is the normaliser of a (cyclic) Sylow $p$ subgroup, $C_p$
and $Q$ is a permutation $G$ lattice, and $\sim$ is flasque equivalence.
Note that the hypothesis of transitivity implies that a Sylow $p$ subgroup
$C_p$ of $G$ is cyclic of order $p$.
So $N=N_G(P)\le N_{S_p}(P)=P\rtimes C_{p-1}$
and so $N=N_G(P)=P\rtimes (C_{p-1}\cap G)$.
Applying $\Ind^G_N\Res^G_N$
we get
$$0\to \Ind^G_N\Res^G_NM\to \Ind^G_N\Res^G_NP\to \Ind^G_N\Res^G_NX\to 0$$
Since $X$ is $p$-torsion,
$\Ind^G_N\Res^G_NX= \Fp[G/N]\otimes_{\Fp}X$.
We have already noted that $\Fp[G/N]=\Fp\oplus \Fp I_{G/N}$.
So $\Ind^G_N\Res^G_NX=X\oplus \Fp I_{G/N}\otimes X$.
So we have the
exact sequence
$$0\to \Ind^G_N\Res^G_NM\to \Ind^G_N\Res^G_NP\to X\oplus \Fp I_{G/N}\otimes X\to 0$$
By hypothesis, $\Fp I_{G/N}\otimes X$ is cohomologically trivial.
$$0\to K\to F\to \Fp I_{G/N}\otimes X\to 0$$
be an exact sequence of $G$ modules with $F$ a free $G$ module.
Then $K$ is also cohomologically trivial and $\Z$-free so
$K$ is $G$ projective by <cit.> and hence stably permutation
by hypothesis.
Adding this sequence to the original we obtain
$$0\to M\oplus K\to P\oplus F\to X\oplus \Fp I_{G/N}\otimes X\to 0$$
and so
$$0\to M\oplus K\to P\oplus F\to X\oplus \Ind^G_N\Res^G_NX\to 0$$
But then by Lemma <ref> and Lemma <ref> and the fact that $\Fp I_{G/N}\otimes X\cong \Ind^G_N\Res^G_NX$, we see that $M\oplus K\sim \Ind^G_N\Res^G_NM$
and since $K$ is stably permutation, we see that $M\sim M\oplus K$
and so $M\sim \Ind^G_N\Res^G_NM$.
For $M\otimes Q$ where $Q$ is $G$-permutation,
we tensor the above sequences by $Q$ to obtain:
$$0\to M\otimes Q\to P\otimes Q\to X\otimes Q\to 0$$
$$0\to \Ind^G_N\Res^G_N(M\otimes Q)\to \Ind^G_N\Res^G_N(P\otimes Q)\to \Ind^G_N\Res^G_N(X\otimes Q)\to 0$$
$$0\to K\otimes Q\to F\otimes Q\to F_p I_{G/N}\otimes X\otimes Q\to 0$$
Noting that $K,F$ are stably permutation and hence so are $K\otimes Q,F\otimes Q$, we add this sequence to the first one to obtain
$$0\to M\otimes Q\oplus K\otimes Q\to P\otimes Q\oplus F\otimes Q\to F_p[G/N]\otimes X\otimes Q\to 0$$
and so by Lemmas <ref> and <ref> as above, we see that
$$M\otimes Q\oplus K\otimes Q\sim \Ind^G_N\Res^G_N(M\otimes Q)$$
and since $K\otimes Q$ is stably permutation, we conclude that
$$M\sim \Ind^G_N\Res^G_N(M\otimes Q)$$
$$J_{S_p/S_{p-1}}\sim \Ind^{S_p}_{N_p}J_{N_p/C_{p-1}}$$
$$J_{S_p/S_{p-1}}\otimes \Z[S_p/A_p]\sim \Ind^{S_p}{N_p}(J_{N_p/C_{p-1}}\otimes \Z[N_p/C_p\times C_{\frac{p-1}{2}}])$$
where $p$ is an odd prime and $N_p=N_{S_p}(C_p)=C_p\rtimes C_{p-1}$.
$$J_{A_5/A_4}\sim \Ind^{A_5}_{D_{10}}\Res^{A_5}_{D_{10}}J_{D_{10}/C_2}$$
and so $J_{A_5/A_4}$ is quasi-permutation.
Note that $S_p$ for an odd prime satisfies the hypotheses of the proposition.
Note that $A_5$ is a transitive subgroup of $S_5$ satisfying the
hypotheses of the proposition and the normaliser of a cyclic
subgroup of order 5 is $D_{10}$, the dihedral group of order 10.
We need only construct an $G$ exact sequence for $M=J_Y$
a transitive $G$ set of size $p$, an odd prime which satisfies the
hypothesis of the proposition.
For any $G$ set $Y$,
there is an inclusion of $G$ lattices
$\alpha: I_{Y}\oplus \Z\to \Z[Y]$
where $\alpha\vert_{I_{Y}}$ is the inclusion and for $n=|Y|$ and $\{\e_i:i=1,\dots,n\}$ a $\Z$-basis of $\Z[Y]$,
$\alpha:\Z\to \Z[Y], 1\to \sum_{i=1}^n\e_i$.
Since $\{\e_1-\e_2,\dots,\e_{n-1}-\e_n,\e_n\}$
is a basis for $\Z[S_n/S_{n-1}]$
is a basis for $I_Y\oplus \Z$,
it is easily checked that
$$0\to I_Y\oplus \Z\to \Z[Y]\to \Z/n\Z\to 0$$
is a short exact sequence of $G$ lattices
with $\Z/n\Z$ having trivial action.
Letting $n=p$, we set $B_Y=I_Y\otimes J_Y=I_Y\otimes (I_Y)^*$
where $Y$ is a transitive $G$ set of size $p$. Then tensoring by $J_{Y}$,
we obtain
$$0\to B_{Y}\oplus J_{Y}\to \Z[Y]\otimes J_Y\to \Fp\otimes J_{Y}\to 0$$
We need to show that $\Fp\otimes J_Y\otimes \F_pI_{G/N}$ is $G$ cohomologically trivial.
Tensoring the following $G$ exact sequence by $\F_pI_{G/N}$:
$$0\to \Fp\to \Fp[Y]\to \Fp\otimes J_Y\to 0$$
we obtain
$$0\to \F_pI_{G/N}\to \Fp[Y]\otimes I_{G/N}\to \Fp\otimes J_Y\otimes I_{G/N}\to 0$$
Now $\Fp[Y]$ is $\Fp[G]$ projective since $Y\cong G/G_Y$ as it is a transitive
$G$ set and $p$ does not divide $|G_Y|$. So $\Fp[Y]\otimes I_{G/N}$ is also
$\Fp G$ projective and so cohomologically trivial as an $\Fp G$ module. We have
already seen that $\F_pI_{G/N}$ is cohomologically trivial as an
$\Fp G$ module. This shows that $(\Fp\otimes J_Y)\otimes (\F_pI_{G/N})$
is cohomologically trivial as an $\Fp G$ module and hence also as a $G$ module as $\Fp G$ is cohomologically trivial as a $G$ module.
Note that we have constructed an exact sequence of the required
form for $B_Y\oplus J_Y$. But since $B_Y$ is $G$ stably
permutation, the fact that $B_Y\oplus J_Y\sim \Ind^G_N\Res^G_N(B_Y\oplus J_y)
=\Ind^G_N\Res^G_N(B_Y)\oplus \Ind^G_N\Res^G_N(J_Y)$ shows that $J_Y\sim
\Ind^G_N\Res^G_N(J_Y)$ since $\Ind^G_N\Res^G_N$ preserve stably permutation
Restricting to $G\le S_p$, we obtain
$$0\to B_G\oplus J_{G/G\cap S_{p-1}}\to P_G\to \Fp\otimes J_{G/G\cap S_{p-1}}\to 0$$
Where $B_G, P_G$ are the restrictions of $B_{S_p/S_{p-1}}$ and $\Z[S_p/S_{p-2}]$
to $G$.
Note the restriction of $J_{S_p/S_{p-1}}$ to $G$ is $J_{G/G\cap S_{p-1}}$
since $G$ is a transitive subgroup of $S_p$.
Tensoring the $\Fp G$ exact sequence
$$0\to \Fp\to \Fp[G/G\cap S_{p-1}]\to \Fp\otimes J_{G/G\cap S_{p-1}}\to 0$$
by $\F_pI_{G/N}$ is exact.
We obtain
$$0\to \F_pI_{G/N}\to \Fp[G/G\cap S_{p-1}]\otimes I_{G/N}\to \Fp\otimes J_{G/G\cap S_{p-1}}\otimes \F_pI_{G/N}\to 0$$
Since $\Fp[G/G\cap S_{p-1}]$ is $\Fp G$ projective, so is
$\Fp[G/G\cap S_{p-1}]\otimes \F_pI_{G/N}$. We also have that
$\F_pI_{G/N}$ is cohomologically trivial.
Then $B_G$ is stably permutation as the restriction of a stably permutation
lattice, $P_G$ is permutation as the restriction of a permutation lattice
and we
$$J_{G/G\cap S_{p-1}}\sim B_G\oplus J_{G/G\cap S_{p-1}}\sim \Ind^G_N(B_N\oplus J_{N/N\cap S_{p-1}})
\sim \Ind^G_{N}J_{N/N\cap S_{p-1}}$$
since induction and restriction preserve stably permutation lattices.
For us,
$G=A_5, G\cap S_4=A_4, N=N_G(P)=D_{10}$ and $N\cap S_4=C_2$
as required.
A flasque resolution of $J_{S_p/S_{p-1}}$ is given
$$0\to J_{S_p/S_{p-1}}\to \Z[S_p/S_{p-2}]\to J_{S_p/S_{p-1}}^{\otimes 2}\to 0$$
This is well-known and was proven in <cit.> using somewhat different
As we require this result, we give a quick self-contained proof.
It suffices to show that $J_{S_p/S_{p-1}}^{\otimes 2}$ is invertible when restricted to Sylow $q$ subgroups of $S_p$. Note that any Sylow $q$ subgroup of $S_p$
for $q\ne p$, must fix $\e_i$ for some $i=1,\dots,p$, where
$\Z[S_p/S_{p-1}]$ has $\Z$-basis $\e_i,i=1,\dots,p$. Then $I_{S_p/S_{p-1}}$
is a permutation lattice for the Sylow $q$ subgroup with $\Z$-basis
$\e_i-\e_j, j\ne i$. So its dual $J_{S_p/S_{p-1}}$ and $J_{S_p/S_{p-1}}^{\otimes 2}$
must also be permutation as Sylow $q$ lattices, $q\ne p$.
As for the Sylow $p$ subgroup $C_p$, a result of Endo-Miyata shows that
for cyclic $p$ groups, every flasque lattice is invertible.
Since $(J_{S_p/S_{p-1}}^{\otimes 2})^*=I_{S_p/S_{p-1}}^{\otimes 2}$, we need only
check that $I_{S_p/S_{p-1}}^{\otimes 2}\vert_{C_p}=I_{C_p}^{\otimes 2}$ is coflasque.
But this follows from the fact that $I_{C_p}^{C_p}=0$
and the exact sequence
$$0\to (I_{C_p})^{\otimes 2}\to \Z[C_p]^{p-1}\to I_{C_p}\to 0$$
Recall the following useful lemma from Florence <cit.>. The original was
stated for lattices for a profinite group. The proof for $G$ lattices
follows immediately.
Let $A_i,B_i,C_i, i=1,2$ be $G$ lattices fitting into
two exact sequences
$$0\to A_i\stackrel{j_i}{\to}B_i\stackrel{\pi_i}{\to}C_i\to 0$$
Assume we are given ax $G$ module map $s_i:C_i\to B_i$, and $d_1,d_2$ two coprime integers, such that $\pi\circ s_i=d_i\id$, $i=1,2$. Let $A_3=A_1\otimes A_2$,
$$B_3=(B_1\otimes B_2)\oplus (C_1\otimes C_2),\qquad
C_3=(C_1\otimes B_2) \oplus (B_1\otimes C_2).$$
Then there is an exact sequence
$$0\to A_3\stackrel{j_3}{\to} B_3\stackrel{\pi_3}{\to} C_3\to 0$$
together with a $G$ module map $s_3:C_3\to B_3$ such that
$\pi_3\circ s_3=d_1d_2\id$.
Observe that this lemma is very handy for showing that
in certain circumstances, the tensor product of two quasi-permutation lattices
is again quasi-permutation. Indeed, with the hypotheses of the Lemma
and the additional assumption that $B_i,C_i,i=1,2$ are all permutation lattices,
then $B_3,C_3$ are also permutation, since the tensor product of permutation
lattices is permutation and the direct sum of permutation lattices is
permutation. Indeed, in Theorem 2.2 of the same paper, he
shows that the tensor product of augmentation ideals of $G$ sets of
pairwise relatively prime order is quasipermutation as a consequence of this
Lemma. He then goes on to give a simple proof of Klyachko's result that
a $k$ torus with character lattice isomorphic to the tensor product
of two augmentation ideals for $G$ sets of relatively
prime order is rational.
We will apply this Lemma to prove that the $A_5\times C_2$ lattice $J_{A_5\times C_2/A_4\times C_2}\otimes \Z^-_{A_5}$ is quasi-permutation.
If $X$ is a $G$ set of order $n$, for the exact sequence
$$0\to \Z\to \Z[X]\stackrel{\pi}{\to} J_X\to 0,$$
there exists a $G$-equivariant map $s:J_X\to \Z[X]$
$s(\pi(\e_x))=n\e_x-\sum_{y\in X}e_y$ such that $\pi\circ s=n\id$.
Tensoring this sequence by $J_X$, we obtain an exact sequence
$$0\to J_X\to \Z[X]\otimes J_X\stackrel{1\otimes \pi}{\to} J_X^{\otimes 2}\to 0$$
and we have a $G$-equivariant map $1\otimes s:J_X^{\otimes 2}\to \Z[X]\otimes J_X$
such that $(1\otimes \pi)\circ (1\otimes s)=n\id$.
If we then modify this sequence for some $G$ lattice $P$ to
$$0\to J_X\to \Z[X]\otimes J_X\oplus P\stackrel{1\otimes \pi\oplus \id}{\to}J_X^{\otimes 2}\oplus P\to 0$$
we have a $G$-equivariant map $s'=1\otimes s\oplus n\id$
such that $\pi'\circ s=n\id$ where $\pi'=(1\oplus \pi)\oplus \id$.
Recall that:
<cit.> Let $K/k$ be a separable extension of prime degree $p$, $p\ge 5$ with Galois closure $L/k$ having Galois group $\Gal(L/k)=C_p\rtimes C_{p-1}$
and $H=\Gal(L/K)=C_{p-1}$. Then $R^{(1)}_{K/k}(\Gm)$ is not a stably
rational variety.
Note that this is equivalent to $J_{C_p\rtimes C_{p-1}/C_{p-1}}$ is not stably permutation as an $C_p\rtimes C_{p-1}$ lattice if $p$ is a prime at least 5.
The $A_5\times C_2$ lattice $ J_{A_5\times C_2/A_{4}\times C_2}\otimes \Z^-_{A_5}$ is
A flasque resolution for $J_X$ for the transitive $A_5$ set $X=A_5/A_4$ can be given by
$$0\to J_X\to \Z[X]\otimes J_X\to J_X^{\otimes 2}\to 0$$
As we have seen, $\Z[X]\otimes J_X$ is permutation as the dual of
$\Z[X]\otimes I_X$. We have shown that $J_{A_5/A_4}$ is quasipermutation.
This implies that $J_X^{\otimes 2}$ is stably permutation so that there
exist permutation $A_5$ lattices $P$ and $Q$ such that
$J_X^{\otimes 2}\oplus P\cong Q$.
Adjusting this sequence to
$$0\to J_X\to \Z[X]\otimes J_X\oplus P\stackrel{\pi_1}{\to} J_X^{\otimes 2}\oplus P\to 0$$
we see from the remark that there exists an $A_5$ equivariant map
$$s_1:J_X^{\otimes 2}\oplus P\to \Z[X]\otimes J_X\oplus P$$
such that $\pi_1\circ s_1=5\id$.
Inflating this sequence from $A_5$ to $C_2\times A_5$ gives us the
same statement for the $C_2\times A_5$ set $X=(C_2\times A_5)/(C_2\times A_4)$.
We remark that the sign lattice $\Z^{-}_{A_5}$ for $C_2\times A_5$ is in
fact the augmentation ideal $I_Y$ for $Y=(C_2\times A_5)/A_5$
and so its augmentation sequence
$$0\to I_Y\to \Z[Y]\stackrel{\pi_2}{\to}\Z\to 0$$
admits an $C_2\times A_5$ equivariant map $s_2:\Z\to \Z[Y]$ such that
$\pi_2\circ s_2=2\id$.
Note that our two sequences have second and last terms permutation lattices.
Then Florence's Lemma gives us a quasi-permutation sequence
for $J_{A_5\times C_2/A_{4}\times C_2}\otimes \Z^-_{A_5}==J_X\otimes I_Y$ as required.
$\Z[F_{20}/D_{10}]\otimes J_{F_{20}/C_4}^{\otimes 2}$ is $\Z F_{20}$
stably permutation. $\Z[S_5/A_5]\otimes J_{S_5/S_4}^{\otimes 2}$ is $\Z S_5$
stably permutation.
Since for $X=F_{20}/C_4$,
$$0\to J_X\to J_X\otimes \Z[X]\to J_X^{\otimes 2}\to 0$$
is a flasque resolution for $J_X$ as an $F_{20}$ lattice
and $J_X^{\otimes 2}$ is in fact invertible,
we see that
$$0\to \Z[F_{20}/D_{10}]\otimes J_X\to \Z[F_{20}/D_{10}]\otimes \Z[X] \to \Z[F_{20}/D_{10}]\otimes J_X^{\otimes 2}\to 0$$
is a flasque resolution for $\Z[F_{20}/D_{10}]\otimes J_X$
with $\Z[F_{20}/D_{10}]\otimes J_X^{\otimes 2}$ also invertible.
In fact, $\Z[F_{20}/D_{10}]\otimes J_X\cong \Ind^{F_{20}}_{D_{10}}\Res^{F_{20}}_{D_{10}}J_X=\Ind^{F_{20}}_{D_{10}}J_{D_{10}/C_2}$
and similarly, $\Z[F_{20}/D_{10}]\otimes J_X^{\otimes 2}\cong \Ind^{F_{20}}_{D_{10}}J_{X}^{\otimes 2}$.
Since $\rho(J_{D_{10}/C_2})=0$, this implies that $J_{D_{10}/C_2}^{\otimes 2}$ is stably
permutation. Then $\Z[F_{20}/D_{10}]\otimes J_X^{\otimes 2}\cong \Ind^{F_{20}}_{D_{10}}\Res^{F_{20}}_{D_{10}}J_X^{\otimes 2}=\Ind^{F_{20}/D_{10}}J_{D_{10}/C_2}$ is also stably permutation. As we have shown earlier that $\Z[S_5/A_5]\otimes J_{S_5/S_4}\sim \Ind^{S_5}_{F_{20}}(\Z[F_{20}/D_{10}]\otimes J_{F_{20}/C_4})$, we similarly see that
$\Z[S_5/A_5]\otimes J_{S_5/S_4}$ is stably permutation.
maximal $S_5\times C_2$ lattice $J_X\times I_Y$ where $Y=C_2\times S_5/S_5$ and
$X=C_2\times S_5/C_2\times S_4$ is quasi-invertible but not quasi-permutation.
From the remark, we have observed that the flasque resolution
for $J_X$ given by
$$0\to J_X\to \Z[X]\otimes J_X\stackrel{1\otimes \pi}{\to} J_X^{\otimes 2}\to 0$$
admits an equivariant map $1\otimes s:J_X^{\otimes 2}\to \Z[X]\otimes J_X$
such that $(1\otimes \pi)\circ (1\otimes s)=5\id$
and the quasi-permutation resolution for $I_Y=\Z_-$
given by
$$0\to I_Y\to \Z[Y]\stackrel{\pi_2}{\to} \Z\to 0$$
which admits an equivariant map $s_2:\Z\to \Z[Y]$ such that
$\pi_2\circ s_2=2\id$.
So the 2 sequences satisfy the hypotheses for Florence's Lemma
and we obtain an exact sequence
$$0\to J_X\otimes I_Y\to \Z[X]\otimes J_X\otimes \Z[Y]\oplus J_X^{\otimes 2}\otimes \Z\to
\Z[X]\otimes J_X\otimes \Z\oplus J_X^{\otimes 2}\otimes \Z[Y]\to 0$$
Recall that $J_X^{\otimes 2}$ is invertible. So there exists a lattice
$F$ such that $J_X^{\otimes 2}\oplus F=Q$ for some permutation lattice $Q$.
Since $L\otimes \Z=L$, we may adjust our above sequence to
$$0\to J_X\otimes I_Y\to \Z[X]\otimes J_X\otimes \Z[Y]\oplus Q\to \Z[X]\otimes J_X\oplus J_X^{\otimes 2}\otimes \Z[Y]\oplus F\to 0.$$
We note that the tensor product of an invertible lattice with a permutation
lattice is invertible and the direct sum of invertible lattices is invertible.
Then the above sequence has middle term permutation and last term invertible,
which shows that $J_X\otimes I_Y$ is quasi-invertible.
Note that $J_X\otimes I_Y\vert F_{20}$ is $J_{F_{20}/C_4}$ which is not quasi-permutation. So $J_X\otimes I_Y$ is not quasi-permutation.
The same sequence applied to the $S_5$ sets $Y=S_5/A_5$ and $X=S_5/S_4$
shows that $I_Y\otimes J_X$ is quasi-invertible as an $S_5$ lattice.
Note though that the term $\Z[Y]\otimes J_X^{\otimes}$ is now stably permutation
and so the flasque class of $I_Y\otimes J_X$ is $\rho(I_Y\otimes J_X)=[F]=-[J_X^{\otimes 2}]=-\rho(J_X)\ne 0$.
The same argument applied to the $F_{20}$ sets $Y=F_{20}/D_{10}$ and $X=F_{20}/C_4$
shows that the flasque class of $\Z_-\otimes J_{F_{20}/C_4}=I_Y\otimes J_X$
is $-\rho(J_{F_{20}/C_4})\ne 0$. (In fact the restriction of the
above sequence to $F_{20}$ gives this fact.)
$\rho(J_{S_5\times C_2/S_4\times C_2}\otimes \Z^-_{S_5})=-\rho(J_{S_5/S_4})\ne 0$
$\rho(J_{F_{20}\times C_2/C_4\times C_2}\otimes \Z^-_{F_{20}})=-\rho(J_{F_{20}/C_4})\ne 0$
Note that the Corollary was obtained computationally in <cit.>.
By results of Hoshi and Yamasaki in <cit.>, there are 7 conjugacy classes of finite subgroups of $\GL_4(\Z)$ which correspond to
retract but not stably rational algebraic tori. It turns out that the DadeGroup(4,7) with GAP ID [4,31,7,2] and corresponding lattice
$(\Lambda(A_4),\Aut(A_4))$ is one such example. All but one of the other such
examples can be seen to be subgroups (not just conjugate subgroups) of this group. Their lattices correspond to restrictions of $\Lambda(A_4)$ to the appropriate subgroup. The justifications are
similar to those in the previous section and are omitted.
The following subgroups with given GAP ID of the DadeGroup(4,7) with GAP ID [4,31,7,2]
and lattice
$$(\Lambda(A_4),\Aut(A_4))=(J_{S_5\times C_2/S_4\times C_2}\otimes \Z^-_{S_4},S_5\times C_2)$$
have lattices given by:
* GAP ID [4,31,1,3]: $(J_{F_{20}/C_4},F_{20})$.
* GAP ID [4,31,1,4]: $(J_{F_{20}/C_4}\otimes \Z^-_{D_{10}},F_{20})$
* GAP ID [4,31,2,2]: $(J_{F_{20}\times C_2/C_4\times C_2}\otimes \Z^-_{F_{20}},F_{20}\times C_2)$.
* GAP ID [4,31,4,2]: $(J_{S_5/S_4},S_5)$.
* GAP ID [4,31,5,2]: $(J_{S_5/S_4}\otimes \Z^-_{A_5},S_5)$.
All of these correspond to retract rational tori which are not stably rational.
Note that those observations for [4,31,1,3] and [4,31,4,2] were made in Hoshi
Yamasaki. It was shown above that $(J_{S_p\times C_2/S_{p-1}\times C_2}\otimes \Z^-_{S_p},S_p\times C_2)$ is quasi-invertible for $p$ prime (or equivalently the corresponding norm one torus is retract
rational). Restrictions of quasi-invertible
lattices are quasi-invertible. So all the lattices on the list above are
quasi-invertible with corresponding tori retract rational.
By the result above, we know that
$(J_{F_{20}/C_4},F_{20})$ is not quasi-permutation. So $\rho(J_{F_{20}/C_4})\ne 0$. Since every lattice in the list above
contains either $(J_{F_{20}/C_4},F_{20})$ or $(J_{F_{20}/C_4}\otimes \Z^-_{D_{10}},F_{20})$,
we see that the result will follow since
$$\rho(J_{F_{20}/C_4}\otimes \Z^-_{D_{10}})=-\rho(J_{F_{20}/C_4})\ne 0$$
We will now address the remaining case of a retract but not stably rational
algebraic $k$ torus of dimension 4.
For any cyclic group $C_n$ and given a primitive
$n$th root of unity, there is a natural
$C_n$ lattice given by $\Z[\omega_n]$, which is the ring of integers of the
field $\Q(\omega_n)$. The generator of $C_n$, $\sigma_n$ then acts
as multiplication by $\omega_n$ on $\Z[\omega_n]$.
As a ring, $\Z[\omega_n]\cong \Z[X]/(\Phi_n(X))$ where $\Phi_n(X)$ is the
$n$th cyclotomic polynomial of degree $\varphi(n)$, the Euler phi function of $n$.
With respect to the basis $1,\omega_n,\dots,\omega_n^{\varphi(n)-1}$,
the matrix of $\sigma_n$ acting on $\Z[\omega_n]$ is the
companion matrix of $\Phi_n(X)$.
The lattice with GAP ID [4,33,2,1] is quasi-invertible
but not quasi-permutation and hence the associated algebraic
torus is retract but not stably rational.
$$G=C_3\rtimes C_8=\langle \sigma,\tau: \sigma^3=1,\tau^8=1,\tau\sigma\tau^{-1}=\sigma^{-1}\rangle$$
Note that this group has centre
$\Z(G)=\langle \tau^2\rangle\cong C_4$ and $P=\langle \sigma\rangle\cong C_3$
is normal.
The group $K=\langle \tau\rangle\cong C_8$ is a Sylow 2-subgroup with
3 distinct conjugates $K=\langle \tau\rangle, K^{\sigma}=\langle \tau\sigma\rangle,K^{\sigma^{2}}=\langle \tau\sigma^2\rangle$ which pairwise intersect
in $\langle \tau^2\rangle$.
There are then $3\varphi(8)=12$ elements of order 8.
Since $\tau^2\sigma$ is an element of order 12 as $\tau^2$ is central,
and the group has order 24, we see that the only proper subgroups of
$G$ are cyclic; there is a unique normal subgroup of each order dividing 12
and the only non-normal subgroups are the 3 conjugates of $K$ which
are cyclic of order 8.
The generators of this subgroup given by GAP for the group with GAP ID
It is easy to check that $A$ has order 8 and $B$ has order 12.
One could then replace the generators by $A$ and $C=B^4$.
and check that $ACA^{-1}=C^{-1}$ which shows that
$\rho:G\to \GL(4,\Z)$ with $\rho(\sigma)=C$ and $\rho(\tau)=A$
gives a faithful representation of the group.
Let $L$ be the rank 4 lattice with the action of $G$ induced by $\rho$.
Then, since the minimal polynomial of $C$ is $x^2+x+1$, we see that $L$
restricted to the Sylow 3 subgroup $H=\langle \sigma\rangle$ is isomorphic
to $\Z[\omega_3]^2\cong I_{C_3}^2$. Since $A$ is a matrix of order 8 in $\GL_4(\Z)$, it must have minimal polynomial $x^4+1$
and $L$ restricted to $K\cong C_8$ must be isomorphic to $\Z[\omega_8]$.
For ease of computations, as we know the rational canonical form
of $C$ is $\diag(C_{x^2+x+1},C_{x^2+x+1})$, where $C_f$ is the companion matrix
of the monic polynomial $f$, we can conjugate the generators
by the change of basis matrix from $\{\e_1,\dots,\e_4\}$ to $\{\e_1,C\e_1,\e_3,C\e_3\}$, i.e. by
and obtain nicer generators
Note that $L_H$ is quasipermutation as a $H$
lattice as the direct sum of 2 quasipermutation lattices
and $L_K$ is quasi-permutation as it is sign-permutation.
This implies that $L$ is quasi-invertible.
Now, we need to construct a flasque resolution of $L$.
Note that $L\cong L^*$ as a $G$ lattice. This means we can construct a coflasque resolution of $L$
and dualise.
It turns out that this is easy. Any non-trivial subgroup $H$ of $G$ contains
$\langle \tau^4\langle \cong C_2$ or $P\cong C_3$.
Now $\rho(\tau^4)=-I_4$ shows that $L^{\langle \tau^4\rangle}=0$
and $L^{C_3}=(IC_3\oplus IC_3)^{C_3}=0$. So for any non-trivial subgroup
$H$ of $G$, we have $L^H=0$.
So to find a coflasque resolution of $L$, we need only find
a permutation lattice which surjects onto $L$.
Since $L$ is an irreducible $G$ lattice,
there is a surjection $\pi:G\to L$. If $C=\ker(\pi)$,
$$0\to C\to G\to L\to 0$$
is a coflasque resolution and its dual is
a flasque resolution of $L\cong L^*$.
Let $F=C^*$.
Then we have the flasque resolution
$$0\to L\to G\to F\to 0$$
Since $G$ is free, we have $\hat{H}^0(H,F)\cong \hat{H}^1(H,L)$ for any subgroup $H$ of
Now, $\hat{H}^1(P,L)\cong \hat{H}^1(C_3,IC_3)^2\cong (\Z/3\Z)^2$
Now, $L_Q=\Z[\omega_8]\cong \Ind^{C_8}_{C_2}\Z_-$ is a sign-permutation lattice.
So $H^1(Q,L)=H^1(Q,\Ind^Q_{C_2}\Z_-)\cong H^1(C_2,\Z_-)=\Z/2\Z$.
Suppose $F\oplus P\cong Q$ for some permutation $G$ lattices $P$
and $G$.
The conjugacy classes of subgroups of $G$
are $H_1=1,H_2=C_2,H_3=C_3,H_4=C_4,H_6=C_6,H_8=C_8,H_{12}=C_{12},H_{24}=G$,
one for every divisor $d$ of $|G|=24$.
$P=\oplus_{d|24}a_d\Z[G/H_d]$, $Q=\oplus_{d|24}b_d\Z[G/H_d]$
We observe that
Indeed, since $L\cong L^*$ as $G$ lattices, $\hat{H}^{1}(G,L)\cong
\hat{H}^{-1}(G,L^*)=\ker_L(N_G)/I_G(L)$.
Since $H\cong C_3$ is a normal subgroup and is generated by $\sigma$
with image having minimal polynomial $1+x+x^2$, it is clear that
$\ker_L(N_H)=L$. As $H$ is normal in $G$, $N_G=\sum_{g\in G/H}gN_H$, and
so $\ker_L(N_G)=L$. It is then easy to check that $I_G(L)=L$ as
$I_G(L)$ contains $I_H(L)+I_K(L)=L$.
Looking at $\hat{H}^0$, and observing that $\hat{H}^0(G,\Z[G/H_i])\cong \Z/|H_i|\Z$,
we see that we have
where $x_d=a_d-b_d\in \Z$.
We may decompose $\Z/d\Z$ into prime powers
and obtain
$x_2+x_6=0, x_4+x_{12}=0,x_8+x_{24}=0, x_3+x_6+x_{12}+x_{24}=0$
from the coefficients of $\Z/2\Z, \Z/4\Z, \Z/8\Z, \Z/3\Z$ respectively.
Now restrict the isomorphism $F\oplus P\cong Q$
to $Q=C_8$.
Note that $\hat{H}^0(F)=\Z/2\Z$,
By Mackey's Theorem,
$$\Res^G_{Q}\Ind^G_{H_d}\Z=\oplus_{QxH_d\in Q\G/H_d}\Z[Q/Q\cap H_d^x]$$
For all $d\ne 8$, $H_d$ is a normal subgroup,
and so $QxH_d=QH_dx$ is a coset of $QH_d$ in $G$
and $H_d^x=H_d$ so we have,
$$\Res^G_Q\Z[G/H_d]\cong \Z[Q/Q\cap H_d]^[G:QH_d]$$
$\Res^G_Q\Z[G/H_1]=\Z[Q]^3, \Res^G_Q\Z[G/H_2]=\Z[Q/H_2]^3,
\Res^G_Q\Z[G/H_3]=\Z[Q], \Res^G_Q\Z[G/H_4]=\Z[Q/H_4]^3, \Res^G_Q\Z[G/H_6]=\Z[Q/H_2], \Res^G_Q\Z[G/H_{12}]=\Z[Q/H_4], \Res^G_Q\Z[G/H_{24}]=\Z[Q/Q]=\Z$.
For $H_8=Q$, $G=H_81H_8\cup H_8\sigma H_8=H_8\cup H_8\sigma H_8$
and $H_8\cap H_8^{\sigma}=H_4$.
$\Res^G_Q\Z[G/H_8]=\Z\oplus \Z[Q/H_4]$.
Taking $\hat{H}^0(Q,\cdot)$ of both sides and comparing coefficients
of $\Z/2\Z, \Z/4\Z, \Z/8\Z$, we obtain
$$3x_2+x_6=1, 3x_4+x_8+x_{12}=0, x_{24}+2x_8=0$$
We get a contradiction because we have $x_2+x_6=0$ and $3x_2+x_6=1$
implies $2x_2=1$ but $x_2\in \Z$.
If $L$ is a $G$ lattice where $G$ is a finite 2 group,
there exists a
$\Q L=\Q B_n$
Let $M$ be a continuous $G_k$ lattice
with continuous representation $\rho_0:G_k\to \GL(M)$.
Suppose that another continuous representation
$\rho$ of $G_k$ has image in $\GL(M)^k\rtimes S_k$
where $S_k$ acts on $\GL(M)^k$ by permuting the copies of $\GL(M)$.
Suppose that the algebraic tori corresponding to all subgroups
of $\rho_0(G_k)$ are rational. Then the algebraic tori corresponding
to all subgroups of $\rho(G_k)$ are rational.
Let $\rho(G_k)=G$. Let $\cN=\ker(\rho)$. Then $L=k_s^\cN$ is a Galois extension of $k$ with Galois group $G$.
$N=\rho(G_k)\cap \GL(M)^k$ is a normal subgroup of
$G$. Then $\rho^{-1}(N)=\cH$ is a normal subgroup of $G_k$ containing
$\cN$ and $\cH/\cN\cong N$. $L^N=(k_s^{\cN})^{\cH/\cN}=k_s^{\cH}$.
Note that $G_{L^N}=G_{k_s^{\cH}}=\cH$ and that $\rho(G_{k_s^{\cH}})=\cH/\cH\cap \cN=
\cH/\cN\cong N$.
That is, there is a $L^N$ torus with continuous representation
$\rho$ restricted to $G_{L^N}$ with image $\rho(G_{L^N})\subset
\GL(M)^k$. By the previous proposition, since the algebraic
$L/L^N=k_s^{\cN}/k_s^{\cH}$ is Galois with Galois group $N\cong \cH/\cN$
and $L^N/k$ is Galois with Galois group $G/N$.
Then $\rho_{G_{L^N}}=(\rho_1,\dots,\rho_k)$ where $\rho_i:G_{L^N}\to \GL(M_i)$
and we write the $i$th copy of $M$ as $M_i$.
The continuous $G_k$ lattice correponding to $\rho$ may be written
as $\oplus_{i=1}^kM_i$ where $M_i$ is the $i$th copy of $M$.
Then the $k$ torus has function field $L(\oplus_{i=1}^kM_i)^G=
where $L_i=k_s^{\ker(\rho_i)}$ is Galois over $L^N$ with Galois group $N_i=G_{L^N}/\ker(\rho_i)$ for each $i=1,\dots,k$
as shown in the previous proposition and induction.
By assumption, each $L_i(M_i)^{N_i}$ is rational over $L^N=L_i^{N_i}$.
Note that $G$ acts by permuting the lattices $M_1,\dots,M_k$.
Assume without loss of generality that $G$ acts transitively on
$\{M_1,\dots,M_k\}$. Let $g_i\in G$ be such that $g_i(M_1)=M_i$.
Let $f_1,\dots,f_r$ be an $L^N$ transcendence basis of $L_1(M_1)^{N_1}$ over
$L^N$. We claim that $\{g_i(f_j): 1\le i\le k, 1\le j\le r\}$
is a transcendence basis for $L(\oplus_{i=1}^kM_i)^N$ over $L^N$
which is permuted by the action of $G/N$.
It is clear that the above is a transcendence basis.
Let $g\in G$. Then $gg_i(f_j)=g_lh(f_j)$ if $gg_i(M_1)=M_l$ for some $h$
stabilizing $M_1$. But $hf_j=f_j$ for any such $h$ and $j$
since $f_j\in L_1(M_1)^{N_1}$. Then $(L(\oplus_{i=1}^kM_i)^N)^{G/N}$
is rational over $(L^N)^{G/N}=L^G=k$ as required.
An algebraic torus corresponding to a $G$ lattice
$L$ which is the restriction of $(G_2^r,W(G_2)^r\rtimes S_r)$
to a subgroup is rational.
The algebraic torus corresponding to $(G_2,W(G_2))$
is rational by an argument of Voskresenskii.
He shows that any subgroup of $W(G_2)$ is conjugate to a subgroup of $\Aut(\PP^1\times \PP^1)$
so that the associated algebraic torus is a $k$ form of $\PP^1\times \PP^1$.
Since all $k$ forms of $\PP^1\times \PP^1$ with a point are rational,
we are done.
But then an algebraic torus corresponding to any
subgroup of $W(G_2)^r\rtimes S_r$ is rational by the previous proposition.
(Klyachko, Florence)
An algebraic torus with character lattice $I_X\otimes I_Y$
for $G_k$ sets $X$ and $Y$ with $\gcd(|X|,|Y|)=1$
is rational.
The subgroup with GAP ID [4,27,4,1] is isomorphic to $D_{10}\times C_2$.
It is conjugate to a subgroup of [4,31,6,2]
and so corresponds to a lattice $J_{
So the lattice determined by [4,27,4,1] is the
However the restriction of this lattice to $D_{10}\times C_2$ is that determined by [4,27,4,1]. This lattice is hereditarily rational.
We will show that the lattice determined by the [4,27,4,1] is isomorphic to
$(\Z A_4,\Aut(A_4))$ restricted to $D_{10}\times C_2$ and so is hereditarily
We will then show that all the tori corresponding to lattices which
are not ocn
From Hoshi and Yamasaki's list of conjugacy classes of finite subgroups
of $\GL(4,\Z)$ corresponding to stably rational tori, the following
are maximal elements of this list;
The lattices corresponding to Dade groups are
DadeGroup(4,1) with GAP ID [4,20,22,1]
corresponds to $(\Z B_2\oplus G_2,W(B_2)\times W(G_2))$.
The generators of this group are
$$B_1=\left[\begin{array}{rr}1&0\\-1&-1\end{array}\right], B_2:=\left[\begin{array}{rr}-1&-1\\1&0\end{array}\right], A_1:=\diag(1,-1),A_2:=\left[\begin{array}{rr}0&1\\-1&0\end{array}\right]$$
Since $B_2^3=A_2^2=-I_2$, we may replace this
set of generators by
For $G=DadeGroup(4,1)$, $M_G=M_1\oplus M_2$
where $M_1=\Z\e_1\oplus \Z\e_2$ and $M_2=\Z\e_3\oplus \Z\e_4$
are both $G$ invariant. In fact $G=W(B_2)\times W(G_2)$.
where $W(B_2)=\langle \diag(A_1,I_2),\diag(A_2,I_2)\rangle$
acts on $M_1$ as $\Z B_2$ and on $M_2$ trivially
and $W(G_2)=\langle \diag(I_2,B_1),\diag(I_2,B_2^2),\diag(I_2,-I_2)\rangle$
acts on $M_2$ as $G_2$ and on $M_1$ trivially.
This is because the lattice corresponding to $H=\langle B_1,B_2^2\rangle$
is $\Z H\cdot \e_1=J_{S_3/S_2}$ since the $H$ orbit of $\e_1$ is $\{\e_1,\e_2,-\e_1-\e_2\}$ and the stabiliser subgroup is $H_{\e_1}=\langle B_1\rangle\cong S_2$.
The corresponding algebraic torus
has function field $L(\oplus_{i=1}^kM_i)^G\cong (L(\oplus_{i=1}^kM_i)^N)^{G/N}$.
Let $N_i$
\end{document}
%We then use GAP to find the GAP IDs of conjugacy classes of subgroups
%of these subgroups corresponding to maximal hereditarily rational lattices.
%Although we will not put this in the paper, we do the following.
%for x in Bad3 do if Intersection(SubConjClass(x[1],x[2],x[3],x[4]),minbad3)=[] then Add(minbad3,x); fi; od;
This produces a list of all GAP IDs for dimension 3 groups in All3,
all those which are rational in Good3, all those which are not in Good3 in Bad3.
The minimal element of Bad3 are given in minbad3. We obtain the same
minimal bad elements as do Kunyavskii and Hoshi,Yamasaki with
GAP IDs [3,3,1,3],[3,3,3,4],[3,4,3,2].
We will not redo the proof given by Kunyavskii to show that these 3 minimal
groups correspond to tori which are not retract rational. We note that our
method for the dimension 4 groups would work here as well and will make
more comments at that time to compare our methods.
Note that Kunyavskii refers to Tahara's labelling of the conjugacy classes
of finite subgroups of $\GL_3(\Z)$.
Here however, we will identify the lattices corresponding to the
groups with GAP IDs [3,3,1,3], [3,3,3,4], [3,4,3,2] so that we will be able
to recognise them up to isomorphism later.
The group with GAP ID [3,3,1,3] is isomorphic to $C_2\times C_2$
and has generators
As we have seen earlier, $\rho_3((13)(24))=A_3$ and $\rho_3((12)(34))=A_4$.
So we have the restriction of $\rho_3:S_4\to \GL_3(\Z)$ determined
by $J_{S_4/S_3}$
determined by
$\langle (13)(24),(12)(34)\rangle\cong C_2\times C_2$.
But this is then $J_{C_2\times C_2}$ since this subgroup is a transitive subgroup
of $S_4$ with trivial intersection with $S_3$.
Note that the observation that this lattice corresponded to $J_{C_2\times C_2}$
was made in Kunyavskii's paper (in different language). It is well known that a torus
with character lattice $J_{C_2\times C_2}$ is not retract rational.
The group with GAP ID [3,3,3,4] is isomorphic to $C_2^3$ and has generators
As we have already seen that the lattice corresponding to $\langle A_3,A_4\rangle$ is $J_{C_2\times C_2}$, the lattice corresponding to $\langle A_1^T,A_2^T\rangle$
is $I_{C_2\times C_2}$ and so the lattice corresponding to [3,3,3,4] is
$I_{C_2^3/C_2}\otimes \Z^-_{C_2^2}$ where the last copy of $C_2$ acts trivially on
$I_{C_2^3/C_2}$ and acts by $-1$ on $\Z^-_{C_2^2}$.
The group with GAP ID [3,4,3,2] is isomorphic to $C_2\times C_4$.
For some reason, GAP lists its generators $\{-I_3,D,D^2\}$
It is easy to see that we may generate this group with $\{-I_3,-D\}$.
The lattice corresponding to $H=\langle -D\rangle$ is isomorphic to
$I_{C_4}\cong J_{C_4}$ (note that lattices for cyclic groups are self-dual)
since the orbit of $\e_1$ under $H$ is $\{\e_1,-\e_2,\e_2-\e_3,-\e_1+\e_3\}$.
Then it is clear that the lattice corresponding to $G=\langle -I_3,-D\rangle$
is isomorphic to $(I_{C_4\times C_2/C_2}\otimes \Z^-_{C_4},C_4\times C_2)$.
\begin{prop}
We have a coflasque resolution of $\Lambda(B_n)$ as a $W(B_n)$ lattice
given by
$$0\to C\to \Z[W(B_n)/W(B_{n-1})]\oplus \Z[W(B_n)/S_n]\to \Lambda(B_n)\to 0$$
\end{prop}
\begin{proof}
Since $\Z B_n$ is a sign permutation lattice, it is quasipermutation
and self-dual.
The stabilizer subgroup of $\e_n$ in $W(B_n)$ is $W(B_{n-1})$
and the $W(B_n)$ orbit of $\e_n$ is $\{\pm \e_i: i=1,\dots,n\}$.
This orbit has $\Z$-span $\Z B_n$ as it contains a $\Z$-basis.
So the $\Z W(B_n)$ map
$$\Z[W(B_n)/W(B_{n-1})]\to \Z B_n, wW(B_{n-1}\to w\e_n$$
is surjective.
Note that $wW(B_{n-1})=w'W(B_{n-1})$ iff $w\e_n=w'\e_n$.
$\{\sigma_i\tau_n^j: 1\le i\le n, 0\le j\le 1\}$
where $\sigma_i=(i,n)$, $i\le n-1$, $\sigma_n=\id$,
is a complete set of coset representatives for $W(B_{n-1})$ in $W(B_n)$
since $\sigma_i(\e_n)=\e_i$ and $\sigma_i\tau_n(\e_n)=-\e_i$.
Let $\x_i=\sigma_iW(B_{n-1})$ and $\y_i=\sigma_i\tau_nW(B_{n-1})$.
Let $\sigma\in S_n$. Then, $\sigma\sigma_i\e_n=\e_{\sigma(i)}=\sigma_{\sigma(i)}\e_n$
and $\sigma\sigma_i\tau_n(\e_n)=-\e_{\sigma(i)}=\sigma_{\sigma(i)}\tau_n(\e_n)$ implies that
$\sigma(\x_i)=\x_{\sigma(i)}$ and $\sigma(\y_i)=\y_{\sigma(i)}$.
Similarly, one can check that $\tau_i$ swaps $\x_i$ and $\y_i$
and $\tau_i$ fixes $\x_j,\y_j, j\ne i$.
Then $\sum_{i=1}^n\sum_{j=0}^1a_{ij}\sigma_i\tau_n^jW(B_{n-1})\in \ker(\pi)$
implies that
$a_{i0}-a_{i1}=0$ for all $i=1,\dots,n$.
So $\ker(\pi)$ has $\Z$-basis
$\sigma_iW(B_{n-1})+\sigma_i\tau_nW(B_{n-1}), i=1,\dots, n$.
In fact, $\ker(\pi)=\Z W(B_n)(\sigma_n+\sigma_n\tau_n)\cong \Z [W(B_n)/C_2^n\rtimes S_{n-1}]$.
So we have a coflasque resolution of $\Z B_n$
Explicitly, we may find its quasipermutation resolution:
$$0\to \Z[W(B_n)/C_2^n\rtimes S_{n-1}]\to \Z[W(B_n)/W(B_{n-1})]\to \Z B_n\to 0$$
We would like to extend this to a coflasque resolution of $\Lambda(B_n)$.
Note that
$0\to \Z B_n\to \Lambda(B_n)\to \Z/2\Z\to 0$
is an exact sequence of $W(B_n)$ lattices where $W(B_n)$ acts trivially
on $\Z/2\Z$.
This follows from the fact that $\Lambda(B_n)=\Z B_n+\Z \omega$
where $\omega=\frac{1}{2}\sum_{i=1}^n\e_i$.
[In fact, it is always true that $W(\Phi)$ acts trivially
on $\Lambda(\Phi)/\Z\Phi$ since $s_i\omega_j=\omega_i-\delta_{ij}\alpha_i$.]
Note that $W(B_n)_{\omega}=S_n$ and so $
\rho:Q=\Z[W(B_n)/S_n]\to \Lambda(B_n),wS_n\to w\omega$
is a $\Z W(B_n)$ map.
This produces the following commutative diagram with exact rows.
We claim that $C$ is $\Z W(B_n)$ coflasque.
It suffices to show that for every $H\le W(B_n)$, $\pi(P^H)\oplus \rho(Q^H)=\Lambda(B_n)^H$. Note that if $\Lambda(B_n)^H=\Z B_n^H$, we have $\pi(P^H)=\Z B_n^H$
as $\ker(\pi)$ is permutation and so coflasque.
If $\Lambda(B_n)^H\ne \Z B_n^H$ then $0\ne \Lambda(B_n)^H/\Z B_n^H$ embeds
into $\Lambda(B_n)/\Z B_n=\Z/2\Z$ which is $H$ trivial
and so $\Lambda(B_n)^H/\Z B_n^H=\Z/2\Z$. This shows that there
exists $\alpha\in \Z B_n$ such that $\omega+\alpha\in \Lambda(B_n)^H$.
We claim that this implies that there exists $\tau\in C_2^n$ such that
$\tau\omega\in \Lambda(B_n)^H$.
We show that the result follows from this claim.
In this case, $H\le W(B_n)_{\tau\omega}=\tau S_n\tau^{-1}$.
Since for any $\sigma\in S_n$, $\tau\sigma\tau^{-1}\tau S_n=\tau S_n$
$\tau S_n\in Q^H$.
Then $\tau \omega=\rho(\tau S_n)\in \rho (Q^H)$.
Let $\lambda\in \Lambda(B_n)^H\setminus \Z B_n^H$. Then
$\lambda-\tau\omega\in \Z B_n^H=\pi(P^H)$ which
shows that $\lambda\in \pi(P^H)+\rho(Q^H)$ as required.
Note that $Q\Lambda(B_n)=\Q B_n$ and $\Lambda(B_n)^H=\Lambda(B_n)\cap \Q B_n^H$.
The condition (*) implies that if $\Lambda(B_n)^H\ne \Z B_n^H$,
then $\pi_i(\Q B_n^H)\ne 0$ for each of the $n$ projection maps $\pi_i(\sum_{j=1}a_j\e_j)=a_i$.
By the condition and the lemma, $\Q B_n^H$ must have a $\Q$ basis of the
form where $\B$ is the set of $\overline{H}$ orbits on
$$\{\sum_{j\in B}\epsilon_j\e_j: B\in \B \}$$
In other words, all orbit sums $\calO_H(\e_i)\ne \0$ and $\cup_{B\in \B}B=\{1,\dots,n\}$.
But then $\frac{1}{2}\sum_{B\in \B}\sum_{j\in B}\epsilon_j\e_j=\tau\omega\in \Lambda(B_n)^H$ where $\tau=\prod_{B\in \B}\prod_{j\in \B}\tau_j$.
By the claim, the result follows.
\end{proof}
%We first examine the claim for a cyclic subgroup $H=\langle h\rangle$.
%We recall that $h$ is a product of signed cycles.
%$h=\tau\sigma$ acts on $Q B_n$ as $\tau\sigma\e_i=\tau_{\sigma(i)}e_{\sigma(i)}$.
%If $\sum_{i=1}^na_ie_i$ is fixed by $h$ then
%$\tau_{\sigma(i)}a_i=a_{\sigma(i)}$. If $(j_1,\dots,j_r)$ is a cycle of $\sigma$,
%This implies that $\prod_{k=1}^r\tau_{j_k}a_{j_1}=a_{j_1}$.
%A cycle $C=(j_1,\dots,j_r)$ is even if $\prod_{k=1}^r\tau_{j_k}=1$
%and odd otherwise. If $C$ is an even cycle,
%$\x_C=\sum_{i=1}^r x_i\e_{j_i}$ where
%$x_{j_1}=1, x_{j_i}=\prod_{p=2}^i\tau_{j_p}, i\ge 2$.
%We see that a $\Q$- basis for the 1-eigenspace of $h=\tau\sigma$
%is given by $\{\x_C: C \mbox{ even cycle of } \sigma\}$.
%So, $\langle h\rangle$ satisfies the condition iff $h$ is
%a product of even cycles (i.e. every $i\in \{1,\dots,n\}$ lies
%in an even cycle). If $h$ has cycles $C_1,\dots,C_t$,
%then $\frac{1}{2}(\sum_{i=1}^t\x_{C_i})=\tau \omega\in \Lambda( B_n)^h$ for some $\tau$.
%Note that for each even cycle of $h$, we have $\x_C=\sum_{i\in C}x_i\e_i$
%where $x_i=\pm 1$.
%For the general case, note that $\Q B_n^H=\cap_{h\in I}\Q B_n^h$
%where $I$ is a generating set of $H$.
%We may show by induction on the size of the generating set of $H$
%that the intersection of $\Q B_n^h$ over $h\in I$ has a
%a $\Q$ basis of the form $\{\y_B: B special orbit of \overline{H}\le S_n$
\begin{lemma}
For any $\Q G$ module $V$ and any subgroup $H\le G$,
$$\calO_H:V\to V^H, \v\to \sum_{\w\in H\cdot\v}\w$$
is surjective.
Let $G=W(B_n)$, $H\le G$ and $V=\Q B_n$.
Then a $\Q$ basis of $V^H$ is given by
$$\{\calO_H(\e_i): i\in I, \calO_H(\e_i)\ne 0\}$$
where $I$ is a set of orbit representatives of the image of $H$
under the surjection $W(B_n)\to S_n$, $\overline{H}\in S_n$,
on $\{1,\dots,n\}$.
If $\calO_H(\e_i)\ne \0$, then $\calO_H(\e_i)=\sum_{j\in \overline{H}\cdot i}\epsilon_j\e_j$
where $\epsilon_j=\pm 1$, $\epsilon_i=1$.
\end{lemma}
\begin{proof}
It is well known that
$e_H=\frac{1}{|H|}\sum_{h\in H}h$ is an idempotent in the group ring $\Q H$
and the map given by
$e_H:V\to V^H, \v\to e_H\v$ is a $\Q$ linear projection.
Note that
$$e_H\v=\sum_{g\in H/H_{\v}}\sum_{h\in H_{\v}}gh\v=\frac{|H_{\v}|}{|H|}\calO_H\v=\frac{1}{|H\cdot \v|}\calO_H\v$$
Now let $V=\Q B_n$, $G=W(B_n)$, $H\le G$.
Let $\w\in V^H$,
Then $\w=\sum_{i=1}^na_ie_H(\e_i)=\sum_{i=1}^na_i\frac{1}{|H\cdot \e_i|}\calO_H(\e_i)$
so $V^H$ is spanned over $\Q$ by $\calO_H(\e_i)$.
Now, $\calO_H(\e_i)=\0$ iff there exists $h\in H$ with $h\e_i=-\e_i$.
If there exists $h\in H$ with $h\e_i=-\e_i$ and $\e_j=g\e_i$ for some $g\in H$,
we see that $-\e_j=gh\e_i$. This implies $\calO_H(\e_i)=\0$.
If there are no $h\in H$ with $h\e_i=-\e_i$, we claim that this
implies that for each $j\in \overline{H}\cdot i$, there exists a unique
$\epsilon_j=\pm 1$ such that $\epsilon_j\e_j\in H\cdot \e_i$.
There exists $h=\tau\sigma\in H$ where $\tau\in C_2^n, \sigma\in S_n$ such that $\sigma(i)=j$ by hypothesis.
So $h\e_i=\tau\e_j=\pm \e_j$.
However if $h\e_i=\e_j$, $h'\e_i=-\e_j$ for some $h,h'\in H$, then
$(h')^{-1}h\e_i=-\e_i$ gives a contradiction.
So $\calO_H(\e_i)=\sum_{j\in \overline{H}\cdot i}\epsilon_j\e_j\ne \0$
in this case.
\end{proof}
\begin{prop} Let $M_1,\dots,M_k$ be $G_k$ lattices
and $M$ be the $G_k$ lattice $M=\oplus_{i=1}^rL_i$. Then $T_M=T_{M_1}\times\cdots \times T_{M_k}$
so that if all $T_{M_i}$ are rational,
so is $T_{M}$.
\end{prop}
\begin{proof}
It suffices by induction to prove this for $r=2$.
The correpondence $T$ to $\hat{T}$ gives a reverse inclusion bijection
between the category of algebraic $k$-tori and the category of
continuous $G_k$ lattices.
Under this correspondence,
exact sequences between algebraic $k$-tori
correspond to exact sequences of continuous $G_k$ lattices
split exact sequences between algebraic $k$-tori correspond to
split exact sequences of continuous $G_k$ lattices.
Here is a more down to earth proof, using the identification
of the function field of an algebraic $k$-torus.
The continuous $G_k$ lattice $M=M_1\oplus M_2$
has representation $\rho:G_k\to \GL(M)$ with image in $\GL(M_1)\times \GL(M_2)$.
That is $\rho=(\rho_1,\rho_2)$ where $\rho_i:\cG_k\to \GL(M_i)$
is the continuous representation of $G_k$ corresponding to $M_i$, $i=1,2$.
Then $N=\ker(\rho)=\ker(\rho_1)\cap \ker(\rho_2)=N_1\cap N_2$.
So $k_s^N=k_s^{N_1}k_s^{N_2}$. Let $K/k$ be Galois with Galois group $G\cong G_k/N$.
$k_s(M_1\oplus M_2)$ is the composite of $G_k$-invariant subfields
$k_s(M_1)$ and $k_s(M_2)$. Then $k_s(M_1\oplus M_2)^{G_k}=k_s(M_1)^{G_k}k_s(M_2)^{G_k}=k_s^{N_1}(M_1)^{G_k/N_1}k_s^{N_2}(M_2)^{G_k/N_2}=K_1(M_1)^{G_1}K_2(M_2)^{G_2}$
where $K_i=k_s^{N_i}$ is Galois over $k$ with Galois group $G_i=G_k/N_i$, $i=1,2$.
So the function field of $T_L$ is the composite of the function fields
of $T_{M_1}$ and $T_{M_2}$ and so $T_M=T_{M_1}\times T_{M_2}$.
It follows that if both $T_{M_1}$ and $T_{M_2}$ are rational then
$K_i(M_i)^{G_i}$ is rational over $K_i^{G_i}=k$ for $i=1,2$ and so
$T_L=T_{M_1}\times T_{M_2}$ with function field $K_1(M_1)^{G_1}K_2(M_2)^{G_2}$
is rational over $k$.
\end{proof}
The tori corresponding to these groups and their subgroups are then rational.
We will then show that the tori corresponding to conjugacy classes of finite
subgroups of $\GL_4(\Z)$ which are not on the list of the rational tori found
above are all not stably rational with 2 exceptions.
We will do this by first finding the minimal conjugacy classes of finite
subgroups of $\GL_4(\Z)$ which have not been found to be rational and showing that all but two of these have tori which are not retract rational and the
last two are retract rational but not stably rational.
\section{Minimal non-rational 2 groups}
We now look at all the tori which are not on the list of rational tori considered above. From this list, we use the lattice of conjugacy classes of finite
subgroups of $\GL_4(\Z)$ to find the minimal indecomposable ones that do not appear on this list.
This gives us the following list:
[4,5,1,12] ($C_2^2$), [4,5,2,5], [4,5,2,9],[4,6,1,6],[4,6,2,6] ($C_2^3$),
[4,12,2,5],[4,12,2,6],[4,13,1,5],[4,13,2,5],[4,18,1,3] ($C_2\times C_4$)
[4,32,1,2] $Q_8$
Since any 2 group is contained in a conjugate of
In fact $G=W(B_2)\times W(G_2)$.
where $W(B_2)=\langle \diag(A_1,I_2),\diag(A_2,I_2),\diag(-I_2,I_2)\rangle$
acts on $M_1$ as $\Z B_2$ and on $M_2$ trivially
and $W(G_2)=\langle \diag(I_2,B_1),\diag(I_2,B_2^2),\diag(I_2,-I_2)\rangle$
acts on $M_2$ as $G_2$ and on $M_1$ trivially.
Note that $B_1,B_2^2$ generate a group $H$ which is
abstractly isomorphic to $S_3$. The $H$ orbit of $\e_3$ is
$\{\e_3,\e_4,-\e_3-\e_4\}$ and the stabilizer subgroup of $\e_3$ is $\langle B_1\rangle=C_1 2$. So $\langle B_1,B_2^2,-I_2\rangle$ has lattice
given by $J_{S_3\times C_2/S_2\times C_2}\otimes \Z^-_{S_3}$ which is congruent to the
lattice $(G_2,W(G_2))\cong (\Z A_2,\Aut(A_2))$.
An algebraic torus is retract rational if and only if its character lattice
is quasi-invertible. A lattice is quasi-invertible if and only
if it is quasi-invertible as a lattice restricted to its Sylow $p$-subgroups.
We should then be able to determine
All 2 subgroups of $\GL_4(\Q)$ are conjugate to a subgroup of $\GL_4(\Z)$
determined by $\Z B_4$ restricted to its Sylow 2-subgroup $P_2$
which is abstractly isomorphic to $D_8^2\rtimes C_2$.
There are 2 $\Z$ forms of $\Q B_4\vert_{P_2}$
given by $M_1=\Z B_4\vert_{P_2}$ and $M_2=\Lambda(B_4)\vert_{P_2}$.
Since the first is sign permutation, it is self-dual and hence the
second lattice is also self-dual.
The latter implies that the previous proposition gives us
a flasque resolution for $\Lambda(B_4)\vert_{P_2}$.
We will analyze the rationality properties of tori corresponding
to 2 subgroups of $\GL_4(\Z)$. The corresponding lattices
are of the form $(L,G)$ where $L$ is the restriction of either $M_1$
or $M_2$ to $G$.
Since the Sylow 2 subgroup of $W(B_4)$
The $k$ forms of $\Gm^r$ are in bijection with elements of
Note that $G_k$ acts trivially on $\Aut(\Gm^r)=\GL(r,\Z)$
and so $H^1(G_k,\GL(r,\Z))=\Hom_{\rm cont}(G_k,\GL(r,\Z))$.
But $G_k$ is a compact topological group and $\GL(r,\Z)$ is discrete,
so that any continuous representation of $G_k$ into $\GL(r,\Z)$
has a finite image.
This means that a $k$ form of $\Gm^r$ determines a lattice of rank $r$
with a continuous action by $G_k$. This lattice is in fact a
$G$ lattice where $G=h(G_k)$ is a finite group. Let $K=k_s^G$.
Then $K/k$ is Galois with Galois group $G$.
If there exists a Galois extension $K/k$ with group $G$,
and there exists a $G$ lattice $L$ of rank $r$, then $L$ is also
a continuous $G_k$ lattice of rank $r$ since $\Gal(k_s/K)$ is a normal subgroup
of $\Gal(k_s/k)$ and $G\cong G_k/\Gal(k_s/K)$. So the $G$ lattice $L$
inflates to a continuous $G_k$ lattice.
So $k$ forms of $\Gm^r$ are in bijection with $\Z\Gal(K/k)$ lattices of
rank $r$ where $K/k$ is a finite Galois extension.
Suppose $T$ is an algebraic $k$ torus with character lattice $\hat{T}$.
Then $\hat{T}$ is a lattice with a continuous $\cG_k$ action.
$T=\Spec(k_s[\hat{T}]^{G_k})$. Let $\rho:\cG_k\to \GL(\hat{T})$
be the continuous representation of $\cG_k$ corresponding to $T$.
Then $G=\rho(G_k)$ is a finite group and so $N=\ker(\rho)$ is
a closed normal subgroup of finite index. Let $L=k_s^N$.
Then $L/k$ is Galois with Galois group $G\cong G_k/N$
and $k_s[\hat{T}]^{G_k}=(k_s[\hat{T}]^{N})^{G_k/N}=L[\hat{T}]^G$
since $N$ acts trivially on $\hat{T}$.
The group associated to the dual lattice
is $\langle A^T,B^T\rangle$.
$\rho_2^-:S_3\times C_2\to \GL_2(\Z)$ is the representation determined by
$J_{S_3\times C_2/S_2\times C_2}\otimes \Z^-_{S_3}$ defined earlier. Note that
But $\rho_2^-((1,2),1)=\rho_2((1,2))=A^T$ and $\rho_2^-((1,2,3),1)=\rho_2((1,2,3))=B^T$ and $\rho_2^-(\id,\gamma)=-I_2$.
It is easy to check that $\langle ((1,2),1),((1,2,3),1),(\id,\gamma)\rangle =
S_3\times C_2$, so we have shown that the dual lattice
determined by [2,4,4,2] is $J_{S_3\times C_2/S_2\times C_2}\otimes \Z^-_{S_3}$.
So the lattice is $I_{S_3\times C_2/S_2\times C_2}\otimes \Z^-_{S_3}$.
This does match the action of $\Aut(A_2)$ on $\Z A_2$
which is equivalent to the action of $W(G_2)$ on $G_2$.
%A basis for $\Z A_2$ is given by $\{\e_2-\e_3,\e_1-\e_3\}$.
%$\Aut(A_2)=S_3\times C_2$. $S_3=\langle (12),(123)\rangle$.
%acts by permuting subscripts and $C_2$ acts by -1.
%Then the matrix of $(12)$ with respect to the above basis is
%$A$ and that of $(123)$ is $B$.
%This shows that the associated lattice is $(\Z A_2,\Aut(A_2))$.
%In terms of $S_3\times C_2$ lattices this
%is $(I_{S_3\times C_2/S_2\times C_2}\otimes \Z^_^{S_3})$.
The subgroup with GAP ID [4,27,4,1] is isomorphic to $D_{10}\times C_2$.
It is conjugate to a subgroup of [4,31,6,2]
and so corresponds to a lattice $J_{
So the lattice determined by [4,27,4,1] is the
However the restriction of this lattice to $D_{10}\times C_2$ is that determined by [4,27,4,1]. This lattice is hereditarily rational.
We will show that the lattice determined by the [4,27,4,1] is isomorphic to
$(\Z A_4,\Aut(A_4))$ restricted to $D_{10}\times C_2$ and so is hereditarily
We will then show that all the tori corresponding to lattices which
are not ocn
From Hoshi and Yamasaki's list of conjugacy classes of finite subgroups
of $\GL(4,\Z)$ corresponding to stably rational tori, the following
are maximal elements of this list;
The lattices corresponding to Dade groups are
DadeGroup(4,1) with GAP ID [4,20,22,1]
corresponds to $(\Z B_2\oplus G_2,W(B_2)\times W(G_2))$.
The generators of this group are
$$B_1=\left[\begin{array}{rr}1&0\\-1&-1\end{array}\right], B_2:=\left[\begin{array}{rr}-1&-1\\1&0\end{array}\right], A_1:=\diag(1,-1),A_2:=\left[\begin{array}{rr}0&1\\-1&0\end{array}\right]$$
Since $B_2^3=A_2^2=-I_2$, we may replace this
set of generators by
For $G=DadeGroup(4,1)$, $M_G=M_1\oplus M_2$
where $M_1=\Z\e_1\oplus \Z\e_2$ and $M_2=\Z\e_3\oplus \Z\e_4$
are both $G$ invariant. In fact $G=W(B_2)\times W(G_2)$.
where $W(B_2)=\langle \diag(A_1,I_2),\diag(A_2,I_2)\rangle$
acts on $M_1$ as $\Z B_2$ and on $M_2$ trivially
and $W(G_2)=\langle \diag(I_2,B_1),\diag(I_2,B_2^2),\diag(I_2,-I_2)\rangle$
acts on $M_2$ as $G_2$ and on $M_1$ trivially.
This is because the lattice corresponding to $H=\langle B_1,B_2^2\rangle$
is $\Z H\cdot \e_1=J_{S_3/S_2}$ since the $H$ orbit of $\e_1$ is $\{\e_1,\e_2,-\e_1-\e_2\}$ and the stabiliser subgroup is $H_{\e_1}=\langle B_1\rangle\cong S_2$.
%by restricting the flasque resolution for $J_{S_5/S_4}$ to $A_5$.
%We have shown that $J_{A_5/A_4}$ is quasi-permutation, which implies
%that $J_{A_5/A_4}^{\otimes 2}$ is stably permutation since the flasque
%class of a lattice is determined up to stable isomorphism.
%The short exact sequence
%$$0\to \Z\to \Z[S_p/S_{p-1}]\to J_{S_p/S_{p-1}}\to 0$$
%admits an $S_p$ invariant map $s:J_{S_p/S_{p-1}}\to \Z[S_p/S_{p-1}]$
%$s(\varphi(\e_i))=p\e_i-(\sum_{i=1}^p\e_i)$ which then
%satisfies $\varphi\circ s=p\id_{J_{S_p/S_{p-1}}}$.
%Then tensoring this sequence with $J_{S_p/S_{p-1}}$, we see that
%$1\otimes \varphi$
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\title
{Sign Permutation Quotients}
\author{Nicole Lemire}
|
1511.00161
|
Mathematics Department
Texas A&M University
College Station, TX 77840
Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing “mesoprimary decompositions” determined by their underlying monoid congruences. These mesoprimary decompositions are highly combinatorial in nature, and are designed to parallel standard primary decomposition over Noetherean rings. In this paper, we generalize mesoprimary decomposition from binomial ideals to “binomial submodules” of certain graded modules over the corresponding monoid algebra, analogous to the way primary decomposition of ideals over a Noetherean ring $R$ generalizes to $R$-modules. The result is a combinatorial method of constructing primary decompositions that, when restricting to the special case of binomial ideals, coincides with the method introduced by Kahle and Miller.
§ INTRODUCTION
Fix a field $\kk$ and a commutative monoid $Q$. A binomial ideal in the monoid algebra $\kk[Q]$ is an ideal $I$ whose generators have at most two terms. The quotient $\kk[Q]/I$ by a binomial ideal identifies, up to scalar multiple, any monomials appearing in the same binomial in $I$. This induces a congruence $\til_I$ on the monoid $Q$ (an equivalence relation perserving additivity), and the quotient module $\kk[Q]/I$ is naturally graded with a decomposition into 1-dimensional $\kk$-vector spaces, at most one per $\til_I$-class. In <cit.>, Kahle and Miller introduce mesoprimary decompositions, which are combinatorial approximations of primary decompositions of $I$ constructed from the congruence $\til_I$.
Mesoprimary decomposition of binomial ideals is motivated by combinatorially constructed primary decompositions of monomial ideals. Any monomial ideal $I$ in the monoid algebra $\kk[Q]$ is uniquely determined by the monomials it contains. Taking the quotient $\kk[Q]/I$ amounts to setting these monomials to 0, and the monomials that lie outside of $I$ naturally grade the quotient $\kk[Q]/I$ with a decomposition into 1-dimensional $\kk$-vector spaces.
An irreducible decomposition for a monomial ideal $I$ whose components are themselves monomial ideals can be constructed by locating witness monomials ${\bf x}^w$ whose annihilator modulo $I$ is prime, and then constructing for each the primary monomial ideal that contains of all monomials not lying below ${\bf x}^w$. The intersection of these ideals (one per witness monomial) equals $I$, and the witnesses are readily identified from the grading on $\kk[Q]/I$. See <cit.> for a full treatment of monomial irreducible decomposition.
Combinatorially constructed irreducible decompositions of monomial ideals have also been shown to live within a larger categorical setting. Much in the way primary decomposition of ideals over a Noetherean ring $R$ generalizes to $R$-modules, combinatorial methods for constructing primary decompositions of monomial ideals can be generalized to certain modules whose gradings resemble the fine gradings of monomial quotients. See <cit.> for an overview of these constructions and <cit.> for consequences.
Kahle and Miller use congruences to extend the above construction from monomial ideals to binomial ideals <cit.>. Given a binomial ideal $I$, they pinpoint a collection of monomials in $\kk[Q]/I$ that behave like witnesses. For each witness ${\bf x}^w$, they construct the coprincipal component at ${\bf x}^w$, a binomial ideal containing $I$ whose quotient has ${\bf x}^w$ as the unique greatest nonzero monomial. The resulting collection of ideals, one for each witness, decomposes $I$, and each component admits a canonical primary decomposition. In this way, mesoprimary decompositions act as a bridge to primary components of a binomial ideal from the combinatorics of its induced congruence.
Mesoprimary decompositions are constructed in two settings: first for monoid congruences, and then for binomial ideals; both are designed to parallel standard primary decomposition in a Noetherian ring $R$. This motivated Kahle and Miller to pose Problems <ref> and <ref> below, which appeared as <cit.> and <cit.>, respectively. These problems, in turn, serve to motivate the results in this paper.
Generalize mesoprimary decomposition of monoid congruences to congruences on monoid modules.
Develop a notion of binomial module over a commutative monoid algebra, and generalize mesoprimary decomposition of binomial ideals to this setting.
One of the largest tasks in generalizing the results of <cit.> to monoid modules is to separate which constructions should happen in the monoid and which should happen in the module, since these coincide for monoid congruences. See Remarks <ref> and <ref> for specific instances of this distinction.
In the first part of this paper (Sections <ref>-<ref>), we introduce the category of modules over a monoid $Q$ (Definition <ref>) and generalize nearly every result from <cit.> on monoid congruences to congruences on monoid modules. We define primary and mesoprimary monoid module congruences (Definition <ref>) and give equivalent conditions for these congruences in terms of associated objects and witnesses (Theorems <ref> and <ref>). We then construct a mesoprimary decomposition, with one component per key witnesses, for any monoid module congruence (Theorem <ref>). The resulting theory completely answers Problem <ref>.
The second part of this paper (Sections <ref>-<ref>) answers Problem <ref>. We introduce the category $\cB_Q$ (Definition <ref>), whose objects are tightly graded modules (Definition <ref>) over the monoid algebra $\kk[Q]$ graded by monoid modules in . It is in this setting that we define binomial submodules (Definition <ref>). We define mesoprimary submodules (Definition <ref>) and associated mesoprime ideals (Definition <ref>), developing a theory of mesoprimary decomposition (Theorem <ref>) that parallels results in <cit.>. In particular, the binomial submodules of the free module $\kk[Q]$ are precisely the binomial ideals; see Example <ref>.
To conclude the paper, we demonstrate in Section <ref> how a binomial primary decomposition may be recovered from a mesoprimary decomposition when the underlying field is algebraically closed.
§.§ Notation
Throughout this paper, assume $Q$ is a Noetherian commutative monoid and $\kk$ is an arbitrary field, and let $\kk[Q]$ denote the monoid algebra over $Q$ with coefficients in $\kk$. Unless otherwise stated, all $\kk[Q]$-modules are assumed to be finitely generated.
§ THE CATEGORY OF MONOID MODULES
In this section, we define the category of modules over a commutative monoid $Q$ and extend some of the fundamental concepts and results from monoid ideals and congruences to the objects of this category. The content of this section (as well as Section <ref>) is motivated by Example <ref>. First, we record some preliminary definitions (see <cit.> for more detail).
Fix a commutative monoid $Q$.
* A $Q$-module $(T,\cdot)$ is a set $T$ together with a left action by $Q$ that satisfies $0 \cdot t = t$ and $(q + q')\cdot t = q\cdot(q' \cdot t)$
for all $t \in T$, $q, q' \in Q$. A subset $T' \subset T$ is a submodule of $T$ if it is closed under the $Q$-action, that is, $Q \cdot T' \subset T'$. The submodule of $T$ generated by elements $t_1, \ldots, t_r \in T$ is $\<t_1, \ldots, t_r\> = \bigcup_{i = 1}^r Q \cdot t_i.$
* A map $\psi:T \to U$ between $Q$-modules $T$ and $U$ is a $Q$-module homomorphism if $\psi(q \cdot t) = q \cdot \psi(t)$ for all $t \in T, q \in Q$. The set of $Q$-module homomorphisms from $T$ to $U$ is denoted by $\Hom_Q(T,U)$, and is naturally a $Q$-module with action $q \cdot \psi$ given by $(q \cdot \psi)(t) = \psi(q \cdot t)$.
* The category of $Q$-modules, denoted , is the category whose objects are $Q$-modules and whose morphisms are $Q$-module homomorphisms.
Direct sums, direct products, and tensor products exist in the category . We now state their constructions explicitly.
Fix two $Q$-modules $T$ and $U$.
* The direct sum $T \oplus U$ is the disjoint union $T \coprod U$ as sets, with the natural $Q$-action on each component.
* The direct product $T \times U$ is the cartesian product of $T$ and $U$ as a set, with componentwise $Q$-action.
* The tensor product $T \otimes_Q U$ is the collection of formal elements $t \otimes u$ for $t \in T$ and $u \in U$ modulo the equivalence relation generated by
$$t \otimes (q \cdot u) \sim (q \cdot t) \otimes u \text{ for } t \in T \text{ and } u \in U$$
The action of $Q$ is given by $q \cdot(t \otimes u) = (q \cdot t) \otimes u$ for $q \in Q$, $t \in T$ and $u \in U$.
Fix a $Q$-module $T$. A congruence on $T$ is an equivalence relation $\til$ on $T$ that satisfies $t \sim t' \Rightarrow q \cdot t \sim q \cdot t'$ for all $q \in Q$ and $t, t' \in T$. The quotient module $T/\til$ is the set of equivalence classes of $T$ under $\til$. The congruence condition on $\til$ ensures that $T/\til$ has a well defined action by $Q$.
A subset $T \subset Q$ is an ideal if it is a $Q$-submodule of $Q$, that is, $Q + T \subset T$. An ideal $P \subset Q$ is prime if its complement in $Q$ is a submonoid of $Q$.
Fix a $Q$-module $T$, a prime ideal $P \subset Q$, and set $F = Q \setminus P$. The localization of $T$ at $P$, denoted $T_P$, is the set $T \times F$ modulo the equivalence relation $\til$ that sets $(t,f) \sim (t',f')$ whenever $w \cdot f' \cdot t = w \cdot f \cdot t'$ for some $w \in Q$. The localization $Q_P$ is naturally a monoid, and $T_P$ is naturally a $Q_P$-module. Write $t-f$ to denote the element $(t,f) \in T \times F$.
Any congruence $\til$ on a $Q$-module $T$ induces a congruence on $T_P$.
Fix a $Q$-module $T$. Green's preorder on $T$ sets $t \preceq t'$ whenever $\<t\> \supset \<t'\>$. Green's relation on $T$ sets $t \sim t'$ whenever $\<t\> = \<t'\>$.
Green's preorder on a monoid orders its elements by divisibility, and this notion extends to $Q$-modules.
Green's relation $\til$ on a $Q$-module $T$ is a congruence on $T$, and the quotient $T/\til$ is partially ordered by divisibility.
For $t, t' \in T$ and $q \in Q$, we can see $\<t\> = \<t'\>$ implies $\<q \cdot t\> = \<q \cdot t'\>$. Each element of the quotient $T/\til$ generates a distinct submodule, so the divisibility preorder is antisymmetric, and thus a partial order.
We now generalize the notion of a nil element of a monoid.
An element $\nil \in T$ in a $Q$-module $T$ is called a nil if it is absorbing, that is, $Q \cdot \nil = \{\nil\}$. The basin of a nil $\nil \in T$ is the set
$$B(\nil) = \{t \in T : qt = \nil \text{ for some } q \in Q\}$$
of elements of $T$ that can be sent to $\nil$ under the action of $Q$. The nil set of $T$, denoted $N(T)$, is the collection of all nil elements in $T$.
Fix a subset $U \subset T$ of a $Q$-module $T$. A $Q$-orbit of $U$ is a connected component of the undirected graph whose vertices are elements of $U$ and whose edges connect two vertices $s, t \in U$ whenever $q \cdot s = t$ for some $q \in Q$. $T$ is connected if it has at most one $Q$-orbit, and $T$ is properly connected if $T \setminus N(T)$ has at most one $Q$-orbit.
Let $T$ and $U$ be connected $Q$-modules with nils $\infty_T$ and $\infty_U$, respectively. If $T \setminus \{\infty_T\}$ and $U \setminus \{\infty_U\}$ are both nonempty, the module $(T \coprod U)/\<\nil_T \sim \nil_U\>$ is connected and has a single nil, but it is not properly connected, since removing the nil produces two distinct $Q$-orbits.
Unlike a monoid, a $Q$-module may have more than one nil element. However, by Lemma <ref>, each $Q$-orbit can have at most one nil element.
The basin of a nil element $\nil \in T$ in a $Q$-module $T$ is the $Q$-orbit of $T$ containing $\nil$.
The basin of $\nil$ is clearly contained in its $Q$-orbit, and whenever $qt = s$ for $q \in Q$ and $s, t \in T$, we have $t \in B(\nil)$ if and only if $s \in B(\nil)$.
Let $Q = \NN^2$, $I = \<x^2,y^2\> \subset \kk[Q]$, $R = \kk[Q]/I$, and
$$M = (R \oplus R)/\<xye_1 - xye_2\>,$$
where $e_1$ and $e_2$ generate the free $\kk[Q]$-module $R \oplus R$. $R$ is graded by the quotient monoid $Q/\til_I$, and $M$ is graded by two disjoint copies of $Q/\til_I$ with both copies of $xy$ and the nil elements identified. Unlike the monoid that grades $R$, this grading does not have a natural monoid structure. It does, however, have a natural action by $Q$, correponding to the action on $M$ by monomials in $\kk[Q]$. See Figure <ref> for an illustration.
[A monoid module over $\NN^2$]The monoid module that grades $M$ in Example <ref>.
There is also a notion of decomposition of $Q$-modules into indecomposables.
Every $Q$-module $T$ has a unique decomposition $T = \bigoplus_i T_i$ as a direct sum of connected modules.
Any $Q$-module is the disjoint union of its $Q$-orbits.
Kernels, in the categorical sense, do not exist in the category . However, there is still a notion of kernel of a $Q$-module homomorphism as a congruence; see Definition <ref>. This definition is justified by Theorem <ref>, a $Q$-module analogue of the first isomorphism theorem for groups.
Fix a homomorphism $\phi:T \to U$. The kernel of $\phi$, denoted $\ker(\phi)$, is the congruence $\til$ on $T$ that sets $t \sim t'$ whenever $\phi(t) = \phi(t')$ for $t, t' \in T$.
If $\phi:T \to U$ is a $Q$-module homomorphism, then $T/\ker(\phi) \iso \image(\phi)$.
The homomorphism $\phi$ is surjective onto its image, and the quotient of $T$ by $\ker(\phi)$ identifies elements with the same image under $\phi$. This ensures that the map $T/\ker(\phi) \too \image(\phi)$ is both injective and surjective.
Any finitely generated $Q$-module $T$ is isomorphic to a quotient of a direct sum of finitely many copies of $Q$.
Fix a finitely generated $Q$-module $T = \<t_1, \ldots, t_r\>$. Let $\phi:\bigoplus_{i=1}^r Q \too T$, where the map on the $i$-th summand is given by $Q \to \<t_i\>$. This map is surjective, so Theorem <ref> implies $T \iso (\bigoplus_{i=1}^r Q)/\ker(\phi)$.
§ PRIMARY AND MESOPRIMARY MONOID MODULES
Mesoprimary decomposition of monoid congruences models primary decomposition of ideals in a Noetherian ring $R$, with mesoprimary congruences playing the role of primary ideals and prime congruences playing the role of prime ideals. In this section, we generalize the notion of mesoprimary monoid congruences to congruences on monoid modules (Definition <ref>), analogous to the way primary decomposition of ideals in $R$ generalizes to finitely generated $R$-modules. The main result is Theorem <ref>, which generalizes <cit.> and characterizes mesoprimary monoid module congruences in terms of their associated prime congruences (Definition <ref>).
Fix a $Q$-module $T$. For each $q \in Q$, let $\phi_q$ denote the map $T \stackrel{\cdot q}\too T$ given by action by $q$.
* An element $q \in Q$ acts cancellatively on $T$ if $\phi_q$ is injective.
* An element $q \in Q$ acts nilpotently on $T$ if for each $t \in T$, $(nq) \cdot t \in N(T)$ for some nonnegative integer $n$.
* An element $t \in T$ is partly cancellative if whenever $a \cdot t = b \cdot t \notin N(T)$ for $a, b \in Q$ that act cancellatively on $T$, the morphisms $\phi_a$ and $\phi_b$ coincide.
Each term in Definition <ref> is also defined in <cit.> for monoid elements. However, we are forced to make a distinction between monoid elements and monoid module elements (these objects coincide in the setting of <cit.>). In particular, “cancellative” and “nilpotent” (Definition <ref>) are properties of monoid elements, whereas “partly cancellative” is a property of monoid module elements. Roughly speaking, “cancellative” and “nilpotent” describe how a particular $q \in Q$ acts on different module elements, whereas “partly cancellative” describes how different monoid elements act on a particular $t \in T$.
A $Q$-module $T$ is
* primary if each $q \in Q$ is either cancellative or nilpotent on $T$.
* mesoprimary if it is primary and each $t \in T$ is partly cancellative.
A congruence $\til$ on $T$ is primary (respectively, mesoprimary) if $T/\til$ is a primary (respectively, mesoprimary) $Q$-module.
Fix a congruence $\til$ on $Q$. The $Q$-module $T = Q/\til$ is (meso)primary in the sense of Definition <ref> if and only if $\til$ is a (meso)primary monoid congruence in the sense of <cit.>.
For $q \in Q$ let $\ol q$ denote the image of $q$ modulo $\til$. An element $q \in Q$ acts cancellatively on $T$ if and only if its image modulo $\til$ is cancellative, and $q$ acts nilpotently on $T$ if and only if it has nilpotent image modulo $\til$. This proves that $T$ is a primary $Q$-module if and only if $\til$ is primary as a monoid congruence. Lastly, assuming $\til$ is $P$-primary, notice that for $a, b \notin P$, $\phi_a = \phi_b$ if and only if $\ol a = \ol b \in T$, so each $\ol q \in T$ is partly cancellative as a monoid element if and only if it is partly cancellative as an element of a $Q$-module. This completes the proof.
We now generalize witnesses and key witnesses from <cit.> to the setting of monoid module congruences. Definition <ref>, while complex, very closely resembles <cit.>; see the original text for several motivating examples. Key witnesses are used to construct the mesoprimary components (Definition <ref>) used to decompose monoid module congruences (Theorem <ref>).
Let $T$ be a $Q$-module, $P \subset Q$ a prime ideal, and $\til$ a congruence on $T$. For $t \in T$, let $\ol t$ denote the image of $t$ in $\ol T_P$, and for $p \in P$, let $\phi_p:\ol T_P \to \ol T_P$ denote the morphism given by the action of $p$.
* An element $w \in T$ is exclusively maximal in a set $A \subset \ol T_P$ if $\ol w$ is the unique maximal element of $A$ under Green's preorder.
* An element $w \in T$ with non-nil image in $\ol T_P$ is a $\til$-witness for $P$ if for each generator $p \in P$, the class of $\ol w$ is non-singleton under $\ker(\phi_p)$ and $\ol w$ is not exclusively maximal in that class.
* An element $w' \in T$ is an aide for a $\til$-witness $w$ for $P$ and a generator $p \in P$ if $w$ and $w'$ have distinct images in $\ol T_P$ but are not distinct under $\ker(\phi_p)$.
* An element $w$ with non-nil image in $\ol T_P$ is a key $\til$-witness for $P$ if $\ol w$ is non-singleton under $\bigcap_{p \in P} \ker(\phi_p)$ and $\ol w$ is not exclusively maximal in this non-singleton class.
* The prime $P$ is associated to $T$ if $T$ has a witness for $P$, or if $P = \emptyset$ and $T$ has a $Q$-orbit with no nil.
Prior to <cit.>, primary decomposition of monoid congruences was developed by Grillet <cit.>, but these decomposition are too course to effectively recover primary components at the level of binomial ideals <cit.>. Nevertheless, in an effort to create a more complete picture, we also generalize primary congruences to our current setting of monoid module congruences.
A finitely generated $Q$-module $T$ is primary if and only if it has exactly one associated prime ideal.
Suppose $T$ is primary. The set of elements with nilpotent action on $T$ is a prime ideal $P \subset Q$. Since $P$ is finitely generated, some non-nil element $w \in T$ satisfies $P \cdot w \subset N(T)$. This means $w$ is a witness for $P$, so $P$ is associated to $T$. Since $Q \setminus P$ acts cancellatively on $T$, any prime associated to $T$ is contained in $P$. Moreover, localizing $T$ at any prime $P'$ contained in $P$ identifies any element $w \in T$ with the nil in its orbit, since some $p \in P \setminus P'$ gives $p \cdot w \in N(T)$. Thus, any associated prime must also contain $P$, which implies $P$ is the only associated prime.
Now suppose $T$ has only one associated prime $P \subset Q$. If $P = \emptyset$, then every element of $Q$ acts cancellatively on $T$. Now suppose $P$ is nonempty, and fix $t \in T$. The submodule $\<t\>$ is isomorphic to $Q$ modulo some congruence. Since each witness in $\<t\>$ is a witness for $P$, $\<t\>$ is $P$-primary by <cit.>. This means each $p \in P$ acts nilpotently on $\<t\>$ and each $f \in Q \setminus P$ acts cancellatively on $\<t\>$. Since $t$ is arbitrary, each $p \in P$ acts nilpotently on $T$ and each $f \in Q \setminus P$ acts cancellatively on $T$, meaning $T$ is $P$-primary.
Lemma <ref> generalizes <cit.> and is central to several proofs, including Theorems <ref> and <ref>.
Fix a connected, $P$-primary $Q$-module $T$, and set $F = Q \setminus P$. Let $T/F$ denote the quotient of $T$ by the congruence
$$t \sim t' \text{ whenever } f \cdot t = g \cdot t' \text{ for } f, g \in F$$
Then Green's preorder on $T/F$ is a partial order, and $T/F$ is finite.
Since $T$ is $P$-primary, the morphisms $T \stackrel{\cdot f}\too T$ are injective for all $f \in F$, so $\til$ is a well-defined congruence. If $\<t\> = \<t'\>$, then $f \cdot t = t'$ and $g \cdot t' = t$ for some $f, g \in Q$. This means $f \cdot g \cdot t = t$, so $f$ and $g$ are not nilpotent and lie in $F$, meaning $t$ and $t'$ are identified in $T/F$. This proves Green's preorder is antisymmetric.
Now, the remaining statement is trivial if $P = \emptyset$, so suppose $P$ is nonempty. $T$ must have a nil $\infty$ since $Q$ contains elements with nilpotent action on $T$. The image of $\infty$ in $T/F$ remains nil as well. Thus, since $Q$ and $T$ are both finitely generated, $T/F$ must be finite.
Fix a $Q$-module $T$, a monoid prime $P \subset Q$, and a non-nil $w \in T$.
* Let $G_P \subset Q_P$ denote the unit group of $Q_P$, and let $K_q^P \subset G_P$ denote the stabilizer of $\ol w \in T_P$ under the action of $G_P$.
* Let $\app$ denote the congruence on $Q_P$ that sets $a \approx b$ whenever
* $a$ and $b$ lie in $P_P$, or
* $a$ and $b$ lie in $G_P$ and $a - b \in K_q^P$.
* The $P$-prime congruence of $T$ at $w$ is given by $\ker(Q \to Q_P/\app)$.
* The $P$-prime congruence at $w$ is associated to $T$ if $w$ is a key witness for $T$.
In Definition <ref>, we are forced to make another distinction between $T$ and $Q$: should an associated prime congruence of $T$ be a congruence on $T$ or on $Q$? The condition for a monoid congruence $\til$ to be $P$-mesoprimary can be characterized in terms of the congruence on $Q \setminus P$ induced by its action on $Q/\til$ <cit.>. The partly cancellative condition is what ensures that each $t \in T$ induces the same congruence, which in our setting is a condition on elements of $T$.
Next, we characterize mesoprimary $Q$-modules in terms of their associated prime congruences, generalizing <cit.> and <cit.>.
For a $Q$-module $T$, the following are equivalent.
$T$ is mesoprimary.
$T$ has exactly one associated prime congruence.
$T$ is $P$-primary, and for $F = Q \setminus P$,
$$\ker(F \to \<t\>) = \ker(F \to \<t'\>)$$
for each non-nil $t, t' \in T$.
From any of these conditions, we conclude that $T$ is primary, say with associated prime $P$. Notice that $\ker(F \to \<t\>)$ is the prime congruence at $t$ restricted to $F$. If these congruences coincide for all $t \in T$, then in particular they coincide for all witnesses, so $T$ has exactly one associated prime congruence. This proves (3) $\implies$ (2).
Now suppose $T$ is mesoprimary, and fix $t, t' \notin N(T)$. Then since $t$ and $t'$ are both partly cancellative, $a \cdot t = b \cdot t$ if and only if $a \cdot t' = b \cdot t'$ for $a, b \notin P$. This means the kernels $\ker(F \to \<t\>)$ and $\ker(F \to \<t'\>)$ coincide. This proves (1) $\implies$ (3).
Lastly, suppose $T$ has exactly one associated prime congruence, and fix $t \in N(T)$. Fix $a, b \notin P$ and let $\phi_a,\phi_b:T \to T$ denote the actions of $a$ and $b$ on $T$, respectively. By Theorem <ref>, $\<t\> \iso Q/\til$ for some congruence $\til$. Since $T$ has only one associated prime congruence, so does $\til$, so by <cit.>, $\til$ is mesoprimary. This means $a \cdot t = b \cdot t$ if and only $a \cdot w = b \cdot w$ for any witness $w \in \<t\>$. Since $T$ has only one associated prime congruence, these actions also coincide for all witnesses in $T$, meaning $\phi_a = \phi_b$. This proves (2) $\implies$ (1), thus completing the proof.
We conclude this section with Theorem <ref>, which ensures that the mesoprimary decomposition constructed in Theorem <ref> has finitely many components.
Any finitely generated $Q$-module $T$ has only finitely many Green's classes of key witnesses.
Fix a generating set $g_1, \ldots, g_k$ for $T$. For each $g_i$, consider the map $\phi_i:Q \to \<g_i\>$ and let $\til_i = \ker\phi_i$. The induced isomorphism $Q/\til_i \to \<g_i\>$ gives a bijection between key $T$-witness and key $\til_i$-witnesses, and by <cit.>, each congruence $\til_i$ has only finitely many Green's classes of key witnesses. Since $g_1, \ldots, g_k$ generate $T$, this bounds the number of Green's classes of key $T$-witnesses.
§ MESOPRIMARY DECOMPOSITION OF MONOID MODULES
In this section, we construct a mesoprimary decomposition for any monoid module congruence $\til$ (with one caveat; see Remark <ref>). First, we construct a mesoprimary component for each $\til$-witness.
Fix a $Q$-module $T$. A cogenerator of $T$ is a non-nil element $t \in T$ with $q \cdot t \in N(T)$ for every nonunit $q \in Q$. A $Q$-module $T$ is coprincipal if it is $P$-mesoprimary and all its cogenerators lie in the same Green's class in $T_P$. A congruence $\til$ on $T$ is coprincipal if $T/\til$ is a coprincipal $Q$-module.
Fix a $Q$-module $T$, a prime $P \subset Q$, and a witness $w \in T$ for $P$. Let $\ol q$ denote the image of $q \in Q$ in $Q_P$, and $\ol t$ denote the image of $t \in T$ in $T_P$.
* The order ideal $T_{\preceq w}^P$ cogenerated by $w$ at $P$ consists of those $a \in T$ whose image $\ol a \in T_P$ precedes $\ol w$ under Green's preorder.
* The congruence cogenerated by $w$ along $P$ is the equivalence relation $\til_w^P$ on $T$ that sets all elements outside of $T_{\preceq w}^P$ equivalent and sets $a \sim_w^P b$ whenever $\ol a$ and $\ol b$ differ by a unit in $T_P$ and $q \cdot \ol a = q \cdot \ol b = \ol w \in T_P$ for some $q \in Q_P$.
Lemma <ref> justifies the nomenclature in Definition <ref>.
The congruence cogenerated by $w$ along $P$ is a coprincipal congruence on $T$ cogenerated by $w$. Furthermore, $T/\til_w^P$ is properly connected, and if $T \setminus T_{\preceq w}^P$ is nonempty, then it is the nil class of $T/\til_w^P$.
Let $T' = T/\til_w^P$. Every non-nil element of $T'$ has the image of $w$ as a multiple, so $T'$ is properly connected, and it is clear that the image of $T \setminus T_{\preceq w}^P$ is nil modulo $\til_w^P$ as long as it is nonempty. Furthermore, $w$ cogenerates $\til_w^P$ since the result of acting by any $p \in P$ lies outside $T_{\preceq w}^P$, and any $t \in T$ with non-nil image in $T'$ satisfies $q \cdot t = w$ for some $q \in Q$, so every cogenerator for $\til_w^P$ lies in the Green's class of $w$ in $T_P$.
It remains to show that $T'$ is mesoprimary. By Lemma <ref>, $T_{\preceq w}^P$ has finitely many Green's classes in $T_P$, so each $p \in P$ acts nilpotently on $T'$ and thus $T'$ is $P$-primary. Furthermore, for each $t \in T$ and for $a, b \in Q \setminus P$, we have $a \cdot t \sim_w^P b \cdot t$ if and only if $a \cdot w \sim_w^P b \cdot w$. In particular, the $P$-prime congruences at the non-nil elements of $T'$ coincide, so by Theorem <ref>, $T'$ is mesoprimary.
Fix a $Q$-module $T$ and a congruence $\til$ on $T$.
* An expression $\til = \bigcap_i \til_i$ of $\til$ as the common refinement of finitely many mesoprimary congruences is a mesoprimary decomposition if, for each component $\til_i$ with associated prime ideal $P \subset Q$, the $P$-prime congruences of $\til$ and $\til_i$ at each cogenerator for $\til_i$ coincide.
* A mesoprimary decomposition $\til = \bigcap_i \til_i$ is key if, for each $P$-mesoprimary component $\til_i$, every cogenerator for $\til_i$ is a key $P$-witness for $\til$.
We are now ready to give the main result of this paper. Theorems <ref> and <ref> together imply, as a special case, that every monoid module with at most one nil element admits a key mesoprimary decomposition (see Remark <ref>).
Fix a congruence $\til$ on a $Q$-module $T$. The common refinement of the coprincipal congruences cogenerated by the key witnesses of $\til$ identifies only the nil elements of $T/\til$.
The nil class of the congruence cogenerated by a witness $w \in T$ for $P$ contains the nil in the connected component of $w$ (if one exists), as well as every element outside of this connected component. This means any $P$-coprincipal component identifies all of the nil elements of $T$.
Now, fix distinct $a, b \in T$ and assume $a$ is not nil. If $a$ and $b$ lie in distinct connected components, then any cogenerated congruence whose order ideal contains $a$ does not identify $a$ and $b$. Assuming $a$ and $b$ lie in the same connected component, it suffices to find a monoid prime $P \subset Q$ and a key witness $w \in T$ for $P$ such that $a$ and $b$ are not equivalent under $\til_w^P$. Fix a prime $P$ minimal among those containing the ideal $I = \{q \in Q : q \cdot a = q \cdot b\}$. Notice that $I$ (and thus $P$) must be nonempty since $a$ and $b$ lie in the same connected component.
Since $P$ contains $I$, the elements $a$ and $b$ have distinct images $\ol a$ and $\ol b$ in $T_P$, and each $\ol q \in I_P$ also satisfies $\ol q \cdot \ol a = \ol q \cdot \ol b$. By minimality of $P_P$ over $I_P$, there is a maximal Green's class among the elements $\{\ol q \in Q_P : \ol q \cdot \ol a \ne \ol q \cdot \ol b\}$. Pick an element $q \in Q$ such that $\ol q$ lies in this Green's class, and set $w = q \cdot a \in T$. Then $w$ is a key witness for $P$ by construction, and the localization of $\til_w^P$ does not equate $\ol a$ and $\ol b$ in $T_P$, so $\til_w^P$ does not equate $a$ and $b$ in $T$. This completes the proof.
Fix a $Q$-module $T$ and a congruence $\til$ on $T$. If $T/\til$ has at most one nil element, then $\til$ admits a key mesoprimary decomposition.
Apply Theorem <ref> and Lemma <ref> to $T/\til$.
Theorem <ref> states that mesoprimary decomposition of monoid modules fails to distinguish nil elements from one another, and that this is the only obstruction to constructing mesoprimary decompositions in this setting. Fortunately, for the purposes of decomposing graded modules over a monoid algebra, these elements all correspond to zero in the module and thus are indistinguishable.
§ THE CATEGORY OF TIGHTLY GRADED $\KK[Q]$-MODULES
Section <ref> defined the category of $Q$-modules, the setting in which mesoprimary decomposition of monoid congruences is generalized in the prior sections of this paper. In this section, we define the category $\cB_Q$ of tightly graded $\kk[Q]$-modules (Definition <ref>), the objects of which are graded by objects of . It is to these graded modules that we generalize mesoprimary decomposition of binomial ideals in the subsequent sections of this paper.
Fix a $Q$-module $T$ and a $\kk[Q]$-module $M$.
* $M$ is graded by $T$ (or just $T$-graded) if there exist a collection of finite dimensional vector spaces $\{M_t\}_{t \in T}$ such that $M \iso \bigoplus_{t \in T} M_t$ as Abelian groups, and for each $q \in Q$, ${\bf t}^q \cdot M_t \subset M_{q \cdot t}$.
* The grading of $M$ by $T$ is fine (or $M$ is finely-graded by $T$) if $\dim_\kk M_t \le 1$ for each $t \in T$.
* A fine grading of $M$ by $T$ is tight (or $M$ is tightly-graded by $T$) if
* $M_t \ne 0$ for each non-nil $t \in T$,
* the orbit of each $\infty \in N(T)$ with $M_\infty = 0$ is properly connected, and
* whenever $m \in M_t$ is nonzero with ${\bf x}^q \cdot m = 0$, we have $\dim_\kk M_{q \cdot t} = 0$.
A tight grading of a $\kk[Q]$-module $M$ by a $Q$-module $T$ ensures that we can determine enough of the structure of $M$ from the grading. The first condition ensures that $T$ does not have any unnecessary elements, and the second ensures each connected component has its own nil (see Proposition <ref>). Example <ref> demonstrates what can cause the third condition to fail.
The $\kk[x]$-module $M = \<x^2\> \oplus (\kk[x]/\<x^2\>)$ is finely graded by $\NN$, but since $x \cdot (0,x)$ is zero, this grading is not tight. However, $M$ is tightly graded by the disjoint union of $\<2\> \subset \NN$ (which tightly grades $\<x^2\> \subset \kk[x]$) and $\NN/\<2\>$ (which tightly grades $\kk[x]/\<x^2\>$). This grading more accurately reflects the algebraic structure of $M$.
In order to study finely graded $\kk[Q]$-modules, it suffices to consider tight gradings. In particular, every tight grading is fine, and Theorem <ref> shows that a tight grading can be recovered from any fine grading by chosing an appropriate $Q$-module.
Fix a $\kk[Q]$-module $M$ finely graded by a $Q$-module $T$. Then there exists a $Q$-module that tightly grades $M$.
We construct the desired $Q$-module in two steps. First, define a $Q$-module $T'$ that, as a set, consists of those $t \in T$ for which $\dim_\kk M_t = 1$, along with a distinguished element $\infty$. Given $t \in T'$ and $q \in Q$, define $q \cdot t \in T'$ by
$q \cdot t = \left\{\begin{array}{ll}
q \cdot t \in T & {\bf x}^q M_t \ne 0 \\
\infty & \text{otherwise}
\end{array}\right.,$
that is, the result of acting on $t$ by $q$ in $T$ if ${\bf x}^q M_t \ne 0$, and $\infty \in T'$ otherwise. The $Q$-module $T'$ also finely grades $M$ since each nonzero $M_t$ for $t \in T$ has a corresponding degree in $T'$. Moreover, the only degree $t \in T'$ with $\dim_\kk M_t = 0$ is $t = \infty$, and whenever ${\bf x}^q M_t = 0$, we have $q \cdot t = \infty$. In particular, $T'$ satisfies the first and third conditions for a tight grading in Definition <ref>.
Next, let $T_1', \ldots, T_r'$ denote the distinct $Q$-orbits of $T' \setminus \{\infty\}$, and let $T''$ denote the the disjiont union of the sets $T_1', \ldots, T_r'$ together with distinguished elements $\infty_1, \ldots, \infty_r$. Define a $Q$-module structure on $T''$ so that $\infty_i$ is nil for each $i \le r$, and
$q \cdot t = \left\{\begin{array}{ll}
\infty_i & q \cdot t = \infty \\
q \cdot t \in T_i' & \text{otherwise}
\end{array}\right.$
for $t \in T_i'$ and $q \in Q$. Since the orbit of each nil $\infty_i$ of $T''$ with trivial support in $M$ is properly connected, $T''$ tightly grades $M$. This completes the proof.
Suppose $M$ and $N$ are $\kk[Q]$-modules, graded by $Q$-modules $T$ and $U$, respectively.
* A homomorphism $\phi:M \to N$ is said to be graded with degree $\psi \in \Hom_Q(T,U)$ if for each $t \in T$, we have $\phi(M_t) \subset N_{\psi(t)}$.
* Let $\underline{\Hom}_R(M,N)_\psi$ denote the set of morphisms $\phi:M \to N$ of degree $\psi$, and write
$$\underline{\Hom}_R(M,N) = \bigoplus_{\psi \in \Hom_Q(T,U)} \underline{\Hom}_R(M,N)_\psi$$
for the set of graded homomorphisms from $M$ to $N$. $\underline{\Hom}_R(M,N)$ is naturally a $\Hom_Q(T,U)$-graded $\kk[Q]$-module with the action of ${\bf t}^q$ given by $({\bf t}^q \cdot \phi)(m) = \phi({\bf t}^q \cdot m)$ for each $m \in M$, $q \in Q$.
* A homomorphism $\phi \in \underline{\Hom}_R(M,N)$ is homogeneous if it is a sum of homomorphisms with homogeneous degree in $\Hom_Q(T,U)$.
The category of tightly graded $\kk[Q]$-modules is the category $\cB_Q$ whose objects are pairs $(M,T)$ consisting of a $Q$-module $T$ together with a $\kk[Q]$-module $M$ tightly graded by $T$, and whose morphisms are graded $\kk[Q]$-module homomorphisms. When there is no confusion, we often write $M \in \cB_Q$ to denote the $\kk[Q]$ module and use $T_M$ to denote the $Q$-module which tightly grades $M$.
The category $\cB_Q$ is closed under taking direct sums and tensor products. More precisely, given two $\kk[Q]$-modules $M$ and $N$ tightly graded by $Q$-modules $T$ and $U$, respectively, the direct sum $M \oplus N$ is naturally graded by $T \oplus U$, and the tensor product $M \otimes_{\kk[Q]} N$ is naturally graded by $T \otimes_Q U$.
The homogeneous elements of $M \oplus N$ have the form $(m,0), (0,n)$ for homogeneous $m \in M_t$, $n \in N_u$, and the degree map is given by $\deg(m,0) = t \in T \oplus U$, $\deg(0,n) = u \in T \oplus U$. The homogeneous elements of $M \otimes_{\kk[Q]} N$ have the form $m \otimes n$ for homogeneous $m \in M_t$, $n \in N_u$, and the degree map is given by $\deg(m \otimes n) = t \otimes u$. Notice that
$$\deg(m \otimes ({\bf t}^q \cdot n)) = t \otimes (q \cdot u) = (q \cdot t) \otimes u = \deg(({\bf t}^q \cdot m) \otimes n)$$
so this degree map is well defined.
§ MESOPRIMARY $\KK[Q]$-MODULES
In this section, we define binomial submodules of tightly graded $\kk[Q]$-modules (Definition <ref>), generalizing the concept of “binomial ideal”. We define mesoprimary binomial submodules (Definition <ref>), which, like mesoprimary binomial ideals, are characterized by their unique associated mesoprime ideal (Theorem <ref>).
Fix a tightly $T$-graded $\kk[Q]$-module $M$ and a nonzero element $m \in M$.
* The element $m$ is a monomial if it is homogeneous under the $T$-grading.
* The element $m$ is a binomial if it is a sum of at most two monomial elements.
* A submodule $N \subset M$ is monomial (resp. binomial) if it is generated by monomial (resp. binomial) elements.
Fix a tightly $T$-graded $\kk[Q]$-module $M$, and a binomial submodule $N \subset M$. Let $\til_N$ denote the equivalence relation on $T$ which sets $a \sim_N b$ whenever $m_a + m_b \in N$ for some nonzero $m_a \in M_a, m_b \in M_b$. Then $\til_N$ is a congruence on $T$, and $\ol M = M/N$ is tightly graded by $\ol T = T/\til_N$.
It is clear that $\til_N$ is a congruence on $T$, and that $\ol T$ finely grades $\ol M$. If $\ol t \in \ol T$ is non-nil, then each representative $t \in T$ for $\ol t$ is non-nil, meaning $\dim_\kk \ol M_\ol t = 1$. Additionally, if ${\bf x}^q \cdot \ol m = 0$ for some nonzero $\ol m \in \ol M_\ol t$, then any nonzero $m \in M$ whose image in $\ol M$ equals $\ol m$ satisfies ${\bf x}^q \cdot m = 0$. Since $T$ tightly grades $M$, this means $\dim_\kk M_{q \cdot t} = 0$, so $\dim_\kk \ol M_{q \cdot \ol t} = 0$. Lastly, if $\ol t \in \ol T$ is nil and $\dim_\kk \ol M_\ol t = 0$, then each representative $t \in T$ of $\ol t$ is nil and satsfies $\dim_\kk M_t = 0$. As such, $N$ cannot contain any nonzero binomials whose monomials have image of degree $\ol t$ in $\ol M$, so $\ol T$ tightly grades $\ol M$, as desired.
A tightly $T$-graded $\kk[Q]$-module $M$ is mesoprimary if $T$ is a mesoprimary $Q$-module and $M_\infty = 0$ for each nil $\infty \in T$. A binomial submodule $N \subset M$ is mesoprimary if $M/N$ is a mesoprimary $\kk[Q]$-module.
If $I \subset \kk[Q]$ is a binomial ideal, then $\kk[Q]/I$ is tightly $T$-graded for $T = Q/\til_I$. Moreover, $\kk[Q]/I$ is mesoprimary when $T$ is mesoprimary and $I$ is maximal among binomial ideals inducing the congruence $\til_I$. By Lemma <ref>, this is precisely when $I$ is mesoprimary; see <cit.>.
Definition <ref> generalizes <cit.>.
Fix a tightly $T$-graded $\kk[Q]$-module $M$, and a binomial submodule $N \subset M$. Fix a monoid prime $P \subset Q$, and let $G$ denote the unit group of $Q_P$.
* The monomial localization of $M$ at $P$, denoted $M_P$, is the $\kk[Q]_P$-module obtained by adjoining to $M$ the inverses of all monomials outside of the monomial ideal $\mm_P = \<{\bf x}^p : p \in P\>$.
* An element $w \in T$ is an $N$-witness for $P$ if $w$ is a $\til_N$-witness for $P$ on $T$, and $w$ is essential if a nonzero element of $M_w$ is minimal (under Green's preorder) among the monomials of some element $m \in M$ annihilated by $\mm_P$ in $M_P/N_P$. A nonzero monomial $m_w \in M_w$ is called a monomial $P$-witness for $N$.
* Fix a monomial $N$-witness $m \in M_w$ for $P$. The stabalizer of $w$ along a prime $P \subset Q$ is the subgroup $K_w^P \subset G_P$ fixing the class of $w$ in $T_P$. The character at $m$ is the homomorphism $\rho:K_w^P \to \kk^*$ such that $(I_{\rho,P})_P = \ann(\ol m) + \mm_P$, where $\ol m$ denotes the image of $m$ in $M_P$. The $P$-mesoprime of $M$ at $m$ is the mesoprime ideal $I_{\rho,P}$.
* If $m \in M_w$ is an essential $N$-witness for $P$, we say the mesoprime $I_{\rho,P}$ is associated to $N$, and $m$ is an $N$-witness for $I_{\rho,P}$.
Proposition <ref> generalizes <cit.>, and can be proven using a similar argument to Theorem <ref>.
Any binomial submodule of a tightly graded $\kk[Q]$-module has finitely many essential witnesses.
Theorem <ref> generalizes <cit.>, and may fail to hold if the grading $Q$-module is not properly connected; see Example <ref>.
Fix a tightly $T$-graded $\kk[Q]$-module $M$ with $T$ properly connected, and fix a binomial submodule $N \subset M$. Then $N$ is mesoprimary if and only if $N$ has exactly one associated mesoprime.
Let $\ol M = M/N$ and $\ol T = T/\til_N$. First, suppose $N$ has exactly one associated mesoprime $I_{\rho,P}$. This means the $P$-prime congruences agree at all $N$-witnesses, so $\ol T$ has exactly one associated prime congruence, and thus is mesoprimary. Furthermore, if $\ol M_\infty \ne 0$ for some nil $\infty \in \ol T$, then the associated mesoprime at any nonzero element of $\ol M_\infty$ differs from the associated mesoprime in any non-nil degree $t \in T$ with $q \cdot \ol t = \infty$ for some $q \in Q$.
Next, suppose $N$ is mesoprimary. Fix $N$-witnesses $w, w' \in T$ with associated mesoprimes $I_{\rho,P}$ and $I_{\rho',P}$, respectively, and suppose $w = q \cdot w'$. Since $T$ has exactly one associated prime congruence, multiplication by ${\bf t}^q$ induces an isomorphism $I_{\rho,P} \to I_{\rho',P}$, that is, the associated mesoprimes at $w$ and $w'$ coincide. Since $T$ is properly connected, this shows that the associated mesoprimes at every $M$-witness coincide.
Resuming notation from Theorem <ref>, the result may fail to hold in general if $T$ is not properly connected. Let $I = \<x - 1, y\>, J = \<x - 2, y\> \subset \kk[x,y]$, $M = \kk[x,y]/I \oplus \kk[x,y]/J$, and $T = Q/\til_I \oplus Q/\til_J$ with nils identified. Even though $M$ is mesoprimary, it has two distinct associated mesoprimes, one for each connected component of $Q/\til_I \oplus Q/\til_J$.
§ MESOPRIMARY DECOMPOSITION OF BINOMIAL SUBMODULES
In this section, we use mesoprimary submodules (Definition <ref>) to construct a mesoprimary decomposition of any binomial submodule of a tightly graded $\kk[Q]$-module (Theorem <ref>), thus completing our answer to Problem <ref>.
Fix a tightly $T$-graded $\kk[Q]$-module $M$, a binomial submodule $N \subset M$, a prime $P \subset Q$, and a monomial $N$-witness $m_w \in M_w$ for $P$.
* The monomial submodule cogenerated by $w$ along $P$ is the submodule $M_w^P(N) \subset M$ generated in those degrees $u \in T$ that lie outside of the order ideal $T_{\le w}^P$ cogenerated by $w$ along $P$ under the congruence $\til_N$.
* The $P$-mesoprime component of $N$ cogenerated by $w$ is the preimage $W_w^P(I)$ in $M$ of the submodule $N_P + C_w^P(N) + M_w^P(N) \subset M_P$, where
$$C_w^P(N) = \<m_a - m_b : {\bf x}^q (m_a - m_b) \in M_w\>.$$
Resume notation from Definition <ref>. If $M = \kk[Q]$ and $N = I$ is a binomial ideal, then $C_w^P(I) = I_{\rho,P} \subset M$, so Definition <ref> is equivalent to <cit.> in this case. In general, we have $I_{\rho,P}M_P \subset C_w^P(I)$, but equality need not hold. Let $Q = \NN^2$, $M = (\kk[x,y]/\<y - xy, y^2\>)^{\oplus 2}$, and $T$ the $Q$-module that tightly grades $M$. Write $e_1$ and $e_2$ for the generators of the summands of $M$, and let $N = \<ye_1 - ye_2\>$, $w = ye_1$. Then the associated mesoprime at $w$ is $I_{\rho,P} = \<1 - x, y\>$ and $W_w^P(N) = \<e_1 - e_2, e_1(1 - x), e_2(1 - x)\>$. Here, $W_w^P(N)$ must contain the binomial $e_1 - e_2$ in order to induce the desired coprincipal congruence on $T$, and this is not captured in the combinatorial data of $I_{\rho,P}$ alone.
Fix a tightly $T$-graded $\kk[Q]$-module $M$, a binomial submodule $N \subset M$, a prime $P \subset Q$, and an $N$-witness $w \in T$ for $P$. The submodule $W_w^P(N)$ is mesoprimary with associated mesoprime $I_{\rho,P}$. In particular, if $N$ induces the congruence $\til$ on $T$, then $W_w^P(N)$ induces the coprincipal congruence $\til_w^P$.
Let $\app$ denote the congruence induced by $W_w^P(N)$.
We can see from the definitions that $\til_w^P$ refines $\app$, so it remains to show that no further relations are added. Since the congruences induced by $N_P$ and $C_w^P(N)$ both refine $\til_w^P$, it suffices to show that the nil class of $\app$ is identical to that of $\til_w^P$. That is, we must check that whenever $a \sim b$ and $a \approx b$ for non-nil $a, b \in T$, these relations are induced by same binomial elements in $N_P$ and $C_w^P(N)$. Suppose $m_a - m_b \in N$ for nonzero $m_a \in M_a$, $m_b \in M_b$ such that $a \approx b$ but $m_a, m_b \notin M_w^P(N)$. Since $a, b \in T_{\preceq w}^P(\til)$, we can find $q \in Q$ so that $q \cdot a$ and $q \cdot b$ are Green's equivalent to $w$ in $T_P$. This means ${\bf x}^q m_a - {\bf x}^q m_b \in M_w \cap N$, and since $w$ is not in the nil class of $\til$, we must have ${\bf x}^q m_a - {\bf x}^q m_b = 0$. In particular, this means $m_a - m_b \in C_w^P(N)$, as desired.
Any binomial submodule $N$ of a tightly $T$-graded $\kk[Q]$-module $M$ equals the intersection of the coprincipal components cogenerated by its essential witnesses.
Pick an element $m \in M$ outside of $N$. The goal is to find an essential witness $w$ and a monoid prime $P$ such that $m$ lies outside of the coprincipal component $W_w^P(N)$.
First, suppose the image of $m$ lies outside of the localization $N_P$ along a maximal prime $P$ of $Q$. Replacing $m$ with a monomial multiple of $m$, it suffices to assume that $\mm_P m \subset N$, that is to say, $m$ is annihilated (modulo $N$) by the maximal ideal $\mm_P$. This means some monomial of $m$ has as its graded degree an essential witness $w$ for $P$. By the minimality of $w$, the image of $m$ modulo $W_w^P(N)$ lies in the image of $M_w$ modulo $W_w^P(N)$, which is nonzero by Proposition <ref>.
Next, suppose the image of $m$ under localization along some non-maximal monoid prime $P$ lies outside of $N_P$. The above argument implies that the localized image of $m$ lies outside of some $P$-coprincipal component of $N_P$, which by Definition <ref> equals the localization $W_P$ of a $P$-coprincipal component $W$ of $N$. Since localizing $W$ along $P$ is injective, this completes the proof.
§ PRIMARY DECOMPOSITION OF BINOMIAL SUBMODULES
In this section, we extend the results of <cit.> to construct a primary decomposition for any binomial submodule of a tightly graded $\kk[Q]$-module. More specifically, the results presented in this section directly parallel those found in <cit.>, <cit.>, and <cit.>, used to construct primary decompositions of mesoprimary binomial ideals over an algebraically closed field. Corollary <ref>, together with Theorem <ref>, yield a combinatorial method of primarily decomposing a binomial submodule whenever $\kk = \ol\kk$ is algebraically closed.
Fix a tightly $T$-graded $\kk[Q]$-module $M$ and a mesoprimary binomial submodule $N \subset M$. The associated primes of $N$ are precisely the associated primes of its unique associated mesoprime $I_{\rho,P}$. In particular, $N$ is primary if and only if its associated mesoprime is prime.
Suppose $\ol T_P = T_P/\til_N$ has unit group $G$. Notice that localizing along $P$ induces an injection $M/N \hookrightarrow (M/N)_P$ since the monomials outside of $\mm_P$ are nonzerodivisors on the quotient modulo any $P$-mesoprimary ideal. Moreover, by Theorems <ref> and <ref>, $(M/N)_P$ has finitely many nonzero $(\ol T_P/G)$-graded pieces, all isomorphic to $(\kk[Q]/I_{\rho,P})_P$. The partial order on $\ol T_P/G$ afforded by Lemma <ref> induces a filtration of $(M/N)_P$ by $M_P$-submodules, each free of finite rank as a module over $(\kk[Q]/I_{\rho,P})_P$.
Suppose $\kk = \ol\kk$ is algebraically closed. Fix a tightly $T$-graded $\kk[Q]$-module $M$ and a mesoprimary binomial submodule $N \subset M$. If $I_{\rho,P}$ is the unique associated mesoprime of $N$ and $I_{\rho,P} = \bigcap_\sigma I_{\sigma,P}$ is the unique primary decomposition of $I_{\rho,P}$ from <cit.>, then
$$N = \textstyle\bigcap_\sigma (N + I_{\sigma,P}M)$$
is the unique minimal primary decomposition of $N$.
Each submodule $N + I_{\sigma,P}M \subset M$ is binomial and mesoprimary, and thus primary by Proposition <ref>. The intersection $\textstyle\bigcap_\sigma (N + I_{\sigma,P}M)$ certainly contains $N$, and the converse follows from the equality $N = N + I_{\rho,P}M$.
Suppose $\kk = \ol\kk$ is algebraically closed. Every binomial submodule $N \subset M$ of a tightly $T$-graded $\kk[Q]$-module $M$ admits a primary decomposition in which each component is again binomial.
Apply Theorem <ref> to construct a mesoprimary decomposition for $N$, then apply Theorem <ref> to each mesoprimary component.
In <cit.>, mesoprimary decomposition is used to combinatorially construct irreducible decompositions of binomial ideals, using “soccular decomposition” as an intermediate step. It remains an interesting question to extend soccular decomposition to tightly graded modules; we record this here.
Extend soccular decomposition <cit.> to tightly graded modules.
§ ACKNOWLEDGEMENTS
The author would like to thank Ezra Miller for his energy and patience during the author's time as a graduate student, and for motivating the work presented here. The author would also like to thank Thomas Kahle and Laura Matusevich for their continued encouragement and for many useful discussions. Much of this work was completed while the author was a graduate student, funded in part by Ezra Miller's NSF Grant DMS-1001437. Some results from Sections <ref>-<ref> also appear in <cit.>.
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Algebra and Number Theory 8 (2014), no. 6, 1297–1364.
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1511.00128
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We propose a new notion called `extremal depth' (ED) for functional data, discuss its properties, and compare its performance with existing concepts. The proposed notion is based on a measure of extreme `outlyingness'. ED has several desirable properties that are not shared by other notions and is especially well suited for obtaining central regions of functional data and function spaces. In particular: a) the central region achieves the nominal (desired) simultaneous coverage probability; b) there is a correspondence between ED-based (simultaneous) central regions and appropriate pointwise central regions; and c) the method is resistant to certain classes of functional outliers. The paper examines the performance of ED and compares it with other depth notions. Its usefulness is demonstrated through applications to constructing central regions, functional boxplots, outlier detection, and simultaneous confidence bands in regression problems.
Keywords: data depth, central regions, functional boxplots, outlier detection, simultaneous inference
§ INTRODUCTION
Ranks, order-statistics, and quantiles have been used extensively for statistical inference with univariate data. Many authors have studied their generalizations for multivariate data using notions of “data depth". The classical measure based on Mahalanobis distance <cit.> is ideally suited for multivariate normal (or more generally elliptical) distributions. Tukey's half-space depth <cit.> appears to be the first new notion for the multivariate case, and there has been a lot of work since then. <cit.> defined a `median' for multivariate data using the $L_1$ metric, and <cit.> extended this to obtain a notion of multivariate depth. Other concepts include simplicial depth <cit.>, geometric notion of quantiles <cit.>, projection depth <cit.>, and spatial depth <cit.>. See <cit.> for a review. Various types of statistical inference have also been based on multivariate depth notions, including classification <cit.>, outlier detection <cit.>, and hypothesis testing <cit.>. <cit.> studied the use of depth-based methods for inference on distributional quantities such as location, scale, bias, skewness and kurtosis.
For functional data, <cit.> proposed integrated data depth (ID); <cit.> introduced band depth (BD) and modified band depth (MBD); and <cit.> proposed a half-region depth (HRD). Several other notions of depth for multivariate data have also been extended to functional data. For instance, <cit.> developed spatial depth (SD) for functional data. One can also extend <cit.>'s projection-based depth functions and multivariate medians to functional data. However, several of these notions and extensions suffer from a “degeneracy" problem pointed out in <cit.>. Specifically, in infinite-dimensional function spaces, with probability one, all the functions will have zero depth <cit.>.
As with multivariate data, functional depth can be used for many applications. <cit.> used ID for constructing trimmed functional mean. <cit.> used BD for classification of functional data, <cit.> proposed functional boxplots based on MBD, <cit.> proposed some notions of extremality for functional data and used them to construct rank tests, and <cit.> considered functional outlier detection based on some measures of depth and outlyingness. Depth notions can also be used to obtain central regions of data which, for instance, form the basis for constructing boxplots.
Both ID and MBD are based on some form of averaging of the depth at different points in the domain and, as a result, their depth level sets are not convex. This has important implications for corresponding central regions as discussed in later sections. In addition, they may not be resistant to functions that are outlying in small regions of the domain.
This paper develops a new notion called Extremal Depth (ED) for functional data. We will show that ED and associated central regions possess several attractive features including:
* ED central regions achieve their nominal coverage exactly due to the convexity of the depth contours;
* There is a direct correspondence between the (simultaneous) ED central regions and the usual pointwise central regions based on
quantiles; as a consequence, the width of the ED simultaneous central regions is, roughly speaking, proportional to a measure of variation at each point; and
* ED central regions are resistant to functions that are `outlying' even in a small region of the domain.
These features lead to desirable properties for corresponding functional boxplots, simultaneous confidence regions for function estimation, and outlier detection.
The rest of the article is organized as follows. Section <ref> introduces ED for a sample of functional data and illustrates it on a real dataset. Section <ref> defines ED for general probability distributions and discusses its theoretical properties. Section <ref> deals with construction of central regions of functional data and develops several results including exact coverage and correspondence to pointwise regions. Section <ref> describes applications to functional boxplots and outlier detection, and the advantages of ED-based methods over others. Section <ref> demonstrates how ED can be used to construct simultaneous confidence bands for functional parameters.
§ EXTREMAL DEPTH
§.§ Depth distribution
Let $S:=\{ f_1(t), f_2(t), \cdots, f_n(t)\}$ be a collection of $n$ functional observations with $t \in {\Tau}$. For ease of exposition, we assume throughout that the functions are continuous and infinite-dimensional and, without loss of generality, we take the domain $\mathscr{\Tau}$ to be $[0, 1]$. However, as with other notions, ED can also be used for functional observations observed at a finite number of points.
Let $g(t)$ be a given function that may or may not be a member of $S$. For each fixed $t \in [0, 1]$, define the pointwise depth of $g(t)$ with respect to $S$ as
\begin{equation}\label{pointwise-depth}
\begin{array}{rll}
\displaystyle D_{g} (t, S) & \displaystyle := 1 - \frac{|\sum_{i=1}^n [\mathbbm{1}\{ f_i(t) < g(t) \} - \mathbbm{1}\{ f_i(t) > g(t) \}] |}{n}. \vspace{-2ex}
\end{array}
\end{equation}
Thus, any given function $g(\cdot)$ is mapped into the pointwise depth function $D_g(\cdot, S)$ whose range is $\mathbb{D}_g \subset\{0, 1/n, 2/n, \cdots, 1 \}$. Let $\mathbb{D}$ be the union of $\mathbb{D}_g $ over all functions $g$. We call $\mathbb{D}$ the set of depth values.
Let $\Phi_g(\cdot)$ be the cumulative distributions function (CDF) of the distinct values taken by $D_g(t, S)$ as $t$ varies in $[0, ~1]$. This will be called the depth CDF or d-CDF and defined formally as
\begin{equation}\label{dcdf}
\Phi_g (r) = \int_{0}^1 \mathbbm{1}\{D_{g} (t, S) \leq r\}dt , \vspace{-2ex}
\end{equation}
for each fixed $r \in \mathbb{D}$. Note that if $ \Phi_g $ has most of its mass close to zero (or one), then $g$ is away from (or close to) the center of the data. (See the illustrative example in Figure <ref> for computation of d-CDFs.)
In this paper, we will focus on depth measures based on d-CDFs (distributions). One concern might be that the d-CDFs do not use the dependence structure of the functional data. However, such information is generally unavailable. Therefore, we will develop methods that do not require the knowledge of the dependence relationships and that can be applied to a broad class of problems. Note that this is also the approach taken in the literature on functional and multivariate depth.
We need an appropriate way to order these d-CDFs to get a one-dimensional notion of depth. (Clearly, there is no single approach that will dominate all others, so one has to decide on the appropriate one by examining its performance under different situations.) First-order stochastic dominance may appear to be the most natural way to order distributions, but it is not useful here except in the trivial case where the functions do not cross. Alternatively, one can use a simple functional of the d-CDFs such as the mean or median. In fact, the integrated depth (or ID) by <cit.> corresponds (approximately) to the average pointwise depth $\int_0^1 D_g(t, S) ~dt,$ which is also the mean corresponding to the depth distribution $\Phi_g (\cdot)$. [We say “approximately” due to the minor difference: we use $[\mathbbm{1}\{ f_i(t) < g(t) \} - \mathbbm{1}\{ f_i(t) > g(t) \}]$ in the definition of $D_g$ while <cit.>'s definition of ID is based on $[\mathbbm{1}\{ f_i(t) \leq g(t) \} - \mathbbm{1}\{ f_i(t) > g(t) \}]$.] Modified Band Depth (MBD) is also related to the d-CDFs although the relationship between d-CDFs and MBD is more complex than the one between d-CDFs and ID. This is because MBD can be expressed as the average of the univariate simplicial depth, and the univariate simplicial depth has a monotone relationship with the univariate depth $D_g(t,S)$. Resultantly, MBD corresponds to the mean of a non-trivial function with respect to the depth distribution $\Phi_g (\cdot)$. We will provide a comparison of various functional depths in Section <ref>.
§.§ Definition of Extremal Depth
Our notion of extremal depth will be based on a comparison of $\Phi_g(r)$, the d-CDFs, for $r$ near zero. It focuses on the left tail of the distribution and can be viewed as left-tail stochastic ordering. The idea can be explained simply as follows. Consider two functions $g$ and $h$ with corresponding d-CDFs $\Phi_g$ and $\Phi_h$. Let $0 \leq d_1< d_2< \cdots < d_M \leq 1$ be the ordered elements of their combined depth levels. If $\Phi_h(d_1) > \Phi_g(d_1)$, then $h \prec g$ (or equivalently $g \succ h$, and is read as $h$ is more extreme then $g$); if $\Phi_g(d_1) > \Phi_h(d_1)$, then $ h \succ g$. If $\Phi_g(d_1) = \Phi_h(d_1)$, we move to $d_2$ and make a similar comparison based on their values at $d_2$. The comparison is repeated until the tie is broken. If $\Phi_g(d_i) = \Phi_h(d_i)$ for all $i = 1, ... M$, the two functions are equivalent in terms of depth and are denoted as $g \sim h$. (This ordering is defined formally in Section <ref> when we consider a more general context with arbitrary function spaces $S$ and distributions.)
The extremal depth (ED) of a function $g$ with respect to the sample $S = \{f_1, \cdots, f_n \}$ can now be defined as
\begin{equation}\label{depth-dpmf}
ED(g, S) = \frac{ \# \{i : g \succeq f_i \}}{n}, \vspace{-2ex}
\end{equation}
where $ g \succeq f_i $ if either $ g \succ f_i $ or $ g \sim f_i$. If $g \in S$, then this is just the normalized rank of $g$; i.e., $ED(g, S) = R(g, S) / n$ where $R(g, S) = \{i: g \succeq f_i \}$ is the rank of $g$. This relationship between ED and its rank is similar to corresponding relationships of normalized rank functions for some other depth notions in the literature <cit.>. The distinguishing feature of ED is the nature of the ordering, i.e., left-tail stochastic ordering of the depth distributions.
The ED median of a set of functional observations $S$ can be defined (in an obvious manner) as the function (or functions) in $S$ that has (or have) the largest depth. ED median also has the following max-min interpretation. For a function $g \in S$,
let $d_{\min}(g) = \inf_{t \in [0, 1]} D_{g}(t, S)$, the pointwise depth in Equation (<ref>). Then, if $g$ is an ED median, $d_{\min}(g)$ attains the maximum: $ d_{\min}(g) = \max_{1 \le k \le n} d_{\min}(f_k)$; i.e., an ED median maximizes the minimum pointwise depth over $t \in [0, ~1]$. This is so because otherwise, there exists a sample function $f_j$ having $d_{\min}(f_j)$ larger than $d_{\min}(g)$. This implies that $\Phi_{f_j} (d_{\min}(g))= 0$ but $ \Phi_{g} (d_{\min}(g)) >0$. The definition of ED then implies that $g \prec f_j$ and $ED(g, S) < ED(f_j, S)$, which is a contradiction because $g$ is a median.
An illustrative example: (a) eight sample functions and (b) their depth CDF's. The columns correspond to each of four depth levels $\{1/8, 3/8, 5/8,7/8 \}$ and the rows correspond to different sample functions.
$r = 1/8$ $ r = 3/8$ $r = 5/8$ $r = 7/8$
$ { \Phi_1 (r)}$ 0.60 0.60 0.70 1.00
$ { \Phi_2 (r)}$ 0.00 1.00 1.00 1.00
$ { \Phi_3 (r)}$ 0.00 0.00 0.80 1.00
$ { \Phi_4 (r)}$ 0.40 0.40 0.50 1.00
$ { \Phi_5 (r)}$ 0.25 0.50 0.60 1.00
$ { \Phi_6 (r)}$ 0.00 0.00 0.40 1.00
$ { \Phi_7 (r)}$ 0.00 0.50 1.00 1.00
$ { \Phi_8 (r)}$ 0.75 1.00 1.00 1.00
We now consider the illustrative example in Figure <ref> (a) with eight sample functions. The d-CDF's of all the functions are shown as a table in Figure <ref> (b). ED gives the ordering $ { f_8} \prec { f_1} \prec f_4 \prec f_5 \prec f_2 \prec f_7 \prec f_3 \prec f_6.$ So $f_8$ is the most extreme observation and $f_6$ is the deepest (median). Note that the ordering $f_8 \prec f_1 \prec f_4 \prec f_5$ is based on a comparison of the d-CDF values at $r = 1/8$ (these values are in bold); the ordering $f_2 \prec f_7$ is based on their d-CDF values at $r = 3/8$; and the ordering $f_3 \prec f_6$ is based on their values at $r = 5/8$. From this, we get the extremal depths of these functions as: $ED(f_8) = 1/8, ED(f_1) = 2/8$ and so on.
We now use the orthosis dataset <cit.> to illustrate ED and visually compare its performance with ID and MBD. This dataset consists of moment of force measured at the knee under four different experimental conditions, measured at 256 equally-spaced time points for seven subjects with ten replications per subject. Figure <ref> shows the results for 240 functional observations from six subjects who have similar range of moment of force values. The x-axis represents time when the measurement is taken and the y-axis shows the resultant moment of force at the knee. The sample functions are plotted in gray, while the deepest function is in blue and the two least deep functions are in red.
The three panels in Figure <ref> correspond to ED, ID and MBD respectively. We restrict attention to ID and MBD in our empirical comparisons because these notions are commonly used, non-degenerate and invariant to monotone-transformations. (These properties are discussed in Section <ref>). The medians for all three notions are qualitatively similar. However, the two extreme functions based on ID and MBD fall well within the boundaries of the entire data cloud while the two for ED are most extreme in at least some part of the domain. As we shall see, this is due to the non-convexity of the depth level sets of ID and MBD. This example also illustrates that as ED is based on “extreme outlyingness", it will penalize functions that are outliers in a short interval even if they are “representative" in the rest of the domain. So, if one is particularly interested in characterizing the overall behavior of the functions, other measures of depth may be preferable.
Orthosis data example: The three panels show the 240 functional observations (in gray) along with their two most outlying functions (in red) and the median (in blue) using ED, ID and MBD, respectively.
§ ED FOR THEORETICAL (POPULATION) DISTRIBUTIONS AND ITS PROPERTIES
There has been discussion of the desirable properties for depth notions in the literature <cit.>. We will examine the performance of ED with respect to these properties and compare it with existing notions. To do this, we first have to extend the notion of ED from sample data to theoretical (population) distributions.
§.§ Definition
Let $\mathbb{P}$ be a distribution on $C[0,1]$ and $X \sim \mathbb{P}$ be a random function. We denote $F_t$ to be the CDF of the random variable $X(t)$, and $\bar{F}_t (\cdot) = 1 - F_t(\cdot)$. For any function $g$, define the depth of $g$ at $t$ as
\begin{equation}
\begin{array}{rll}
D_g(t, X) &:=& 1 - \left|\mathbb{P} \left[ X(t) > g(t) \right] - \mathbb{P} \left[ X(t) < g(t) \right] \right|\\
& =& 1 - |\bar{F}_t(g(t)) - F_t (g(t)-) |. \vspace{-2ex}
\end{array}
\end{equation}
When the univariate distributions $F_t$ are continuous, $D_g(t, X) = 1 - |1 - 2 F_t (g(t)) |$, and is monotonically related to univariate half-space depth and simplicial depth, which are given by $\min (F_t (\cdot), 1- F_t (\cdot)),$ and $F_t (\cdot) (1 - F_t (\cdot)),$ respectively.
The d-CDF of the function $g$ is defined, similar to the finite-sample case, as
\begin{equation}\label{depth-dist}
\Phi_g (r) = \int_{[0,1]} \mathbbm{1}\{D_g(t, X) \leq r\} \, d t, \vspace{-2ex}
\end{equation}
for $r \in [0,1]$. Note that, if necessary, one can replace the uniform weight distribution in the definition of $\Phi_g (r) $ by a weighted measure to give higher or lower importance to certain regions of the domain.
As in the finite-sample case, we use the d-CDFs to obtain an ordering of functions. Because the d-CDFs now can be continuous, we need a slightly more general definition. Consider a pair of functions $g, h \in C[0,1]$, and define
\begin{equation}\label{dijplus}
r^* = \inf \{r \in [0,1]: \Phi_{g}(r) \neq \Phi_{h}(r) \}, \vspace{-2ex}
\end{equation}
the infimum of values at which d-CDFs of $g$ and $h$ differ. Then, we say $h \prec g$ ($h$ more extreme than $g$) if there exists $\delta >0$ such that $\Phi_{h}(r) > \Phi_g(r)$ for all $r \in (r^*, ~r^* + \delta).$ If $r^* <1,$ such a $\delta$ exists as long as $ \Phi_{g}$ and $ \Phi_{h}$ have finitely many crossings (see Appendix for a more general definition when such $\delta$ may not exist). If $r^* = 1,$ we say that $g \sim h$.
Extremal depth of a function $g$ w.r.t. the distribution $\mathbb{P}$ is now defined as
\begin{equation}\label{depth-def}
ED (g, \mathbb{P}) :=1 - \mathbb{P} \left[ g \prec X \right] = \mathbb{P} \left[ g \succeq X \right], \text{ where } X \sim \mathbb{P}. %\vspace{-2ex}
\end{equation}
§.§ Properties
<cit.> proposed several desirable properties for multivariate depth notions, and <cit.> extended them for functional depth. The first four properties below are satisfied by ED, ID, BD and MBD but not by some others. The next two concepts discussed below, convexity and `null at the boundary' (NAB), are satisfied by ED but not by ID and MBD. The convexity property leads to desirable shapes for central regions as shown in the next section. The NAB property is also important and is related to being resistant to outliers.
Transitivity (if $f_1 \prec f_2$ and $f_2 \prec f_3$, then $f_1 \prec f_3$) and invariance under monotone transformations (order preserving as well as order reversing) are two well-known properties. It can be easily shown that ED satisfies them, as do ID, BD and MBD (where the ordering $f_1 \prec f_2$ for ID and MBD is interpreted as $f_2$ deeper than $f_1$). However, spatial depth (SD) <cit.> does not satisfy the invariance property. The details are omitted.
Maximality of the center property requires that if there exists a natural center for the distribution of interest such as a center of symmetry, then it should have the highest depth. A center of symmetry can be formally defined as $s \in C[0,1]$ satisfying: $\mathbb{P} \left[ X(t) > s(t) \right] = \mathbb{P} \left[ X(t) < s(t) \right], \forall t \in [0,1] $. ED has a depth of one at the center of symmetry if it exists. More generally, consider the following set
\begin{equation} \label{med}
\mathscr{M}:= \left \{m \in C[0,1] : m(t) = \argmin \limits_{y} \left|\mathbb{P} \left[ X(t) > y \right] - \mathbb{P} \left[ X(t) < y \right] \right| \right \}.
\end{equation}
When $\mathscr{M}$ is nonempty, any function $m \in \mathscr{M}$ has extremal depth equal to one because $m \succeq f, ~ \forall ~ f \in C[0,1]$. This follows easily because at any point $t \in [0,1]$, $D_t(m, X) \geq D_t(f, X), \forall f$. Note that $\mathscr{M}$ contains center of symmetry when it exists and generalizes center of symmetry when it doesn't exist. However, $\mathscr{M}$ is not guaranteed to be non-empty only in the irregular case that none of the functions satisfying the “argmin condition” in Equation (<ref>) are continuous.
When a center of symmetry exists, ID and MBD also have it as the median but it may not necessarily be the median for SD. However, under the stronger notion of symmetry that $X - s$ and $s - X$ have the same distribution, SD has $s$ as the median. While BD also has the center of symmetry as its median, <cit.> showed that, for many common stochastic processes, BD assigns a depth of zero to the center of symmetry, making it not deeper than any other function.
Monotonicity from the center requires that if $m$ is a median and functions $f$, $g$ are such that $\forall t$, either $m(t) \leq g(t) \leq f(t)$ or $m(t) \geq g(t) \geq f(t),$ then we require $g$ to be at least as deep as $f$.
For any median $m \in \mathscr{M}$, the monotonicity from the median property is satisfied for ED; the extremal depth does not increase as we move away from $m$. ED, ID, BD and MBD all satisfy monotonicity from the center of symmetry, when it exists. The proof is omitted.
Maximality of the center property requires that if there exists a natural center for the distribution of interest, such as a center of symmetry, then it should have the highest depth. This holds for ED and that point is the ED median. When a center of symmetry exists, it has the highest depth for ED, ID and MBD; this is not necessarily true for SD. While BD also has the center of symmetry as its median under some conditions (<cit.>), <cit.> showed that, for many common stochastic processes, BD assigns a depth of zero to the center of symmetry, making it not deeper than any other function.
Monotonicity from the center requires that, if $m$ is a median and two functions $f$ and $g$ are such that either $m(t) \leq g(t) \leq f(t)$ or $m(t) \geq g(t) \geq f(t)$ for all $t$, then $g$ should be at least as deep as $f$. ED, ID, BD and MBD all satisfy monotonicity from the center of symmetry, when it exists. The proof is omitted.
Convex depth level sets:
For a given function $h$ and fixed $\alpha \in (0, ~1)$, define the ED level set as $\{h: ED(h, \mathbb{P}) \geq \alpha \}.$
Under a mild condition (Condition <ref>(b) in Appendix), the ED level sets are convex for each $\alpha \in (0,1)$.
This property is highly desirable for constructing central regions of a desired coverage $(1 - \alpha)$ (developed in the next section). Neither ID nor MBD is guaranteed to have convex depth level sets, which was already suggested by Figure <ref>. The proof is provided in the Appendix.
Null at the Boundary: <cit.> considered a depth notion to satisfy the `null at infinity' (NAI) property if $D(h, \mathbb{P}) \rightarrow 0$ as $\| h\| \rightarrow \infty$. It is shown in the Appendix that ED satisfies the NAI property. Neither ID nor MBD satisfies the NAI property. This can be seen, for instance, by taking functions that go to infinity in a small interval but are near the center in the rest of the domain.
The NAI notion is not very informative if $\| X\|$ is bounded with $ \mathbb{P}-$probability one. Therefore, we generalize it to the concept of `null at the boundary' (NAB) which is defined in terms of quantiles rather than norms of the functional observations. The formal definition is given in Appendix where it is also shown that ED satisfies NAB property. Although BD may satisfy convexity and NAB properties, it may do so trivially due to the degeneracy problem noted earlier. ID and MBD do not satisfy NAB, since they do not satisfy the weaker NAI property.
§.§ Convergence of Sample ED
<cit.> showed that, under suitable regularity conditions, the finite-sample versions of ID converge to the population quantity. The following proposition establishes the analogous consistency result for ED under suitable regularity conditions. The conditions and proof are given in the Appendix.
Let $C_n$ be the total number of functional crossings by any pair of functions, where $n$ is the number of sample functions. We assume that $C_n = \exp\{ o(n) \}$.
Let $\mathbb{P}$ be a stochastic process on $C[0,1]$ whose univariate CDF at $t \in [0,1]$ is denoted by $F_t$. Define $R(\delta, u) = \sup \limits_{|t-s| < \delta} |F_t(u) - F_s(u)|$. Then we assume that for any $u_0$, there is a neighborhood $B(u_0, \epsilon)$ such that $R(\delta_n,u) \rightarrow 0$ uniformly in $u \in B(u_0, \epsilon)$ as $\delta_n \rightarrow 0$. Also, we assume $\mathbb{P}$ to have Glivenko-Cantelli (GC) property uniformly over convex sets.
Condition <ref> assumes that the number of crossings of sample functions is at most exponential in sample size. Condition <ref> assumes that the CDF's of neighboring points in the domain are close. The GC property of $\mathbb{P}$ requires that the empirical distributions corresponding to $\mathbb{P}$ converge uniformly to $\mathbb{P}$ over convex sets. GC property for convex sets holds under general conditions for finite dimensional distributions <cit.>. <cit.> provides a GC type result for spatial distributions of infinite-dimensional spaces. For our following result, we assume this as a technical condition.
Let $\mathbb{P}$ be a stochastic process satisfying the regularity conditions <ref> - <ref> in the Appendix. Let $\mathbb{P}_n$ be the empirical distribution based on $n$ samples from $\mathbb{P}$. Then,
\[ \lim_{n \rightarrow \infty} \sup_{f \in C[0,1]} |ED(f, \mathbb{P}_n) - ED(f, \mathbb{P})| \rightarrow 0,\]
§.§ Non-Degeneracy of ED
<cit.> showed that several existing notions of functional depth suffer from the following degeneracy problem. For a general class of continuous time Gaussian processes, with probability one, the depth of every function is zero. This is true for BD and the extensions of projection depth and half-region depth to functional data in the literature. ID, MBD, and SD do not suffer from these problems. Proposition <ref> shows that extremal depth is non-degenerate for a general class of stochastic processes.
Consider $X = \{ h(t, Y_t)\}, t \in [0,1] $, where: i) $Y_t$ is a mean zero Gaussian process having continuous sample paths, bounded variance function $0 < \sigma^2(t) := E(Y^2(t)) <\infty$, and $\sup \{Y_t/\sigma(t), {t \in [0,1]} \}$ has a continuous distribution, ii) the function $h:[0, 1]\times \mathbb{R}$ is continuous, and iii) $h(t, .)$ is strictly increasing with $h(t, s) \rightarrow \infty$ as $s \rightarrow \infty$ for each $t \in [0, 1].$ Let $X \sim \mathbb{P}$, and define the range of ED for $X$ as $R:= \{\alpha \in [0,1]: ED(f, \mathbb{P}) = \alpha, \text{ for some } f \in C[0, ~1] \}$.
The range of ED for $X$ is $(0,1]$.
The result is proved in the Appendix.
§ CENTRAL REGIONS BASED ON ED
This section deals with construction of ED-based central regions, their theoretical properties and comparison with central regions based on other depth notions.
§.§ Definition and Properties
Consider a function space $S$ of interest (such as $C[0,1]$ or a sample of $n$ functional observations), and let $\mathbb{P}$ be the associated distribution of interest.
Let $(1- \alpha)$ be the desired coverage level, where coverage of a region $C$ is given by $\mathbb{P}[f: \inf \limits_{g \in C} g(t) \leq f(t) \leq \sup \limits_{g \in C} g(t), \forall t \in [0,1]$.] Define the lower and upper $\alpha-$envelope functions as
\begin{equation} \label{central-eq}
\begin{array}{clll}
f_L(t) := \inf \{f(t): f \in S,~ ED(f, \mathbb{P}) > \alpha \},\\ f_U(t) := \sup \{f(t): f \in S,~ ED(f, \mathbb{P}) > \alpha \}, \vspace{-2ex}
\end{array}
\end{equation}
respectively. Then, the $(1 - \alpha)$ ED central region is given by
\begin{equation} \label{envelopes}
C_{1 - \alpha} = \{f \in S: f_L(t) \leq f(t) \leq f_U(t), \forall t \in [0,1]\}. \vspace{-3ex}
\end{equation}
When $S$ is a finite set of functions and $\mathbb{P}$ is the empirical distribution, then $C_{1 - \alpha}$ is just the convex hull formed by all the sample functions having depth larger than $\alpha$. When $S$ is $C[0,1]$, and the marginal distribution of $\mathbb{P}$ at $t$ has zero mass to the right of $f_L(t)$ or to the left of $f_U(t),$ we take $f_L (t)$ and $f_U (t)$ to be the largest and smallest possible values (which retain the marginal probability of the interval $[f_L(t), f_U(t)]$), respectively.
The following proposition shows that the central region of level $\alpha$ contains at least the desired amount of coverage $(1- \alpha)$. Further, when the boundary of the central region does not have any mass, the actual coverage equals the desired coverage exactly. This property is not shared by ID or MBD, and they often tend to have over-coverage problem. The proof is provided in the Appendix.
Fix $\alpha$ in the range of ED. Define the boundary set of $C_{1 - \alpha}$ as $\partial C_{1 - \alpha} = \{f \in C_{1 - \alpha}: f(t) = f_L(t) \text{ or } f_U(t) \text{ for some } t \in [0,1]\}$. Then:
We have
\begin{equation}
1 - \alpha \leq \mathbb{P} \left[ f \in C_{1 - \alpha} \right] \leq (1 - \alpha) + \mathbb{P} \left[f \in \partial C_{1- \alpha} \right]. \vspace{-2ex}
\end{equation}
In particular, if $\mathbb{P} \left[f \in \partial C_{1 - \alpha} \right] = 0$, we have $\mathbb{P} \left[ C_{1 - \alpha} \right] = 1 - \alpha$.
As we shall see in Section <ref>, this property is very useful in achieving desired coverage in simultaneous inference problems. When $S$ is the set of $n$ sample functions, the boundary set $\partial C_{1 - \alpha}$ is the same as the set of functions in $C_{1 - \alpha}$ that equal $f_L$ or $f_U$ (defined in Equation (<ref>)) for a part of the domain. The probability $\mathbb{P} \left[f \in \partial C_{1 - \alpha} \right]$ may not be exactly zero in finite samples if there are one or more functions $f_i(t)$ which coincide with the upper or lower envelopes of the central region over an interval. However, in most situations of interest, this probability goes to zero as $n \rightarrow \infty$.
\begin{equation}
\mathbb{P} \left[ f \in C_{1 - \alpha} \right] = \mathbb{P} \left[f_L(t) \leq f(t) \leq f_U(t) , \forall t \in [0,1] \right] \geq 1 - \alpha.
\end{equation}
In particular, when $\mathbb{P} \left[\partial C_{1 - \alpha} \right] = 0$, we have $\mathbb{P} \left[ C_{1 - \alpha} \right] = 1 - \alpha$.
ED central regions have another interesting and attractive property: there is a close relationship between the ED (simultaneous) regions and the usual pointwise central regions. Specifically, for a fixed $\gamma \in (0, 1)$, let $Q_{1 - \gamma}$ be the $(1 - \gamma)$- pointwise central region given by
\begin{equation}
Q_{1 - \gamma} = \{f \in S: q_{\gamma/2} (t) \leq f(t) \leq q_{1 - \gamma/2} (t), \forall t \in [0,1]. \vspace{-2ex}
\end{equation}
Here $q_{\eta} (t)$ is the $\eta-$th quantile of the univariate distribution of $\mathbb{P}$ at $t$. Then, it is shown below that for every $\gamma \in [0, 1]$, $Q_{1 - \gamma}$ corresponds to an ED central region for some $\alpha$. Thus, every pointwise central region is an ED central region.
Let $\mathbbm{P}$ be the stochastic process of interest. For any $\gamma \in [0,1]$, there exists an ED central region $C_{1 - \alpha}$ for some $\alpha$ such that
$\mathbbm{P} \left[ Q_{1- \gamma} ~ \Delta ~ C_{1- \alpha} \right] = 0 $, where $\Delta$ denotes set difference. That is, up to a set of $\mathbbm{P}-$measure zero, the two sets are the same.
Note that while $Q_{1-\gamma}$ corresponds to an ED central region for each $\gamma$, the converse may not be true in general. However, there is indeed a one-to-one correspondence for most continuous stochastic processes. For example, let $X = \{ h(t, Y_t)\}, t \in [0, 1],$ where $Y_t$ and $h( \cdot, \cdot)$ satisfy the conditions in Proposition <ref>.
For every ED central region $C_{1 - \alpha}$ of $X$, there exists $\gamma \in [0,1]$ such that $\mathbbm{P} \left[ C_{1 - \alpha}~ \Delta ~ Q_{1 - \gamma} \right] = 0 $. In particular, all the ED central regions for $Y := \{Y_t, t \in [0,1] \}$ take the form $\{ f: -w \sigma(t) \leq f(t) \leq w\sigma (t) , \forall t \in [0,1] \}$, for some $w >0$.
The last statement of Corollary <ref> implies that ED central regions for the Gaussian process $Y$ have width proportional to the standard deviation, which are perhaps the most natural central regions. Although ED and resultant central regions do not take explicitly into account the dependence structure of the underlying processes, the properties of the ED central region will still depend on the covariance structure. That is, even though two different Gaussian processes may have the same point-wise variance, their $(1 - \alpha)$ ED central regions would be different depending on their covariance structure. This is because the covariance structure would determine how much width $w$ is needed to have $(1 - \alpha)$ coverage.
§.§ Comparison of Central Regions
Central regions of Orthosis data set: 90 % and 50 % central regions in the upper and lowe panels, respectively.
We use the orthosis dataset considered earlier to compare the central regions formed by ED with those from other functional depths. Figure <ref> compares the $90 \%$ (upper panel) and $50 \%$ (lower panel) regions formed by ED, ID and MBD. The ID and MBD central regions are defined in a similar way as the ED central regions: the convex hull formed by the deepest $(1 - \alpha) \times 100 \%$ of the sample functions.
In the upper panel, both ID and MBD regions include the peak (at the top) at around the value of 180 on x-axis while ED does not. The ID region in the lower panel ($50\%$) also includes some of this peak. Of course, one does not know the “right” answer in this case. However, the connection with pointwise intervals would suggest that behavior of the ED regions is more reasonable.
Figure <ref> is a plot of the widths of the ED, ID, and MBD central regions against the pointwise standard deviations of the data. We see that the ED central regions scale (approximately) proportionally to the pointwise standard deviations. This is not the case for the regions based on ID or MBD.
Width of the 90 % and 50 % central regions using different approaches: The blue dots are the widths versus standard deviation and the solid black line is the least squares line. It can be seen that the ED has width mostly proportional to the standard deviation while having relatively smaller or comparable width.
§ FUNCTIONAL BOXPLOTS AND OUTLIER DETECTION
§.§ Boxplots
Central regions can be readily used to construct functional boxplots that provide a summary of the data. <cit.> used MBD to develop functional boxplots that are analogous to classical boxplots for univariate data. The plot includes middle $50\%$ central region (the `box') and an envelope obtained by inflating the central 50% central region by 1.5 times its pointwise range, the boundaries of which are referred to as `whiskers'. Functions outside this envelope are considered potential candidates for outliers.
We use a simulation study to compare the performance of ED-based functional boxplots to those based on MBD (<cit.>) and ID. The models considered below in our analysis are the same as those in <cit.>.
Model 1: Baseline: $X_i (t) = 4t + e_i (t ), 1 \leq i \leq n$, where $e_i (t)$ is a Gaussian process with mean zero and covariance function $\gamma (s, t) = \exp \{-|t -s|\}$. This is the baseline model for the subsequent models.
Models $2 - 5$ include outliers. Here $\{ c_i, i =1 \leq i \leq n \}$ are indicator functions of outliers and are $i.i.d$ Bernoulli with $ p =0.1.$ That is, on average $10 \%$ of the observations are outliers. $\{ \sigma_i, i =1 \leq i \leq n \}$ are variables that take on values $\pm 1$ with equal probability and indicate the direction of the outliers; $K = 6$ is the magnitude of the outlier.
Model 2: Symmetric contamination: $Y_i (t) = X_i (t) + c_i \sigma_i K$.
Model 3: Partial contamination: Let $T_i$ be randomly generated from uniform distribution on $[0, 1]$. Then, $Y_i (t) = X_i (t) + c_i \sigma_i K$, if $t \geq T_i$ , and $Y_i (t) = X_i (t)$, if $t <T_i$.
Model 4: Contaminated by peaks: Let $T_i$ be randomly generated from uniform distribution on $[0, 1-\ell]$. Then, $Y_i (t) = X_i (t) + c_i \sigma_i K$, if $T_i \leq t \leq T_i + \ell$, and $Y_i (t) = X_i (t)$ otherwise. In the simulation, we fixed $\ell = 0.08$.
Model 5: Shape contamination with different parameters in the covariance function: $Y_i(t) = 4 t + \tilde{e}_i(t)$, where $\tilde{e}_i$ is a mean zero Gaussian process with covariance $\gamma (s, t) = k \exp \{-|t - s|^{\mu}\}$, with $k =8, \mu = 0.1$.
For the simulation, we generated $n = 100$ functional observations from the above models and evaluated them on a grid of size $50$. Only a summary of the results is given here. For the baseline model with no outliers, all of the depths lead to `well-behaved' boxplots. With outliers, ID and MBD-based boxplots exhibited undesirable features, and this was most evident for Models 3 and 4. Figure <ref> shows a sample dataset. For Model 3 (upper panel), the middle $50\%$ of the central region is affected by the $10\%$ contamination. The problem is less so for MBD but it is still evident. The issue is more serious for Model 4 where the performances of both ID and MBD are badly affected. As noted, part of the reason is that both ID and MBD rely on some type of averaging. The ED plots, which rely on the extremal property, are unaffected by the outliers in these examples.
Functional boxplots: The top and bottom panels correspond to data from Models 3 and 4, respectively. In each plot, the region in blue is the central $50 \%$ region and the lines in red are the whiskers.
§.§ Outlier Detection
This section provides a formal comparison of the performance of boxplots as outlier-detection tools. We use the same measures in <cit.> for comparison:
i) $p_c$: percentage of correctly identified outliers, and
ii) $p_f$: percentage of incorrectly detected outliers (equals the number of incorrectly identified outliers divided by total number of non-outlying functions). The standard errors of the percentages are given in parenthesis.
Table <ref> shows the results based on 100 data sets simulated using the Models 1-5 described above. We see that $p_f-$values of ED are much lower across all models. The values of $p_c$ are generally similar for the different depth notions except for model 4, where ED outperforms by a clear margin. This is not surprising as model 4 is contaminated by peaks; ID and MBD fail to find the outliers due to their “averaging" property as was evident in Figure <ref>.
These results suggest that when the outlying functions are consistently outlying in the whole domain, all three notions – ED, ID and MBD – perform well. However, when there are functions that are outlying in a subset of the domain as in Models 3 and 4, ED performs better while ID and MBD can do poorly.
Outlier detection using Functional Box-Plots: $p_c$ is the percentage of correctly identified outliers; $p_f$ is the proportion of incorrectly identified outliers. Numbers in brackets indicate their standard errors.
ED ID MBD
Model 1 $p_f$ 0.03 (0.17) 0.06 (0.25) 0.07 (0.27)
2*Model 2 $p_c$ 98.52(4.42) 98.89(3.49) 99.15(3.03)
$p_f$ 0.01 (0.10) 0.03 (0.20) 0.04 (0.21)
2*Model 3 $p_c$ 86.43(13.64) 77.24
(16.72) 83.17(13.77)
$p_f$ 0.01 (0.12) 0.03 (0.18) 0.03 (0.21)
2*Model 4 $p_c$ 84.42 (13.29) 41.06
(18.90) 45.94(18.99)
$p_f$ 0.01 (0.17) 0.04 (0.21) 0.04 (0.22)
2*Model 5 $p_c$ 75.74(16.15) 74.97
(16.91) 78.17 (15.79)
$p_f$ 0.01 (0.11) 0.03 (0.19) 0.04 (0.24)
The above discussion indicates that the corresponding estimators, such as functional trimmed means, based on ED will be more resistant to outliers. Specifically, let $m(\alpha)$ is the trimmed mean based on the sample functions in $(1-\alpha)$ ED central region. Then, the simulation results suggest that $m(\alpha)$ may remain bounded even as the outliers increase in magnitude while the corresponding trimmed means for ID and MBD can be unbounded. This result can be established formally and we plan to pursue this in the future.
§ SIMULTANEOUS INFERENCE
In problems involving functional inference, such as regression and density estimation, it is often difficult to obtain exact simultaneous confidence bands. In such cases, one can combine resampling methods, such as the bootstrap <cit.>, with central regions using functional depth to obtain simultaneous confidence regions. Under the asymptotic validity of the resampling technique, we can get approximate simultaneous confidence regions of desired coverage. This section demonstrates the application for the case of polynomial regression and compares it with other methods.
§.§ Polynomial and Other Parametric Regression
Consider the polynomial regression problem $Y(x_i) = \mu(x_i)+ \epsilon_i$ with $\mu(x_i) = \theta_0 + \theta_1 x_i + \cdots+ \theta_q x_i^q$.
The covariates $x_i$'s are fixed and the error terms $\epsilon_i$'s are $i.i.d$ with the standard regression assumptions. The goal is to get a simultaneous confidence region for $\mu(x)$ for all $x$.
It is known that there is no `exact' method for this general problem. Scheffe's method <cit.> leads to conservative regions since a polynomial of a variable $x$ of degree $q$ does not span the full $(q+1)-$dimensional Euclidean space.
The level of conservatism gets higher as the degree $q$ increases. Exact methods have been developed in special cases. <cit.> considered quadratic regression and provided confidence bands sharper than Scheffe's bands. <cit.> proposed exact bands for quadratic and cubic polynomial regressions. <cit.> developed exact bands when the errors are normally distributed using special properties of normality. We describe here a general re-sampling based approach using ED central regions.
Let $\theta = (\theta_0,\theta_1, ... \theta_q)$ denote the vector of parameters, $\hat{\theta}$ denote the usual least-squares estimator, and $\hat{\mu}(x)$ be the corresponding predictor. Consider the residuals $r_i = Y(x_i) - \hat{\mu}(x_i)$ and $\hat{s}$, the residual standard error. Generate $B$ bootstrap samples from the residuals to obtain bootstrap estimates $\hat{\theta}^*_1, \hat{\theta}^*_2, \cdots, \hat{\theta}^*_B$ of $\theta$, and $\hat{s}^*_1, \hat{s}^*_2, \cdots, \hat{s}^*_B$, of $\sigma$. Define an estimate of the polynomial mean function $\hat{\mu}(x|\hat{\theta}^*)$ in the obvious manner and the normalized (centered and scaled) version of this function as
\begin{equation} \label{bootstrap-fn}
m^*_j (x)= \frac{\hat{\mu}(x|\hat{\theta}^*_j) - \hat{\mu}(x|\hat{\theta})}{\hat{s}^*_j},
\end{equation}
for $j = 1, 2, \cdots B$.
These are pivotal quantities: their distribution is free of $\theta$ and $\sigma$. The set of normalized bootstrapped functions $S^*: = \{m^*_1, m^*_2, \cdots, m^*_B \}$ can now be treated as our functional data, and they can be used to construct the ED central region. Specifically, let $f_L^*(x)$ and $f_U^*(x)$ be the lower and upper envelopes of this region. Then, the $(1 - \alpha)-$level simultaneous confidence band for $\mu(x)$ is given by
\begin{equation}
C_n^{\alpha} = \{\mu(x): \hat{\mu}(x) + \hat{s} f_L^*(x) \leq \mu(x) \leq \hat{\mu}(x)+ \hat{s} f_U^*(x), \forall x \}. \vspace{-2ex}
\end{equation}
Based on the results in Section <ref>, and the bootstrap validity for parametric regression models <cit.>, we get $P[\mu(x) \in C_n^{\alpha} ~\forall ~x] \rightarrow (1 - \alpha)$ as $n \rightarrow \infty$.
We use a limited simulation study to examine the finite sample performance of this band and compare it with bands based on Scheffe's method and a Kolmogorov-like sup-norm statistic. The sup-norm statistic is $K_j^* = \sup_{x} (|\hat{\mu}_j^*(x) - \hat{\mu}(x)|)/\hat{s}_j^*$. The Scheffe's band is obtained in the usual manner assuming normality. The simulation was done for a degree five polynomial $\mu(x) =192 (x - 0.5)^5$; the coefficient 192 was chosen so that the absolute mean function integrates to one. This is the dashed function in the right panel of Figure <ref>. We simulated $n=100$ observations with $i.i.d.$ normal error terms having standard deviation 5; the covariate $x$ was randomly generated from $U[0, 1]$. We used $B = 2000$ bootsrap samples for obtain ED confidence bands.
Figure <ref> shows the confidence bands and the true mean function (dashed line) for one data set. The confidence band based on ED are tighter than both Scheffe's and K- bands (the band using $K_j^*$'s).
Simultaneous confidence bands: The figure on the left plots all the bootstrapped functions along with 90 % ED central region and the plot on the right gives confidence bands from the three different methods
Table <ref> gives the numerical results from the simulation study. The first row is the coverage probability and the next five rows show the power values for five different alternative polynomials. The first two alternatives are given by $P_k = C_k ~ sign(x-0.5)~ (x - 0.5)^k $ for degrees $k = 4, 6$, and $C_k$ is a constant such that $|P_k|$ integrates to one. The next three alternatives are additive shifts from the original mean function $P_5.$
As expected, the Scheffe-band is very conservative (actual coverage is $99 \%$ while the nominal coverage is only $90 \%$). The ED-band has coverage very close to $90 \%$ as desired. The coverage proportion of the K-band is close to the nominal. However, the band is wide in the middle and narrow in the tails. This leads to lower power than the ED-band for a large class of alternatives which have shift in the middle of the domain. This can be seen in last three rows of Table <ref>, where K-band has substantially lower power for the three shift alternatives. Power of ED-bands for the $P_4$, $P_6$ alternatives, which mostly differ from K-band at the tails, also remains competitive.
Level (row 1) and Power (rows 2 - 6) for 90 % simultaneous confidence bands using different methods
Scheffe K-band ED
Level ($P_5$) 0.01 0.10 0.10
$P_4$ 0.02 0.14 0.16
$P_6$ 0.03 0.17 0.19
$0.2+ P_5 $ 0.08 0.21 0.32
$0.2+ 0.2x + P_5$ 0.31 0.32 0.66
$0.2~ sign(x-0.5) + P_5$ 0.09 0.22 0.38
This application to polynomial regression can be readily extended to more general models of the form
$Y_i = \theta_0 + \phi_1({\bf x}_i) \theta_1 + \cdots + \phi_q({\bf x}_i) \theta_q + \epsilon_i,$ where $\phi_1, \cdots, \phi_q$ are splines or other known basis functions. The covariates can also be multidimensional in this framework.
§.§ Other Functional Inference Problems
The resampling approach described in the last section can be readily extended to other problems. These include the goodness-of-fit testing problem where one wants to determine if the generative model belongs to a certain parametric family of distributions. One can combine the bootstrapping technique (parametric or nonparametric) with ED central regions to construct acceptance or confidence regions. (See <cit.>, <cit.> and <cit.> for some related discussion.) While this is a classical problem, our initial studies suggest that the ED-based approach has some advantages over methods based on weighted Kolmogorov statistics.
The approach can also be used to construct confidence bands in nonparametric functional estimation problems, such as regression or density estimation. However, the justification of the ED-based regions in general function estimation problems will depend on the limiting distributions of the functional estimators and the asymptotic validity of the bootstrap. Simulation results in finite samples suggest that the convergence of the actual level to the nominal one is slow in fully nonparametric inference problems. A more extensive study is needed to understand the behavior, both theoretically and empirically.
§ CONCLUDING REMARKS
An important class of problems deals with the case where the underlying functions of interest are observed with error. In other words, instead of observing random functions $X_i(t)$ from a generative model of interest, we observe $Y_i(t) = X_i(t) + \epsilon_i(t), ~~~ i = 1, \cdots, n$. A natural approach is to use some type of smoother to `recover' $X_i(t)$ and then use the techniques discussed so far. If there is some information of the error structure in $\epsilon(t)$, this can be used to guide the smoothing algorithm or the `reconstruction' methods for $X_i(t)$.
In summary, we have developed a new notion of functional depth, studied its properties, and demonstrated its usefulness through several applications. While no single notion of functional depth will do uniformly better than others, we hope that the results here suggest that the extremal-depth concept has many attractive properties and is a useful tool for exploratory analysis of functional data. It can also be used in other applications, such as the construction of simultaneous confidence bands.
§ APPENDIX
More general definition of extremal depth:
The depth ordering defined in the paper can be generalized as follows. Consider a pair of functions $g, h \in C[0,1]$, and define
\begin{equation}\label{dijplus}
r^* = \inf \{r \in [0,1]: \Phi_{g}(r) \neq \Phi_{h}(r) \}.
\end{equation}
Then, we say $h \prec g$ if for any sequence $r_m \downarrow r^*$, we have $\Phi_{h}(r_m) > \Phi_g(r_m)$ for all $m \geq M,$ for some large enough constant $M$. Similarly, we say $h \succ g$ if for any sequence $r_m \downarrow r^*$, we have $\Phi_{h}(r_m) < \Phi_g(r_m)$ for all $m \geq M.$ Otherwise, if neither of these cases holds, we say $h \sim g$. This generalizes the definition that assumes the existence of a neighborhood in which either $\Phi_{h}(r_m) > \Phi_g(r_m)$ or $\Phi_{h}(r_m) < \Phi_g(r_m)$ holds true.
§.§.§ Condition and proof for Proposition <ref>:
Assume that
(a) $\mathbbm{P}[d_f = 0] = 0$, and (b) $\mathbbm{P}[d_f = d_g, f \neq g] = 0,$
where $f, g$ are independent random functions from $ \mathbbm{P}$ and $d_f : = \inf \limits \{r \in [0,1]: \Phi_f(r ) >0 \}$.
Proof of Proposition <ref>: We shall show that, if $ED(f_1, \mathbb{P}), ED(f_2, \mathbb{P}) \geq \alpha$, and $f_1(t) \leq f(t) \leq f_2(t)$ $\forall t \in [0,1]$, then $ED(f, \mathbb{P}) \geq \alpha.$ Note that $\forall t$, $D_{f} (t, \mathbb{P}) \geq \min (D_{f_1} (t, \mathbb{P}), D_{f_2} (t, \mathbb{P}))$, and hence $d_f \geq \min(d_{f_1}, d_{f_2})$. Therefore either $f \succeq f_1$ or $f \succeq f_2$ w.p.1 and $f \in C_{\alpha}$ due to Condition <ref> (b).
Condition <ref> is a mild condition on $\mathbbm{P}$. For instance, this holds if $ \mathbb{P}$ is the distribution of $X$ in Proposition <ref>.
§.§.§ NAB property:
Denote the pointwise quantile functions of $\mathbb{P}$ as $q_{\alpha}$, i.e., for each $t$, $\mathbb{P}[X(t) < q_{\alpha}(t)] \leq \alpha$, and $\mathbb{P}[X(t) \leq q_{\alpha}(t)] \geq \alpha$ (for uniqueness we take the smallest one).
Let for $\alpha_n \downarrow 0$, $f_n(t) \leq q_{\alpha_n}(t), \forall t \in U$, and for $\beta_n \uparrow 1$, $g_n(t) \geq q_{\beta_n}(t), \forall t \in U$, where $U$ is some open interval in $[0,1]$. Then we say the depth notion $D$ to have NAB property if $D(f_n, X) \rightarrow 0$ and $D(g_n, X) \rightarrow 0$.
We now show that ED satisfies NAB under Condition <ref> (a). Since $f_n(t) \leq q_{\alpha_n}(t), \forall t \in U$, we have $\forall n \geq N$,
\begin{equation*}
\begin{array}{lllllllll}
\mathbb{P}[f_{n+1} \succeq X] & \leq & 1 - \mathbb{P}[ q_{\alpha_n} < X < q_{1 - \alpha_n}]\\
& =& 1 - \mathbb{P}[ \cup_{k \leq n} \{ q_{\alpha_k} < X < q_{1 - \alpha_k} \}]. \vspace{-2ex}
\end{array}
\end{equation*}
$\lim \sup \mathbb{P}[f_{n+1} \succeq X] \leq 1 - \mathbb{P}[\Omega:= \cup_{1 < k < \infty} \{q_{\alpha_k} < X < q_{1 - \alpha_k} \}] = 0$, as the set $\Omega$ has probability one due to Condition <ref> (a). Therefore, $D(f_n, X) \rightarrow 0$ and similarly $D(g_n, X) \rightarrow 0$.
§.§.§ Conditions and proof of Proposition <ref>:
Let $C_n$ be the total number of functional crossings by any pair of functions, where $n$ is the number of sample functions. We assume that $C_n = \exp\{ o_{P}(n) \}$.
Let $\mathbb{P}$ be a stochastic process on $C[0,1]$ whose univariate CDF at $t \in [0,1]$ is denoted by $F_t$. Define $R(\delta, u) = \sup \limits_{|t-s| < \delta} |F_t(u) - F_s(u)|$. Then we assume that for any $u_0$, there is a neighborhood $B(u_0, \epsilon)$ such that $R(\delta_n,u) \rightarrow 0$ uniformly in $u \in B(u_0, \epsilon)$ as $\delta_n \rightarrow 0$. Further, we assume $\mathbb{P}$ to have Glivenko-Cantelli (GC) property uniformly over convex sets.
Condition <ref> assumes the number of crossings is at most exponential in sample size, and is related to the smoothness of the process. Condition <ref> assumes that the CDF's of neighboring points in the domain are close. The GC property of $\mathbb{P}$ requires that the empirical distributions corresponding to $\mathbb{P}$ converge uniformly over convex sets. GC property for convex sets holds under general conditions for finite dimensional distributions <cit.>. <cit.> provides a GC type result for spatial distributions of infinite-dimensional spaces. For our result, we assume this as a technical condition.
Let $f \succeq_n g$ and $f \succeq g$ denote that $f$ is deeper than or equal to $g$ using ED w.r.t. the empirical distribution $\mathbb{P}_n$ and the true distribution $\mathbb{P}$, respectively. Then,
\begin{equation} \label{split-eq}
\begin{array}{llllllll}
\sup \limits_{f \in C[0,1]} |ED(f, \mathbb{P}_n) - ED(f, \mathbb{P})| & = & \sup \limits_{f \in C[0,1]} |\mathbb{P}_n \left[ f \succeq_n X_n \right] - \mathbb{P} \left[ f \succeq X \right]|\\
& = & \sup \limits_{f \in C[0,1]} |\mathbb{P}_n \left[ f \succeq_n X_n \right] - \mathbb{P} \left[ f \succeq_n X \right]| \\
& & \qquad + \sup \limits_{f \in C[0,1]} |\mathbb{P} \left[ f \succeq_n X \right] - \mathbb{P} \left[ f \succeq X \right]|, \vspace{-2ex}
\end{array}
\end{equation}
where $X_n \sim \mathbb{P}_n$ and $X \sim \mathbb{P}$.
The first term in RHS of (<ref>) can be shown to go to zero because of the Glivenko-Cantelli (GC) property assumed by Condition <ref>. That is because the sets $\{ f \succeq_n X \}$ are convex and the GC type result holds over all convex subsets. We then only need to show that $\sup_{f } |\mathbb{P} \left[ f \succeq_n X \right] - \mathbb{P} \left[ f \succeq X \right]|\rightarrow 0$.
We will now show that the second term in RHS of (<ref>) goes to zero. Define $d_f : = \inf \limits_{y \in [0,1]} \{ \Phi_f(y, \mathbb{P}) >0 \}$, and $d_f^n: = \inf \limits_{y \in [0,1]} \{ \Phi_f(y, \mathbb{P}_n) >0\}.$ We shall first show that $\sup_{f} |d_f^n - d_f| \cprob 0$ as $n \rightarrow \infty$.
Due to the rate of Glivenko-Cantelli of empirical distributions <cit.>, we have for any $t$,
\begin{equation} \label{gc-ineq}
\mathbb{P} \left[ \sup \limits_{u} |F^n_t(u) - F_t(u)| > \epsilon \right] \leq \exp \{ -c \epsilon^2 n \}. \vspace{-2ex}
\end{equation}
Let $D = \{d_1, d_2, \cdots, \}$ be a countable dense subset of $[0,1]$. Define $T_n = \{ t_1, \cdots, t_{k_n}\}$ be the set containg all the points in $[0,1]$ where $n$ sample functions cross and along with $\{ d_1, d_2, \cdots, d_n \}$. As $n \rightarrow \infty$ we have $T:= \cup_{n} T_n $ is the union of all the crossing points and $D$. Due to Condition <ref>, we have $\log |k_n| = {o_P(n)}.$
Due to Equation (<ref>), we have
\begin{equation}\label{unif-conv}
\mathbb{P} \left[\sup \limits_{t \in T_n} \sup \limits_{u} |F^n_t(u) - F_t(u)| > \epsilon \right] \leq \exp \{ -c' \epsilon^2 n + \log k_n \}. \vspace{-2ex}
\end{equation}
Now, note that $d_f^n= \inf \limits_{t \in [0,1]} D_f(t, \mathbb{P} _n) = \min \limits_{t \in T_n} D_f(t, \mathbb{P} _n),$ because the univariate depths in $T_n$ have the same range as that of the whole interval $[0,1]$. We shall first show that
\begin{equation} \label{dfeq}
d_f = \inf \limits_{t \in T} D_f(t, \mathbb{P} ) =\lim_{n} \inf \limits_{t \in T_n} D_f(t, \mathbb{P} ), \vspace{-2ex}
\end{equation}
using the facts that $\cup_n T_n = T$, $T$ is dense and Condition <ref>. To see this, first note that $d_f \leq \inf \limits_{t \in T} D_f(t, \mathbb{P} )$. For the reverse inequality, consider a $y_0$ such that $D_{y_0}(f, \mathbb{P} ) = d_f$ (this exists due to continuity of $F_t$ in $t$). Since $T$ is dense, we have a sequence $y_n \in T$ such that $y_n \rightarrow y_0$. Due to continuity of $F$ and Condition <ref>, we have
\begin{equation*}
\begin{array}{lllllll}
| D_{y_n} (f, \mathbb{P} ) - D_{y_0} (f, \mathbb{P} ) |& = &| |1 - 2 F_{y_n}(f(y_n))| - |1 - 2 F_{y_0}(f(y_0))||\\
& \leq &2| F_{y_n}(f(y_n)) - F_{y_0}(f(y_0))|\\
&\leq& 2 |F_{y_n}(f(y_n)) - F_{y_0}(f(y_n))| + 2 | F_{y_0}(f(y_n)) - F_{y_0}(f(y_0))| \rightarrow 0, \vspace{-2ex}
\end{array}
\end{equation*}
which implies (<ref>). Now, using (<ref>), we have
\begin{equation*}
\begin{array}{lllllll} \mathbb{P} [\sup \limits_{f} |d_f^n - d_f| > \epsilon_n ] \leq \mathbb{P} \left[\sup \limits_{t \in T_n} \sup \limits_{u} |F^n_t(u) - F_t(u)| > \epsilon_n/4 \right] \rightarrow 0,
\end{array}
\end{equation*}
if $\epsilon_n \rightarrow 0$ and $ c' \epsilon_n^2 n - \log k_n \rightarrow \infty$. In particular, when $\epsilon_n =4 \max(\left(3 \log k_n/c'n \right)^{1/2}, 1/\sqrt{\log n})$, $ \mathbb{P} [\sup \limits_{f} |d_f^n - d_f| > \epsilon_n] < C n^{-1 - \epsilon}$, for some $C, \epsilon >0$. Then using Borel-Cantelli lemma, we obtain $\sup \limits_{f} |d_f^n - d_f| \rightarrow 0$ almost surely. Now, consider the events $A_n= \{d_f^n \geq d_g^n \}$ and $B_m= \{d_f < d_g - \delta_m \}$, where $\delta_m \rightarrow 0 $ as $m \rightarrow \infty$. Note that $A_n$ and $B_m$ depend on the functions $f$ and $g$. Then,
$\mathbb{P}[\cup_{f, g} A_n \cap B_m ] \leq \mathbb{P} [\sup \limits_{h} |d_h^n - d_h| > \epsilon_m] \rightarrow 0$ as $n \rightarrow \infty$. Therefore, we have
\begin{equation*}
\begin{array}{lllll}
\limsup_n \sup_{f } |\mathbb{P} \left[ f \succeq_n X \right] - \mathbb{P} \left[ f \succeq X \right]| & \leq &
\limsup_n \mathbb{P} \left[ \cup_{f} \{ f \succeq_n X\} \Delta \{ f \succeq X\} \right]\\
& \leq & \limsup_n \lim_m \mathbb{P} \left[ \cup_{f, g} A_n \cap B_m \right]\\
&\leq & \lim_m \limsup_n \mathbb{P} \left[ \cup_{f, g} A_n \cap B_m \right] = 0. \qed \vspace{-2ex}
\end{array}
\end{equation*}
§.§.§ Proof of Proposition <ref>:
As the process $Y= \{Y_t \}, t \in [0,1]$ has continuous sample paths, the sample paths of the process $X=\{X_t \}, t \in [0,1]$ also lie in $C[0, 1]$ almost surely. Due to the monotone invariance property of ED, we only need to show that ED of $Y$ takes all the values in $[0,1]$.
Consider the sets $Q_{1-\gamma} := \{f: q_{\gamma/2}(t) \leq f(t) \leq q_{1 - \gamma/2}(t), \forall t\},$ for $\gamma \in [0,1]$, where $q_{\alpha}$ is the $\alpha$-th pointwise quantile of $Y$. Note that $q_{\gamma/2} \preceq f,$ for any $f \in Q_{1- \gamma}$ and $q_{\gamma/2} \succ g$, for $g \in Q_{1 - \gamma}^c$. Therefore, $ED(q_{\gamma/2},\mathbb{P} ) = \mathbb{P} [ Q_{1-\gamma}]$. By noting that $Q_{1 - \gamma} = \{f: \sup \limits_{t}| f(t)/\sigma(t)| \leq c \}$ and that $\sup \limits_{t}| f(t)/\sigma(t)|$ has a continuous distribution, $\mathbb{P} [ Q_{1 - \gamma}]$ takes all the values in $(0,1]$.
§.§.§ Proof of Proposition <ref>:
To prove the lower bound, consider a function $g$ having ED equal to $\alpha$, that is, $g$ is such that $ED(g, \mathbb{P}) = \alpha$. Consider the set $A:= \{ f: f \succ g, \text{ and } f \in C_{1 - \alpha}^C \}$, then $\mathbb{P}[A] = 0$. This is because, otherwise if $\mathbb{P}[A] >0$, there exists a function $f_* \in A$ such that $ED(f_*, \mathbb{P}) > \alpha$, which is a contradition as $f_* \not \in C_{1 - \alpha}$. This implies that $\alpha = \mathbb{P}[X: g \succeq X] \geq \mathbb{P} [ C_{1 - \alpha}^C ] $, and hence $ \mathbb{P} [ C_{1 - \alpha}] \geq (1 - \alpha) $.
To prove the upper bound, we first note that the set $\{f: g \succeq f \}$ is contained in the union of the sets $C_{1 - \alpha}^C$ and $ \partial C_{1 - \alpha}.$ This is because, for any function $h \in C_{1-\alpha} - \partial C_{1 - \alpha},$ $d_{min}(h, \mathbb{P}) > d_{min}(g, \mathbb{P}),$ where $d_{\min}(h, \mathbb{P}) = \inf_{t \in [0, 1]} D_{h}(t, \mathbb{P})$ as in Section <ref>. Otherwise, we have a function $f$ with ED larger than $\alpha$ but $d_{min}(f, \mathbb{P}) <d_{min}(g, \mathbb{P}),$ which is a contradiction. Therefore, $\alpha = \mathbb{P}[X: g \succeq X ] \leq \mathbb{P}[C_{1 - \alpha}^C \cup \partial C_{1 - \alpha} ].$ This implies that $\mathbb{P}[C_{1 - \alpha} - \partial C_{1 - \alpha} ] \leq (1 - \alpha)$ and $ \mathbb{P}[C_{1 - \alpha}] \leq (1 - \alpha) + \mathbb{P} [\partial C_{1 - \alpha} ] $, and the result follows.
§.§.§ Proof of Proposition <ref> & Corollary <ref>:
We shall show that the ED central region $C^*$ formed by the functions $\{f: ED(f, \mathbbm{P}) \geq ED(q_{\gamma/2}, \mathbbm{P})\}$ proves the proposition. Although this central region is not in the form defined by Equation (<ref>) (due to “$\geq$” instead of a “$>$”), this does not make a difference when $\mathbbm{P}$ is a continuous stochastic process, and this same set can be written with a “$>$” when $ \mathbbm{P}$ is an empirical distribution. We have $ f \succeq q_{\gamma/2} \sim q_{1 - \gamma/2} \succ g$, for any $f \in Q_{1 - \gamma}$, and $g \in Q_{1 - \gamma}^C$. Therefore, $Q_{1 - \gamma} \subset C^*$ and it remains to show that $\mathbbm{P}[C^* - Q_{1 - \gamma}] = 0$. However, $C^* - Q_{1 - \gamma} \subset B:= \{f \not \in Q_{1 - \gamma}: ED(f, \mathbbm{P}) = ED(q_{\gamma/2}, \mathbbm{P})\}$. As all the functions in $B$ have the same ED, $ \mathbbm{P}[f \in B]= \mathbbm{P}[f \in B: f \sim q_{\gamma/2}] =0. $ Therefore, $\mathbbm{P}[C^* \Delta Q_{1 - \gamma}] = 0$. The corollary follows directly because ED is a decreasing function of $\sup \limits_t |f(t)|/{\sigma(t)}$.
§.§.§ Proof of Proposition <ref>:
Let us denote the new functional data as $S^* = S \cup \{ g \}$ having (n+1) functions. Any function not in $b (S)$ can not be more outlying than the function $g$ because the d-CDF of $g$ has positive mass at the least depth $1/(n+1)$. Therefore, the trimmed mean remains bounded if there are $ (n+1) (1 - \alpha)$ sample functions deeper than $g$. This is the case if $(n+1) (1 - \alpha) \leq n - b(S),$ which proves part (i).
Let us now consider the same for ID. We can choose a function $g \in B_{\delta}$ such that its ID is arbitrarily close to $(1 - \delta)$ by letting $g$ stay at the median except for an interval of length close to $\delta$. Therefore, for $g$ not to be included for computing trimmed mean $m_2^{\alpha}$, we need at least $(n+1) (1- \alpha)$ sample functions having ID (w.r.to. the $(n+1)$ functions including the outlier) to be at least $(1 - \delta)$. That is, $\mathbb{P}_{(n+1)}[f: ID(f, S^*) \geq (1 - \delta)] \geq (1- \alpha). $ By noting that
\begin{equation*} ID (f, S^*) \leq \frac{n}{n+1} ID(f, S) + \frac{2}{n+1}, \vspace{-2ex}
\end{equation*}
we have
$n\mathbb{P}_{ n} \left[f: ID(f, S) \geq \left((1 - \delta) - 2/(n+1) \right) \frac{n+1}{n} \right] \geq (n+1) (1- \alpha).$ This implies that, $g$ is not included as a central $(1-\alpha)$ function only if
\begin{equation}\label{alpha-cond}
\begin{array}{clll}
\alpha &\geq 1 - \frac{n}{ n+1 } \mathbb{P}_{ n} \left[f: ID(f, S) \geq \left((1 - \delta) - \frac{2}{n+1} \right) \frac{n+1}{n} \right]\\
&\approx 1 - \mathbb{P}_{ n} \left[f: ID(f, S) \geq (1 - \delta) \right]. \vspace{-2ex}
\end{array}
\end{equation}
A similar argument can be used for MBD. For a fixed point, the largest MBD (assuming $n$ to be odd)
\[ \frac{ (\frac{n+1}{2})^2 - 1}{{n \choose 2}} = \frac{n+3}{2 n}. \vspace{-2ex} \]
Therefore, $g$ can be chosen such that its MBD is almost as large as $\frac{(n+3)(1 - \delta)}{2n}$. As before, $MBD(f, S^*) \leq \frac{n-1}{n+1} MBD(f, S) + \frac{2}{n+1}$. Therefore, central $(1-\alpha)$ functions using MBD doesn't include $g$ only if
\begin{equation}\label{alpha-condbd}
\begin{array}{clll}
\alpha &\geq 1 - \frac{n}{ n+1 } \mathbb{P}_{n } \left[f: MBD(f, S^*) \geq \left( \frac{(n+3 )(1 - \delta)}{2 n} - \frac{2}{n+1} \right) \frac{n+1}{n-1}\right]\\
&\approx 1 - \mathbb{P}_{ n} \left[f: (1 - \delta)/2 \leq MBD(f, S) \leq 1/2 \right]. \vspace{-2ex}
\end{array}
\end{equation}
Equations (<ref>) and (<ref>) imply parts (ii) and (iii), respectively.
|
1511.00165
|
Positive configurations of points in the affine building were introduced in <cit.> as the basic object needed to define higher laminations. We start by giving a self-contained, elementary definition of positive configurations of points in the affine building and their basic properties. Then we study the geometry of these configurations. The canonical functions on triples of flags that were defined by Fock and Goncharov in <cit.> have a tropicalization that gives functions on triples of points in the affine Grassmannian. One expects that these functions, though of algebro-geometric origin, have a simple description in terms of the metric structure on the corresponding affine building.
We give a several conjectures describing the tropicalized canonical functions in terms of the geometry of affine buildings, and give proofs of some of them. The statements involve minimal networks and have some resemblance to the max-flow/min-cut theorem, which also plays a role in the proofs in unexpected ways. The conjectures can be reduced to purely algebraic statements about valuations of lattices that we argue are interesting in their own right.
One can view these conjectures as the first examples of intersection pairings between higher laminations. They fit within the framework of the Duality Conjectures of <cit.>.
§ INTRODUCTION
This paper has two goals: the first is to give a simple exposition of positive configurations of points in the affine building; the second is to begin the study of the rich geometry of this object. In particular, we lay out several conjectures that give a geometric interpretation of the beautiful set of canonical functions that were defined by Fock and Goncharov in <cit.>. These conjectures serve as the first step towards defining the intersection pairing between higher laminations, which is naturally a pairing between higher laminations for Langlands dual groups.
We start with some motivation. The canonical functions were originally defined on the space of configurations of three principal affine flags associated to the group $G=SL_n(\R)$ (with appropriate adjustments, one can let $G=PGL_n(\R)$ or $GL_n(\R)$). There is a canonical function $f_{ijk}$ for every triple of non-negative integers $i, j, k$ such that $i+j+k=n$.
To a principal affine flag, one can associate a horocycle in the symmetric space $X=G/K$, where $K \subset G$ is the maximal compact subgroup of $G$. One can interpret the value of the canonical functions on a triple of affine flags in terms of the minimal total weighted distance of a spanning network between the three corresponding horocycles.
We are interested in the tropicalization of these functions, explained in <cit.> and <cit.>. Whereas the canonical functions parameterize configurations of principal affine flags, the tropicalized canonical functions parameterize (positive) virtual configurations of points in the affine building associated to $G$. The adjective “virtual” is a technicality that will not be relevant for this paper, while “positivity” will only be mentioned in passing. The affine building associated to $G$ is the natural tropical analogue of the symmetric space.
These tropicalized functions again should have a geometric interpretation: we expect that they are given by the minimal total weighted distance of a spanning network between three horocycles in the affine building. However, we would rather work with points in the affine building than with horocycles. For this reason, we conjecture that there is a more refined and simpler description of these functions in terms of the geometry of configurations of points in the affine building.
For our purposes, we will restrict our attention to configurations of points in the affine building, which is the most important case. The extension to virtual configurations is straightforward. So the question becomes: given a configuration of three points in the affine building, how do we calculate, in a geometric way, the value of the tropicalized functions $f_{ijk}^t$ on this configuration? We conjecture that they are given by the minimal total weighted distance of a spanning network between the three points in the affine building.
Let us elaborate on the conjecture. We will see below that the affine building has a (non-symmetric) distance function valued in the coweight lattice. If $x_1$ and $x_2$ are points in the affine building, $d(x_1,x_2)$ will be a dominant coweight.
Let $x_1, x_2, x_3$ be any configuration of points in the affine Grassmannian. Let $\omega_i, \omega_j, \omega_k$ be fundamental weights of $SL_n$ with $i+j+k=n$. Then we conjecture that
$$f_{ijk}^t (x_1,x_2,x_3) = \min_{p} \{ \omega_i \cdot d(p,x_1) + \omega_j \cdot d(p,x_2) + \omega_k \cdot d(p,x_3) \},$$
where the minimum is taken over all $p$ in the affine building.
The functions $f_{ijk}^t$ are defined in terms of finding the most negative value of some determinant expression. On the other hand, the conjecture states that the same quantity is computed by minimizing some weighted network. Thus the maximum of one type of quantity is the minimum of another, and the statement resembles max-flow/min-cut.
Below, we prove two special cases of the conjecture, which are in some sense orthogonal:
* the configuration of points $x_1, x_2, x_3$ is small in a precise sense described below
* the points $x_1, x_2, x_3$ all lie in an apartment of the building
We think that combining the approaches of these two special cases could possibly prove the entire conjecture. It is very interesting to us that our proofs of both cases use max-flow/min-cut, but in each case in a very different way. We think of these conjectures as a kind of max-flow/min-cut for affine buildings.
§ BACKGROUND
§.§ Affine Grassmannian and affine buildings
We now begin by laying out the necessary definitions. First we will give an introduction to the affine Grassmannian and the affine building.
Let us define the affine Grassmannian. Let $G$ be a simple, simply-connected complex algebraic group and let $\vG$ be its Langlands dual group. Let $\F$ be a field, which for our purposes will always be $\R$ or $\C$. Let $\cO = \F[[t]]$ be the ring of formal power series over $\F$. It is a valuation ring, where the valuation $\val(x)$ of an element
$$x = \sum_k a_k t^k \in \F((t))$$
is the minimum $k$ such that $a_k \neq 0$.
Let $\cK = \F((t))$ be the fraction field of $\cO$. Then
$$\Gr(\F) = \Gr(G) = G(\cK)/G(\cO)$$
is the set of $\F$-points of the affine Grassmannian for $G$. It can be viewed as a direct limit of $\F$-varieties of increasing dimension.
For $G=SL_n$, a point in the affine Grassmannian corresponds to a finitely generated, rank $n$, $\cO$-submodule of $\cK^n$ such that if $v_1, \dots, v_n$ are generators for this submodule, then
$$v_1 \wedge \dots \wedge v_n=e_1 \wedge \dots \wedge e_n,$$
where $e_1, \dots, e_n$ is the standard basis of $\cK^n$. We will often call such full rank $\cO$-submodules lattices. $G(\cK)$ acts on the space of lattices with the stabilizer of each lattice being isomorphic to $G(\cO)$, which acts by changing the basis of the submodule while leaving the submodule itself fixed. We will later make use of this interpretation.
The affine Grassmannian $\Gr$ also has a metric valued in dominant coweights: the set of pairs of elements of $\Gr$ up to the action of $G(\cK)$ is exactly the set of double cosets
$$G(\cO) \backslash G(\cK) / G(\cO).$$
These double cosets, in turn, are in bijection with the cone $\Lambda_+$ of dominant coweights of $G$. Recall that the coweight lattice $\Lambda$ is defined as $\mathrm{Hom}(\mathbf{G}_m,T)$. The coweight lattice contains dominant coweights, those coweights lying in the dominant cone. For example, for $G=GL_n$, the set of dominant coweights is exactly the set of $\mu=(\mu_1, \dots, \mu_m)$, where $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_n$ and $\mu_i \in \Z$. Let us explain why the set of double cosets is in bijection with the set of dominant coweights.
Given any dominant coweight $\mu$ of $G$, there is an associated point $t^\mu$ in the (real) affine Grassmannian: to a coweight $\mu=(\mu_1, \dots, \mu_m)$ we associate the element of $G(\cK)$ with diagonal entries $t^{\mu_i}$, and then project to the affine Grassmannian. Any two points $p$ and $q$ of the affine Grassmannian can be translated by an element of $G(\cK)$ to $t^0$ and $t^\mu$, respectively, for some unique dominant coweight $\mu$. This gives the identification of the double coset space with $\Lambda_+$.
Under this circumstance, we will write
$$d(p,q) = \mu$$
and say that the distance from $p$ to $q$ is $\mu$.
Let us collect some facts about this distance function $d$. Note that this distance function is not symmetric; one can easily check that
$$d(p,q)=-w_0 d(p,q)$$
where $w_0$ is the longest element of the Weyl group of $G$ (recall that the Weyl group acts on both the weight space $\Lambda^*$ and its dual $\Lambda$). However, there is a partial order on $\Lambda$ defined by $\lambda > \mu$ if $\lambda - \mu$ is positive (i.e., in the positive span of the positive co-roots). Under this partial ordering, the distance function satisfies a version of the triangle inequality. By construction, the action of $G(\cK)$ on the affine Grassmannian preserves this distance function.
We are interested in the affine Grassmannian, but not in its finer structure as a variety. In fact, we will only consider properties of the affine Grassmannian that depend on the above distance function, and possibly on some positive structure. For this reason, we will introduce affine buildings, a sort of combinatorial skeleton of the affine Grassmannian.
Let us first introduce the affine building for $G=PGL_n$. The affine Grassmannian for $G=PGL_n$ consists of lattices (finitely generated, rank $n$ $\cO$-submodules of $\cK^n$) up to scale: two lattices $L$ and $L'$ are equivalent if $L=cL'$ for some $c \in \C((t))$. The set of vertices of the affine building for $PGL_n$ is precisely given by the points of the affine Grassmannian $\Gr(PGL_n)$.
For any lattices $L_0, L_1, \dots, L_k$, there is a $k$-simplex with vertices at $L_0, L_1, \dots, L_k$ if and only if (replacing each lattice by an equivalent one if necessary)
$$L_0 \subset L_0 \subset \cdots \subset L_k \subset t^{-1}L_0.$$
This gives the affine building the structure of a simplicial complex. The affine building for $G=SL_n$ is the same simplicial complex, but where we restrict our attention to those vertices that come from the affine Grassmannian for $G=SL_n$.
The non-symmetric, coweight-valued metric we defined above descends from the affine Grassmannian to the affine building. The notion of a geodesic with respect to his metric is sometimes useful. For our purposes, a geodesic in the building is a path that travels along edges in the building from vertex to vertex, such that the sum of the distances from vertex to vertex is minimal (with respect to the partial order defined above). It is a property of affine buildings that geodesics exist. Note that in general there will be many geodesics between two any points.
§.§ Canonical functions
We now define the canonical functions of triples of affine flags, which will lead up to the definition of the associated functions on triples of points in the affine Grassmannian.
Let $G=SL_n$ and let $U \subset G$ be the subgroup of unipotent upper triangular matrices. An element of $G/U$ is called a principal affine flag. In concrete terms, a principal affine flag is given by a set of $n$ vectors $v_1, \dots, v_n$ where we only care about the forms
$$v_1 \wedge \dots \wedge v_k$$
for $k=1, 2, \dots, n-1$. We will require that
$$v_1 \wedge \dots \wedge v_n$$
is the standard volume form.
We are interested in the space of (generic) triples of flags up to the left translation action of $G$. Suppose we have three flags $F_1, F_2, F_3$ which are represented by $u_1, \dots, u_n$, $v_1, \dots, v_n$ and $w_1, \dots, w_n$ respectively. Fock and Goncharov define a canonical function $f_{ijk}$ of this triple of flags for every triple of non-negative integers $i, j, k$ such that $i+j+k=n$ and $i, j, k < n$. It is defined by
$$f_{ijk}(F_1, F_2, F_3)=\det(u_1, u_2, \dots, u_i, v_1, v_2, \dots v_j, w_1, w_2, \dots, w_k),$$
and it is $G$-invariant by definition. Note that when one of $i, j, k$ is $0$, these functions only depend on two of the flags. We can call such functions edge functions, and the remaining functions face functions.
Given a cyclic configuration of $m$ flags, imagine the flags sitting at the vertices of an $m$-gon, and triangulate the $m$-gon. Then taking the edge and face functions on the edges and faces of this triangulation, we get a set of functions on a cyclic configuration of flags.
For any triangulation, the edge and face functions form a coordinate chart. Different triangulations yield different functions that are related to the original functions by a positive rational transformation (a transformation involving only addition, multiplication and division) <cit.>.
We will now analogously define the triple distance functions $f_{ijk}^t$ on a configuration of three points in the affine Grassmannian for $SL_n$. The functions $f_{ijk}^t$ are the same as the functions $H_{ijk}$, which were defined in a slightly different way in <cit.>. Recall that the affine Grassmannian is given by $G(\cK)/G(\cO)$. For $G=SL_n$, a point in the affine Grassmannian can be thought of as a finitely generated, rank $n$ $\cO$-submodule of $\cK^n$ such that if $v_1, \dots, v_n$ are generators for this submodule, then
$$v_1 \wedge \dots \wedge v_n=e_1 \wedge \dots \wedge e_n$$
where $e_1, \dots, e_n$ is the standard basis of $\cK^n$.
Let $x_1, x_2, x_3$ be three points in the affine Grassmannian, thought of as $\cO$-submodules of $\cK^n$. For $i, j, k$ as above, we will consider the quantity
\begin{equation}\label{def fijk}
f_{ijk}(x_1,x_2,x_3) = -\val(\det(u_1, \dots, u_i, v_1, \dots v_j, w_1, \dots, w_k))
\end{equation}
as $u_1, \dots, u_i$ range over elements of the $\cO$-submodule $x_1$, $v_1, \dots v_j$ range over elements of the $\cO$-submodule $x_2$, and $w_1, \dots, w_k$ range over elements of the $\cO$-submodule $x_3$. Define $f_{ijk}^t (x_1,x_2,x_3)$ as the maximum value of of this quantity, i.e., the largest value of
$$-\val(\det(u_1, \dots, u_i, v_1, \dots v_j, w_1, \dots, w_k))$$
as all the vectors $u_1, \dots, u_i,$ $v_1, \dots v_j,$ $w_1, \dots, w_k$ range over elements of the respective $\cO$-submodules $x_1, x_2, x_3$.
There is a more invariant way to define $f_{ijk}^t$. Lift $x_1, x_2, x_3$ to elements $g_1, g_2, g_3$, of $G(\cK)$, then project to three flags $F_1, F_2, F_3 \in G(\cK)/U(\cK)$. Then define $f_{ijk}^t$ to be the maximum of $-\val(f_{ijk}(F_1,F_2,F_3))$ over the different possible lifts followed by projection from $G(\cK)/G(\cO)$ to $G(\cK)/U(\cK)$.
It is not hard to check that the edge functions recover the distance between two points in the affine Grassmannian (and hence also the affine building). More precisely, $f_{ij0}^t (x_1,x_2,x_3)$ is given by $\omega_j \cdot d(x_1,x_2)=\omega_i \cdot d(x_2,x_1)$ where $\omega_i$ is a fundamental weight for $SL_n$.
§.§ Positive configurations and conjectures
Now we may define positive configurations of points in the affine building.
Let $x_1, x_2, \dots x_m$ be $m$ points of the real affine Grassmannian. Then $x_1, x_2, \dots x_m$ will be called a positive configuration of points in the affine Grassmannian if and only if there exist ordered bases for $x_i$,
$$v_{i1}, v_{i2}, \dots, v_{in},$$
such that for each $1 \leq p < q <r \leq m$, and each triple of non-negative integers $i, j, k$ such that $i+j+k=n$,
* $f_{ijk}^t (x_p,x_q,x_r) = -\val(\det(v_{p1}, \dots, v_{pi}, v_{q1}, \dots v_{qj}, v_{r1}, \dots, v_{rk}))$
* the leading coefficient of $\det(v_{p1}, \dots, v_{pi}, v_{q1}, \dots v_{qj}, v_{r1}, \dots, v_{rk})$ is positive.
Note that it is important in the above definition that we are taking the valuations of the determinants of the first $i$ (respectively $j, k$) vectors among the bases for $x_p$ (respectively $x_q, x_r$), and not just any $i$ (respectively $j, k$) vectors.
By the results of <cit.>, it is sufficient to verify the two conditions above for only those triples $p, q, r$ occuring in any one triangulation of the $m$-gon. The valuation condition and the positivity condition for one triangulation implies the these conditions for any other triangulation, and hence for an arbitrary triple $p, q, r$.
We can now introduce our conjectures on the tropical functions $f_{ijk}^t$. We need some notation first. Let $\omega_i$ be the $i$-th fundamental weight for $SL_n$: $\omega_i = (1, \dots, 1, 0, \dots, 0)$ where there are $i$ $1$'s and $n-i$ $0$'s. Recall that for any two points $p, q$ in the affine Grassmannian, $d(p,q)$ is an element of the coweight lattice for $SL_n$.
(Weak form) Let $x_1, x_2, x_3$ be a positive configuration of points in the affine Grassmannian. Then
$$f_{ijk}^t (x_1,x_2,x_3) = \min_{p} \omega_i \cdot d(p,x_1) + \omega_j \cdot d(p,x_2) + \omega_k \cdot d(p,x_3),$$
where the minimum is taken over all $p$ in the affine Grassmannian.
There is a stronger, perhaps bolder, form of the conjecture, which is also interesting, although it is less related to the geometry of laminations:
(Strong form) Let $x_1, x_2, x_3$ be any configuration of points in the affine Grassmannian. Then
$$f_{ijk}^t (x_1,x_2,x_3) = \min_{p} \omega_i \cdot d(p,x_1) + \omega_j \cdot d(p,x_2) + \omega_k \cdot d(p,x_3),$$
where the minimum is taken over all $p$ in the affine Grassmannian.
We can make a few elementary observations.
On positive configurations of points in the affine building, the functions $f_{ijk}^t$ only depend on metric properties of the configuration within the building <cit.>
Thus our conjecture is giving a more precise description of how the functions $f_{ijk}$ measure metric properties of the configuration.
One inequality in the conjecture is clearly true. We have that for any configuration of points $x_1, x_2, x_3$ in the affine Grassmanian,
$$f_{ijk}^t (x_1,x_2,x_3) \leq \min_{p} \omega_i \cdot d(p,x_1) + \omega_j \cdot d(p,x_2) + \omega_k \cdot d(p,x_3),$$
This observation follows from the fact that $f_{ijk}^t$ (in fact, $f_{ijk}$) is invariant under the diagonal action of $G(\cK)$ on $x_1, x_2, x_3$, so it is enough to verify the inequality in the case when $p$ is the trivial lattice spanned by $e_1, \dots, e_n$. Thus the conjecture reduces to showing the other inequality.
If one of $i, j, k$ is equal to $0$, then the conjecture holds. In particular, the conjecture is true for $SL_2$. For example, if $k=0$,
$$f_{ij0}^t (x_1,x_2,x_3) = \omega_i \cdot d(p,x_1) + \omega_j \cdot d(p,x_2)$$
for any $p$ lying on a geodesic between $x_1$ and $x_2$.
Let $x_1, x_2, \dots x_m$ be any configuration of points in the affine Grassmannian, with ordered bases for $x_i$,
$$v_{i1}, v_{i2}, \dots, v_{in},$$
$$f_{ijk}^t (x_p,x_q,x_r) = -\val(\det(v_{p1}, \dots, v_{pi}, v_{q1}, \dots v_{qj}, v_{r1}, \dots, v_{rk})).$$
For any set of dominant coweights $(\lambda_1, \lambda_2, \dots, \lambda_m)$, where
$$\lambda_i=(\lambda_{i1}, \lambda_{i2}, \dots, \lambda_{in}), \lambda_{i1} \geq \lambda_{i2} \geq \dots \geq \lambda_{in},$$
we can form the configuration of points
$$x'_1, x'_2, \dots x'_m$$
where $x'_i$ has basis
$$t^{-\lambda_{i1}}v_{i1}, t^{-\lambda_{i2}}v_{i2}, \dots, t^{-\lambda_{in}}v_{in}$$
(it is easy to check the positivity of the configuration $x'_1, x'_2, \dots x'_m$). We have the following observation <cit.>:
For sufficiently large coweights $(\lambda_1, \lambda_2, \dots, \lambda_m)$, and any $1 \leq p, q, r \leq m$, we have that
$$f_{ijk}^t (x'_p,x'_q,x'_r) = \min_{p} \omega_i \cdot d(p,x'_1) + \omega_j \cdot d(p,x'_2) + \omega_k \cdot d(p,x'_3).$$
In other words, the conjecture is true asymptotically.
Let us note that the previous observation was originally shown in <cit.> for positive configurations, but the same proof works for all configurations.
§.§ Relationship to the Duality Conjectures
The duality conjectures of Fock and Goncharov posit a relationship between the spaces $\A_{G,S}$ and $\X_{\vG,S}$ where $\vG$ is the Langlands dual group to $G$. In particular, the main part of the conjecture state that there should be a bijection between $\X_{\vG,S}(\Zt)$ (the tropical points of $\X_{G,S}$) and a basis of functions for $\A_{G,S}$. This bijection should satisfy many compatibility relations which we will not discuss here.
This pairing further specializes to a pairing
$$\X_{\vG,S}(\Zt) \times \A_{G,S}(\Zt) \rightarrow \Z.$$
The pairing works as follows: a point $l \in \X_{\vG,S}(\Zt)$ corresponds to a function $f_l$ on $\A_{G,S}$. A point $l' \in \A_{G,S}(\Zt)$ comes from a taking valuations of some Laurent-series valued point $x_{l'} \in \A_{G,S}(\cK)$. Then the pairing between $l$ and $l'$ is defined by
$$(l,l')=-\val f_l(x_{l'}).$$
The value of $(l,l')$ is independent of the choice of the point $x_{l'}$, as the conjectures state that $f_l$ should be a positive rational function on $\A_{G,S}$ (in fact, it should be a Laurent polynomial in the cluster co-ordinates).
Equivalently, we can describe the pairing as follows. Given a point $l \in \X_{\vG,S}(\Zt)$, take the corresponding function $f_l$ on $\A_{G,S}$. Then if $l' \in \A_{G,S}(\Zt)$, then
Points of $\X_{\vG,S}(\Zt)$ and $\A_{G,S}(\Zt)$ correspond to higher laminations for the groups $\vG$ and $G$, respectively. The pairing between $\X_{\vG,S}(\Zt)$ and $\A_{G,S}(\Zt)$ should realize the intersection pairing between higher laminations. When $G=SL_2$, this construction reduces to the usual intersection pairing between laminations on a surface <cit.>.
For each $i, j, k$, the function $f_{ijk}$ is a cluster function on $\A_{G,S}$. As such, it should be part of the basis parameterized by $\X_{\vG,S}(\Zt)$. In particular, $f_{ijk}$ is associated to the tropical point of $\X_{\vG,S}(\Zt)$ where the corresponding tropical cluster $x$-variable is $1$ and all other $x$ variables are set to $0$. (For every cluster and cluster variable for $\A_{G,S}$ one can canonically associate a cluster and cluster variable for $\X_{\vG,S}$. This is partly a reflection of the fact that the dual pair of spaces forms a cluster ensemble.)
Our conjectures give a way of computing $f_{ijk}(l')$ for $l' \in \A_{G,S}$. Thus they give a geometric interpretation of the intersection pairings. The pairing extends linearly to a pairing between $l \in \X_{\vG,S}(\Zt)$ and $l' \in \A_{G,S}(\Zt)$ whenever $l$ has positive co-ordinates in one of the cluster co-ordinate systems for $\X_{\vG,S}$ associated to a triangulation of $S$ constructed in <cit.>. Thus they give the pairing
$$\X_{\vG,S}(\Zt) \times \A_{G,S}(\Zt) \rightarrow \Z.$$
for any $l' \in \A_{G,S}(\Zt)$ and $l$ contained in a union of open cones inside $\X_{\vG,S}(\Zt)$.
§ MAIN THEOREM
In this section we prove some partial results towards the strong version of the conjecture.
Because we will be dealing with the case of $G=SL_n, GL_n$ or $PGL_n$, and because we would like to deal with all these cases uniformly we will reformulate the conjectures in terms of lattices. By a lattice we mean a $\C[[t]]$-submodule of the vector space $\C((t))^n$ generated by $n$ vectors linearly independent over $\C((t))$. The simplest lattice is the “elementary” lattice $E = \C[[t]]^n$.
Let $i_1,\ldots,i_k$ be nonnegative integers that sum to $n$, and let $L_1, \ldots ,L_k$ be lattices. Define the determinantal valuation
\[
f^t_{i_1,\ldots,i_k}(L_1,\ldots,L_k) = \max_{v_ij \in L_i} -\val \det(v_{11},\ldots,v_{1i_i},v_{21},\ldots,v_{2i_2},\ldots,v_{k1},\ldots,v_{ki_k}).
\]
We are primarily interested in the cases $k=1$, $2$, and $3$.
The unary determinantal valuation
\[
f^t_n(L) = \max_{v_j \in L} -\val \det(v_1,\ldots,v_n)
\]
can easily be computed by choosing any $\C[[t]]$-basis $v_1,\ldots,v_n$ for $L$.
As we saw previously, the binary determinantal valuation $v_{ij}(L,M)$ is also well understood:
Let $L$ and $M$ be lattices. Then there exists $g \in \mathrm{GL}_n(\C((t)))$ and integers $a_1 \geq a_2 \geq \cdots \geq a_n$ such that
\begin{align*}
gL &= \<e_1, e_2, \ldots, e_n\> \\
gM &= \<t^{-a_1}e_1, t^{-a_2}e_2, \ldots, t^{-a_n}e_n\>.
\end{align*}
Moreover, $f_{ij}(gL,gM) = a_1 + \ldots + a_j$ and
\[
f^t_{ij}(L,M) = a_1 + \ldots + a_j + \frac{i f_n^t(L) + j [f_n^t(M) - a_1 - \cdots - a_n]}{n}.
\]
We see that the numbers $a_1,\ldots,a_n$ are unique up to adding a fixed constant to all of them (though $g$ is far from unique).
We reformulate our main conjecture as saying that the ternary determinantal valuation can be expressed in terms of the binary one.
Let $L$, $M$, $N$ be lattices and $i$, $j$, $k$ nonnegative integers, $i+j+k=n$. Then
\[
f^t_{ijk}(L,M,N) = \min_{\text{lattices }P} (f^t_{i,j+k}(L,P) + f^t_{j,i+k}(M,P) + f^t_{k,i+j}(N,P) - 2f^t_n(P))
\]
which can also be written as
\[
f^t_{ijk}(L,M,N) = \min_{\text{lattices }P} (f^t_{ijk}(L,P,P) + f^t_{ijk}(P,M,P) + f^t_{ijk}(P,P,N) - 2f^t_{ijk}(P,P,P))
\]
The inequality $\leq$ between the two sides is not difficult to prove (Observation <ref>); hence the content of the conjecture is the existence of a lattice $P$ for which equality holds.
§.§ The case of three “close” lattices
Our first partial result is as follows:
Conjecture <ref> holds when $E \subseteq L,M,N \subseteq t^{-1}E$. In particular, one of the choices
\[
tE, L, M, N, L+M, L+N, M+N, L+M+N
\]
works for $P$.
If $E \subseteq L \subseteq t^{-1}E$, then $L$ is determined by its projection $U_1$ to the space $\F^n \equiv t^{-1}E/E$. Likewise $M \simeq E \oplus U_2$ and $N \simeq E \oplus U_3$, where $U_1$, $U_2$, and $U_3$ are subspaces of $V = \F^n$. Now a system of three subspaces of an ambient space
\[
\xymatrix@1{
& U_1\ar[d] & \\
U_2\ar[r] & V & U_3\ar[l]
\]
forms a representation of the quiver $\xymatrix@!0{
& \bullet\ar[d] & \\
\bullet\ar[r] & \bullet & \bullet\ar[l]
}$ of Dynkin type $D_4$ and thus can be expressed as a direct sum of its $12$ irreducible representations, of which $3$ are excluded since they correspond to non-injective maps. In the following diagram, we assign names to the remaining $9$ representation types and their basis vectors:
\begin{equation}
\begin{tabular}{c|ccccccccc}
\multicolumn{1}{c}{Rep.} & $A$ & $A'$ & $A''$ & $B$ & $B'$ & $B''$ & $C$ & $D$ & $S$ \\ \hline
$U_1$ & $a_\ell$ &&& $b_\ell$ & $b_\ell'$ && $c_\ell$ & $u_\ell$ \\
$U_2$ && $a_\ell'$ && $b_\ell$ && $b_\ell''$ & $c_\ell$ & $v_\ell$ \\
$U_3$ &&& $a_\ell''$ && $b_\ell'$ & $b_\ell''$ & $c_\ell$ & $u_\ell + v_\ell$ \\
$V$ & $a_\ell$ & $a_\ell'$ & $a_\ell''$ & $b_\ell$ & $b_\ell'$ & $b_\ell''$ & $c_\ell$ & $u_\ell, v_\ell$ & $s_\ell$
\end{tabular}
\end{equation}
To compute $f_{ijk}(L,M,N)$, we must search for a choice of vectors $u_1, \dots, u_i \in L$, $v_1, \dots, v_j \in M$, $w_1, \dots, w_k \in N$ minimizing the valuation of the determinant in (<ref>). There is no reason not to choose vectors in either $t^{-1}V$ or $V$, and then the valuation of the determinant depends only on the number of $t$'s involved, as long as the $n$ underlying vectors in $V$ are linearly independent. So we have the following interpretation.
We have $f_{ijk}(L,M,N) = g$, where $g$ is the maximum number of linearly independent vectors that may be chosen from $U_1$, $U_2$, $U_3$, with the restriction that at most $i$ vectors from $U_1$, $j$ from $U_2$, $k$ from $U_3$ may be chosen.
Moreover, we may limit our vectors to the bases of $U_1$, $U_2$, $U_3$ constructed above. Now $g$ depends in a purely combinatorial way on $i$, $j$, $k$, and the multiplicities of the irreducible representations.
We will now interpret $g$ as the maximum flow in a certain graph. Draw four layers of vertices as follows:
* A single source vertex;
* One representation vertex for each of the irreducible components of the representation of $D_4$ determined by $U_1$, $U_2$, $U_3$;
* Three $U$-vertices, labeled $U_1$, $U_2$, and $U_3$;
* A single sink vertex.
Then draw arrows from each level to the next as follows:
* Each representation vertex is joined to the source with an arrow whose capacity is the dimension of the portion of $V$ that it corresponds to (always $1$ except for the representation $D$, where it is $2$);
* Each representation vertex is joined to each $U$-vertex $U_m$ with an arrow whose capacity is the dimension of the portion of $U_m$ that it corresponds to (always $0$ or $1$); for future reference, the arrows with capacity $1$ are labeled with the appropriate basis vector of $U_i$.
* The three $U$-vertices $U_1$, $U_2$, and $U_3$ are joined to the sink with arrows of capacity $i$, $j$, and $k$, respectively.
Now it is readily verified that any choice of basis vectors satisfying the conditions of Lemma <ref> can be represented by a flow in this graph, where for each vector chosen, one unit of fluid flows from the source through the appropriate representation-to-$U$ edge and then out the sink. Hence the maximum flow is $g$.
We turn our attention to the cuts of the constructed graph. Note that once each of the $U_m$ is either cut from or “soldered” to the sink (the vertices may be identified once the decision is made not to cut the edge), the graph becomes a union of noninteracting subgraphs, one for each representation vertex. So the minimal cut in each of the eight cases is easily determined:
* If all three $U_m$ are cut, no further cuts are necessary, and we obtain a cut of capacity $i+j+k=n$.
* If two of the $U_m$ are cut, say $U_2$ and $U_3$, then the representation vertices with connections to $U_1$ (namely, those of type $A$, $B$, $B'$, $C$, and $D$) can be dealt with by cutting these connections, which are labeled by the basis vectors of $U_1$. Hence there are cuts of capacity $j+k+\dim U_1$ and, symmetrically, $i+k+\dim U_2$ and $i+j+\dim U_3$.
* If only one $U_m$ is cut, say $U_3$, then we must further cut one unit for each representation vertex of type $A$, $A'$, $B$, $B'$, $B''$, or $C$, and two units for each representation vertex of type $D$. This amounts to one unit for each vector in a basis of $U_1 + U_2$. So we get cuts of capacity $k + \dim(U_1 + U_2)$ and, symmetrically, $j + \dim(U_1 + U_3)$ and $i + \dim(U_1 + U_2)$.
* Finally, if none of the $U_m$ are cut, then there is no better option than cutting the inflow to each representation vertex, excepting those of type $S$, for which of course no cut is necessary. So we get a cut of capacity $\dim(U_1 + U_2 + U_3)$.
So we have reached a second checkpoint in the computation of $f_{ijk}$:
$f_{ijk}(L,M,N) = \min\{i+j+k,j+k+\dim U_1, i+k+\dim U_2, i+j+\dim U_3, k + \dim(U_1 + U_2), j + \dim(U_1 + U_3), i + \dim(U_1 + U_2), \dim(U_1 + U_2 + U_3)\}$.
Finally, we must relate the eight terms on the right-hand side of this lemma to the right-hand side of Conjecture <ref> for the lattices $P$ listed in Theorem <ref>. The key is that in max-flow/min-cut, there is always a flow that uses every edge of the minimal cut at full capacity. We will use this flow to bound the $f_{\bullet,\bullet}$ terms from above, proving that the right-hand side is at most $f_{ijk}(L,M,N)$, since as previously remarked the reverse inequality is trivial (Observation <ref>).
* If $i+j+k$ is the minimal cut, then there are $i$ vectors from $U_1$, $j$ vectors from $U_2$, and $k$ vectors from $U_3$, all linearly independent. Picking $P = E$, we find $f_{i,j+k}(L,E) \leq i$ by taking the $i$ linearly independent vectors from $L$ and filling out with vectors from $V$. Calculating the other two terms by symmetry, we get
\begin{align*}
&f_{i,j+k}(L,tE) + f_{j,i+k}(M,tE) + f_{k,i+j}(N,tE) - 2f_n(tE) \\
&\leq i + j + k \\
&= f_{i,j,k}(L,M,N).
\end{align*}
* If $j+k+\dim U_1$ is the minimal cut, then there are $\dim U_1 \leq i$ vectors from $U_1$, $j$ vectors from $U_2$, and $k$ vectors from $U_3$, all linearly independent. We will pick $P = L$. The most difficult term is $f_{j,i+k}(M,L)$ which can be bounded by picking the $j$ vectors from $U_2$ and the basis for $U_1$, filling out with vectors from $V$, getting a bound of $ j + \dim{U_1}$. The other terms are either symmetric or trivial, and we get
\begin{align*}
&f_{i,j+k}(L,L) + f_{j,i+k}(M,L) + f_{k,i+j}(N,L) - 2f_n(L) \\
&\leq \dim U_1 + j+\dim U_1) + k + \dim U_1) - 2(\dim U_1) \\
&= j + k + \dim U_1 = f_{i,j,k}(L,M,N).
\end{align*}
* If $k + \dim(U_1 + U_2)$ is the minimal cut, there is a basis for $U_1 + U_2$ consisting of at most $i$ vectors from $U_1$ and at most $j$ vectors from $U_2$, and also $k$ vectors chosen from $U_3$ that are linearly independent from $U_1 + U_2$. We pick $P = L+M$. For $f_{i,j+k}(L,L+M)$, it is possible to pick all the vectors in the basis of $U_1 + U_2$ (at most $i$ from $L$ and $j$ from $M$) before filling out with $V$. The term $f_{j,i+k}(M,L+M)$ is symmetric, while in $f_{k,i+j}(N,L+M)$ we can get the $k$ linearly independent vectors in $U_3$ as well as a basis of $L+M$. So in all, we get
\begin{align*}
&f_{i,j+k}(L,L+M) + f_{j,i+k}(M,L+M) + f_{k,i+j}(N,L+M) - 2f_n(L+M) \\
&\leq \dim (U_1+U_2) + \dim (U_1+U_2) + k + \dim (U_1+U_2)) - 2(\dim (U_1+U_2)) \\
&= k + \dim (U_1+U_2) = f_{i,j,k}(L,M,N).
\end{align*}
* Finally, if $\dim(U_1 + U_2 + U_3)$ is the minimal cut, then $U_1 + U_2 + U_3$ has a basis consisting of at most $i$ vectors from $U_1$, $j$ vectors from $U_2$, and $k$ vectors from $U_3$. This can be used to bound all the terms if we pick $P = L+M+N$:
\begin{align*}
&f_{i,j+k}(L,L+M+N) + f_{j,i+k}(M,L+M+N) + f_{k,i+j}(N,L+M+N) - 2f_n(L+M+N) \\
&\leq \dim (U_1+U_2+U_3) + \dim (U_1+U_2+U_3) + \dim (U_1+U_2+U_3) \\
&\quad - 2(\dim (U_1+U_2+U_3)) \\
&= \dim (U_1+U_2+U_3) = f_{i,j,k}(L,M,N).
\end{align*}
Of course, the other $4$ possibilities for the minimal cut are symmetric.
§.§.§ Connection to Konig's theorem
The portion of our proof of Theorem <ref> lying between Lemmas <ref> and <ref> can be thought of as a combinatorial problem in linear algebra. It can be related to some familiar theorems in combinatorics in a way which we now describe.
Hall's theorem or Hall's Marriage Lemma is frequently described in terms of the following story: $n$ boys are to be married off to $m$ girls, and each boy-girl pair either likes or dislikes one another. A matching in which all the boys are paired exists if and only if no subset of the boys likes a strictly smaller subset of the girls. Or, in the inanimate language favored by mathematicians:
If $S_1,S_2,\ldots,S_r$ are sets, then a system of distinct representatives of the $S_i$ (one from each set) exists if and only if for each subset $I \subseteq [r]$,
\[
\left|\bigcup_{i \in I} S_i\right| \geq |I|.
\]
A refinement of Hall's theorem is Konig's theorem. Instead of giving conditions for a perfect matching to exist, it provides a formula for the maximum number of disjoint pairs that may be made:
If $S_1,S_2,\ldots,S_r$ are sets, the maximum number of distinct representatives of the $S_i$ (at most one from each set) is
\[
\min_{I \subseteq [r]} \left( \left| \sum_{i\in I} S_i \right| + r - |I| \right).
\]
In <cit.>, Theorem 2, Moshonkin “linearized” Hall's theorem in the sense of replacing sets by vector spaces and adjusting conditions accordingly. His result is:
If $V_1,V_2,\ldots,V_r$ are subspaces of an ambient vector space $V$, then a system of linearly independent representatives of the $V_i$ exists if and only if for each subset $I \subseteq [r]$,
\[
\dim \sum_{i \in I} S_i \geq |I|.
\]
In a similar vein, we would like to state and prove the following linearization of Konig's theorem:
If $V_1,\ldots,V_r$ are subspaces of an ambient space $V$, the maximum number of linearly independent representatives from different $V_i$'s is
\begin{equation}\label{eq:vsKonig}
\min_{I \subseteq [r]} \left[ \dim \left( \sum_{i\in I} V_i \right) + r - |I| \right].
\end{equation}
This result is applicable to our investigations in the following manner: if $U_1$, $U_2$, $U_3$ are subspaces, then the maximum number of linearly independent representatives from different terms of the multiset
\[
\underbrace{U_1,\ldots,U_1}_{i}, \underbrace{U_2,\ldots,U_2}_{j}, \underbrace{U_3,\ldots,U_3}_{k}
\]
is the quantity $g$ in the statement of Lemma <ref>. On the other hand, the expression (<ref>) clearly can only reach its minimal value when the index set $I$ includes either all or none of each of the three strings of repeated $U_i$. Thus Theorem <ref> provides an alternative proof of Lemma <ref> from Lemma <ref>.
We now deduce Theorem <ref> from Theorem <ref>. In fact, the two theorems are readily found to be equivalent.
Let $M$ be the subset $I \subseteq [r]$ that minimizes (<ref>). Denote
\begin{align*}
m &= |M|,\\
K &= [r]\backslash M,\\
k &= |K| = r-m,\\
W &= \sum_{i\in M} V_i,\\
n &= \dim W.
\end{align*}
We will construct the requisite $n+k$ linearly independent representatives in the following way: we will find a basis for $W$ whose elements come from distinct $V_i$, and we will supplement this basis with vectors in the $k$ spaces $V_i$, $i \in K$, which are all linearly independent in the quotient space $V/W$. The proof will apply the minimality condition to sets $I$ which are respectively subsets and supersets of $M$.
We begin with the second step. Let $\tilde{V}_i$ be the image of $V_i$ in $V/W$. For each $I \subseteq K$, we have the condition
\[
\dim \left( \sum_{i\in M\cup I} V_i \right) + r - (m + |I|) \geq n + k
\]
which simplifies to
\[
\dim \sum_{i\in I} \tilde{V}_i \geq |I|.
\]
So the $\tilde{V}_i$, $i \in K$ satisfy precisely the condition of Theorem <ref> and hence a system of linearly independent representatives exists.
The first step is only slightly trickier. For each $I \subseteq M$, we have
\[
\dim \left( \sum_{i \in I} V_i \right) + r - |I| \geq n + k
\]
which simplifies to
\[
\dim \sum_{i\in I} V_i \geq |I| \geq n - m + |I|.
\]
This would be the condition of Theorem <ref> were it not for the summand $n-m$. So we use a trick. Plugging $I = \emptyset$, we see that $m\geq n$. Let $V' = V \oplus \F^{m-n}$ and $V'_i = V_i \oplus \F^{m-n}$. Then the $V'_i$, $i \in M$, satisfy the conditions of Theorem <ref> and we can find a basis of $V'$ with one vector from each $V'_i$. Projecting down to $V$, we have a spanning set, of which some $n$ vectors form a basis.
Both Konig's and Moshonkin's theorems as well as our proof of Theorem <ref> rely on max-flow/min-cut-type results. We believe that this is not accidental, and that in general, defining intersection pairings between higher laminations will involve proving statements about affine buildings that have the flavor of max-flow/min-cut.
§.§ Apartments
Let $X = \{x_1, \ldots, x_n\}$ be a basis for $\cK^n$ over $\cK$. For any choice integers $c_1, \dots, c_n$, the set of lattices of the form $t^{c_j} x_j$, $j = 1,2,\ldots,n$ form a subset of the affind building called an apartment.
Our second result is a generalization of our conjecture in the situation where $L_1,\ldots,L_k$ all lie in the same apartment:
If $L_1,\ldots,L_k$ lie in the same apartment and $i_1,\ldots,i_k$ nonnegative integers with sum $n$, then there exists a lattice $P$ such that
\[
f^t_{i_1,\ldots,i_k}(L_1,\ldots,L_k) = \sum_j f^t_{i_j,n-i_j}(L_j,P) - (n-1)f^t_n(P).
\]
(Remark: We expect that the above generalization holds for general lattices $L_1,\ldots,L_k$.)
Use the following combinatorial result:
Let $[c_{ij}]$ be a real $n\times n$ matrix. Then the maximal sum of a transversal of $[c_{ij}]$ equals the minimal sum $\sum_i a_i + \sum_j b_j$ where the $a_i$ and $b_j$ satisfy $a_i + b_j \geq c_{ij}$ for all $i,j$. If the $c_{ij}$ are integers, the $a$'s and $b$'s can be taken integral as well.
In our situation, we can first reduce to the case that $k=n$ and all $i_j$ are $1$. Then write $L_i = \<t^{-c_i1} x_1,\ldots, t^{-c_in} x_n\>$. Using the multilinearity of the determinant, we can assume that the value of $f_{i_1,\ldots,i_k}$ is attained by taking the valuation of the determinant of generators $t^{-c_ij} x_j$ of the respective lattices. Since all the $i$'s must be distinct as must the $j$'s, the valuation of the determinant is a transversal $-\sum_i c_{i\sigma(i)}$ for some $\sigma \in S_n$. Therefore $v_{1\cdots1}(L_1,\ldots,L_n)$ is the maximal transversal sum in the matrix $[c_{ij}]$.
By Kuhn-Munkres, there exist integers $a_i$ and $b_j$ such that $a_i + b_j \geq c_{ij}$ and
\[
f^t_{1\cdots1}(L_1,\ldots,L_n) = \sum_i a_i + \sum_j b_j.
\]
Choose the lattice
\[
P = \<t^{-b_1} x_1, \ldots, t^{-b_n} x_n \>.
\]
Obviously, equality $a_i + b_j = c_{ij}$ must hold whenever $c_{ij}$ belongs to the winning transversal. So
\[
f^t_{1,n-1}(L_i,P) = \max_k \Big(c_{ik} + \sum_{j \neq k} b_j \Big)
= \sum_j b_j + \max_k(c_{ik} - b_k) = \sum_j b_j + a_i.
\]
Since $f^t_n(P) = \sum_{j} b_j$, we have the desired equality.
§ GENERALIZATIONS
Theorem <ref> hints at some generalizations of conjectures. It is known that the functions $f_{i_1,\ldots,i_k}$ are cluster co-ordinates for $k=2, 3, 4$, so that the corresponding functions $f^t_{i_1,\ldots,i_k}$ also give intersection pairings between higher laminations. In fact, we expect that this is true for all $k$.
Thus, in general, we conjecture that
$$f^t_{i_1,\ldots,i_k}(L_1,\ldots,L_k) = \sum_j f^t_{i_j,n-i_j}(L_i,P) - (k-1)f^t_n(P).$$
Note that if $L_1, \dots L_k$ lie in the same apartment, then Theorem <ref> gives the above result. This gives one interpretation of the functions $f^t_{i_1,\ldots,i_k}$ in terms of the geometry of the affine building.
Let us another interpretation when $k=4$. We would like to calculate. We conjecture $f^t_{i_1, i_2, i_3, i_4}(L_1, L_2, L_3, L_4)$ is also given by the minimum of the two following expressions:
$$f^t_{i_4, n-i_4}(L_4, P) + f^t_{i_1, n-i_1}(L_1, P) + f^t_{i_4+i_1, i_2+i_3}(P, Q) + f^t_{i_2, n-i_2}(L_2, Q) + f^t_{i_3, n-i_3}(L_3, Q) - 2f^t_n(P) - 2f^t_n(Q)$$
where $P$ and $Q$ range over all lattices. If $P$ and $Q$ are normalized to have determinant $1$ (possibly by introducing fractional powers of $t$), we get the minimum over $P$ and $Q$ of
$$f^t_{i_1, n-i_1}(L_1, P) + f^t_{i_2, n-i_2}(L_2, P) + f^t_{i_1+i_2, i_3+i_4}(P, Q) + f^t_{i_3, n-i_3}(L_3, Q) + f^t_{i_4, n-i_4}(L_4, Q)$$
$$f^t_{i_4, n-i_4}(L_4, P) + f^t_{i_1, n-i_1}(L_1, P) + f^t_{i_4+i_1, i_2+i_3}(P, Q) + f^t_{i_2, n-i_2}(L_2, Q) + f^t_{i_3, n-i_3}(L_3, Q).$$
If $P=Q$, these expressions reduce to the previous expression
$$f^t_{i_1, n-i_1}(L_1, P) + f^t_{i_2, n-i_2}(L_2, P) + f^t_{i_3, n-i_3}(L_3, P) + f^t_{i_4, n-i_4}(L_4, P) - 3f^t_n(P).$$
The content of our conjecture is that allowing $P \neq Q$ does not change the minimum.
Equivalently, it is given by the minimum over $P$ and $Q$ of (again assuming that $P$ and $Q$ are normalized to have determinant $0$)
$$d(P,L_1) \cdot \omega_{i_1} + d(P, L_2) \cdot \omega_{i_2} + d(P, Q) \cdot \omega_{i_3+i_4} + d(Q, L_3) \cdot \omega_{i_3} + d(Q, L_4) \cdot \omega_{i_4}$$
$$d(P,L_4) \cdot \omega_{i_4} + d(P, L_1) \cdot \omega_{i_1} + d(P, Q) \cdot \omega_{i_2+i_3} + d(Q, L_2) \cdot \omega_{i_2} + d(Q, L_3) \cdot \omega_{i_3}.$$
Thus we conjecture that $f^t_{i_1, i_2, i_3, i_4}(L_1, L_2, L_3, L_4)$ is calculated by the minimum distance of a weighted network connecting the $L_i$. However, this network can take one of two shapes (three if one counts the degenerate case $P=Q$ separately). One recognizes that these two different networks, along with the weights along the networks, are identical to the spin networks that calculate the untropicalized function $f_{i_1, i_2, i_3, i_4}$.
We believe that in many other more general cases, the functions corresponding to points of $\X_{\vG,S}(\Zt)$ are calculated by some set equivalent spin networks which calculate the same function, and that the associated tropical functions are calculated by the minimal distance over these various weighted networks inside the affine building. For example, when $G=SL_4$ there are two inequivalent spin networks with four leaves with weights $\omega_2, \omega_1, \omega_2, \omega_3$ in that cyclic order. We believe that the corresponding tropical functions are given by the minimums of
$$f^t_{2,2}(L_1, P) + f^t_{1,3}(L_2, P) + f^t_{3, 1}(P, Q) + f^t_{2,2}(L_3, Q) + f^t_{3, 1}(L_4, Q) - 2f^t_n(P) - f^t_n(Q)$$
$$f^t_{3,1}(L_4, P) + f^t_{2,2}(L_1, P) + f^t_{1, 3}(P, Q) + f^t_{1,3}(L_2, Q) + f^t_{2, 2}(L_3, Q) - f^t_n(P) - 2f^t_n(Q)$$
respectively. This points towards a general geometric interpretation of intersection pairings between higher laminations.
[FG1]FG1 V.V. Fock, A.B. Goncharov. Moduli spaces of local systems and higher Teichmuller theory. Publ. Math. IHES, n. 103 (2006) 1-212. math.AG/0311149.
[GS]GS A.B. Goncharov, L. Shen. Geometry of canonical bases and mirror symmetry. arXiv:1309.5922
[K]K J. Kamnitzer. Hives and the fibres of the convolution morphism, Selecta Math. N.S. 13 no. 3 (2007), 483-496.
[Le]Le I. Le. Higher Laminations and Affine Buildings. arXiv:1209.0812
[M]M A.G. Moshonkin. Concerning Hall's Theorem, from Mathematics in St. Petersburg, eds. A. A. Bolibruch, A.S. Merkur'ev, N. Yu. Netsvetaev. Amererican Mathematical Society Translations, Series 2, Volume 174, 1996.
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1511.00434
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In any flux-density limited sample of blazars, the distribution of the timescale modulation factor $\Delta t'/\Delta t$, which quantifies the change in observed timescales compared to the rest-frame ones due to redshift and relativistic compression follows an exponential distribution with a mean depending on the flux limit of the sample. In this work we produce the mathematical formalism that allows us to use this information in order to uncover the underlining rest-frame probability density function of measurable timescales of blazar jets. We extensively test our proposed methodology using a simulated FSRQ population with a 1.5 Jy flux-density limit in the simple case (where all blazars share the same intrinsic timescale), in order to identify limits of applicability and potential biases due to observational systematics and sample selection. We find that for monitoring with time intervals between observations longer than $\sim$30% of the intrinsic timescale under investigation the method loses its ability to produce robust results. For time intervals of $\sim$3% of the intrinsic timescale the error of the method is as low as 1% in recovering the intrinsic rest-frame timescale. We applied our method to rotations of the optical polarization angle of blazars observed by RoboPol. We found that the intrinsic timescales of the longest-duration rotation event in each blazar follows a narrow distribution, well-described by a normal distribution with mean 87 days and standard deviation 5 days. We discuss possible interpretations of this result.
galaxies: active – galaxies: jets – galaxies: blazars
§ INTRODUCTION
Blazars are among the most active of galaxies. Their jets are known to show unique properties, such as superluminal motion, boosted emission, and high variability throughout the entire electromagnetic spectrum from $\gamma$-rays to radio. These unique properties are attributed to the preferential alignment of their jet <cit.>. Due to beaming, small differences in the rest frame result in large scatter of observables, hampering phenomenological studies and dispersing correlations and behavior trends. This also constitutes the reason why there are still many open questions regarding blazars, despite many years of systematic study. Even today very little is known about blazar properties in their rest frame. In this work we focus on studies of blazar jets in the time domain.
Timescales in blazars are compressed due to Doppler boosting, but are also elongated due to the expansion of the Universe. This distortion is quantifiable by the timescale modulation factor $m$,
\begin{equation}
m=\frac{\Delta t'}{\Delta t}=\frac{1+z}{D},
\label{tscale_mod}
\end{equation}
where $\Delta t'$ is the timescale in the observer's frame, $\Delta t$ is the timescale in the rest frame of the jet, $z$ is the redshift and $D=\sqrt{1-\beta^2}/(1-\beta\cos\theta_j)$ is the Doppler factor, where $\theta_j$ is the angle to the line of sight, and $\beta$ the velocity of the jet in units of speed of light. Even though redshift is, in most cases, a well measured quantity, measuring the amount of boosting in a jet is not straight forward <cit.>. Although several methods have been proposed (e.g. ), the amount of available data in the literature is limited, making a more broad statistical study of timescales unfeasible. In addition, current Doppler factor estimates involve large uncertainties <cit.>, so “correcting” the measured timescales on a source-by-source basis using available Doppler factor estimates may in fact introduce even more dispersion in the distribution of observables. In order to overcome such limitations, we are proposing a methodology that can estimate the effect of boosting at a population level by using the Doppler factor distribution in flux-density limited samples predicted by blazar population models.
Information on rest-frame characteristic timescales can provide important information on blazar jets and their emission processes. For example rest-frame time delays between frequencies are indicative of the distance between different emission regions <cit.>. Rest-frame durations of flares can test different models of the evolution of jet disturbances <cit.>. Rest-frame durations of EVPA rotations can help us distinguish between mechanisms proposed to interpret them <cit.>.
This paper is organized as follows. In <ref> we describe our model for the blazar population. In <ref> we present our mathematical formalism for uncovering the probability density function of rest-frame timescales. In <ref> we benchmark our method with respect to the time interval between observations and sample size. In <ref> we apply our method to timescales associated with EVPA rotations as seen by the RoboPol survey as well as possible interpretations of the results, and in <ref> the conclusions derived from this work.
The cosmology we have adopted throughout this work is $H_0=71$ ${\rm km \, s^{-1} \, Mpc^{-1}}$, $\Omega_m=0.27$ and $\Omega_\Lambda=1-\Omega_m$ <cit.>.
§ BLAZAR POPULATION MODEL
Although blazars exhibit very diverse behavior on a source-by-source basis, they can be statistically treated as a population of relativistically boosted sources with relatively simple (power-law) distributions of rest-frame properties (luminosities and Lorentz factors) viewed at random angles. Typically such population models (see e.g ) are developed to fit the observed luminosity and flux-density distributions.
However, because blazars are known to show high variability throughout the electromagnetic spectrum, single-epoch flux densities can be unreliable observables. For this reason, in a recent work we used observables that are not as affected by variability: the observed apparent velocity and redshift distributions of the MOJAVE [Monitoring Of Jets in Active galactic nuclei with VLBA Experiments, http://www.physics.purdue.edu/MOJAVE/] sample <cit.>. MOJAVE uses a statistically complete flux-density limited sample, selected at 15 GHz <cit.>, ideal for population studies of blazars. Below we present a summary of the our population models (, hereafter Paper I, , hereafter Paper II). For a more detailed description of the model, results and applications see Paper I, <cit.>, and Paper II.
We assumed single power-law distributions for the Lorentz factor and the unbeamed luminosity for the simulated parent population. We treated BL Lacs and the FSRQs as separate cases in order to assess the difference in beaming between the two classes. We adopted pure luminosity evolution <cit.>, where sources become brighter with look-back time while maintaining a constant comoving volume density, making the number of sources in any redshift bin proportional to the comoving volume element in that bin ($dN\propto dV$). This scaling is normalized so that the final simulated sample for the comparison with the observed distributions will consist of $ \sim 10^3$ sources. We used a Monte-Carlo approach to calculate the simulated distributions of the observables we consider: we drew values for the intrinsic luminosity $L_v$ and Lorentz factor $\Gamma$ from power-law distributions, and for the viewing angle $\theta_j$ from a uniform distribution. We then calculated a flux density using,
\begin{equation}\label{flux_dens}
S_\nu=\frac{L_\nu D^p}{4\pi{d_L^2}}(1+z)^{1+s},
\end{equation}
where $D=1/\Gamma(1-\beta\cos\theta)$, $\beta=(1-\Gamma^{-2})^{1/2}$, $d_L$ is the luminosity distance, $s$ the spectral index, and $p$ is defined as $p=2-s$; and we finally applied the 1.5 Jy flux-density limit of the MOJAVE sample.
Following the procedure described above, we produced simulated samples for the blazar populations (Paper I), from which we extracted information regarding the Doppler factor distribution, the distribution of $\Gamma\theta$ which quantifies how beamed is a source within a flux-density limited sample, and the distribution of the timescale modulation factor in each class.
In the original version of the model, we used a single value for the spectral index for each population. In the most recent version (Paper II) we have included a spectral index distribution in the models. The spectral index for both source classes is normally distributed with mean and standard deviation determined using the maximum likelihood analysis presented in <cit.> and data from <cit.>. In Paper II, before calculating a flux density, each source is also assigned a random value for the spectral index from the distribution of the corresponding source class.
A major result of Paper I and Paper II, which is directly releveant to the work we present in this paper is that the timescale modulation factor (Eq. <ref>) follows an exponential distribution,
\begin{equation}
\label{expo_eq}
\end{equation}
where $C$ is a normalization constant to account for the truncated range and $\lambda$ is the inverse mean of the distribution, with the mean depending on the source class and the flux limit of the sample. Using the upgraded models from Paper II (which we use throughout this work), the timescale modulation factor has a similar mean (0.381 for the BL Lacs and 0.318 for the FSRQs, for 1.5 Jy flux-density limited samples) for both classes.
§ INTRINSIC PROBABILITY DENSITY FUNCTION OF TIMESCALES
Our aim in this work is to extract information about the intrinsic (rest-frame) probability density function (pdf) of timescales for any class of events characterized by a single timescale per blazar. Examples of such classes of events include: shortest flare rise time in a blazar; the duration of the largest (in amplitude) flare ever observed in each source; the duration of the longest (in angle) rotation of the polarization angle in each blazar; and the average time delays between flares in two different frequencies in each blazar.
In order to achieve our goals, we first treat the inverse problem: given a rest-frame pdf of timescales in the blazar population for events of a particular class, what would the observed timescale pdf be, after applying the modulation induced by relativistic effects? In the following, we will use $t_i$ to denote intrinsic (source rest-frame) timescale, and $t_o$ to denote observed (observer-frame) timescales.
Assuming that the physical processes in the rest frame have no knowledge of redshift or Doppler factor, the intrinsic timescales of any event are independent of the timescale modulation factor $m=(1+z)/D$. As a result,
\begin{equation}
\label{intrinsic_pfd}
\end{equation}
where $P(t_i,m)$ is the probability of the intrinsic timescale, $t_i$, to be modified by $m$; $P(t_i)$ is the intrinsic timescale probability density; and $P(m)$ is the timescale modulation factor probability density (Eq. <ref>).
The observed timescales are connected to the intrinsic through $t_o=mt_i$. Thus Eq. <ref> can be transformed as:
\begin{equation}
\label{trans}
\end{equation}
where $P(t_o,m)$ is the probability of observing $t_o$ given $m$, and $J(t_i,m)$ is the Jacobian of the transformation. From the relation between observed and intrinsic timescales, the Jacobian will be:
\begin{equation}
\begin{bmatrix}
& \frac{1}{m} & 0\\
& & & \\
& 0 & 1\\
\end{bmatrix}
\end{equation}
Thus the (absolute value of the) determinant of the Jacobian is equal to $|J(t_i,m)|=1/m$, and Eq.<ref> becomes $P(t_o,m)=P(t_i)P(m)/m$. The probability density function of the observed timescales is thus:
\begin{equation}
P(t_o)=\int^{m_{max}}_{m_{min}}P(t_o,m)dm=\int^{m_{max}}_{m_{min}}P(t_i)C(\lambda e^{-\lambda m})\frac{1}{m}dm,
\label{pdf}
\end{equation}
where $m_{min}$, $m_{max}$ are the minimum and maximum timescale modulation factors.
Since we know the functional form of $P(m)$ from the results of Papers I and II, we can assume a family of distributions for $P(t_i)$ and use Eq. <ref> to obtain the resulting $P(t_o)$. The parameters of $P(t_i)$ can then be recovered by requiring that the resulting $P(t_o)$ fits the observed data best. In this work we provide in Appendix <ref> some examples of this process for the most commonly used and/or encountered distributions in astrophysics in general.
§ METHOD BENCHMARKING
In order to benchmark the accuracy of our method under realistic observing conditions, we generate a simulated survey and explore the accuracy of the method with respect to the cadence of observations and sample size.
§.§ Simulated sample
For our benchmarking experiments, we have chosen to use the simple case where the class of events under investigation have the same intrinsic characteristic timescale in the rest-frame of all blazars. Then the rest-frame distribution can be described by a $\delta$-function and the observer-frame distribution in the ideal case of perfect sampling will be an appropriately normalized exponential distribution (Eq. <ref>). We constructed our simulated samples by drawing a random value for the timescale modulation factor from an exponential distribution (Eq. <ref>), and multiplying it with the same characteristic intrinsic timescale (set to $t_i=$100 days to match our real-data application, see <ref>).
The minimum modulation factor we accept is $10^{-2}$ following the results of Papers I and II. The mean of the timescale modulation factor distribution depends on the object class and the flux limit of the sample. For this particular experiment, we choose to use the model for the FSRQ population with an 1.5 Jy flux-density limit ($\lambda^{-1}$ was set to 0.318). Since the mean of the BL Lac population for the 1.5 Jy flux limit is very similar with that of the FSRQs, we expect the conclusions derived for the latter to extend to the former. However, this is not necessarily true for different flux limits. We discuss the possible effects of a different flux limit to the results of our benckmarking in <ref>.
§.§ Effects of sample size and cadence
Estimated best-fit $t_i$ versus interval between observations $\Delta t_o$ (both in days) for samples without pileup. The error bars represent the uncertainty of the best-fit $t_i$. The green line at 100 days shows the value of the “true” intrinsic timescale, while the black dashed lines at 50 and 150 days show the limits of the parameter space scanned.
Upper panel (a): Estimated best-fit $t_i$ versus sample size ranging from 30 to 1500 sources. The error bars represent the uncertainty of the best-fit $t_i$. Lower panel (b): Fractional bias between the estimated best-fit and the “true” $t_i$ ($|t_{i,fit}-t_i|/t_i$) versus sample size. Symbols as above.
The finite cadence of any monitoring program can affect measured timescales in three ways. First, it can introduce an artificial cut-off at the lower end of the observed timescale distribution: timescales shorter than $\Delta t_o$ (the time interval between observations) will not be measured depending on $m$. Second, it can also introduce an uncertainty in measured timescales. The measured timescale will lie between $t_{observed}$ and $t_{observed}+2\Delta t_o$. This is because in the worst case scenario, both the beginning/end of an “event” can happen exactly after/before the first/last datapoints. For this reason, once we generate our simulated observer-frame sample, we create an “observed” sample by drawing a random value from a uniform distribution with range [$t_o-2\Delta t_o$,$t_o$], where $t_o$ is the actual value of the observer-frame timescale.
Third, it may introduce a “pileup” of events close to the cadence limit depending on how events faster than $\Delta t_o$ are treated. One possibility is that every timescale smaller than $\Delta t_o$ is rejected as not observed. In this case there is no pileup. Another possibility is that timescales smaller than $\Delta t_o$ are set as observed, with value equal to that of $\Delta t_o$. Physically, the latter case corresponds to observations of events of various timescales (e.g flares in a particular frequency), of which we are interested in the smallest duration event occurring in a particular source. In this scenario faster-than-observed events may occur in a source, but the “fastest” one we record is set by the survey cadence, leading to a pileup resulting in a systematic offset in our statistical analysis. For the remainder of this section we will only consider the first possibility (no pileup). For a discussion of the possible effects of pileup, see Appendix <ref>.
We perform our benchmarking experiments as follows:
* We generate an “observed” sample using the procedure described above for a given $\Delta t_o$.
* We construct the cumulative distribution function of Eq. (<ref>) for different assumed values of $t_i$ and compare it with the simulated “observed” sample with the use of the Kolmogorov Smirmov test (K-S test). The K-S test provides a straightforward way of rejecting distributions that are inconsistent with our simulated dataset. We accept that for a probability value larger than $p=0.05$ we can not reject the null hypothesis that the two samples are drawn from the same distribution. Throughout this work, we refer to this p-value as the “probability of consistency”.
In order for our distribution to be normalized in the observer's frame, we need to calculate the normalization constant C (see Appendix A) which depends on $t_{o,min}$ and $t_{o,max}$. In practice, $t_{o,min}$ is established by $\Delta t_o$ and $t_{o,max}$ by the length of the observing season. For our benchmarking experiments, we simulated an observing experiment with an observing season set to ten times the longest observed timescale in the simulated sample. The value of $t_i$ which yielded the highest p-value of the K-S test is the value we considered the best-fit parameter value.
* We repeat the above process 100 times in order to evaluate the statistical spread of the results, for a given sample size and $\Delta t_o$. We quote the mean of these best-fit parameters as the most likely estimate of the rest-frame timescale and the standard deviation of the estimates in different iterations as the uncertainty of the final estimate.
Due to relativistic compression, the shortest rest-frame timescale can either be equal or longer than the shortest one observed. The longest rest-frame timescale can be up to the maximum observed divided by the smallest modulation factor ($\sim 10^{-2}$). Thus in the application of the method, the expected range of the rest-frame timescale estimate should be at least from the shortest to 100 times the longest observed timescale.
For this particular exercise, since we have a priori knowledge of the $t_i$, the range is set from 50 to 150 days. If the best-fit $t_i$ lies outside of that range, then the estimate will take the value of the closest boundary value. Any best-fit $t_i$ that is either $\leq 50$ or $\geq 150$, would correspond to an at least $50\%$ error in the estimate.
We examined three sample sizes (30, 60, 90 sources) and $\Delta t_o$ starting from 1 day, up to 60 days with a step of 1 day. We draw sources until we reach the desired sample size in the observer's frame ignoring cases where $t_o$ is smaller than $\Delta t_o$.
Figure <ref> shows the estimated best-fit $t_i$ versus $\Delta t_o$. The error bars are the spread of the estimated rest-frame timescale. As expected, increasing the number of sources in our samples results in a more narrow spread of estimates. There is a negative bias (systematic shift towards lower values of the estimated intrinsic characteristic timescale) with the increase of the number of days between observations. In many cases for $\Delta t_o\geq 30$ days, the best-fit estimate is more than $1\sigma$ away from the true value. This leads us to the conclusion that for a $\Delta t_o$ of more than 30 days ($\sim$1/3 the actual timescale we are trying to measure), the method loses its ability to accurately estimate the best-fit parameter of the intrinsic probability density function.
In addition, we explore the effect of the size of the sample at a fixed interval between observations. We follow the procedure described above with $\Delta t_o$ fixed to a predefined value, and a varying number of sources in the sample. We chose three values for $\Delta t_o$; 3, 7, and 14 days, all below the 30 days limit determined above (Fig. <ref>). The different samples have a size starting from 30 sources up to 150 with a 20 source step, from 150 up to 400 with a 50 source step, from 400 up to 700 with a 100 source step, and from 700 up to 1500 with a 200 source step.
Figure <ref>a shows the estimated $t_i$ versus the sample size with the error bars being the uncertainty of the method, while figure <ref>b shows the fractional bias of the estimated best-fit and the “true” $t_i$ ($|t_{i,fit}-t_i|/t_i$) versus sample size. It is obvious that the larger the sample the smaller the spread (Fig <ref>a), and the smaller the interval between observations, the smaller the bias (Fig. <ref>b). There is no significant change in the fractional bias for a given $\Delta t_o$ regardless of sample size. The method has an accuracy of $\leq 8\%$ in the estimates (Fig. <ref>b) as long as $\Delta t_o \leq 14\%$ of the intrinsic timescale value. For a large number of sources ($\geq 200$) and small $\Delta t_o$ (3 days, i.e 3% of the timescale we are trying to determine) the bias is smaller than $\sim 1\%$.
§.§ Testing for the type of distribution
Estimated $\delta$-function best-fit parameter ($t_i$) for an intrinsic uniform distribution versus $\Delta t_o$. The error bars represent the uncertainty of the best-fit $t_i$.
The best-fit intrinsic distribution family is not necessarily unique. Different families of distributions of intrinsic timescales can produce similar results in the observer's frame. For example, an intrinsic sharp distribution would appear consistent with a $\delta$-function as well as with a narrow normal distribution. Physically this is not necessarily problematic, since all the well-fitting distributions should describe approximately the same physical reality (in our example, a “preferred” timescale in the rest-frame). A more worrisome situation would be if, because of uncertainties or systematic effects, a single dataset appeared consistent with families of distributions describing a very different physical reality. To demonstrate how this effect can cause confusion, we tested whether a finite-width intrinsic distribution can appear consistent with a $\delta$-function. To this end, we performed the following experiment.
We created simulated samples of $t_o$ using an intrinsic uniform distribution from $t_i=$50 to $t_i=$500 days. The sample size was fixed to 200 sources, a size large enough to ensure accurate estimates (as seen in Fig. <ref>b). Using the procedure described in <ref> we fitted an intrinsic $\delta$-function for intervals between observations ($\Delta t_o$) from 1 to 60 days. We assumed no a priori knowledge of the intrinsic distribution or the range of the parameter space. Figure <ref> shows the $\delta$-function best-fit parameter $t_i$ versus $\Delta t_o$. The best-fit estimates of the characteristic timescale fall approximately at the median value of the intrinsic distribution with a systematic shift towards higher values for increasing time intervals between observations. The K-S test yields a probability of consistency that ranges from $45\%$ to $\leq 65\%$. In this case we could naively assume that all blazars have the same characteristic timescale in the jet rest frame.
However, there is a simple test for this particular case that can clarify the nature of the intrinsic distribution. If indeed the intrinsic distribution of timescales is a $\delta$-function, or at least a sharp distribution, then by fitting a normal distribution, we should in principle find that the best-fit distribution is very sharp. For simplicity we tested only two intervals, 7 and 30 days. The 7 day interval was chosen because it is the limit at which the method becomes very accurate ($\leq$ 4% error) and the 30 day interval because it is the limit after which results are unreliable (see Fig. <ref>).
For the 7 day interval we find that the best-fit intrinsic normal distribution would have mean $\mu=253.0$ and standard deviation $\sigma=168.0$ with a 97% probability of consistency between samples. For the 30 day interval the best-fit normal distribution has $\mu=168.8$ and $\sigma=239.0$ with a 76% probability of consistency. In both cases the resulting best-fit normal distribution is far from consistent with a $\delta$-function. The fact that they are significantly wider and with more than 30% higher probability of consistency is suggestive that the “true” intrinsic distribution is much wider than a $\delta$-function. Such simple tests can provide invaluable insights on the “true” shape of the intrinsic distribution in the application of the method.
§.§ K-S test versus conventional fitting methods
Fractional error between the maximum likelihood and K-S test fitting methods versus sample size for the 3 day, the 7 day, and the 14 day time interval between observations ($\Delta t_o$).
The K-S test provides a convenient way to automatically reject families of rest-frame distributions that are a poor fit to the data, especially in cases when there is no analytical solution. However, the K-S statistic (or the associated p-value) are not formally appropriate for fitting. To examine how much error we introduce by optimizing our parameters through K-S statistic minimization, we perform the same analysis in section <ref> for fixed time intervals between observations, but instead of using the K-S test fitting method, we use a maximum likelihood estimation (MLE) of parameters. Figure <ref> shows the fractional error between the K-S statistic and maximum likelihood fitting methods. Even for the longest time interval (14 days) between observations, the fractional error is $\lesssim 6\%$. In cases with small $\Delta t_o$ and a large number of sources, the fractional error is only $\sim 1\%$. We thus conclude that the K-S test method used throughout this work can adequately mimic a formally appropriate fitting method and provide robust results without the computational cost and complexity required by other methods. For a discussion of K-S versus MLE fitting in the case of population models themselves, see Paper I.
§ APPLICATION TO BLAZAR POLARIZATION ANGLE SWINGS
Source count distribution (number of sources N with flux density larger than S) for the RoboPol main sample sources. The black dotted line marks the assumed radio flux-density limit.
Distribution of the simulated Doppler factors for the RoboPol flux-density limited sample. Solid red is for the BL Lacs, and dashed green is for the FSRQs.
Distribution of the simulated timescale modulation factor for the RoboPol flux-density limited sample. Solid red is for the BL Lacs, and dashed green is for the FSRQs.
Best-fit distribution, for various distribution families, of the intrinsic timescales of the longest EVPA rotations in a radio flux-density limited subsample of RoboPol blazars.
Having benchmarked our method, we present the first real-data application to timescales associated with the rotation of the polarization plane (Electric Vector Position Angle (EVPA) rotation), in optical, seen in blazars. The physical mechanism of these rotations is, to this day, unknown, with models ranging from random walk processes <cit.> to shocks propagating in the helical magnetic field of the jet <cit.> and bends in the jet <cit.>. Although some rotations have been associated with $\gamma-ray$ flares and ejections of radio components <cit.> this is not always the case <cit.>.
In order to uncover the underline intrinsic timescale distribution of these rotations, we use data from the RoboPol survey <cit.>. The RoboPol survey uses a $\gamma$-ray flux-limited sample, specifically designed for rigorous statistical studies <cit.>. Since our method works for radio flux-density limited samples, the first step to applying our method is to identify an appropriate subset of the RoboPol main sample that is an unbiased (randomly drawn) subsample of a radio flux-density limited sample. All the RoboPol sources have been monitored by the Owens Valley Radio Observatory (OVRO) blazar program <cit.>. We use the maximum likelihood mean flux-densities <cit.> to construct the source count distribution for the RoboPol main sample sources (N(>S) as a function of S, Fig <ref>) and we approximate it with a horizontal part at low flux densities S and a declining power-law at higher S. We take sources brighter than 0.446 Jy (the transition point between the two approximations, see Fig. <ref>) to be the desired subsample, which we consider to be approximately unbiased compared to a parent flux-density-limited sample with a limit of 0.446 Jy. We have verified that our results are not sensitive to the exact location of the flux-density limit (i.e the inclusion or the omission of one more source). The number of sources in our radio flux-density limited sample is 31.
Figure <ref> shows the simulated distribution of Doppler factors and Fig. <ref> the simulated distribution of the timescale modulation factor for a 0.446 Jy radio flux-density limit, as derived from the population models in Paper II. Due to the low flux-density limit, we expect that the Doppler factor values will be, on average, smaller than the ones derived from VLBI or variability studies of brighter sources <cit.> and the ones in the higher flux-density–limit simulated samples we investigated in Papers I and II. The mean is $\sim 6$ for the BL Lacs and $\sim 10$ for the FSRQs. The mean of the timescale modulation factor is 0.475 for the FSRQs, and 0.49 for the BL Lacs. Since the two values are very similar, we adopted their mean (0.4825) as the common value. This way we avoid splitting the sample and reducing our statistics. We have verified that using either time scale modulation factor mean (FSRQ or BL Lac population) for the value of the whole sample does not result in any significant change on our results or our conclusions.
§.§ Observations
The data were collected during the 2013, 2014 and 2015 observing seasons of the RoboPol survey. The definition of the EVPA swings we adopt is similar to the definition of an EVPA rotation from <cit.>. We define an EVPA rotation as any continuous change of EVPA with total amplitude $\Delta \theta_{total}> 90^o$ comprised of at least four consecutive observations with statistically significant swings ($\Delta\theta>\sqrt{\sigma^2_{\theta_{i+1}}+\sigma^2_{\theta_{i}}}$, where $\sigma_\theta$ is the uncertainty in the EVPA) between them. The start and end points of a rotation are defined by a factor of 5 change of the slope of the EVPA time series, $\Delta \theta / \Delta t$ or a change of its sign. Contrary to <cit.> we do not apply any limits in the number of points or the length (rotation angle $\Delta \theta_{total}$) of an event. Following the definition in <cit.> we find 29 rotations in 14 sources, whereas with the definition adopted in this work we find 570 rotations in 31 sources making a statistical approach feasible.
Our method is applicable when each blazar is characterized by a single observed timescale. In the case of EVPA rotations, for each blazar we select the rotation (as defined above) with the longest duration (longest rotation timescale $\Delta T_{max}$). The shortest time interval between observations, $\Delta t_o$, is $\sim$1 day with an average of $\sim$9 days, while the events (see Fig. <ref>) last from 3 to 108 days.
§.§ Results
Upper panel: Distribution of $\Delta T_{max}$ in our sample. Lower panel: Cumulative distribution function of the $\Delta T_{max}$ of the EVPA rotations.
Following the procedure described in <ref>, we apply our method to the radio flux-density limited RoboPol subsample testing the six families of distributions described in Appendix <ref>. For the intrinsic timescales, we explored a parameter space from 1 day to 10$\times$ the maximum observing season length (244 days) of the RoboPol survey. In the case of the power-law index the range of the parameter space was set to [-7,7]. The best fit parameter values for each family of distributions and the K-S test p-value for each are shown in Table <ref>. The uncertainties in the parameters given in Table <ref> indicate the range in which for those parameters the K-S test yielded a $>5\%$ probability of consistency between observed and simulated samples.
Best-fit parameters and probability of consistency (K-S test) values for each family of distributions. The uncertainties indicate the range of the parameter that can produce an acceptable fit (i.e K-S test probability $>5\%$). In the case of the log-normal distribution, $t_i$ and $\sigma_{t_i}$ are related to $\mu$ and $\sigma_{sc}$ (see <ref>) through $t_i=exp(\mu+\sigma_{sc}^2/2)$ and $\sigma_{t_i}=\sqrt{ exp(\sigma_{sc}^2-1)exp(2\mu+\sigma_{sc}^2)}$
Distribution $t_i$ $\sigma_{t_i}$ p-value
(days) (days) (%)
Normal $87_{-15}^{+26}$ $5_{-4}^{+35}$ 31.5
$\delta$-function $88_{-19}^{+16}$ - 29.7
Log-Normal $92_{-18}^{+89}$ $18_{-17}^{+220}$ 25.5
Exponential $137_{-15}^{+28}$ - 11.8
Distribution $t_{i,min}$ $t_{i,max}$ slope p-value
(days) (days) (%)
Uniform $80_{-10}^{+14}$ $96_{-16}^{+33}$ - 31
Power-law $80_{-20}^{+20}$ $101_{-21}^{+399}$ $-4.4_{-1.1}^{+0.8}$ 30
All families of distributions, with the exception of the exponential, converge to approximately the same range of intrinsic timescales (Fig. <ref>) with very similar K-S test probabilities. The probability of consistency is highest for the normal and uniform distributions (31.5% and 31% respectively). For these two intrinsic (rest-frame) distributions, it is clear even by inspection that the resulting observer-frame distributions are almost indistinguishable (Fig <ref>). We observe a lack of very short and very long duration events. Both however can be attributed to observational constrains. For example, the former could be due to the cadence being too low to observe shorter smooth variations. Indeed, there are blazars for which even the longest observed smooth variation does not last much more than the time interval between observations. The current (season 2016) RoboPol observing strategy (i.e monitoring of the most variable sources with nightly observations, $\Delta t_o=1$) will be able to resolve faster (shorter duration) events.
The latter could be due to the length of the observing seasons ($\sim 240$ days) or due to the source not being available for observations throughout the observing season. In Appendix <ref>, we test whether observational constrains could significantly alter our results by shortening long-duration events. We find that it is highly unlikely that the observed event timescale distribution is significantly wider than the observed, thus our results are independent of such constrains.
For the best-fit intrinsic $t_i$ (87 days) we estimate, based on our benchmarking of <ref>, that the bias is $\leq$8% since the average $\Delta t_o$ is only $\sim$10% of the rest-frame $t_i$. From Fig. <ref>b we can conclude that the results of our analysis are robust independent of our sample size since the average $\Delta t_o$ is $<14\%$ of the $t_i$. Given that we are looking for the longest, rather than the shortest, event timescale in each blazar, a systematic pileup of timescales close to the average distance between observations (see Appendix <ref>) is not expected in this particular application. It should be noted that although the narrowly peaked intrinsic distributions have a systematically higher probability of consistency than the much wider exponential distribution, the exponential remains an acceptable fit, and our sample size is not large enough to conclusively reject a significantly wider distribution of rest-frame timescales (it is clear from Table <ref> that the lognormal and power-law distributions can also yield acceptable fits for a wider distribution). However, the exponential distribution is a monoparametric family, and none of the multi-parametric families prefers a shape similar to the best-fit exponential, indicating a preference of the data for the narrower distributions, according to the test we discussed in <ref>.
To check whether the “true” intrinsic distribution is an exponential, and our results arose from random sampling, we performed the following test. We created $10^4$ simulated datasets from an intrinsic exponential distribution with the same mean as in Table <ref> in a simulated observing survey with the same characteristics ($\Delta t_o$, observing length etc.) as RoboPol. We then used the distributions discussed above (including an exponential) with the same parameters from Table <ref> to fit the simulated samples and calculate the K-S test statistic. Out of the $10^4$ simulated samples, we calculated the percentage of trials that achieved a K-S test p-value equal or higher than the one reported in Table <ref> for each family of intrinsic distributions (other than the exponential). The probability values range between $<10^{-4}$ and 25%. However, there was no case where at the same time an exponential distribution would yield a $<11.8\%$ probability while all the other families yield a $>25.5\%$. For this reason, we formally reject the exponential distribution as a acceptable model.
§.§ Possible interpretations
Our findings suggest that the intrinsic timescales of the longest EVPA events in the radio flux-density limited subsample of RoboPol blazars that we have examined are confined to a relatively small range (spread of intrinsic timescales $\sim 30$ days).
It is possible that a subgroup in our sample has a small range of intrinsic timescales and is dominating over the rest of the sample.
We have tested for this scenario in the following way. We simulated a physical situation where our sample was an admixture of sources with a narrow distribution of timescales (distributed according to the best-fit uniform distribution describing the RoboPol sample) and sources with a wide distribution of timescales (uniform [50,500] days). We created a simulated sample following the procedure described in <ref> and proceeded in fitting an intrinsic uniform distribution. We repeated the same procedure 31 times, and each new simulated sample had a larger (by one) number of sources that were drawn from the wider distribution. Even with a small number of sources from a wide uniform distribution (<5 sources) the method is unable to produce a distribution with such a small range. We conclude that if there were two underlining distributions of significantly different widths, the wider of the two would dominate even if it only contributed a small number of sources.
If we assume that the EVPA rotations are produced by a shock propagating downstream in a jet, we should observe a rotation when its tracing the spirals of the helix of the jet <cit.>. In this case our results will imply that the maximum length of the jet traveled during a rotation is similar for all blazars and is roughly between 75 and 100 light days. In this case, the sources would have a narrow distribution of intrinsic timescales but due to different Lorentz factors and viewing angles the observed distribution is much wider. This would suggest that the intrinsic size of the jets is very similar for all blazars. However, given the range of jet sizes seen in radio galaxies (i.e the unbeamed parent distribution of blazars) this is highly unlikely.
If the EVPA rotations are caused by a random walk processes, the range of intrinsic timescales found in this work would represent a typical timescale of the smooth variation of turbulent plasma cells. To test if random walk processes can produce such a small range of timescales we perform the following experiment. We use the random walk model described in <cit.> to create a simulated EVPA curve comparable to the EVPA curve of each source. We create a simulated distribution of the longest EVPA rotations and we repeat the process to recover the intrinsic timescale distribution as described above. The simulated sample created via the random walk process has a 56% probability of consistency with the observed according to the K-S test. We found that the best-fit distribution is a normal distribution with mean 69 days and standard deviation 5 days with a 82% probability of consistency followed by a uniform with $t_{i,min}=66$ and $t_{i,max}=73$ and 81.5% probability of consistency. The results are very similar, showing that the random walk process can indeed reproduce such a small range and cannot be ruled out as a potential mechanism. It is interesting that the probability of consistency is more than twice as much as the one of the observed data. This could be coincidental, due to the random processes responsible for the simulated distribution, or could indicate the existence of two EVPA rotation mechanisms. It is argued in <cit.> and <cit.> that not all rotations can be explained by a random walk or a deterministic event. Therefore it is not unlikely that rotations created by random walks dominate our observed sample, yet there are still deterministic events present creating the large difference between K-S test p-values of random walk simulated and observed samples.
The assumption implied in treating the RoboPol data is that the beaming between the optical and radio regions is the same. If this is not the case, then the timescale modulation factor distribution would change, altering our results. The fact that the intrinsic distribution is narrow suggests that the difference in beaming is likely to be systematic, with all sources having either higher or lower Doppler factors. Otherwise the induced spread would create the appearence of a much wider distribution. If this is indeed the case, the width of the best-fit distributions will not change, but the distribution will be shifted to higher or lower values. Finally, we caution that, although disfavoured by the data, distributions allowing for a much wider range of rest-frame timescales are not formally rejected by our relatively small source sample (Table <ref>).
§ SUMMARY AND CONCLUSIONS
In Papers I and II, we found that the distributions of observer-frame timescales of the blazar populations are modulated by an exponential distribution compared to their rest-frame counterparts. This exponential has a mean depending on source class and the flux-density limit of the sample.
In this work, assuming that the intrinsic timescales of blazars are independent of redshift and Doppler boosting, we have developed a novel method of uncovering the underlying rest-frame probability density function of timescales of blazar jets using the observer-frame probability density.
We caution that the independence of rest-frame timescales from redshift and Doppler boosting is not obvious. There may be a dependence on either redshift or Doppler boosting, or both. However such a dependence can best be addressed in a model-dependent fashion (i.e. by testing an explicit model proposing such a relationship). Our formalism can be extended in a straightforward fashion to test such models.
In addition we have benchmarked our method in a realistic observing scenario using the timescale modulation factor distribution for the FSRQ population with a flux-density limit of 1.5 Jy and we assessed the impact of various systematic effects.
We found that when the interval $\Delta t_o$ between observations is longer than 30% of the timescale we try to measure, there is a systematic negative bias in the estimates that can lead to best-fit parameters deviating more than 1$\sigma$ from the characteristic timescale. Low cadence of observations can thus prevent our method from producing robust results.
Exploring the method's sensitivity to sample size we found that all estimates for sample sizes $\geq 30$ sources have an accuracy of $\leq 8\%$ if $\Delta t_o$ is sufficiently low. Monitoring programs with small $\Delta t_o$ and large number of sources such as the Owens Valley Radio Observatory (OVRO)[http://www.astro.caltech.edu/ovroblazars] blazar monitoring program <cit.> would make an ideal candidate for the application of our method.
In practice, observational constraints of specific experiments may induce additional effects other than the ones discussed in our general benchmarking. For this reason, we encourage the users of our method to conduct additional program-specific simulations that can uncover such effects, as we have also done in Appendix <ref> to address the specific effect of interrupted seasonal observations on EVPA rotation lengths.
The family of distributions that best fits a particular observer-frame dataset does not need to be unique. Different families can provide similarly fitting results. A test that can help determine whether the deduced rest-frame distribution is an acceptable description of the physical reality of the source population is to compare the best fits from different families of distributions. All distributions with the same number of parameters should converge to a similar answer (for example, regarding distribution width) for the obtained fit to be considered reliable. Distributions with more parameters should converge to the simpler distribution shapes, if the simpler distributions are acceptable descriptions of reality, otherwise the simpler distributions should be rejected.
The results of the analysis presented in this work during the benchmarking of our method are representative for a simulated 1.5 Jy flux-density limited FSRQ sample. For the same flux-density limit the mean of the timescale modulation factor distribution for the BL Lac population is fairly similar. Thus we would not expect any significant differences with the conclusions derived for the FSRQ sample. However, for a sample with a different flux-density limit the overall amount of beaming will be in principle different for different classes. The flux-density limit will affect the regimes of robustness of our method.
Samples with flux-density limit lower than 1.5 Jy will have a larger mean of the timescale modulation factor distribution, whereas a higher flux-density limit will result in a smaller mean. In the first case (lower flux-density limit) there will be a shift towards higher values of the observed timescales which in turn will push the limit ($\Delta t_o=30$ days) for the method to produce robust results to higher values. The opposite effect will be true for the higher flux-density limit. Shorter observed timescales will bring that limit to smaller $\Delta t_o$ values.
We have applied our method to the maximum duration of optical EVPA rotations observed in each blazar in a radio flux-density limited sample observed by RoboPol. We have found that the best-fit intrinsic distribution is a normal distribution with mean 87 days and standard deviation 5 days with a 31.5% probability of consistency between observed and simulated samples. An intrinsic uniform distribution with $t_{i,min}=80$ and $t_{i,max}=96$ has a 31% probability of consistency making it an equally preferred candidate since the two simulated observer-frame distributions are indistiguisable (Fig. <ref>). We have examined several interpretations of our results. If a significant fraction of our events are the result of a random-walk process, a result similar to the one of this work would be obtained. However, without better statistics we cannot exclude other interpretations.
In the case of timescale distributions in different wavelengths (e.g optical, $\gamma$-rays), whether the same boosting (Doppler factor) applies is still an open question. Deceleration in the jet from the $\gamma$-ray to the radio emission region <cit.> may result in the underestimation of the intrinsic timescales. If such a deceleration does not exist or its effect is not significant, timescales derived from the Fermi Gamma-Ray Space observatory all sky survey <cit.> will also constitute an ideal dataset for the application of our method due to the high cadence of observations and sample size, although one should keep in mind that a flux-density limited sample in $\gamma$-rays does not translate directly into a flux-density limited sample in radio <cit.>, which is the basis of our population models.
On the other hand, applications of our method to timescales extracted by radio monitoring of flux-density limited samples do not suffer from either of these potential problems, and thus constitute prime candidates where our method could be applied with maximum confidence, provided that the cadence of the monitoring is sufficiently high.
§ ACKNOWLEDGMENTS
The authors would like to thank Talvikki Hovatta, Dimitrios Giannios and Sebastian Kiehlmann for comments that helped improve this work. This research was supported by the “Aristeia” Action of the “Operational Program Education and Lifelong Learning” and is co-funded by the European Social Fund (ESF) and Greek National Resources, and by the European Commission Seventh Framework Program (FP7) through grants PCIG10-GA-2011-304001 “JetPop” and PIRSES-GA-2012-31578 “EuroCal”.
§ OBSERVED PROBABILITY DENSITY FUNCTIONS
§.§ Intrinsic Delta Function Distribution
If the intrinsic timescale distribution is a Delta function (all events share the same duration in the rest frame in all blazars) in the form of $\delta(t_{i}-\frac{t_o}{m})$, where $t_{i}$ is the characteristic timescale of the $\delta$-function, Eq. <ref> becomes:
\begin{equation}
\end{equation}
In order to solve the integral we have to transform $\delta(t_{i}-\frac{t_o}{m})$ to $\delta(m-\frac{t_o}{t_{i}})$ taking into account that, in general,
\begin{equation}
\delta(g(x))=\frac{\delta(x-x_0)}{|g'(x)|_{x_0}|}.
\label{delta}
\end{equation}
In our case $g(m)=t_{i}-\frac{t_o}{m}$ and $\delta(m-x_0)=\delta(m-\frac{t_o}{t_{i}})$. From Eq. <ref> we have:
\begin{equation}
\delta(t_{i}-\frac{t_o}{m})=\frac{t_o}{t_i^2}\delta(m-\frac{t_o}{t_i})
\end{equation}
The probability density function of the observed timescales will be:
\begin{eqnarray}
\end{eqnarray}
The value of C can be calculated as a function of $t_i$ and $t_{o,min}$, $t_{o,max}$ (the bounds of the observed timescales) by requiring $P(t_o)$ to be normalized:
\begin{equation}
\end{equation}
Thus the probability density function will be:
\begin{equation}
\label{prob_delta}
\end{equation}
Equation <ref> can be fitted to the observed data, in order to optimize the characteristic timescale $t_i$ of the intrinsic $\delta$-function
§.§ Intrinsic Uniform Distribution
If all timescales in blazar jets in the rest-frame are equally probable, we can assume an intrinsic uniform distribution in the form of:
\begin{equation}\label{intri_unif}
\begin{tabular}{lr}
$\frac{1}{t_{i,max}-t_{i,min}}$, & $t_{i,min}\leq t_i\leq t_{i,max}$\\
$0$,& $ t_i\leq t_{i,min}$ or $t_i\geq t_{i,max}$
\end{tabular}\right.
\end{equation}
Using the Heaviside step function defined as:
\begin{equation}\label{heaviside}
\begin{tabular}{lr}
$0$, & $x<0$\\
$1$,& $ x>0$
\end{tabular}\right.
\end{equation}
we can re-write Eq. <ref>:
\begin{eqnarray}
\label{intri_unif2}
\end{eqnarray}
Then Eq. <ref> becomes:
\begin{eqnarray}
P(t_o)&=&\frac{C\lambda}{t_{i,max}-t_{i,min}}\int^{m_{max}}_{m_{min}}\frac{1}{m}e^{-\lambda m}\nonumber\\
&\times &H(\frac{t_o}{m}-t_{i,min})H(t_{i,max}-\frac{t_o}{m}) dm.
\label{prob_uni}
\end{eqnarray}
Due to the properties of the Heaviside step function (Eq. <ref>) the observed probability density will be non-zero for $\frac{t_o}{m}-t_{i,min}>0\Rightarrow m<\frac{t_o}{t_{i,min}}$ and $t_{i,max}-\frac{t_o}{m}>0\Rightarrow m>\frac{t_o}{t_{i,max}}$. For the bounds of the integral there are four cases, two for the upper and two for the lower bound. For the upper bound either $m_{max}>\frac{t_o}{t_{i,min}}$ and the bound is $\frac{t_o}{t_{i,min}}$, or
$m_{max}<\frac{t_o}{t_{i,min}}$ and the bound is $m_{max}$. For the lower bound either $m_{min}>\frac{t_o}{t_{i,max}}$ and the lower bound is $m_{min}$, or $m_{min}<\frac{t_o}{t_{i,max}}$ and the lower bound is $\frac{t_o}{t_{i,max}}$:
If $m_{max}<\frac{t_o}{t_{i,min}}$ then:
\begin{equation}
\begin{tabular}{lr}
$\frac{C\lambda}{t_{i,max}-t_{i,min}}\int^{m_{max}}_{m_{min}}\frac{1}{m}e^{-\lambda m}dm$, & $t_o\leq t_{i,max}m_{min}$\\
& \\
$\frac{C\lambda}{t_{i,max}-t_{i,min}}\int^{m_{max}}_{t_o/t_{i,max}}\frac{1}{m}e^{-\lambda m} dm$,& $ t_o\geq t_{i,max}m_{min}$
\end{tabular}\right.
\label{prob_uni2}
\end{equation}
If $m_{max}>\frac{t_o}{t_{i,min}}$ then:
\begin{equation}
\begin{tabular}{lr}
$\frac{C\lambda}{t_{i,max}-t_{i,min}}\int^{t_o/t_{i,min}}_{m_{min}}\frac{1}{m}e^{-\lambda m}dm$, & $t_o\leq t_{i,max}m_{min}$\\
& \\
$\frac{C\lambda}{t_{i,max}-t_{i,min}}\int^{t_o/t_{i,min}}_{t_o/t_{i,max}}\frac{1}{m}e^{-\lambda m} dm$,& $ t_o\geq t_{i,max}m_{min}$
\end{tabular}\right.
\label{prob_uni3}
\end{equation}
Fitting Eq. <ref> and <ref> to the observed data, we can optimize for $t_{i,min}$ and $t_{i,max}$ that enter Eq. <ref>.
§.§ Intrinsic Power Law Distribution
We now assume that the intrinsic timescales in all blazars follow a power law distribution with slope k in the form of:
\begin{equation}
P(t_i)=C_1 t_i^{k}H(t_i-t_{i,min})H(t_{i,max}-t_i).
\label{intri_power}
\end{equation}
$H(t_i-t_{i,min})$ and $H(t_{i,max}-t_i)$ are Heaviside step functions (Eq. <ref>) in order to account for the truncated range of the intrinsic timescales. Equation <ref> becomes:
\begin{equation}
P(t_o)=\int^{m_{max}}_{m_{min}}C_1 (\frac{t_o}{m})^{k}H(\frac{t_o}{m}-t_{i,min})H(t_{i,max}-\frac{t_o}{m})C(\lambda e^{-\lambda m})\frac{1}{m}dm
\end{equation}
Setting $C_2=C_1C$, and following the procedure described in <ref>:
If $m_{max}<\frac{t_o}{t_{i,min}}$ then:
\begin{equation}
\begin{tabular}{lr}
$C_2\lambda t_{o}^k\int^{m_{max}}_{m_{min}}\frac{1}{m^{k+1}}e^{-\lambda m}$, & $t_o\leq t_{i,max}m_{min}$\\
& \\
$C_2\lambda t_{o}^k\int^{m_{max}}_{t_o/t_{i,max}}\frac{1}{m^{k+1}}e^{-\lambda m} $,& $ t_o\geq t_{i,max}m_{min}$
\end{tabular}\right.
\label{prob_power2}
\end{equation}
If $m_{max}>\frac{t_o}{t_{i,min}}$ then:
\begin{equation}
\begin{tabular}{lr}
$C_2\lambda t_{o}^k\int^{t_o/t_{i,min}}_{m_{min}}\frac{1}{m^{k+1}}e^{-\lambda m}$, & $t_o\leq t_{i,max}m_{min}$\\
& \\
$C_2\lambda t_{o}^k\int^{t_o/t_{i,min}}_{t_o/t_{i,max}}\frac{1}{m^{k+1}}e^{-\lambda m} $,& $ t_o\geq t_{i,max}m_{min}$
\end{tabular}\right.
\label{prob_power3}
\end{equation}
We can thus fit a function of the form given in equations <ref> and <ref> to the observed data to obtain the optimal $k$, $t_{i,min}$, and $t_{i,max}$ that enter Eq. <ref>.
§.§ Intrinsic Exponential Distribution
Assuming that the intrinsic timescales in blazar jets follow an exponential distribution in the form of:
\begin{equation}
P(t_i)=C_3\nu e^{-\nu t_i},
\label{intri_exp}
\end{equation}
where $C_3$ is the normalization constant, and $\nu$ is the inverse mean of the distribution, equation <ref> becomes:
\begin{equation}
P(t_o)=C C_3\nu\lambda\int^{m_{max}}_{m_{min}}\frac{1}{m}\exp\left[-(\lambda m+\nu t_o/m)\right] dm
\label{prob_expo}
\end{equation}
Fitting Eq. <ref> to the observed data we can optimize $\nu$ that enters Eq. <ref>.
§.§ Intrinsic Normal and Log-Normal Distributions
Assuming that the intrinsic timescales in all blazars are normally distributed with mean $\mu$, and standard deviation $\sigma$ in the form of:
\begin{equation}
\label{intri_normal}
\end{equation}
equation <ref> becomes:
\begin{eqnarray}
P(t_o)&=&C\lambda\int^{m_{max}}_{m_{min}}\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{(\frac{t_o}{m}-\mu)^2}{2\sigma^2}-\lambda m\right]\frac{1}{m}dm\nonumber\\
&=&\frac{C\lambda}{\sigma\sqrt{2\pi}}\int^{m_{max}}_{m_{min}}\exp\left[-\frac{(\frac{t_o}{m})^2+\mu^2-2\mu\frac{t_o}{m}}{2\sigma^2}-\lambda m\right]\nonumber\\
&\times &\frac{1}{m}dm.
\label{prob_normal}
\end{eqnarray}
If the logarithm of the intrinsic timescales is normally distributed with mean $\tilde\mu$ and scale $\sigma_{sc}$ then:
\begin{equation}
P(t_i)=\frac{1}{t_i\sigma_{sc}\sqrt{2\pi}}\exp\left[-\frac{(\ln t_i-\tilde{\mu})^2}{2\sigma_{sc}^2}\right],
\label{intri_lognormal}
\end{equation}
The probability density function of observed timescales will be:
\begin{eqnarray}
&P(t_o)&=\frac{C\lambda \exp\left[-\frac{(\ln t_o-\tilde{\mu})^2}{2\sigma_{sc}^2}\right]}{t_o\sigma_{sc}\sqrt{2\pi}} \nonumber\\
&\times &\int^{m_{max}}_{m_{min}}\exp\left[-\frac{(\ln m)^2+2\ln m(\tilde{\mu}+\ln t_o)}{2\sigma_{sc}^2}-\lambda m\right]dm.\nonumber\\
\label{prob_lognormal}
\end{eqnarray}
By fitting equations <ref> and <ref> to the observed data we can optimize for the mean ($\mu$) and standard deviation ($\sigma$) entering Eq. <ref> and the mean ($\tilde{\mu}$) and scale ($\sigma_{sc}$) entering Eq. <ref>.
The derivation of an analytical solution was only possible for the case of the $\delta$-function. The rest of the cases have to be solved either numerically or with the use of Monte-Carlo sampling.
§ ASSESSING FINITE-SAMPLING SYSTEMATIC EFFECTS
Estimated best-fit $t_i$ versus interval of observations (both in days) for samples with pileup. The black “+” is for the 30 source sample, red “x” for the 60 source sample and the green “$\star$” for the 90 source sample. The error bars represent the uncertainty of the best-fit $t_i$. The green line at 100 days shows the position of the “true” intrinsic timescale, while the black dashed lines at 50 and 150 days show the limits of the parameter space.
As discussed in <ref>, the finite sampling of a light curve could lead to a systematic offset in the observed timescale distribution, especially in cases were we are interested in the fastest event observed in a source (e.g flares). The time duration of events shorter than the time interval between observations will be observed with duration equal to that time interval, thus creating pileups. For this reason, knowledge of the survey $\Delta t_o$ is important in understanding such observational bias. A simple comparison between $\Delta t_o$ and the minimum observed timescale can provide insight on the existence (or not) of pileups in the data. The situation becomes even more complicated in highly variable sources where it is possible to observe multiple overlapping flares within a short period of time. The blending of the flares will lead to overestimating their duration according to the survey $\Delta t_o$. In this case, a more sophisticated approach is required, one that will take into account such overlap, mitigating the effects of finite sampling <cit.>.
Here we focus only on the simple case where the fastest events observed are those with duration equal to the survey $\Delta t_o$. To examine the effects and systematic shifts induce by the sampling of our survey in this case, we repeat the procedure described in <ref>. This time, when creating the “simulated-observed” sample, timescales shorter than the time interval of observations are not discarded, but instead are set equal to that time interval. It is clear from Fig. <ref> that event pileup results in a significantly larger scatter in the estimated best-fit $t_i$ with a negative bias for small values of $\Delta t_o$ and a positive bias for larger values. There are also cases (samples produced with large $\Delta t_o$) where the estimated best-fit parameter is $t_i\approx 150$ (Fig. <ref>), which is the upper end of the parameter space. This means that these estimates have a $\gtrsim 50\%$ error. This effect is to be expected, since we are contaminating our simulated samples with longer timescales according to the chosen $\Delta t_o$. Moreover, the majority of cases have their 1$\sigma$ error reaching that level regardless of $\Delta t_o$. We conclude that the presence of pileup prevents the method from providing an accurate estimate and will lead to the overestimation of the intrinsic timescales.
Since many events in the time domain of blazar jets are connected to the size of the emission region through causality arguments, such an overestimation would also lead to the overestimation of that region's size and induce scatter or create artificial correlations between events in the time domain and physical properties of blazar jets. Thus datasets should be treated with great caution with respect to the $\Delta t_o$ of the survey in the application of the method.
§ ASSESSING THE EFFECTS OF LIMITED OBSERVING SEASON LENGTH
Distribution of the ratio of durations ($T_i/T_{i+1}$) for all the EVPA events in each source observed with the definition adopted in this work for the RoboPol flux-limited subsample.
Our simulations show a surplus of long EVPA rotations (Fig. <ref>). As discussed in <ref> this is due to observational constrains related to the amount of time a source is available, with respect to the total RoboPol observing season length. Although the RoboPol sample was selected so that the sources would be available for the majority of the Skinakas observatory's observing period, it is not always the case. A limit on the observing season length sets an upper limit on the longest rotation the survey is able to detect, which in the best case scenario is the length of that observing season. Moreover, the time gap between observing seasons can affect the observed duration of any time-like event including an EVPA rotation. It is often unknown whether a rotation has begun prior to the beginning of the observing season, or if it continues after its end. Thus the observed time duration of an event can be significantly shorter than the “true” duration.
Figure <ref> (upper panel) shows the observed distribution of EVPA rotations. There is a peak at $\sim$ 40 days after which the distribution rapidly declines for longer events. Here we examine the possibility that the “true” observed distribution extends to 250 days (approximately the length of the RoboPol observing season) or longer, but the limited availability of the sources prevents us from observing longer rotations than the ones in Fig. <ref>.
For every source, we sort the observed events by increasing duration, calculate the ratios $T_i/T_{i+1}$, and construct the distribution of ratios (Fig. <ref>) for the whole sample. This ratio is indicative of the spread of the observed durations between one event and the following longer/shorter event. Since each event in a given source is modulated by the same modulator factor, the ratio is independent of any relativistic effects, which allows us to combine all the ratios for individual sources in one distribution.
Using the above distribution, and assuming that the “true” duration of events is uniformly distributed from 3 (the shortest observed duration) to 250 days, we use random sampling to create a simulated dataset of the longest events that would have been observed given all the observational constrains (observing gap, limited availability, random event starting time with respect to beginning and end of the observing season) of the RoboPol survey. We then compare the distribution of simulated durations to the observed (Fig. <ref>) using the K-S test. We repeat the process $10^6$ times and calculate the number of trials for which the K-S test could not reject the null hypothesis that the samples are drawn from the same distribution (i.e. the probability value was $>$5%). We find that $\sim 0.5\%$ of the trials resulted in a distribution consistent with the observed. If we extend the “true” duration of the events to 500 days, the number of trials drops to $<10^{-6}$. Thus it is unlikely that the “true” observer's frame duration of EVPA events is much longer, yet due to observational constrains we are not able to observe them. However, the fact that our method predicts the existence of events longer than what is observed in the data suggests that, although not very significant, there is a bias towards shorter in time events due to the availability of a source. Our findings stress the importance of long term uninterrupted observations in uncovering the true nature of EVPA rotations.
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1511.00309
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A Deep Observation of Gamma-ray Emission from Cassiopeia A using VERITAS
Augusto Ghiotto, for the VERITAS Collaboration
Columbia University,
Supernova remnants (SNRs) have long been considered the leading candidates for the accelerators of cosmic rays within the Galaxy through the process of diffusive shock acceleration. The connection between SNRs and cosmic rays is supported by the detection of high energy (HE; 100 MeV to 100 GeV) and very high energy (VHE; 100 GeV to 100 TeV) gamma rays from young and middle-aged SNRs. However, the interpretation of the gamma-ray observations is not unique. This is because gamma rays can be produced both by electrons through non-thermal Bremsstrahlung and inverse Compton scattering, and by protons through proton-proton collisions and subsequent neutral-pion decay. To disentangle and quantify the contributions of electrons and protons to the gamma-ray flux, it is necessary to measure precisely the spectra and morphology of SNRs over a broad range of gamma-ray energies. Cassiopeia A (Cas A) is one such young SNR ($\sim 350$ years) which is bright in radio and X-rays. It has been detected as a bright point source in HE gamma rays by Fermi-LAT and in VHE gamma rays by HEGRA, MAGIC and VERITAS. Cas A has been observed with VERITAS for more than 60 hours, tripling the published exposure. The observations span 2007-2013, and half of the data were taken at large zenith angles to boost the effective area above a few TeV. We will present the detailed spectral and morphological results from the complete dataset.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
Cassiopeia A (Cas A) is a young supernova remnant ( $\sim 350$ years <cit.>) at distance of $\sim 3.4$ kpc <cit.> from Earth. It is currently thought, based on light echo observations, that the remnant was created by a Type IIb supernova of 15-25 solar masses <cit.>. This type of supernova occurs when a red supergiant that has already lost its hydrogen envelope undergoes core collapse.
Objects such as Cas A have been considered among the best candidates to explain the acceleration of cosmic rays, whose sources cannot be resolved due to diffusion by magnetic fields. Gamma rays, on the other hand, can be traced back to their point of origin, offering an avenue to understanding particle acceleration in supernova remnants (SNRs) and other astrophysical objects.
Gamma rays in a SNR, however, can be produced either leptonically or hadronically. Leptonic processes include non-thermal bremsstrahlung and inverse-Compton scattering, while hadronic collisions lead to neutral-pion production and decay. Looking for the neutral-pion decay signature in the SNR spectrum can give strong evidence for cosmic ray acceleration in the expanding shell.
The emission from Cas A has been well studied from radio to gamma rays <cit.>. Cas A is a faint optical source, due mostly to thermal emission in the reverse-ejecta and fast moving knots <cit.>. In the milimeter wavelength range, the Heinrich Hertz Submillimeter Telescope found a broadening of CO emission lines in the south and west regions of Cas A, indicating interaction between the shock front and nearby molecular clouds. Infrared observations from the Spitzer Space Telescope suggested interaction with molecular clouds in the northern region <cit.>.
Cas A has also been extensively studied in X-rays by XMM-Newton and Chandra (0.1 to 10 keV) and by NuSTAR (3 to 79 keV). A model with two coexisting X-ray emission mechanisms, thermal and non-thermal, is consistent with data. Thermal X-ray emission originates in the reverse-shocked ejecta, rich in highly ionized atoms <cit.>. Non-thermal - mostly synchrotron - radiation is produced at both the forward and reverse shocks <cit.>. NuSTAR observations also showed interior knots as sources for non-thermal X-ray emission above 15 keV <cit.>.
Very high energy (VHE) gamma rays from Cas A were first detected by HEGRA in 2001 <cit.>, and later confirmed by MAGIC <cit.> and VERITAS <cit.>. The spectral index $( \gamma = 2.4 - 2.6)$ and the fluxes at $3\% $ of the Crab Nebula published by these three groups are in good agreement within errors. High energy (HE) gamma rays (MeV to GeV) were first detected by Fermi-LAT in 2010 <cit.>. Subsequent Fermi-LAT data indicated a break in the spectrum around 1.72 GeV, favoring a hadronic emission model over a leptonic one <cit.>.
In this study with VERITAS data, we extend the VHE spectrum up to 7 TeV while setting an upper limit in the 10 TeV bin and we look at the centroid location of Cas A. After collecting more than 60 hours of data from 2007 to 2013, we were able to reduce the statistical errors in the spectral index and the centroid below the level of their systematic errors. These updated results have also recently been presented at the 34th ICRC <cit.>.
§ THE VERITAS EXPERIMENT
The Very Energetic Radiation Imaging Telescope Array System (VERITAS) consists of 4 ground-based telescopes located in southern Arizona ($31\degree $ 40'N, $110\degree $ 57' W, 1.3 km a.s.l.). Each 12-m diameter telescope contains a camera with 499 photomultiplier tubes (PMTs), yielding a $3.5 \degree $ field of view. From 2007 to 2012, there were two major upgrades:
* in 2009, a telescope was moved to make the array more symmetric and increase the typical telescope baselines;
* 2011-12: installation of FPGA-based camera trigger system and high-efficiency PMTs <cit.>.
Currently, a source with a flux level of $1\%$ of the Crab Nebula can be detected in less than 25 h. The angular resolution for gamma rays at 1 TeV is $0.08\degree $ and the sensitivity range spans from 85 GeV to 30 TeV. There are currently 54 sources detected by VERITAS.
After data are collected, they are analyzed in the following steps:
* Image is calibrated and cleansed, selecting pixels with Cherenkov light and removing the ones with night sky background <cit.>;
* Hillas parameters are calculated (length, width and size of the image), and used to differentiate showers originated by gamma rays from those originated by cosmic rays <cit.>;
* The intersection of major axes of the shower images in the camera plane provides a geometric technique to locate the origin of the gamma-ray.
§ CAS A DATA
From 2007 to 2013, more than 60 hours of Cas A data was taken by VERITAS. In this work, we re-analyzed the previously published data <cit.>, taken at small zenith angles (SZA), and added post-upgrade data taken both at SZA and large zenith angles (LZA). Data were selected undeer dark and clear sky conditions, with all four telescopes functioning and set at $0.5 \degree $ wobble from the source location. LZA data gives a higher effective area for events above 1 TeV in comparison with SZA. Separately, using only post-upgrade SZA data, we looked for the centroid location of the remnant. We summarize our data in Table <ref>.
[]Summary of Cas A data taken by VERITAS from 2007 to 2013.
Date $ \theta _Z $ range Average $ \theta _Z $ Live Time Mean Trigger Rate
(deg) (deg) (Hours) (Hz)
09/07 - 11/07 27-40 34 18 250
12/11 - 12/11 33-43 38 2 350
09/12 - 12/12 24-39 30 19 400
09/12 - 12/13 40-64 56 25 300
§ RESULTS AND DISCUSSION
We obtained the spectrum for the entire data set and fitted it with a power-law in the energy range from 300 GeV to 7 TeV, giving a $\chi ^2 $ of 2.22 for 5 degrees of freedom and resulting in a good fit probability of $ 81\% $ (Figure <ref> ). The differential energy spectrum for the whole data set is given below:
\begin{equation}
\frac{dN}{dE} = (1.45\pm 0.11) \times 10^{-12} (E/1 TeV)^{-2.75\pm 0.10_{stat} \pm 0.20 _{sys}} cm^{-2}s^{-1}TeV^{-1}
\end{equation}
Cas A differential energy spectrum: published VERITAS data <cit.> and new data fit to a power-law. The Crab spectrum is also shown for comparison.
Both the index and the normalization are in agreement with previous results by HEGRA <cit.>, MAGIC <cit.> and VERITAS <cit.>.
Figure <ref> shows the combined spectrum with Fermi-LAT <cit.> and the complete VERITAS data set. Fitting the broad-band spectrum above 2 GeV, a broken power-law model is favored at the 4.9 sigma level over a single power-law when only statistical errors are considered, and at $>3.5$ sigma level when considering systematic errors as well. This suggests a softening of the spectrum above a few hundred GeV.
In order for the data to be consistent with the current hadronic emission model (Figure <ref>), the cut off energy would have to be higher than 10 TeV.
for Cas A: Combined spectral points: Fermi-LAT and VERITAS current results. VERITAS points shown are for the entire data set. An upper limit is set at the 10 TeV bin <cit.> .
Combined Fermi-LAT <cit.> and new VERITAS spectrum in comparison with leptonic and hadronic emission models <cit.>.
Figure <ref> (left frame) shows a skymap generated by a reflected-region background model <cit.>. It is derived from 18 h of post-upgrade observations at small zenith angles, yielding a significance of $11 \sigma $. Cas A is consistent with a point source for our point spread function (PSF) shown in white. The centroid is marked by a blue cross at $ RA=23\textrm{h}\
23\textrm{m}\ 20.4\textrm{s}\ \pm \ 0\degree .006_{stat} \pm 0\degree .014_{sys} $ and $ Dec= 58.817 \pm 0\degree .006_{stat} \pm 0\degree .014_{sys} $. Figure <ref> (right) compares the centroid positions from Fermi (yellow, <cit.>), VERITAS (green, <cit.>) and MAGIC (red, <cit.>) with the new VERITAS (white). This updated result is consistent with both the Fermi-LAT centroid location as well as our previously published centroid. This updated result does not confirm the speculation in <cit.> that the GeV emission is associated with a north-central region bright in the infrared while the TeV emission is associated with bright synchrotron knots on the western side of the SNR.
Left: skymap with Cas A SZA data only, yielding a significance of $11 \sigma $. Point spread function (PSF) is shown as a white circle and the centroid location as a blue cross. Right: Chandra image of Cas A <cit.> with overlayed comparison of centroid positions from Fermi (yellow, <cit.>), VERITAS (green, <cit.>) and MAGIC (red, <cit.>) with the new VERITAS (white).
§ CONCLUSIONS AND PROSPECTS
We were able to refine the VHE spectrum, both at lower and higher energy. Statistical errors in the index and in the centroid location are now below the current systematic uncertainties, motivating further work to improve our systematics. There are prospects for a better analysis process for the Large Zenith Angle data, important for the TeV-range spectrum.
Finally, in the near future the Cherenkov Telescope Array (CTA) will offer better angular resolution and pointing, which may refine the location of the centroid or even resolve the emission. CTA should also be able to extend the spectrum both to lower energy to overlap with Fermi-LAT and to energies beyond 10 TeV.
This research is supported by grants from the U.S. Department of Energy Office of Science, the U.S. National Science Foundation and the Smithsonian Institution, and by NSERC in Canada. We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument. The VERITAS Collaboration is grateful to Trevor Weekes for his seminal contributions and leadership in the field of VHE gamma-ray astrophysics, which made this study possible.
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1511.00286
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The Higgs Physics Program at the International Linear Collider
Jan Strube
for the ILC detector and physics community
The International Linear Collider (ILC) is a proposed electron – positron collider with a collision energy of $\sqrts=\unit[500]{GeV}$ in the baseline configuration. The ILC physics program takes full advantage of the fact that the machine can be operated at arbitrary energy from the maximum down to the peak of the $\PZ\PH$ production cross section near $\sqrts=\unit[250]{GeV}$ or below. It will advance our understanding of nature through precision measurements of Standard Model parameters, detailed study of the Higgs sector, and a comprehensive search for new phenomena that extends beyond the purely kinematic reach. This note gives an overview of the ILC Higgs program.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
The discovery of a Higgs boson by the LHC experiments completes the Standard Model of Particle Physics, which has enjoyed tremendous success in collider-based particle physics experiments.
However, it cannot explain a number of established phenomena, like cosmological Dark Matter, the accelerating expansion of the Universe, or the observed Matter – Anti-Matter asymmetry.
New physical models that can explain at least some of these lead to predictions for the coupling between the Higgs boson and the other known fundamental particles that deviate to varying degrees from Standard Model predictions. Measuring the deviations at the per cent level would show the direction of new physics from the size of the deviations.
§ THE ILC ACCELERATOR
Candidate scenario for operating the ILC in the baseline configuration and the luminosity upgrade.
The International Linear Collider is a proposed 31 km electron – positron accelerator with a center-of-momentum energy of 500 GeV in the baseline.
A candidate site has been identified in the Japanese Kitakami region.
The time lines for different deliverables of the physics program in a number of candidate operating scenarios have been evaluated to guarantee the highest possible impact at the earliest time<cit.>. The preferred candidate for an operating scenario is shown in Figure <ref>.
§ THE ILC DETECTORS
Detector concepts that can deliver the required precision have been studied in the ILD and SiD groups. They are designed around the particle flow paradigm and feature highly granular calorimeters that are contained in the solenoid. They feature high precision silicon vertex detectors with a trigger-less readout, a low-mass tracking system and nearly $4\pi$ coverage in solid angle. They feature complementary technologies for the main tracker and the hadronic calorimeter. The operating scenario foresees them sharing beam time in a push–pull scenario. The performance of the baseline design has been studied in extensive simulation campaigns. The ability to carry out precision Higgs physics measurements has a large influence on the optimization of the detector design, and studies to improve the understanding of how detector parameters impact the physics performance continue.
§ THE RECOIL TECHNIQUE
The low background and accurate knowledge of the collision energy at the ILC allows measuring the cross section of $\PZ\PH$ production independently of the $\PH$ decay in a so-called recoil technique. The $\PZ$ decay to a pair of leptons can be reconstructed very cleanly, due to the well-known mass of the $\PZ$ boson and the high momentum resolution of the tracking detectors. A fit to the recoil mass $m_{\text{rec}} = ((\sqrts - E_{\PZ})^2-p_{\PZ}^2)^{1/2}$ allows the measurement of the $\PZ\PH$ cross section without reconstructing decays of the Higgs boson. This technique is applicable at any collision energy, but it has the lowest uncertainty near threshold, where the effect of the finite spread of the collision energy due to beam–beam interaction is smallest.
§ HIGGS DECAYS TO BOTTOM AND CHARM QUARKS, AND TO GLUONS
Higgs decay $\PQb\PAQb$ $\PQc\PAQc$ $\Pg\Pg$ $\PQq\PAQq$
SM prediction 57.8% 2.68% 8.56% $<0.05\%$
Branching ratios of a Standard Model Higgs Boson with a mass of [125]GeV to hadrons<cit.>. In this table $\PQq$ is an alias for the light quark flavors $\PQs,\PQu,\PQd$. Decays to $\PQt$ quarks are kinematically prohibited.
The Standard Model prediction for Higgs decays are listed in Table <ref>. A large range of models that have survived the discovery and subsequent study of the scalar boson by the LHC experiments predict only per-cent level deviations from these branching ratios.
The Higgs boson decay to a pair of b quark jets has the largest branching ratio, yet is one of the most challenging decays to measure at the LHC due to the large background from b quark jets. The vertex resolution of the detectors is not high enough to tag c quark jets, thus the branching ratio of Higgs decays to c quark jets and gluons cannot be measured at the LHC or its upgrades.
At the ILC, Higgs decays to b and c quark jets, and gluon jets are measured in the signature $\PGn\PGn \PH$ with large missing energy, or $\PZ\PH$ with hadronic or leptonic $\PZ$ decays.
At the ILC, vertices are found from all charged tracks. Charged tracks, reconstructed vertices and neutral calorimeter clusters are combined into two (four) jets in events with large missing energy (hadronic $\PZ$ decays). The jets are then flavor-tagged based on the number of reconstructed vertices and a multi-variate classifier that uses additional kinematic variables. The main background in the channel with missing energy is from hadronic $\PZ$ decays, while the background in the four-jet channel consists mainly of $\PZ\PZ$ and $\PW\PW$ decays. It is mostly reduced by cuts on the invariant mass of jet pairs. Events that remain after selection cuts are fit simultaneously using histogram templates of the flavor tag output.
The achievable final precision in the ILC physics program is a statistical uncertainty of 0.7% on the measurement of the b Yukawa coupling, 1.2% on the measurement of the c Yukawa coupling, and 1.0% on the measurement to the loop-induced gluon coupling<cit.>.
§ HIGGS DECAYS TO $\TAU$ LEPTONS
The $\tau$ lepton is the heaviest lepton of the Standard Model and as such has the largest branching fraction, with the Standard Model prediction of 6.37%. This would yield a sizable number of events that can be probed further for CP properties in an angular analysis.
Events are reconstructed in the channels $\PQq\PQq\PGtp\PGtm$ and $\Plp\Plm\PGtp\PGtm$. The analysis includes a $\tau$ jet finder that uses the jet charge to reduce background. Using the approximation that the visible $\tau$ decay products and the neutrinos are collinear, with no other invisible decay products, the ILC program can achieve a precision on the $\tau$ Yukawa coupling of 1.9% in the baseline and 0.9% in a luminosity upgrade (extrapolated from the analysis at [250]GeV<cit.>).
§ INVISIBLE HIGGS DECAYS
Higgs decays to invisible final states proceed in the Standard Model through the decay $\PH\to\PZ\PZ$, where each $\PZ$ boson in turn decays to a pair of neutrinos. The branching fraction for this decay is 0.1%. Weakly interacting massive particles, that are of yet unobserved, but are hypothesized as candidates for cosmological dark matter, would lead to a signature of large missing energy in a collider detector. Higgs decays to these particles are kinematically allowed, if their mass is $m \leq m_{\PH}/2$; this would significantly alter the measured branching ratio of invisible Higgs decays.
The recoil mass technique allows the measurement of the branching fraction of these decays in the ILC baseline program with a precision of 0.29%. This is more than one order of magnitude improvement over LHC predictions and allows for accurate verification of the Standard Model prediction.
§ HIGGS TOTAL WIDTH
The Standard Model prediction for the natural width of the Higgs boson is [4]MeV, too small to be measured directly by analyzing a reconstructed mass distribution, or a production cross section threshold. Under the assumption that the off-shell couplings are identical to the on-shell couplings, the LHC experiments can measure the width of the Higgs boson using off shell decays $\PH\to\PZ\PZ$ with a precision of [22]MeV.
At the ILC, the $\PH$ width can be measured in channels where Higgs production and decay are mediated by the same coupling (Equation <ref>), e.g. a heavy gauge boson coupling. The decay $\PH\to\PW\PW$ can be measured with greater precision than $\PH\to\PZ\PZ$, owing to the larger branching ratio. Using the definition of a branching ratio in Equation <ref>, one can use a high-precision measurement of a branching ratio, such as $\PH\to\PQb\PQb$ as a crutch to substitute the coupling $g^2_{\PH\PW\PW}$ with the coupling $g^{2}_{\PH\PZ\PZ}$. The latter can be measured in the recoil analysis, independent of the Higgs boson decay.
The $\PZ\PH$ production cross section can be measured with an uncertainty of less than 2% using the recoil mass technique<cit.>.
\begin{equation}
\label{eq:definitionBR}
\Gamma_{\PH} = \frac{\Gamma(\PH\to\PW\PW)}{\mathcal{BR}(\PH\to\PW\PW)}\propto\frac{g^2_{\PH\PW\PW}}{\mathcal{BR}(\PH\to\PW\PW)}
\end{equation}
\begin{equation}
\frac{g^2_{\PH\PW\PW}}{g^2_{\PH\PZ\PZ}}\propto\frac{\sigma_{\PGn\PGn\PH}\times\mathcal{BR}(\PH\to\PQb\PQb)}{\sigma_{\PZ\PH}\times\mathcal{BR}(\PH\to\PQb\PQb)}
\end{equation}
§ TOP YUKAWA COUPLING
Production cross section of the $\PQt\PAQt\PH$ process and projected measurement uncertainty as function of the collision energy, scaled relative to the value at $\sqrts=\unit[500]{GeV}$.
The main challenge for measuring the top Yukawa coupling at the ILC is the low cross section at the collision energy of $\sqrts=\unit[500]{GeV}$. An increase of 10% in the energy reach of the baseline leads to a nearly four-fold increase of in the $\PQt\PAQt\PH$ production cross section (see Figure <ref>).
The analysis strategy of the measurement of the top Yukawa coupling in direct production at the ILC is independent of the collision energy. Isolated tracks from leptonic $\PW$ decays are found and removed from the event. The rest of the event is clustered into four, six or eight jets, depending on whether two, one, or zero isolated leptons were found. Top quarks and the Higgs boson are reconstructed from the jets and leptons using a chi-squared minimization technique with the nominal masses and mass resolution terms found from simulation. Background is reduced using flavor tagging information on the jets, jet size and event shape information. The remaining background is predominantly due to top quark pair production, and to b quark pairs and Z bosons produced in association with top quark pairs.
The projected precision on the top Yukawa coupling in $\unit[1]{ab^{-1}}$ at an ILC upgraded to $\unit[1]{TeV}$ is 4%<cit.>. When combined with the analysis at [500]GeV<cit.>, this projects to a precision of 2% in the full ILC program.
§ HIGGS SELF-COUPLING
The self-coupling term of the Higgs potential can be probed in multi-Higgs production at high energy colliders. The quartic coupling involving triple Higgs production is most likely inaccessible for the foreseeable future due to the small cross section of the process. Double Higgs production can be measured at the ILC in the $\PGn\PGn\PH\PH$ channel. The measurement depends on the excellent performance of b-tagging and jet energy resolution to reduce background from $\PZ\PH$ and double $\PZ$ production. With jet clustering in addition to intrinsic detector resolution being a major source of systematic uncertainty, the current estimate for the achievable uncertainty on the tri-linear self-coupling constant in the baseline program is 27%<cit.>. An energy upgrade to [1]TeV will reduce this uncertainty to $\approx 10\%$.
§ CONCLUSIONS
Time development of the precision of the Higgs couplings in scenario “H” (Figure <ref>).
$\Gamma_{\PH}$ $g(\PH\PZ\PZ)$ $g(\PH\PW\PW)$ $g(\PH\PQb\PAQb)$ $g(\PH\Pg\Pg)$ $g(\PH\PQc\PAQc)$,$g(\PH\PQt\PAQt)$ $g(\PH\gamma\gamma)$ $g(\PH\PGt\PGt)$,$g(\PH\PGm\PGm)$
[500]GeV ILC (%) 0.96 0.2 0.24 0.49 0.95 1.1 3.4 0.73
Precision on Higgs width and couplings to SM particles in a global fit to the measurements in the ILC program of the $\sqrts=\unit[500]{GeV}$ baseline configuration (top row) and for the total program including the luminosity upgrade (bottom row). The measurement of $g(\PH\gamma\gamma)$ can be improved to 1.2% (1.0%) in the baseline configuration (luminosity upgrade) when taking into account the projected LHC measurements of the ratio of $g(\PH\PZ\PZ)/g(\PH\gamma\gamma)$.
The ILC baseline program offers a comprehensive picture of the Higgs sector and allows for a self-consistent global fit of all couplings to achieve the greatest precision. Figure <ref> shows the development of the coupling measurements over time in Scenario “H”. The precision for the measurements of the properties of the LHC Higgs boson in the full ILC program in that scenario is described in the previous sections, with additional improvements of a [1]TeV ILC option, where this option leads to a significant improvement. The uncertainties achievable in a the 20-year program of a [500]GeV ILC in a global fit using the Snowmass prescription<cit.> are listed in Table <ref>.
This precision is not optional: Different models that can explain some of the obvious discrepancies between the Standard Model and cosmological observations can lead to subtly different predictions in the interactions between the Higgs boson and the known fundamental Standard Model particles. The per-cent-level precision required to distinguish these models can be achieved in the ILC program.
This work was presented on behalf of the global ILC detector and physics community.
|
1511.00551
|
Department of Physics and Institute of Theoretical Physics and Astrophysics,
Xiamen University, Xiamen 361005, Fujian, China
Collaborative Innovation Center of Chemistry for Energy Materials, Xiamen
University, Xiamen 361005, Fujian, China
We propose an effective phonon treatment in one dimensional
momentum-conserved lattice system with asymmetric interparticle interaction
potentials. Our strategy is to divide the potential into two
segments by the zero-potential point, and then approximate them by piecewise harmonic potentials with effective force constants $\tilde{k}_L$ and $\tilde{k}_R$ respectively. The effective
phonons can then be well described by $\omega_c=\sqrt{2(\tilde{k}_L+\tilde{k}_R)}%
|sin(\frac{1}{2}aq)|$. The numerical verifications show that this treatment works very well.
Effective phonon treatment(EPT) has significant importance in condensed matter physics
) and it seems they do effective in many situations. In the past two decades, the heat
conduction in low-dimensional systems has attracted intensive studies<cit.>(see also references therein),
and the EPT is applied also to this problem since phonons act as the predominant heat carrier. The
scaling behaviors of heat conduction have been successfully explained <cit.>, and the sound speed is predicted <cit.>.
Recently, Y. Zhang et al.<cit.> pointed out that while EPT works well in lattices with symmetric interaction potential,
significant divergence occurs in lattices with asymmetric interaction potential even in predicting the sound velocity.
In the present paper, we present a simple but very effective treatment of effecive phonons. Our idea is
to divide the interaction potential into two segments by the minimum potential point. Then the
left and the right segments are equivalent to two segments of harmonic potential respectively. The piecewise harmonic potential is applied to derive the properties of the unharmonic system analysis. We will varify our idea by analytically and numerically using several typical one-dimentional lattice models, including the Fermi-Pasta-Ulam-$\alpha$-$\beta$ (FPU-$\alpha$-$\beta$) model <cit.> and the Lennard-Jones (L-J) model.The Hamiltonian we study reads
\begin{equation} \label{eq-Hamiltonian}
\end{equation}
where $p_i$ is the momentum and $x_i$ is the displacement from equilibrium
position for the $i$th particle, $N$ is the total number of particles that
equals system's size (the lattice constant is set be unit in our studies below), $V$ is the
interaction potential between nearest neighbour lattices.
For the FPU-$\alpha$-$\beta$ lattice, $V$ is given as
\begin{equation} \label{eq-FPU-V}
\end{equation}
where $\alpha$ controls the degree of asymmetry (see Fig. <ref>
(b) and (c)). While $\alpha=0$, $\beta\ne0$, it becomes symmetric FPU-$\beta$ model. On the contrary, if $\alpha\ne0$, $\beta=0$, it
appears as the FPU-$\alpha$ model (see Fig. <ref>(c)) <cit.>. The potential of the L-J model is
\begin{equation}
\end{equation}
where the parameter set ($m$, $n$) control the degree of asymmetry (see Fig.
<ref>(d)). This model has important practical implications
because it can well approximate the inter-particle interactions in many real
As the harmonic potential model is applied as the templet model in traditional EPTs, we also need a templet. It is the piecewise harmonic model defined as
\begin{equation} \label{eq-asy-harmonic-V}
\begin{cases}
\frac{k_{L}}{2}x^2, & x<0 \\
\frac{k_{R}}{2}x^2, & x\ge0 \\
\end{cases}%
\end{equation}
where $k_{L}$/$k_{R}$) is the left/right force constant to
the equilibrium position. The different set ($k_L$, $k_R$) control
the degree of asymmetry (see Fig. <ref>(a) for several
examples). It returns to the classical harmonic model when $%
k_{L}=k_{R}\ne0$. If one of the constants equals zero, it
will works as a collision model. To establish our EPT theory, we need a full study of the piecewise harmoic model.
(Color lines) (a) to (d) are plots of potential of piecewise model, FPU-$\protect\alpha $-$\protect\beta $ model, FPU-$\protect\alpha $
model, and L-J model, respectively.
We first study a two-particle system with the piecewise harmonic potential for the illuminating purpose.
The equations of motion with periodic boundary conditions are
\begin{equation} \label{eq-twoparticle-motion}
\begin{cases}
\ddot{x}_1=k_{L,R}(x_2-x_1)-k_{L,R}(x_1-x_2) \\
\ddot{x}_2=k_{L,R}(x_1-x_2)-k_{L,R}(x_2-x_1)%
\end{cases}%
\end{equation}
and we assume $x_1(0)-x_2(0)<0$ while $t=0$, then Eq.<ref> can be rewritten as
\begin{equation} \label{eq-twoparticle-motion-reW}
\begin{cases}
\ddot{x}_1=(k_R+k_L)x_2-(k_R+k_L)x_1 \\
\ddot{x}_2=(k_L+k_R)x_1-(k_L+k_R)x_2%
\end{cases}%
\end{equation}
In addition, momentum and energy are conserved, i.e., $p_{1}(t)+p_{2}(t)=0~%
\&~\frac{1}{2}p_{1}(0)^2+\frac{1}{2}p_{2}(0)^2+\frac{1}{2}%
(here $\varepsilon$ is mean energy of each particle). So we get $%
x_1(0)=-x_2(0)$, and $\varepsilon=\frac{1}{2}p_1(0)^2+(k_L+k_R)x_1(0)^2=%
\frac{1}{2}p_2(0)^2+(k_L+k_R)x_2(0)^2$. Then the results can been easily
obtained and the simplified form is
\begin{equation} \label{eq-twoparticle-result}
\begin{cases}
x_{1}(t)=\Lambda cos(2\sqrt{\bar{k}}t+\varphi) \\
x_{2}(t)=-\Lambda cos(2\sqrt{\bar{k}}t+\varphi)%
\end{cases}%
\end{equation}
where $\bar{k}=\frac{k_L+k_R}{2}$, $\Lambda=\sqrt{\frac{\varepsilon}{2\bar{k}%
}}$, $\varphi=-arctan(\frac{\sqrt{2}}{2\sqrt{\bar{k}}}\frac{p_{1}(0)}{%
x_{1}(0)})$. It's easy to see the result does not depend on the initial
conditions (if we assume $x_1(0)\ge x_2(0)$, we will get a same result).
We can clearly see the frequency of the system is $\omega=2\sqrt{\bar{k}}=2%
\sqrt{\bar{k}}sin(\frac{\pi}{2})$, which only relies on the mean value of $%
k_L$ and $k_R$ but has no direct dependency upon themselves. Besides, the
frequency is completely same as that of the pure harmonic lattice <cit.>. It means that the piecewise linear two-particle system can be
strictly described by a harmonic one with force constant $\bar{k}$.
For multi-particle situation
\begin{equation} \label{eq_omegaC_AsyH}
\omega=2\sqrt{\bar{k}}|sin(\frac{1}{2}aq)|
\end{equation}
where $a$ is the lattice constant that is set be unit in our study, and $q$
is the wave-vector.
To verify our treatment for multi-particle cases, we immediately analyze the
power spectrum of the time series of the particles' instantaneous momentum
by fast Fourier transformation (FFT) with molecular dynamics simulation
(MDS). And the results are presented in Fig.<ref>.
In Fig.<ref>(a), the red solid line is power spectra of
pure harmonic chain, i.e., to set $k_L=k_R$ in our model (here we set $%
). The green dot line is the theoretical position of frequency. It is
clearly seen that our numerical results and existing theoretical results
<cit.> are consistent, namely, our calculation program is
completely accurate. Then we calculated our model and the results are shown
in Fig.<ref>(b) to (d). The blue solid (or red dash)
line in Fig.<ref>(b) is normalized power corresponding
to $k_L=0.75~\&~k_R=0.85$ (or $k_L=0.85~\&~k_R=0.75$), and the green dot
line is the theoretical value of frequencies' centre position that is
calculated by Eq.<ref>. It is clearly seen that the
theoretical and numerical results fit perfectly in all frequency regime. An
interesting is that the two lines are almost complete overlap.
This is due to exchanging the order of $k_L$ and $k_R$ does not affect the
mean value of $\bar{k}$ (the two systems are mirror symmetrical). We also
note that $\bar{k}$ does not rely on the temperature of system. The effect
can be observed in Fig.<ref>(c), where we did numerical
experiments with fixed $k_L=1.8~\&~k_R=1.2$ under different energy density $%
\varepsilon=1.0,~10.0$. But the centre position of frequencies are still
identical. At the same time, we find that the normalized (by energy density)
power spectra overlap completely. This is because the model owns a similar
certain scale that is pointed in Ref.<cit.>. Fig.<ref>(d) shows an extremely asymmetric case ($%
k_L=3.0~\&~k_R=0.0$) that is very like a collision model. Strictly speaking,
it is not a lattice but a fluid. But Eq.<ref> is still able
to give a precise prediction for the centre positions of some low
frequencies (the centre position of lowest frequency is identical with $%
k_L=1.8~\&~k_R=1.2$ because they have same mean value). It states that the
piecewise linear system can be equivalent to a harmonic lattice with force
constant $\bar{k}$, seriously.
To make a comparison from (a) to (d) in Fig.<ref>, we
can easily see that the phonon peak is broadening with asymmetry increasing
(see the most typical examples Eq.<ref>(c) and(d)). A similar
phenomenon is also reported in Refs.<cit.>. Therefore, here we
emphasize that an asymmetric piecewise linear system can be well described
by a pure harmonic lattice, only indicate that the centre positions of
phonons peaks are same as an equivalent harmonic lattice, but there is
essential difference comes from asymmetry between them, e.g., the phonon
peak will broaden in true asymmetric piecewise linear lattice, which means
that there exist stronger interactions between phonons. Besides, normal heat
conduction is also observed in asymmetric harmonic model <cit.>
, but these are absent in the equivalent pure harmonic chain.
All above presented results, the system size is fixed to $N=64$, so that we
can clearly see all frequencies. But the correctness has been proved by our
numerical experiment (not shown here) that is independent with system size.
(Color lines) Normalized power spectra of the momentum time series.
(a) The red solid line is corresponding to pure harmonic chain with $%
k_L=k_R=1.0$. (b) The blue solid (or red dash) line is corresponding to $%
k_L=0.75~\&~k_R=0.85$ (or $k_L=0.85~\&~k_R=0.75$). (d) the red solid
corresponding $k_L=3.0~\&~k_R=0.0$. Energy density $\protect\varepsilon$ is
set be unit in these three simulations. (c) The blue solid line and red dash
line are, respectively, corresponding to $\protect\varepsilon=1.0,~10.0$
with fixed $k_L=1.8~\&~k_R=1.2$. The vertical green dot lines in all are
predicted centre position of frequency. System size $N=64$ for all.
Up to now, we established our template model. For an unharmonic lattice, our goal is to obtain the
two effective force constants.
It is well known that the potential of a linear force ($F=-kx$) is $V=\frac{1%
}{2}kx^2$, which can easily obtained by $V=-\bar{F}x$ (here $\bar{F}=\frac{1%
}{2}F$ is the average force), and only the linear force has this nature.
Based on this principle, we think that if a nonlinear force can be
equivalent to a linear one, the equivalent potential should be also obtained
by the same way, i.e., $\tilde{V}_h=\langle\frac{1}{2}\tilde{k}%
x^2\rangle=\langle-\bar{F}x\rangle$, and $\bar{F}=-\frac{1}{2}\frac{\partial
V}{\partial x}$, the $\langle\cdots\rangle$ indicates the ensemble average.
Further, we can get an specific equivalent approach as
\begin{equation} \label{eq_KLR_FPU}
\begin{cases}
\tilde{k}_{L}=\frac{\langle \frac{\partial V}{\partial x}x\rangle}{\langle
x^2\rangle}, & x<0 \\
\tilde{k}_{R}=\frac{\langle \frac{\partial V}{\partial x}x\rangle}{\langle
x^2\rangle}, & x\ge0 \\
\tilde{k}=\frac{\tilde{k}_L+\tilde{k}_R}{2} &
\end{cases}%
\end{equation}
and above equations can be further rewritten as a general integral form
\begin{equation} \label{eq_K_FPU}
\tilde{k}=\frac{1}{2}\big(\frac{\int_{-\infty}^{0}\frac{\partial V}{\partial
\partial V}{\partial x}x\rho(x)dx}{\int_{0}^{\infty}x^2\rho(x)dx}\big),
\end{equation}
where $\rho(x)=e^{-\frac{V(x)+Px}{k_BT}}$ is distribution function of
relative displacement, $p=-\langle\frac{\partial V}{\partial x}\rangle$ is
the thermodynamic pressure<cit.>, and $k_B$ is
Boltzmann constant that set to be unit, and $T$ is the temperature of
system. Note that $\tilde{k}$ has a certain relationship with the speed of
sound $c_s=\sqrt{\tilde{k}}$ in our dimensionless models. It is not a fixed
constant but a function of temperature $T$ and system parameters. As a
concrete example, FPU-$\alpha\beta$ model, $\tilde{k}=f(T,\alpha,\beta)$.
For symmetric FPU-$\beta$ model ($\alpha=0$) the pressure $p=0$, $V(x)$ and $%
\rho(x)$ will become even function, at this moment Eq.<ref> can be
simplified as
\begin{equation} \label{eq_K_FPU_B}
\tilde{k}=\frac{\int_{-\infty}^{\infty}\frac{\partial V}{\partial x}%
\end{equation}
A interesting observation is that Eq.<ref> is completely same as the form
given in Ref.<cit.>. So our tactics at least is
effective for symmetric nonlinear models. In addition, from Eq.<ref> we can see that if the pressure is known, we can get the
equivalent force constant $\tilde{k}$ through integration directly instead
of MDS. Hence, it is necessary for us to find an approach to calculate the
pressure. This goal is achived based on Spohn's recent works <cit.>. Following these works
we develop an algorithm for calculating the pressure of a momentum-conserved lattice as
\begin{equation} \label{eq_pressure}
\begin{cases}
p=\frac{V(-\lambda)-V(\lambda)}{2\lambda} & (a) \\
\phi_{+}(\lambda)=\int_{0} ^{\infty} x e^{-\frac{V(x)+px}{T}}dx & (b) \\
\phi_{-}(\lambda)=\int_{0} ^{\infty} x e^{-\frac{V(-x)-px}{T}}dx & (c) \\
\phi(\lambda)=\phi_{+}(\lambda)-\phi_{-}(\lambda)\equiv 0 & (d)%
\end{cases}%
\end{equation}
where $\lambda$ is an arguments that is governed by (d). Specific steps is
to replace (a) into (b) and (c) in turn, then to solve the equation (d),
after $\lambda$ is received the pressure will be easily obtained only need
to put $\lambda$ back into (a). It is clearly seen that $\lambda$ is not
unique but arbitrary for symmetric model. Nevertheless, arbitrary nonzero
values of $\lambda$ put into the equation (a) will get fixed zero pressure,
which just agree well with expectation of symmetric model. We should point
out that if $\int_{-\infty}^{\infty}\rho(x)dx\rightarrow\infty$, the
algorithm will do not work. The internal pressure and $\tilde{k}$ can only
be acquired by MDS with Eq.<ref> (e.g., FPU-$\alpha$ model and
L-J model).
In the following, we test whether Eq.<ref> is suitable for the
asymmetric situation by the numerical integration. Here, we only need to
check whether the prediction of the sound velocity accurately instead of
calculating the centre position of lowest frequency by FFT. This is because
Ref.<cit.> have proved that the shift of lowest frequency
and the speed of sound are equal in dimensionless models. We compare our
theoretical results (red uptriangle lines in Fig.<ref>) with
the standard sound velocity computed via the method developed by Spohn<cit.> (blue square lines in Fig.<ref>, which can also be calculated by numerical integration). In
order to make a comparison, we also show the result obtained with the traditional EPT<cit.> (shorted as TEPT in the figure). All these are calculated by numerical integration
instead of MDS, so they are independent of system size. One can see that the
three methods get same results in low temperature regimes where the velocity
is tending to one (gray dash line) and high temperature regimes $c_s\propto
T^{1/4}$ (gray dot line, this can be also observed in Ref.<cit.>).
In the moderated temperature regimes, significant deviation appears with the traditional treatment, especially for large $\alpha$.
In all the regimes, our theoretical predictions show no deviations.
(Color lines) The speed of sound $c_s$ against temperature $T$ with
different $\protect\alpha=0.0 (a), 1.0 (b), 1.5 (c) and 2.0 (d)$. Blue square line is calculated by Spohn's formula, green
circles are obtained by traditional EPT, and
red uptriangles are achieved by our formula. $\protect\beta=1.0$ is fixed throughly.
Next, we test it in FPU-$\alpha$ model and L-J model. For FPU-$\alpha$, we
apply a small energy density $\varepsilon=0.01$ to avoid runaway instability
of trajectories. The results are drawn in Fig. <ref>(a) for $%
\alpha=-0.5$, and Fig. <ref>(b) for $\alpha=-1.0$. For L-J
model, $\varepsilon=0.5$ is fixed, and the results are shown in Fig.<ref>(c) for $(m,n)=(12,6)$, and Fig.<ref>(d) for $%
(m,n)=(2,1)$. From these results, we clearly see that our approach is very
workable. And phonons peaks are broadening with asymmetry increasing as well
in these models. Especially, it is clearly seen that the results of L-J
model are vary similar to the result of a fluid (compare with Fig. <ref>(d), normal heat conduction is observed in both of them
<cit.> ).
(Color lines) Power spectra of the momentum time series. (a) and
(b) for FPU-$\protect\alpha$ model, $\protect\varepsilon=0.01$. (c) and (d)
for L-J model, $\protect\varepsilon=0.5$. The vertical green dot lines are predicted centre positions of frequencies. System size is fixed at $N=64$.
To summarize, we present an effective phonon treatment that works very well in
one dimensional momentum-conserved lattice with either symmetric or
asymmetric interaction potentials. In
asymmetric models, all phonons peaks are broaden, and the degree of the broaden depends on the degree of the potential asymmerty. The more stronger the asymmetry, the more stronger the phonon scattering. This effect may explain why the normal heat conduction
behavior appears in certain lattice models <cit.>. We shall discuss this problem in a forthcoming paper <cit.>.
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1511.00230
|
Abstract. The properties of Volterra-composition operators on the weighted Bergman space with exponential type weights are investigated in this paper. For a certain class of subharmonic function $\psi:\mathbb{D}\rightarrow\mathbb{R}$, we state some necessary and sufficient conditions that a Volterra-composition operator $V_{\varphi}^{g}$ from the weighted Bergman space $AL_{\psi}^{2}(\mathbb{D})$ to Bloch type space $B_{\psi}(\mathbb{D})$ ( or little Bloch type space $B_{\psi,0}(\mathbb{D})$) must satisfy for $V_{\varphi}^{g}$ to be bounded or compact.
Keywords The weighted Bergman space; Bloch type space(little Bloch type space); Volterra-composition operator; bounded; compact.
MR(2010) Subject Classification 74D05; 35Q72; 35L70; 35B40
This research was supported by
the National Natural Science Foundation of China under Grant (No.11301224,11401256,11501249)
Let $\mathbb{D}=\{z:~|z|<1\}$ be the open unit disk in the complex plane $\mathbb{C}$ and $dA$ be the normalized area measure on $\mathbb{D}$ .
Let $L^{2}(\mathbb{D},dA)$ be the space of square integrable functions and let $L^{\infty}(\mathbb{D},dA)$ be the space of essential bounded measurable functions. We will use abbreviations $L^{2}(\mathbb{D})$ for $L^{2}(\mathbb{D},dA)$ and $L^{\infty}(\mathbb{D})$ for $L^{\infty}(\mathbb{D},dA)$.
For a subharmonic function $\psi:\mathbb{D}\rightarrow\mathbb{R}$, let $L^{\infty}_{\psi}(\mathbb{D})$ be the space of measurable functions $f$ on $\mathbb{D}$ such that $e^{-\psi}f\in L^{\infty}(\mathbb{D})$
and let $L^{2}_{\psi}(\mathbb{D})$ be the Hilbert space of measurable function $f$ on $\mathbb{D}$ such that
The inner product of $L^{2}_{\psi}(\mathbb{D})$ is given by
$$\langle f, g\rangle_{L_{\psi}^{2}}=\int_{\mathbb{D}}f\overline{g}e^{-2\psi}dA$$
Let $H^\infty_\psi(\mathbb{D})$ be the subspace of $L^\infty_\psi(\mathbb{D})$ consisting of analytic functions and $AL^{2}_{\psi}(\mathbb{D})$ be the closed subspace of $L^{2}_{\psi}(\mathbb{D})$ consisting of analytic functions.
$AL^{2}_{\psi}(\mathbb{D})$ is called the weighted Bergman space with exponential type weights(see [3],[4]).
An analytic function $f$ is said to belong to the Bloch-type space $B_{\psi}(\mathbb{D})$ if
Let $B_{\psi,0}(\mathbb{D})$ denote the subspace of $B_{\psi}(\mathbb{D})$ such that
$$\lim_{|z| \rightarrow 1}e^{-2\psi(z)}|f'(z)|=0.$$
This space is called little Bloch-type space.
$^{[1],[2]}$ For real valued function $\psi\in C^{2}(\mathbb{D})$ with $\Delta\psi>0$,
where $\Delta$ is the Laplace operator.
Let $\tau(z)=(\Delta\psi(z))^{-\frac{1}{2}}$. We say that $\psi\in\mathcal{D}$ if the following conditions are satisfied.
$($1$)$ There exists a constant $C_{1}>0$ such that $|\tau(z)-\tau(\xi)|\leq C_{1}|z-\xi|$ for $z,\xi\in\mathbb{D}$.
$($2$)$ There exists a constant $C_{2}>0$ such that $\tau(z)\leq C_{2}(1-|z|)$ for $z\in\mathbb{D}$.
$($3$)$ There exist constants $0<C_{3}<1$ and $a>0$ such that $\tau(z)\leq \tau(\xi)+C_{3}(1-|z|)$ for $|z-\xi|>a\tau(\xi)$.
Let $H(\mathbb{D})$ be the space of analytic functions on $\mathbb{D}$ and suppose that $g:\mathbb{D}\rightarrow\mathbb{C}$ is an analytic function, $\varphi$ is an analytic self-map of the unit disk and $f\in H(\mathbb{D})$,
the Riemann-Stieltjes operator is defined by
The composition operator $C_{\varphi}$ is defined as $C_{\varphi}f\triangleq f\circ \varphi$.
Let $V_{\varphi}^{g}$ be the Volterra-composition operator which is defined as
$$V_{\varphi}^{g}f(z)\triangleq (T_{g}\circ C_{\varphi}f)(z)=\int_{0}^{z}f(\varphi(t))g'(t)dt,\quad\forall f\in H(\mathbb{D}), ~z\in\mathbb{D}.$$
Note that when $\varphi$ is an analytic self-map of the unit disk and $f,g\in H(\mathbb{D})$, we have
In this paper, we will characterize the boundedness and compactness of the operator $V_{\varphi}^{g}$ from the weighted Bergman space with exponential type weights to the Bloch-type space (or little Bloch-type space). These properties of Hankel operators have been considered(see [1],[2]).
We assume that $H_{\psi}^{\infty}(\mathbb{D})$ is dense in $AL_{\psi}^{2}(\mathbb{D})$, the set of all polynomials is dense in $AL_{\psi}^{2}(\mathbb{D})$.
In order to prove the main results, we need the following lemmas.
then Bloch–type space $B_{\psi}(\mathbb{D})$ is a Banach space with the norm $\|\cdot\|_{B_{\psi}}$.
At first, we will show that $\|\cdot\|_{B_{\psi}}$ is a norm.
For a function $f\in B_{\psi}(\mathbb{D})$, we define
$$b_{\psi}( f)=\sup_{z\in\mathbb{D}}e^{-2\psi(z)}|f^{'}(z)|.$$
It is easy to see that
$$\|\alpha f\|_{B_{\psi}}=|\alpha f(0)|+b_{\psi}(\alpha f)=|\alpha||f(0)|+|\alpha|b_{\psi}(f)=|\alpha|\| f\|_{B_{\psi}}.$$
\begin{eqnarray*}
\|f+g\|_{B_{\psi}}&=&|f(0)+g(0)|+b_{\psi}(f+g)
\leq|f(0)|+b_{\psi}(f)+|g(0)|+b_{\psi}(g)\\
&=&\| f\|_{B_{\psi}}+\| g\|_{B_{\psi}}.
\end{eqnarray*}
So $\|\cdot\|_{B_{\psi}}$ is a semi-norm.
Assume that $\|f\|_{B_{\psi}}=|f(0)|+b_{\psi}(f)=0$, we have $f(0)=0$ and $b_{\psi}(f)=0$, which means that
Since $e^{-2\psi(z)}\neq0$ when $z\in\mathbb{D}$, it must be $|f'(z)|\equiv 0$. It follows that $f(z)\equiv C $ for some constant $C$. Since $f(0)=0$, we obtain $C=0$. It implies that $f=0$.
Now we are going to prove the completeness of $\|\cdot\|_{B_{\psi}}$.
Suppose that $\{f_{n}\}_{n\in\mathbb{N}}$ is a Cauchy sequence of $B_{\psi}(\mathbb{D})$, i.e.
$$\|f_{n}-f_{m}\|_{B_{\psi}}=|f_{n}(0)-f_{m}(0)|+\sup_{z\in\mathbb{D}}e^{-2\psi(z)}|f_{n}^{'}(z)-f_{m}^{'}(z)|\rightarrow 0 ~~~~(n,m\rightarrow\infty).$$
\begin{eqnarray*}
\end{eqnarray*}
for each $z\in \mathbb{D}$ , there exists a $f\in H(\mathbb{D})$ such that
$f_{n}$ uniformly converges to $f$ on any compact subsets of $\mathbb{D}$ .
By Cauchy integral formula, we have $f^{'}_n$ uniformly converges to $f^{'}$ on any compact subsets of $\mathbb{D}$ .
Since $b_{\psi}(f_{n}-f_{m})\rightarrow 0$, for any $\varepsilon>0$, there exists a positive integer $N$, such that when $n,m>N$, we have
$$e^{-2\psi(z)}|f_{n}^{'}(z)-f_{m}^{'}(z)|<\varepsilon,~~~~for ~all ~z\in\mathbb{D}.$$
Letting $m\rightarrow\infty$ in the above inequality, we can see that
$$e^{-2\psi(z)}|f_{n}^{'}(z)-f^{'}(z)|\leq\varepsilon,~~~~for~all~ z\in \mathbb{D},$$
this implies that $b_{\psi}(f_{n}-f)\rightarrow 0~~(n\rightarrow \infty)$.
It follows that $\|f_{n}-f\|_{B_{\psi}}\rightarrow 0~(n\rightarrow\infty).$
It is easy to see that there exists a positive integer $n$ such that for all $z\in \mathbb{D}$
i.e. $f\in B_{\psi}$, thus the completeness is proved.
Therefore, $B_{\psi}(\mathbb{D})$ is a Banach space with the norm $\|\cdot\|_{B_{\psi}}$.
Notation: we write $f(z,w)\sim g(z,w)$ if there exist positive constants $C$ and $C'$ such that
$$C g(z,w)\leq f(z,w)\leq C' g(z,w)$$
Let $\psi\in\mathcal{D}$, then we have
where $K(z,w)$ is the reproducing kernel of $AL_{\psi}^{2}(\mathbb{D})$.
Let $f\in AL_{\psi}^{2}(\mathbb{D})$ and $\psi\in\mathcal{D}$, then
$|f(z)|=|\langle f,K_{z}\rangle|\leq\|f\|_{L_{\psi}^{2}}\|K_{z}\|_{L_{\psi}^{2}}=\sqrt{K(z,z)}\|f\|_{L_{\psi}^{2}}$.
Let $\varphi$ be an analytic self-map of the unit disk $\mathbb{D}$ and $g\in H(\mathbb{D})$. Assume that
$V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is bounded. Then $V_{\varphi}^{g}$ is compact if and only if $V_{\varphi}^{g}f_{k}$ converges to zero as $k\rightarrow\infty$, for any bounded sequence $\{f_{k}\}_{k\in\mathbb{N}}$ in $AL_{\psi}^{2}(\mathbb{D})$ which uniformly converges to zero on compact subsets of $\mathbb{D}$.
First, assume that $V_{\varphi}^{g}$ is compact. Let $\{f_{k}\}_{k\in\mathbb{N}}$ be any bounded sequence in $AL_{\psi}^{2}(\mathbb{D})$ and uniformly converges to zero on compact subsets of $\mathbb{D}$. Since $V_{\varphi}^{g}$ is a compact operator , there exists a subsequence of $\{f_{k}\}_{k\in\mathbb{N}}$ ( without loss of generality, we assume it is $\{f_{k}\}_{k\in\mathbb{N}}$ ) and a function $h\in B_{\psi}(\mathbb{D})$ such that
$$\|V_{\varphi}^{g}f_{k}-h\|_{B_{\psi}}\rightarrow0 (k\rightarrow\infty).$$
That is,
$$|V_{\varphi}^{g}f_{k}(0)-h(0)|+\underset {z\in\mathbb{D}}{\textup{sup}}e^{-2\psi(z)}|f_{k}(\varphi(z))g^{'}(z)-h^{'}(z)|\rightarrow0(k\rightarrow\infty).$$
By the definition of $V_{\varphi}^{g}$, it is obvious that $V_{\varphi}^{g}f_{k}(0)=0$. It follows that $h(0)=0.$
It is not difficult to see that $f_{k}(\varphi(z))g^{'}(z)$ uniformly converges to $h^{'}(z)$ on any compact subsets of $\mathbb{D}$.
Besides, $\{f_{k}\}_{k\in\mathbb{N}}$ uniformly converges to zero on any compact subsets of $\mathbb{D}$ and $g\in H(\mathbb{D})$,
it follows that $h^{'}(z)\equiv0.$ Also, as $h(0)=0$, we have $h(z)\equiv0$, that is
$$\|V_{\varphi}^{g}f_{k}\|_{B_{\psi}}\rightarrow0 (k\rightarrow\infty).$$
Conversely, let $\{f_{k}\}_{k\in\mathbb{N}}$ be any bounded sequence in $AL_{\psi}^{2}(\mathbb{D})$. By Lemma 2.3, we have $|f_{k}(z)|\leq\sqrt{K(z,z)}\|f_{k}\|_{L_{\psi}^{2}}.$
therefore, $\{f_{k}\}_{k\in\mathbb{N}}$ is uniformly bounded on any compact subset $K$
of $\mathbb{D}$.
By Montel Theorem, there exists a subsequence $\{f_{k_{n}}\}_{k_n\in\mathbb{N}}$ of $\{f_{k}\}_{k\in\mathbb{N}}$ and an analytic function $f$, such that $\{f_{k_{n}}\}$ converges to
$f$ uniformly on any compact subsets $K$ of the unit disk $\mathbb{D}$.
According to the Fatou Lemma, $f\in AL_{\psi}^{2}(\mathbb{D})$. Together with the assumption, we have $\|V_{\varphi}^{g}(f_{k_{n}}-f)\|_{B_{\psi}}\rightarrow0 ~(n\rightarrow\infty).$ Therefore,
$\{V_{\varphi}^{g}f_{k}\}_{k\in\mathbb{N}}$ has subsequence $\{V_{\varphi}^{g}f_{k_{n}}\}_{n\in\mathbb{N}}$ which converges to $V_{\varphi}^{g}f\in AL_{\psi}^2(\mathbb{D})$. From the characterization of compact operators, we see that $V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is compact.
Let $\psi\in\mathcal{D}$, then the bounded closed set $K$ of $B_{\psi,0}(\mathbb{D})$ is compact if and only if $\underset {|z|\rightarrow1}{\textup{lim}}\underset {f\in K}{\textup{sup}}e^{-2\psi(z)}|f^{'}(z)|=0.$
Assume that the bounded closed set $K$ of $B_{\psi,0}(\mathbb{D})$ is compact. Let $\varepsilon>0$, we have $\underset{f\in K}{\bigcup}B(f,\frac{\varepsilon}{2})\supset K$, where $B(f,\frac{\varepsilon}{2})$ is a ball with center $f$ and radius $\frac{\varepsilon}{2}$.
Because $K$ is compact, there exist $f_{1},f_{2},\cdot\cdot\cdot,f_{n}\in K$, such that for any $f\in K$, we have
$$\|f-f_{j}\|_{B_{\psi}}<\frac{\varepsilon}{2} ~~~~(1\leq j\leq n,j\in\mathbb{N}^{*}).$$
Hence, for any $z\in\mathbb{D}$, we have, for $1\leq j\leq n$,
$$e^{-2\psi(z)}|f^{'}(z)|\leq e^{-2\psi(z)}|f_{j}^{'}(z)|+\frac{\varepsilon}{2}.$$
Since $f_{j}\in B_{\psi,0}(\mathbb{D})$, for each positive integer $j$, there exists a positive real number $r_{j}\in(0,1)$ such that $e^{-2\psi(z)}|f_{j}^{'}(z)|<\frac{\varepsilon}{2}$ when $r_{j}<|z|<1$. Let $r=\textup{max}\{r_{1},r_{2},\cdot\cdot\cdot,r_{n}\}$, then we have, when $r<|z|<1$,
$$\underset {|z|\rightarrow1}{\textup{lim}}\underset {f\in K}{\textup{sup}}e^{-2\psi(z)}|f^{'}(z)|=0.$$
Conversely, suppose that the sequence $\{f_{n}\}_{n\in\mathbb{N}}\subset K$. Since $K$ is bounded, there is a positive constant $M$ such that for any $n\in \mathbb{Z^{+}} $, $\|f_{n}\|_{B_{\psi}}\leq M$.
By Lemma 2.3, it is easy to see $\{f_{n}\}_{n\in\mathbb{N}}$ is uniformly bounded on any compact subsets of $\mathbb{D}$.
By Montel Theorem, there exists subsequence of $\{f_{n}\}_{n\in\mathbb{N}}$ ( without loss of generality , we suppose it to be $\{f_{n}\}_{n\in\mathbb{N}}$ ) and an analytic function $f$, such that $\{f_{n}\}_{n\in\mathbb{N}}$ converges to $f$ uniformly on any compact subsets of $\mathbb{D}$.
$$\underset {|z|\rightarrow1}{\textup{lim}}\underset {f\in K}{\textup{sup}}e^{-2\psi(z)}|f^{'}(z)|=0$$
and $\{f_{n}\}_{n\in\mathbb{N}}\subseteq K$, for any $\varepsilon>0$, there exists an $r\in(0,1)$, such that $e^{-2\psi(z)}|f_{n}^{'}(z)|<\frac{\varepsilon}{2}$ for all $n$, when $r<|z|<1$.
Furthermore, $e^{-2\psi(z)}|f^{'}(z)|<\frac{\varepsilon}{2}$ when $r<|z|<1$, therefore we have
when $r<|z|<1$.
Combines with Cauchy Integral Theorem, $f_{n}^{'}$ converges to $f^{'}$ uniformly on any compact subset of $\mathbb{D}$. Note that $e^{-2\psi(z)}$ is continuous on $r\overline{\mathbb{D}}$, hence for any $z\in r\overline{\mathbb{D}}$, we have $|e^{-2\psi(z)}|\leq C$ for some constant $C$. As $f_{n}^{'}$ converges to $f^{'}$ uniformly on $r\overline{\mathbb{D}}$, there exists an integer $G$, when $n\geq G$, $|f_{n}^{'}(z)-f^{'}(z)|<\frac{\varepsilon}{C}$ holds for any $z\in r\overline{\mathbb{D}}$.
Thus, it is easy to see, when $n\geq G$, that $e^{-2\psi(z)}|f_{n}^{'}(z)-f^{'}(z)|<\varepsilon$ holds for any $z\in \mathbb{D}$. This implies $\|f_{n}-f\|_{B_{\psi}}\rightarrow0~~(n\rightarrow\infty).$ Since $K$ is closed, we have $f\in K$, that means $K$ is compact.
§ MAIN RESULTS
Let $\varphi$ be an analytic self-map of the unit disk $\mathbb{D}$ and $g\in H(\mathbb{D})$, $\psi\in\mathcal{D}$. Then $V_{\varphi}^{g}: AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is bounded if and only if
By Lemma 2.2,
$$e^{\psi(\varphi(z))}\sqrt{\Delta\psi(\varphi(z))}\sim \sqrt{K(\varphi(z),\varphi(z))},$$
therefore, it suffices for us to prove that
$V_{\varphi}^{g}$ is bounded if and only if
$$\underset {z\in\mathbb{D}}{\textup{sup}}\sqrt{K(\varphi(z),\varphi(z))}e^{-2\psi(z)}|g'(z)|<\infty.$$
First, assume that $V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is bounded. For every $w\in\mathbb{D}$, let $f_{w}(z)=k_{w}(z)=\frac{K(z,w)}{\sqrt{K(w,w)}}.$ It is easy to check that $f_{w}\in AL_{\psi}^{2}(\mathbb{D})$ and $\|f_{w}\|_{L_{\psi}^{2}}=1$ for any $w\in\mathbb{D}$.
Hence, for a fixed $w\in \mathbb{D}$,
\begin{eqnarray*}
&=&\|V_{\varphi}^{g}\|< \infty
\end{eqnarray*}
for any $z\in \mathbb{D}.$
Noticing that
$$f_w(\varphi(z))=\frac{K(\varphi(z),w)}{\sqrt{K(w,w)}},~~~~\mbox{for all}~z\in \mathbb{D}, $$
by setting $w=\varphi(z)$, we have
It follows that
$$\underset {z\in\mathbb{D}}{\textup{sup}}\sqrt{K(\varphi(z),\varphi(z))}|g'(z)|e^{-2\psi(z)}<\infty.$$
Conversely, assume that
$$\underset {z\in\mathbb{D}}{\textup{sup}}\sqrt{K(\varphi(z),\varphi(z))}|g'(z)|e^{-2\psi(z)}=M<\infty.$$
For any $f\in AL_{\psi}^{2}(\mathbb{D})$. By Lemma 2.3, we have
\begin{eqnarray*}
\|V_{\varphi}^{g}f\|_{B_{\psi}}&=&|V_{\varphi}^{g}f(0)|+\underset {z\in\mathbb{D}}{\textup{sup}}|(V_{\varphi}^{g}f)'(z)|e^{-2\psi(z)}\\
&=&\underset {z\in\mathbb{D}}{\textup{sup}}|f(\varphi(z))g'(z)|e^{-2\psi(z)}\\
&\leq&\underset {z\in\mathbb{D}}{\textup{sup}}\|f\|_{L_{\psi}^{2}}\sqrt{K(\varphi(z),\varphi(z))}e^{-2\psi(z)}|g'(z)|\\
\end{eqnarray*}
Therefore, $V_{\varphi}^{g}$ is bounded.
Let $\varphi$ be an analytic self-map of the unit disk $\mathbb{D}$ and $g\in H(\mathbb{D})$, $\psi\in\mathcal{D}$. Then $V_{\varphi}^{g}: AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi,0}(\mathbb{D})$ is bounded if and only if $V_{\varphi}^{g}: AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is bounded and
$$\lim_{|z| \rightarrow 1}e^{-2\psi(z)}|g'(z)|=0.$$
Assume that $V_{\varphi}^{g}: AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi,0}(\mathbb{D})$ is bounded. It is clear that
$V_{\varphi}^{g}: AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is bounded. Taking $f(z)=1\in AL_{\psi}^{2}(\mathbb{D})$ and $V_{\varphi}^{g}f\in B_{\psi,0}(\mathbb{D})$, then
\begin{eqnarray*}
0=\underset {|z|\rightarrow1}{\textup{lim}}|(V_{\varphi}^{g}f)^{'}(z)|e^{-2\psi(z)}&=&\underset {|z|\rightarrow1}{\textup{lim}}|f(\varphi(z))||g'(z)|e^{-2\psi(z)}\\
&=&\underset {|z|\rightarrow1}{\textup{lim}}|g'(z)|e^{-2\psi(z)}
\end{eqnarray*}
Conversely, suppose that $V_{\varphi}^{g}: AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is bounded and $\lim_{|z| \rightarrow 1}e^{-2\psi(z)}|g'(z)|=0$. For each polynomial $p(z)$, the following inequality holds
\begin{eqnarray*}
&\leq& M_{p}|g'(z)|e^{-2\psi(z)}.
\end{eqnarray*}
where $M_{p}=\underset {z\in\mathbb{D}}{\textup{sup}}|p(z)|$. Since $M_{p}<\infty$ and $\underset {|z|\rightarrow1}{\textup{lim}}|g'(z)|e^{-2\psi(z)}=0$, then
$$\underset {|z|\rightarrow1}{\textup{lim}}|(V_{\varphi}^{g}p)^{'}(z)|e^{-2\psi(z)}=0.$$
That means for each polynomial $p$, $V_{\varphi}^{g}p(z)\in B_{\psi,0}(\mathbb{D})$. Since the set consisting of polynomials is dense in $AL_{\psi}^{2}(\mathbb{D})$, for every $f\in AL_{\psi}^{2}(\mathbb{D})$, there is a sequence of polynomials $\{p_{k}\}_{k\in\mathbb{N}}$ such that
$$\|f-p_{k}\|_{L_{\psi}^{2}}\rightarrow 0~~~~(k\rightarrow\infty).$$
\rightarrow0(k\rightarrow\infty).$$
Since $V_{\varphi}^{g}p_{k}\in B_{\psi,0}(\mathbb{D})$ and $B_{\psi,0}(\mathbb{D})$ is the the closed subset of $B_{\psi}(\mathbb{D})$,
we have $V_{\varphi}^{g}(AL_{\psi}^{2}(\mathbb{D}))\subset B_{\psi,0}(\mathbb{D})$.
Since $V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is bounded , we see that $V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi,0}(\mathbb{D})$ is bounded.
Let $\varphi$ be an analytic self-map of the unit disk $\mathbb{D}$ and $g\in H(\mathbb{D})$, $\psi\in\mathcal{D}$. Then $V_{\varphi}^{g}: AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is compact if and only if
$$\lim_{|\varphi(z)|\rightarrow 1}e^{\psi(\varphi(z))-2\psi(z)}\sqrt{\Delta\psi(\varphi(z))}|g'(z)|=0.$$
By Lemma 2.2,
$$e^{\psi(\varphi(z))}\sqrt{\Delta\psi(\varphi(z))}\sim \sqrt{K(\varphi(z),\varphi(z))},$$
therefore, we should only show that $V_{\varphi}^{g}$ is compact if and only if
$$\lim_{|\varphi(z)|\rightarrow 1}\sqrt{K(\varphi(z),\varphi(z))}e^{-2\psi(z)}|g'(z)|=0.$$
First, assume that $\lim_{|\varphi(z)|\rightarrow 1}\sqrt{K(\varphi(z),\varphi(z))}e^{-2\psi(z)}|g'(z)|=0$, then for any $\varepsilon>0$, there is a positive real number $r_{0}\in(0,1)$ such that
when $r_{0}<|\varphi(z)|<1$.
Besides, $\sqrt{K(\varphi(z),\varphi(z))}e^{-2\psi(z)}|g'(z)|$ is bounded when $|\varphi(z)|\leq r_{0}$, it is easy to see that
$$\underset {z\in\mathbb{D}}{\textup{sup}}\sqrt{K(\varphi(z),\varphi(z))}e^{-2\psi(z)}|g'(z)|<\infty.$$
By Theorem 3.1, $V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is bounded.
Let $\{f_{k}\}_{k\in\mathbb{N}}$ be a bounded sequence in $AL_{\psi}^{2}(\mathbb{D})$ which uniformly converges to zero on any compact subset of $\mathbb{D}$ as $k\rightarrow\infty$.
Assume that for any $k\in\mathbb{N}$,$\|f_{k}\|\leq C,$ for some positive constant $C$.
Note that
$$\underset {|\varphi(z)|\leq r_{0}}{\textup{sup}}|g'(z)|e^{-2\psi(z)}\leq M$$
for some constant $M$.
It follows that
\begin{eqnarray*}
&\leq&\underset {|\varphi(z)|\leq r_{0}}{\textup{sup}}|f_{k}(\varphi(z))||g'(z)|e^{-2\psi(z)}\\
\underset {|\varphi(z)|> r_{0}}{\textup{sup}}|f_{k}(\varphi(z))||g'(z)|e^{-2\psi(z)}\\
&\leq& M\underset {|\varphi(z)|\leq r_{0}}{\textup{sup}}|f_{k}(\varphi(z))|\\
\|f_{k}\|_{L_{\psi}^{2}}\underset {|\varphi(z)|> r_{0}}{\textup{sup}}\sqrt{K(\varphi(z),\varphi(z))}|g'(z)|e^{-2\psi(z)}\\
&\leq& M\underset {|\varphi(z)|\leq r_{0}}{\textup{sup}}|f_{k}(\varphi(z))|+\varepsilon\|f_{k}\|_{L_{\psi}^{2}}\\
&\leq& M\underset {|\varphi(z)|\leq r_{0}}{\textup{sup}}|f_{k}(\varphi(z))|+\varepsilon C.
\end{eqnarray*}
Since $\{f_{k}\}_{k\in\mathbb{N}}$ uniformly converges to zero on any compact subset of $\mathbb{D}$, there exists a $ K\in\mathbb{Z}^{+}$ such that if $k>K,$ we have $ \underset {|\varphi(z)|\leq r_{0}}{\textup{sup}}|f_{k}(\varphi(z))|<\varepsilon. $
Therefore, $\|V_{\varphi}^{g}f_{k}\|_{B_{\psi}}\rightarrow0$ as $k\rightarrow\infty$. By Lemma 2.4, we see that $V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is compact.
Conversely, suppose that $V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is compact, then it is clear that $V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is bounded. Let $\{z_{k}\}_{k\in\mathbb{N}}$ be sequence in $\mathbb{D}$ such that $\underset {k\rightarrow\infty}{\textup{lim}}|\varphi(z_{k})|=1$. Let
then, $f_{k}\in AL_{\psi}^{2}(\mathbb{D})$ and $\|f_{k}\|_{L_{\psi}^{2}}=1$.
Since the set consisting of polynomials is dense in $ AL_{\psi}^{2}(\mathbb{D})$, for any $\varepsilon>0$ and $f\in AL_{\psi}^{2}(\mathbb{D})$, there exists a polynomial $P_{f,\varepsilon}(z)\in AL_{\psi}^{2}(\mathbb{D})$ such that
\begin{eqnarray*}
|\langle f_{k},f\rangle|&\leq& |\langle f_{k},f-P_{f,\varepsilon}\rangle|+|\langle P_{f,\varepsilon},f_{k}\rangle|\\
&\leq&\|f_{k}\|_{L_{\psi}^{2}}\|f-P_{f,\varepsilon}\|_{L_{\psi}^{2}}+|\langle P_{f,\varepsilon},f_{k}\rangle|
\end{eqnarray*}
by Lemma 2.2 and Definition 1.1, we have
$$\sqrt{K(z,z)}\geq C_{1}\tau(z)^{-1}e^{\psi(z)}\geq\frac{C_{2}e^{\psi(z)}}{1-|z|}$$
for some positive constants $C_{1}$ and $C_{2}$.
Notice that $\lim_{k\rightarrow \infty}|\varphi(z_k)|=0$, we have
$$\langle P_{f,\varepsilon},f_{k}\rangle=\langle P_{f,\varepsilon},\frac{K_{\varphi(z_{k})}}{\|K_{\varphi(z_{k})}\|_{L_{\psi}^{2}}}\rangle
=\frac{1}{\|K_{\varphi(z_{k})}\|_{L_{\psi}^{2}}}P_{f,\varepsilon}(\varphi(z_{k}))\rightarrow 0~~~~
That means that $f_{k}$ weakly converges to zero as $k\rightarrow\infty$.
Because $V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is compact, we see that $\|V_{\varphi}^{g}f_{k}\|_{B_{\psi}}\rightarrow 0 ~~~~(k\rightarrow\infty)$. From the following fact
\begin{eqnarray*}
\|V_{\varphi}^{g}f_{k}\|_{B_{\psi}}&\geq&\underset {z\in\mathbb{D}}{\textup{sup}}|f_{k}(\varphi(z))||g'(z)|e^{-2\psi(z)}\\
\end{eqnarray*}
we immediately obtain that $\underset {|\varphi(z)|\rightarrow1}{\textup{lim}}\sqrt{K(\varphi(z),\varphi(z))}|g'(z)|e^{-2\psi(z)}=0$.
The proof is completed.
Let $\varphi$ be an analytic self-map of the unit disk $\mathbb{D}$ and $g\in H(\mathbb{D})$, $\psi\in\mathcal{D}$. Then the operator $V_{\varphi}^{g}: AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi,0}(\mathbb{D})$ is compact if and only if $V_{\varphi}^{g}: AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi,0}(\mathbb{D})$ is bounded and
$$\lim_{|z|\rightarrow 1}e^{\psi(\varphi(z))-2\psi(z)}\sqrt{\Delta\psi(\varphi(z))}|g'(z)|=0.$$
At first, we note that
$$\lim_{|z|\rightarrow 1}e^{\psi(\varphi(z))-2\psi(z)}\sqrt{\Delta\psi(\varphi(z))}|g'(z)|=0$$
equals to
$$\underset {|z|\rightarrow1}{\textup{lim}}\sqrt{K(\varphi(z),\varphi(z))}e^{-2\psi(z)}|g'(z)|=0.$$
Firstly, we prove the sufficiency . Let $K=\{f:f\in AL_{\psi}^{2}(\mathbb{D}), \|f\|_{L_{\psi}^{2}}\leq1\}$. As $V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi,0}(\mathbb{D})$ is bounded, $\{V_{\varphi}^{g}f:f\in K\}$ is the bounded closed set of $B_{\psi,0}(\mathbb{D})$. It suffices to show that $\{V_{\varphi}^{g}f:f\in K\}$ is compact in $B_{\psi,0}(\mathbb{D})$. By Lemma 2.5, it is only to prove
$$\underset {|z|\rightarrow1}{\textup{lim}}\underset {\|f\|_{L_{\psi}^{2}}\leq1}{\textup{sup}}e^{-2\psi(z)}|(V_{\varphi}^{g}f)^{'}(z)|=0.$$
By lemma 2.3, for any $ f\in K,$ we have
\begin{eqnarray*}
\end{eqnarray*}
Note that the condition $\underset {|z|\rightarrow1}{\textup{lim}}\sqrt{K(\varphi(z),\varphi(z))}e^{-2\psi(z)}|g'(z)|=0$, we have
$$\underset {|z|\rightarrow1}{\textup{lim}}\underset {\|f\|_{L_{\psi}^{2}}\leq1}{\textup{sup}}e^{-2\psi(z)}|(V_{\varphi}^{g}f)^{'}(z)|=0.$$
Therefore, the operator $V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi,0}(\mathbb{D})$ is compact.
Secondly, we will prove the necessity. Suppose $V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi,0}(\mathbb{D})$ is compact, it is obvious that
$V_{\varphi}^{g}:AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi}(\mathbb{D})$ is compact. By Theorem 3.3, we have
$$\underset {|\varphi(z)|\rightarrow1}{\textup{lim}}\sqrt{K(\varphi(z),\varphi(z))}e^{-2\psi(z)}|g'(z)|=0.$$
That is, for any $\varepsilon>0$, there exists an $r\in(0,1)$, such that
when $r<\varphi(z)<1$.
Since $V_{\varphi}^{g}: AL_{\psi}^{2}(\mathbb{D})\rightarrow B_{\psi,0}(\mathbb{D})$ is bounded,
by Theorem 3.2, we have
$$\underset {|z|\rightarrow1}{\textup{lim}}e^{-2\psi(z)}|g'(z)|=0.$$
Let $\varepsilon^{'}=\frac{\varepsilon}{C_{r}}$, there exists a positive real number $\sigma>0$, such that
when $\sigma<|z|<1$, where $C_{r}$ is the upper bound of $\sqrt{K(\varphi(z),\varphi(z))}$ when $|\varphi(z)|\leq r$.
Therefore, we have
when $\sigma<|z|<1$ and $|\varphi(z)|\leq r$.
Combines with $(\ast)$, we see that $\sqrt{K(\varphi(z),\varphi(z))}e^{-2\psi(z)}|g'(z)|<\varepsilon$ when $\sigma<|z|<1$.
$$\underset {|z|\rightarrow1}{\textup{lim}}\sqrt{K(\varphi(z),\varphi(z))}e^{-2\psi(z)}|g'(z)|=0.$$
The proof is completed.
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exponential type weights, Integr. Equ. Oper. Teory, 21(1995), 460-483.
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the weighted Bergman spaces with exponential type weights, Pacific Journal of Mathematics, Vol.173(1996), 127-146.
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type spaces, J. Korean Math. Soc, 45(2008), 229-248.
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|
1511.00401
|
icrar]Richard Dodsoncor1
[icrar]International Centre for Radio Astronomy Research (ICRAR),
The University of Western Australia,
M468, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia
[cor1]Corresponding Author
icrar]Kevin Vinsen
icrar]Chen Wu
caastro]Attila Popping
icrar]Martin Meyer
icrar]Andreas Wicenec
icrar]Peter Quinn
[caastro] Australian Research Council, Centre of Excellence for All-sky Astrophysics (CAASTRO)
[columbia]Department of Astronomy,
Columbia University,
Mail Code 5246, 550 West 120th Street,
New York, New York 10027, USA
columbia]Jacqueline van Gorkom
[nrao]National Radio Astronomy Observatory,
1003 Lopezville Rd., P. O. Box O,
Socorro, NM 87801, USA
nrao]Emmanuel Momjian
We present the results of our investigations into options for the computing platform for the imaging pipeline in the project, an ultra-deep HI pathfinder for the era of the Square Kilometre Array. pushes the current computing infrastructure to its limits and understanding how to deliver the images from this project is clarifying the Science Data Processing requirements for the SKA.
We have tested three platforms: a moderately sized cluster, a massive High Performance Computing (HPC) system, and the Amazon Web Services (AWS) cloud computing platform.
We have used well-established tools for data reduction and performance measurement to investigate the behaviour of these platforms for the complicated access patterns of real-life Radio Astronomy data reduction.
All of these platforms have strengths and weaknesses and the system tools allow us to identify and evaluate them in a quantitative manner.
With the insights from these tests we are able to complete the imaging pipeline processing on both the HPC platform and also on the cloud computing platform, which paves the way for meeting big data challenges in the era of SKA in the field of Radio Astronomy.
We discuss the implications that all similar projects will have to consider, in both performance and costs, to make recommendations for the planning of Radio Astronomy imaging workflows.
methods: data analysis Parallel Architectures: Multicore architectures Distributed architectures: Cloud computing
§ INTRODUCTION: DEVELOPMENTS TOWARDS DATA-DRIVEN RADIO INTERFEROMETRIC PROCESSING
In this paper we present part of our on-going efforts towards prototyping `Data-Driven Processing' for the Square Kilometre Array (SKA) Science Data Processor (SDP).
The SKA will require advances of several orders of magnitude in the processing of data, pushing Radio Astronomy into the forefront of the `Big-Data challenge' <cit.>.
The SKA Phase 1 data-rates out of the correlator will approach a Terabyte/Sec
for SKA-MID and SKA-LOW <cit.> combined.
The final stage, with the complete collecting area, will be an order of magnitude higher.
The calculation of the required data processing rates for the SDP is highly dependent on the science case, but will be several hundred PetaFLOPS in the most extreme cases.
To deliver such performance, highly parallel computer processing solutions are required and in this paper we set out to explore some of the options.
In this investigation we are testing pipeline solutions for the calibrated and flagged datasets from the Karl G. Jansky Very Large Array (VLA) deep HI survey , the COSMOS HI Large Extragalactic Survey <cit.>.
This survey aims to study the neutral atomic hydrogen (HI) content of galaxies over 4 billion years of cosmic time, approximately 1/3 the history of the Universe and twice the lookback time of any previous emission-line survey. HI is a crucial ingredient to study for understanding galaxy evolution as it is the dominant baryonic fuel out of which stars and galaxies are ultimately made, as well as being an important tracer of galaxy kinematics. Such surveys have previously been too expensive to carry out due to limitations in telescope technology and back-end processing resources.
The survey will be one of the prime pathfinders for the data processing on SKA scales.
(Although not alone in this; the LOFAR project <cit.> also has similar challenges.)
The data volumes and processing requirements mean this project will stretch the bounds of current computing capability.
The project therefore is performing a crucial role in our prototyping investigations for the key SDP concepts and approaches.
§.§ The project
is running at the VLA, which is a 27 antenna array.
The new, upgraded front-end (wide-band L-band receivers) and back-end (the WIDAR correlator) <cit.>, provided through the Expanded Very Large Array project <cit.>, can now provide instantaneous coverage for spectral line observing between $\sim$940 and $\sim$1430-MHz on the sky (15 spectral windows of 32MHz, each giving a total of 480MHz in each session).
The observations are dithered in frequency to smooth out the edges of the spectral windows.
The antennas (being 25m in diameter) see about 0.5 degrees ($\sim$2000 arcsecond) across the sky at the pointing centre.
The array configuration is VLA-B, which has 11 km baselines and a typical beam size of $\sim$5x7 arcsec at 1.4 GHz, assuming natural weighting. We are currently oversampling this with 2048 pixels of 1 arcsecond in size during this development phase.
This data is in 15.625kHz channels (representing about 3km/s at the rest frequency of the HI line being observed).
Therefore there are 351 baselines, a little less than 31,000 channels per polarization product to be processed, and a full field of view of 2048x2048 pixels in the image plane.
These datasets are therefore much larger than those normally analysed and therein lies the challenge.
We expect about five epochs of observing (the epochs are defined by when the VLA is in the correct configuration for the science, approximately every 15 months).
The first epoch was of 178 hours in total, broken into 42 days worth of observing, with each day's observation between 1 and 6 hours long.
The mean size of the flagged and calibrated dataset from a days observations in the first epoch are 330GB, with the maximum size being 803GB and the minimum being 45GB.
The second epoch of 213 hours has been observed but not yet calibrated.
One of the main challenges for the project will be to produce the image cubes from the observations, post-flagging and calibrating.
The processing steps required to complete this analysis will be presented in a forthcoming paper; here we limit ourselves to the investigations on the computing resources required to undertake the analysis.
§ ISSUES FOR SKA-SCALE DATASETS
§.§ Compute environments
To investigate the application and total costs of operation for the workflow in a range of indicative environments we have repeated the same data-reduction pipeline on three very different computing infrastructures: a moderate sized cluster (), such as a group like ICRAR could (and does) host and control; a high performance computing cluster () that would be provided by a national facility
such as the Pawsey centre <cit.> and a cloud computing environment, such as provided by the Amazon Web Service ().
This allowed us to explore three very different approaches, all of which would be of the scale accessible to groups such as ours via in-house capital expenditure, via competitive applications for resources on national infrastructure or via cumulative operational expenditure, respectively.
It should be noted that the SKA infrastructure is not necessarily limited to the above three candidate environments, so these may not represent the final choice of SKA architecture.
§.§ Data Transfer
We have found that the copying stage is an important work item, which is not normally considered part of the data reduction pipeline.
Given the size of the input and output data items, we tried to keep the data movement to a minimum.
The data was moved from the CHILES project's data repository at NRAO (Socorro) to Perth for every individual observation, after the flagging and calibration steps had been completed.
Once the data was in Perth it was stored on a large dedicated storage pod and then copied to the target test environments of and .
The storage pod is connected via a 10G ethernet connector.
The copying stage, as it is immediately followed with the splitting, could be combined into a single process or be staged via temporary short term storage.
§.§ Data reduction tools
The data-reduction tools we use are exclusively from <cit.> Version 4.3 and therefore the issues that arose
all revolve around limitations in the CASA performance, as discussed in the following sections. Note that `operations' are printed in small capitals; `tasks' used to perform the operations are printed in small capitals with brackets appended.
The data provided by NRAO has been calibrated and the target field selected, therefore the remaining operations to be performed in the HI-data reduction are: to copy the data to an accessible point, to split the data into manageable sizes whilst correcting for the station doppler shifts of that day and selecting specific frequency ranges and to Fourier invert the observed data (taken in the reciprocal of the image domain) into a 3-D image cube (these dimensions being Right Ascension, Declination and Velocity). Additionally one should deconvolve the image to correct the initial `dirty' image for the spatially extended point spread function (PSF), which is the Fourier Transform of the points in reciprocal space where the antenna pairs measured the correlated signal. After the deconvolution the PSF is replaced with a compact Gaussian representing the maximum resolution to form the `clean' image. See <cit.> for a full discussion of these concepts.
Traditionally one would read all the data files simultaneously and invert to produce an image cube, but this is not possible as the task clean() <cit.> fails due to the extreme size of the input datasets.
This is why we must split the input data into smaller frequency ranges, or sub-bands, and perform on all of the days simultaneously, but with fewer channels.
The copying does not involve data reduction, so is discussed in detail later.
Once the data is accessible we need to pre-select the data from the input (henceforth referred to as the operation ).
For this we have trialed the tasks split(), cvel() and finally mstransform() and have settled on the latter, as that was the fastest.
We note that the doppler shift correction process involves FFTing the entire selected frequency range (or spectral window) before applying the doppler shift, then selecting only those channels which fall in the requested frequency range.
This process has a strong potential for improved efficiency.
The operation takes the frequency split data and combines the many days into images for that frequency range. In these investigations we have not deconvolved the images, except to investigate the residual noise as discussed in Section <ref>.
We found that with the limited frequency ranges we were using, the noise levels per channel were very sensitive to the weighting scheme. For example uniform weighting (or Briggs weight -2) causes large increases of noise at the edges of the sub-cube. This is due to the implementation of the Briggs' scheme in , which depends on the total data selected not just the data channel being imaged. Obviously in this case, where we have a limited input frequency span of data points from which to derive the weights, the edge data displays enhanced noise.
However with natural weighting (equivalent to Briggs +2), or any Briggs weight greater than $\sim$1 this does not occur. Alternative weighting schemes are being discussed, both in the team and within the literature (see <cit.>) so this should not be an issue for the SDP continuum imaging pipeline architectures.
The final operation is to combine the individual small image cubes into the required full sized cube.
The final cube is 2048$^2$ pixels by 30,720 channels, single stokes, resulting in a $\sim$500GB cube when using 4 bytes per voxel.
In these tests we have not explored the range of options for deconvolution and continuum subtraction, as the detailed plan is still being developed.
This latter step could be post-clean (distributed but frequency independent) or post-combination.
In the latter case the parallelisation could come from subtraction along individual pixels of the signal averaged (perhaps with a spectral dependence) along frequency.
§.§ Parallelism
The obvious issue for massive datasets is how to process them in a parallel fashion.
It is an often-stated fact that “Astronomy data is embarrassingly parallel”; image cubes can be formed per frequency channel and calibration can be performed per solution interval. These are to a large extent completely independent of each other.
However when working with the standard data formats, be they FITS or , the combination of the visibility data into a single file or file structure limits the immediate implementation of the natural parallelism.
Therefore we have derived a workflow, see Figure <ref>, which would divide the data between work-units in a usable fashion.
The input data was per day, and this was used as the division in the first stages (copy and ).
After splitting the day's data into multiple frequency sub-bands the images can be formed in parallel by combining all the days across the different frequency sub-bands simultaneously.
§ METHODS
In this section, we discuss the workflow for the analysis pipeline and how we assessed it.
§.§ The proposed data flow
The detailed workflow model we are following is shown in
Figure <ref>, which represents the data distribution breakdown.
For each day (that is N$_{\rm day}$ parallel processes) there is a copy process (or the ingest stage), followed by a split process to separate each frequency.
The ingested could be retained in a temporary archive.
The split includes the data re-ordering, as the outputs are files that are frequency sorted as well as divided by day.
The process (that is N$_{\rm split}$ parallel processes) takes the frequency split data and combines the many days into images for that frequency range.
The final process is the combination of the frequency ranges into a concatenated cube.
A data distribution view of this workflow. This also shows all information on data dimensions and indices that are important for our scheduling data partitioning, distribution and gathering operations. \(M\) represents the number of visibilities collected per time step, ranging from $\sim$4,000 to $\sim$6,000. The number of sub-bands \(S\) is determined by the instantaneous bandwidth (i.e. 480 MHz) and each subband's frequency width \(K\) (e.g. 4MHz).
§.§ The script layout
The work flow is controlled by a set of python and shell scripts.
The shell scripts contain the setup information (and are provided to the queue) and the python scripts extract that setup information to drive the process.
The pipeline scripts for the three environments are extremely similar, but have to be independent because of the different processing environments.
§.§ Results Capture
We use two different methods to collect performance metrics for both compute (e.g. CPU and memory usage) and I/O (e.g. I/O operations, throughput, inter-arrival time, etc.).
In the first method, we periodically measure a list of process-specific kernel counters available in the Linux file system <cit.> while the processing tasks were running.
The sample interval is currently one measurement per second.
While this method provides useful measurements on CPU and memory usage, some detailed I/O metrics cannot be directly derived from the file system. Therefore, we used a second method — the <cit.> linux system tool — to access more advanced I/O performance indicators, such as whether the I/O requests were sequential or random and the size of each I/O requests issued to the underlying file system. is able to capture all system calls and signals. However, since we are only interested in I/O requests made by the application as system calls, we instruct to only measure four types of system calls — file descriptor related, process management-related (in order to track sub-processes), socket-related and those which take a file name as an argument such as open, close, read, write, etc. One issue of is the tracing overhead associated with frequent context switching, which can vary between less than 10% (for hundreds of system calls per second) and over 100% (for tens of thousands of system calls per second). This overhead in turn substantially prolongs the application completion time. However, this is not an issue for profiling the I/O access patterns (random and sequential), which are basically time invariant. The and measurements are both provided in the additional data products available in the online version of this paper and only the results from are plotted in the printed version.
§ TEST ENVIRONMENTS
Three test environments were selected to represent three different approaches to the data reduction.
Our goal is to present options for deciding which model for sourcing computational resources would best match both the specific case we address (i.e. ) and guide the resourcing decisions that will be raised for other computing problems.
§.§ Moderate Size Departmental Cluster Platform
was specifically designed and built to provide a development platform for ICRAR's HPC projects. It is a six-compute node HPC cluster (+1 head node) located at ICRAR.
Each of the compute nodes currently contains a dual Intel Xeon X5650 2.66GHz CPU, 64-192 GB of RAM, one Tesla or two K10 or K20 GPUs, and a Mellanox MT26428 QDR (40Gbps) Infiniband interconnect.
In addition, a dedicated storage node provides persistent data across the QDR Infiniband fabric.
The dataset is provided on a triple RAID-6 striped volume that is 147TB in size.
§.§ High Performance Computing Platform
The HPC cluster is provided by the Pawsey Centre, which plays a key role in the Australian Government’s strategy to provide high level scientific computing resources for the Australian research community.
It is sited in Perth, owned by CSIRO and managed by the Pawsey Supercomputing Centre.
comprises 1536 nodes in 384 blades. Each compute node hosts two 12-core, Intel Xeon E5-2690V3 “Haswell” processors running at 2.6 GHz, for a total of 35,712 cores, delivering in excess of 1 PetaFLOP of computing power. Each node hosts 64GB of RAM.
The nodes communicate amongst themselves over Cray's high-speed, low-latency Aries interconnect.
Global storage (also known as the scratch file system) is provided by a three-cabinet Cray Sonexion 1600 Lustre appliance, with a usable capacity of 3PB and a sustained read/write performance of 70 GB/sec.
In the November 2014 Top500 list, debuted at #41, achieving 1,097 TeraFLOPS (1 PetaFLOP+).
At the time of writing, this makes the most advanced scientific supercomputer in the Southern Hemisphere.
For our investigations we requested 44-nodes of ,
about 3% of the total computing power. This ratio is carried forward for the calculation of the fractional capital expenditure.
§.§ Cloud Computing Platform
The cloud environment allows for considerable flexibility, which is discussed below;
The main constraint is how much one is willing to pay for the performance.
The code to run the Pipeline on is written in Python using the boto package <cit.>.
This allows us to start many servers with different configurations on demand when we need them.
Our python scripts will always look for the cheapest spot price in the regions specified.
§.§.§ Disk Storage
Disk storage is provided by Amazon Elastic Block Store (EBS).
Amazon EBS volumes are network-attached, and persist independently from the life of an Amazon Elastic Compute Cloud (EC2) instance.
Amazon provides three volume types: General Purpose (SSD), Provisioned IOPS[Input/Output Operations Per Second] (SSD), and Magnetic.
General Purpose (SSD) is the SSD-backed general purpose EBS volume type.
IO rates are primarily controlled by the instance types generic network capacity.
Provisioned IOPS (SSD) volumes offer storage with consistent and low-latency performance.
These were used for all EBS instances to improve the IOPS required by .
Initial experiments showed the general purpose SSD gave about 20-30 IOPS with .
Provisioned IOPS improved this to between 90-100 IOPS.
Magnetic storage was not used.
Many Amazon EC2 instance types can also access disk storage located on SSD disks that are physically attached to the host computer and do not persist. This disk storage is referred to as instance store or ephemeral storage.
This was used for scratch storage for some processing tasks and when data was to be written to long term storage in the Amazon Simple Storage Service (S3).
§.§.§ Long Term Storage
Amazon S3 provides access to a reliable data storage infrastructure.
S3 stores data as objects within resources called “buckets”.
One can store as many objects as required within a bucket, and write, read, and delete objects in the bucket.
Objects can be up to 5TB in size.
S3 is designed for 99.999999999% durability and 99.99% availability of objects over a given year.
There is also a low-cost Reduced Redundancy Storage option for less critical data, and Amazon Glacier for long term storage where access time is not important.
All our work used the reduced redundancy S3 storage.
§.§.§ Spot instance vs On-Demand Instance pricing
AWS has two relevant pricing models.
A third option exists call Reserved Instances, but that requires the purchase of 1 or 3 year contracts and was not used for these tests.
On-Demand Instances: These provide the purchase of compute capacity by the hour with no long-term commitments or upfront payments.
One can increase or decrease the compute capacity depending on the demands of the application and only pay the specified hourly rate for the instances used.
Spot Instances: These provide the ability to purchase compute capacity at hourly rates, usually at lower cost than the On-Demand rate.
Spot Instances allow us to specify the maximum hourly price that we are willing to pay to run a particular instance type.
EC2 sets a Spot Price for each instance type in each Availability Zone, which is the price all customers will pay to run a Spot Instance for that given period.
The Spot Price fluctuates based on supply and demand for instances, but customers will never pay more than the maximum price they have specified.
If the Spot Price moves higher than a customer’s maximum price, the customer’s instance will be shut down after a two minute warning.
Table <ref> shows the difference in cost between on demand and spot prices at the AWS Sydney data centre during the test runs. For the final processing only spot instances, being significantly cheaper, were used.
Instance On demand (AUD) Spot Price (AUD)
m3.medium $0.098 $0.01
m3.xlarge $0.392 $0.04
r3.2xlarge $0.840 $0.09
r3.4xlarge $1.680 $0.20
A table showing the typical difference in cost between on demand and spot prices on the AWS cloud. These numbers are for the Sydney data centre on 6 Mar 2015
§ OPERATIONAL FLOW
§.§ Cluster and HPC Environments
Both the cluster and HPC environments were sufficiently similar that they can be described as a common work flow.
The hardware resources required for each step of the work flow are requested via PBS or SLURM (for or respectively).
The work is then divided between the requested cores.
This is one of the major issues for the workflow with a conventional cluster, as there is no flexibility in the differing hardware requirements during the workflow.
The parallel resources can either be set to match the number of days, or the number of frequency sub-bands.
If either is smaller than that required, the work-units will cycle over all those stages until all the tasks are completed.
Whilst waiting for a particularly slow work-unit or for a work-unit which has been allocated more work than the others the requested resources are held without being active.
We can sub-divide the stages for more efficient a-priori allocation of resources, but this still falls foul of work-units which take longer than the average.
§.§.§ Step 1 - Copy
For the cluster/HPC environments we had all the initial data products in place on globally accessible storage.
Given that the first epoch (including the pre-calibrated data) is around 80TB, and future epochs will be larger, this is a significant storage requirement.
The cluster/HPC environments are a fixed hardware configuration and the configuration of the required software (mainly the setup and the cloning of the pipeline software from github) is straight forward.
As described in section <ref>, has a luster file system, however has stored the data on an external raid network attached disk. This limited filesystem was a significant roadblock for the parallel implementation of the data distribution in this environment.
§.§.§ Step 2 - Splitting the files
The list of days to be processed is counted and these are divided between the work-units. Where the number of days is greater than the number of work-units, each work-unit is allocated multiple days to process.
Each work-unit is responsible for splitting one , as by default locks a dataset from further access. The work-unit cycles over the requested frequency range, breaking the data into the smaller more manageable sub-bands. These sub-bands are stored for the next stage.
As does not support concurrent access to large it would be possible to significantly improve the performance of this step, for example with a parallel splitting implementation of mstransform().
§.§.§ Step 3 - Inverting the image
The stage combined all the days for a particular frequency sub-band into a single image and (if requested), deconvolves this image with a conventional clean() <cit.> step.
Until the work flow is finalised we have only performed a limited deconvolution of the datasets; currently we only clean the image in each individual frequency channel (i.e. 15.625 kHz) rather than attempting to subtract continuum sources over the entire frequency range.
§.§.§ Step 4 - Image Concatenation
All the images of the frequency sub-bands are recombined into a larger image cube. We have found that is unable to combine the entire frequency range for the entire Field of View.
We add together only 12 sub-bands as the resulting is 48GB; anything larger becomes too large to open in the standard visualisation tools and to combine all the datasets causes to crash.
§.§ EC2 Configuration
Table <ref> shows the 4 different types of EC2 instance that were used in the workflow.
The main drivers for selecting a particular instance type where the memory required to run and the ephemeral storage required to hold the intermediate data.
For example clean() cannot use 16 cores, but we required the 320GB of fast, direct attached SSD to make the process run quickly.
Model Used on vCPU Memory (GB) Instance Storage (GB)
m3.medium copying 1 3.5 1 x 4
m3.xlarge splitting 4 15 2 x 40
r3.2xlarge concatenation 8 61 1 x 160
r3.4xlarge cleaning 16 122 1 x 320
The instance types used for various stages of the work flow: Copying, Splitting, Cleaning and Concatination. A vCPU is equivalent to a single CPU thread. Instance Storage is the SSD ephemeral disk storage.
§.§.§ Step 1 - Data transfer - Initial Setup
To transfer the 10+TB of calibrated visibility datasets, over WAN to AWS Sydney data centre in an efficient manner, we created a dedicated “copy machine" (using an m3.medium instance).
We then fine tuned the Linux kernel configuration on both ends of the link based on <cit.>.
In particular, we increased the net.core.rmem-max value to 1600 MB in order to accommodate the large bandwidth delay over the WAN link from ICRAR to AWS Sydney.
The network iperf tests (Figure <ref>) show that 2-4 parallel streams will saturate the ICRAR-AWS network bandwidth, which appears to be 1Gb.
We therefore employed the bbcp <cit.> transfer tool to send two parallel data streams for all the subsequent data transfer.
iPerf network throughput tests for 1, 2 and 4 parallel streams. It shows that 2 or more parallel streams saturate the connection to , suggesting the limit of the bandwith is 1Gb. Therefore for the data transfer we limited ourselves to 2 streams.
EBS snapshots of each day's observations were created by copying the from the data storage area in ICRAR onto an EBS volume, whose size had been pre-determined to accommodate the files.
Once the data transfer was complete the EBS volume was detached from the copy machine, the snapshot was created and the EBS volume was deleted.
As bill for the EBS volumes per GB, once a volume was no longer required it was deleted.
We chose to create EBS snapshots rather than placing the data in the S3 archive because creating a new EBS volume from a snapshot takes $\sim$3 seconds, so we were able to create multiple parallel instances and attach different volumes to each, to split the 480 MHz into smaller spanning 4MHz.
We could not mount the on a shared disk and have parallel access to it because does not support concurrent access to large .
§.§.§ Step 2 - Splitting the files
As we are paying for the spot instance by the hour (or part thereof) it is more cost efficient to split the file into 4-5 separate one after the other.
The system creates an EC2 instance in the spot market and then uses cloud-init and bash scripts to attach a new EBS volume created from the appropriate snapshot.
Each instance then splits the days into the smaller frequency sub-bands.
The degree of parallelism is controlled by the user and is driven by how quickly the results are required.
Anything under 3 sub-bands is not particularly economic as 100GB measurements sets can be done in under an hour.
The 700GB could take an hour to process.
The resulting are then copied to S3 for loner term storage, as that provided the best balance of cost and accessibility for our requirements.
§.§.§ Step 3 - Inverting the image
The EC2 operation downloads the 4MHz frequency for each of the daily datasets from the S3 archive and stores it on a large ephemeral disk for local processing.
After all the days have been combined into one image cube spanning the frequency sub-band, it was written back to S3.
Because the requires a big instance it affects the spot market price quite quickly;
for these tests the scheduling of the runs on was done by hand.
The spot price was checked using the dashboard and the parallel tasks launched one by one.
Running the processes strongly affected the spot price in the Sydney region as can be seen by Figure <ref>.
This meant we had to manually throttle the starting of tests to let the price return to an acceptable level.
The variations in Spot Price for March 2015 in the Sydney as the multiple instances of were launched, as provided by the monitoring tools. The baseline price (y-axis) for a r3.4xlarge instance was about $0.16, but when we submitted multiple parallel tasks the price was driven almost double. We therefore were careful to submit the clean jobs in a staggered fashion, manually
§.§.§ Step 4 - Image Concatenation
The cleaned are copied back from S3 and concatenated, in 48MHz sections for the reasons above. Once these are formed we archive the final data products on S3 from where they can be downloaded to the host, or any other collaborating, institution.
§ RESULTS AND DISCUSSION
The pipelines were deployed on all three platforms, both as single broken-out work-units and as parallel work-units. The single work-units functioned as expected on all platforms and provided the following measurements and analysis of the system performance metrics. It is important to stress that the parallel processing did not proceed on , as we immediately hit the network-attached disk access limits. The complete process can be performed on , but only in a serial fashion. Therefore this platform must be considered to be unable to deliver the required workflow. This is of course compatible with the role of this cluster, which is only designed for workflow prototyping before implementation on larger HPC platforms. On the total completion time was $\sim$110 hours of run time, after $\sim$170 hours in the queue. On the total completion time was $\sim$96 hours of run time, manually staggered to avoid driving up the spot market.
Comparison of three facilities on the two main CASA tasks: (i.e. mstranform()) and (i.e. clean())
These tasks were submitted as single work items, so avoided the known I/O bottleneck for accessing multiple datasets on . The completion time for for the operation was the worst for the three platforms and is dominated by the slow network reads for the EBS storage, whereas for the operation the fast random access for the attached SSD disks give the best performance of the three platforms. and have very similar performance for both tasks as they have similar disk I/O performance.
The scatter, as derived from from the multiple passes through the processing, are indicated as ranges around the average values. They arise from the variations of workload and resource provisioning in the shared environments. Note the reduction of scatter on between run time for and due to the highly self-contained nature of the latter instance, and the increase in scatter for and due to the greater random access requirements for and respectively.
We have compared the mean completion time of both Step 2 () and Step 3 () on three computing environments as shown in Figure <ref>.
The comparison of the three operations (on , , and ) take as input the same visibility dataset - a single days observation of 400 GB, and produce as output a single 4MHz sub-band with an arbitrarily selected frequency range between 1020 and 1024 MHz. We repeated the same tests in three different days (over two weeks) in order to accommodate variations of workload and resource provisioning in the shared environments.
For a single instance of the operation, the mean completion time (5346 seconds) is an order of magnitude longer than (437 seconds) and (624 seconds). We hypothesised that this significant difference is attributed to the distinctive underlying I/O sub-systems in the three computing environments. Recall that both and use Infiniband fabric (QDR) to interconnect compute nodes and storage nodes. In the case of , 70 GB/s read/write performance has been previously reported <cit.>.
Since both input and output datasets are placed on these high performance storage nodes, the time spent on I/O is significantly reduced. On the contrary, the input for the operation on is placed on the network-attached, general-purpose EBS disk storage volumes, which have no guaranteed I/O performance. To verify this conjecture, we used the system tool to measure the I/O throughput on the operations.
Analysis of the data
was from time series of I/O throughput (bandwidth) sampled every second for four types of I/O activities — Sequential Write, Random Read, Random Write, and Sequential Read.
Since the operation involves scanning the by , sequential read/writes dominate these I/O activities.
However, ' peak sequential read throughput ($\sim$100 MB/s) for the operation is almost eight times greater than that on AWS EBS volumes ($\sim$12 MB/s). It is this difference of I/O performance that leads to the significant difference in the final completion time. The variance of the SPLIT completion time on is significantly higher than and . This is again caused by the network-attached EBS volumes, whose I/O performance is neither stable nor guaranteed in a multi-tenant network infrastructure. Likewise, the reason that the scatter of INVERT completion time on is far greater than and is attributed to the shared file system (i.e. the scratch space) relying on the resource-limited global storage network, where hundreds of users could concurrently perform I/O intensive workloads, significantly affecting one another.
A plot of the cpu load (user and kernel, green and red solid line respectively) and memory usage (resident and virtual, dashed and dotted line respectively) as a function of time for the computing task. The breakdown into different stages of the task are indicated with the numbers and arrows. For the operation the relatively lower CPU usage (40%) on suggests that the CPU was idling, waiting for I/O requests in the queue to be completed
The relatively higher CPU usage (100%) on for the operation suggests that CPU was busy, and I/O requests are dealt with fast enough to feed data to CPU. The labelling is as in Figure <ref>.
The relatively higher CPU usage (100%) on for the operation suggests that CPU was busy, and I/O requests are are dealt with fast enough to feed data to CPU most of the time. The labelling is as in Figure <ref>.
To further examine the impact of I/O system on the completion time, we plot the CPU and memory usage of the operation on , and in Figure <ref>, <ref>, and <ref> respectively. Each figure shows CPU and memory usage for a particular run rather than the aggregated performance values across multiple runs over the two-week test period.
Compared to and , the CPU usage on is low — less than 40% (vs. 100%), suggesting the CPU spends 60% of its time waiting for I/O requests to be completed.
Since the I/O is not fast enough to feed data to the CPU and memory, the actual (resident) memory usage on is lower (less than 400 MB) than that on both and , which reaches almost 600 MB at its peak.
The underlying I/O system also caused noticeable differences between and .
As shown in Figure <ref> an extremely sharp “trough" of CPU usage (as low as 0%) appears frequently on every 100 seconds or so.
However, CPU usage on (Figure <ref>) does not exhibit such spikes, only fluctuating between 40% to 100% after 90 seconds. This difference suggests that the I/O system may have caused I/O waits that potentially created those sudden plunges on the CPU usage curve. Consequently, a higher usage of CPU on has led to a slightly shorter overall completion time.
The three operations take as input the same eight 4MHz sub-bands from 8 observations with an arbitrarily selected frequency range between 1136 and 1140MHz. (at 2967 seconds) performs better than both (6156 seconds) and (5541 seconds) for two reasons. First, the operation on reads data directly from the “local" SSD-backed ephemeral disks that are physically attached to the r3.4xlarge instance.
Since reading data from directly-attached disks is much faster than from network-attached EBS volumes (as in the operation), the performance has benefited greatly from the underlying I/O system.
For single-node applications like our tests, -based measurements
show that the sequential I/O throughput of ephemeral disks is comparable to storage (the Lustre global file system) and slightly better than storage (NFS V4-mounted, infiniband disk arrays).
The random I/O performance on ephemeral disks is significantly better than and since SSD disks are optimised for random access.
We found that the intensity of random I/O accesses in the operation is much higher than that in the operation.
This can be verified by the Inter-Arrival Time (IAT), which is defined as the time duration between the end of the previous I/O request and the start of the next I/O request. A smaller IAT corresponds to a more intensive series of I/O requests. IAT captures some intrinsic characteristics in an I/O workload regardless of its hosting platforms.
In the operation the majority of random accesses have an IAT greater than 0.1 seconds, which is a thousand times longer than IATs of many sequential reads that reach \(10^{-4}\) seconds. On the contrary, small IATs in the operations are primarily dominated by random reads and writes.
The random accesses are almost three orders of magnitude more intensive in the operation than in the operation.
This explains why SSD-backed ephemeral disks perform better than HDD and network-backed storage on and , leading to a shorter completion time on .
The second reason that performs better on than is due to the CPU usage — four cores are fully utilized on for the operation (see Figure <ref>), and we were only able to use a single core on each node (See Figure <ref>). This is due to the mismatch between the SLURM resource manager and the way program utilises multicores. Therefore it is not surprising to see that on is twice as fast as on . This also explains why , which has a slightly weaker I/O system than , still performs better than as the operation can fully utilise four cores on , as shown in Figure <ref>.
Four cores are utilised for the operation on , the memory consumption has peaked at 50GB. The labelling is as in Figure <ref>.
Only a single core is utilised for the operation on . The labelling is as in Figure <ref>.
on can also easily exploit four cores, showing a 400% CPU usage. The labelling is as in Figure <ref>.
Note that for a “fair" comparison, the completion time on also includes the data transfer time (287 seconds) between S3 and the ephemeral disks. This is because any practical use of ephemeral disks involves copying data from/to some non-volatile storage “remote" to the instances.
Therefore, we need to account for data transfer to and from local storage for any useful computation. This is not the case for and , where both input and output are persistent on global file systems accessible by other applications. Nevertheless the results show still performs better, even after data transfer is accounted for. Furthermore, the S3 storage guarantees an availability of 99.999%. Data on scratch space (without any fault-tolerance mechanisms) and disks (RAID6) are subject to corruption and loss at a much higher rate.
Lastly, we note that performance on is much more stable than on (with a standard deviation of 877 seconds). We believe such stability arises from the directly-attached ephemeral SSD disks used exclusively by the r3.4xlarge instance. This is in contrast to a shared global file system (e.g. that used in ), where performance is susceptible to the constantly changing impact of jobs submitted by other users. In particular, in a workload dominated by random I/O accesses,
a large number of I/O operations have to travel across the storage network and get processed on a single Meta Data Server (i.e. the MDS in the Lustre filesystem), which can quickly become a bottleneck for processing I/O requests from thousands of running jobs[On a normal day, a “squeue" command on shows more than 2000 jobs are either in the queue or running] at any particular point in time. Directly-attached SSD could also explain why performance is more stable than , which uses network-attached EBS volumes subject to network traffic fluctuations within the AWS data centre. Moreover, the m3.xlarge virtual instance, on which runs, is more likely to share the same physical hardware resources with other virtual instances than is the r3.4xlarge instance, on which runs. Therefore the completion time is less likely to be affected by dynamically provisioned virtual machines.
§ VALIDATION
To confirm that the pipeline was functioning correctly we plot the rms of the image residuals in Figure <ref> as a function of the number of visibilities (with 8 second integrations) included in the inversion.
For a Gaussian distribution of noise we expect the rms of the residuals to decrease with the square root of the number of visibilities. In these tests we firstly ran the step with 10 iterations of deconvolution included. We then calculated the rms of the residual image (i.e. the source subtracted, nominally noise-only cube). We performed this measurement on the residual image for one days worth of data and one sub-band and then repeated it, doubling the number of days thereafter.
The slope in this case is -0.38 and is noticeably non-linear, which indicates that the residuals are non-Gaussian. This we hypothesised was from the residual sources uncovered as the cube reaches a higher sensitivity.
Therefore we repeated the analysis with the number of clean iterations set to 100 and then 1000. For these cases the slope was -0.45 and -0.49, respectively. This gives us confidence that firstly the pipeline is operating correctly and also that deconvolution will be required in the final workflow. The noise level achieved is in line with
the expected sensitivity, based on the VLA calculator.
The log of the number of visibilities against the log of the residual RMS per channel (15.625kHz, but Hanning smoothed in the calibration pipeline) after including cleaning, with the number of iterations being 10, 100 and 1000, plotted in red with a cross, blue with a diamond and green with a circle respectively. The first point in each case is for a single days worth of data included in the imaging, the subsequent ones double until all the days are included.
With a limited clean deconvolution of only 10 iterations the noise does not average down at the expected rate (rms $\propto$N$^{-0.5}$), only achieving a factor of -0.38; when more deconvolution is included this is significantly improved, reaching -0.45 and -0.49 for 100 and 1000 iterations, respectively.
Additionally we checked the completion time for the stage against the number of datasets. We find that it scales very close to the square root of the number of visibilities, underlining the advantage of imaging all days together rather than each day individually and summing these for the final data product. On the other hand we find that the completion time as a function of the number of channels imaged is linear, so doubling the number of channels doubles the completion time.
§ SUMMARY OF METRICS AND OPERATIONS
In the following sections we summarise the contributions to the total time taken to complete the different operations on the different environments.
§.§ Inter-arrival times metric
The Inter-Arrival Time (IAT) is the time between I/O requests. The lowest values therefore represent the most intensive I/O workloads. In the majority of cases the I/O request is only issued once the preceding work-task is completed, therefore it also represents the non-I/O limited computational speed. For the sequential reads are the dominant load, arriving on all machines every 0.1 mseconds. has a low CPU intensity, and deals with the data sequentially, so this behaviour is as expected. In comparison the operation places the highest demand on random reads, which is why computing environments with SSD disks (i.e. our selected configuration in the , but not in the , operations) perform so much better than systems with global file-systems.
§.§ I/O Throughput metric
When massive datasets are being accessed, as for this project, the task completion times become extremely sensitive to I/O requirements of the various sub-tasks. We find the throughput has very similar behaviour to that of the IAT; is dominated by sequential reads, and the infiniband backed global filesystems work very well. On and the read maximum throughput ($\sim$100MB/s) approached the theoretical performance. On , however, the maximum throughput was almost an order of magnitude less. This is a consequence of running on network disk instances. To have a large ephemeral disk on is expensive, therefore we chose not to use this option for the stage.
The last stage of the on involves the transfer of the flag tables, which involves a massive read of the whole table for a minuscule write of the relevant section. This could easily be improved by dropping flagged data at ingest.
Generally the writing of output is less important than the reading, as much more data is read in than written out. This is discussed in Section <ref> and could be improved in future operations, which could potentially make the write speeds an issue.
The operation, for which random reads as significant, performed poorly on the Infiniband based disk solutions but extremely well on the SSD-backed processing. In this case we did require a high powered compute platform so it made sense to use one with a large SSD scratch space (i.e. the r3 instance).
§.§ CPU load metric
does not support massively parallel operations, although some tasks support OpenMP for up to four cores. However we failed to get this working on the system because of difficulties parsing environment variables under the SLURM queue manager. Therefore we found that it was impossible to achieve the CPU-loads on for that we could achieve on and . For the operation the computational load is single threaded, so the maximum load would be 100%, but this is potentially throttled by the I/O throughput. For the the limited network bandwidth to the EBS disks prevents us achieving more than 40% CPU load. In comparison the CPU load on and reaches 100% as the sequential reads over the Infiniband are well provisioned.
§.§ mstransform() Task
The mstranform() task has four separate stages, all of which have significantly different access and usage patterns. These stages are:
* Selection: The requested data (one or more spectral windows) is selected.
* Regrid: The requested data is FFT-ed, shifted and inverse FFT-ed.
* Apply: The data channels required are selected from the regridded data.
* Flagging: Any flagging updates required are applied
1) The selection step (typically about 100 seconds) reads the input file information and prepares the new MS for output. Processing is dominated by sequential reads.
2) The regrid and 3) apply processes are CPU bound, unless the sequential reads for new data are provisioned slowly, as is the case on .
4) The final $\sim$100 seconds updates the new MS flagging table from the old one. This is a read only process in our investigation as there were no flags to be transferred.
§.§ clean() Task
The clean() task has upto five separate stages, all of which have significantly different access and usage patterns. These stages are:
* Creation: The requested image is created and prepared.
* Gridding: The channels to be imaged are gridded
* Major Cycle: If there are clean iterations requested there will be a loop where the models are converted to the visibility sampling (degridding), subtracted from the original visibilities, after which the residuals are gridded again for a further cycle of clean.
* Deconvolve: The image plane is iteratively deconvolved with a peak search and subtraction.
* Restore: The image plane is restored by replacing the removed model components with a Gaussian beam convolved with those model components.
* Finalise: The image is written out.
1) The creation step (typically about 70 seconds) prepares the new image file for writing and clears the model fields. Processing is dominated by sequential reads and writes.
After this the memory is flushed.
2) The antenna response is computed and the required gridding array allocated. This involves building the dirty image, whether or not any clean iterations are performed. This stage is normally CPU limited with moderate sequential and low random read requirements. 3) If clean iterations are requested additional cycles of processing occur in this section, as the model components are stored in memory.
4) Next the deconvolution and 5) restore steps are performed, which occurs even if no clean iterations are requested. This step is characterised by both sequential reads and random writes; on the machines with infini-band provisioned disk the writes are several fold slower than the reads, unlike on the SSD-provisioned machine. This accounts for the major performance gain of over the other systems.
6) The final stage is to write out the results, and is characterised by a high sequential write and low CPU demand.
The validation shows that i) the time for completion is linear with the number of channels and follows the square root of the number of visibilities and ii) to achieve the theoretical noise we will require deconvolution in the imaging stage. The former point led us to image all days together but divide the data into manageable sized sub-bands, the latter point will influence the final workflow design.
§ CONSIDERATIONS FOR WORKFLOW DESIGN
We have successfully explored the options for the computing platform for the imaging pipeline, and constructed a workflow solution.
The exact details of this implementation are hard to convert into definitive long-living prescriptions for success for other projects because of: the short term nature of computing infrastructure, the different access patterns for other computing tasks and even the improvements which will be made to our own computing tasks based on the identification of the processing bottle necks in these investigations.
Nevertheless we believe the result and methods presented could be useful for other investigators to configure their in-house computing/storage environment or to formulate cost-effective Cloud strategies suited to workloads similar to the imaging pipeline described in this paper. Therefore, we provide in Table <ref> a summary of Section 6 and Figures 5 — 10.
The considerations will include the resources available to the user, the scale of the compute required and the difficulty of transferring the data to and from the compute environment.
Therefore we have ranked the following considerations and performance measures on a scale from 0 to 5 (unacceptable, poor, passable, acceptable, good, excellent). For costs, high costs are considered poor and lower costs would be scaled higher.
§.§ Costs
The total cost of ownership in our three environments are very different.
For a fully-owned cluster the purchase has to be made ahead of time, and then the system needs to be maintained.
The ICRAR cluster, , was purchased 4 years ago at a cost of $\sim$AUD$50K for the six compute nodes and we have one full time staff who is responsible for the management.
It supports all the users in ICRAR, but not beyond. However, as it could not complete the pipeline in parallel (which therefore means the system could not perform the required data reduction) we cannot measure hours of operation required and therefore cannot estimate its operational cost. If the tasks were perform sequentially we estimate it would take 1,060 hours.
was purchased one year ago at a cost of $\sim$AUD$12M for the 1536 compute nodes and 3PB of disk storage. For total cost of ownership, the Pawsey Supercomputing Centre uses a figure of AUD$0.67 per node hour for compute jobs <cit.>.
It supports many users who apply for time in regular calls for proposals; our tasks would request 44 nodes, $\sim$3% of the total capacity, so we use 3% of the total costs for comparison.
The system, on the other hand has no setup costs for the user, but computational usage is billed monthly.
For the data reduction of the first epoch including debugging, the total AWS bill for computation and storage was $\sim$AUD$2K. This breaks down into $225 for $\sim$3,000 hours of computation, $1.2K for 7TB of on-demand EBS storage and $40 for 65TB of long term S3 storage. This demonstrates the potential for low cost computing provided by cloud facilities, but also the potential for cost blow out if large amounts of data are kept long term in high availability storage.
We have ranked the three systems (, and ) for capital costs as `excellent', `passable' and `good' respectively and for operational costs as `excellent', `excellent' and `unacceptable' respectively.
§.§ Usability
The issue of `Usability' and `Control' in some fraction reflects on our limitations as computer users, rather than the intrinsic capabilities of the machines.
This we break down into two branches: the amount of control we had over the system and the ease of use of the system.
The importance of control is demonstrated by the significant hit in performance we had on the Pawsey machines, because we could not reconfigure nor test the systems. Without root access, which is the case with , we could not fully implement the performance measurements or resolve the environment variable problems, for example. On both our own system and on the cloud we had full root access, which allowed us to maximize the performance.
On the other hand the setup and submission of the jobs on and were much simpler compared to those on , where we have to search for the best moment to launch instances, to configure those instances on the fly and to access data products spread over a complex zoo of support infrastructure. We have ranked the three systems (, and ) for control as `excellent', `acceptable' and `excellent' respectively and for usability as `passable', `good' and `good' respectively.
§.§ Data Transfer
It is a major overhead to transfer the massive datasets from NRAO onto the local clusters and the supercomputing centre in ICRAR, Pawsey and in Sydney, where it can be processed by the compute platform. Both and have physical 10Gb network interfaces to the outside world, which provides reasonably good data transfer infrastructure. has set up a data transfer service on two dedicated data nodes with optimal configuration for both inbound and outbound traffic. The cluster also has a dedicated data transfer node with a 10 Gb NIC, but with only two CPUs and 8 GB of RAM, which means a relatively smaller buffer capacity during data transfer compared to . Nevertheless, we found that data transferring time between ICRAR and Pawsey to be acceptable as long as we managed to saturate the link. However, data throughput recorded on our copy machine has only reached 1Gb for parallel streams during data transfer. Moreover, it is unknown (i) whether this link/NIC is exclusively used by our application and (ii) whether the bandwidth has been deliberately “shaped” by AWS, since the throughput also fluctuates considerably when we increase the number of parallel stream from 1 to 2, then to 4 as shown in Figure <ref>. Compared to , an advantage of and is the root access, which allows us to fine tune the Linux kernel (e.g. increase the default TCP window buffer) for optimal data transfer. Given the above analysis, we have ranked the three systems (, and ) for data transfer as `Acceptable', `Good' and `Good' respectively.
§.§ I/O Performance
For workloads (such as ) dominated by sequential I/O read/writes, the EBS volumes used in performed almost ten times worse than both and in terms of throughput during the processing. The same is true for IOPS achieved by on (EBS-backed) which is again an order of magnitude smaller than those on and . However for random I/O intensive workloads such as , the SSD storage used in instances has definitely shown the best throughput results (up to 500 MB/s). Similarly, the IOPS for has peaked at \(10^4\), five or ten times higher than the other two. Overall, we have ranked the three systems based on peak performance. For the bandwidth we ranked , and as `Good', `Acceptable' and `Acceptable' respectively and for I/O performance as `Excellent', `Good' and `Good' respectively.
§.§ Radar Analysis
Table <ref> summarizes these performance considerations for the platforms and plots them on a radar plot in Figure <ref>.
Moderate Size Departmental Computing , red dashed line with diamonds.
The cluster was unable to perform the parallel pipeline analysis, as highly parallel tasks fail because of the I/O limits. Notwithstanding it was essential for testing the work-units.
High Performance Computing , green line with circles.
Not surprisingly the Cray XC40 was the fastest and the per-node compute costs are very reasonable. Nevertheless it is an inflexible environment, because it is a shared national resource and is a monolithic architecture. We can not adjust the computing to the problem and we have no root access to tune the performance to our needs. These are natural consequences of using such a shared resource.
Cloud Computing , blue line with squares.
The ability to tune the hardware to the particular problem was the strongest advantage of the system.
For example the data reduction bottle neck was clearly the I/O to the single disk when running parallel tasks.
This was initially also true in the implementation, but by upgrading the instance for that work-unit (to provisioned SSD disks with high IOPS) we were able to easily improve the performance.
The range of available hardware options for the implementation of different aspects of the workflow is one standout advantage of cloud computing approaches.
Consideration 2|c|| 2|c|| 2|c
Completion Time 96hr 5 110hr 5 1,060 hr (est.) 0
Capital Costs $0 5 $340,000 2 $50,000 4
Operational Costs $2,000 5 $3,240 5 - 0
Data Transfer 1Gb (high variance) 3 10Gb 4 10Gb 4
Typical Bandwidth $\sim$300MB/s 4 $\sim$100MB/s 3 $\sim$100MB/s 3
Typical IOPS $\sim$1,000 5 $\sim$100 4 $\sim$100 4
Control Root Access 5 Limited Access 3 Root Access 5
Usability Python/Boto 2 Python 4 Python 4
Product ($\Pi/5^8$) 0.15 0.07 0
The performance rankings for the workflow items on the three platforms under test, , and respectively. The metric is given for each aspect, and is ranked, from 5 to 0, as `Excellent', `Good', `Acceptable', `Passable', `Poor' or `Unacceptable'.
Spider plot for the workflow on the three platforms under test, (blue line with squares), (green line with circles) and (red dashed line with diamonds) respectively. The ranking in Table <ref>, from 5 to 0, is used along the corresponding labelled axes.
Operation Platform Peak Memory I/O Throughput CPU Usage I/O Characteristics
(EBS) 420 MB $<$10MB/s 40% Sequential
545 MB 40 $\sim$ 100 MB/s 100% read/write
390 MB 60 $\sim$ 100 MB/s 100% dominated
(SSD) 60 GB 70 $\sim$ 500MB/s 400% Random writes
30 GB 50 $\sim$ 400 MB/s 100% and sequential
35 GB 50 $\sim$ 300 MB/s 400% reads dominated
Performance summary broken down for the and operations across three measured metrics — peak memory usage, I/O throughput, and CPU usage. Inherent I/O characteristics for each operation are also summarised in the last column. The profiling information in this summary constitutes an essential input for optimal resource provisioning and job scheduling.
§.§ Future Developments
We are using these studies to refine and develop our operating infrastructure. We list here the improvements we are making for processing the second epoch of data, as informed by the performance measurements made. These are improvements which are probably of interest to all facility managers.
* SSD for local high speed scratch space. We are installing local SSD disks on all nodes of as that will allow a high-speed random access on locally-hosted data files. With this we maybe able to complete the processing on a moderate sized cluster.
* Improved I/O performance. Conversations with AWS are underway to improve the I/O limitations we have been struggling with.
* Trialling the Next Generation Archive System ngas <cit.> for the transfer of the data from NRAO to the infrastructure. We will attempt to perform the entire data reduction chain, including flagging and calibration, on the cloud computing platform.
* Developing a data-driven workflow for the project, which will be able to prototype many of the SDP concepts and pipeline designs.
* A new task is being developed in , uvgridder() can cumulatively grid all days onto one uv-grid, which may prove to be the best approach <cit.>.
§ ACKNOWLEDGEMENTS
The Karl G. Jansky Very Large Array and the National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement with Associated Universities Inc.. We wish to thank the CHILES team for flagging and calibrating the data used in these tests.
This work was supported by grants from Amazon Web Services, the AstroCompute project and by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia. The supercomputer <cit.> is managed by The Pawsey Supercomputing Centre. The CHILES project is supported by a collaborative research grant from NSF.
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§ INTRODUCTION
Galaxy clusters (GCls) are the largest gravitationally bound structures in the Universe. Their baryonic content, primarily in the form of a hot, dense, gas which emits thermal X-rays, is bound together by a much more massive halo of dark matter (DM).
It is typically found that 80% of the total mass is comprised of DM, which is further supported by gravitational lensing analysis of GCls (see, e.g., <cit.>). To date, DM has only been observed through its gravitational interactions (see e.g., <cit.> for a review). Weakly interacting massive particles (WIMPs) make a good DM candidate largely because their thermal production in the early Universe naturally results in an abundance matching what we observe today (see, e.g. <cit.>). The self-annihilation that guided this process could still carry on in regions of high DM density, producing standard-model particles including (see, e.g. <cit.> for a review). Such can be detected but, as baryonic matter falls into the potential well formed by DM, these dense DM structures provide in addition the host environment for interactions of cosmic rays with ambient gas which in turn can yield a sizable foreground.
One way to disentangle such potential foreground emissions is to focus on a subdominant,
but unambiguous, WIMP process — the direct annihilation into monochromatic photons (see, e.g. <cit.>).
The flux ($\rmn{ph\,cm^{-2}\,s^{-1}}$) at Earth from a direct-to-photon annihilation is described by the formula,
ϕ_s(ΔΩ)=1/4π/m^2_DM_Φ_PP×∫_ΔΩ∫_l.o.s.ρ^2 dl dΩ_J-factor,
where is the thermally-averaged cross section for DM annihilation into two photons, $m_\rmn{DM}$ is the WIMP mass, $\rho$ is the DM density, and the integration is carried out over the solid angle and line of sight. A related process is the DM annihilation into $\gamma Z$ or $\gamma h$. In these cases one would expect to observe two lines, one located at the energy $E$ that corresponds to the mass of the WIMP and one at $E'=E\left(1-\frac{m^2}{4E^{2}}\right)$, where $m$ corresponds to the mass of the Z and Higgs bosons, respectively. [Note that we assume the WIMP to be a Majorana particle, which normally includes a factor $\frac{1}{2}$ in Eq. <ref>, however, since we assume the WIMP to annihilate into two photons, this factor is cancelled out.]
It is convenient to separate the flux calculation of Eq. <ref> into particle physics ($\Phi_\rmn{PP}$) and astrophysical (J-factor) components. Since the branching ratios DM annihilation into two photons are small however ($10^{-4}-10^{-1}$) <cit.>, in order to achieve a flux detectable by the Fermi Large Area Telescope (LAT), the low-probability process must be compensated by a large J-factor. The Galactic center (GC) of the Milky Way is commonly expected to have the largest J-factor (see, e.g. <cit.> for a recent review). Although the Galactic center of the Milky Way has the largest J-factor, GCls, though much more distant than the GC, have two advantages. First, there are many of them, and the J-factor can be increased by analyzing them jointly, using a joint likelihood method <cit.>. The second stems from the fact that cold DM clumps together hierarchically, with large structures being comprised of small ones. These small DM clumps residing inside larger halos, also known as sub-halos, were formed earlier and are thus highly concentrated, which in turn means important contributions to the total DM annihilation flux from them. Indeed, because the annihilation rate depends on $\rho^{2}$, the J-factor is very sensitive to the detailed structure of these DM sub-halos. However, the DM (sub)halo hierarchy is only partially resolved by the cosmological N-body simulations used to infer DM density distributions (e.g. <cit.>), and the relevant quantities must be extrapolated over many orders of magnitude in mass down to the smallest predicted substructures. When enclosing entire GCls in our regions of interest (ROIs), we observe the full effect of this extrapolation and the `boost' factor could potentially increase the expected line feature flux by factors into the thousands (e.g., <cit.>; see however <cit.>).
Recent works have found hints for a line-like feature around 130 in both the GC (see, e.g., <cit.>) and the most promising GCls (<cit.>, hereafter referred to as HRT13). At such energies, the astrophysical foregrounds become negligible, making alternative (to new physics) explanations difficult. Nevertheless, the GC region has undergone particular scrutiny with detailed follow-up analyses suggesting the feature to be less significant than originally claimed <cit.> while the most recent re-analysis finds no supporting evidence for monochromatic lines from the GC <cit.>.
One of the last remaining undisputed supports of the DM interpretation for the $\sim$130 feature is the claim by HRT13 that it is also present in the regions surrounding 18 of the most massive nearby GCls. There, a stacking analysis revealed hints for a double-peaked line at 130 and 110 with a global significance of 3.6. In light of the reduced significance of the GC excess <cit.>, we revisit the case of GCls, making use of additional data, an unbinned fit using both spatial and spectral information, and the technique of joint likelihood <cit.>.
§ METHOD
§.§ Data Selection
The LAT, main instrument aboard the Fermi satellite, is a pair-conversion telescope sensitive to in the energy range from 20 to $>300~\gev$. For a more detailed description, the reader is referred to <cit.>, and for the on-orbit performance to <cit.>. We analyze five years of public Pass 7 reprocessed data taken between 2008-08-04 and 2013-08-04 in the energy range between 10 and 400 . Above 10 the otherwise structured Galactic diffuse emission is less important and the LAT point spread function (PSF) is relatively narrow, with the on-axis 68% containment radius being $<0.2\deg$ <cit.>. Above 400 the number of detected photons is very low so we limit ourselves to lower energies. In order to reduce the residual contamination of misidentified cosmic rays, we selected events passing the CLEAN-class selection cuts and use the associated P7REP_CLEAN_V15 instrument response functions (IRFs). We avoid contamination generated by cosmic rays hitting the atmosphere of the Earth by removing events with a zenith angle $\theta_{z}>100\deg$ and excluding data taken during times when the field of view of the LAT came too close to the Earth limb (specifically we apply a cut on the LAT rocking angle $|\theta_{r}|\leq52\deg$). In addition we excise time periods of bright solar flares or bursts and only select nominal science operations data. For the data preparation and analysis we use the publicly available Fermi Science Tools version v9r34p0.[The data, associated software packages, and templates used to model the and extragalactic emission are publicly distributed through the Fermi Science Support Center (FSSC) available at <http://fermi.gsfc.nasa.gov/ssc/data/>.]
Following earlier works <cit.>, we employ a sliding-window technique, in which we split the analysis into 128 energy windows. Each fit occurs within a sliding energy window of $\pm 6\sigma$, typically corresponding to steps of half the LAT energy resolution $\sigma$ at the center of each window energy. This window size is wide enough to contain an instrument response-convolved line, but narrow enough that the background can be simply approximated using a power law.
§.§ Background Model
Within our energy windows, the diffuse background can be described by a power law with free index $\Gamma$, and normalization, $n_{b}$ <cit.>,
In order to improve convergence, we introduce the reasonable physical constraint $\Gamma \geq 0$. The diffuse background is a combination of emission from Galactic cosmic-ray interactions and unresolved extragalactic sources. It is smoothly varying, and so for the small scales associated with our ROIs, we model it as an isotropic background. The effects of these simplifying assumptions are discussed in §<ref>.
In addition to the diffuse emission, we model sources from the third Fermi source catalog (3FGL, <cit.>) that are contained within our ROIs.[We model point sources up to $0.3\deg$ outside of our ROIs to account for the point spread function at 10 .] We let the normalization $\boldsymbol{n}_{\rm{src}}$ of sources with detection significance greater than $10\sigma$ float, and fix those below.
§.§ Signal Model
Since WIMPs are non-relativistic, direct annihilations into photons produce gamma rays that are monochromatic. From there, we first account for the small cosmological redshifts to the GCls. Second, we incorporate the LAT energy dispersion.
To do this, we simulate the redshifted lines (note that this effect is generally small since the farthest GCl is at $z<0.028$) according to the parametrized IRFs, and use the output as our spectral model. This method takes advantage of the standard tools, in particular the likelihood fitting tool gtlike, while also including the spatial information of an extended source.
Spatial GCl DM signal templates are derived by using the results from cosmological N-body DM simulations and linking those to the total mass determined from X-ray observables or gravitational lensing of each individual GCl (see, e.g. <cit.> for a review). In general, the distribution consists of a gravitationally binding host halo which is itself partly comprised of self-bound subhalos at a variety of mass scales. Given their formation times and accretion history, these subhalos are expected to be highly DM concentrated. As the annihilation rate is proportional to the square of the DM density, estimates of the total GCl flux are extremely sensitive to the level of this halo substructure. Incorporating local over-densities increases the total annihilation rate by a `boost' factor, $b$, compared with the prediction of the spherically smoothed DM distribution of the main halo. The determination of $b$ requires assumptions on both the relative abundance and concentration of all substructure masses (mass and concentration functions, respectively), see e.g. <cit.>.
Although the DM density distribution of each individual sub-halo can be well-approximated by the Navarro-Frenk-White (NFW) <cit.> parametrization,
with $r_s$ being the scale radius and $\rho_{0}$ the central density, there are uncertainties on the internal structure of sub-halos. We commonly introduce the concentration parameter $c_{200} \equiv r_{s}/R_{200}$ to describe the internal structure of DM halos, where $R_{200}$ is defined as the radius of the GCl where the enclosed density equals 200 times the critical density of the Universe. The uncertainty on this concentration parameter stems from the fact that N-body simulations can only resolve structure down to a few hundred times the size of their test particles, currently on the order of $10^{8}\msol$ for GCl-sized simulations <cit.>.[High-resolution simulations can resolve these halo-mass scales at high redshifts <cit.>, but the lack of GCl-size N-body simulation resolving the whole substructure hierarchy and the required extrapolations to present time still imply substantial uncertainties.] Depending on the details of the chosen model, substructures could exist at masses as low as $10^{-12}\msol$, meaning that the mass-concentration relation must be extrapolated over many orders of magnitude. The exact value of this cutoff is set by the kinetic-decoupling and baryonic-acoustic-oscillation damping, which ultimately depend on the particle physics nature of the DM candidate <cit.>. Here we adopt a common cutoff value of $10^{-6}\msol$ which has become a standard value in the field.
To account for the mass concentration uncertainty, we perform our analysis using two boost models — generated by adopting bracketing extrapolations for the mass-concentration function. For the first we employ a power-law relation between DM-halo mass and concentration <cit.>. This results in boost factors for individual GCls ranging from 240 to nearly 2000, and is hereafter referred to as our optimistic configuration.
Alternatively, we follow the fiducial model from <cit.> to obtain boost factors from 24–37 in our conservative configuration.[We note though that power-law concentration models are strongly disfavored by recent developments in both the simulation side <cit.> and in our theoretical understanding of halo concentration at the smallest scales, e.g. <cit.>. Yet, we decided to include the optimistic boost in this work for a direct comparison with the results in HRT13. Note that the conservative model is not particularly sensitive to the precise choice of the substructure mass cutoff-value.] Compared to the pure NFW-case, which corresponds to a centrally peaked flux annihilation profile, the existence of substructure increases the expected intensity towards the outskirts of the GCl. For the projected luminosity profile for annihilating DM, $I_\rmn{sub}$ from substructure as function of subtended angle $\theta \leq \theta_{200}$ from the center of the GCl, we adopt the functional form given by Eq. 2 in <cit.>:
I_sub(θ) = 16b ×L_γ^NFW/πln(17) 1/θ_200^2 + 16θ^2.
In the above equation we have introduced $\theta_{200}$ as the angle substended by the GCl virial radius, i.e. $\theta_{200}=\mathrm{arctan}(R_{200}/D_{a})\times180\deg/\pi$ where $D_{a}$ is the angular distance to the GCl. $L_{\gamma}^{\mathrm{NFW}}$ is the total luminosity of the host halo, assuming an NFW profile (without substructure), in units of $\mathrm{ph\,s^{-1}\,sr^{-1}}$. We note that in this definition the non-boosted (NFW) scenario corresponds to $b=0$. Also, beyond $R_{200}$ we assume the predicted signal to be negligible. We note that the fiducial model in <cit.> does not provide a parametrization of the predicted luminosity profile. Yet, as it was shown in <cit.>, even for moderate values of $b$, such as the ones proposed in <cit.>, the flattening occurs in a similar way. We have compared the predicted profiles with those given by Eq. <ref> and find that they agree well. Hence we assume the same functional form for the different boost factor scenarios with only the value of $b$ varying.
§.§ Target Selection
Starting from the extended HIFLUGCS <cit.> catalog of X-ray flux-limited GCls, we first order potential targets by their J-factor (highest first), taking the first 16 GCls (note the discussion on the sample size in §<ref>). The location in the sky is show in Fig. <ref>.
Hammer-Aitoff projection (in Galactic coordinates) of the Fermi-LAT counts map above $10\,\gev$ for 5 years of LAT exposure. Blue crosses represent the position of the 106 GCls that are contained in the extended HIFLUCGS catalog with the cross size proportional to $\log_{10}{M_{200}/D_{L}^{2}}$ of the GCl (larger crosses being more massive, nearby systems), where $D_{L}$ refers to the luminosity distance in units of Mpc. The colored red triangles indicate the targets used in this study.
Since the predicted number of photons from astrophysical backgrounds is small, and we can localize the signal with its expected spatial information, the primary concern when determining the ROIs surrounding each GCl is to avoid overlap. Performing a joint likelihood with overlapping signal regions can lead to over-estimation of significance <cit.>.
There are infinite combinations of angular radii $R$ which yield a non-overlapping
set of ROIs, so we must impose an additional constraint. Because we only model
the central target in each fit, expected DM emission from nearby GCls is considered
background. We therefore choose to require that our set of ROIs maximizes
the ratio of total target emission to nearby GCl contamination. Switching
between boost scenarios changes the relative GCl DM emission levels and results
in two unique sets of optimal ROI.
The respective values of $R$ for each ROI and each substructure scenario, $R_\rmn{opt}$ (optimistic) and $R_\rmn{cons}$ (conservative), along with other GCl parameters are summarized in Table <ref>.[In regards to the aforementioned free sources, note the following exception to this procedure concerns A2877: for the larger configuration, $R_\rmn{opt}$, we include four sources towards the edge of the ROI which have $TS>100$. Freeing all of these led to covergence issues which we mitigated by fixing the normalization of the weakest of these sources. Generically, the numbers of free sources is typically low: between 1–5 per ROI for both configurations.]
GCl $\alpha_{2000}$ $\delta_{2000}$ $z$ $M_{200}$ $R_{200}$ $\theta_{200}$ $c_{200}$ ${J_{\mathrm{NFW}}}$ $b_{\mathrm{cons}}$ $b_{\mathrm{opt}}$ $R_{\mathrm{cons}}$ $R_{\mathrm{opt}}$
($^{\circ}$) ($^{\circ}$) ($10^{14}\,M_{\odot}$) (Mpc) ($^{\circ}$) ($\mathrm{GeV^{2}\,cm^{-5}}$) ($^{\circ}$) ($^{\circ}$)
3C 129 72.29 45.01 0.021 5.90 1.78 1.12 5.030 16.19 34 922 6.0 5.8
A 1060 159.20 -27.53 0.013 2.72 1.37 1.38 5.180 16.38 32 681 2.6 2.7
A 1367 176.10 19.84 0.022 8.13 1.98 1.19 5.000 16.25 34 1045 3.0 3.0
A 2877 17.45 -45.90 0.025 7.54 1.93 1.02 5.000 16.11 34 1014 9.4 9.9
A 3526 192.20 -41.31 0.011 3.72 1.52 1.80 5.110 16.62 33 770 4.0 2.1
A 3627 243.90 -60.91 0.016 5.38 1.72 1.41 5.050 16.40 34 889 2.5 1.0
AWM7 43.63 41.59 0.017 5.38 1.72 1.33 5.050 16.35 34 889 1.0 1.3
Coma 195.00 27.98 0.023 10.92 2.18 1.25 4.970 16.30 35 1172 6.2 3.3
Fornax 54.63 -35.45 0.005 1.39 1.10 2.84 5.350 17.02 30 525 5.2 3.1
M 49 187.40 8.00 0.003 0.72 0.88 3.79 5.580 17.27 28 405 1.5 1.6
NGC 4636 190.70 2.69 0.003 0.19 0.56 2.43 6.150 16.88 23 240 1.2 1.7
NGC 5813 225.30 1.70 0.007 0.46 0.76 1.40 5.750 16.40 26 340 0.5 0.1
Ophiuchus 258.10 -23.38 0.028 42.44 3.43 1.63 5.020 16.55 36 1990 5.0 3.1
Perseus 49.65 41.52 0.018 6.66 1.85 1.35 5.020 16.37 34 966 1.4 1.1
S 636 157.50 -35.32 0.009 1.69 1.17 1.69 5.300 16.56 30 566 3.5 2.7
Virgo 187.70 12.34 0.004 5.60 1.70 6.28 4.210 17.41 34 1299 1.0 0.5
List of GCl parameters. The columns from left to right are: right ascension and declination in J2000 epoch, redshift, mass contained in virial radius $R_{200}$, angular radius, $\theta_{200}$, NFW halo concentration parameter, integrated $J$-factor for a smooth NFW halo (logarithm), boost factor for conservative and optimistic substructure models and the associated individually optimized radii for each ROI according to the prediction from the substructure model. We derive both $M_{200}$ and $R_{200}$ from the reported values for $M_{500}$ and $R_{500}$ from the HIFLUGCS catalog <cit.> and assuming a value of the concentration parameter $c_{200}$, given by the model of <cit.>. The mass of Virgo is taken from <cit.>.
§.§ Joint Likelihood
Following the precedent set by the Fermi-LAT Collaboration's searches for DM in dwarf galaxies (<cit.>, <cit.>), we maximize our sensitivity by combining the information from multiple targets in a joint likelihood.
Beginning with each individual target, we compute the unbinned Poisson likelihood as
ℒ_t(,_t|_t) = exp(-∫∫λ dE dr) ∏_i λ(r_i,E_i).
In the above equation we introduced $\lambda$, the count-distribution function for all observed photons, as the product of $\sigp$ (our parameters of interest, $\sigp=\{\mdm,\langle \sigma v \rangle_{\gamma\gamma}\}$) and $\nup$ (the list of nuisance parameters $\nup_t=\{n_{b,t},\Gamma_t,\boldsymbol{n}_{\rmn{src},t}\})$. The integrals cover the ROI and energy window, respectively. We label the normalizations of the background sources $\boldsymbol{n}_\rmn{src}$, diffuse background $n_{b}$, as well as the power-law index for the diffuse background Gamma, with an index $t$ to indicate that these quantities vary and are determined for each target individually. We calculate $\lambda$ by summing our model components (i.e. background source normalizations, power-law spectral index and normalization as well as the normalization of the DM target), each convolved with the exposure and instrument response functions.[Done with the tool gtdiffrsp.] We find the maximum likelihood $\mathcal{L}_{t}(\sigp,\hat{\nup}_t|\data_t)$ with respect to the nuisance background parameters and define the joint likelihood as the product over each individual target likelihood, $\mathcal{L}_{t}$:
(|)=∏_t ℒ_t(,_t|_t).
Our measure of significance is the test statistic (TS),
TS = 2 ln((|)/(_0|)),
or twice the difference in log-likelihood between the best-fit and null ($\sigv=0$) hypotheses. According to the asymptotic theorem of Chernoff <cit.>, for fixed WIMP mass, the TS should be $\chi^{2}$-distributed with a single degree of freedom (we scan over a series of fixed values of $m_\rmn{DM}$ with being the only free parameter), and we set our confidence intervals accordingly.
For global significance, the DM mass and the boost setup become free parameters, increasing the degrees of freedom and trials factor. We assess both coverage and global significance through Monte-Carlo (MC) experiments and outline these results in Section <ref>.
could in principle be comprised of every possible GCl with no loss of sensitivity. Targets from which we expect little to no flux are insensitive to changes in and cancel out of the TS. However, to minimize both ROI overlap and computational burden, we form with a truncated set. We then compute TS as a function of the cumulative set for ten MC experiments that include a very strong ($\sigv=1.1\times10^{-22}\mathrm{cm^{3}\,s^{-1}}$) simulated line at 133 .[Using the gtobssim package.] We place a cut at 16 targets (see Fig. <ref>), where more GCls would only contribute insignificantly to the TS.
Evolution of the mean joint TS value derived from the combined analysis of GCls (in order of decreasing J-factor). Based on ten simulations with the conservative boost factor setup and a $\sigv=1.1\times10^{-22}\mathrm{cm^{3}\,s^{-1}}$ for a 133 monochromatic line with a background model comprised of large-scale diffuse emission templates and 3FGL catalog sources, respectively. The dashed line indicates the number of targets used in this analysis. Injecting a signal with lower $\sigv$ causes the TS to plateau earlier.
§ PERFORMANCE AND SYSTEMATIC UNCERTAINTIES
§.§ Monte-Carlo Performance
We set our significance and confidence interval for each value of under the assumption that the TS is distributed according to the asymptotic theorem around the maximum likelihood estimate of . A test of this assumption is warranted, in particular, given our simplified background model and the fact that many of our energy windows contain very few or even zero counts. To do this, we simulate a representative background which includes structured Galactic diffuse, isotropic, and point source emission.[To simulate the structured diffuse background we use the Galactic (gll_iem_v05_rev1) and isotropic diffuse (iso_clean_v05) templates that are produced by the Fermi-LAT collaboration and are distributed through the FSSC <http://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html>] By performing our analysis on a set of these simulations we calibrate our significance threshold. Then, following the spatial templates described in Section <ref>, we add DM lines resulting from a variety of to the simulation and assess our sensitivity and coverage.
To determine the null distribution, and thus calibrate our significance, we perform a set of 500 background-only MC experiments for all 16 of our targets. For each one, we extract the maximum significance ($\smax = \sqrt{\rm TS}$) from all energy windows and both optimistic and conservative boost scenarios. The resulting distribution is depicted in Figure <ref>.
Probability distribution function (PDF) of maximum local significance ($s_\rmn{max}=\sqrt{\rmn{TS}}$) over all masses and boost scenarios for 500 MC experiments. Fit with a trial-corrected $\chi^{2}$ function, Eq. (<ref>).
To convert local to global significances we fit this with a trial-corrected $\chi^{2}$ distribution with both a free number of bounded degrees of freedom $k$, and trials $n_{t}$ <cit.>:
f() ≡d/dxCDF(χ^2_k;x)^n_t
CDF here means the cumulative distribution function. The best-fitting parameters are $k=3.8\pm0.3$ and $n_{t}=14.1\pm1.9$.[Note that based on theoretical grounds, the expected number of degrees of freedom is two <cit.>.]
Using the calibration of global significance, we gauge our sensitivity to a weak DM signal, equivalent to the HRT13 claim. To simulate this, we begin with a signal model which uses the spatial distribution and relative J-factor weighting of our conservative setup. We then tune the value of $\sigv$ such that the total expected counts, including background, matches that reported in HRT13.[A value of $\sigv=1.7\times10^{-24}\,\mathrm{cm^{3}\,s^{-1}}$ yields approximately 15 photons within a 5radius for the stack of 18 HRT13 clusters with $125\,\gev\leq E\leq135\,\gev$. Note that roughly half of these events are signal photons.] We perform 10 MC experiments, keeping the total counts fixed, and find that we detect the feature with a local (global) significance of $4.0\pm0.6\sigma$ ($2.4\pm0.7\sigma$).[For the background component we use the same set of 500 MC simulations that were used to gauge the significance but was adjusted accordingly; however only a small fraction of the original number of MC simulations satisfies the selection criteria we use to map the analysis of HRT13.]
To confirm that our technique yields proper coverage, we further simulate lines for ranging from $3\times10^{-26}$ to $1\times10^{-21}$ cm$^{3}$ s$^{-1}$. To make the calculation feasible, we perform 100 MC experiments for each line using only the four highest J-factor targets, and set joint upper limits.[These targets are Virgo, Ophiuchus, Fornax, and M49. ]
At high , we recover the expected 95% containment. Below a certain threshold in signal strength, however, we expect over-coverage as the sources predict no photons. This occurs at approximately $3\times10^{-25}\mathrm{cm^{3}\,s^{-1}}$ (again for the conservative boost setup), and below that our method rises to 100% coverage. We conclude that our confidence intervals are conservative where they are not accurate.
§.§ Systematics
In this analysis we make use of the approximate representation of the LAT response via the IRFs. In order to assess how our results are affected by the uncertainties in both point spread function and effective area, we repeat our baseline calculation for the conservative setup but use custom IRFs that bracket the associated uncertainties. These IRFs represent the minimal or maximal variations in the computation of the effective area and PSF within the systematic uncertainties of our chosen IRF (P7REP_CLEAN_V16). Specifically, these IRFs are chosen to maximize and minimize effective area and PSF, respectively (c.f. <cit.> for details on the bracketing of PSF and effective area, respectively). Previous searches for lines have shown that the uncertainties associated with the energy dispersion are much smaller than the statistical uncertainties of the analysis we report here <cit.>. Hence we neglect this uncertainty in our bracketing IRF approach. The effect on our upper limits is minor, ranging between 10–20%, with no obvious trend in energy. TS increases when all targets prefer the same value of the joint parameter, . Since the expected flux in Equation <ref> also depends degenerately on J-factor, TS is sensitive to this relative weighting of targets. We explore our sensitivity to this fact by calculating for both the HRT13-like signal simulations, and our LAT data, using identical J-factors for each target. We find our local significance to be affected on average by $0.2\pm0.2\sigma$, with a largest individual change of $-1.2\sigma$ (compared with the conservative J-factor result) at $\mdm=52~\gev$.
A potential additional background source in each of the clusters is its brightest cluster galaxy (BCG). In some clusters, individual cluster member galaxies have been detected in (e.g. NGC 1275 in Perseus <cit.> or M87 in Virgo <cit.>). However, searches for emission from a sample of 114 radioselected BCGs have only yielded null results <cit.>. Even though the sample studied in <cit.> differs from the one presented in this paper, we can use the observed average flux limit from <cit.> to derive a conservative estimate of the BCG flux within our energy (and time) range. We find $\overline{F}_{\gamma,\mathrm{BCG}}(E>10\gev)\lesssim2.0\times10^{-12}\,\mathrm{ph\,cm^{-2}\,s^{-1}}$. This flux upper limit is comparable to the constraints on diffuse emission $\overline{F}_{\gamma,\mathrm{ICM}}(E>10\gev)\lesssim2.7\times10^{-12}\,\mathrm{ph\,cm^{-2}\,s^{-1}}$ <cit.>, as it is expected from cosmic-ray interactions in the intracluster medium (ICM) of the galaxy clusters <cit.>. The sum of these two contributions amounts to less than 0.1% of the isotropic background flux measured by the LAT in each ROI <cit.>. Converting these flux limits into photon counts, we find that the combined contribution corresponds to a total of $\sim20$ photons for the entire sample (with Virgo being the dominant cluster contributing $\sim8$ photons). Note that both scenarios (BCG and cosmic-ray induced emission) assume a power-law like spectrum, in which case the dominant contribution would be expected towards the lowest energies and thus negligible compared to the observed number of photons in all ROIs (c.f. Fig. <ref>).
§ RESULTS
Shown is the stacked spectrum from photons sub-selected within each optimal radius of the 16 clusters analyzed in this work. The inset shows a zoom in the energy window (light-gray shaded area) that coincides with the discussed line signals at 110, 133, and 158 , respectively (dashed-gray lines). Note that displaying the data as stack of binned histograms is for visualization purposes only, while the underlying analysis employs an unbinned joint likelihood method (see text).
Applying our analysis to a five-year dataset, we observe no significant excess over the energy range from 10 to 400 GeV and place 95% confidence upper limits on . The limits are plotted in Fig. <ref> for both boost scenarios, along with 68/95% bands containing background-only MC limits.
Observed 95% confidence upper limits for both conservative (left) and optimistic (right) boost scenarios. Yellow and green bands represent 68% and 95% containment of limits obtained by MC simulations. Note that the upper limits and TS are not always correlated – the upper limit value by itself gives no information about the relative likelihoods of the null and best-fit hypotheses.
The stacked spectrum of all included photons can be seen in Fig. <ref>. The highest local significance occurs for the optimistic setup at 75 with a value of $1.5\sigma$, corresponding globally to $<0.1\sigma$. These results are in agreement with recent updates on the GC <cit.>, which also find no significant line emission.
It should be noted that the most recent of these GC studies <cit.>, utilized a new data set (Pass 8) which benefits from an improved instrument acceptance and PSF. Performing our analysis with 6 years of P8_SOURCE_V6 data set yields results qualitatively compatible to those from Pass 7 reprocessed (this work), with a maximum local significance of $1.5\sigma$ at 181 . Determining the global significance would require repeating the studies of §<ref>, and would yield a lower value. We also neglected to take advantage of a new feature, known as event types, which subdivide the data based on the uncertainty of the energy reconstruction and provide corresponding IRFs. Although this could have improved our sensitivity by approximately 15% <cit.>, the lack of significant features in the data did not warrant its implementation.
§.§ The Feature at $\mathbf{133\,\rmn{\mathbf{GeV}}}$ and Double Lines
Although within the 95% containment band expected from random background fluctuations, we note two upturns (around 133 GeV and 158 GeV) in both our observed limits between the energies of 100 and 200 .
If the 133 GeV feature is indeed of the same origin as previously been seen in the GC <cit.>, where statistical fluctuations in Pass 7 data seem to have played a major role <cit.>, we expect to see a significant decrease in TS since the HRT13 claim.
Repeating our analysis with exposure periods of two, three and four years, respectively, we find the $\sim133\,\gev$ feature to decrease monotonically from $\mathrm{TS}\simeq1.8$ with 2 years of exposure to $\mathrm{TS}\simeq0.3$ with five years.
Inspired by the the HRT13 statements of a potential double peaked feature, from a $\sim$110 GeV and $\sim$130 GeV line, we also investigated if such a setup could increase the TS. With slightly wider energy windows,[We take the full energy range from the joint sliding windows from the two lines.] we repeat our analysis for the conservative boost scenario[Note that the optimistic boost factor scenario should give similar results.] with a second lower energy line together with a 133 GeV line.
We vary the relative intensity and take various energies of a second line around 110 GeV. We test 110 GeV as well as 106 GeV and 120 GeV, where the latter two energies are the expected energies if a 133 GeV line from DM annihilates into $\gamma\gamma$ is adjoined with a second line from $\gamma h$ or $\gamma Z$ final state, respectively. The TS for such double lines stays below $1$ in our joint-cluster analysis, and we conclude that with our analysis setup we see no significant single nor double line associated with a 133 GeV line. Similarly, we tested for a double line at 158 GeV and 133 GeV. These energies coincide with our observed two bumps in cross-section limits and can be theoretically expected if the DM particles annihilate into $\gamma\gamma$ and $\gamma h$. Such a double line can give a TS of 1.7, which is however still not statistically significant (especially if taking into account the “look elsewhere” penalty and that a larger energy window is used).F
§ DISCUSSION AND CONCLUSIONS
A detection of a monochromatic line from the most massive and nearby GCls would be extremely intriguing. The prime candidate for causing such a signal could only be the existence of DM particle annihilations or decays.
After a tentative line-like feature at 133 GeV has been reported in Fermi-LAT data in the GC region <cit.>, evidence was also presented that it was seen in GCls (HRT13).
In some DM particle models such strong line-like feature can indeed be expected in this energy region (see, e.g., <cit.>). Moreover, if a DM annihilation signal is confirmed towards GCls it would not only reveal important properties of the DM particle but also about how DM clusters — a detectable line signal in GCls can namely only be excepted if a large number of highly concentrated DM substructures exist down to small halo masses.
Comparison of results presented in this work with other line searches; we only consider works in which constraints on $\sigv$ were calculated that overlap with the energy range used here. Green-solid and Blue-dashed lines correspond to results obtained analyzing data from the GC assuming an NFW profile using LAT data <cit.> while the squared markers correspond to recently published preliminary observed limits from 2.8 hours of observing the GC with the H.E.S.S. II telescope <cit.>. The gray-dashed vertical line corresponds to the energy at which the initial excess was reported $\sim130\,\gev$.
Subsequent Fermi-LAT data have revealed that the statistical significance has dropped for the $\sim$133 GeV line signal from the GC <cit.>.
In this study we analyzed 5 years of Pass 7 reprocessed data to scrutinize the auxiliary support of the line feature from GCls. We find no remaining indications of a monochromatic line in our stacked cluster analysis. For different J-factor assumptions, we instead derive upper bounds on DM annihilation cross sections for DM masses in the range from 10 to 400 GeV. These limits (see Fig. <ref> for a comparison) are weaker than those derived from the Galactic line search program by the Fermi-LAT collaboration <cit.>, but they nevertheless constitute a very important complementary probe as long as the nature and clustering properties of the DM remain unknown.
§ ACKNOWLEDGMENTS
BA is supported by a VR excellence grant from the Swedish National Space Board (PI: Jan Conrad). MG acknowledges partial support from the European Union FP7 ITN Invisibles (Marie Curie Actions, PITN-GA-2011-289442). MSC is a Wenner-Gren Fellow and acknowledges the support of the Wenner-Gren Foundations to develop his research.
The Fermi LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden.
Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'Études Spatiales in France.
|
1511.00506
|
igc,isu]R. Shendrik
igc]A. S. Myasnikova
igc]T. Yu. Sizova
igc,isu]E. A. Radzhabov
[igc]Vinogradov Institute of geochemistry SB RAS, Favorskogo 1a, Irkutsk,Russia, 664033
[isu]Physics department of Irkutsk state university, Gagarina blvd 20, Irkutsk, Russia, 664003
We report data on the luminescence spectra associated with photochromic centers in X-ray irradiated calcium fluoride crystals doped with Lu ions. Irradiation in low energy photochromic centers absorption band excites emission, which can be identify with transitions into photochromic centers. Ab initio calculation of absorption spectrum of photochromic center agrees rather well with experimental data.
photochromic center rare earth luminescence ab initio F-center TD-DFT
§ INTRODUCTION
Calcium fluoride crystals doped with certain rare earth ions (CaF$_{2}$:$RE$, $RE$=La$^{3+}$, Ce$^{3+}$, Gd$^{3+}$, Tb$^{3+}$, Lu$^{3+}$, Y$^{3+}$) demonstrate the photochromic behavior under x-ray or gamma irradiation and additive coloration. Photochromic centers are responsible for the intense absorption bands in visible wavelength range. On basis of an extensive study of optical and EPR behavior <cit.> of this centers the two models of photochromic centers were proposed.
In the first model photochromic effect occurs under thermally or optically stimulated electron transition from the divalent rare earth ion to the nearest neighboring anion vacancy. Thus, PC center is F-center disturbed by the nearest-neighbor trivalent rare earth ion, but ionized PC center (PC$^{+}$) is charged anion vacancy near the divalent rare earth ion <cit.>.
In the second and the widely accepted model thermally stable photochromic centers in CaF$_{2}$ crystals consist of one PC$^{+}$(RE) or two electrons PC(RE) bound at the anion vacancy adjacent to the trivalent rare earth ion RE <cit.>. Otherwise, PC(RE) center is F$'$-center, F-center having two electrons in the ground state, and PC$^{+}$(RE)-center is F-center disturbed by nearest-neighboring RE$^{3+}$ ion. This model is also confirmed by ENDOR for the Ce$^{3+}$ photochromic center in CaF$_{2}$ <cit.>. Colored crystals exhibit a photochromic effect, i.e. they change color under exposure to light. This process is accompanied by a reversible transformation of PC(RE) center <cit.>.
The data on the PC center luminescence could clarify the model of the photochromic center and explain the mechanism of its formation. However, no luminescence of F-like centers in alkaline earth fluorides has been observed yet. Furthermore, according to Bartram and Stoneham excited F-centers in fluoride crystals decay non-radiatively <cit.>. Nevertheless, <cit.> and <cit.> observed partially polarized broad band luminescence corresponding to PC(Y) and PC(La) centers in additively colored CaF$_{2}$-Y and CaF$_{2}$-La crystals.
Attempts of theoretical calculations of photochromic centers were made to clarify mechanism of its formation and structure. <cit.> showed in semiempirical calculation that bands in optical absorption spectra of PC$^{+}$(Ce) center were associated with transitions in a lowering of symmetry of F-center by neighbor rare earth ion. Overlap of 5d orbitals of rare earth ion with orbitals of F-center formed excited states of photochromic center. Ab initio calculations of PC(Y) and PC$^{+}$(Y) centers were performed by <cit.>. The authors calculated position of absorption bands of PC(Y) and PC$^{+}$(Y) centers. Unfortunately, the agreement of calculated optical absorption bands with experimental was worse than they expected. In the article by <cit.> optical absorption and luminescence spectra of perturbed F-center were calculated using ab initio method. Authors predicted luminescence of this centers in near infrared wavelength range. We employed a similar calculation method of hybrid embedded cluster.
In this article, we observe near infrared broadband luminescence associated with photochromic centers in the irradiated CaF$_{2}$ crystals doped with Lu ions. Our optical spectroscopy and theoretical calculation data give further information about electronic states of PC(Lu) centers.
§ METHODOLOGY
§.§ Experimental technique
Crystals of CaF$_{2}$ were grown from the melt by the Bridgman-Stockbarger method in graphite crucibles in vacuum and were doped with 0.1 mol.% of LuF$_{3}$. In alkaline earth fluoride single crystal growth a small amount of CdF$_{2}$ was generally used as a scavenger in order to remove oxides contained in the raw materials. The crystals were irradiated at 300 K by x-rays from a Pd tube operating at 35 kV and 20 mA for one hour. After irradiation, the crystals acquired a golden-yellow color. Undoped crystals had no color after irradiation.
The optical absorption spectra were obtained on a Perkin-Elmer Lambda 950 UV/VIS/NIR spectrophotometer at 80 and 300 K at the Baikal Analytical Center for Collective Use, Siberian Branch, Russian Academy of Sciences. Photoluminescence (PL) measurements were conducted using a 700 W xenon arc lamp at 80 and 300 K in vacuum cold-finger cryostat. The spectra were detected with a MDR2 grating monochromator, a photomultiplier FEU-83 with Ag-O-Cs photocathode, and a photon-counter unit. The luminescence spectra were corrected for spectral response of detection channel. The photoluminescence excitation (PLE) spectra were measured with a grating monochromator MDR12 and Xe arc lamp. Excitation spectra were corrected for the varying intensity of exciting light due.
§.§ Calculation details
The ab initio calculations were performed using a hybrid embedded cluster method which allowed to combine quantum mechanical (QM) cluster calculation and classically atoms described with shell model. The QM cluster with defect was surrounded by a large number ($\approx$700) of atoms which were described classically with the pair potentials. We used the pair potential parameters of Bukingem form which were the same as by <cit.>. About 50 cations between QM cluster and classical region were replaced with interface atoms with LANL1 ECP pseudopotentials specially optimized to minimize distortion of edge of QM cluster. All atoms of the QM, interface and classical region were allowed to relax during geometry optimization step. The described method allowed to represent the lattice distortion around defect with taking into account deformation and polarization of lattice. About 6000 fixed atoms surrounded classical region for the representation the correct Madelung potential inside classical region. For the density functional theory (DFT) calculations we used the modified B3LYP functional containing 40% of Hartree-Fock and 60% of DFT exchange energies which showed most adequate electron state localization and was successfully employed for DFT calculations of defects in fluoride crystals. Optical energies and dipole matrix elements of transitions were calculated with the time-dependent DFT (TD-DFT) method applied.
We used the GUESS computer code by <cit.> for geometry optimization step and Gaussian 2003 code <cit.> for TD DFT calculations. The applicability of the embedded cluster calculation method for point defects in ionic crystals was described by <cit.> and <cit.> in more details.
The calculations were performed in a cluster Ca$_{16}$F$_{33}$. The central fluorine atom was deleted for modeling vacancy and one nearest cation was replaced by lutetium ion. We used SDD basis setted on Lu$^{3+}$ ion and 6-311$^{+}$G$^{*}$ basis on fluorine and calcium ions. Moreover for correct representation of the F-center density we added the diffuse d-shell to calcium basis.
§ RESULTS AND DISCUSSION
If colorless CaF$_{2}$-Lu crystal is irradiated at room temperature it acquires golden-yellow. Its optical absorption spectrum is given in Fig. <ref>, solid curve. The color of crystals is due to an intense absorption in visible wavelength range. The absorption bands at 2.54; 3.25 and 3.83 eV are resolved. The bands do not resolved well at low temperatures down to 7.5 K. This bands are attributed to transition in photochromic PC(Lu) center. The ones are thermally bleached at about 600 K (see curve 2 of inset to the fig. <ref>).
The crystals irradiated at 80 K demonstrate other bands in optical absorption spectrum. The intense absorption bands at 2.44; 3.8 and 4.5 eV correspond to transition in ionized photochromic PC$^{+}$(Lu) center (Fig. <ref>, dashed curve). The spectrum of the PC(Lu) center at 80 K is identified by then photo-ionizing the aligned PC$^{+}$(Lu) center with visible (VIS) light and vise versa by photo-ionizing with ultraviolet (UV) light. Ionized photochromic centers become unstable at temperatures higher than 250 K (curve 1 of inset to the fig. <ref>). That is why only PC centers are observed in irradiated at room temperature crystals. Photochromic effect is observed only at low temperatures. The similar results were observed by <cit.> in the additively colored crystals.
We should also mention a band with energy 1.7 eV in the optical absorption spectra of PC(Lu) centers (see fig. <ref>, solid curve). In additively colored crystals in this band linear dichroism due to orientation of the PC(Lu) center was not observed by <cit.>. Also, this band is not involved in the reversible transformation of PC(Lu) center (photochromic effect) at low temperature. Therefore, we, as before <cit.>, do not attribute this band for transition into the PC(Lu) center.
Optical absorption spectra of CaF$_{2}$-Lu$^{3+}$ irradiated at room temperature (solid curve) and 80 K (dashed curve). In the inset temperature behavior of ionized photochromic center (curve 1) and photochromic center (curve 2) is shown
We find an intense luminescence in X-ray irradiated at 300 K CaF$_{2}$ crystals doped with Lu$^{3+}$ ions. The luminescence under lamp excitation in green wavelength range peaked at about 1.23 eV with a halfwidth of 0.21 eV (Fig. <ref>, solid curve 2). The excitation spectrum of the luminescence contains two bands peaking at 2.54 and 3.15 eV (Fig. <ref>, dashed curve 2). The low energy excitation band has more intensity than the higher energy one. The measured excitation spectrum also correlates well with the absorption spectrum of the PC(Lu) centers (Fig. <ref>, dashed curve 3). Therefore, we can conclude that the luminescence is due to radiative transitions into PC(Lu) center.
Measured at 80 K excitation (curve 1), emission (curve 2), and optical absorption (curve 3) spectra of PC(Lu) centers in irradiated at 300 K crystals of CaF$_{2}$-Lu$^{3+}$. Vertical solid lines and curve 4 demonstrate calculated optical absorption spectrum of PC(Lu) center
Intensity of the luminescence increases with decrease temperature (Fig. <ref> (a)). The temperature dependence of the luminescence intensity is explained in terms of the probability of nonradiative transitions by Mott's equation <cit.>:
\begin{equation}
\label{Mott_law}
I(T)=\frac{1}{1+w_{0}exp(-\Delta E/k_{B}T)},
\end{equation}
with frequency factor $w_{0}=1.7\cdot10^{8}$. The activation barrier for thermal quenching can be estimated at the value of $\Delta E$=0.47 eV.
The calculated lattice distortion of photochromic center is small. So in the fully relaxed configuration the three nearest cations displace inward about 0.04 Å, and the displacements of six nearest cations do not exceed 0.11 Å. However, lutetium ion is displaced about 0.13 Å to the fluorine vacancy direction from the starting position. The small deformation around F-center is quite typical for this defect as was mentioned by <cit.>. So the calculated structure of photochromic center is correspond to F$'$-center having two electrons in ground state which is slightly disturbed by rare earth ion. One-electron ground state is shown in Fig. <ref> (a). It is clear, that the ground level is a spin-singlet 1A state in which both electrons are in the lowest-energy spatial state (1s) with antiparallel spins.
Excited state with energy 2.54 eV above the ground state is given in Fig. <ref> (b). This state is formed by admixture of low-lying 5d states of rare earth ion and 2s-like states of F$'$-center. Higher energy states corresponding to transition with energy 3.25 eV is shown in Fig. <ref> (c). It is formed by admixture of low-lying 5d states and 2p-like levels of F$'$-center. This result is in agreement with model proposed by <cit.> for PC(Ce) center and results of calculation PC(Y) center reported by <cit.>.
In the calculated optical absorption spectrum three absorption bands are clearly observed (Fig. <ref>, vertical lines)). The smooth solid curve 4 shown in Fig. <ref> is a convolution of Gaussian-type functions, each centered at the excitation energy of the corresponding transition and weighed with the value of the corresponding oscillator strength. The full width at half maximum (FWHM) for all Gaussians is 1 eV. The band position is in rather well agreement with experimental data (within 0.1 eV).
Temperature dependence of luminescence intensity of PC(Lu) centers. Red curve is the Mott's equation fit (a). In subfigure (b) configuration coordinate model of absorption and emission processes associated with PC(Lu) centers is shown, here $E_{exc1}$=3.15 eV; $E_{exc2}$=2.54 eV; $E_{em}$=1.23 eV and $\Delta E$=0.47 eV estimated from eq. (<ref>)
Calculated one-electron states participating in the most intense optical absorption transition in PC(Lu) center
Based on experimental and theoretical results we can construct the simple configurational coordinate diagram of luminescence process in PC(Lu) center (Fig. <ref> (b)). In absorption of Lu-doped CaF$_{2}$, the allowed 1A$\to$1E and 1A$\to$2A transitions occur at 2.54 and 3.15 eV. Emission should then take place at longer wavelengths because of the expected Stokes shift. Thus, the 1.23 eV luminescence band is candidate for this emission. The absence of fine structure in the absorption (excitation) bands and strong temperature dependence of luminescence imply strong electron-phonon coupling for this transitions. Thermal activation energy of non-radiative intracenter process can be estimated about 0.47 eV. The relative weakness of the 3.15 eV band means a preferential population of the 1E level after excitation into optical absorption band of the PC(Lu) center.
<cit.> observed polarized luminescence of PC(Y) center in Y$^{3+}$ doped CaF$_{2}$ and SrF$_{2}$ crystals. <cit.> found weak luminescence in CaF$_{2}$ doped with lanthanum ions related to PC(La) centers. The luminescence was not detected at temperatures higher 40 K for the PC(Y) centers and higher 78 K for the PC(La) centers. That strong temperature dependence can be due to dominating non-radiative mechanism. PC center can be considered as F-like center, therefore <cit.> estimation of the condition for luminescence to be observed is applied:
\begin{equation}
\label{bertr_law}
\Lambda'=(1-E_{emission}/E_{abs})/2\leq0.3.
\end{equation}
Here $E_{emission}$ is energy for center of luminescence band and $E_{abs}$ is mean energy for optical absorption.
For PC(Lu) centers we can calculate the ratio $\Lambda'$ from luminescence spectrum in Fig <ref>. It is approximately equal to 0.25. For PC(Y) and PC(La) centers using data of <cit.> and <cit.> the ratios are 0.4 and 0.28, respectively.
The value of $\Lambda'$ is the lowest for PC(Lu) center and it demonstrates the brightest luminescence among all investigated PC centers due to lower probability of non-radiative transition. For PC(La) centers non-radiative recombination becomes more probable, however the thermal energy barrier $\Delta E$ is still high to observe luminescence at 78 K. In the case of PC(Y) centers the value of $\Lambda'$ is the largest, therefore the weak luminescence can be detected only at low temperature. PC(Ce) center has close to PC(La) energies for optical absorption bands, therefore, we can expect that PC(Ce) centers would demonstrate luminescence properties only at about 78 K. For other "photochromic" impurities the $\Lambda'$ value would be higher and the luminescence observes at lower than 77 K temperatures.
§ CONCLUSION
We studied optical properties of photochromic centers in lutetium doped CaF$_{2}$ crystals. The photoluminescence in near IR wavelength range is attributed to photochromic centers. The theoretical calculation shows that excited states of PC(Lu) center have low-lying d orbitals of rare earth ion which overlap the F$'$-center wave functions. Therefore, the luminescence occurs due to transition from higher energy levels formed by admixture of p-like F$'$-center wavefunctions and low-lying d orbitals of rare earth ion to slightly disturbed 1s level of F$'$-center.
§ ACKNOWLEDGMENTS
The authors gratefully acknowledge A. V. Egranov for the fruitful discussions. The work was partially supported by RFBR grants 15-02-06666a and 15-02-06514a. The authors appreciate the use of Blackford computational cluster located at the Institute of System Dynamics and Control Theory SB RAS.
|
1511.00274
|
Generalisation of the Yang-Mills Theory
George Savvidy
Institute of Nuclear and Particle Physics
${}^+$ Demokritos National Research Center, Ag. Paraskevi, Athens, Greece
We suggest an extension of the gauge principle which includes tensor gauge fields.
In this extension of the Yang-Mills theory the vector gauge boson becomes a
member of a bigger family of gauge bosons of arbitrary large integer spins.
The proposed extension
is essentially based on the extension of the Poincaré
algebra and the existence of an appropriate transversal representations.
The invariant Lagrangian is expressed in terms of new
higher-rank field strength tensors. It does not contain higher derivatives of
tensor gauge fields and all interactions take place through three- and four-particle
exchanges with a dimensionless coupling constant.
We calculated the scattering amplitudes of non-Abelian tensor gauge bosons
at tree level, as well as their one-loop contribution into the Callan-Symanzik beta function.
This contribution is negative and
corresponds to the asymptotically free theory. Considering the contribution
of tensorgluons of all spins into the beta function we found that it is
leading to the theory which is conformally invariant at very high energies.
The proposed extension may
lead to a natural inclusion of the standard theory of fundamental forces
into a larger theory in which vector gauge bosons, leptons and quarks represent
a low-spin subgroup. We consider a possibility
that inside the proton and, more generally, inside hadrons there are
additional partons - tensorgluons, which can carry a part of the proton
momentum. The extension of QCD influences the unification scale at which
the coupling constants of the Standard Model merge, shifting its value to
lower energies.
Talk given at the "Conference on 60 Years of Yang-Mills Gauge Field Theories"
Singapore 2015
§ INTRODUCTION
It is well understood that the concept of local gauge
invariance formulated by Yang and Mills <cit.>
allows to define the non-Abelian gauge fields ,
to derive their dynamical field equations
and to develop a universal point of view on matter interactions
as resulting from the exchange of gauge quanta of different forms.
The fundamental forces - electromagnetic, weak and strong interactions
are successfully described by the non-Abelian Yang-Mills fields.
The vector-like gauge particles - the photon, $W^{\pm},Z$ and
gluons mediate interaction
between smallest constituents of matter - leptons and quarks.
The non-Abelian local gauge invariance, which was formulated
by Yang and Mills <cit.>,
requires that all interactions must be invariant under
independent rotations of internal
charges at all space-time
points. The gauge principle allows very little arbitrariness: the interaction of matter
fields, which carry non-commuting internal charges, and the nonlinear
self-interaction of gauge bosons are essentially fixed by the requirement
of local gauge invariance, very similarly to the self-interaction of
gravitons in general relativity <cit.>.
It is therefore appealing to extend the gauge principle, which was elevated by Yang and
Mills to a powerful constructive principle, so that it will define the
interaction of matter fields which carry
not only non-commutative internal charges, but
also arbitrary large spins[The research in high spin field theories
has long and rich history. One should mention the early works of
Majorana <cit.>, Dirac <cit.>, Fierz <cit.>, Pauli <cit.>, Schwinger <cit.>, Singh and Hagen <cit.>, Fronsdal <cit.>,
Weinberg <cit.>, Minkowski <cit.>, Brink et.al. <cit.>,
Berends, Burgers and Van Dam <cit.>,
Ginsburg and Tamm <cit.>, Nambu <cit.>,
Ramond <cit.>, Brink <cit.>, Fradkin<cit.>,
Vasiliev <cit.>,
Sagnotti, Sezgin and Sundel <cit.>, Metsaev <cit.>,
Gabrielli <cit.>, Castro <cit.>, Manvelyan et.al. <cit.>, and many other works (see also the references in <cit.>). ]. It seems that this will naturally
lead to a theory in which fundamental forces will be mediated by
integer-spin gauge quanta and
that the Yang-Mills vector gauge boson will become a member of a bigger
family of tensor gauge bosons <cit.>
The proposed extension of Yang-Mills theory
is essentially based on the extension of the Poincaré
algebra and the existence of an appropriate transversal representations
of that algebra. The tensor gauge fields take value in extended
Poincaré algebra.
The invariant Lagrangian is expressed in terms of new
higher-rank field strength tensors. The Lagrangian does not contain higher derivatives of
tensor gauge fields and all interactions take place through three- and four-particle
exchanges with a dimensionless coupling constant <cit.>.
It is important to calculate the scattering amplitudes of non-Abelian tensor gauge bosons
at tree level, as well as their one-loop contribution into the Callan-Symanzik beta function.
This contribution is negative and
corresponds to the asymptotically free theory. The proposed extension may
lead to a natural inclusion of the standard theory of fundamental forces
into a larger theory in which vector gauge bosons, leptons and quarks represent
a low-spin subgroup <cit.>.
In the line with the above development we considered a possible extension of QCD.
In so extended QCD the spectrum of the theory contains new bosons, the tensorgluons,
in addition to the quarks and gluons. The tensorgluons have
zero electric charge, like gluons, but have a larger spin.
Radiation of tensorgluons by gluons leads to a
possible existence of tensorgluons inside the proton and,
more generally, inside the hadrons. Due to the emission
of tensorgluons part of the proton momentum which is
carried by the neutral constituents can be shared between
gluons and tensorgluons. The density of neutral partons is
therefore given by the sum: $G(x,t)+T(x,t)$, where $T(x,t)$
is the density of the tensorgluons<cit.>.
To disentangle these contributions and to
decide which piece of the neutral partons is the contribution of gluons $G(x,t)$
and which one is of the tensorgluons one should measure the helicities of the neutral
components, which seems to be a difficult task.
The extension of QCD influences the unification scale at which
the coupling constants of the Standard Model merge.
We observed that the unification scale at which
standard coupling constants are
merging is shifted to lower energies telling us that it may be that a
new physics is round the corner.
Whether all these phenomena are consistent with experiment is an open question.
The paper is organised as follows. In Section 2 we shall define the
composite gauge field ${\cal A}_{\mu}(x,e)$,
which depends on the space-time coordinates $x_{\mu}$ and the new space-like vector variable $e^{\lambda}$. The high-rank tensor gauge fields $A^{a}_{\mu\lambda_1 ... \lambda_{s}}(x)$ appear in the expansion of ${\cal A}_{\mu}(x,e)$ over the vector variable.
We introduce a corresponding
extension of the Poincaré algebra $L_{G}(\CP)$ and consider the high-rank fields as
tensor gauge fields taking value in algebra $L_{G}(\CP)$.
In Section 3 we shall describe the transversal representation
$L^{\bot}$ of the generators of the algebra $L_{G}(\CP)$, their helicity content and their invariant scalar products. The fact that the representation of the generators is transversal plays an important role in the definition of the gauge field $\CA_{\mu}(x,e)$.
In transversal representation the tensor gauge fields are
projecting out into the plane transversal to the momentum and contain only
positive space-like components of a definite helicity.
In Section 4 we shall define the gauge
transformation of the gauge fields, the field
strength tensors and the invariant Lagrangian. The kinetic term describes the propagation
of positive definite helicity states. The helicity spectrum of the propagating modes
is consistent with the helicity spectrum which appears in the projection of the tensor
gauge fields into transversal generators $L^{\bot}$.
The Lagrangian defines not only a free propagation of tensor gauge bosons, but also their
interactions. The interaction diagrams for the lower-rank bosons are
presented on Fig.<ref>-<ref>. The high-rank bosons interact
through the triple and quartic interaction vertices with a dimensionless coupling constant.
In Section 5 we shall calculate and study
the scattering amplitudes of the vector and tensor gauge bosons and their splitting
amplitudes by using spinor representation of the momenta and polarisation tensors.
In Section 6 we shall consider a possibility that inside the proton
and, more generally, inside hadrons there are additional partons - tensorgluons,
which can carry a part of the proton momentum. We generalise the DGLAP equation
which includes the splitting probabilities of the gluons into tensorgluons and calculated
the one-loop Callan-Simanzik beta function. This contribution is negative and
corresponds to the asymptotically free theory. Considering the contribution
of tensorgluons of all spins into the beta function we found that it is
leading to the theory which is conformally invariant at very high energies.
In Section 7 we observed that the
unification scale at which standard coupling constants are
merging is shifted to lower energies.
In conclusion we summarise the results and discuss
the challenges of the experimental verification of the suggested model.
§ TENSOR GAUGE FIELDS AND EXTENDED POINCARÉ ALGEBRA
The gauge fields are defined as rank-$(s+1)$ tensors
A^a_μλ_1 ... λ_s(x), s=0,1,2,...
and are totally symmetric with respect to the
indices $ \lambda_1 ... \lambda_{s} $. A priory the tensor fields
have no symmetries with
respect to the first index $\mu$. The index $a$ numerates the generators $L_a$
of the Lie algebra $L_G$ of a compact Lie group G with totally antisymmetric
structure constants $f_{abc}$.
The tensor fields (<ref>) can be considered as the components of a composite
gauge field ${\cal A}_{\mu}(x,e)$ which depends on additional translationally
invariant space-like unite vector <cit.>:
e_λ e^λ=-1.
A similar vector variable, in addition to the space-time coordinate $x$, was introduced
earlier by Yakawa <cit.>, Fierz <cit.>, Wigner<cit.>, Ginzburg and Tamm <cit.> and
others <cit.>. The variable $e^\lambda$ is also
reminiscent to the Grassmann variable $\theta$ in supersymmetric
theories where the superfield $\Psi(x,\theta)$ depends on two variables $x$ and $\theta$
We shall consider all tensor gauge fields (<ref>) as the components
appearing in the expansion over the above mentioned
vector variable <cit.>:
A_μ(x,e)=∑_s=0^∞ 1s! A^a_μλ_1...
λ_s(x) L_ae^λ_1...e^λ_s.
The gauge field $A^{a}_{\mu\lambda_1 ... \lambda_{s}}$ carries
indices $a,\lambda_1, ..., \lambda_{s}$ which are labelling the
generators $L_{a}^{\lambda_1 ... \lambda_{s}} = L_a e^{\lambda_1}...e^{\lambda_s}$
of extended current
algebra $L_{\CG}$ associated with the Lie algebra $L_G$ <cit.>.
The algebra $L_{\CG}$
has infinitely many generators
$L_{a}^{\lambda_1 ... \lambda_{s}} $ and
is given by the commutator <cit.>
[L_a^λ_1 ... λ_k, L_b^λ_k+1 ... λ_s]=if_abc
L_c^λ_1 ... λ_s , s=0,1,2....
The generators $L_{a}^{\lambda_1 ... \lambda_{s}}$ commute to themselves forming an infinite series of commutators of current algebra $L_{\CG}$ which
cannot be truncated, so that the index s runs from zero to infinity.
Because the generators $L_{a}^{\lambda_1 ... \lambda_{s}}$ are
space-time tensors, the full algebra should include the Poincaré generators $P^{\mu},~M^{\mu\nu}$ as well. This naturally leads to the
extension $L_G (\CP )$ of the Poincaré algebra $L_{\CP}$
[P^μ, P^ν]=0,
[M^μν, P^λ] = η^νλ P^μ
- η^μλ P^ν ,
[M^μν, M^λρ] = η^μρ M^νλ
-η^μλ M^νρ +
η^νλ M^μρ -
η^νρ M^μλ ,
[P^μ, L_a^λ_1 ... λ_s]=0,
[M^μν, L_a^λ_1 ... λ_s] =
η^νλ_1 L_a^μλ_2... λ_s
-η^μλ_1 L_a^νλ_2... λ_s
η^νλ_s L_a^μλ_1... λ_s-1 -
η^μλ_s L_a^νλ_1... λ_s-1 ,
[L_a^λ_1 ... λ_k, L_b^λ_k+1 ... λ_s]=if_abc
L_c^λ_1 ... λ_s.
We have here an extension of the Poincaré algebra by generators
$L_{a}^{\lambda_1 ... \lambda_{s}}$ which carry the
internal charges and spins.
The algebra $L_G (\CP )$ incorporates the Poincaré algebra $L_{\CP}$ and an
internal algebra $L_G$ in a nontrivial way, which is different from the direct
There is no conflict with the Coleman-Mandula theorem
<cit.> because
the theorem applies to the symmetries that act on S-matrix elements and not on all
the other symmetries that occur in quantum field theory.
The above symmetry group (<ref>) is the
symmetry which acts on the gauge field
$\CA_{\mu}(x,e)$ and is not the symmetry of the S-matrix.
The theorem
assumes among other things that the vacuum is nondegenerate and
that there are no massless particles in the spectrum. As we shall see, the
spectrum of the extended Yang-Mills theory is massless.
In order to define the gauge field $\CA_{\mu}(x,e)$ in (<ref>) and find out
its helicity content one should specify the representation of the generators
$L_{a}^{\lambda_1 ... \lambda_{s}}$ in algebra (<ref>).
In the next section we shall describe the
so called transversal representation, which is used to define the tensor gauge fields in the
decomposition (<ref>).
§ TRANSVERSAL REPRESENTATION OF ALGEBRA $L_G (\CP )$
The important property of the algebra (<ref>) is its
invariance with respect to the following "gauge"
transformations <cit.>:
L_a^λ_1 ... λ_s →L_a^λ_1 ... λ_s
+ ∑_1 P^λ_1L_a^λ_2 ... λ_s+
∑_2 P^λ_1 P^λ_2 L_a^λ_3 ... λ_s +...+
P^λ_1... P^λ_s L_a
M^μν →M^μν,
P^λ →P^λ,
where the sums $\sum_{1},\sum_{2},... $ are over all inequivalent index permutations.
The above transformations contain polynomials of the momentum operator $P^{\lambda}$
and are reminiscent of the gauge field transformations.
This is “off-shell" symmetry
because the invariant operator $P^2$ can have any value.
As a result, to any given representation of
$L_{a}^{ \lambda_1 ... \lambda_{s}},~s=1,2,...$
one can add the longitudinal terms, as it follows from the transformation (<ref>).
All representations are therefore defined modulo longitudinal terms,
and we can identify these generators as "gauge generators".
The second general property of the extended algebra is that each gauge generator
$ L_{a}^{ \lambda_1 ... \lambda_{s}}$ cannot be
realised as an irreducible representation of the Poincaré algebra of a definite helicity,
i.e. to be a symmetric and traceless tensor. The reason is that the commutator of
two symmetric traceless generators in (<ref>) is not any more
a traceless tensor. Therefore the generators $L_{a}^{ \lambda_1 ... \lambda_{s}}$
realise a reducible representation of the Poincaré algebra and each of
them carries a spectrum of helicities, which we shall describe below.
The algebra $L_G(\CP)$ has representation in terms of differential operators of the
following general form:
P^μ = k^μ ,
M^μν = i(k^μ ∂∂k_ν
- k^ν ∂∂k_μ) + i(e^μ ∂∂e_ν
- e^ν ∂∂e_μ),
L_a^λ_1 ... λ_s =e^λ_1...e^λ_s ⊗L_a,
where $e^{\lambda} $ is a translationally invariant space-like unite vector (<ref>).
The vector space of a representation is parameterised
by the momentum $k^{\mu}$ and translationally invariant
vector variables $e^{\lambda}$:
Ψ(k^μ, e^λ ) .
The irreducible representations can be obtained from (<ref>)
by imposing invariant constraints on the
vector space of functions (<ref>)
of the following form <cit.>:
k^2=0, k^μ e_μ=0, e^2=-1 .
These equations have a unique solution <cit.>
e^μ= χk^μ + e^μ_1cosφ+e^μ_2sinφ,
where $e^{\mu}_{1}=(0,1,0,0),~ e^{\mu}_{2}=(0,0,1,0)$ when $k^{\mu}=\omega(1,0,0,1)$.
The $\chi$ and $\varphi$ remain as independent variables
on the cylinder $ \varphi \in S^1, \chi \in R^1 $. The invariant subspace of functions (<ref>) now reduces to
the following form:
Ψ(k^μ, e^ν ) δ(k^2) δ(k·e) δ(e^2 +1)
= Φ(k^μ, φ, χ) .
If we take into account (<ref>) the generators
$L_{a}^{ \lambda_1 ... \lambda_{s}}= e^{\lambda_1}...e^{\lambda_s} \otimes L^a $,
it takes the following form:
L_a^ λ_1 ... λ_s= ∏^s_n=1 ( χk^λ_n + e^λ_n_1cosφ+e^λ_n_2sinφ) ⊗L_a.
This is a purely transversal representation because of (<ref>):
k_λ_1L_a^λ_1 ... λ_s=0, s=1,2,...
The generators
$L_{a}^{\bot~ \lambda_1 ... \lambda_{s}}$ carry helicities in the following range:
h=(s,s-2,......, -s+2, -s),
in total $s+1$ states. Indeed,
this can be deduced from the explicit representation (<ref>) by using
helicity polarisation vectors $e^{\lambda}_{\pm}= (e^{\lambda}_1 \mp i e^{\lambda}_2)/2$:
L_a^ λ_1 ... λ_s= ∏^s_n=1 ( χk^λ_n + e^i φ e^λ_n_+
+e^-i φ e^λ_n_-)⊕L_a.
Performing the multiplication
in (<ref>) and collecting the terms of a given power
of momentum we shall get the following expression:
L_a^ μ_1 ... μ_s= ∏^s_n=1 (e^i φ e^μ_n_+
+e^-i φ e^μ_n_-)⊕L_a +
+∑_1 χk^λ_1 ∏^s-1_n=1 (e^i φ e^μ_n_+
+e^-i φ e^μ_n_-)⊕L_a +...+χk^λ_1... χk^λ_s⊕L_a ,where the first term
$\prod^{s}_{n=1} (e^{i \varphi} e^{\mu_n}_{+}
+e^{-i \varphi} e^{\mu_n}_{-})$ represents the helicity generators
$(L^{+\cdot\cdot\cdot+}_{a},...,L^{-\cdot\cdot\cdot-}_{a})$, while their
helicity spectrum is described by the formula (<ref>).
The rest of the terms are purely
longitudinal and proportional to the increasing powers of momentum $k$.
The last formula also illustrates the
realisation of the transformation (<ref>), that is, the
helicity generators $(L^{+\cdot\cdot\cdot+}_{a},...,L^{-\cdot\cdot\cdot-}_{a})$
are defined modulo longitudinal terms proportional to
$k^{\lambda_1}...k^{\lambda_n}, n=1,...,s$.
The very fact that the representation of the generators $L_{a}^{\bot \lambda_1 ... \lambda_{s}}$ is transversal plays an important
role in the definition of the gauge field $\CA_{\mu}(x,e)$ in (<ref>).
Indeed, substituting the transversal representation (<ref>) of the
generators $L_{a}^{\bot \lambda_1 ... \lambda_{s}}$ into the
expansion (<ref>) and collecting the terms in front of the helicity generators
$(L^{+\cdot\cdot\cdot+}_{a},...,L^{-\cdot\cdot\cdot-}_{a})$ we shall get
A_μ(x,e) = ∑_s=0^∞ 1s!
(Ã^a_μλ_1 ... λ_s e^λ_1_+...e^λ_s_+
⊕L_a +...+
Ã^a_μλ_1 ... λ_s e^λ_1_-...e^λ_s_-⊕L_a )
= ∑_s=0^∞ 1s!
(Ã^a_μ+···+ L^+···+_a +...+
Ã^a_μ-···- L^-···-_a ),
where s is the number of negative indices.
This formula represents the projection $\tilde{A}^{a}_{\mu \lambda_1 ... \lambda_{s}}$
of the components of
the non-Abelian tensor gauge field $A^{a}_{\mu\lambda_1 ... \lambda_{s}} $ into the plane
transversal to the momentum. The projection contains only positive definite space-like
components of the helicities <cit.>:
h = ±(s+1), ±(s-1)
±(s-1) ,
±(s-3) , ....,
where the lower helicity states have double degeneracy. The analysis of the
kinetic terms of the Lagrangian and of the
corresponding equation of
motions, which will be considered in the next section, confirms that indeed
the propagating degrees of freedom are described by helicities (<ref>).
In order to define the gauge invariant Lagrangian one should know the Killing
metric of the algebra $L_G(\CP)$.
The explicit transversal representation of the $L_G(\CP)$ generators given above
(<ref>), (<ref>) and (<ref>)
allows to calculate the corresponding Killing metric <cit.>:
L_G: L_a; L_b ⟩=δ_ab,
L_: P^μ ; P^ν ⟩ =0
M_μν ; P_λ ⟩ =0
M^μν ; M^λρ ⟩=η^μλ η^νρ
-η^μρ η^νλL_G(): P^μ;L_a^ λ_1 ... λ_s⟩=0,
M^μν;L_a^ λ_1 ... λ_s⟩=0,
L_a; L^ λ_1_b ⟩=0,
L^ λ_1_a; L^ λ_2_b ⟩= δ_ab η̅^λ_1 λ_2 ,
L_a; L^ λ_1λ_2_b ⟩= δ_ab η̅^λ_1 λ_2 ,
L^ λ_1_a; L^ λ_2 λ_3_b ⟩=0,
L^ λ_1...λ_n_a; L^ λ_n+1....λ_2s+1_b ⟩= 0, s=0,1,2,3,...
L^ λ_1...λ_n_a; L^ λ_n+1....λ_2s_b ⟩=
δ_ab s! (η̅^λ_1 λ_2 η̅^λ_3 λ_4...
η̅^λ_2s-1 λ_2s +perm),where $\bar{\eta}^{\lambda_1\lambda_2}$ is the projector into
the two-dimensional plane transversal to the momentum $k^\mu$ <cit.>:
η̅^λ_1λ_2 = k^λ_1k̅^λ_2
+k̅^λ_1k^λ_2 k k̅- η^λ_1 λ_2 ,
k_λ_1 η̅^λ_1λ_2=
k_λ_2 η̅^λ_1λ_2=0,
and $\bar{k}^{\mu}=\omega(1,0,0,-1)$.
It follows then that the transversality conditions (<ref>) are fulfilled:
k_λ_i L^ λ_1...λ_n_a; L^ λ_n+1....
λ_2s_b ⟩=0, i=1,2,...2s.
The Killing metric on the internal $L_G$ and on the Poincaré $L_{\CP}$
subalgebras (<ref>), (<ref>) are well known.
The important conclusion which follows from the above result is that the
Poincaré generators $P^{\mu}, M^{\mu\nu}$ are orthogonal to the gauge generators
$L_{a}^{ \lambda_1 ... \lambda_{s}}$ (<ref>). The last formulas
(<ref>) represent the Killing metric on the $L_{\CG}$ current
algebra (<ref>),(<ref>) and will be used
in the definition of the Lagrangian in the next section. It should be stressed that
the metric (<ref>) is defined modulo longitudinal terms. This is
because under the "gauge" transformation of the generators (<ref>)
the metric will receive terms which are polynomial in momentum.
The provided metric (<ref>) is written in a particular gauge.
This peculiar property of the metric is mirrored in the definition of the Lagrangian
which can be written in different gauges. The spectrum of the propagating
modes does not depend on the gauges chosen, as one can get convinced
by inspecting the expression (<ref>).
Notice that
the reducible representation (<ref>), without
any of the constraints (<ref>), should also be considered,
as well as the representation in which only
the last constrain in (<ref>) is imposed. In that cases the transversality of the
representation (<ref>) will be lost, but instead one arrives to the homogeneous
Killing metric in (<ref>) $\bar{\eta}^{\lambda_1\lambda_2}\rightarrow \eta^{\lambda_1\lambda_2}$ and the longitudinal terms which can be gauged away.
With this Killing metric in hands one
can define the Lagrangian of the theory.
§ THE LAGRANGIAN
The gauge transformation of the field $\CA_{\mu}(x,e)$ is defined as <cit.>
^'_μ(x,e) = U(ξ) _μ(x,e) U^-1(ξ) -ig
∂_μU(ξ) U^-1(ξ),
where the group parameter $\xi(x,e)$
U(\xi)=e^{i \xi(x,e)}
has the decomposition <cit.>
\xi(x,e)= \sum_s {1\over s!}~\xi^{a}_{\lambda_1 ... \lambda_{s}}(x) ~~L_{a}e^{\lambda_{1}}...e^{\lambda_{s}}
and $\xi^{a}_{\lambda_1 ... \lambda_{s}}(x)$ are totally symmetric gauge parameters.
Using the commutator of the covariant derivatives
\nabla^{ab}_{\mu} = (\partial_{\mu}-ig \CA_{\mu}(x,e))^{ab}
[∇_μ, ∇_ν]^ab = g f^acb ^c_μν ,
we can define the extended field strength tensor
_μν(x,e) = ∂_μ _ν(x,e) - ∂_ν _μ(x,e) -
i g [ _μ(x,e) _ν(x,e)],
which transforms homogeneously:
^'_μν(x,e)) = U(ξ) _μν(x,e) U^-1(ξ).
It is useful to have an explicit expression for the transformation law of the field components
δA^a_μ = ( δ^ab∂_μ
+g f^acbA^c_μ)ξ^b ,
δA^a_μν = ( δ^ab∂_μ
+ g f^acbA^c_μ)ξ^b_ν + g f^acbA^c_μνξ^b,
δA^a_μνλ = ( δ^ab∂_μ
+g f^acb A^c_μ)ξ^b_νλ +
g f^acb( A^c_μνξ^b_λ +
A^c_μλξ^b_ ν+
......... . ............................ These extended gauge transformations generate a closed algebraic structure.
The component field strengths tensors take the following form
G^a_μν =
∂_μ A^a_ν - ∂_ν A^a_μ +
g f^abc A^b_μ A^c_ν,
G^a_μν,λ =
∂_μ A^a_νλ - ∂_ν A^a_μλ +
g f^abc( A^b_μ A^c_νλ + A^b_μλ A^c_ν ),
G^a_μν,λρ =
∂_μ A^a_νλρ - ∂_ν A^a_μλρ +
g f^abc( A^b_μ A^c_νλρ +
A^b_μλ A^c_νρ+A^b_μρ A^c_νλ
+ A^b_μλρ A^c_ν ),
...... . ............................................and transform homogeneously with respect to the
transformations (<ref>):
δG^a_μν = g f^abc G^b_μν ξ^c ,
δG^a_μν,λ = g f^abc ( G^b_μν,λ ξ^c
+ G^b_μν ξ^c_λ ),
δG^a_μν,λρ = g f^abc
( G^b_μν,λρ ξ^c
+ G^b_μν,λ ξ^c_ρ +
G^b_μν,ρ ξ^c_λ +
G^b_μν ξ^c_λρ ),
...... . ..........................The field strength tensors are
antisymmetric in their first two indices and are totally symmetric with respect to the
rest of the indices. The symmetry properties of the field strength $G^{a}_{\mu\nu,\lambda_1 ... \lambda_s}$ remain invariant in the course of these transformations.
The first gauge invariant density is given by the expression <cit.>
L(x)= L(x,e)⟩= -14 ^a_μν(x,e)^a μν(x,e)⟩,
where the trace of the generators is given in (<ref>).
One can get convinced that the variation of the (<ref>) with respect to the
gauge transformations (<ref>)
and (<ref>) vanishes:
\delta {{\cal L}}(x,e) = -{1\over 2}\CG^{a}_{\mu\nu}(x,e)~ g f^{abc}~
\CG^{b \mu\nu}(x,e) ~\xi^{c}(x,e) =0.
The invariant density (<ref>) allows to extract
gauge invariant, totally symmetric, tensor densities
$\CL_{\lambda_1 ... \lambda_{s}}(x)$ by
using expansion with respect to the vector variable $e^{\lambda}$:
(x,e) = ∑^∞_s=0 1s!
_λ_1 ... λ_s(x) e^λ_1...e^λ_s .
In particular, the expansion term which is quadratic in powers of $e^{\lambda}$ is
_λ_1λ_2 = -14G^a_μν,λ_1G^a_μν,λ_2
The gauge invariant density thus can be represented in the following form
(x) = (x,e)⟩= ∑^∞_s=0 1s!
_λ_1 ... λ_s(x) e^λ_1...e^λ_s⟩and the density for the lower-rank tensor fields is
{{\cal L}}_2 =-{1\over 4}G^{a}_{\mu\nu,\lambda}G^{a}_{ \mu\nu,\lambda}
-{1\over 4}G^{a}_{\mu\nu}G^{a}_{ \mu\nu,\lambda \lambda}.
Let us consider the second gauge invariant density of the form <cit.>
L^'(x)=L^'(x,e)⟩= 14
^a_μρ_1(x,e) e^ρ_1 ^a μ_ ρ_2(x,e) e^ρ_2⟩^'.
It is gauge invariant because its variation is also equal to zero:
δL^'(x,e) =14g f^acb
^c_μρ_1(x,e)e^ρ_1 ξ^b(x,e)
^aμ_ ρ_2(x,e)e^ρ_2+
+14^a_μρ_1(x,e)e^ρ_1 g f^acb
^cμ_ ρ_2(x,e) e^ρ_2 ξ^b(x,e) =0.
The Lagrangian density (<ref>) generates the
second series of gauge invariant tensor densities
$(\CL^{'}_{\rho_1\rho_2})_{\lambda_1 ... \lambda_{s}}(x)$
when we expand it in powers of the vector variable $e^{\lambda}$:
L^'(x)=L^'(x,e)⟩= ∑^∞_s=0 1s!
(^'_ρ_1ρ_2)_λ_1 ... λ_s(x) e^ρ_1e^ρ_2
e^λ_1...e^λ_s ⟩^'.
The term quartic in variable $e^{\lambda}$ after contraction of the vector
variables takes the following form:
L^'_2 = 14G^a_μν,λG^a_μλ,ν
+12 G^a_μνG^a_μλ,νλ .
One can get convinced that it is gauge invariant under the
transformation (<ref>) and (<ref>).
The total Lagrangian density is a sum of two invariants (<ref>)
and (<ref>):
L = + ^'=-14 ^a_μν(x,e)^a μν(x,e)⟩+
^a_μρ_1(x,e) e^ρ_1 ^a μ_ ρ_2(x,e) e^ρ_2⟩^'.
The Lagrangian for the lower-rank tensor gauge fields has the following form:
L= L_1 + L_2 + L^'_2 +...=
- 14G^a_μνG^a_μν
- 14G^a_μν,λG^a_μν,λ
+ 14G^a_μν,λG^a_μλ,ν
+12G^a_μνG^a_μλ,νλ +...The above Lagrangian defines the kinetic operators for the rank-1 $A^a_{\mu}$
and rank-2 $A^{a}_{\mu\lambda_1}$ fields,
as well as trilinear and quartic interactions
with the dimensionless coupling constant g (see Fig.<ref>-<ref>) .
The interaction vertex for the vector gauge boson V and two
tensor gauge bosons T - the VTT vertex -
in non-Abelian tensor gauge
field theory <cit.>.
Vector gauge bosons are conventionally drawn
as thin wave lines, tensor gauge bosons are thick wave lines.
The Lorentz indices $\alpha\acute{\alpha}$ and momentum $k$ belong to the
first tensor gauge boson, the $\gamma\acute{\gamma}$ and momentum $q$
belong to the second tensor gauge boson, and Lorentz index $\beta$ and
momentum $p$ belong to the vector gauge boson.
As we found in
the corresponding free field equations coincide with the equations introduces in
the classical works <cit.> and
describe the propagation of the helicity-two and zero $h = \pm 2,0 $
massless charged
tensor gauge bosons, and there are no propagating negative norm states.
This is in agreement with the spectrum presented in (<ref>).
The next term in expansion of the Lagrangian density has the following
form <cit.>:
L_3 + L^'_3
= - 14G^a_μν,λρG^a_μν,λρ
-12G^a_μν,λ G^a_μν,λρρ
-18G^a_μν G^a_μν,λλρρ+
+ 13G^a_μν,λρG^a_μλ,νρ+
13 G^a_μν,νλG^a_μρ,ρλ+
+ 13G^a_μν,λG^a_μλ,νρρ
+13G^a_μνG^a_μλ,νλρρand the corresponding free field equations for the tensor
gauge field $A_{\mu\lambda_1\lambda_2}$
in four-dimensional space-time
describe the propagation of helicity-three and one
$h= \pm 3, \pm 1,\pm 1 $ massless charged gauge bosons in
agrement with the spectrum (<ref>).
There are no propagating negative norm states. The comparison
of these equations with the Schwinger-Fronsdal equations
can be found in <cit.>.
Considering the free field equation for the general
rank-(s+1) tensor gauge field one can find that
the quadratic part of the Lagrangian has the following form <cit.>:
L_s+1 + L^'_s+1 |_quadratic=
1 2 A^a_αλ_1...λ_s
^αλ_1...λ_s γλ_s+1...λ_2s
and is invariant with respect to the group of gauge transformations
=∂_α ξ^a_λ_1...λ_s,
δ̃ A^a_αλ_1...λ_s =
∂_λ_1 ζ^a_λ_2...λ_sα+...
+∂_λ_s ζ^a_λ_1...λ_s-1α,
which should fulfil the following constraints:
1 s-2(∂_λ_1 ζ^a_λ_2...λ_s-1 ρρ+
...+ ∂_λ_s-1 ζ^a_λ_1...λ_s-2 ρρ)=0,
∂_λ_1 ζ^a_λ_2...λ_s-1 ρρ-
∂_λ_2 ζ^a_λ_1...λ_s-1 ρρ=0.
The quartic vertex with two vector gauge bosons and two
tensor gauge bosons - the VVTT vertex -
${{\cal V}}^{abcd}_{\alpha\beta\gamma\acute{\gamma}\delta\acute{\delta}}(k,p,q,r)$
in non-Abelian tensor gauge
field theory <cit.>.
Vector gauge bosons are conventionally drawn
as thin wave lines, tensor gauge bosons are thick wave lines.
The Lorentz indices $\gamma\acute{\gamma}$ and momentum $q$ belong to the
first tensor gauge boson, $\delta\acute{\delta}$ and momentum $r$ belong to the
second tensor gauge boson, the index $\alpha$ and momentum $k$
belong to the first vector gauge boson and Lorentz index $\beta$ and
momentum $p$ belong to the second vector gauge boson.
In momentum representation the kinetic operator has the following general form:
_αλ_1...λ_s γλ_s+1...λ_2s =
+ 1 s!
(∑_p η_λ_i_1 λ_i_2 .......
η_λ_i_2s-1 λ_i_2s)
(-k^2 η_αγ + k_αk_γ)
+ 1 (s+1)! (∑_P η_ αλ_i_1
η_λ_i_2 λ_ i_3 .......
η_ λ_ i_2s-2 λ_i_2s-1
η_ γλ_i_2s) k^2
- 1(s+1)!
(∑_P η_ρλ_i_1
η_λ_i_2 λ_i_3 .......
η_ λ_ i_2s-2 λ_i_2s-1
η_ γλ_i_2s)
k_α k_ρ
- 1(s+1)!
(∑_P η_ρλ_i_1
η_λ_i_2 λ_i_3 .......
η_ λ_ i_2s-2 λ_i_2s-1
η_ αλ_i_2s)
k_ρ k_γ
+ 1(s+1)! η_αγ
η_λ_i_2 λ_i_3 .......
η_ λ_ i_2s-2 λ_i_2s-1
η_ σλ_i_2s)
k_ρ k_σ
,where the sum $\sum_P$ runs over all non-equal permutations of
$\lambda_i~'s$. The solution of the free field equation for the rank-(s+1)
field <cit.>
^αλ_1...λ_s γλ_s+1...λ_2s
A_γλ_s+1...λ_2s =0.
describes the propagation of the helicities:
h= ±(s+1), ±(s-1)
±(s-1) ,
±(s-3) , ....
It is convenient to represent the spectrum (<ref>) of tensor gauge bosons
in the form which combines the helicity spectrum of all bosons. It is unbounded
and has the following form <cit.>:
±2, 0
±3, ±1, ±1
±4, ±2, ±2, 0
±5, ±3, ±3, ±1, ±1
±6, ±4, ±4, ±2, ±2, 0
In summary, we defined the composite gauge field (<ref>) which takes a value in
the transversal representation (<ref>), (<ref>),
(<ref>) of the extended Poincaré algebra $L_G(\CP)$. We
constructed the invariant Lagrangian (<ref>), (<ref>), (<ref>)
which contains infinity many tensor gauge fields (<ref>) and found their
helicity content (<ref>), (<ref>).
The theory has unexpected symmetry with respect to the duality transformation of the gauge
fields <cit.>. The complementary gauge transformation $\tilde{\delta} $ is defined as:
δ̃ A^a_μ = ( δ^ab∂_μ
+g f^acbA^c_μ)η^b ,
δ̃ A^a_μλ_1 = ( δ^ab∂_λ_1
+ g f^acbA^c_λ_1)η^b_μ + g f^acbA^c_μλ_1η^b,
δ̃ A^a_μλ_1λ_2 = ( δ^ab∂_λ_1
+g f^acb A^c_λ_1)η^b_μλ_2 +( δ^ab∂_λ_2
+g f^acb A^c_λ_2)η^b_μλ_1 +
+g f^acb( A^c_μλ_1η^b_λ_2 +
A^c_μλ_2 η^b_λ_1+
A^c_λ_1λ_2η^b_μ +
A^c_λ_2 λ_1η^b_μ +A^c_μλ_1λ_2η^b),
......... . ............................The transformations $\delta$ in (<ref>) and $\tilde{\delta} $ in
(<ref>) do not
coincide and are complementary to each other in the following sense:
in $\delta$
the derivatives of the gauge parameters $\{ \xi \}$ are over the first
index $\mu$, while in $\tilde{\delta} $ the derivatives of the
gauge parameters $\{ \eta \}$ are over the rest of the totally symmetric
indices $\lambda_1 ... \lambda_{s}$.
One can construct the
new field strength tensors
$\tilde{G}^{a}_{\mu\nu,\lambda_1 ... \lambda_s}$
which are transforming homogeneously
with respect to the $\tilde{\delta}$ transformations and then
to construct the corresponding gauge invariant Lagrangian
$\tilde{L}(A)$ <cit.>. The relation between these two Lagrangians
was found in the form of duality transformation <cit.>:
Ã_μλ_1 = A_λ_1μ ,
Ã_μλ_1λ_2 =
12(A_λ_1μλ_2 + A_λ_2μλ_1)
-12 A_μλ_1λ_2,
Ã_μλ_1λ_2λ_3 =
+ A_λ_2μλ_1λ_3
+ A_λ_3μλ_1λ_2)
-23 A_μλ_1λ_2 λ_3 ,
which maps the Lagrangian $L(\tilde{A})$
into the Lagrangian $\tilde{L}(A)$. This takes place
G_{\mu\nu,\lambda_1 ... \lambda_s}(\tilde{A})=
\tilde{G}_{\mu\nu,\lambda_1 ... \lambda_s}(A)
and therefore
L(\tilde{A})~= ~\tilde{L}(A) .
The Lagrangian (<ref>) defines not only a free propagation
of tensor gauge bosons, but also their
interactions. The interaction diagrams for the lower-rank bosons are
presented on Fig.<ref>-<ref>. The high-rank bosons also interact
through the triple and quartic interaction vertices. It is therefore important
to calculate and study
the scattering amplitudes, the quantum loop corrections and their high energy
behaviour. By using the diagram technique it is possible to
calculate the scattering amplitude, but the difficulties
lie in the evaluation and contraction of high-rank tensors structures appearing in the diagram
approach. In the next section we shall use alternative approach based on spinor representation
of amplitudes developed recently in <cit.>.
§ SCATTERING AMPLITUDES AND SPLITTING FUNCTIONS
A scattering amplitude for the
massless particles of momenta $p_i$ and polarisation tensors $\varepsilon_i$ $(i=1,...,n)$,
which are described by irreducible massless representations of the Poincaré group,
can be represented in the following form:
M_n = M_n(p_1,\varepsilon_1;~p_2,\varepsilon_2;~...;~p_n,\varepsilon_n).
It is more convenient to represent the momenta $p_i$ and polarisation tensors $\varepsilon_i$ in terms of spinors. In that case
the scattering amplitude $M_n$ can be considered
as a function of spinors $\lambda_i$, $\tilde{\lambda}_i$ and helicities $h_i$
M_n=M_n(λ_1,λ̃_1,h_1; ...; λ_n,λ̃_n,h_n) .
The advantage of the spinor representation is that introducing a complex deformation of
the particles momenta
one can derive a general form for the three-particle interaction vertices
M_3(1^{h_1} ,2^{h_2},3^{h_3} ).
The dimensionality of the three-point vertex $M_3(1^{h_1} ,2^{h_2},3^{h_3} )$ is
In the generalised Yang-Mills theory <cit.>, which we described in the previous sections,
all interaction vertices
between high-spin particles have dimensionless coupling constants,
which means that the helicities of the interacting particles in the vertex are
constrained by the relation
D=\pm(h_1+h_2+h_3)= 1~.
Therefore the interaction vertex between
massless tensor-bosons, the TTT-vertex, has the following
general form <cit.>:
M_3 = g f^abc <1,2>^-2h_1 -2h_2 -1 <2,3>^2h_1 +1 <3,1>^2h_2 +1, h_3= -1 - h_1 -h_2,
M_3 = g f^abc [1,2]^2h_1 +2h_2 -1 [2,3]^-2h_1 +1 [3,1]^-2h_2 +1, h_3= 1 - h_1 -h_2,
where $f^{abc}$ are the structure constants of the internal gauge group G.
In particular, considering the interaction between a boson of helicity $h_1 = \pm 1$ and
a tensor-boson of helicity $h_2 = \pm s$, the VTT-vertex, one can find from (<ref>) that
h_3 = ±|s-2 |, ±s,
±|s+2 |
and the corresponding vector-tensor-tensor interaction vertices VTT have
the following form:
M^a_1a_2a_3_3(1^-s ,2^-1 ,3^+s ) = g f^a_1 a_2 a_3 <1,2>^4 <1,2> <2,3> <3,1>
(<1,2> <2,3> )^2s-2,
M^a_1a_2a_3_3(1^-s ,2^+1,3^s-2 ) = g f^a_1 a_2 a_3 <1,3>^4 <1,2> <2,3> <3,1>
(<1,2> <2,3> )^2s-2.
These are the vertices which reduce to the standard triple YM vertex when $s=1$.
Using these vertices one can compute the scattering amplitudes of vector and tensor bosons.
The colour-ordered
scattering amplitudes involving two tensor-bosons of helicities $h =\pm s$,
one negative helicity vector-boson
and $(n-3)$ vector-bosons of positive helicity were found in <cit.>:
M̂_n(1^+,..i^-,...k^+s,..j^-s,..n^+)=i g^n-2 (2π)^4 δ^(4)(P^aḃ)
<ij>^4/∏_l=1^n <l l+1> ( <ij>/<ik>)^2s-2,
where $n$ is the total number of particles and the dots stand for any number of
positive helicity vector-bosons, $i$ is the position of the negative-helicity vector,
while $k$ and $j$ are the positions of the tensors with helicities $+s$ and $-s$ respectively.
The expression (<ref>) reduces to the
famous Parke-Taylor formula <cit.> when $s=1$.
In particular, the five-particle amplitude takes the following form:
M̂_5(1^+,2^-,3^+,4^+s,5^-s)= i g^3 (2π)^4 δ^(4)(P^aḃ)
<i i+1> (<25> /<24>)^2s-2,
P^{a\dot{b}} = \sum^n_{m=1} \lambda^a_m \tilde{\lambda}^{\dot{b}}_m
$ is the total momentum. Notice that the scattering amplitudes (<ref>) and (<ref>)
have large validity area: in the limit $s \rightarrow 1/2$ they
reduce to the tree level gluon scattering amplitudes into a quark pair and
into a pair of scalars as $s\rightarrow 0$.
The scattering amplitudes (<ref>) and (<ref>)
can be used to extract splitting amplitudes
of vector and tensor bosons <cit.>. The collinear behaviour
of the tree amplitudes has the following factorised form
M^tree_n(...,a^λ_a,b^λ_b,...) a ∥b → ∑_λ=±1
Split^ tree _-λ(a^λ_a,b^λ_b) × M^tree_n-1(...,P^λ ,...),
where $Split^{tree}_{-\lambda}(a^{\lambda_a},b^{\lambda_b})$ denotes the splitting amplitude and the
intermediate state $P$ has momentum $k_P=k_a +k_b$ and helicity $\lambda$.
Considering the amplitude (<ref>) in the limit when the
particles 4 and 5 become collinear, $k_4 \parallel k_5$, that is,
$k_4 = z k_P,~k_5 = (1-z) k_P$, $k^2_P \rightarrow 0$ and $z$ describes the
longitudinal momentum sharing, one can deduce that the corresponding behaviour of spinors is
\lambda_4 = \sqrt{z} \lambda_P,~~~\lambda_5 = \sqrt{1-z} \lambda_P,
and that the amplitude (<ref>) takes the following factorisation
form <cit.>:
= A_4(1^+,2^-,3^+,P^-) × Split_+(a^+s,b^-s),
Split_+(a^+s,b^-s) = (1-z/z )^ s-1 (1-z)^2/√(z(1-z))
1/ <a, b>.
In a similar way one can deduce that
Split_+(a^-s,b^+s) = (z /1-z )^ s-1 z^2/√(z(1-z))
1/ <a, b>.
Considering different collinear limits $k_1 \parallel k_5$ and $k_3 \parallel k_4$
one can get <cit.>
Split_+s(a^+,b^-s) = (1-z)^s+1/√(z(1-z))
1/ <a, b>,
Split_+s(a^-s,b^+) = z^s+1/√(z(1-z))
1/ <a, b>
Split_-s(a^+s,b^+) = z^-s+1/√(z(1-z))
1/ <a, b>,
Split_-s(a^+,b^+s) = (1-z)^-s+1/√(z(1-z))
1/ <a, b>.
The set of splitting amplitudes (<ref>)-(<ref>)
$V\rightarrow TT$, $T \rightarrow VT$ and
$T \rightarrow TV$ reduces to the full set of gluon splitting amplitudes
when $s=1$.
Since the collinear limits of the scattering amplitudes
are responsible for parton evolution <cit.>
we can extract from the above expressions the
Altarelli-Parisi splitting probabilities for
tensor-bosons. Indeed, the residue of the collinear
pole in the square (of the factorised
amplitude (<ref>)) gives Altarelli-Parisi splitting probability $P(z)$:
P(z)= C_2(G) ∑_h_P , h_a, h_b |Split_-h_P(a^h_a,b^h_b) |^2 s_ab,
where $s_{ab}=2 k_a \cdot k_b= <a,b>[a,b]$.
The invariant operator $C_2$ for the representation R is defined by the equation
$ t^a t^a = C_2(R)~ 1 $ and $tr(t^a t^b) = T(R) \delta^{ab}$.
Substituting the splitting amplitudes (<ref>)-(<ref>)
into (<ref>) we are getting
P_TV(z) = C_2(G)[ z^4 z(1-z)( z1-z )^2s-2
+(1-z)^4 z(1-z) ( 1-zz)^2s-2 ],
P_VT(z) = C_2(G)[ 1z(1-z)( 11-z )^2s-2
+(1-z)^4 z(1-z) (1-z)^2s-2 ],
P_TT(z) = C_2(G)[ z^4 z(1-z) z^2s-2
+1 z(1-z) ( 1z)^2s-2 ].
The decay of a gluon of helicity $h_A$ into
the tensorgluons of helicities $h_B$ and $h_C$. The arrows
show the directions of the helicities.
The corresponding splitting probability is defined as $P_{BA}$.
The momentum conservation in the vertices clearly fulfils because these
functions satisfy the relations
P_TV(z)=P_TV(1-z), P_VT(z)=P_TT(1-z), z < 1.
In the leading order the kernel $P_{TV}(z)$ has a meaning of variation per unit
transfer momentum of the probability
density of finding a tensor-boson inside the vector-boson, $P_{VT}(z)$ - of finding
a vector inside
the tensor and $P_{TT}(z)$ - of finding a tensor inside the tensor.
For completeness we shall present also quark and vector-boson kernels
P_qq(z) = C_2(R)1+z^2 1-z ,
P_Vq(z) = C_2(R)1+(1-z)^2 z ,
P_qV(z) = T(R)[z^2 +(1-z)^2],
P_VV(z) = C_2(G)[1 z(1-z)+ z^4 z(1-z)+(1-z)^4 z(1-z)],where $C_2(G)= N, C_2(R)={N^2-1 \over 2 N}, T(R) = {1 \over 2}$ for the SU(N) groups.
Having in hand the new set of splitting probabilities
for tensor-bosons (<ref>)
we can consider a possible generalisation of quantum chromodynamics <cit.>.
In so generalised theory in addition to the quarks and gluons there should be
tensorgluons. We can hypothesise that a possible emission of tensorgluons
by gluons, as it is shown on Fig.<ref>,<ref> should produce a non-zero
density of tensorgluons
inside the proton in additional to the quark and gluon densities.
Our next goal is to derive DGLAP equations
<cit.> which will take into account these new emission processes.
§ GENERALIZATION OF DGLAP EQUATION.
CALCULATION OF CALLAN-SIMANZIK BETA FUNCTION
In this section we shall consider a possibility that inside the proton
and, more generally, inside hadrons there are
additional partons - tensorgluons, which can carry a part of the
proton momentum <cit.>.
Tensorgluons have
zero electric charge, like gluons, but have a larger spin. Inside the proton
a nonzero density of the tensorgluons can be generated by the emission of tensorgluons
by gluons <cit.>.
The last mechanism is typical for non-Abelian tensor gauge
theories, in which there exists a gluon-tensor-tensor vertex of order g
(see Fig.<ref>-<ref>) <cit.>.
Therefore a number of gluons changes not only
because a quark may radiate a gluon or because a gluon may split into a quark-antiquark
pair or into two gluons <cit.>,
but also because a gluon can split into two tensorgluons
The process of gluon splitting into tensorgluons suggests that part of the proton momentum
which was carried by neutral partons can be shared between vector and tensorgluons.
Our aim is to calculate the scattering amplitudes and splitting function in QCD generalised in this way.
It is well known that the deep inelastic structure functions
can be expressed in terms of quark distribution densities.
If $q^i(x)$ is the density of quarks of type i (summed over
colours) inside a proton target with fraction x of the proton longitudinal momentum
in the infinite momentum frame <cit.> then the scaling structure functions can
be represented in the following form:
2F_1(x)= F_2(x)/x= ∑_i Q^2_i [q^i(x)+q̅^i(x)].
The scaling behaviour of the structure functions is broken and the results
can be formulated by assigning a well determined $Q^2$
dependence to the parton densities. This can be achieved by
introducing integro-differential equations which describe the
$Q^2$ dependence of quark $q^i(x,t)$ and gluon densities $G(x,t)$,
where $t=\ln(Q^2/Q^2_0)$ <cit.>.
Let us see what will happen if one supposes that
there are additional partons - tensorsgluons - inside the proton.
In accordance with our hypothesis there
is an additional emission of tensorgluons in the proton, therefore one should introduce
the corresponding density $T(x, t)$ of tensorgluons (summed over colours)
inside the proton in the $P_{\infty}$ frame <cit.>. We can derive
integro-differential equations that describe the $Q^2$ dependence
of parton densities in this general case <cit.>:
d q^i(x,t)dt = α(t) 2 π ∫^1_x dy y[∑^2 n_f_j=1 q^j(y,t)
P_q^i q^j(x y)+ G(y,t) P_q^i G(x y)] ,
d G(x,t)dt = α(t) 2 π ∫^1_x dy y[∑^2 n_f_j=1 q^j(y,t)
P_G q^j(x y)+ G(y,t) P_G G(x y)+ T(y,t) P_G T(x y) ],
d T(x,t)dt = α(t) 2 π ∫^1_x dy y[
G(y,t) P_T G(x y) + T(y,t) P_T T(x y)].The $\alpha(t)$ is the running coupling constant ($\alpha = g^2/4\pi$).
In the leading logarithmic approximation $\alpha(t)$ is of the form
αα(t) = 1 +b α t ,
where $\alpha = \alpha(0)$ and $b$ is the one-loop Callan-Simanzik coefficient,
which, as we shall see below, receives an additional contribution from the tensorgluon
Here the indices i and j run over quarks and antiquarks of all flavors. The number
of quarks of a given fraction of momentum changes when a quark looses momentum by
radiating a gluon,
or a gluon inside the proton may produce a quark-antiquark pair <cit.>.
Similarly the number of gluons changes
because a quark may radiate a gluon or because a gluon may split into a quark-antiquark
pair or into two gluons or into two tensorgluons. This last possibility is realised,
because, as we have seen, in non-Abelian tensor gauge
theories there is a triple vertex VTT (<ref>) of a gluon and two tensorgluons of order g
This interaction should be taken into consideration, and we added the
term $T(y,t) ~P_{G T}({x \over y})$ in the second
equation (<ref>). The density of tensorgluons $T(x,t)$ changes when
a gluon splits into two tensorgluons or when a tensorgluon radiates a gluon. This
evolution is described by the last equation (<ref>).
In order to guarantee that the total momentum of the proton, that is, of
all partons is unchanged, one should impose the following constraint:
ddt∫_0^1 dz z [∑^2n_f_i=1q^i(z,t)+G(z,t)+T(z,t)]=0.
Using the evolution equations (<ref>) one can express the derivatives
of the densities in (<ref>) in terms of kernels and to see that the following
momentum sum rules should be fulfilled:
∫_0^1 dz z [P_qq(z)+P_Gq(z) ]=0,
∫_0^1 dz z [2 n_f P_qG(z)+P_GG(z)+P_TG(z)]=0,
∫_0^1 dz z [ P_GT(z)+P_TT(z)]=0.
Before analysing these momentum sum rules let us first
inspect the behaviour of the gluon-tensorgluon kernels (<ref>)
at the end points $z=0,1$. As one can see,
they are singular at the boundary values similarly to the
case of the standard kernels (<ref>).
Though there is a difference here: the singularities are of higher order compared to the standard case
Therefore one should define the regularisation procedure for the singular factors
$(1 - z)^{-2s+1}$ and $ z^{-2s+1}$ reinterpreting them as the distributions $(1 - z)^{-2s+1}_{+}$ and
$z^{-2s+1}_{+}$, similarly to the Altarelli-Parisi regularisation <cit.>.
We shall define them in the following
∫_0^1 dz f(z)(1 - z)^2s-1_+ =
∫_0^1 dz f(z)- ∑^2s-2_k=0 (-1)^k k! f^(k)(1) (1-z)^k (1 - z)^2s-1,
∫_0^1 dz f(z)z ^2s-1_+ =
∫_0^1 dz f(z)- ∑^2s-2_k=0 1 k! f^(k)(0) z^k z^2s-1,
∫_0^1 dz f(z)z_+ (1-z)_+ =
∫_0^1 dz f(z)- (1-z)f(0) - z f(1) z (1-z) ,where $f(z)$ is any test function which is sufficiently regular at the end points
and, as one can see, the defined substraction guarantees the convergence of the integrals.
Using the same arguments as in the standard case <cit.> we should add the delta function
terms into the definition of the diagonal kernels so that they will completely determine
the behaviour of $P_{qq}(z)$ , $P_{GG}(z)$ and $P_{TT}(z)$ functions. The first equation
in the momentum sum rule (<ref>)
remains unchanged because there is no tensorgluon contribution into the quark evolution. The second
equation in the momentum sum rule (<ref>) will take the following form:
∫_0^1 dz z [2 n_f P_qG(z)+P_GG(z)+P_TG(z) + b_G δ(z-1)]=
=∫_0^1 dz z [2 n_f T(R)[z^2 +(1-z)^2]+C_2(G)[1 z(1-z)
+ z^4 z(1-z)+(1-z)^4 z(1-z)]+
+C_2(G)[ z^4 z(1-z)( z1-z )^2s-2
+(1-z)^4 z(1-z) ( 1-zz)^2s-2 ] ] +b_G=
=23 n_f T(R) - 11 6C_2(G)- 12 s^2 -16 C_2(G) + b =0.
From this result we can extract an additional contribution
to the one-loop Callan-Symanzik beta function
arising from the tensorgluon loop.
Indeed, the first beta-function coefficient enters into this expression because the momentum sum
rule (<ref>) implicitly comprises unitarity, thus the one-loop effects <cit.>.
In (<ref>) we have three terms which come from gluon and quark loops:
b_1 = 11 6C_2(G) - 2 n_f 3T(R),
and from the tensorboson loop of spin s:
b_T = 12 s^2 -16 C_2(G), s=1,2,3,4,....
It is a very interesting result because at s=1 we are rediscovering the
asymptotic freedom result <cit.>.
For larger spins the tensorgluon contribution into the Callan-Simanzik beta function
has the same signature as the standard gluons, which means that tensorgluons
"accelerate" the asymptotic freedom (<ref>) of the strong
interaction coupling constant $\alpha(t)$.
The contribution is increasing quadratically
with the spin of the tensorgluons, that is, at large transfer momentum
the strong coupling constant tends to zero faster compared to the standard case:
α(t)= α1+ b α t ,
b = (12s^2 -1) C_2(G) - 4 n_f T(R) 12 π, s=1,2,...
Surprisingly, a similar result based on the parametrization of the
charge renormalization taken in the form $b = (-1)^{2s}(A+Bs^2)$
was conjectured by Curtright
<cit.>. Here $A$ represents an orbital contribution and
$B s^2$ - the anomalous magnetic moment contribution
<cit.>. The unknown coefficients A and B were found
by comparing the suggested parametrisation with the known results for s= 0, 1/2 and 1.
It is also possible to consider a straitforward generalisation of the result obtained
for the effective action in Yang-Mills theory long ago
to the higher spin gauge bosons. With the spectrum of the tensorgluons in
the external chromomagnetic field $\lambda = (2n+1 + 2s)gH +k^2_{\parallel}$
one can perform a summation of the modes and get an exact result for the one-loop effective action similarly to <cit.>:
ϵ= H^2 2 +(gH)^2 4π b [lngH μ^2-12],
b = -2 C_2(G)π ζ(-1, 2s+12)=
12s^2-112 πC_2(G),
and $\zeta(-1, q)=-{1\over 2}(q^2 -q +{1\over 6})$ is the generalised zeta
function[The generalised zeta function is defined as $\zeta(p, q)=\sum^{\infty}_{k=0}{1\over (k+q)^p}
={1\over \Gamma(p)} \int^{\infty}_{0} dt t^{-1+p} { e^{-qt} \over 1-e^{-t}} $ .].
Because the coefficient in front of the logarithm defines the beta function
<cit.>, one can see that (<ref>) is
in agrement with the result (<ref>).
It is also natural to ask what will happen if one takes into consideration the contribution
of tensorgluons of all spins into the beta function[I would like to thank
John Iliopoulos and Constantin Bachas for raising this question.]. One can suggest two scenarios.
In the first one the high spin gluons, let us say, of $s \geq 3$, will get large mass
and therefore they can be ignored at a given energy scale. In the second case,
when all of them remain massless, then one can
suggest the Riemann zeta function regularisation, similar to the Brink-Nielsen regularisation
<cit.>. The summation over the spectrum in (<ref>) gives<cit.>:
b_11 = C_2(G) [ ∑^∞_s=1 ( 12s^2 -1) 12 π+
∑^∞_s=0 ( 12s^2 -1) 12 π +
∑^∞_s=1 ( 12s^2 -1) 12 π+
∑^∞_s=0 ( 12s^2 -1) 12 π +.....]
= C_2(G)[1π ζ(-2)- 112π ζ(0)-112π
+1π ζ(-2)- 112π ζ(0) +...]
= C_2(G)[124π - 112π +
where $\zeta(-2)=0,~ \zeta(0)=-1/2$, leading to the theory which is conformally invariant at very high energies. The above
summation requires explicit regularisation and further justification.
§ UNIFICATION OF COUPLING CONSTANTS OF STANDARD MODEL
It is interesting to know how the contribution of tensorgluons changes the high energy
behaviour of the coupling constants of the Standard Model <cit.>.
The coupling constants are evolving in accordance with the formulae
1 α_i(M) = 1 α_i(μ)+ 2 b_i lnMμ, i=1,2,3,
where we shall consider only the contribution of the lower $s=2$ tensorbosons:
2b = 58 C_2(G) - 4 n_f T(R) 6 π.
For the $SU(3)_c \times SU(2)_L \times U(1)$ group with its coupling constants $\alpha_3, \alpha_2$ and $\alpha_1$
and six quarks $n_f=6$ and $SU(5)$ unification group we will get
2 b_3= {1 \over 2\pi} 54,~~~ 2 b_2= {1\over 2 \pi} {104\over 3},~~~2 b_1= -{1\over 2 \pi} 4,
so that solving the system of equations (<ref>) one can get
lnMμ = π58 (1α_el(μ)- 83 1α_s(μ)),
where $\alpha_{el}(\mu)$ and $\alpha_s(\mu)$ are the electromagnetic and strong coupling constants at
scale $\mu$. If one takes $\alpha_{el}(M_Z)= 1/128$ and $\alpha_s(M_Z) =1/10$ one can get that
coupling constants have equal strength at energies of order
M \sim 4 \times 10^4 GeV = 40~ TeV,
which is much smaller than the scale $M\sim 10^{14} GeV$ in the absence of the tensorgluons contribution.
The value of the weak angle <cit.> remains intact :
sin^2θ_W = 16 +59 α_el(M_Z)α_s(M_Z),
as well as the coupling constant at the unification scale remains of the same order
§ CONCLUSION
In the present article we describe a possible extension
the Yang-Mills gauge principle <cit.> which includes tensor gauge fields.
In this extension of the Yang-Mills theory the vector gauge boson becomes a
member of a bigger family of gauge bosons of arbitrary large integer spins.
The proposed extension of Yang-Mills theory
is essentially based on the existence of the enlarged Poincaré
algebra and on an appropriate transversal representations
of that algebra.
The invariant Lagrangian is expressed in terms of new
higher-rank field strength tensors. The Lagrangian does not contain higher derivatives of
tensor gauge fields and all interactions take place through three- and four-particle
exchanges with a dimensionless coupling constant (see Fig.<ref>-<ref>).
We calculated the scattering amplitudes of non-Abelian tensor gauge bosons
at tree level, as well as their one-loop contribution into the Callan-Symanzik beta function.
This contribution is negative and
corresponds to the asymptotically free theory. The proposed extension may
lead to a natural inclusion of the standard theory of fundamental forces
into a larger theory in which vector gauge bosons, leptons and quarks represent
a low-spin subgroup.
In the line with the above development we considered a possible extension of QCD.
In so extended QCD inside the proton and, more generally, inside hadrons there should be
additional partons - tensorgluons, which can carry a part of the proton
momentum. Among all parton distributions, the gluon density $G(x,t)$ is one of the least
constrained functions since it does not couple directly to the
photon in deep-inelastic scattering measurements of the proton $F_2$ structure function.
Therefore it is only indirectly constrained by scaling violations and by the momentum sum rule
which resulted in the fact that only half of the proton momentum is carried by charged
constituents - the quarks - and that the other part is ascribed to the
neutral constituents.
As it was suggested, the process of gluon splitting leads to the emission
of tensorgluons and
therefore a part of the proton momentum which is carried by the neutral constituents
can be shared between gluons and tensorgluons. The density of neutral partons in the proton is
therefore given by the sum of two functions: $G(x,t)+T(x,t)$, where $T(x,t)$
is the density of the
tensorgluons. To disentangle these contributions and to decide which
piece of the neutral partons is the contribution of gluons and which one
is of the tensorgluons one should measure the helicities of the neutral
components, which seems to be a difficult task.
The gluon density can be directly constrained by jet production <cit.>.
In the suggested model the situation is such that the standard quarks
cannot radiate tensorgluons
(such a vertex is absent in the model
therefore only gluons are radiated by quarks.
A radiated gluon then can split into a pair of tensorgluons
without obscuring the structure of the
observed three-jet final states. Thus it seems that there is no obvious contradiction with the existing
experimental data. Our hypotheses may be wrong, but the uniqueness and simplicity of
suggested extension seems to be the reasons for serious consideration.
This extension of QCD influences the unification scale at which
the coupling constants of the Standard Model merge.
In the last section we observed that the unification scale at which
standard coupling constants are
merging is shifted to lower energies telling us that it may be that a
new physics is round the corner.
Whether all these phenomena are consistent with experiment is an open question.
§ ACKNOWLEDGEMENT
I would like to thank the organisers of the "Conference on 60 Years of Yang-Mills Gauge Field Theories" for their kind hospitality in the Institute of Advanced Studies of the
Nanyang Technological University
in Singapore and Prof. Kok-Khoo Phua and Prof. Yong Min Cho for invitation.
This work was supported in part by the General Secretariat for Research and
Technology of
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1511.00594
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Extreme Black holes are an important theoretical laboratory for exploring the nature of entropy. We suggest that this unusual nature of the extremal limit could explain the entropy of extremal Kerr black holes. The time-independence of the extremal black hole, the zero surface gravity, the zero entropy and the absence of a bifurcate Killing horizon are all related properties that define and reduce to one single unique feature of the extremal Kerr spacetime. We suggest the presence of a true geometric discontinuity as the underlying cause of a vanishing entropy.
PACS: 04.62.+v, 04.70.Dy, 04.20.Dw
Keywords: Bifurcate Killing Horizon, Extremal Kerr Black hole; Black hole Entropy
§ INTRODUCTION
One of the most remarkable ideas in black hole theory is the
analogy between the laws of classical black hole mechanics and the laws of thermodynamics.
Black hole thermodynamics has become an active area of research since
Bekenstein showed that the entropy of a black hole is proportional
to the area of the horizon. It is a well known fact now that that a black hole exhibits
an unusual similarity to a thermodynamic system.
Our analysis reveals a purely geometrical disparity between the extreme
and near extreme Kerr geometries, due to the singular nature of extreme regime.
In other words, the approach to extremality is not continuous.
The nature of the extremal Kerr metric is very different from other stationary solutions.
We focus on relating the entropy of extremal Kerr black holes strictly to their geometric structure.
Any classical method involving a finite number of steps used for the extremal case leads to subtle inconsistencies like a vanishing entropy and zero surface gravity while the area of the event horizon still remains positive. Using the near-extremal limit to evaluate the black hole entropy leads to major discontinuities.
Our aim is to understand this discontinuity working solely on pure geometric grounds.
The discontinuous nature of entropy during the transition from non-extremal to extremal black hole is directly connected with a discontinuous topological nature of the horizon. The entropy of extremal black holes can't be determined as a limit
of the non-extremal case. The geometry of the extreme black holes could shed some light on understanding black hole entropy in general.
Extremal black holes can't be regarded as limits of non-extremal
black holes due to this discontinuity. We suggest that the reason for this discontinuity is that non-extremal
and extremal black holes are topologically different and the switch from
one to another can't be done in a continuous manner. In this paper we study various properties of extreme
Kerr black holes to expose the underlying topological nature of the discrepancy between extreme and non-extreme regimes.
§ NO EVOLUTION OR TIME REVERSIBILITY
It is well known that Einstein's field equations are time-reversal invariant. A maximally extended spacetime includes apart from the black hole solution, its "time-reverse" case. In the non-extreme case, the extended spacetime ($\kappa\ne0$) possesses a bifurcate Killing horizon. In the extreme case (surface gravity $\kappa\eq0$), no distinct time-reverse equivalent exists, the black hole is time-independent everywhere and possesses a single degenerate Killing horizon. The surface gravity $\kappa$ of a black hole cannot be reduced to zero within a finite time. In the extended Schwarzschild spacetime, a white hole region becomes the time-reverse of the black hole region. The future event horizon, which separates two regions, is distinct from the past event horizon. The two regions intersect at the bifurcation two-sphere. The time-reverse of one region yields another region. In the non-extremal Kerr extended case, a new region is the time-reverse of the black hole, generating distinct patches in between the two horizons. The extremal Kerr extended spacetime does not contain such a distinct time-reverse patch but always duplicates of the same patch. There is no distinct time-reverse region and no distinct event horizon. The extremal Kerr black hole has no time-reverse equivalent or, in other words, it is time-independent everywhere. Furthermore, a naked singularity is composed of one single region and has no time-reverse region, has no event horizon and zero entropy.
In the extremal case, the Killing vector field on the horizon is
null on a timelike hypersurface intersecting the horizon and it is
spacelike on both sides.
The event horizon is determined by a Killing vector field whose causal properties
change from timelike to spacelike across the horizon. This Killing horizon becomes null on a timelike hypersurface surrounding the horizon. The presence of a Killing vector field which is timelike
in a region around the event horizon is a very peculiar and puzzling feature.
The horizon Killing field is spacelike except at the horizon itself.
In Boyer-Lindquist coordinates, the Kerr metric is given by
\begin{equation}
\begin{split}
ds^{2} = -\left(1-\frac{2Mr}{{\rho}^2}\right)dt^2 - \frac{4Mar{\sin^2{\theta}}}{\rho^2}dtd\phi \\
+ \frac{\Sigma}{\rho^2}{{\sin^2{\theta}}}d\phi^2 +
\frac{\rho^2}{\Delta}dr^2 + {\rho^2}d\theta^2. \hspace*{+2.4cm}
\end{split}
\end{equation}
\begin{eqnarray}
\rho^2 &=& r^2 + a^2{\cos}^2\theta. \\
\Delta &=& r^2 + a^2 - 2Mr.\\
\Sigma &=& \left(r^2 + a^2\right)^2 - a^2\Delta{\sin}^2\theta.
\end{eqnarray}
The horizons are situated at $\Delta = 0 $, i.e at,
\begin{equation}
r_{\pm} = M \pm \sqrt{M^2 - a^2}.
\end{equation}
The Kerr spacetime gets divided into three regions:
\begin{eqnarray}
\mbox{I}: \ r_+ < r < \infty \ , \\
\mbox{II}: \ r_- < r < r_+ \ , \\
\mbox{III}: - \infty < r < r_- \ .
\end{eqnarray}
Region I is the exterior, region II lies between
the two Killing horizons at $\Delta = 0$, and region III is an
asymptotic region. The mass of the black hole is $m$, its angular momentum $J = am$,
and its event horizon happens at $r = r_+$. The extremal case is obtained by setting
$a = m$, in which case there is no region II since the two horizons coincide. The
$a < m$ case describes the generic black hole, while $a > m$ describes a naked singularity.
The solutions for $\Delta\eq 0$ generate the inner and outer horizons:
$r_{\pm}=m \pm\sqrt{m^2-a^2}$.
The metric has two Killing vectors, $K^{\mu}\eq (\partial_t)^{\mu}\eq (1,0,0,0)$ and $R^{\mu}\eq (\partial_{\phi})^{\mu}\eq (0,0,0,1)$. We can construct a Killing vector $\chi^{\mu}$ as a linear combination of two vectors: $\chi^{\mu}\eq K^{\mu} +\omega_0\,R^{\mu}$ with $\omega_0$ being a constant.
Both $\rho^2$ and $\Sigma$ are positive but on the ring singularity they vanish.
In the non-extremal case, $\Delta$=$(r-r_{\p})(r-r_{\m})$.
$\Delta$ is positive for $r\!>\! r_{\p}$ or $r\!<\! r_{\m}$ and zero for $r\eq r_{\p}$ and $r\eq r_{\m}$.
But $\Delta$ is negative in the region $\Re$ representing $r_{\m}\!<\!r\!<\!r_{\p}\,\,$, always remaining between the horizons. Since any linear combination of the two Killing vectors remains spacelike, $\Re$ is nonstationary.
In contrast, the extreme case ($a\eq m$) has $\Delta$=$(r-a)^2$, positive everywhere except at the event horizon ($r \eq a$) where it is zero. If $\omega_0$=$\omega$ at $r\eq r_0$, we obtain the angular velocity of a zero angular momentum observer (ZAMO). A ZAMO observer exists at every point in the spacetime. The Killing vector $\chi^{\mu}$ is timelike or null. At each point, one can have a different timelike Killing vector (except at the horizon where it vanishes). There is no point where all Killing vectors are spacelike and no region is non-stationary. A general Killing vector field of the generic Kerr solution
is given by the linear combination
\begin{equation} \xi = \partial_t + K\partial_\phi \ , \end{equation}
where $K$ is a constant. Its norm is
\begin{eqnarray} ||\partial_t + K\partial_\phi ||^2 = \frac{1}{\rho^2}[
-\Delta + \sin^2{\theta}(a^2- \nonumber \\
4marK + AK^2)] \ \hspace*{+3.5cm}
\end{eqnarray}
This can vanish for all $\theta$ only if $\Delta = 0$, that is
on one of the horizons. It vanishes on the outer horizon $r = r_+$ if and only if
\begin{equation} K = \frac{a}{2mr_+}. \ \end{equation}
The linear combination defines the horizon Killing
vector field $\xi_{\rm hor}$. We have
\begin{equation} || \xi_{\rm horizon}||^2 = - \frac{r-r_+}{4m^2\rho^2r_+^2}f(a,r,\theta) \ ,
\label{norm} \end{equation}
\begin{eqnarray} f(a, r,\theta) = - 4m^2a^2\sin^2{\theta}(r-r_+) + \nonumber \\
+( 4m^2r_+^2 - a^2(r^2+2mr+a^2)\sin^2{\theta} + \nonumber \\
a^4\sin^4{\theta})(r-r_-) \ .
\hspace*{+3.5cm}
\end{eqnarray}
In $3+1$ decomposition, the Kerr metric becomes:
\begin{dmath}
ds^2 = N^2\,dt^2 + h_{ab}\,(dy^a +N^a \,dt)(dy^b + \\
N^b\,dt) = N^2\,dt^2 +h_{\phi\phi}(d\phi +N^{\phi}\,dt)(d\phi + \\
N^{\phi}\,dt) +h_{rr}\,dr^2 +h_{\theta\theta}\,d\theta^2 \\
=\dfrac{\rho^2 \,\Delta}{\Sigma}\, dt^2 - \dfrac{\Sigma\sin^2\theta}{\rho^2} \,(d\phi-\omega\,dt)^2 -
\dfrac{\rho^2}{\Delta}\,dr^2 -\rho^2\,d\theta^2 \\
\end{dmath}
\begin{eqnarray}
N^2 = \dfrac{\rho^2 \,\Delta}{\Sigma}, h_{\phi\phi}=
- \dfrac{\Sigma\sin^2\theta}{\rho^2}, h_{rr} \eq -\dfrac{\rho^2}{\Delta}, \nonumber \\
h_{\theta\theta}\eq -\rho^2 \word{and} N^{\phi} = -\,\omega\,. \nonumber \\
\hspace*{+5cm}
\end{eqnarray}
where we introduced the lapse function $N$, the shift vector $N^a$ and the induced metric $h_{ab}$.
Depending on $\Delta$, the solution depicts a black hole or a naked singularity. The lapse function is never negative. Since $\rho^2$, $\Sigma$ and $\Delta$ depend only on $r$ and $\theta$, the lapse function $N$ also depends only on $r$ and $\theta$ and is time-independent.
The induced $h_{ab}$ and the momentum conjugate $P^{ab}$ are time-independent everywhere. Thus, there is no time evolution of the phase space $[h_{ab},P^{ab}]$.
The coefficients $h_{\phi\phi}, h_{rr}$ and $h_{\theta\theta}$ can't be positive and they are independent of time. They all depend on $r$ and $\theta$: $\dot{h}_{ab}$ is zero everywhere. $N^{\phi}$ is also time-independent and it depends only on $r$ and $\theta$. The momentum conjugate to $h_{ab}$, $P^{ab}$ is also time independent. The extrinsic curvature $K_{ab}$ is:
K_ab≡12N (ḣ_ab-∇_b N_a-∇_a N_b) . gfd
with the components:
K_ϕθ=K_θϕ=12N(-∂_θ N_ϕ +12 N^ϕ∂_θ h_ϕϕ)
K_ϕ r=K_r ϕ=12N(-∂_r N_ϕ +12 N^ϕ∂_rh_ϕϕ)
N_ϕ=h_ϕϕ N^ϕ = Σ ω sin^2θρ^2
K≡h^ab K_ab =0 .
The momentum conjugate is:
P^ab=√(-h)16π (K^ab - K h^ab)= √(-h)16π K^ab
with the components:
P^ϕθ=P^θϕ= √(-h)16π K^ϕθ
P^ϕ r=P^r ϕ= √(-h)16π K^ϕ r
K^ϕθ = K_ϕθh_ϕϕh_θθ and K^ϕ r = K_ϕ rh_ϕϕ h_r r .
Obviously $P^{ab}$ is only dependent on $r$ and $\theta$ and consequently it is independent of time. There is only one single classical microstate $[h_{ab}(r,\theta), P^{ab}(r,\theta)]$ available.
In the non-extremal case, the region $\Re$ is non-stationary and the general $3+1$ decomposition in this region has $N^2$ positive. However, all $\Re$ components are time dependent. The phase space in $\Re$ is time dependent. The entropy is not vanishing, since there is more than one classical microstate hidden from the outside observer.
§ NO THERMALITY
A non-extremal spacetime with an outer and an inner horizon becomes
extremal as the outer horizon approaches the inner one.
The two horizons are in equilibrium at two different temperatures and
the temerature of the outer horizon approaches zero. Consequently,
no thermality is observed at the outer horizon.
In the extremal regime, the flux is the same
both outside and inside the horizon and approaches zero on the horizon,
with a vanishing temperature in the extremal case. There is a finite
discontinuity of flux in the extremal limit of the non-extremal
regime. There is no physically acceptable smooth transition from
the extremal regime as a limiting case of the non-extremal one and
the pure extremal case.
To understand the thermal nature of an extreme black hold, let us firstly find the probability flux across the horizon. To achieve this, we write the Kerr metric in Kruskal coordinates and set $a = m$ for the extreme regime:
ds^2 = - (1-2mr/ρ^2)dt^2 - 4m^2rsin^2θ/ρ^2dtdϕ+
Σ/ρ^2sin^2θdϕ^2 +
ρ^2/Δdr^2 + ρ^2dθ^2.
\begin{eqnarray}
\rho^2 &=& r^2 + m^2{\cos}^2\theta. \\
\Delta &=& \left(r-m\right)^2 \\
\Sigma &=& \left(r^2 + m^2\right)^2 - m^2\Delta{\sin}^2\theta
\end{eqnarray}
The horizon is located at $r = m$. Introducing the null coordinates $u = t - r_*, v = t + r_*$ with
\begin{eqnarray}
u &=& -M\cot U.\\
v &=& - M\tan V.
\end{eqnarray}
\begin{eqnarray}
r_* = \int\frac{r^2 + m^2}{\left(r-m\right)^2}dr = \nonumber \\
r + 2m\ln|r - m| - \frac{2m^2}{r-m}.\\ \hspace*{+5cm} \nonumber
\end{eqnarray}
\begin{equation}
\tilde\phi = \phi - \frac{1}{2m}t = \phi - {\Omega}t
\end{equation}
the surface $r = m$ appears at $v-u = -\infty$. The Kruskal coordinates $U,V$ are:
The future horizon is located at:
$U = 0, V < \frac{\pi}{2}$.
The Killing vector $\partial_t + \Omega\partial_{\phi}$ becomes
\begin{eqnarray}
\partial_t + \Omega\partial_{\phi} &=& \partial_u + \partial_v + \frac{\partial\tilde\phi}{\partial t}\partial_{\tilde\phi} + \Omega\partial_{\tilde\phi} \nonumber \\
&=&\frac{1}{m}\left[\sin^2{U}\partial_U - \cos^2{V}\partial_V\right]. \nonumber \\
\end{eqnarray}
To find the outgoing probability flux across the horizon, we find:
j^out/Ψ_ω(U)Ψ_ω(U) =
-iU^2/m∂_U[lnΨ_ω(U) - lnΨ_ω(U)] =
= ωU^2[1/(U +iϵ)^2 + 1/(U-iϵ)^2]
We have $\left(U\pm i\epsilon\right)^{-2} = PV(1/U^2) \pm i\pi\delta^\prime(U)$ and $U^2\delta^\prime(U) = -2U\delta(U) = 0$.
Taking logarithm of both sides we get the flux:
\begin{equation}
\ln j^{out} = \ln |N_\omega|^2 - 2\pi\omega m\delta(U).
\end{equation}
For any $N\neq 0$, $j^{out} = |N_\omega|^2$, no thermality is observed. The horizon behaves like a transparent membrane. The flux at $U=0$, is obtained by regularizing the delta-function. Using the limit $\epsilon \to 0$, we obtain:
\begin{equation}
j^{out} = |N_\omega|^2\exp\left(-2m\omega\frac{\epsilon}{U^2 + \epsilon^2}\right).
\end{equation}
The flux $j^{out} = |N_\omega|^2\exp\left(-2m\omega/\epsilon\right)\to 0 $ when $\epsilon\to 0$. A finite discontinuity of flux and no thermality is present at the horizon. This discontinuity is not coordinate related. For an extremal black hole, the proper radial distance from the horizon to any point close to the horizon outside or inside, is infinite. It is impossible for any incident particle state to cross the horizon. An extremal black hole cannot absorb or emit particle states. The coordinates $U_+,V_+$ remove the coordinate singularity at the horizon. However $U_+$ fails to be a proper distance. The discontinuity in $U_+$ is not physical. The flux needs an infinite proper time to become zero at the horizon when it is incident either from inside or outside. When equated to the Boltzmann factor, it implies an infinite $\beta$. This is equivalent to setting a zero temperature for the black hole.
In the extremal limit, when $r_+ \to r_-$, the effective temperature is zero and the emission probability is also zero. The flux is the same both outside and inside the horizon and zero at the horizon. This means a vanishing temperature, therefore a finite discontinuity in flux.
§ VANISHING ENTROPY
An interesting question appears. Where should we calculate the entropy: on (or nearby) the horizon or within the black hole (disc) itself?
Is the entropy created immediately after the gravitational collapse, or later during the black hole
evolution? The extremal entropy is independent of the black hole evolution and its internal configuration.
Can such a function be purely derived from geometric or topological considerations?
The third law of thermodynamics states that the surface gravity vanishing limit cannot be reached within a finite time. Cosmic censorship Conjecture forbids reducing the surface gravity to zero. It is impossible through a finite number of physical processes to reach the
zero limit surface gravity. However, the extremal Kerr black hole has a vanishing surface gravity therefore zero temperature.
Statistical mechanics describes the entropy of a system by the natural
logarithm of internal states count:
S = lnQ.
A microstate QŁ is a function of the system's macrostate. The entropy
is a function of these variables. An interesting
relation between (macroscopic) entropy and statistical thermodynamics (number of microscopic states) becomes evident.
Where there is only one microstate, Q = 1 and the entropy is zero, no disorder is present.
One single state corresponds to the extremal regime.
Extremal black holes can't be viewed as limits of non-extremal regime
because of this discontinuity.
Our assessment is that the non-extremal and
extremal regimes are topologically different and the discontinuity itself can be explained on geometric grounds.
An infinitesimal perturbation in mass, spin and horizon area can be written in Kerr metric:
\begin{equation}
dm~=~\frac{k}{8\pi} dA + \Omega_J da,
\label{bhtd}
\end{equation}
where $m$ and $a$ are the mass and the spin of the black hole.
The horizons form at $r_\pm = m \pm \sqrt{m^2 -
a^2}$. The surface gravity $\kappa$ is
$(r_+-r_-)/2r_+^2$. The Hawking temperature of
the black hole is $T_H=\kappa/2\pi$. If we compare it with the first law of
\begin{equation}
dE~=~ TdS - PdV~,
\label{td}
\end{equation}
and replace the term $PdV$ by $-\Omega_J da$, we find that the entropy $S$
must be identified with $A/4$ up to an affine
constant, which we set to zero by introducing the limit $S_{BH}
\rightarrow 0$ when $m \rightarrow 0$.
The entropy becomes:
\begin{equation}
S_{BH}~=~ \frac{A}{4G}~,
\end{equation}
where $S_{BH}$ is the Bekenstein-Hawking entropy, $G$ is Newton's constant and $A$ is
the event horizon area.
We see that in extremal regime, $T_H=0$ because $r_+=r_-$. The analogy between the equations (<ref>) and (<ref>) can't be done anymore. We also have
\begin{equation}
T_H^{-1}~=~\left(\frac{\partial S_{BH}}{\partial m}\right)_a~,
\end{equation}
where the right hand side diverges at the limit $T_H=0$.
The entropy can be now described a function of $m$ with a singularity at the
extremal limit $m=a$.
Let us assume the entropy being an arbitrary function of the area:
\begin{equation}
\end{equation}
\begin{equation}
\Delta S_{BH}~=~\frac {d~f}{d A} \Delta A~,
\label{sbh}
\end{equation}
where $\Delta S_{BH}$ and $\Delta A$ are the
change in entropy and area of the black
hole when the internal configuration is changed (for example, when a particle falls into it). After the particle crosses
the horizon, there is no information about
its state or status. We therefore assume that
it is equally probable for it to exist
or not. The minimum entropy change can be written
\begin{equation}
(\Delta S)_{min}~=~\sum_n~p_n \ln p_n~=~ \ln2~,
\end{equation}
where the summation over $n$ represents all possible
states of the particle.
The proper radius
$b$ for an incident particle with mass $\mu$ and
center of mass at $r_+ + \delta$ is $\int_{r_+}^{r_+ + \delta} \sqrt {g_{rr}} dr$. The
minimum change of black hole area
can be written
\begin{eqnarray}
(\Delta A)_{min}~=~ 2 \mu b~, ~~~~ {\mbox with }\nonumber\\
b~=~2 \delta^{1/2} \frac {r_+}{\sqrt {r_+-r_-}}~.
\end{eqnarray}
$g_{rr}~=~ (r-r_-)(r-r_+)/r^2$,
under the non-extremal condition $(r_+ - r_-\gg \delta)$.
However, in the extremal limit $(r_+\rightarrow r_-)$, the radius b becomes:
\begin{equation}
b~=~\delta~+~r_+\ln(r-r_+)|_{r_+}^{r_+ + \delta}~,
\label{bmin}
\end{equation}
diverging for any $\delta >0$.
For any small finite $\delta$, the proper
radius of the particle becomes infinite. Therefore we need to have $\delta~=~0$. For
$b~=~0$(point particle), we have $(\Delta A)_{min}~=~0$.
The change in entropy requires that
\begin{equation}
\left (\frac {\partial f}{\partial A} \right )_{r_+=r_-} \longrightarrow \infty~,
\end{equation}
with a discontinuity in the
extremal limit.
A semi-classical picture evaluates the gravitational path integral by employing the euclideanized black hole geometry.
If we consider the euclideanised case, the metric in $n$ dimensions near the
horizon is
\begin{equation}
ds^2~=~ N^2 d\tau^2 + N^{-2} dr^2 + r^2~d \Omega_{n-2}^2~.
\end{equation}
and the proper angle $\theta $
in the $r- \tau$ plane near the horizon becomes
θ =
∫_t_1^t_2 √(g_ττ) dτ/∫_r_+^r √(g_rr) dr =
= (N N')|_r_+ (t_2-t_1)
where $N'$ is the N differentiated with respect to $r$.
$N$ depends on the proper angle $\theta$:
\begin{equation}
(t_2-t_1) N^2~=~2\theta~(r-r_+) + O [(r-r_+)^2]~.
\end{equation}
In two dimensions, the metric near the horizon reduces to:
\begin{equation}
ds^2~=~d\rho^2 + \rho^2 d\theta^2~,
\end{equation}
with $\rho \equiv \sqrt{2 (r-r_+)/NN'}.$
To avoid a conical singularity at the horizon, the period of
$\theta$ is identified with $2\pi$, which corresponds to the
topology of a disc with zero deficit angle
in the $r-\tau$ plane. This can always be done for non-extremal
black holes, as $(NN')|_{r_+}$ is non-zero.
In the extremal case, the proper radius diverges and the proper angle tends to zero.
The disk topology is replaced with an annulus one. The topology of the transverse section in
either case is $S^{n-2}$. For a wrong periodicity, the geometry would have a singularity at the origin linked to the excess angle, like a cone structure that we may create out of a sheet of paper.
The conical excess angle becomes $2\pi$ and the topology is that of an annulus. The topology of the transverse section in
either case is $S^{n-2}$. An interesting feature of the solution is that the interior of the black hole is completely absent in the euclidean case. To know what is happening inside the black hole, the euclideanized spacetime is continued to an imaginary value of the radial coordinate near the horizon.
The Euclidean action leads to the entropy
\begin{equation}
S = \left( \beta \frac{d}{d\beta} - 1 \right) I_E .
\end{equation}
The euclideanized version imposes $t_Euclid = i t$ in the metric, with $r > r_+$ leading to a $R^2 \times S^{2}$ manifold topology. The polar coordinates $\{r, t_Euclid \}$ are defined on the $R^2$ with origin $r_+$ and periodicity $\beta$ of the Euclidean time angle $t_Euclid$. An interesting fact is that the Lorentzian manifold $r < r_+$ regions can't be included in the Euclidean solution.
The boundary contributions from the vicinity of the origin for the non-extremal regime are independent of $\beta$ and the canonical action is proportional to $\beta$, leading to Bekenstein-Hawking entropy $S = A / 4 G$. The $S^2$ contribution does not degenerate at the origin because $r = r_+$.
In the extremal regime, $r_+$ is infinitely far away from any point outside the horizon and consequently this point must be removed from the whole Euclidean manifold. The extremal black hole horizon is an infinite proper distance from any
stationary observer. There is an absence of a conical singularity at the origin of the manifold. The topology of the Euclidean extremal solution becomes $R \times S^1 \times S^2$. The periodicity of the the Euclidean time is not fixed. Because the origin doesn't exist any more, for any periodicity of the Euclidean time, no conical singularity is formed. The origin is not part of the manifold and the contribution from the vicinity of the origin vanishes. Consequently the entropy is zero.
We believe that this topology can explain what is actually happening with the entropy. If we consider the black hole as a
microcanonical ensemble, in a
Hamiltonian formulation, the action $I$ is proportional
to the entropy. A dimensional continuation of Gauss-Bonnet
theorem to $n$ dimensions gives us the action:
\begin{equation}
I~\propto~ \chi ~ A_{n-2}
\label{act}
\end{equation}
with $\chi$ the Euler characteristic of the euclideanised
$r- \tau$ plane and $A_{n-2}$ the area of the transverse
The black hole entropy becomes
\begin{equation}
S~=~\frac{ \chi A}{4G}~.
\end{equation}
In the non-extremal regime, $\chi =1$ corresponding to a disc and leading to the regular area law.
In the extremal case, we have $\chi=0$ (annulus), implying a
vanishing entropy.
§ DISCUSSION
Cosmic Censorship tries to solve the following problem: can a singularity exist in the absence of a horizon (naked singularity, not a topic of this paper)? Similarly, the existence of extreme black holes relies upon answering the question: can a spinning singularity exist without two separate horizons? This is an interesting problem.
Extremal Kerr black holes are stationary black holes whose inner and outer horizons coincide.
An extremal black hole has $\kappa \eq 0$ and no bifurcate Killing horizon. Also, the past and future horizons never intersect.
There is no physical process that can make an extremal black hole out of a non-extremal black hole.
A near-extremal black hole has a sort of potential barrier close to the horizon that prevents to reach extremal regime. These type of black holes represent the asymptotic limit of physical black holes.
Extremal black holes never behave as thermal objects. Their temperature is always undefined and their emission spectrum is non-planckian. Their non-thermal nature is a consequence of the geometric nature of the horizon.
All characteristics of the stress-energy tensor are different from
those in the non-extremal regime. A final thermal macrostate can't be achieved in a smooth continuous
way without violating the energy conditions. The extremal Kerr black hole cannot be produced through a process involving any finite number of steps without violating the weak energy condition. Consequently an extremal black hole cannot be produced through a finite number of processes or through standard gravitational collapse.
However, black hole entropy can be also defined as a measure of the observer's accessibility to information about the internal configuration hidden behind the event horizon <cit.>. This internal configuration can be depicted as the sum of points in the phase space, defined by a number of classical microstates $[h_{ab},P^{ab}]$. We can say that extremal Kerr black hole has zero entropy because it possesses a single classical microstate. Extremal black holes have zero temperature, surface gravity $\kappa\eq 0$ and zero entropy therefore they obey the strong version of the third law of thermodynamics.
We could say that an extremal Kerr black hole has zero entropy only because it has one single classical microstate. There is no continuous set of classical states and no time evolution. All metric components are time independent. Any observer should have complete access to the unique classical state found within the region beyond the event horizon.
A regular black hole has non-zero entropy because an event horizon hides its internal configuration and there are more than one internal states within its configuration. An extremal black hole has an event horizon but because the phase space is time independent, it doesn't hide more than one internal configuration. The extremal solution doesn't have a time-reverse equivalent or a bifurcate Killing horizon. The condition for a non-zero entropy is the existence of a bifurcate Killing horizon in the extended spacetime. Wald's general formula for the entropy of a black hole should be calculated at the bifurcation two-sphere not on the event horizon. The entropy is calculated as the integral of a geometric quantity over the spacelike cross section of the event horizon. The generated entropy at the event horizon represents the Noether charge of the Killing isometry that generates the event horizon itself. This geometric origin of the entropy suggest a deeper connection between the gravitational entropy, the topological structure of the spacetime and the nature of gravity. We suggest a purely local and geometrical character of the entropy. The different nature of the event horizon in the regular and extreme case requires different calculations of the entropy. The thermodynamic features are consequences of the topological structure of the space-time. The geometric nature of the boundary of the manifold determines the character of the entropy of the black hole. In the non-extremal regime, the proper distance and the coordinate distance between the inner and outer horizons are finite. In the extremal case, the proper distance between the event and Cauchy horizons becomes infinite, even though the coordinate distance vanishes. All these peculiar features are connected to a major property of extremal geometry: the absence of outer trapped surfaces within the horizon, which will be the subject of further work.
Wald also assumed a local, geometrical character of the concept
of black hole entropy, when he calculated the entropy at the bifurcation two-sphere. Entropy is depicted as a local geometrical quantity
integrated over a spacelike region of the horizon.
For a regular black hole, the zeroth law imposes that the
horizon of a black hole must be bifurcate and the surface gravity must be must constant and non-vanishing.
In the case of a degenerate Killing horizon there is no such bifurcation between the horizons. The area between horizons is completely absent.
§ CONCLUSION
The primary condition for a non-zero entropy is the existence of a bifurcate Killing horizon. This single criteria is enough. We analyzed a few features of the Kerr black hole that distinguish the extremal regime from the near-extremal one. From a thermodynamic point of view, there is a discontinuous nature of the entropy between the non-extremal and extremal cases, as the entropy of extremal regime is not the limit of the non-extremal one. While dual and string theory dual microstate counting predict non-vanishing entropy solutions for extreme regime, we suggested that the unusual nature encountered in the extremal limit using semi-classical methods represents a genuine relevant topological discontinuity and therefore the origin of a vanishing entropy. The entropy is zero, in agreement with semi-classical solutions, due to a degeneracy of the horizon geometry. The spacetime topology plays an essential role in the explanation
of intrinsic thermodynamics of the extreme black hole solution.
We conclude that the topology itself of the extreme black hole is enough to explain entropy in this regime. Moreover, the study of extreme black holes could play a crucial role in understanding gravitational entropy in general.
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hhr S. W. Hawking, G.T. Horowitz and S. F. Ross,Phys. Rev. D 51 (1995) 4302.
bek J. D. Bekenstein, Phys. Rev. D 7 (1973) 2333.
isr W. Israel, Phys. Rev. Lett. 57 (1986) 397.
swh S. W. Hawking, Phys. Rev. D 13191 (1976).
c1 B. Carter, Phys. Review. Lett. 141(4) (1966) 1242.
c2 B. Carter, Phys. Review. Lett. 174(5) (1968) 1559.
sc S. Chandrashekar, The Mathematical Theory of Black Holes,
Clarendon Press, Oxford (1983).
he S.W. Hawking and G.F.R. Ellis, The Large Scale Structure
of Space-Time, Cambridge University Press, Cambridge (UK) 1973
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of Chicago Press, Chicago 1984
bw J.M. Bardeen and R.V. Wagoner, Astrophys. J. 167 (1971) 359
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|
1511.00155
|
A first look at data from the NO$\nu$A upward-going muon trigger
R. Mina$^1$, E. Culbertson$^1$, M. J. Frank$^1$, R. C. Group$^1$, A. Norman$^2$, and
I. Oksuzian$^1$
The NO$\nu$A collaboration has constructed a 14,000 ton, fine-grained, low-Z, total absorption tracking calorimeter at an off-axis angle to an upgraded NuMI neutrino beam. This detector, with its excellent granularity and energy resolution and relatively low-energy neutrino thresholds, was designed to observe electron neutrino appearance in a muon neutrino beam, but it also has unique capabilities suitable for more exotic efforts. In fact, if sufficient cosmic ray background rejection can be demonstrated, NO$\nu$A will be capable of a competitive indirect dark matter search for low-mass Weakly-Interacting Massive Particles (WIMPs). The cosmic ray muon rate at the NO$\nu$A far detector is approximately 100 kHz and provides the primary challenge for triggering and optimizing such a search analysis. The status of the NO$\nu$A upward-going muon trigger and a first look at the triggered sample is presented.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
WIMPs captured by the gravitational field of the Sun that are slowed
through collisions with solar matter can accumulate in the solar core.
There, WIMP annihilation may produce neutrinos with much larger energy
than solar neutrinos. The signal would be an excess of high-energy
($>0.5$ GeV) neutrino events pointing back to the Sun <cit.>. The cleanest
signature at NO$\nu$A will be from $\nu _{\mu}$ charged-current scattering (CC) events producing
upward-going muons that can be reconstructed in the NO$\nu$A detector.
The large and unique NO$\nu$A far detector, with its excellent granularity
and energy resolution, and relatively low-energy neutrino thresholds,
is an ideal tool for these indirect dark matter searches.
Only the upward-going flux will be considered in order to suppress
the cosmic ray-induced muon background. The downward-going muon rate in the NO$\nu$A
far detector is approximately 100,000 Hz.
The neutrino flux from dark matter annihilation is model
dependent; however, energies from $\sim$0.5 GeV to many TeV should
be detected with high acceptance. For high-mass signal hypotheses,
NO$\nu$A will not be able to compete with the high acceptance of the IceCube
detector <cit.>. For lower-mass scenarios (below
$\sim$20 GeV) the Super-Kamiokande experiment currently has the
best sensitivity <cit.>. If an efficient upward-going
muon trigger and sufficient cosmic ray background rejection can be
achieved, NO$\nu$A will be competitive with Super–Kamiokande
for WIMP mass hypotheses below 20 GeV/c$^2$.
Neutrinos are produced by interactions of high-energy cosmic rays in the atmosphere,
and these atmospheric neutrinos comprise the primary component of the upward-going $\nu _{\mu}$ flux
at the NO$\nu$A far detector. It is well known that such neutrinos undergo
oscillations as they travel through the Earth <cit.>.
Therefore, isolating the upward-going muon signal should allow for an atmospheric
neutrino oscillation study at NO$\nu$A. Since atmospheric neutrinos represent a background
in the dark matter annihilation search, such a study is a natural preliminary step.
At NO$\nu$A, the neutrino analyses simply store events synchronous
with the NuMI beam. For non-beam exotic physics searches, so-called
data-driven triggers <cit.> are required to select events of interest.
Two data-driven triggers were developed for use at the NO$\nu$A
far detector: one that searches for long muon tracks originating from
outside the detector, and another that searches for shorter tracks
that are fully contained in the detector volume. Both triggers use
timing information for detector cell hits along the length of the track
to determine its direction. The effectiveness of the trigger at differentiating
upward- from downward-going muons in simulated events has been
demonstrated <cit.>.
This note will examine data from the first trigger which searches for
through-going muon tracks, and will demonstrate the effectiveness of
combining hit timing with event geometry information to isolate a small,
likely signal-rich component of the triggered sample. Producing such a
subsample is a necessary preliminary step to making an atmospheric neutrino
oscillation measurement and to performing the indirect dark matter search.
§ TRIGGERED SAMPLE
The upward-going muon trigger was first implemented and tested in August 2014,
but did not run in a stable configuration until December 2014. The triggered sample
examined in this note covers a period of 164 days from December 2014 to May 2015.
The total livetime of this sample is $\sim$84 days.
Over the period of this sample, the through-going trigger fired at a consistent rate of $\sim$1 Hz.
Each triggered event is 50 $\mu$s in length, so that the triggered sample
contains approximately 1 part in 20,000 of the total background activity during the exposure time.
Activity in the NO$\nu$A far detector is dominated by muons from cosmic ray interactions above and
around the detector <cit.>.
NO$\nu$A reconstruction software was run on the triggered sample to produce the
desired track and hit objects and to perform the necessary timing calibrations.
The reconstructed sample contained 4.36 track objects.
§ HIT TIMING AND LLR
NO$\nu$A's cm-scale spatial resolution allows determination of particle direction
by comparing timing for detector cell hits along a track. By applying several
timing calibration techniques, single hit time resolution for the Far Detector
has been improved to $\sim$10 ns <cit.>. This timing resolution is sufficient
to allow effective directionality determination using a timing-based classifier
called LLR <cit.>.
Cleanup requirements including track length, track linearity, and number of hits
in the track object improve the reliability of the LLR as a discriminator,
and are used in the trigger to improve the determination of directionality.
Applying those requirements used in the trigger to the reconstructed sample
with full timing calibration produced a timing-based candidate subsample of 16,000 tracks
that appear to be from upward-going muons.
§ EVENT GEOMETRY
The distribution of sine of the elevation
angle for each track in the timing-based candidate subsample (red) and all
tracks excluded from the subsample (black). Almost all candidates have
an elevation angle near 0, indicating they are nearly parallel with the ground.
A negative elevation angle indicates a downward-going track, while a positive angle
indicates an upward-going track.
The tracks passing cleanup and timing requirements are predominantly horizontal or slightly
upward-going, as shown in Fig. <ref>. The abundance of
mostly-horizontal tracks in this subsample is explained by the position
of the far detector on the surface, where energetic cosmic ray-induced muons travelling
slightly upward can penetrate the walls of the detector hall and
the thin layer of the Earth's crust surrounding it. At steeper angles
the horizon provides shielding from these upward-going cosmic ray muons,
explaining the fall-off in the subsample elevation angle distribution.
An example of an event that is prone to
a possible misreconstruction. Note the two overlapping muon tracks (the two long
colored lines in each plot) with similar
extent in the $z$-dimension (horizontal axis) and near coincidence in time (hits are colored by time). The detector produces
separate two-dimensional views of each event, and the 2D track objects from each view must
be merged to produce a full 3D object. Cases such as this produce ambiguity in
matching the 2D components between the views; is the correct matching (A1,B2) or
(A2,B1)? This class of events represents the largest component of the subsample when
timing and elevation angle requirements are applied.
Placing an additional requirement on the elevation angle at 10 degrees further reduced
the size of the candidate subsample to 1,051 tracks. Of those that remain, $\sim$75%
are not conclusively upward-going due to a possible misreconstruction in which two
unrelated but overlapping muon tracks create ambiguity in the reconstruction,
as shown in Fig <ref>. A simple geometric requirement was then applied to eliminate
this component while preserving the neutrino-induced muon signal. 255 candidates remained
after this requirement.
§ EVENT CATEGORIZATION
Through-going 105
Stopping 75
3D mismatch 34
Up-scattered cosmics 23
In-produced 1
Likely downward-going 1
Likely caused by timing miscalibration 14
Event topologies in the candidate subsample.
Each of the remaining candidate events was then examined visually and categorized. With the
exception of 2 events that were difficult to categorize because they had attributes of multiple
event topologies, all the events could be placed into seven categories based on event topology.
The categories are summarized in Table <ref>.
This event contains a long upward-going muon track
that appears to have been caused by a $\nu _{\mu}$ CC interaction
within the detector.
The most common event topology was through-going tracks, indicating muons that originated
outside the detector and traveled all the way through without stopping. Both tracks
in Fig. <ref> exemplify this topology. Stopping tracks
caused by muons originating below or to the sides of the detector
that stop within the detector volume are the second most abundant
component. The final signal-like event in the subsample contains a track that appears
to be caused by a neutrino interaction within the detector. This event is shown
in Fig. <ref>.
In this event, a slightly downward-going muon,
likely originating from a cosmic ray interaction outside the detector, enters from
the high-z extreme of the detector, scatters upward within it, and produces an
upward-going track.
The other categories correspond to events that are not signal-like.
34 events exemplifying the possible misreconstruction discussed in the previous section
passed the requirements (Fig. <ref>). 23 events appear to contain tracks from downward-going cosmic ray
muons that scattered upwards within the detector, as shown in Fig. <ref>.
14 of the through-going events were placed
into a separate category because they share extraordinary features
that may indicate a temporary problem in the timing system for one portion
of the detector; namely, they all have candidate tracks that have both ends
in one particular portion of the detector, and they all occured
during an isolated, continuous running period in which the rate of through-going
events was many times higher than the average. Finally, one event appears to contain a downward-going muon.
The distribution of sine of the
elevation angle for some of the categorized candidate events. The black vertical
line indicates the requirement at 10 degrees. Note that above 20 degrees (the blue vertical line),
there are no events containing cosmic ray muons that scattered upward within the detector.
The existence of events that probably contain upward-going tracks from
scattering of cosmic ray muons within the detector indicates that these cosmic ray muons
are also scattering outside the detector. The through-going and stopping muon subsamples
are likely contaminated by this background process, but it is not possible
to distinguish between signal muons created by neutrino interactions
outside the detector and those that were scattered upward in the rock around the detector.
However, all of the candidate tracks from cosmic muons scattering in the detector
had elevation angles below 20 degrees, so by requiring all tracks to have an elevation angle above 20 degrees, almost all
contamination by up-scattering cosmic ray muons should be eliminated, as shown in Fig. <ref>.
The remaining signal-like events after this requirement was applied were 43 through-going and 5 stopping muon events.
§ CONCLUSIONS
A first look at the triggered sample from the NO$\nu$A upward-going muon trigger
showed that in its first six months of stable running, the trigger selected
dozens of events with signal-like muon tracks. The extraction of this subsample
from the triggered sample involved a reduction by 6 orders of magnitude in the number of tracks,
and this was accomplished by combining techniques that use hit timing to determine track directionality
with simple geometry-based requirements and a visual scan. This effort revealed that
several backgrounds other than downward-going cosmic ray muons contaminate the sample,
and new requirements were developed to minimize the acceptance of these backgrounds. These techniques
will allow studies of atmospheric neutrino oscillations and, ultimately, an indirect dark matter
search at NO$\nu$A.
This work demonstrates that NO$\nu$A is capable of isolating a sample
that is likely rich in neutrino-induced upward-going muons for the through-going and
stopping muon event topologies. A similar effort that examines data from the other upward-going
muon trigger will reveal whether a signal-rich sample can be isolated for the fully-contained
event topology.
This conference presentation was made possible by a travel award from the
American Institute of Physics Society of Physics Students. Additional
financial support was provided by the Jefferson Trust, the UVa Physics
Department, and the Fermilab Particle Physics Division. The authors
also acknowledge that support for this research was carried out by the
Fermilab scientific and technical staff. Fermilab is Operated by Fermi
Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the
United States Department of Energy. The University of Virginia
particle physics group is supported by DE-SC0007838.
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|
1511.00436
|
Radio properties of CSS and GPS radio sources
M. Orienti
INAF-IRA, Via Gobetti 101, 40129 Bologna, Italy
Compact steep spectrum (CSS) and GHz-peaked spectrum (GPS) radio
sources represent a large fraction of the extragalactic objects in
flux density-limited samples. They are compact, powerful radio
sources whose synchrotron peak frequency ranges between a few
hundred MHz to several GHz. CSS and GPS radio sources are currently
interpreted as
objects in which the radio emission is in an early evolutionary
stage. In this contribution I review the radio properties and the
physical characteristics of this
class of radio sources, and the interplay between their radio emission and the
ambient medium of the host galaxy.
§ INTRODUCTION
Compact steep spectrum (CSS) and GHz-peaked spectrum (GPS) radio
sources are powerful (P$_{\rm 1.4\ GHz}>$10$^{25}$ W/Hz) and compact
objects with angular sizes not exceeding 1 – 2 arcsec.
The main peculiarity of these
objects is the convex synchrotron radio spectrum that peaks
around 100 MHz in the case of CSS sources, and at about 1 GHz in the
case of GPS objects, or even up to a few GHz in the sub-population of
high frequency peakers (HFP) defined by Dallacasa et al. (2000).
Above the peak frequency the spectrum is steep
with a spectral index $\alpha \sim
0.7$ ($S_{\nu} \propto \nu^{- \alpha}$).
Depending on both the frequency and the flux-density limit
of the catalogues used, the CSS/GPS samples are dominated by different
sub-classes of objects. Bright CSS and GPS samples have been selected
from the 3C, PW and 1-Jansky catalogues (see, e.g., Spencer et al. 1989,
Fanti et al. 1990, Stanghellini et al. 1998). On the other hand, deep
catalogues, like B3, FIRST, WENSS and AT20GH, were used for selecting weak
samples (e.g., Fanti et al. 2001, Snellen et al. 1998, Kunert et
al. 2002, Hancock et al. 2010). In the last decades other samples
of CSS/GPS candidates were constructed using different selections tools like the radio
morphology, optical counterpart, compact linear sizes (e.g., COINS sample, Peck &
Taylor 2000; CSS-VIPS sample,
Tremblay et al. 2009; CORALZ, Snellen et al. 2004), as well as
polarization properties (Cassaro, Dallacasa & Stanghellini 2009).
Statistical analysis of CSS/GPS samples pointed out an empirical
anti-correlation between the peak frequency of the spectrum and the
linear size (O'Dea & Baum 1997). This has been interpreted either in terms
of synchrotron-self absorption related to the compact dimension of the
sources (e.g., Snellen et al. 2000, Fanti 2009, Orienti & Dallacasa
2008a) or due
to free-free absorption
from an ionized medium enshrouding the radio emission (e.g., Bicknell
et al. 1997, Tingay et al. 2015, Callingham et al. 2015), although a
combination of both
mechanisms may take place
(e.g., Orienti & Dallacasa 2008a).
CSS and GPS radio sources represent a significant fraction (15% – 30%
depending on the frequency) of the sources in flux-density limited
catalogues, opening a debate about their nature.
Fanti et al. (1990) investigated
whether the compact size is a result of projection effects, but they
concluded that this was unlikely, leaving room only for a minority
($<$25%) of large objects foreshortened by geometrical effects.
The intrinsically compact size of CSS/GPS is interpreted mainly in
terms of Youth: these sources are small because they are still
in an early stage of their evolutionary path, and may become/develop
into Fanaroff-Riley type-I/II (FRI/FRII, Fanaroff & Riley 1974) radio
sources (e.g., Fanti et al. 1995, Snellen et
al. 2000, Alexander 2000, Perucho 2015). Strong supports to the Youth Scenario came from the estimate of the kinematic
age by the determination of the hot spot separation velocity in a
handful of the most compact objects (Polatidis 2009, Giroletti &
Polatidis 2009, and references therein, Polatidis & Conway 2003, and
references therein), as well as from the radiative age (Murgia 2003,
Murgia et al. 1999, Nagai et al. 2006).
The alternative scenario that postulated the presence of an
exceptionally dense medium able to frustrate the jet growth was not
supported by multiband observations which pointed out that the gas of
their host galaxies are similar to those of extended FRII sources
(e.g., Fanti et al. 1995,
Fanti et al. 2000, Siemiginowska et al. 2005).
In the next Sections I will briefly review the observational and
physical properties of CSS and GPS radio sources. Radio properties
are presented in
Sections 2. In Section 3 I describe the physical parameters, such as the
luminosity and the magnetic field, and how they evolve.
In Section 4 I discuss the duty cycle of the
radio emission, while in Section 5 and 6 I present the characteristics
of the ambient medium and their role in producing the observed source
asymmetries. A brief Summary is
presented in Section 7.
Throughout the paper I assume H$_0$ = 71 km s$^{-1}$ Mpc$^{-1}$,
$\Omega_{\rm M}$ = 0.27, $\Omega_{\Lambda}$ = 0.73 in a flat
Universe. The spectral index is defined as $S_{\nu} \propto \nu^{- \alpha}$.
§ RADIO PROPERTIES
§.§ Morphology
Due to their compact size, CSS/GPS sources appeared unresolved in
single-dish observations, and only the advent of interferometers with
sub-arcsecond resolution could pinpoint their radio morphology.
CSS and GPS are divided into three main morphological classes: 1)
symmetric (i.e. two-sided) structures; 2) core-jet structures; 3)
complex morphology.
Symmetric objects have a two-sided radio structure resembling
a scaled-down FRII radio source (Fig. <ref>).
The main ingredients are mini-lobes
and hot spots. Sometimes, a weak component hosting the core is
present, and, depending on their size, CSS/GPS may be
termed as “compact symmetric objects” (CSO) if they are smaller
than 1 kpc,
or “medium-sized symmetric objects” (MSO) if they extend up to 10 –
15 kpc (Fanti et al. 2001).
However, a large fraction of “symmetric” sources have a very
asymmetric two-sided morphology, where one side of the source is much
brighter than the other (e.g. Saikia et al. 2003, Rossetti et
al. 2006). When the core is
detected it is usual to find that the brighter lobe is the one closer
to the core. This is opposite to what is expected in presence of
geometrical effects, suggesting some interaction between the jet and
the ambient medium (see Section <ref>).
In two-sided objects the core
usually represent a very small fraction (a few per cent) of the total
radio emission of the source indicating the absence of beaming effects
(e.g. Wilkinson et al. 1994).
On the contrary, in radio sources with a core-jet and complex structure the core
dominates the radio emission, suggesting the presence of significant
boosting effects. In addition core-jet structures are usually found in
CSS/GPS which are optically identified with quasars (e.g., Rossetti et
al. 2004, Orienti et al. 2006), supporting the role of projection effects.
As in the case of core-jet structures, also complex morphologies are
usually caused by boosting effects, at least in high-luminosity
sample. This does not seem to hold in low-luminosity samples where a high
fraction of sources ($\sim$30%) have weak
extended emission and distorted structures which are likely intrinsic
(Kunert-Bajraszewska et al. 2010).
psfile=morpho_1.ps voffset=-170 hoffset=20 vscale=50
psfile=morpho_2.ps voffset=-170 hoffset=280 vscale=50
Examples of a symmetric “two-sided” CSO (left), and
of an asymmetric one (right). Adapted from Orienti et
al. 2004
§.§ Variability
The spectral behaviour and the flux density variability is
investigated by multifrequency observations, closely simultaneous when
possible, carried out at various epochs. In past works CSS/GPS objects
considered the least variable extragalactic radio sources (O'Dea
1998). However, long-term monitoring campaigns found different results
depending on the different sub-classes considered (CSS, GPS, HFP, quasars,
It turned out that many GPS/HFP sources are picked up with a convex
spectrum only during flaring events, when the radio emission is
dominated by the jet, while in their average state they possess a flat
spectrum (Torniainen et al. 2005, Tinti et al. 2007).
In particular, high fraction of CSS radio galaxies are not variable, while
only $\sim$30% of GPS/HFP galaxies preserve the convex spectrum
(Torniainen et al. 2007, Orienti et al. 2010, Hancock et al. 2010).
The majority of the CSS, GPS, and HFP quasars show
significant flux density and spectral variability
(Mingaliev et al. 2012, Orienti et
al. 2007).
Spectral changing and flux density variability do not always imply
that the source is part of the blazar population, rather than a
genuine CSS/GPS/HFP object. In fact, changes in the radio spectrum may
be a direct consequence of the source expansion (e.g., Tingay & de
Kool 2003). In newly born radio sources, the evolution time-scales can
be of the order of a few tens of years. Changes in the radio spectrum
of such young objects can be appreciable after the short time (5 – 10
yr) elapsing between the observing epoch. If the variability is due
to the source expansion we expect that the peak
shifts to lower frequencies, the flux density in the optically-thin
regime decreases, while that in the optically-thick part of the
spectrum increases. This behaviour has been observed in the HFP
RXJ1459+3337 (Orienti & Dallacasa 2008b), as well as in a handful of
HFP/GPS sources (Dallacasa &
Orienti 2015, Orienti et al. 2010), although some additional
variations in the free-free optical depth may be present (e.g., the GPS
PKS 1718-649, Tingay et al. 2015).
§.§ Polarization
CSS/GPS objects are weakly polarized. Multifrequency
polarimetric measurements of a sample of CSS/GPS radio sources (Fanti
et al. 2004, Cotton et al. 2003) show that very compact objects ($<$1
kpc) are unpolarized or strongly depolarized, and the fractional
polarization is strictly related to the frequency: the lower the
frequency, the stronger the depolarization. Furthermore, the
fractional polarization does not increase gradually with the source
size, but there seems to be a discontinuity at a “critical size”
(the so-called Cotton effect),
that is about 6 kpc at
1.4 GHz and moves down to about 1 kpc at higher frequencies.
The strong
depolarization of the most compact objects may be related to the
interstellar medium of the narrow line region (NLR) which acts as a
Faraday Screen, depolarizing and/or rotating the polarized signal. The
“amount” of depolarization and/or rotation depends on both the
inhomogeneities of the ambient medium and the distribution of the
magnetic field. Support to the presence of a dense and inhomogeneous
medium in front of the radio source comes from the large rotation
measure (RM) estimated for those sources with some polarized
emission. RM of the order of 1000 rad m$^{-2}$ or even higher are
commonly observed (e.g., O'Dea 1998, Cotton et al. 2006, Rossetti et
al. 2008, Mantovani et al. 2013).
A different result was found by Mantovani et al. (2009) who studied a
complete sample of CSS with multifrequency single-dish
observations. In this case no drop in the polarization was found for
the most compact sources. However, this apparent contradiction is
likely due to the contamination by geometrically-foreshortened
quasars. In fact, if the galaxies are considered separately, the Cotton effect is visible again, while the fractional polarization
in quasars seems independent from the linear size. In addition, quasars
have higher fractional polarization than galaxies, while galaxies
experience larger RM. In HFP quasars the high
fractional polarization is associated with low RM and high flux
density variability (Orienti & Dallacasa 2008c), similar to what is
found in blazars (Mantovani, Bondi, & Mack 2011).
Sub-arcsecond resolution
observations point out higher RM than those estimated by
low-resolution single-dish observations suggesting the presence of
blended components.
Another intriguing aspects pointed out by multifrequency observations
is an increase of the polarized emission at low frequency
(re-polarization). This may be explained either in terms of multiple
unresolved components with different spectral characteristics that
dominate at different frequencies, or in presence of variability
(Mantovani et al. 2009, Orienti & Dallacasa 2008c). In this case the
RM does not follow the $\lambda^{2}$ correlation (see e.g., Burns 1966 ).
A remarkable result found is that CSS are more asymmetric in the
polarization of the outer
lobes than the extended galaxies (Saikia & Gupta 2003).
The high incidence of polarization
and morphological asymmetries observed in CSS and GPS is likely an
indication of jet-gas interaction which is more probable when the
radio jet is still piercing its way through the dense medium of the
host galaxy (see Section <ref>).
§ PHYSICAL PROPERTIES
In a scenario where radio sources grow in a self-similar way, the
evolution of each radio object originated by an AGN depends on its
linear size.
The determination of the physical properties in objects at the
beginning of their evolution is crucial for setting tight constraints
on the initial conditions of the radio emission.
The source size, flux density, and the peak frequency are parameters
that can be easily derived from the observations and then used as
a starting point to determine the physical properties of the
The existence of a
relation between the rest-frame peak frequency and the projected
linear size (e.g. O'Dea & Baum 1997) indicates that the mechanism responsible
for the curvature of the spectrum is related to the source dimension,
and thus to the source age. Interestingly, some of the most compact
and asymmetric sources seem to depart from this relation. However, this
is likely due
to the presence of several sub-components that are responsible for different
part of the total radio spectrum: one bright and compact hot spot
dominates the radio emission, overwhelming the contribution from the
extended structures (e.g. J1335+5844, Orienti & Dallacasa 2014).
Another important relationship to investigate is between the peak flux
density and the peak frequency, since they provide constraints on the
magnetic field in case the spectral curvature is due to SSA.
The analysis of the peak flux density as a
function of the peak frequency in CSS/GPS/HFP sources from bright
samples suggests a segregation between sources identified with galaxies
and quasars. When only galaxies are considered
there seems to be an anticorrelation between the peak flux density and
the peak frequency, as expected if the spectral turnover is due to
SSA, although FFA cannot be completely discarded. On the contrary,
this anticorrelation does not hold in case of
quasars. They have peak frequency in GHz regime, while the
peak flux density covers three order of magnitudes, independently of
the peak frequency, suggesting a different nature for the majority of
galaxies and quasars (Orienti et al. 2010).
§.§ Magnetic field
The direct measurement of the magnetic field in extragalactic radio
sources is a difficult task to carry out. An indirect way to estimate
the magnetic field is to assume that the radio source is in
minimum energy condition corresponding to a near equipartition of
energy between the
radiating particles and the magnetic field (Pacholczyk 1970).
Although this condition is assumed in many evolutionary models,
there is no a priori reason for believing that magnetic fields in
radio sources are in equipartition.
As mentioned earlier, a direct measurement of the magnetic field from
observable quantities is obtained by means of the spectral parameters. If the
spectral peak is produced by SSA, we can compute the magnetic
field $H$ by using observable quantities only:
\begin{equation}
H \sim f(\alpha)^{-5} \theta^{4} S_{\rm p}^{-2} \nu_{\rm p}^{5} (1+z)^{-1}
\label{magnetic}
\end{equation}
where $\theta$ is the source solid angle, $\nu_{\rm p}$ and
$S_{\rm p}$ are the peak frequency and peak flux density,
respectively, $z$ is the redshift, and $f$($\alpha$) is a function
that depends weakly on the spectral index (Kellermann & Pauliny-Toth
Scott & Readhead (1977) and Readhead (1994) computed the magnetic
field for sources of low-frequency spectral turnovers close in value
to the observing frequency and found that the magnetic fields inferred
directly from the spectrum were within a factor of 16 of the
equipartition values. However, there are no systematic studies of
sources with spectra peaking at higher frequencies, i.e. objects
younger than those in Scott & Readhead (1977).
The main difficulty in applying this method has been the
uncertainty in determining source component parameters at the turnover
frequency, which results in a limited accuracy of the magnetic field
estimates. However this method may be used for GPS/HFP sources. The
peak frequency around a few GHz gives the possibility to sample both
the optically-thick and -thin part of the spectrum by multifrequency
high resolution VLBA observations, leading to a fairly accurate
estimate of the peak parameters.
The peak magnetic fields estimated for the components of HFP radio
sources turned out to
be in good agreement with those derived by assuming minimum energy
conditions, supporting the idea that in general young radio sources
are in minimum energy conditions and their spectral turnover is caused
by SSA (Orienti & Dallacasa 2008a, 2008b, 2014). However, there are a
few exceptions where the peak magnetic field is orders of
magnitude higher than the equipartition value. The analysis of the
optically-thick part of the spectrum turned out to be more inverted
than the limit value for SSA ($\alpha < - 2.5$), indicating that the
spectral peak is due to FFA. Therefore the magnetic field determined
by the peak values is meaningless.
Depending on the size of the sources, the estimated magnetic fields
range from a few 150 mG in the most compact components of HFP radio
sources down to 0.1 mG in the lobes of CSS (Fanti et al. 2001,
Dallacasa et al. 2002, Orienti et al. 2006). The anticorrelation
between the magnetic field strength and the linear size is in
agreement with what is expected
in case the source is adiabatically expanding. However, when the
single source components are considered, it emerges that the field
intensities found in the various components of the same object can
vary up to an order of magnitude (Orienti & Dallacasa 2012). Such differences may arise from
asymmetries in the source propagation, for example when
the two sides
experience a different environment (see Section <ref>),
indicating that simple
self-similar evolution models may be not adequate to describe the
radio source growth (e.g., Sutherland & Bicknell 2007).
§.§ Luminosity evolution
Several evolutionary
models[The evolutionary models considered in this review try
to explain how an
individual radio source grows and evolves, without taking into account any cosmological implication.] (e.g. Fanti et al. 1995; Readhead et al. 1996; Snellen et
al. 2000) were proposed to describe
how the physical parameters (i.e. luminosity, linear size and
velocity) evolve as the radio emission grows within the host galaxy.
The majority of the proposed models predict an increase of the
luminosity and a decrease of the jet advance speed when the radio
emission is still embedded within the dense medium of the NLR. Then,
as the radio emission emerges from the NLR the luminosity is expected
to decrease, while the jet advance speed should not vary
significantly. These predictions should be validated by statistical studies of
samples of CSS/GPS/HFP spanning a large
range of linear size.
O'Dea (1998) studied the radio luminosity at 5 GHz as a function of
the projected linear size, but no correlation was found. This may be
related to the similar radio power of the selected sources.
To determine how the luminosity evolves as the source grows, Orienti
& Dallacasa (2014) studied a sample of 51 bona fide young radio
sources with an unambiguous detection of the core region, and spanning
a wide range of linear size, from a few pc up to tens of kpc, for
high-luminosity radio sources. To get rid of possible
boosting/projection effects
that may contaminate the estimate of the physical parameters, only
objects optically associated with galaxies were considered when
searching for empirical relations.
The analysis of the source luminosity at 375 MHz versus the linear size points
out two different relations depending on the source size: sources
smaller than a few kpc increase their luminosity as they grow, while
larger sources progressively decrease their luminosity, in agreement
with the evolutionary models (Fig. <ref>).
The smallest sources reside within the
innermost region of the host galaxy, where the dense and inhomogeneous
ambient medium favours radiative losses. As the radio source expands
on a kpc scale, it experiences a smoother and less dense ambient
medium and adiabatic losses dominate.
Kunert-Bajraszewska et al. (2010) extended the analysis of the
radio power as a function of linear size by including a sample of
low-luminosity CSS objects having 1.4-GHz luminosity comparable to
that of FRI. They found a clear distinction between high-power and
low-power objects: the former seem to follow an evolutionary path
similar to that expected by the models. The latter are located below
this path and they may represent short-lived objects with a different
psfile=lum_ls_css.ps voffset=-250 hoffset=-5 vscale=58 hscale=58
Total luminosity at 375 MHz versus the linear size for the
sources of the sample selected by Orienti & Dallacasa
(2014). Filled circles are galaxies, while empty triangles are
quasars. Crosses represent the median values of the total luminosity
and linear size, for galaxies only, separated into different bins. As a
comparison, in the bottom right corner there is the evolutionary
trend expected by the model developed in Snellen et al. (2000).
§ THE LIFE-CYCLE OF THE RADIO EMISSION
It is nowadays clear that powerful (L$_{\rm 1.4~GHz} >
10^{25}$ W/Hz) radio sources are a small fraction of the
AGN generally associated with ellipticals, suggesting that the radio activity is
a transient phase in the life of these systems. The typical age of
active phase in radio sources is about 10$^{7}$ – 10$^{8}$ years,
which is followed by a relic phase which is roughly one order of
magnitude shorter (Parma et al. 2007).
The onset of radio emission is currently thought to be
related to mergers which provide fuel to the central
AGN. However, the reason why and when the radio emission switches off
is still an open question. The excess of young objects in flux-limited
samples suggests the existence of short-lived objects
unable to become FRII, and additional ingredients,
like the recurrence of the radio emission (e.g., Czerny et al. 2009), or the
interplay between the source and the environment,
must be considered (see Section <ref>).
Support to the existence of short-lived objects
come from statistical studies of the ages of CSO which peak at about
500 years (Gugliucci et al. 2005), and of the sub-class of
low-luminosity CSS radio sources (Kunert-Bajraszewska et al. 2010).
If the supply of new relativistic particles turns
off, the radio emission fades rapidly due to the severe energy
losses and the radio spectrum steepens fast making these sources
under-represented in flux-limited catalogues. Indeed, only a few
objects have been suggested as faders so far, based on the absence
of active regions (Kunert-Bajraszewska et al. 2005, 2006), and the
distribution of spectral index found steep across the whole source,
like in the case of PKS
1518+047 (Orienti, Murgia, & Dallacasa 2010).
A different situation is the recurrence of the radio emission in an
AGN. In this case a “young” radio source, with new activity regions
like the core and hot spots, is present close to the fossil of a
previous epoch of the radio emission. A clear example of intermittent
radio activity is the FRII radio galaxy B0925+420 where three
different episodes of jet activity have been observed (Brocksopp et
al. 2007).
Extended emission on the kpc-scale and beyond was discovered in the
GPS galaxy J0111+3906 (Baum 1990), and interpreted in terms of the
relic of a past radio activity which occurred about 10$^{7}$-10$^{8}$ years ago.
Recently, a remnant of about 160 kpc in size from an earlier stage of
activity was found in the
CSS galaxy B2 0258+35, suggesting that the time between subsequent
phases of activity in this source is about 10$^{8}$ (Shulevski et
al. 2012, Brienza et al. 2015).
Following the model by Czerny et al. (2009), the restarting of the
radio emission may occur on much shorter (a few
thousand years) time scales. This is the case of the HFP
sources J1511+0518 and OQ 208 where a relic from the past activity is
found at about 50 pc from the reborn object (Orienti & Dallacasa
2008, Luo et al. 2007), indicating that, at least, in some objects the
duty-cycle of the radio emission occurs on time-scale of 10$^{3}$ –
10$^{4}$ years.
Following the evolutionary path, CSO should evolve into FRII radio
galaxies with ages of 10$^{7}$ –10$^{8}$ years before the radio emission
enters in the relic phase. However, it is
possible that not all the CSO would become FRII, and a population of
fading short-lived objects under-represented in flux-limited catalogue
is expected. If the interruption of the radio activity is a temporary
phase and the radio emission from the central engine will restart
soon, it is possible that the source will appear again as a CSO
without the severe steepening at high frequencies. If this does not
happen, the fate of the fading radio source is to emit at lower and lower
frequencies, until it disappears at frequencies well below the MHz
regime (Fig. <ref>).
psfile=evolutionary_path.ps voffset=-270 hoffset=110 vscale=50
The evolutionary path of the radio emission. Young CSO (top left, image adapted from Orienti et al. 2004) may become
either a classical large FRII (bottom
left, image adapted from Mack et al. 2009) or a relic in the case
of the activity phase switches off
soon after its onset (bottom right, image adapted from Murgia
et al. 2011). If the central engine
goes through another active phase, a newly-born bright and compact
object can be observed close to the relic of the previous activity
(top right, image adapted from Tinti et al. 2005).
§ AMBIENT MEDIUM
CSS and GPS radio sources are usually found in early-type gas-rich
galaxies. The presence of significant amount of gas in young radio
galaxies is supported by a larger incidence of HI absorption in these
objects (Vermeulen et al. 2003, Pihlström, Conway, & Vermeulen 2003, Gupta et
al. 2006) compared to
what is typically found in old and larger radio sources (Morganti et
al. 2001).
This dense medium is likely the result of the merger that
triggered the radio source (Morganti et al. 2004a). The knowledge of
the distribution of the gas, either settled in a circumnuclear structure like a disk or
a torus, or inhomogeneously distributed in clouds, provides important
information on the environment in the innermost region
of AGN and its role in the radio jet evolution.
Pihlström et al. (2003) found an empirical anti-correlation between
the HI column density, $N_{\rm HI}$ and the source linear size in a sample of 41
CSS/GPS sources: smaller sources ($<$0.5 kpc) have larger HI column
densities than the larger sources ($>$0.5 kpc). This result suggests
that the HI gas is likely settled in a circumnuclear disk/torus with a
radially declining density, and
the absorption takes place only
against the receding lobe. It is worth noting that the $N_{\rm
HI}$-$LS$ anticorrelation does not seem to hold in the most compact
HFP objects (Orienti, Morganti, & Dallacasa 2006), which highly
deviate from the trend by Pihlström et
al. (2003). The absence of high HI column density in very compact
sources can be explained by both the orientation and the extreme
compactness of the sources in a disk/torus scenario, in which our line
of sight intersects the circumnuclear structure in its inner region
where the low optical depth may be due to high spin and kinetic
The presence of a gas distribution consistent with a circumnuclear
disk/torus is supported by high-resolution VLBI observations that
could trace either the atomic hydrogen across the source (e.g., 4C 31.04,
Conway 1999; PKS 1946+708, Peak & Taylor 1999; B2352+495, Araya et
al. 2010)
or the ionized gas via free-free absorption (J1324+4048 and
J0029+3457, Marr et al. 2014; J0111+3906, Marr, Taylor, & Crawford
2001), and by high-sensitivity observations of key molecular species
such as CO, HCN, and HCO$^{+}$ (e.g., 4C 31.04, Garcia-Burillo et
al. 2009; OQ 208, Ocaña-Flaquer et al. 2010).
Not all the ionized, atomic, and molecular gas is organized in a
circumnuclear structure. As a consequence of the recent
merger/accretion events experienced by the host galaxy, we expect the
presence of unsettled, clumpy gas inhomogeneously distributed
particularly in the innermost region. However, a secure prove of such
clouds is difficult to obtain usually due to their low optical depth and
small filling factor. However, if the cloud impacts with the jet
temporarily confining its expansion. Shallow and broad HI absorption
lines suggesting a jet-cloud interaction, were found in young or
restarted objects, but the lack of angular resolution prevented a
reliable interpretation of their origin, either circumnuclear or
off-nuclear (Morganti, Tadhunter & Oosterloo 2005).
An outstanding example is
represented by the young, restarted GPS
4C 12.50, where VLBI observations locate the HI outflow of about
10$^{4} M_{\odot}$ at
about 100 pc from the
nucleus where the radio jet interacts with the ISM, as well as around
the associated radio lobe (Morganti et al. 2013). A possible molecular
counterpart of the atomic outflow was found by Dasyra & Combes (2012)
by deep CO observations. Interestingly, in
the same source HI is observed in absorption also against the other
lobe and may cause the bending of the jet (Morganti et
al. 2004b). These results indicate that, despite the small filling
factor of the clumpy medium, jet-cloud interaction can take place and
may be responsible for some asymmetric radio sources observed. This
may be the case of the two highly asymmetric CSS sources 3C 49 and 3C 268.3,
in which HI absorption is observed only against the brightest lobe,
which is also the closest to the core, although the non-detection in
the other lobe might be due to sensitivity limitation (Labiano et
al. 2006).
§ ASYMMETRIES
As mentioned in Section <ref>, a large fraction of CSS/GPS
sources have a very asymmetric structures (e.g., Saikia et
al. 2003). A significant fraction
($\sim$50%) of
asymmetric sources have the brighter lobe that is also the closer to
the core, which is opposite to what is expected if the source is not
on the plane of the sky and some beaming effects and path delay are
present. The brighter-when-closer trend
suggests that the two jets are piercing their way through an
inhomogeneous medium as pointed out by HI observations (see Section
<ref>). The interaction between the advancing jet and a
clumpy medium may enhance the luminosity due to high radiative losses
which become predominant with respect to the adiabatic
ones. Jet-cloud interactions should be more frequent during the first
stages of the radio emission, when the jet is piercing its way
through the dense and inhomogeneous gas of the host
galaxy. Fig. <ref> shows the flux density ratio versus the
linear size of a sample of CSS/GPS and FRII (Orienti & Dallacasa
2008d). Sources larger than $\sim$15 kpc are more symmetric than
the smaller ones.
The enhancement of the flux density may explain part of the high
number counts of CSS/GPS objects in flux density limited
samples. Furthermore, although the jet-medium interaction may not
frustrate the source expansion for its whole lifetime, it may severely
slow down its growth. The high fraction of CSS sources with a
brighter-when-closer behaviour suggests that jet-cloud interactions
are not so unlikely and it may cause an underestimate of the source
psfile=mediana.eps voffset=-230 hoffset=0 vscale=43 hscale=43
The flux density ratio of the lobes versus the source linear
size. Crosses represent the median value computed on various
range of linear sizes. Adapted from Orienti & Dallacasa (2008d).
§ CONCLUDING REMARKS
It has been more than two decades that compact steep spectrum and
GHz-peaked spectrum radio sources are under investigation. During this
time some questions got an answer, perhaps not unique, and new
questions raised.
It is now fairly recognized that the majority of CSS and GPS radio
sources, at least
those optically associated with galaxy
represent an early stage in the radio source evolution. These objects
show two-sided structures similar to FRII radio galaxies, but on much
smaller scales. No significant variability is observed. However, some
level of variability explained in terms of adiabatic expansion in very
young objects have been observed in a few radio sources.
The distinctive characteristic of CSS/GPS objects, i.e. the presence of a
spectral turnover at
low frequencies, is mainly due to SSA, although an additional
contribution from FFA is detected in the most compact objects.
ambient medium enshrouding the radio emission is rich and inhomogeneous,
favouring interactions between the jet and dense clouds. The high
fraction of asymmetric sources, that is much higher than in extended
FRII galaxies, points out how frequently a cloud can impact with one
jet, temporarily confining, or at least slowing down, its advance
speed, and enhancing the synchrotron emission. This may explain part
of the excess in the number counts in flux limited sample. In fact,
CSS/GPS are too many, even if we assume a decrease of the luminosity
as they grow. This is predicted by the evolutionary models developed
taking into account the ambient medium, and is confirmed by
observations of samples of CSS/GPS spanning a large range of linear
Following the evolutionary scenario, GPS radio sources are the
progenitors of CSS objects, which then should evolve in FRII radio
galaxies. However, it seems that some CSS/GPS are short-lived
objects, with ages up to 10$^{3}$ – 10$^{4}$ years, which may never become
large sources with size of hundred of kpc or more. Only a handful of
CSS/GPS have been recognized to be in a relic phase on the basis of
the absence of active regions. These objects are under-represented in
flux-limited catalogues due to the steepness of their synchrotron
spectrum. The advent of the new radio facilities, like LOFAR, Merkaat,
ASKAP, and SKA, will provide a step forward in estimating the
incidence of short-lived and/or recurrent objects, providing a further
piece in the puzzle of our understanding of the radio emission
I would like to thank Daniele Dallacasa for all the helpful and
constructive discussions about the topics presented in this work.
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|
1511.00002
|
This study will explicitly demonstrate by example that
an unrestricted infinite and forward recursive hierarchy of
differential equations must be identified as an unclosed system of
equations, despite the fact that to each unknown function in the
hierarchy there exists a corresponding determined equation to
which it can be bijectively mapped to. As a direct consequence,
its admitted set of symmetry transformations must be identified as
a weaker set of indeterminate equivalence transformations. The
reason is that no unique general solution can be constructed, not
even in principle. Instead, infinitely many disjoint and thus
independent general solution manifolds exist. This is in clear
contrast to a closed system of differential equations that only
allows for a single and thus unique general solution manifold,
which, by definition, covers all possible particular solutions
this system can admit. Herein, different first order Riccati-ODEs
serve as an example, but this analysis is not restricted to them.
All conclusions drawn in this study will translate to any first
order or higher order ODEs as well as to any PDEs.
Keywords: Ordinary
Differential Equations, Infinite Systems, Lie Symmetries and
Equivalences, Unclosed Systems, General Solutions, Linear and Nonlinear Equations,
Initial Value Problems$\,$;
MSC2010: 22E30, 34A05, 34A12, 34A25, 34A34,
34A35, 40A05
§ INTRODUCTION, MOTIVATION AND A BRIEF OVERVIEW WHAT TO EXPECT
Infinite systems of ordinary differential equations (ODEs) appear
naturally in many applications (see e.g.
<cit.>). Mostly
they arise when regarding a certain partial differential equation
(PDE) as an ordinary differential equation on a function space.
This idea ranges back to the time of Fourier when he first
introduced his method of representing PDE solutions as an infinite
series of trigonometric basis functions, through which he
basically formed the origin of functional analysis which then a
century later was systematically developed and investigated by
Volterra, Hilbert, Riesz, and Banach, to name only a few.
The key principle behind the idea that a PDE can be viewed as a
ordinary differential equation within an infinite dimensional
space, is that every PDE by construction describes the change of a
system involving infinitely many coupled degrees of freedom, where
each degree then changes according to an ODE. In this regard, let
us sketch five simple examples to explicitly see how naturally and
in which variety such infinite dimensional ODE systems can
arisefrom a PDE:
Example 1 (Partial
Discretization): Using the numerical method of finite
differences, e.g. on the nonlinear initial-boundary value problem
for the 2-dimensional function $u=u(t,x)$
\begin{equation}
\partial_t u=\partial^2_x u+ u^2,\;\;\text{with}\;\;
u(0,x)=\phi(x),\;\;\text{and}\;\; u(t,0)=\phi(0)=\phi(1)=u(t,1),
\label{150427:1245}
\end{equation}
in respect to the spatial variable $x$, one formally obtains the
following infinite hierarchy of coupled ODEs for the 1-dimensional
functions $u(t,k\cdot\Delta)=:u_k(t)$
\begin{equation}
\frac{du_k}{dt}=
\left(\frac{u_{k+1}-2u_k+u_{k-1}}{\Delta^2}\right)+u_k^2,\;\;
\text{for all}\;\; k=0,1,2,\dotsc, N\rightarrow\infty,
\label{150427:1243}
\end{equation}
when approximating the second partial derivative by the
equidistant central difference formula, where $\Delta=1/N$ is the
discretization size of the considered interval $0\leq x\leq 1$.
Through the given initial and boundary conditions, this system is
restricted for all $k$ and $t$ by
\begin{equation}
u_k(0)=\phi_k, \;\;\text{and}\;\; u_0(t)=\phi(0)=\phi(1)=u_N(t),
\;\;\text{for}\;\; N\rightarrow\infty.
\end{equation}
To obtain numerical results, the infinite system
(<ref>) needs, of course, to be truncated, which then
leads to an approximation of the original PDE initial-boundary
value problem (<ref>). The higher $N$, or likewise,
the smaller $\Delta$, the more exact the approximation (depending
on the numerical rounding errors of the computational system,
which itself will always be limited).
Example 2 (Power Series
Expansion): Let's assume that the linear initial value
\begin{equation}
\partial_t u+\alpha_1\cdot\partial_x u +\alpha_2\cdot u=0, \;\;\text{with}\;\;
u(0,x)=\phi(x), \label{150427:1332}
\end{equation}
allows for an analytical function as solution with respect to $x$
in the interval $a\leq x\leq b$. Then the solution may be expanded
into the power series
\begin{equation}
u(t,x)=\sum_{n=0}^\infty p_n(t)\cdot x^n,
\end{equation}
which, when applied to (<ref>), results into the
following restricted infinite ODE system for the expansion
\begin{equation}
\frac{d p_n}{dt}+\alpha_1\cdot (n+1)\cdot p_{n+1}+\alpha_2\cdot
p_n =0,\;\; \text{for all}\;\; n\geq 0, \;\;\text{with}\;\;
\sum_{n=0}^\infty p_n(0)\cdot x^n=\phi(x).
\end{equation}
Example 3 (Fourier's Method): Consider
the initial value problem of the Burgers' equation
\begin{equation}
\partial_t u + u\cdot \partial_x u = \nu\cdot\partial_x^2 u,\;\;
\text{with}\;\; u(0,x)=\phi(x), \label{150427:1440}
\end{equation}
where $\phi(x)$ is periodic of period $2\pi$. If we are looking
for such a $2\pi$-periodic solution $u(t,x)$, we can write
\begin{equation}
u(t,x)=\sum_{n=-\infty}^\infty u_n(t)e^{i\cdot n\cdot x}.
\end{equation}
Substitution in (<ref>) and equating the coefficients
leads to the infinitely coupled ODE relations
\begin{equation}
\frac{d u_n}{dt}+ \sum_{k+l=n} i\cdot k\cdot u_k \cdot u_l = -\nu
\cdot n^2 \cdot u_n , \;\; \text{for}\;\; (n,k,l)\in\mathbb{Z}^3,
\end{equation}
being restricted by the initial conditions $u_n(0)=\phi_n$, which
are determined by the Fourier expansion
$\phi(x)=\sum_{n=-\infty}^\infty \phi_n e^{inx}$.
Example 4 (Vector Space
Method): This example is the generalization of the two
previous ones. Let's consider the linear initial-boundary value
problem for the diffusion equation
\begin{equation}
\partial_t u=\partial^2_x u,\;\;\text{with}\;\;
u(0,x)=\phi(x),\;\;\text{and}\;\; u(t,a)=\phi(a),\;\;
u(t,b)=\phi(b), \label{150427:1822}
\end{equation}
where $a<b$ are two arbitrary boundary points of some interval
$\mathcal{I}\subset\mathbb{R}$, which both can be also placed at
infinity. Consider further the vector space
\begin{equation}
\mathcal{V}=\Big\{\, v(x),\: \text{differentiable functions on
$\mathcal{I}$, with $v(a)=\phi(a)$, $v(b)=\phi(b)$} \,\Big\},
\label{150427:1843}
\end{equation}
with $\{w_n\}_{n=1}^\infty$ as a chosen basis, for which at this
point we do not know their explicit functional expressions, and
assume that the solution $u(t,x)$ for each value $t$ of the
initial-boundary value problem (<ref>) is an element
of this space $\mathcal{V}$. Then we can write the solution as an
element of $\mathcal{V}$ in terms of the chosen basis vectors
\begin{equation}
u(t,x)=\sum_{n=1}^\infty u_n(t)\cdot w_n(x), \label{150427:2145}
\end{equation}
which, when inserted into (<ref>), induces the
following two infinite but uncoupled sets of linear eigenvalue
\begin{equation}
\frac{du_n(t)}{dt}=-\lambda_n^2\cdot u_n(t),\;\;\;\;
\frac{d^2w_n(x)}{dx^2}=-\lambda_n^2\cdot w_n(x), \;\; \text{for
all}\;\; n\geq 1, \label{150427:2042}
\end{equation}
one infinite (uncoupled) set for the expansion coefficients $u_n$,
and one infinite (uncoupled) set for the basis functions $w_n$,
where $-\lambda_n^2$ is the corresponding eigenvalue to each order
$n$. Both systems are restricted by the given initial and boundary
conditions in the form
\begin{equation}
\sum_{n=1}^\infty u_n(0)w_n(x)=\phi(x);\;\; w_n(a)=\phi(a),\;\;
w_n(b)=\phi(b),\;\;\text{for all $n\geq 1$}, \label{150429:1220}
\end{equation}
and, if the boundary values $\phi(a)$ or $\phi(b)$ are non-zero,
then the system for $u_n$ is further restricted by
\begin{equation}
\sum_{n=1}^\infty u_n(t)=1,\;\; \text{for all possible $t$}.
\label{150429:1221}
\end{equation}
This situation can now be generalized further when using a
different basis $\{\psi_n\}_{n=1}^\infty$ of $\mathcal{V}$
(<ref>) than the PDE's optimal basis
$\{w_n\}_{n=1}^\infty$, which itself is defined by the PDE induced
infinite (uncoupled) ODE system (<ref>). For that, it
is helpful to first introduce an inner product on the outlaid
vector space $\mathcal{V}$ (<ref>). Given two
functions $f,g\in\mathcal{V}$, we will define the inner product
between these two functions as the following symmetric bilinear
form on $\mathcal{I}\subset\mathbb{R}$
\begin{equation}
\L f,g\R=\int_a^b f(x)\, g(x)\, dx.
\end{equation}
Now, to construct the corresponding induced infinite ODE system
for the new representation of the PDE's solution
\begin{equation}
u(t,x)=\sum_{n=1}^\infty \tilde{u}_n(t)\cdot \psi_n(x),
\label{150427:2126}
\end{equation}
it is expedient to identify the differential operator
$\partial_x^2$ in the PDE (<ref>) as a linear operator
$A$ acting on the given vector space $\mathcal{V}$
\begin{equation}
\partial_t u=\partial_x^2 u\;\;\; \simeq\;\;\; \partial_t u=A(u),
\label{150521:2254}
\end{equation}
which then can also be written in explicit matrix-vector form as
\begin{equation}
\frac{d \tilde{\vu}}{dt}=\vA\cdot\tilde{\vu}, \label{150429:1222}
\end{equation}
where the infinite vector
is composed of the expansion coefficients of the solution vector
$u(t,x)$ (<ref>), and where the matrix elements of the
infinite matrix $\vA$ are given as
\begin{equation}
A_{mn}=\L \psi_m, A(\psi_n)\R=\int_a^b \psi_m(x)\, \partial_x^2
\psi_n(x)\, dx . \label{150429:1210}
\end{equation}
The explicit element values of both $\tilde{\vu}$ and $\vA$ depend
on the choice of the basis
$\{\psi_n\}_{n=1}^\infty\subset\mathcal{V}$, while the solution
vector $u(t,x)$ of the underlying PDE (<ref>) itself
stays invariant under a change of base, i.e., for example, the
expansion (<ref>) relative to the basis
$\{w_n\}_{n=1}^\infty$ represents the same solution as the
expansion (<ref>) relative to the basis
Of particular interest is now to investigate whether
$\vA=(A_{nm})_{n,m\geq 1}$ (<ref>) represents a
symmetric matrix or not. If not, are there then any conditions
such that a symmetric matrix can be obtained? Performing a double
partial integration in (<ref>) will yield the
\begin{align}
A_{mn} &= \int_a^b \psi_m(x)\,
\partial_x^2 \psi_n(x)\, dx\nonumber\\[0.25em]
& = \int_a^b \psi_n(x)\, \partial_x^2 \psi_m(x)\, dx
\psi_n(x)-\psi_n(x)\partial_x\psi_m(x)\Big]_{x=a}^{x=b}\nonumber\\[0.25em]
& = A_{nm}+\Big[\psi_m(x)\partial_x
\psi_n(x)-\psi_n(x)\partial_x\psi_m(x)\Big]_{x=a}^{x=b},
\label{150429:1132}
\end{align}
which implies that in order to obtain a symmetric
matrix, we have to choose the boundary values $\phi(a)$ and
$\phi(b)$ such that they satisfy the symmetry relation
\begin{equation}
\Big[\psi_m(x)\partial_x
\psi_n(x)-\psi_n(x)\partial_x\psi_m(x)\Big]_{x=a}^{x=b}=0,\;\;\text{for
all $\, n,m\geq 1$}. \label{150429:1157}
\end{equation}
For example, the simplest choice is to enforce vanishing boundary
values $\phi(a)=\phi(b)=0$. This will satisfy the condition
(<ref>) independently on the choice of the basis
functions, with the advantageous effect then that the matrix $\vA$
is a real symmetric matrix to any chosen basis of the vector
space $\mathcal{V}$ (<ref>). In particular, since the
eigenvectors of a real symmetric matrix are orthogonal, the matrix
elements $A_{mn}$ for the optimal basis $\{w_n\}_{n=1}^\infty$,
according to (<ref>), form the diagonal matrix
\begin{equation}
A_{mn}=-\lambda^2_n \cdot \|w_n\|^2\,\delta_{mn}=\int_a^b w_m(x)\,
\partial_x^2 w_n(x)\, dx,
\end{equation}
where no summation over the repeated index $n$ is implied. Hence,
if we choose the alternative basis $\{\psi_n\}_{n=1}^\infty$ such
that it's orthonormal[If $\{\psi_n\}_{n=1}^\infty$ is
an orthonormal basis of $\mathcal{V}$, then each $\psi_n$ must be
collinear to $w_n$, in particular $\psi_n=w_n/\|w_n\|$.]
\begin{equation}
\L \psi_m, \psi_n\R = \delta_{mn},
\end{equation}
then, according to (<ref>) and (<ref>),
the PDE's induced infinite ODE system (<ref>) will
only be restricted by the initial conditions
\begin{equation}
\tilde{u}_n(0)=\int_a^b \phi(x)\, \psi_n(x)\, dx,\;\; \text{for
all} \;\; n\geq 1.
\end{equation}
Example 5 (Method of
moments): Consider the initial value problem (Cauchy
problem) of the more generalized linear diffusion equation
\begin{equation}
\partial_t u = a\cdot\partial^2_x u + b\cdot x\cdot\partial_x u +
(b+c\cdot x^2)\cdot u, \;\;\text{with}\;\; u(0,x)=\phi(x).
\label{150430:1313}
\end{equation}
If one is interested in the moments
\begin{equation}
u_n(t)=\int_{-\infty}^\infty x^n\cdot u(t,x)\, dx,\;\; n\geq 0,
\end{equation}
by multiplying the PDE (<ref>) with $x^n$ and
integrating over $\mathbb{R}$, and if one assumes that partial
integration is justified, i.e. when assuming the natural boundary
\begin{equation}
\lim_{x\rightarrow \pm\infty} u(t,x)=0,\quad \lim_{x\rightarrow
\pm\infty} \partial_x u(t,x)=0,
\end{equation}
then one obtains the following infinite system of coupled ODEs for
all $n\geq 0$
\begin{equation}
\frac{d u_n}{dt} = a\cdot n\cdot(n-1)\cdot u_{n-2}-b\cdot n\cdot
u_n +c\cdot u_{n+2},\;\; \text{with}\;\;
u_n(0)=\int_{-\infty}^\infty x^n\cdot \phi(x)\, dx.\;\;
\label{150501:1415}
\end{equation}
Note that for $c=0$ the solution $u=u(t,x)$ of (<ref>)
has the property of a probability measure, for example in that it
can be interpreted as a probability density of a particle
undergoing Brownian motion. Only for $c=0$ the PDE
(<ref>) attains the structure of a Fokker-Planck
equation with the drift coefficient $D_1(t,x)=-b\cdot x$ and the
diffusion coefficient $D_2(t,x)=a>0$
\begin{equation}
\partial_t u = \partial_x \big(-D_1\cdot u+D_2\cdot\partial_x
u\big), \label{150504:1046}
\end{equation}
which describes the time evolution of the probability distribution
$u(t,x)\geq 0$ such that no probability is lost, i.e. conserved
for all $t\geq 0$
\begin{equation}
\int_{-\infty}^\infty u(t,x)\, dx =1, \label{150501:1337}
\end{equation}
due to the defining structure of equation (<ref>)
being a conservation law with the probability current $J=-D_1\cdot
u+D_2\cdot\partial_x u$. Hence, for $c=0$ the corresponding
infinite ODE system (<ref>) is thus further
(automatically) restricted by $u_0(t)=1$ for all times $t\geq 0$.
However, if $c\neq 0$ then the zeroth moment (<ref>)
in general is not constant in time; it will rather evolve
according to some prescribed 'normalization' function $N(t)$
\begin{equation}
\int_{-\infty}^\infty u(t,x)\, dx =N(t). \label{150504:1112}
\end{equation}
To exogenously enforce a certain function $N(t)=N_0(t)$ as an
additional (non-local) boundary condition onto the Cauchy problem
(<ref>) would result into an overdetermined system,
for which no solutions may exist. The Cauchy problem itself, e.g.
of equation (<ref>), is well-posed and allows, up to a
normalization constant, for a unique solution. The normalization
constant can be fixed by posing at $t=0$ a normalized initial
condition, e.g. $\int_{-\infty}^\infty
\phi(x)dx=N(0)=1$.Note that although such integral
overdetermination conditions as (<ref>) are widely
used to study associated inverse problems, e.g. to find
corresponding heat sources and diffusion coefficients (see e.g.
<cit.>), we will not consider them
[4] These five examples discussed above show that there
is a multitude of possibilities in how a PDE can induce an ODE in
an infinite dimensional space. If the PDE is restricted by initial
or boundary conditions they are transcribed to the infinite system
accordingly. Mostly, only the initial conditions get directly
transferred, while the boundary conditions are only needed as
auxiliary conditions to actually perform the reduction process,
e.g. as in the case of Example [E4]4. Note that all
examples only considered the reduction of $(1+1)$-dimensional
parabolic PDEs, but it's obvious that this concept extends to any
type of PDEs of any dimension. The result is then not a single
infinite system, but rather a collective hierarchy of several
infinite systems of coupled ODEs.
As all examples showed, it should be clear that
the associated infinite system of ODEs is not identical to
the PDE. It only represents a reduction of the PDE, since always a
certain Ansatz of the PDE's solution manifold has to be made in
order to obtain its associated infinite ODE system. Stated
differently, the PDE operates on a higher level of abstraction
than its induced infinite system of (lower level) ODEs, which,
although infinite dimensional, nevertheless depends on assumptions
and in particular on the choice of the reduction method used. That
is, a single PDE can always be reduced to a multitude of
functionally and structurally different infinite systems of
ODEs depending on the choice of method. This insight can be
transferred to differential equations which operate on an even
higher abstraction level than PDEs, e.g. so-called functional
equations which involve functional derivatives. Then an infinite
hierarchy of PDEs instead of ODEs takes the place of the reduced
A natural question which arises is whether the infinite set of
reduced equations is easier to analyze than the original PDE? In
general, the answer to this question is "no". However, if the
infinite system is truncated and approximated to a low-dimensional
form, then often qualitative analysis is possible, and useful
insights into the dynamics of the original system can be obtained.
Also from a numerical point of view many interesting stability
questions arise when the system is truncated, because, in order to
obtain numerical results from an infinite system, some method of
truncation must be employed. Surely, the quality of the subsequent
approximation towards a consistent finite dimensional system
strongly depends on this method in how the system was truncated,
which is part of the theories of closure and
differential approximation.
The formal mathematical environment to study and analyze an
infinite (non-truncated) sequence of differential equations is set
by the infinite-dimensional theory of Banach spaces. Questions
regarding existence and uniqueness of solutions can only be
properly dealt with from the perspective of a Banach space in
defining and constructing appropriate functional norms. Such
systematic investigations, however, are beyond the scope of this
article; for that, the rich literature on this topic has to be
consulted (see e.g.
Instead, we will only make a small excursion into the uniqueness
issue of these solutions when restricting the infinite system by a
sufficient set of initial conditions, but only to show where still
the problems lie and not on how to solve these problems.
The main focus of this article will be based on the unrestricted infinite set of ODEs, and to primarily study the
general solutions they admit. By taking the perspective of a
Lie group based symmetry analysis
<cit.>, we
can demonstrate by example that eventually any unrestricted
infinite set of differential equations, which is based on a forward recurrence relation, must be identified as an unclosed
and thus indeterminate system, although, in a formal one-to-one
manner, one can associate to each equation in the hierarchy a
corresponding unknown function. As a consequence, such unclosed
differential systems do not allow for the construction of a unique
general solution. Any desirable general solution can
be generated. The side-effect of this result is that each symmetry
transformation then only acts in the weaker sense of an
equivalence transformation
<cit.>. Such an
identification is necessary in order to allow for a consistent
invariance analysis among an infinite set of differential
At first sight it may seem to be a trivial
observation that an unrestricted infinite set of ODEs has
the property of an underdetermined system. Because if it
represents a specific reductionof an unrestricted
PDE, i.e. of a PDE which is not accompanied by any initial or
boundaryconditions, its general solution is only unique
up to certain integration functions. And since this arbitrariness
on the higher abstraction level of the PDE is transferred down to
the lower abstraction level of the reduced ODE system, it is not
surprising that the latter system is somehow arbitrary as well.
But, by closer inspection there is no one-to-one correspondence,
because, for example, for any evolutionary PDE with fixed spatial
boundary conditions, the degree of arbitrariness in its general
solution only depends on the order of the time derivative, which
in turn is directly linked to the number of initial condition
functions needed to generate a unique solution from the general
one. However, for its reduced ODE system the degree of
arbitrariness is differently larger in that it not only depends on
the temporal differential order as the underlying PDE does, but
also, additionally, on the direction and the order of the spatial
recurrence relation which this system inherently defines. For
example, if the recurrence relation is a forward recurrence
of order one, then, independent of the temporal differential
order, one unknown function anywhere in the ODE hierarchy can be
specified freely; if its a forward recurrence of order two,
then two unknown functions can be specified freely, and so on.
This freedom in choice has no correspondence on the higher
abstraction level of the PDE. Appendix <ref> and <ref>
provide a preview demonstration of these statements by considering
again Example [E5]5.In Section <ref> and
Section <ref> this insight will then be investigated in more
detail by also involving different examples.
In particular, it was exactly Example [E5]5 with its
properties discussed in Appendix <ref> and <ref> which
motivated this study. The inherent principle that the (higher
abstraction level) PDE (<ref>) represents a closed
system while the correspondingly reduced (lower abstraction level)
ODE system of its moments (<ref>), although being
infinite in dimension, constitutes an unclosed system if the
recurrence is of forward direction (see Appendix <ref>),
obviously transfers to an even higher abstraction level of
description, as seen, for example, when formulating the
statistical description of Navier-Stokes turbulence. There the
functional Hopf equation formally serves as the (higher
abstraction level) closed equation while its correspondingly
induced (lower abstraction level) infinite Friedmann-Keller PDE
system of multi-point moments is unclosed; for more details on
this issue, see <cit.> and <cit.>.
Part of the current study is to mathematically clarify this point
in statistical turbulence research, namely where any formally closed set of equations which operates on a higher statistical
level always induces an unclosed infinite system on the
lower statistical level of the moment equations, and, where
thus, both levels of description are not equivalent. The reason
for this is that due to the nonlinearity of the Navier-Stokes
problem a forward recurrence relation is always generated on
the lower abstraction level of the statistical moments (in the
sense similar to the problem demonstrated in Appendix <ref>),
turning thus the corresponding infinite Friedmann-Keller PDE
hierarchy inherently into an unclosed system. It is necessary to
clarify this point, because it seems that in the relevant
literature on turbulence there still exists a misconception on
this issue, in particular in the studies of
Oberlack et al. <cit.>.
A detailed discussion on this misconception is given in
The paper is organized as follows: Section <ref> first considers
a single (closed) ODE to define the concept of a unique
general solution manifold from the perspective of an invariance
analysis. In Section <ref> this concept will be applied to an
(unclosed) infinite ODE system based on a forward recurrence
relation. Both a linear (Section <ref>) as well as a nonlinear
system (Section <ref>) will be investigated, which both stem
as special cases from a generalized hierarchy of first order
Riccati-ODEs. To which higher level PDE this infinite ODE system
belongs to is an inverse problem, which will not be investigated
since it's clearly beyond the scope of this article. Based on the
results presented herein, we can conclude that any unrestricted
infinite system which follows a forward recurrence relation must
be identified as an unclosed system, which then, as consequence,
only leads to non-unique general solution manifolds and which,
instead of symmetry transformations, only admits the weaker
equivalence transformations. This twofold conclusion is
independent of whether an infinite hierarchy of ODEs or whether an
infinite hierarchy of PDEs is considered.
§ LIE-POINT SYMMETRIES AND GENERAL SOLUTION OF A SINGLE RICCATI-ODE
In general a Riccati equation is any first-order ODE that exhibits
a quadratic nonlinearity of the form
\begin{equation}
y^\prime(x)=q_0(x)+q_1(x)y(x)+q_2(x)y^2(x), \label{150401:1222}
\end{equation}
with $q_i(x)$ being arbitrary functions (see e.g.
<cit.>).[Note that the nonlinear Riccati-ODE
(<ref>) can always be reduced to a linear ODE of
second order by making use of the transformation $y(x)=-z^\prime
(x)/(q_2(x)\cdot z(x))$.] In this section, however, we only want
to consider the following specific Riccati-ODE
\begin{equation}
y^\prime -\frac{y}{x}-\frac{y^2}{x^3}=0, \label{150323:1732}
\end{equation}
which is also categorized as a specific Bernoulli differential
equation of the quadratic rank (see e.g. <cit.>). Its
unique general solution[Transforming the nonlinear ODE
(<ref>) according to $y(x)=x/z(x)$ will reduce it to a
linear ODE of first order which can be solved then by a simple
integration.] is given by
\begin{equation}
y(x)=\frac{x^2}{1+c\cdot x},\quad c\in\mathbb{R},
\label{150323:2207}
\end{equation}
involving a single free integration parameter $c$. Since
(<ref>) is a single first order ODE it has the
special property of admitting an infinite set of Lie-point
symmetries (see e.g. <cit.>). The
symmetries are generated by the tangent field
$X=\xi(x,y)\partial_x +\eta(x,y)\partial_y$, which, in the
considered case (<ref>), satisfies the following
underdetermined relation for the infinitesimals $\xi=\xi(x,y)$ and
\begin{align}
\!\!\!\!\! 0= &\; \xi\cdot(3y^2x^2+yx^4)-\eta\cdot (2yx^3+x^5)\nonumber\\
& -\partial_x\xi\cdot (y^2x^3+yx^5)-\partial_y\xi\cdot
(y^4+2y^3x^2+y^2x^4)+\partial_x\eta\cdot (x^6)
+\partial_y\eta\cdot (y^2x^3+yx^5). \label{150323:1836}
\end{align}
Note that in constructing the general solution of equation
(<ref>) only one function can be chosen arbitrarily,
either $\xi$ or $\eta$, but not both. Without restricting the
general case, we will choose $\xi$ as the free infinitesimal,
which, once chosen in (<ref>), then uniquely fixes the
second infinitesimal $\eta$. For the present, it is sufficient to
only consider monomials in the normalized form $x^n$ with $n\geq
0$ as a functional choice for $\xi$. According to
(<ref>), the corresponding tangent field $X$ up to
order $n$ is then given by
\begin{align} \mathsf{T}_n: & \;\;\;
X_n =x^n\partial_x +
\left[x^{n+2}\cdot\left(\frac{y}{x^3}+\frac{y^2}{x^5}\right)
\quad n\geq 0, \label{150323:1920}
\end{align}
where the $F_n$ are arbitrary integration functions with argument
$(y-x^2)/yx$. For the sake of simplicity it is convenient to
choose these functions such that for each order $n$ the lowest
degree of complexity is achieved. For example, for the first four
elements in this chain (<ref>) we choose the $F_n$
such that
\begin{equation}
\left. \begin{aligned} \mathsf{T}_0: & \;\;\;
\mathsf{T}_1: & \;\;\; X_1=x\partial_x+2y\partial_{y},\\[0.75em]
\mathsf{T}_2: & \;\;\; X_2=x^2\partial_x+ xy\partial_y,\\[0.75em]
\mathsf{T}_3: & \;\;\;
\end{aligned}
~~~ \right \} \label{150323:1919}
\end{equation}
which, according to Lie's central theorem (see e.g.
<cit.>), are equivalent to the 1-parameter symmetry group
\begin{equation}
\left. \begin{aligned} \mathsf{T}_0: & \;\;\;
\tilde{x}=x+\varepsilon_0,\;\;\;
\tilde{y}=\left(\frac{y-x^2}{x^3}+\frac{1}{x+\varepsilon_0}\right)
\mathsf{T}_1: & \;\;\; \tilde{x}=e^{\varepsilon_1} x,\;\;\;
\tilde{y}=e^{2\varepsilon_1}y,\\[0.75em]
\mathsf{T}_2: & \;\;\; \tilde{x}=
\frac{x}{1-\varepsilon_2\, x},\;\;\; \tilde{y}=\frac{y}{1-\varepsilon_2\, x},\\[0.75em]
\mathsf{T}_3: & \;\;\; \tilde{x}=\frac{x}{\sqrt{1-2\varepsilon_3\,
x^2}},\;\;\; \tilde{y}=\frac{yx^2}{y\cdot (1-2\varepsilon_3\,
x^2)-(y-x^2)\cdot\sqrt{1-2\varepsilon_3\, x^2}}.
\end{aligned}
~~~ \right \} \label{150323:2000}
\end{equation}
By construction each of the above transformations leaves
the considered differential equation
(<ref>) invariant, i.e. when transforming
(<ref>) according to one of the transformations
(<ref>) will thus result into the invariant form
\begin{equation}
\tilde{y}^\prime
\label{150323:2126}
\end{equation}
Note that for any point transformations, as in the case
(<ref>), the transformation for the first order
ordinary derivative is induced by the relation
\begin{equation}
\tilde{y}^\prime=\frac{d
\tilde{y}}{d\tilde{x}}=\frac{\frac{\partial \tilde{y}}{\partial
x}dx+\frac{\partial \tilde{y}}{\partial y}dy}{\frac{\partial
\tilde{x}}{\partial x}dx+\frac{\partial \tilde{x}}{\partial y}dy}
=\frac{\frac{\partial \tilde{y}}{\partial x}+\frac{\partial
\tilde{y}}{\partial y}y^\prime}{\frac{\partial \tilde{x}}{\partial
x}+\frac{\partial \tilde{x}}{\partial
y}y^\prime}=\left(\frac{\partial\tilde{x}}{\partial x}\right)^{-1}
\left(\frac{\partial \tilde{y}}{\partial x}+\frac{\partial
\tilde{y}}{\partial y}y^\prime\right), \label{150404:1245}
\end{equation}
where the last equality only stems from the fact that
$\partial{\tilde{x}}/\partial y =0$ for all transformations
Now, if the general solution (<ref>) would not be
known beforehand, then the symmetries (<ref>) can be
used to construct it. For any first order ODE, as in the present
case for (<ref>), at least one symmetry is necessary
to determine its general solution, which, for example, can be
achieved by making use of the method of canonical variables (see
e.g. <cit.>). But, instead of performing this
construction, the opposite procedure will be investigated, namely
to validate in how far the function (<ref>) represents
a unique general solution[By definition a unique
general solution of a differential equation or a system of
differential equations should cover all particular (special)
solutions this system can admit. In other words, every special
solution that can be constructed must be covered through this
general solution by specifying a corresponding initial condition,
otherwise the given general solution is not complete or unique.]
of (<ref>) when transforming it according to the
symmetries (<ref>). The result to expect is that if
function (<ref>) represents the unique general
solution of (<ref>), then it either must map to
\begin{equation}
\tilde{y}(\tilde{x})=\frac{\tilde{x}^2}{1+\tilde{c}\cdot
\tilde{x}},\quad \tilde{c}\in\mathbb{R}, \label{150323:2318}
\end{equation}
with a new free transformed parameter $\tilde{c}=\tilde{c}(c)$ as
a function of the old untransformed parameter $c$, or it must
invariantly map to
\begin{equation}
\tilde{y}(\tilde{x})=\frac{\tilde{x}^2}{1+c\cdot \tilde{x}},\quad
c \in\mathbb{R}, \label{150323:2319}
\end{equation}
with an unchanged free parameter $c$ before and after the
transformation. The reason is that since all transformations
(<ref>) form true symmetries, which map solutions to
new solutions of the underlying differential equation
(<ref>), any solution which forms a unique general
solution of this equation can thus only be mapped into itself,
either into the non-invariant form (<ref>) or into the
invariant form (<ref>), because no other functionally
independent solution exists to which the symmetries can map to. If
this is not the case, we then have to conclude that either the
considered transformation is not a symmetry transformation or that
the given solution is not the general solution.
Hence, since we definitely know that (<ref>) is the
unique general solution of equation (<ref>), which
again admits the symmetries (<ref>), these symmetry
transformations only need to be classified into two categories,
namely into those which reparametrize the general solution
(<ref>) and into those which leave it invariant
(<ref>). For example, the symmetries $\mathsf{T}_0$,
$\mathsf{T}_1$ and $\mathsf{T}_2$ reparametrize the general
solution (<ref>) with
$\tilde{c}=e^{-\varepsilon_1}c$, and $\tilde{c}=c+\varepsilon_3$
respectively, while symmetry $\mathsf{T}_3$ keeps it invariant.
This game can then be continued for all higher orders of $n$ in
(<ref>), or even for any other functionally different
symmetry using the general determining relation
To conclude this section, it is helpful to formalize the above
insights: Let $f_\lambda$ formally be a parameter dependent
solution of a differential equation $E$, and $\mathsf{S}$ any
transformation which leaves this differential equation invariant
$\mathsf{S}(E)=E$. The transformation $\mathsf{S}$ on $f_\lambda$
is called a reparametrization if
$\mathsf{S}(f_\lambda)=f_{s(\lambda)}$, which includes the special
case of an invariant transformation if $s(\lambda)=\lambda$, where
the parameter mapping $s$ is induced by the variable
mapping $\mathsf{S}$.Then, based on these conditions,
the following two statements are equivalent:
\begin{equation}
\left .
\begin{aligned}
\quad\quad\;\text{$f_\lambda$ is a unique general solution}\quad &
\Rightarrow & \;\,
\mathsf{S}(f_\lambda)=f_{s(\lambda)},\hspace{3.68cm}\\[0.5em]
\mathsf{S}(f_\lambda)\neq f_{s(\lambda)}\quad & \Rightarrow & \;\,
\text{$f_\lambda$ is not a unique general solution},
\end{aligned}
~~~ \right\}\label{150425:1121}
\end{equation}
where in each case the opposite conclusion is, of
course, not valid, i.e.
\begin{equation}
\left .
\begin{aligned}
\mathsf{S}(f_\lambda)=f_{s(\lambda)}\quad & \nRightarrow & \;\,
\text{$f_\lambda$ is a unique general solution},\\[0.5em]
\text{$f_\lambda$ is not a unique general solution}\quad &
\nRightarrow & \;\,\mathsf{S}(f_\lambda)\neq
\end{aligned}
~~~~~~~~ \right\}\!\!\!\label{150425:1142}
\end{equation}
§ LIE-POINT SYMMETRIES AND GENERAL SOLUTION FOR AN INFINITE
SYSTEM OF ODES
Let's consider the following unrestricted infinite and forward recursive hierarchy of ordinary differential equations
based on the Riccati ODE (<ref>)
\begin{equation}
n=1,2,3,\,\dotsc \label{150401:1710}
\end{equation}
A solution of such a system is defined as an infinite set of
functions $\{y_1(x),y_2(x),\ldots,y_n(x),\ldots\}$ for which all
the equations of the system hold identically. Without restricting
the general case, we will consider two specifications: a linear
and a nonlinear one.
§.§ Infinite linear hierarchy of first order ODEs
In this section we will consider the linear specification
$q_0=q_1=q_2=0,q_3=-1$ of (<ref>)
\begin{equation}
y_n^\prime = -y_{n+1},\quad n=1,2,3,\,\dotsc ,\label{150401:1859}
\end{equation}
which also can be equivalently written in vector form as
\begin{equation}
\vy^\prime=-\vA\cdot\vy,\label{150401:1830}
\end{equation}
where $\vA$ is the infinite but bounded bi-diagonal matrix
\begin{equation*}
\vA=\begin{pmatrix}
\,\, 0 & 1 & 0 & 0 & 0 & \cdots & 0 & \cdots\,\, \\
\,\, 0 & 0 & 1 & 0 & 0 & \cdots & 0 & \cdots\,\, \\
\,\, 0 & 0 & 0 & 1 & 0 & \cdots & 0 & \cdots\,\, \\
\,\, \vdots & \vdots & \vdots & \ddots & \ddots & \vdots & \vdots
& \vdots\,\,
\end{pmatrix},
\end{equation*}
along with the infinite dimensional solution vector
$\vy^T=(y_1,y_2,y_3,\dots ,y_n,\dots)$ of (<ref>).
Naively one would expect that the unique general solution to
(<ref>) is given by[The operator
$e^{-x\vA}$ is called the flow of the differential equation
(<ref>), as it takes the initial state $\vy=\vc$ at
$x=0$ into the new state $\vy=e^{-x\vA}\cdot \vc$ at position
$x\neq 0$. If (<ref>) represents an evolution equation
with its forward marching time $t\geq 0$ as the independent
variable, then the set of all operator elements $e^{-t\vA}$ only
forms a semi-group. The operator $\vA$ is then said to be the
infinitesimal generator of this semi-group.]
\begin{equation}
\vy =e^{-x\vA}\cdot \vc, \label{150401:1848}
\end{equation}
where $\vc$ is the infinite dimensional integration constant
$\vc^T=(c_1,c_2,c_3,\dots c_n,\dots)$. When evolving the
exponential function into its power series with its infinite
radius of convergence, the general solution (<ref>)
can be equivalently written as
\begin{equation}
\left.
\begin{aligned}
y_1(x)=c_1-c_2\cdot x+ \frac{1}{2!}\, c_3\cdot x^2 -
\frac{1}{3!}\, c_4\cdot x^3 + \cdots \\
y_2(x)=c_2-c_3\cdot x+ \frac{1}{2!}\, c_4\cdot x^2 -
\frac{1}{3!}\, c_5\cdot x^3 + \cdots \\
y_3(x)=c_3-c_4\cdot x+ \frac{1}{2!}\, c_5\cdot x^2 -
\frac{1}{3!}\, c_6\cdot x^3 + \cdots \\
\vdots\hspace{3.2cm}\vdots\hspace{3cm}\vdots\hspace{0.15cm}
\end{aligned}
~~~ \right \} \label{150401:1909}
\end{equation}
or compactly as
\begin{equation}
y_n(x)=\sum_{k=0}^\infty c_{n+k}\frac{(-1)^k}{k!}\, x^k,\quad
n=1,2,3,\,\dotsc ,\label{150401:1925}
\end{equation}
which, naively considered, might then serve as the unique general
solution for (<ref>). Of course, the precondition for
it is that for any given initial condition the constant component
values of the infinite dimensional vector $\vc$ must be given such
that the matrix product (<ref>) is converging. That
(<ref>), or equivalently (<ref>),
represents a general solution to (<ref>) is
obvious, because to every (first order) differential equation of
the hierarchy (<ref>) one can associate a solution
$y_n$ involving a free integration parameter $c_n$.
Now, let us see in how far the general solution
(<ref>) represents a unique general solution of
(<ref>). For that we first consider one of the
equations' scaling symmetries admitted by (<ref>)
\begin{equation}
\mathsf{L}_1: \;\;\; \tilde{x}=e^{-\varepsilon}x,\;\;\;
\tilde{\vy}=\vD(\varepsilon)\cdot\vy,\label{150403:1113}
\end{equation}
where $\vD$ is the infinite diagonal matrix
\begin{equation*}
\vD(\varepsilon)=\begin{pmatrix}
\,\, e^{\varepsilon} & 0 & 0 & 0 & \cdots\,\, \\
\,\, 0 & e^{2\varepsilon} & 0 & 0 & \cdots\,\, \\
\,\, 0 & 0 & e^{3\varepsilon} & 0 & \cdots\,\, \\
\,\, \vdots & \vdots & \cdots & \ddots & \vdots\,\,
\end{pmatrix}.
\end{equation*}
Transforming the general solution (<ref>) according to
$\mathsf{L}_1$ (<ref>) will map the solution up to a
reparametrization in the integration constant $\vc\mapsto
\tilde{\vc}$ into itself$\,$[To obtain the second
relation in (<ref>) one has to use the non-commutative
property $\vD\cdot \vA^n=e^{-n\varepsilon} (\vA^n\cdot \vD)$.]
\begin{equation}
\tilde{\vy}=\vD(\varepsilon)\cdot
\Big(e^{-e^{\varepsilon}\tilde{x}\vA}\cdot\vc\Big)
=e^{-\tilde{x}\vA} \cdot \big(\vD(\varepsilon)\cdot\vc\big)=
\end{equation}
That means, regarding symmetry transformation $\mathsf{L}_1$
(<ref>) the general solution (<ref>)
represents itself as unique general solution indeed. But this is
no longer the case if we consider for example the following
symmetry transformation
\begin{align}
\mathsf{L}_2: \;\;\; &
& \;\;\text{with}\;\;\xi=x^2,\;\;\text{and}\;\;
\eta_n=y_n+(n-1)(n-2)\, y_{n-1}-2\,(n-1)\, x\, y_{n},
\hspace{3.35cm}\nonumber
\end{align}
which in global form reads as (see derivation
\begin{equation}
\left.
\begin{aligned}
\negthickspace\negthickspace\negthickspace\!\mathsf{L}_2: \;\;\; &
\tilde{x}=\frac{x}{1-\varepsilon
& \tilde{y}_n=\, \sum_{k=1}^{n-1}
B_{n,k}\,\varepsilon^{n-k-1}(1-\varepsilon x)^{n+k-1}\,
e^{\varepsilon}\, y_{k+1},\;\;\text{for all}\;\; n\geq
\end{aligned}
~~~ \right \} \label{150403:1804}
\end{equation}
where e.g. the first three explicit elements in this hierarchy are
given as
\begin{equation}
\left.
\begin{aligned}
\negthickspace\negthickspace\negthickspace\,\!\! \mathsf{L}_2:
\;\;\; & \tilde{x}=\frac{x}{1-\varepsilon
x},\;\;\;\tilde{y}_1=e^{\varepsilon}y_1,\\ &
\tilde{y}_2=(1-\varepsilon x)^2\, e^\varepsilon y_2,\;\;\;
\tilde{y}_3=2\varepsilon(1-\varepsilon x)^3\, e^\varepsilon y_2 +
(1-\varepsilon x)^4\, e^\varepsilon y_3,\;\;\; \cdots
\qquad\quad\;\;\;\;\;\,
\end{aligned}
~~~ \right \} \label{150403:1855}
\end{equation}
Because when transforming the general solution (<ref>)
according to the above symmetry transformation
(<ref>), which in matrix-vector form reads as
\begin{equation}
\mathsf{L}_2: \;\;\; \tilde{x}=\frac{x}{1-\varepsilon
\label{150407:1002}
\end{equation}
where $\vG$ is the infinite group matrix
\begin{equation*}
\vG(x,\varepsilon)=e^{\varepsilon}\begin{pmatrix}
\,\, 1 & 0 & 0 & 0 & 0 & \cdots\,\, \\
\,\, 0 & B_{2,1}\varepsilon^0(1-\varepsilon x)^2 & 0 & 0 & 0 & \cdots\,\, \\
\,\, 0 & B_{3,1}\varepsilon^1(1-\varepsilon x)^3
& B_{3,2}\varepsilon^{0}(1-\varepsilon x)^4 & 0 & 0 &\cdots\,\, \\
\,\, \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \,\, \\
\;\; 0 & B_{n,1}\varepsilon^{n-2}(1-\varepsilon x)^n & \cdots &
B_{n,n-1}\varepsilon^{0}(1-\varepsilon x)^{2(n-1)} & 0 &
\cdots\,\,\\
\;\; \vdots & \vdots & \vdots & \vdots & \ddots & \vdots\,\,
\end{pmatrix},
\end{equation*}
we obtain a fundamentally different general solution
$\tilde{\vy}$, which, for all
$x\in\mathbb{R}\backslash\{\frac{1}{\varepsilon}\}$, can not
be identified as a reparametrization of the integration constant
$\vc\mapsto\tilde{\vc}$ of the primary solution
(<ref>) anymore:
\begin{equation}
\tilde{\vy}=\vG\!\left({\textstyle\frac{\tilde{x}}{1+\varepsilon
\tilde{x}}}\, ,\varepsilon\right)\cdot
\Big(e^{-\frac{\tilde{x}}{1+\varepsilon
\tilde{x}}\vA}\cdot\vc\Big)\neq e^{-\tilde{x}\vA}\cdot\tilde{\vc}.
\label{150407:1130}
\end{equation}
The fundamental difference between the solutions $\tilde{\vy}$
(<ref>) and $\vy$ (<ref>) already shows
itself in the fact that the former one
$\tilde{\vy}=\tilde{\vy}(\tilde{\vx},\varepsilon)$ has a permanent non-removable singularity at $\tilde{x}=-1/\varepsilon$
independently of how $\vc$ is chosen, which thus implies that the
transformed solution $\tilde{\vy}$ has one essential
parameter more than the primary solution $\vy$, namely the group
parameter $\varepsilon$, which can not be generally absorbed
into the integration constant $\vc$.[Note that
although the transformation for $x\mapsto \tilde{x}$ in
$\mathsf{T}_2$ (<ref>) is identical to the one in
$\mathsf{L}_2$ (<ref>) by also showing a non-removable
singularity at the inverse value of the group parameter, the full transformation $\mathsf{T}_2$, however, does not induce this
singularity into the transformed general solution
(<ref>), simply because the corresponding
transformation $y\mapsto \tilde{y}$ in $\mathsf{T}_2$
(<ref>) annihilates this singularity. Hence, in
contrast to transformation $\mathsf{L}_2$ (<ref>), the
group parameter in $\mathsf{T}_2$ (<ref>) is non-essential since it can be absorbed into the integration
constant to give the reparametrized general solution
Yet, $\tilde{\vy}$ (<ref>) is not to
(<ref>) the only functionally different general
solution which can be constructed by a symmetry transformation.
Infinitely many different general solutions can be obtained by
just relaxing the specification $\xi=x^2$ in $\mathsf{L}_2$
(<ref>) and considering, for example, the more general
symmetry transformation
\begin{align}
\mathsf{L}_2^f: \;\;\; &
& \;\;\text{with}\;\;\xi=f(x),\;\;
\eta_n=y_n+\sum_{k=1}^{n-1}(-1)^{n-k}\binom{n-1}{k-1}\frac{d^{n-k}f(x)}{dx^{n-k}}y_{k+1},
\hspace{3.75cm}\nonumber
\end{align}
where $f$ is some arbitrary function. Hence, no unique and thus no
privileged general solution can be found for the infinite
hierarchy of differential equations (<ref>). As a
consequence, the infinite hierarchy (<ref>) must be
identified as an unclosed and thus indeterminate set of equations,
irrespective of the fact that to every differential equation in
the hierarchy (<ref>) one can formally associate
a solution function to it, which then, in a unique way, is coupled
to the next higher order equation.
And, once accepted that the hierarchy (<ref>) is
unclosed, all invariant transformations which are admitted by this
system, as e.g. $\mathsf{L}_1$ (<ref>), $\mathsf{L}_2$
(<ref>) and $\mathsf{L}^f_2$ (<ref>),
must then be identified not as symmetry transformations, but only
as weaker equivalence transformations which map between unclosed
(see e.g. <cit.>);
in this case they even constitute indeterminate transformations.
This identification is clearly supported when studying the most
general invariant transformation which the system
(<ref>) can admit. It is given by two arbitrary
functions $f$ and $g$, one for the independent infinitesimal
$\xi=f(x,y_1)$ and one for the lowest order dependent
infinitesimal $\eta_1=g(x,y_1)$, which then both uniquely assign
the functional structure for all remaining infinitesimals $\eta_n$
in the form $\eta_n=\eta_n(f(x,y_1),g(x,y_1),y_2,y_3,\dots,y_n)$,
for all $n\geq 2$.[Note that each dependent
infinitesimal $\eta_n$ only shows a dependence up to order $n$ and
not beyond, i.e. it only depends on all dependent variables $y_m$
which appear below a considered level $n$, i.e. where $m\leq n$.]
This result shows complete arbitrariness in the choice for the
transformation of $x$ and $y_1$ to invariantly transform system
(<ref>), which, after all, is actually a trivial
result since the complete infinite hierarchy (<ref>)
can also be equivalently written in the form of an underdetermined
solution as
\begin{equation}
y_{n+1}=(-1)^n\frac{d^n y_1}{dx^n},\;\; n=1,2,3,\dots,
\label{150415:1027}
\end{equation}
where it's more than obvious now that the considered system
(<ref>) is not closed, since, through relation
(<ref>), all higher-order functions $y_{n+1}$ are
predetermined by the lowest order function $y_1$, but which itself
can be chosen completely arbitrarily. However, note that $y_1$ is
not privileged in the sense that only this function can be chosen
arbitrarily. Any function $y_{n^*}$ in the hierarchy
(<ref>) can be chosen freely, where $n=n^*$ is some
arbitrary but fixed order in this hierarchy. Its[4]
underdetermined general solution can then be written as
\begin{equation}
\left.
\begin{aligned}
& y_{1} = (-1)^{n^*-1}\int y_{n^*}\, d^{n^*-1} x\\
& y_{n^*-k} = (-1)^k\int y_{n^*}\, d^k x\\
& y_{n^*-1} = -\int y_{n^*}\, dx\\
& y_{n^*+1} = - \frac{d y_{n^*}}{dx}\\
& y_{n^*+2} = \frac{d^2 y_{n^*}}{dx^2}\\
& \;\vdots\\
& y_{n^*+l} = (-1)^l\frac{d^l y_{n^*}}{dx^l}\\
& \;\vdots
\end{aligned}
~~~ \right \} \label{150411:1120}
\end{equation}
A corresponding invariance analysis certainly sees the same
effect, namely that one function, anywhere in the infinite
hierarchy (<ref>), can be chosen freely. Hence, since
through (<ref>) any arbitrary general solution $\vy$
can be constructed, system (<ref>) does not allow for
a unique general solution. The primary general solution
(<ref>) is thus only one among an infinite set of
other, different possible general solutions which this system can
It should be noted here that our study only reveals the property
of global non-uniqueness when constructing a general
solution for an infinite system of differential equations which is
unrestricted$\,$; for example as for the plain system
(<ref>) when no restrictions or any further conditions
on the solution manifold are imposed. In particular, our
statements do not invalidate the local uniqueness principle which
may exist for a system of ODEs once its restricted to
satisfy an initial condition.[It is not exactly clear
yet in how far the well-defined local uniqueness principle for a
system of ODE initial value problems (Picard-Lindelöf theorem)
applies to systems which are infinite in dimension. Because, for
example, since for $\vA$ (<ref>) not all matrix norms
are finite, they cannot be regarded as equivalent anymore. That
means, in order to guarantee the necessary Lipschitz continuity
for the function $\vA\cdot\vy$ on some interval, the infinite
matrix $\vA$ needs to satisfy the condition $\|\vA\|\leq L$ for
some finite Lipschitz constant $L$, thus leading to a conclusion
which now depends on the matrix norm used: For the maximum, row
and column norm, which give
respectively, the function $\vA\cdot\vy$ is Lipschitz continuous,
while for a norm which gives an infinite value, e.g. like the
Euclidean norm $\|\vA\|_{2}\rightarrow\infty$, the function
$\vA\cdot\vy$ is not Lipschitz continuous.] Independent of
whether this principle (Picard-Lindelöf theorem) uniquely applies
to infinite dimensional ODE initial value systems or not, for the
simple linear and homogeneous ODE structure (<ref>),
however, it is straightforward to show that local uniqueness
in the solution for this particular infinite system must
exist, when specifying an initial condition at
$x=x_0\in\mathcal{I}$ inside some given local interval
$\mathcal{I}\subset\mathbb{R}$ (for the proof, see Appendix
<ref>). But, this local uniqueness interval $\mathcal{I}$ can be
quite narrow, and, depending on the chosen functions, can be even
of point-size only. In how narrow this local interval
$\mathcal{I}$ can be successively made is studied at a simple
example in Appendix <ref>.
Besides this, when specifying a particular initial condition, say
$\vy(x_0)=\vy_0$, or in component form $y_n(x_0)=y_{(0)n}$ for all
$n\geq 1$, then infinitely many and functionally independent
invariant (equivalence) transformations can be constructed which
all are compatible with this arbitrary but specifically chosen
initial condition. Because, since e.g. the infinitesimals
$\xi=f(x,y_1)$ and $\eta_1=g(x,y_1)$ can be chosen arbitrarily,
one only has to guarantee that the initial condition
$y_n(x_0)=y_{(0)n}$, for all $n\geq 1$, gets mapped invariantly
into itself. This is achieved by demanding all infinitesimals to
satisfy the restrictions
\begin{align}
\xi(x,y_1)\Big\vert_{\{x=x_0;\vy=\vy_0\}}=0,\qquad
\eta_1(x,y_1)\Big\vert_{\{x=x_0;\vy=\vy_0\}}=0,\hspace{1cm}
\label{150513:0759}\\[0.75em]
\eta_n=\eta_n\Big(\xi(x,y_1),\eta_1(x,y_1),y_2,y_3,\dots,y_n\Big)
\Big\vert_{\{x=x_0;\vy=\vy_0\}}=0,\;\; n\geq 2,
\label{150513:0800}
\end{align}
where only the two infinitesimals $\xi$ and $\eta_1$ can
be chosen freely, while the remaining infinitesimals $\eta_n$, for
all $n\geq 2$, are predetermined differential functions of their
indicated arguments. The conditions (<ref>), in
accordance with (<ref>), can be easily fulfilled
e.g. by restricting the arbitrary functions $\xi$ and $\eta_1$ to
\begin{equation}
\xi(x,y_1)=f_0(x,y_1)\cdot
\eta_1(x,y_1)=g_0(x,y_1)\cdot
e^{-\frac{\gamma_g^2}{(y_1-y_{(0)1})^2}}, \label{150513:0906}
\end{equation}
where $f_0$ and $g_0$ are again arbitrary functions, however, now
restricted to the class of functions which are increasing slower
than $e^{1/r^2}$ at $r=0$, where
$r=\sqrt{(x-x_0)^2/\gamma_f^2+(y_1-y_{(0)1})^2/\gamma_g^2}$. And,
since in this case all differential functions $\eta_n$, for $n\geq
2$, have the special non-shifted affine property
$\eta_n|_{\{\xi=0;\,\eta_1=0\}}=0$, the conditions
(<ref>) all are automatically satisfied by the above
restriction (<ref>). Hence, an infinite set of
functionally independent (non-privileged) invariant solutions
$\vy=\vy(x)$ can be constructed from (<ref>) which all
satisfy the given initial condition $\vy(x_0)=\vy_0$.
Remark on partial overlapping and
analytic continuation:
Before closing this section let's briefly revisit the
result (<ref>). Important to mention here is that if
we expand the function $\frac{\tilde{x}}{1+\varepsilon \tilde{x}}$
into a power series around some arbitrary point $\tilde{x}=a\neq
-1/\varepsilon$, then the alternative general solution
$\tilde{\vy}$ (<ref>) will map into a
reparametrization of the primary solution (<ref>), but
only in a very restrictive manner due to the existence of three
restrictions in order to ensure overall convergence (see
derivation (<ref>) and (<ref>)):
If, in $\mathbb{R}$, the chosen values for $\tilde{x}$,
$\varepsilon$ and $a$ satisfy the following three restrictions
simultaneously[Note that since there are three
restrictions for three values, $\tilde{x}$, $\varepsilon$ and $a$,
they can not be chosen arbitrarily and independently anymore, i.e.
all three values depend on each other according to
(<ref>). In general this combined set of restrictions
leads to a very narrow radius of convergence.]
\begin{equation}
\bigg\vert\,
\frac{(\tilde{x}-a)\,\varepsilon}{1+a\,\varepsilon}\,\bigg\vert <
1 ,\;\;\; \bigg\vert\,
\frac{a\,\varepsilon}{1+a\,\varepsilon}\,\bigg\vert < 1 ,\;\;
\text{and}\;\; \vert\, \tilde{x}\,\varepsilon\,\vert < 1,
\label{150409:2242}
\end{equation}
then, and only then, the symmetry transformation $\mathsf{L}_2$
(<ref>) allows for a reparametrization of the primary
solution (<ref>)
\begin{equation}
\tilde{\vy}=\vG\!\left({\textstyle\frac{\tilde{x}}{1+\varepsilon
\tilde{x}}}\, ,\varepsilon\right)\cdot
\Big(e^{-\frac{\tilde{x}}{1+\varepsilon
\tilde{x}}\vA}\cdot\vc\Big)= e^{-\tilde{x}\vA}\cdot\tilde{\vc},
\label{150407:1136}
\end{equation}
where the reparametrized integration constant $\tilde{\vc}$ is
then given by (<ref>), or equivalently
by (<ref>), which both take the same form
\begin{equation}
\tilde{\vc}=\vG(0,\varepsilon)\cdot \vc. \label{150413:1036}
\end{equation}
However, for the remaining wide range of values $\tilde{x}$,
$\varepsilon$ and $a$ within $\mathbb{R}$, namely for all those
values in which at least one of the three restrictions
(<ref>) is violated, we still have, as was discussed
before, a second, fundamentally different general solution
$\tilde{\vy}$ (<ref>) than as given by the primary
general solution $\vy$ (<ref>).
Further note that this particular example even allows for an
interesting special case when choosing $x_0=a=0$ as the
position, and $\vy_0=\tilde{\vc}$ (<ref>) as the value
for an initial condition $\vy(x_0)=\vy_0$ of
system (<ref>). Because, since on the one side the
symmetry transformation (<ref>) invariantly maps the
initial condition's space point from $x_0=0$ to $\tilde{x}_0=0$,
and since on the other side there exist a common domain
(<ref>) in which the transformed general solution
(<ref>) matches the primary general solution
(<ref>), the former (transformed) solution serves as
the functional continuation of the latter (primary) solution; but
only if, of course, both solutions originate from the same initial
condition, and if the primary solution (<ref>)
converges on a given (untransformed) integration constant $\vc$.
To be explicit, we state that when considering the initial value
\begin{equation}
\vy^\prime(x)=-\vA\cdot\vy(x),\;\;\text{with}\;\; \vy(0)=\vy_0,
\label{150519:1034}
\end{equation}
where $\vA$ is the infinite matrix given by (<ref>)
and $\vy_0$ some arbitrary constant, then two solutions $\vy^A$
and $\vy^B$ exist, namely the primary solution $\vy^A$
(<ref>) and the transformed solution $\vy^B$
\begin{align}
\vy^A(x) & = e^{-x\vA}\cdot\vy_0,\label{150413:1430}\\
\vy^B(x) & = \vG\!\left({\textstyle\frac{x}{1+\varepsilon x}}\,
,\varepsilon\right)\cdot \bigg[e^{-\frac{x}{1+\varepsilon
\label{150413:1431}
\end{align}
which both satisfy the same initial condition
$\vy^A(0)=\vy^B(0)=\vy_0$, but where, according to the local
uniqueness principle for ODE initial value problems, each solution
serves as the functional continuation of the other solution on a
domain where it is not converging anymore. This domain depends on
the explicit initial value $\vy_0$ — here $\vG$ is the infinite
group matrix (<ref>) with its inverse
$\vG^{-1}(0,\varepsilon)=\vG(0,-\varepsilon)$ (see
(<ref>)). In particular, if we choose the initial
value as given in (<ref>), i.e. if
$\vy_0=\tilde{\vc}$, with $\tilde{\vc}=\vG(0,\varepsilon)\cdot
\vc$, and where the primary (untransformed) integration constant
is e.g. fixed as $\vc\sim\boldsymbol{1}$, then the transformed
solution $\vy^B$ significantly extends the primary solution range
of $\vy^A$. The reason is that, since according to
(<ref>) we are considering the special case $a=0$, the
primary solution $\vy^A$ (<ref>) for this initial
condition only converges in the restricted domain $\vert
x\,\varepsilon\vert <1$, while the transformed solution $\vy^B$
(<ref>) converges in the complete range
$x\in\mathbb{R}\backslash\{{\textstyle -\frac{1}{\varepsilon}}\}$.
Symbolically we thus have the relation $\vy^A\subset\vy^B$. For
more details and for a graphical illustration of these statements,
see Appendix <ref>. Surely, if we choose the initial value
$\vy_0$ such that on a specific domain the solution $\vy^B$ is not
converging, then $\vy^A$ serves as its continuation, i.e. we then
have the opposite relation $\vy^B\subset\vy^A$. See Table <ref>
in Appendix <ref> for a collection of several choices in the
initial value $\vy_0$ and the corresponding domains for which the
solutions $\vy^A$ (<ref>) and $\vy^B$
(<ref>) are converging.
§.§ Infinite nonlinear hierarchy of first order ODEs
In this section we will consider the nonlinear specification
$q_0=q_3=0,q_1=1/x,q_2=1/x^3$ of (<ref>):
\begin{equation}
\end{equation}
This coupled system represents a genuine system of nonlinear
equations, since it cannot be reduced to a linear set of equations
as in the case of the corresponding single Riccati-ODE
(<ref>) via the transformation $y(x)=x/z(x)$. In
addition, its underdeterminate solution cannot be written
compactly in closed form anymore as it was possible for the
previously considered linear system (<ref>), either
through (<ref>), or, more generally, through
(<ref>). But nevertheless, since (<ref>)
is an infinite forward recurrence relation of first
order, i.e. where each term $y_{n+1}$ in the sequence depends on
the previous term $y_n$ in the functional form
$y_{n+1}=\mathcal{F}[y_n]=\pm\sqrt{x^3y_n^\prime-x^2y_n}$,it naturally acts again as an unclosed (underdetermined) system,
where, on any level in the prescribed hierarchy
(<ref>), exactly one function can be chosen freely.
This statement is again supported when performing an invariance
analysis upon system (<ref>), namely in the same way
as it was already discussed in the previous section: To transform
system (<ref>) invariantly, complete arbitrariness
exists in that two arbitrary functions are available in order to
perform the transformation, one for the independent variable $x$
and one for any arbitrary but fixed chosen dependent variable
$y_{n^*}$, i.e. where in effect the transformation of one function
$y_{n^*}=y_{n^*}(x)$ can be chosen absolutely freely. Hence, again
as in the previous case, infinitely many functionally independent
equivalence transformations can be constructed in the sense of
(<ref>), all being then compatible with any
specifically chosen initial condition $\vy(x_0)=\vy_0$.
As explained in the previous section in detail, an unclosed set of
differential equations, e.g. such as (<ref>) or
(<ref>), does not allow for the construction of a unique general solution that can cover all possible special solutions these systems can admit. To explicitly
demonstrate this again for the nonlinear system
(<ref>), we first construct its most obvious general
solution based on the power series
\begin{equation}
y_n(x)=x^2\cdot \sum_{k=0}^\infty \lambda_{n,k}\, (x-a)^k,\;\;
n\geq 1, \label{150423:1555}
\end{equation}
where $a\in\mathbb{R}$ is some arbitrary expansion point. To be a
solution of (<ref>), the expansion coefficients have
to satisfy the following first order recurrence relation (see
Appendix <ref>)
\begin{equation}
k\cdot \lambda_{n,k}+a\cdot (k+1)\cdot
\lambda_{n,k+1}+\lambda_{n,k} = \sum_{l=0}^k
\lambda_{n+1,k-l}\cdot \lambda_{n+1,l}\, ,\;\text{for all}\;\;
n\geq 1,\, k\geq 0. \label{150423:1603}
\end{equation}
For arbitrary but fixed given initial values $c_n:=\lambda_{n,0}$,
this recurrence relation can be explicitly solved for all higher
orders relative to the expansion index $k\geq 1$. If $a\neq 0$,
the first four expansion coefficients, for all $n\geq 1$, are then
given as
\begin{equation}
\left .
\begin{aligned}
\lambda_{n,0} & = c_n,\\[0.5em]
\lambda_{n,1} & = -\frac{1}{a}\cdot
\Big(c_n-c_{n+1}^2\Big),\\[0.5em]
\lambda_{n,2} & =
\frac{1}{a^2}\cdot\Big(c_n-2c_{n+1}^2+c_{n+1}\cdot c_{n+2}^2\Big),\\[0.5em]
\lambda_{n,3} & =
c_{n+2}^2-2c_{n+1}\cdot c_{n+2}\cdot
& \;\;\vdots
\end{aligned}
~~~~~~~~ \right\}\!\!\!\!\!\! \label{150423:1617}
\end{equation}
while if $a=0$, they are given as[For $a=0$
the recurrence relation (<ref>) can be solved by
making use for example of a generating function or the more
general $Z$-transform (see e.g. <cit.>). Hereby should
be noted that relation (<ref>) is a 1-dimensional
recurrence relation of first order for any arbitrary but fixed
order $n$.]
\begin{equation}
\left .
\begin{aligned}
\negthickspace \negthickspace\negthickspace
\phantom{c_n =} \lambda_{n,0} & = e^{2^{1-n}\cdot\, \sigma_1}=c_n,\\[0.5em]
\lambda_{n,1} & = e^{(2^{1-n}-1)\cdot\sigma_1}\cdot
\sigma_2,\\[0.5em]
\lambda_{n,2} & = 2^{-n}\cdot
3^{-1+n}\cdot e^{\sigma_1}\cdot\sigma_3\right),\\[0.5em]
\lambda_{n,3} & = 2^{-1-n}\cdot
(2^{1+n}+2^{2+2n}-2\cdot 3^{1+n})\\
&\qquad\qquad\qquad\qquad\qquad\quad\;\; +(4\cdot
3^n-4^{1+n})\cdot e^{\sigma_1}\cdot \sigma_2\cdot\sigma_3+4^n\cdot
& \;\;\vdots
\end{aligned}
~~ \right\} \label{150423:1618}
\end{equation}
where $\vsigma=(\sigma_1,\sigma_2,\dotsc,
\sigma_n,\dotsc)$ is a new infinite set of integration constants,
which now, instead of $\vc=(c_1,c_2,\dotsc,c_n,\dotsc)$, take the
place for the freely selectable parameters in the general solution
(<ref>) as soon as the expansion point $a$ turns to
zero; simply because in this singular case all constants $c_n$
within (<ref>) cannot be chosen independently anymore
as they would all depend on the single choice of the first
parameter $\sigma_1$ as given above in the first line of
That solution (<ref>), along with either
(<ref>) or (<ref>), represents a general
solution is obvious, since to each solution $y_n$ of the first order system (<ref>) one can associate to it a
free parameter for all $n\geq 1$, either $c_n$ if $a\neq 0$, or
$\sigma_n$ if $a=0$. But this general solution is not unique
as it does not cover all possible special solutions which
that system (<ref>) can admit. For example, if we
consider the following independent[The only dependence
which exists between these three different solutions is that
$\lim_{n\to\infty}y_n^{(3)}=y_{n^*}^{(1)}=x^2$ for all $n^*\geq
1$.] special solutions of (<ref>)
\begin{align}
y_n^{(1)}(x) & = x^2, \;\;\text{for all}\;\; n\geq 1,\label{150423:1914}\\[0.5em]
y_n^{(2)}(x) & = \begin{cases}\, -1, \;\;\text{for}\;\;
\;\;\; x, \;\;\text{for}\;\; n=2,\\
\;\;\:\, 0, \;\;\text{for all}\;\; n\geq 3, \end{cases}\label{150423:1946}\\[0.5em]
y_n^{(3)}(x) & = \frac{1}{5^{2^{-n}}}\left[\,\prod_{k=0}^{n-1}
\left(1+\frac{1}{2^{k-2}}\right)^{2^{k-n}}\,\right]\cdot
x^{2+\frac{1}{2^{n-2}}}, \;\;\text{for all}\;\; n\geq
\end{align}
only solution $y_n^{(1)}$ is covered by the general
solution (<ref>) for all $n\geq 1$, in choosing either
$c_n=1$ (if $a\neq 0$), or $\sigma_n=0$ (if $a=0$). However, the
special solution $y_n^{(2)}$, and in general also
$y_n^{(3)}$,are not covered. The reason for $y_n^{(2)}$
(<ref>) is obvious, because it's a polynomial with a
smaller degree than $y_n$ (<ref>), which itself is
always at least of second order for all $n\geq 1$. And to
see the reason for $y_n^{(3)}$ (<ref>), it's
sufficient to explicitly evolve it up to third order in $n$:
\begin{equation}
y_1^{(3)}=x^4,\;\;\;\; y_2^{(3)}=\sqrt{3}\cdot x^3,\;\;\;\;
y_3^{(3)}=\sqrt{2}\cdot\sqrt[4]{3}\cdot x^{5/2},\;\;\;\cdots
\end{equation}
Because, if $a=0$, one has to choose $\sigma_1\rightarrow
-\infty$ in order to obtain the lowest order particular solution
$y_1^{(3)}$, which then turns into a contradiction when trying to
determine $\sigma_2$ for the next higher order particular solution
$y_2^{(3)}$. This indeterminacy will then propagate through all
remaining orders $n\geq 3$, i.e. for $a=0$ no consistent set of
expansion coefficients $\lambda_{n,k}$ (<ref>) can be
determined to generate the special solution (<ref>)
from the general solution (<ref>). If, however, the
expansion point of the general solution (<ref>) is
chosen to be $a\neq 0$, then one obtains a coverage for the
special solution $y_n^{(3)}$ (<ref>), but only
partially, namely only in the domain $|x-a|<|a|$ (for the proof,
see Appendix <ref>). Hence, the general solution
(<ref>) is not unique, since other general solutions
of (<ref>) must exist in order to completely cover for
example the special solutions $y_n^{(2)}$ (<ref>) and
$y_n^{(3)}$ (<ref>) for all $x\in\mathbb{R}$.
To close this investigation it is worthwhile to mention that the
infinite nonlinear hierarchy of coupled equations
(<ref>) allows for two invariant Lie group actions
which are uncoupled. For all $n\geq 1$, these are:
\begin{equation}
\left. \begin{aligned} \mathsf{T}_1^\infty: & \;\;\;
\tilde{x}=e^{\varepsilon_1} x,\;\;\;
\tilde{y}_n=e^{2\varepsilon_1}y_n,\\[0.75em]
\mathsf{T}_2^\infty: & \;\;\; \tilde{x}=
\frac{x}{1-\varepsilon_2\, x},\;\;\;
\tilde{y}_n=\frac{y_n}{1-\varepsilon_2\, x},
\end{aligned}
~~~ \right \} \label{150424:2033}
\end{equation}
being the equivalent invariances to the scaling symmetry
$\mathsf{T}_1$ and projective symmetry $\mathsf{T}_2$ of the
corresponding single Riccati-ODE (<ref>) in
(<ref>) respectively. Their infinitesimal form has the
\begin{equation}
\left. \begin{aligned} \mathsf{T}_1^\infty: & \;\;\; X_1^\infty=\,
\mathsf{T}_2^\infty: & \;\;\; X_2^\infty=\,
\end{aligned}
~~~ \right \} \label{150424:2258}
\end{equation}
and, as proven in Appendix <ref>, they are the two
only possible non-coupled Lie-point invariances which the
coupled hierarchy (<ref>) of first order Riccati-ODEs
can admit. As was demonstrated in the previous section, these
invariant (equivalence) transformations (<ref>) can
now be used to either generate new additional special
solutions or to generate functionally different general
solutions by just transforming (<ref>) respectively.
Hereby note that the special solution $y_n^{(1)}$
(<ref>) is an invariant solution with respect to
the scaling invariance $\mathsf{T}_1^\infty$ (<ref>),
which even can be prolonged to the more general invariant solution
\begin{equation}
y_n^{(1)}(x;\tau)=e^{\tau\cdot2^{1-n}}\cdot x^2,
\end{equation}
which then involves a free parameter $\tau\in\mathbb{R}$ for all
$n\geq 1$.
§ CONCLUSION
At the example of first order ODEs this study has shown that an
infinite and forward recursive hierarchy of differential equations
carries all features of an unclosed system, and that,
conclusively, all admitted invariance transformations must be
identified as equivalence transformations only. To obtain from
such systems an invariant solution which shows a certain
particular functional structure is ultimately without value, since
infinitely many functionally different and non-privileged
invariant solutions can be constructed, even if sufficient initial
conditions are additionally imposed. In order to obtain valuable
results, the infinite system needs to be closed by posing
modelling assumptions which have to reflect the structure of the
underlying (higher abstraction level) equations from which the
infinite system emerges.
It is clear that this insight is not restricted to ODEs, but that
it holds for differential equations of any type as soon as the
infinite hierarchy is of a forward recursive nature. For example,
as it's the case for the infinite Friedmann-Keller hierarchy of
PDEs for the multi-point moments in statistical turbulence theory.
As it is discussed in detail in <cit.>and
further in <cit.>, this infinite system is undoubtedly
unclosed and that it's simplywithout any value,
therefore, to determine particular invariant solutions if no prior
modellingassumptions are invoked on that system.
§ INFINITE BACKWARD VERSUS INFINITE FORWARD DIFFERENTIAL
RECURRENCE RELATIONS
§.§ Example for an infinite backward differential recurrence relation
(Closed system with unique solution manifold)
Let us consider the Cauchy problem (<ref>) of
Example [E5]5 in the simplified form $a=1$ and $b=c=0$,
along with an initial function $\phi$ which is normalized to
$\int_{-\infty}^\infty \phi(x)dx=1$. Then this initial value
problem (<ref>) has the unique solution
\begin{equation}
u(t,x) = \frac{1}{\sqrt{4\pi t}}\int_{-\infty}^\infty
e^{-\frac{(x-x^\prime)^2}{4t}}\phi(x^\prime) \,
dx^\prime,\;\;\text{for}\;\; t\geq 0, \label{150501:1915}
\end{equation}
which by construction, due to $c=0$, automatically satisfies the
normalization constraint (<ref>). To study this
uniqueness issue on the corresponding moment induced ODE system
(<ref>), let usfirst consider the unrestricted system
\begin{equation}
\frac{d u_n}{dt} = n\cdot(n-1)\cdot u_{n-2},\;\; n\geq
\end{equation}
which, as an infinite backward recursive system, can be
written into the equivalent form
\begin{equation}
\left .
\begin{aligned}
& \frac{du_0(t)}{dt}=0,\quad \,\,\frac{d^{n+1}
=(2n+2)!\cdot u_0(t),\;\; n\geq 0,\\[0.5em]
\text{and} \;\;\, &\frac{du_1(t)}{dt}=0,\quad \frac{d^m
u_{2m+1}(t)}{dt^m}=(2m+1)!\cdot u_1(t),\;\; m\geq 1.
\end{aligned}
~~~~ \right\} \label{150504:1341}
\end{equation}
This system can then be uniquely integrated to give the
general solution
\begin{equation}
\!\!\!\!\!\!\left .
\begin{aligned}
u_0(t)& =c_0,\\[0.5em]
dt^\prime\, dt_0\cdots
dt_{n-1}+\sum_{k=0}^{n}\frac{q^{(1)}_{n,k}}{k!}\, t^k,\; n\geq 0,\\[0.5em]
\quad u_1(t)&=c_1,\\[0.5em]
dt^\prime\, dt_1\cdots
dt_{m-1}+\sum_{k=0}^{m-1}\frac{q^{(2)}_{m,k}}{k!}\, t^k,\; m\geq
\end{aligned}
\right\}\label{150501:1916}
\end{equation}
with the expansion coefficients given as
\begin{equation}
\left .
\begin{aligned}
q^{(1)}_{n,k}&=\frac{(2n+2)!}{(2n+2-2k)!}\, c_{2n+2-2k},\;\; n\geq
0;\;\; 0\leq k\leq n,\\[0.5em]
q^{(2)}_{m,k}&=\frac{(2m+1)!}{(2m+1-2k)!}\, c_{2m+1-2k},\;\; m\geq
1;\;\; 0\leq k\leq m-1,
\end{aligned}
~~~ \right\}
\end{equation}
where all $c_n$ for $n\geq 0$ are arbitrary integration
constants. Hence we see that the unrestricted system
(<ref>) provides a general solution
(<ref>) which only involves arbitrary constants, i.e.
the unrestricted system (<ref>) provides a unique general solution. Because, when restricting this system to
the underlying PDE's initial condition $u(0,x)=\phi(x)$, with
$\int_{-\infty}^\infty\phi(x)dx=1$, which for the ODE system
(<ref>) takes the form
\begin{equation}
u_n(0)=\int_{-\infty}^\infty x^n\cdot \phi(x)\, dx,\;\; n\geq
0,\;\;\text{with}\;\; u_0(0)=1, \label{150501:1934}
\end{equation}
it will turn the general solution (<ref>) into a
unique and fully determined solution, where the integration
constants are then given by
\begin{equation}
c_n=u_n(0), \;\; n\geq 0,\;\;\text{with}\;\; c_0=1.
\end{equation}
§.§ Example for an infinite forward differential recurrence relation
(Unclosed system with non-unique solution manifold)
Now, let's consider the case $a=1$, $b=0$ and $c=-1$, where again
the initial condition function $\phi$ is normalized to
$\int_{-\infty}^\infty \phi(x)dx=1$. To solve this initial value
problem (Cauchy problem)
\begin{equation}
\partial_t u=\partial_x^2 u-x^2 u,\;\:\text{for}\;\: t\geq
0,\;\;\text{with}\;\; u(0,x)=\phi(x), \label{150504:2203}
\end{equation}
it is necessary to realize that the following nonlinear point
transformation <cit.>[This continuous
point transformation is not a group transformation, as it
neither includes a group parameter nor does it include the unique
continuously connected identity transformation from which any
infinitesimal mapping can emanate.]
\begin{equation}
\tilde{t}=\frac{1}{4}\cdot\left(e^{4t}-1\right),\quad
\tilde{x}=x\cdot e^{2t},\quad \tilde{u}=u\cdot
e^{-\frac{1}{2}x^2-t}, \label{150504:2314}
\end{equation}
which has the unique inverse transformation
\begin{equation}
u=\tilde{u}\cdot \sqrt[4]{1+4\tilde{t}}\cdot
\end{equation}
maps the original Cauchy problem (<ref>) into the
following Cauchy problem for the standard diffusion equation with
constant coefficients:
\begin{equation}
\partial_{\tilde{t}\,}
\tilde{u}=\partial_{\tilde{x}\,}^2\tilde{u},\;\:\text{for}\;\:
\tilde{t}\geq 0,\;\;\text{with}\;\;
\tilde{u}(0,\tilde{x})=\phi(\tilde{x})\cdot
e^{-\frac{1}{2}\tilde{x}^2}. \label{150504:2210}
\end{equation}
Important to note here is that the initial time $t=0$ as well as
the relevant time range $t\in [0,\infty)$ both get invariantly
mapped to $\tilde{t}=0$ and $\tilde{t}\in [0,\infty)$
respectively. Hence, the unique solution of the transformed Cauchy
problem (<ref>) is thus again given by
(<ref>), but now in the form
\begin{equation}
\tilde{u}(\tilde{t},\tilde{x}) =
\frac{1}{\sqrt{4\pi^{\vphantom{A^1}}
\tilde{t}}}\int_{-\infty}^\infty
\, e^{-\frac{1}{2}\tilde{x}^{\prime\, 2}} \,
d\tilde{x}^\prime,\;\;\text{for}\;\; \tilde{t}\geq 0,
\label{150504:2302}
\end{equation}
which then, according to transformation (<ref>), leads
to the unique solution for the original Cauchy
problem (<ref>)
\begin{equation}
\frac{e^{\frac{1}{2}x^2+t}}{\sqrt{\pi\big(e^{4t}-1^{\vphantom{1}}\big)}}
\int_{-\infty}^\infty
\,\, e^{-\frac{1}{2}x^{\prime\, 2}} \,
dx^\prime,\;\;\text{for}\;\; t\geq 0. \label{150504:2344}
\end{equation}
Considering, however, the corresponding moment induced infinite
ODE system (<ref>) for (<ref>), we will
now show that this system is not uniquely specified and thus
has to be identified as an unclosed system even if sufficient
initial conditions are imposed. In clear contrast to its
associated higher level PDE system (<ref>), which, as
a Cauchy problem, is well-posed by providing the unique solution
(<ref>). To see this, let us first again consider the
unrestricted ODE system
\begin{equation}
\frac{du_n}{dt} = n\cdot (n-1)\cdot u_{n-2}-u_{n+2},\;\; n\geq 0,
\label{150507:1032}
\end{equation}
which can be rewritten into the equivalent and already solved
form[In the following we agree on the definitions that
$\frac{d^0}{dt^{0}}=1$, $\frac{d^{q<0}}{dt^{q}}=0$, and
\begin{equation}
\left.
\begin{aligned}
A^{(1)}_i(n)\frac{d^{n+1-2i}}{dt^{n+1-2i}}\, u_0(t),\;\; n\geq
A^{(2)}_j(m)\frac{d^{m+2-2j}}{dt^{m+2-2j}}\, u_1(t),\;\; m\geq 1,
\end{aligned}
~~~ \right\}\label{150507:1030}
\end{equation}
where the coefficients $A^{(1)}_i(n)$ and $A^{(2)}_j(m)$ are
recursively defined as:
\begin{equation}
\left .\!\!\!\!\!\!
\begin{aligned}
\bullet &\;\:\text{{\it Initial seed for $A^{(1)}_i(n)$}:}\;\;\;
A^{(1)}_0(-1)=1,\;\text{and}\;\: A^{(1)}_0(n)=1,\;\text{for all $n\geq0$},\hspace{1.4cm}\\[0.5em]
&\;\;\: A^{(1)}_i(n) = \sum_{k=0}^{n-(2i-1)} (2n-2k)\cdot
A^{(1)}_{i-1}(n-2-k),\;\;\; i\geq 1,\;\; n\geq 0,\hspace{0.8cm}\\[0.5em]
\bullet &\;\:\text{{\it Initial seed for $A^{(2)}_j(m)$}:}\;\;\;
A^{(2)}_1(m)=1,\;\text{for all $m\geq 1$},\hspace{1.375cm}\\[0.5em]
&\;\;\: A^{(2)}_j(m) = \sum_{k=1}^{m-(2j-3)} (2m-2k)\cdot
(2m+1-2k)\cdot A^{(2)}_{j-1}(m-1-k),\;\;\; j\geq 2,\;\; m\geq 1.
\end{aligned}
\right\}
\end{equation}
In contrast to the general solution
(<ref>) of the previously considered unrestricted system (<ref>), we see that the degree
of underdeterminedness in the above determined general
solution (<ref>) is fundamentally different and higher
than in (<ref>). Instead of integration constants
$c_n$, we now have two integration functions $u_0(t)$ and $u_1(t)$
which can be chosen freely. Their (arbitrary) specification will
then determine all other solutions for $n\geq 0$ and $m\geq 1$
according to (<ref>). The reason for having two free
functions and not infinitely many free constants is that system
(<ref>) defines a forward recurrence relation
(of order two)[The order of the recurrence relation is
defined relative to the differential operator.] that needs not to be integrated in order to determine its general solution,
while system (<ref>), in contrast, defines a backward recurrence relation (of order two) which needs to be
integrated to yield its general solution.
To explicitly demonstrate that (<ref>) is not a unique general solution, we have to impose the corresponding
initial condition $u(0,x)=\phi(x)$, with $\int_{-\infty}^\infty
\phi(x)\, dx =1$, which led to the unique solution
(<ref>) of the underlying PDE system
(<ref>). For the current ODE system
(<ref>) this condition takes again the same form
(<ref>) as it did for previous ODE system
\begin{equation}
u_n(0)=\int_{-\infty}^\infty x^n\cdot \phi(x)\, dx,\;\; n\geq
0,\;\;\text{with}\;\; u_0(0)=1. \label{1505017:1438}
\end{equation}
The easiest way to perform this implementation is to choose the
two arbitrary functions $u_0(t)$ and $u_1(t)$ as analytical
functions which can be expanded as power series
\begin{equation}
u_0(t)=\sum_{k=0}^\infty \frac{c^{(1)}_k}{k!}\, t^k,\quad\;
u_1(t)=\sum_{k=0}^\infty \frac{c^{(2)}_k}{k!}\, t^k,
\label{150508:0818}
\end{equation}
where $c^{(1)}_k$ and $c^{(2)}_k$ are two different infinite sets
of constant expansion coefficients. By inserting this Ansatz into
the general solution (<ref>) and imposing the initial
conditions (<ref>) will then uniquely specify these
coefficients in a recursive manner as
\begin{equation}
\left .
\begin{aligned}
c^{(1)}_k =0,\;\; k<0; \quad\;\; c^{(1)}_k & =(-1)^k\cdot
A^{(1)}_i(k-1)\cdot c^{(1)}_{k-2i},\;\;k\geq 0,\\[0.5em]
c^{(2)}_k =0,\;\; k<0; \quad\;\; c^{(2)}_k & =(-1)^k\cdot
u_{2k+1}(0)-\sum_{i=1}^\infty A^{(2)}_{i+1}(k)\cdot
c^{(2)}_{k-2i},\;\;k\geq 0.
\end{aligned}
~~~ \right\}\label{150507:2005}
\end{equation}
Indeed, the two solutions (<ref>) with the
above determined coefficients (<ref>) form the
analytical part of the corresponding unique PDE moment solutions
relative to (<ref>). For example, choosing the
non-symmetric and to one normalized initial condition function
$\phi(x)=\frac{1}{\sqrt{\pi}}e^{-(x-1)^2}$ will give the first two
unique PDE moment solutions as
\begin{equation}
\left .
\begin{aligned}
u_0(t)&=\int_{-\infty}^\infty x^0\cdot u(t,x)\, dx = \frac{2\,
t\geq 0,\\[0.75em]
u_1(t)&=\int_{-\infty}^\infty x^1\cdot u(t,x)\, dx =\frac{8\,
t\geq 0,
\end{aligned}
~~~ \right\}\label{150508:0825}
\end{equation}
and, if these were Taylor expanded around $t=0$, they
would exactly yield the first two power series solutions
(<ref>) of the associated infinite ODE system
(<ref>). But, the Taylor expansions of both functions
(<ref>) only converge in the limited range $0\leq t <
\frac{1}{4}\sqrt{\pi^{2^{\vphantom{.}}}+(\ln 3)^2}\sim 0.83$.
That means, our initial assumption that the first two ODE
solutions $u_0(t)$ and $u_1(t)$ are analytical functions on the
global and unlimited scale $t\in[0,\infty)$ is thus not correct.
Only for a very limited range this functional choice
(<ref>) is valid. But, if we don't know the full scale
PDE solutions (<ref>) beforehand, how then to choose
these two unknown functions $u_0(t)$ and $u_1(t)$ for the infinite
ODE system (<ref>)? The clear answer is that there is
no way without invoking a prior modelling assumption on the ODE
system itself. Even if we would choose specific functions $f_0(t)$
and $f_1(t)$, which for $u_0(t)$ and $u_1(t)$ are valid on any
larger scale than the limited analytical Ansatz
(<ref>), we still have the problem that this
particular solution choice is not unique, because one can always
add to this choice certain independent functions which give no
contributions when evaluated at the initial point $t=0$. For
example, if $u_0(t)=f_0(t)$ and $u_1(t)=f_1(t)$, and if both
functions $f_0$ and $f_1$ satisfy the given initial conditions at
$t=0$, then
\begin{equation}
u_0(t)=f_0(t)+ \psi_0(t)\cdot e^{-\frac{\gamma_0^2}{t^2}},\qquad
u_1(t)=f_1(t)+ \psi_1(t)\cdot e^{-\frac{\gamma_1^2}{t^2}},
\label{150508:1335}
\end{equation}
is also a possible solution choice which satisfies the same
initial conditions, where $\psi_0(t)$ and $\psi_1(t)$ are again
arbitrary functions, with the only restriction that, at the
initial point $t=0$, they have to increase slower than
$e^{\gamma_0^2/t^2}$ and $e^{\gamma_1^2/t^2}$ respectively.
That no unique solution can be constructed a priori provides the
reason that the PDE induced ODE system (<ref>),
although infinite in dimension, has to be treated as an unclosed
system. It involves more unknown functions than there are
determining equations, although formally, in a bijective
manner, to each function within the hierarchy a corresponding
equation can be mapped to. But, since the hierarchy
(<ref>) can be equivalently rewritten into the form
(<ref>), it explicitly reveals the fact that exactly
two functions $u_0(t)$ and $u_1(t)$ in this hierarchy remain
unknown, and without the precise knowledge of their global
functional structure all remaining solutions $u_n(t)$ for $n\geq
2$ then remain unknown too. And, since the equivalently rewritten
form (<ref>) already represents the general
solution of the original infinite ODE system (<ref>),
the general solution itself is unclosed as well. In other words,
the general solution (<ref>) is not a unique
general solution. The degree of arbitrariness in having two
unknown functions cannot be reduced, even when imposing initial
conditions, simply due to the existing modus operandi in the sense
of (<ref>) when constructing possible valid solutions.
Hence, posing any initial conditions are thus not sufficient to
yield a unique solution for the (lower abstraction level) ODE
system (<ref>) as they are for the (higher abstraction
level) PDE equation (<ref>). Without a prior modelling
assumption on the ODE system (<ref>), this system
remains unclosed. Fortunately, the solutions of this particular
case (<ref>) possessed an analytical part in their
functions for which the assumed Ansatz (<ref>)
expressed the correct functional behavior, though only in a very
narrow and limited range. But, of course, for more general cases
such a partial analytical structure is not always necessarily
provided, and an Ansatz as (<ref>) would then be
§ ALTERNATIVE METHOD IN CONSTRUCTING A GLOBAL
Besides Lie's central theorem, a more efficient way to determine
the global 1-parametric symmetry transformation of $y_n$ from its
infinitesimal form (<ref>) for $n\geq 2$ is, in this
particular case, to make use of the underlying recurrence relation
(<ref>) along with the transformation rule
(<ref>) in its more general form:
\begin{align}
\tilde{y}_n &
=-\left(\frac{\partial\tilde{x}}{\partial x}\right)^{-1}
\left(\,\sum_{q=1}^{n-1}\frac{\partial \tilde{y}_{n-1}}{\partial
y_{q}}y^\prime_{q}+\frac{\partial \tilde{y}_{n-1}}{\partial
\end{align}
\begin{align}
\tilde{y}_n & =\left(\frac{\partial\tilde{x}}{\partial
x}\right)^{-1} \left(\,\sum_{q=1}^{n-1}\frac{\partial
\tilde{y}_{n-1}}{\partial y_{q}}y_{q+1}-\frac{\partial
\tilde{y}_{n-1}}{\partial x}\right) = (1-\varepsilon
\tilde{y}_{n-1}}{\partial y_{q}}y_{q+1}-\frac{\partial
\tilde{y}_{n-1}}{\partial x}\right)\nonumber\qquad\qquad\\
& = \, \sum_{k=1}^{n-1} B_{n,k}\,\varepsilon^{n-k-1}(1-\varepsilon
x)^{n+k-1}\, e^{\varepsilon}\, y_{k+1} ,\;\;\text{for all}\;\;
n\geq 2, \label{150404:1253}
\end{align}
where the coefficients $B_{n,k}$ are defined via the
following 2-dimensional recurrence relation[In order
to obtain from a (1+1)-dimensional recurrence relation a unique
solution it has to be supplemented by one initial condition and
two zero-dimensional boundary conditions; in full analogy to the
situation for PDEs.]
\begin{equation}
B_{n,k}=(n-2+k)\cdot B_{n-1,k}+B_{n-1,k-1},\;\; \text{for $n\geq
3$ and $k=1,2,\dotsc,n-1$}, \label{150518:1452}
\end{equation}
with the initial condition $B_{2,1}=1$, and the boundary
conditions $B_{n,0}=0$ (left boundary) and $B_{n,n}=0$ (right
boundary) for all $n\geq 2$.
§ LOCAL UNIQUENESS PROOF AND AN EXAMPLE ON ITS RANGE
Proposition: Given is the following initial value
problem for the infinite ODE system (<ref>)
\begin{equation}
\vy^\prime=-\vA\cdot\vy,\;\;\text{with}\;\; \vy(x_0)=\vy_0,
\label{150417:0935}
\end{equation}
in some local interval $\mathcal{I}\subset\mathbb{R}$, where
$x_0\in\mathcal{I}$. Then this (restricted) differential system
(<ref>) only has the one solution
\begin{equation}
\vy(x)=e^{-(x-x_0)\vA}\cdot \vy_0,\;\;\text{for all}\;\; x\in
\mathcal{I}.
\end{equation}
In particular, if $\vy_0=\boldsymbol{0}$ then
$\vy(x)=\boldsymbol{0}$ is the only solution for all $x\in
\mathcal{I}$.
Proof: Let $\vy=\vy(x)$ be any
solution which satisfies the initial value problem
(<ref>) in the given interval $\mathcal{I}$. Then we
can formulate the obvious relation
\begin{equation*}
\frac{d}{dx}\left(e^{(x-x_0)\vA}\cdot\vy\right)= \left(\vA\cdot
e^{(x-x_0)\vA}\right)\cdot\vy + e^{(x-x_0)\vA}\cdot\vy^\prime =
\left(\vA\cdot e^{(x-x_0)\vA}\right)\cdot\vy -
e^{(x-x_0)\vA}\cdot\left(\vA\cdot\vy\right) =\boldsymbol{0},
\end{equation*}
since the infinite matrix $\vA$ commutes with itself, i.e.
$[\vA,\vA^n]=\boldsymbol{0}$ for all $n\in\mathbb{N}$. This
relation then implies that
\begin{equation}
\end{equation}
where $\vc$ is some integration constant. But since $\vy$
satisfies the initial condition $\vy(x_0)=\vy_0$, we obtain the
result that $\vc=\vy_0$ and that thus the considered solution
$\vy$ can only have the unique form:
\begin{equation}
\hspace{4.6cm}\vy(x)=e^{-(x-x_0)\vA}\cdot \vy_0,\;\;\text{for
all}\;\; x\in \mathcal{I}.\hspace{3.25cm}\square
\end{equation}
$n=1,\;\: \gamma=1,\;\: |\mathcal{I}_1|\sim 0.9$
$n=2,\;\: \gamma=1,\;\: |\mathcal{I}_2|\sim 0.6$
$n=3,\;\: \gamma=1,\;\: |\mathcal{I}_3|\sim 0.4$
$n=4,\;\: \gamma=1,\;\: |\mathcal{I}_4|\sim 0.3$
Plots of the first four solutions of the initial value
problem (<ref>). The solid lines display the solutions
$y_n^\text{I}$ (<ref>) and the dashed lines the
solutions $y_n^\text{II}$ (<ref>) for $\gamma=1$. For
each order $n$, the highlighted region on the $x$-axis indicates
the local uniqueness interval $\mathcal{I}_n$ where
$y_n^\text{I}=y_n^\text{II}$. The size of each interval
$|\mathcal{I}_n|$ decreases as the order $n$ of the solution
increases. In the limit $n\rightarrow\infty$ the corresponding
interval narrows down to point-size, i.e.
$\lim_{n\to\infty}|\mathcal{I}_n|\rightarrow 0$. Hence, the size
of the common uniqueness interval $\mathcal{I}$ of the
initial value problem (<ref>), which is the
intersection of all intervals $\mathcal{I}=\bigcap_{n=1}^\infty
\mathcal{I}_n$, thus converges to point-size too.
$n=1,\;\: \gamma=0.1,\;\: |\mathcal{I}_1|\sim 0.2$
$n=1, \;\: \gamma=0.001,\;\: |\mathcal{I}_1|\sim 0.02$
Plots of the first order solutions $y_1^\text{I}$
((<ref>), solid lines) and $y_1^\text{II}$
((<ref>), dashed lines) for decreasing $\gamma$.
Hence, for $\gamma\rightarrow 0$ the size of the local uniqueness
interval $\mathcal{I}_{n^*}$ for each arbitrary but fixed order
$n=n^*$ diminishes to point-size.
[4]Example: In how narrow this local
uniqueness interval $\mathcal{I}$ can be made up to point-size, we
want to demonstrate at the following specific initial value
problem (<ref>)
\begin{equation}
\vy^\prime=-\vA\cdot\vy,\;\;\text{with}\;\; \vy(0)=\boldsymbol{1},
\label{150417:1123}
\end{equation}
in some local interval $\mathcal{I}\subset \mathbb{R}$ for
$x$ around the initial point $x_0=0$. Of course, globally,
i.e. for all $x\in\mathbb{R}$, the solution of the initial value
problem (<ref>) is not necessarily unique. Indeed, at
least two global solutions $\vy=(y_n)_{n\in\mathbb{N}}$ can be
found, e.g.
\begin{align}
y_n^\text{I} & =e^{-x}, \;\;\text{for all}\;\; n\geq 1,\label{150418:1428}\\[0.75em]
y_{n}^{\text{II}} & = \begin{cases}\,
e^{-x}+e^{-\frac{\gamma}{x^2}}, \;\;\text{for}\;\; n= 1,\; \gamma
\, (-1)^{n-1}{\displaystyle\frac{d^{n-1}
y_1^{\text{II}}}{dx^{n-1}}}, \;\;\text{for}\;\; n\geq
2,\end{cases} \label{150418:1429}
\end{align}
which both satisfy (<ref>) for all
$n\in\mathbb{N}$, and all $x\in\mathbb{R}$. Figure <ref> and
<ref> shows this for constant and different $\gamma$
§ REMAPPING OF THE GENERAL SOLUTIONS' INTEGRATION CONSTANT
§.§ Reparametrization of solution $n=1$
\begin{align}
\tilde{y}_1 & = e^{\varepsilon}y_1\nonumber\\
& =e^{\varepsilon}\sum_{k=0}^\infty c_{1+k}\frac{(-1)^k}{k!}\, x^k
= e^{\varepsilon}\sum_{k=0}^\infty
& =e^{\varepsilon}\sum_{k=0}^\infty
\frac{\tilde{x}^k(\tilde{x}-a)^{l}}{(1+a\,\varepsilon)^{k+l}}\right),\;\;
\text{for}\;\; \bigg\vert\,
\frac{(\tilde{x}-a)\,\varepsilon}{1+a\,\varepsilon}\,\bigg\vert <
1 ,\;\;
a\neq -\frac{1}{\varepsilon},\nonumber\\
& = e^{\varepsilon}\sum_{k=0}^\infty
\frac{(-1)^{l+q}\, a^q}{(1+a\,\varepsilon)^{k+l}}
\binom{l}{q}\frac{k\,(k+l)!}{k!\,(k+l)}\frac{\varepsilon^l}{l!} \,
\tilde{x}^{k+l-q}\right)
\nonumber\\
& = e^{\varepsilon}\sum_{k=0}^\infty
\frac{(-1)^{m}\, a^r}{(1+a\,\varepsilon)^{k+m+r}}
\binom{m+r}{r}\frac{k\,(k+m+r)!}{k!\,(k+m+r)}\frac{\varepsilon^{m+r}}{(m+r)!}
\, \tilde{x}^{k+m}\right)
\nonumber\\
& = \sum_{k=0}^\infty\, \sum_{m=0}^\infty\, \sum_{r=0}^\infty
a^r\,\varepsilon^{m+r}}{(1+a\,\varepsilon)^{k+m+r} k!\,
& = \sum_{i=0}^\infty \, \sum_{j=0}^i \, \sum_{r=0}^\infty
e^{\varepsilon} c_{1+j} \frac{(-1)^{i}\,
a^r\,\varepsilon^{i-j+r}}{(1+a\,\varepsilon)^{i+r} j!\,
(i-j+r)!}\binom{i-j+r}{r}\frac{j\, (i+r)!}{j!\,
& = \sum_{i=0}^\infty \left(\,\sum_{j=0}^i \, \sum_{r=0}^{\infty}
e^{\varepsilon} c_{1+j} \frac{i!\,
a^r\,\varepsilon^{i-j+r}}{(1+a\,\varepsilon)^{i+r} j!\, r!\,
(i-j)!}\frac{j\, (i+r)!}{j!\,
& = \sum_{i=0}^\infty \left(\,\sum_{j=0}^i e^{\varepsilon} c_{1+j}
\frac{i!\, \varepsilon^{i-j}}{j!\, (i-j)!}\frac{i!\, j}{i\, j!}
\sum_{r=0}^\infty
\frac{(a\,\varepsilon)^r}{(1+a\,\varepsilon)^{i+r}\, r!}
\frac{i\, (i+r)!}{i!\, (i+r)}\right)\frac{(-1)^i}{i!}\,\tilde{x}^i\nonumber\\
& = \sum_{i=0}^\infty \left(\,\sum_{j=0}^i e^{\varepsilon} c_{1+j}
\frac{i!\,\varepsilon^{i-j}}{j!\, (i-j)!}\frac{i!\, j}{i\,
j!}\cdot 1\right)\frac{(-1)^i}{i!}\,\tilde{x}^i,\;\;
\text{for}\;\; \bigg\vert\,
\frac{a\,\varepsilon}{1+a\,\varepsilon}\,\bigg\vert <
1 ,\nonumber\\
& =: \sum_{i=0}^\infty
\tilde{c}_{1+i}\frac{(-1)^i}{i!}\,\tilde{x}^i,\;\; \text{for}\;\;
\vert\, \tilde{x}\varepsilon\,\vert < 1 , \label{150409:1333}
\end{align}
where we made use of the Cauchy product rule in both directions:
\begin{equation}
\sum_{k=0}^\infty\, \sum_{l=0}^\infty f_k\cdot g_l \cdot h_{k+l} =
\sum_{i=0}^\infty \, \sum_{j=0}^i f_j\cdot g_{i-j}\cdot h_i.
\end{equation}
Note that the reparametrization of the integration constant
\begin{equation}
c_{1+i}\mapsto \tilde{c}_{1+i}=
\begin{cases}
\; e^\varepsilon c_1,\;\;\text{for}\;\; i=0,\\[0.4em]
\; {\displaystyle\sum_{j=0}^i e^{\varepsilon} c_{1+j}
\frac{i!\,\varepsilon^{i-j}}{j!\, (i-j)!}\frac{i!\, j}{i\, j!}},
\;\;\text{for}\;\; i\geq 1,
\end{cases}
\label{150409:1325}
\end{equation}
is independent of the expansion point $a$ for all $i\geq 0$. In
particular, relation (<ref>) represents the
reparametrization for $a=0$, which explains the third and last
constraint $\vert\, \tilde{x}\varepsilon\,\vert < 1$ in
(<ref>). Note that in vector form relation
(<ref>) can be condensed to
\begin{equation}
\vc\mapsto\tilde{\vc}=\vG(0,\varepsilon)\cdot \vc,
\label{150502:1359}
\end{equation}
where the infinite matrix $\vG$ is defined by (<ref>).
§.§ Reparametrization of all remaining solutions $n\geq 2$
\begin{align}
\tilde{y}_n & = \sum_{k=1}^{n-1}
B_{n,k}\,\varepsilon^{n-k-1}(1-\varepsilon x)^{n+k-1}\,
e^{\varepsilon}\, y_{k+1},\;\;\text{for all}\;\; n\geq 2,\nonumber\\
& = \sum_{k=1}^{n-1} B_{n,k}\,\varepsilon^{n-k-1}(1-\varepsilon
x)^{n+k-1}\, e^{\varepsilon}\,
\sum_{l=0}^\infty c_{k+1+l}\frac{(-1)^l}{l!}x^l\nonumber\\
& = \sum_{k=1}^{n-1} B_{n,k}\,\varepsilon^{n-k-1}
\left(1-\frac{\varepsilon
\tilde{x}}{1+\varepsilon\tilde{x}}\right)^{n+k-1}
e^{\varepsilon}\, \sum_{l=0}^\infty
& = \sum_{k=1}^{n-1}\, \sum_{l=0}^\infty
B_{n,k}\,\varepsilon^{n-k-1} e^{\varepsilon} c_{k+1+l}
\frac{(-1)^l}{l!}
\frac{\tilde{x}^l}{(1+\varepsilon\tilde{x})^{n+k+l-1}}
\nonumber\\
& =\sum_{k=1}^{n-1}\, \sum_{l=0}^\infty
B_{n,k}\,\varepsilon^{n-k-1} e^{\varepsilon} c_{k+1+l}
\frac{(-1)^l}{l!}\nonumber\\
& \hspace{0.5cm}\cdot \left(\,\sum_{m=0}^\infty
\frac{\tilde{x}^{l}(\tilde{x}-a)^m}{(1+a\,\varepsilon)^{n+k+l+m-1}}\right),\nonumber\\
& \hspace{1.275cm}\text{for}\;\; \bigg\vert\,
\frac{(\tilde{x}-a)\,\varepsilon}{1+a\,\varepsilon}\,\bigg\vert <
1 ,\;\; a\neq -\frac{1}{\varepsilon},\nonumber\\
& =\sum_{k=1}^{n-1}\, \sum_{l=0}^\infty
B_{n,k}\,\varepsilon^{n-k-1} e^{\varepsilon} c_{k+1+l}
\frac{(-1)^l}{l!}\nonumber\\
& \hspace{0.5cm}\cdot \left(\,\sum_{m=0}^\infty\, \sum_{q=0}^m
\frac{(-1)^{m+q}\,
\frac{\varepsilon^m}{m!}\,\tilde{x}^{l+m-q}\right)\qquad\qquad\quad\nonumber
\end{align}
\begin{align}
\tilde{y}_n & =\sum_{k=1}^{n-1}\, \sum_{l=0}^\infty
B_{n,k}\,\varepsilon^{n-k-1} e^{\varepsilon} c_{k+1+l}
\frac{(-1)^l}{l!}\nonumber\\
& \hspace{0.5cm}\cdot \left(\,\sum_{p=0}^\infty\,
\sum_{r=0}^\infty \frac{(-1)^{p}\,
\frac{\varepsilon^{p+r}}{(p+r)!}\,\tilde{x}^{l+p}\right)\nonumber\\
& =\sum_{i=0}^\infty\,\sum_{j=0}^i\,\sum_{k=1}^{n-1}
B_{n,k}\,\varepsilon^{n-k-1} e^{\varepsilon} c_{k+1+j}
\frac{(-1)^j}{j!}\nonumber\\
& \hspace{0.5cm}\cdot \left(\, \sum_{r=0}^\infty
\frac{(-1)^{i-j}\,
\frac{\varepsilon^{i-j+r}}{(i-j+r)!}\,\tilde{x}^{i}\right)\nonumber\\
& =\sum_{i=0}^\infty\left(\,\sum_{k=1}^{n-1}\,\sum_{j=0}^i
B_{n,k}\,\varepsilon^{n+i-j-k-1} e^{\varepsilon} c_{k+1+j}
\frac{i!}{j!\,(i-j)!}\frac{(n+k+i-2)!}{(n+k+j-2)!}\right)\frac{(-1)^i}{i!}\,\tilde{x}^i\nonumber\\
& \hspace{0.5cm}\cdot \left(\, \sum_{r=0}^\infty \frac{
r!}\frac{(n+k+i+r-2)!}{(n+k+i-2)!} \right),\;\; \text{and if}\;\;
\bigg\vert\, \frac{a\,\varepsilon}{1+a\,\varepsilon}\,\bigg\vert <
1, \;\;\text{then:}\nonumber\\
& =\sum_{i=0}^\infty\left(\,\sum_{k=1}^{n-1}\,\sum_{j=0}^i
B_{n,k}\,\varepsilon^{n+i-j-k-1} e^{\varepsilon} c_{k+1+j}
\frac{i!}{j!\,(i-j)!}\frac{(n+k+i-2)!}{(n+k+j-2)!}\right)\frac{(-1)^i}{i!}\,\tilde{x}^i\nonumber\\
& =: \sum_{i=0}^\infty
\tilde{c}_{n+i}\frac{(-1)^i}{i!}\,\tilde{x}^i, \;\; \text{for}\;\;
\vert\, \tilde{x}\varepsilon\,\vert < 1,\;\; \text{and}\;\; n\geq
\end{align}
Note that the reparametrization of the integration
\begin{equation}
c_{n+i}\mapsto \tilde{c}_{n+i}= \sum_{k=1}^{n-1}\,\sum_{j=0}^i
B_{n,k}\,\varepsilon^{n+i-j-k-1} e^{\varepsilon} c_{k+1+j}
\frac{i!}{j!\,(i-j)!}\frac{(n+k+i-2)!}{(n+k+j-2)!},\;\; n\geq 2,
\label{150409:2319}
\end{equation}
is again independent of the expansion point $a$ for all $i\geq 0$
and $n\geq 2$, and that it basically represents the exact
reparametrization for $a=0$, which thus again explains the third
and last constraint $\vert\, \tilde{x}\varepsilon\,\vert < 1$ in
(<ref>). Obviously, when evaluated,
(<ref>) must give the same result for the transformed
integration constant
\tilde{c}_n,\dots)$ as (<ref>), which is
§ INVERSE INFINITE GROUP MATRIX
This section demonstrates how the inverse of the infinite group
matrix $\vG$ (<ref>) is constructed. Since, by
construction, the group transformation $\mathsf{L}_2$
(<ref>) is based on an additive composition law of the
group parameter $\varepsilon$, the inverse transformation of
$\mathsf{L}_2$ (<ref>) is thus given as
\begin{equation}
\mathsf{L}_2^{-1}: \;\;\; x=\frac{\tilde{x}}{1+\varepsilon
\tilde{x}},\;\;\;\vy=\vG(\tilde{x},-\varepsilon)\cdot\tilde{\vy}.
\label{150413:1359}
\end{equation}
And since the transformation $\vy\mapsto\tilde{\vy}$ in
$\mathsf{L}_2$ (<ref>) can be formally written as
\begin{equation}
\vy=\vG^{-1}(x,\varepsilon)\cdot\tilde{\vy},
\end{equation}
the infinite inverse matrix $\vG^{-1}$ is thus defined as
\begin{equation}
\vG^{-1}(x,\varepsilon)=\vG\big({\textstyle\frac{x}{1-\varepsilon
x}},-\varepsilon\big). \label{150519:0947}
\end{equation}
§ EXPLICIT FORMS AND GRAPHS OF THE SOLUTIONS $\VY^A$ AND $\VY^B$
The explicit componential form of the primary solution $\vy^A$
(<ref>) is given by (<ref>)
\begin{equation}
y^A_n(x)=\sum_{k=0}^\infty y_{(0)n+k}\frac{(-1)^k}{k!}\, x^k,\;\;
n\geq 1,
\end{equation}
while to bring the transformed solution $\vy^B$
(<ref>) into its corresponding componential form, one
first has to recognize that its matrix-vector structure is
iteratively composed as
\begin{equation}
\vy^B(x)=\vG(x^* ,\varepsilon)\cdot\vy^*(x^*),\;\;\text{with}\;\;
x^*=\frac{x}{1+\varepsilon x}, \label{150413:1547}
\end{equation}
\begin{equation}
\vy^*(x^*)= e^{-x^*\vA}\cdot\vy_0^*, \label{150413:1548}
\end{equation}
\begin{equation}
\vy_0^*=\vG(0,-\varepsilon)\cdot \vy_0. \label{150413:1549}
\end{equation}
Then, according to (<ref>), the componential form of
(<ref>) is given as
\begin{equation}
\begin{cases}
\; e^{\varepsilon}y_1^*(x^*),\;\;\text{for}\;\; n=1, \\[0.5em]
\; {\displaystyle\sum_{k=1}^{n-1}}
B_{n,k}\,\varepsilon^{n-k-1}(1-\varepsilon x^*)^{n+k-1}\,
e^{\varepsilon}\, y^*_{k+1}(x^*),\;\;\text{for}\;\; n\geq 2,
\end{cases}
\label{150413:1647}
\end{equation}
where (<ref>), according to
(<ref>), has the form
\begin{equation}
y^*_q(x^*)=\sum_{l=0}^\infty y^{*}_{(0)q+l}\frac{(-1)^l}{l!}\,
(x^*)^l,\;\; q\geq 1, \label{150413:1646}
\end{equation}
and (<ref>), again according to (<ref>),
but now for $\vy=\vy_0$, goes over into
\begin{equation}
\begin{cases}
\; e^{-\varepsilon}y_{(0)1},\;\;\text{for}\;\; r=1, \\[0.5em]
\; {\displaystyle\sum_{m=1}^{r-1}}
B_{r,m}\,(-\varepsilon)^{r-m-1}\, e^{-\varepsilon}\,
y_{(0)m+1},\;\;\text{for}\;\; r\geq 2,
\end{cases}
\end{equation}
which then needs to be inserted back into
(<ref>), and this result again back into
(<ref>) to finally give the componential form of
$\vy^B$ (<ref>). For a fixed set of initial
conditions, Figure <ref> displays the solutions $\vy^A$ and
$\vy^B$. The convergence domain for each solution for different
initial conditions is given in Table <ref>.
$n=1,\;\: y_1^A\:(\text{solid-line})\subset y_1^B$
$n=2,\;\: y_2^A\:(\text{solid-line})\subset y_2^B$
$n=3,\;\: y_3^A\:(\text{solid-line})\subset y_3^B$
$n=4,\;\: y_4^A\:(\text{solid-line})\subset y_4^B$
Plots of the first four solutions of the initial value
problem (<ref>). The solid lines display the solutions
$y_n^A$ (<ref>), while the solutions $y_n^B$
(<ref>) are given by the solid lines along with the
extensions displayed by the dashed lines. The initial condition
was set $\vy_0=\vG(0,\varepsilon)\cdot \vc$, with
$\vc=e^{-\varepsilon}\cdot\boldsymbol{1}$, and $\varepsilon=1$.
$\vy_0=(y_{(0)i})_{i\in\mathbb{N}}$ $\;\vy^A\;$ $\;\vy^B\;$
$y_{(0)i}={\displaystyle\sum_{j=0}^\infty} G_{ij}(0,\varepsilon)\, c_j$,
with $c_j=\begin{cases}\, \alpha_1 j^n,\;\text{for any finite}\;
n\in\mathbb{R}\negthickspace\negthickspace\negthickspace\\[0.5em] \, \alpha_2$, for all
$j\in\mathbb{N}\\[0.5em] \, {\displaystyle\frac{\alpha_3}{i!}}
\end{cases}$ $\vert\, x\,\varepsilon\,\vert <1$
$x\in\mathbb{R}\backslash\{{\textstyle -\frac{1}{\varepsilon}}\}$
$y_{(0)i}=\alpha_4\, i!$ $\vert\, x\,\vert <1$ $\,\bigg\vert\,
{\displaystyle\frac{x(1-\varepsilon)}{1+\varepsilon\, x}}
\,\bigg\vert <1$
$y_{(0)i}=\begin{cases}\, \alpha_5\, i^n,\;\text{for any finite}\;
n\in\mathbb{R}\\[0.5em] \, \alpha_6,\;\text{for all}\; i\in\mathbb{N} \\[0.5em] \,
{\displaystyle\frac{\alpha_7}{i!}}\end{cases}$ $x\in\mathbb{R}$
x}}\,\bigg\vert <1$
$y_{(0)i}=\alpha_8 (i!)^2$ x = 0 x = 0
Convergence domains for the solutions $\vy^A$
(<ref>) and $\vy^B$ (<ref>) for a
collection of various different initial values $\vy_0$, where all
$\alpha$'s are arbitrary global constants. The domains were
determined by using the Cauchy-Hadamard root test.
§ DERIVATION OF A GENERAL SOLUTION FOR THE NONLINEAR
Given is the infinitely of first order coupled system of
Riccati-ODEs (<ref>)
\begin{equation}
y_n^\prime-\frac{y_n}{x}=\frac{y_{n+1}^2}{x^3},\quad n\geq
\end{equation}
which, if a power series solution around some arbitrary expansion
point $x=a\in\mathbb{R}$ is sought, first should be transformed
into an adequate form. This is achieved by transforming the
function values as $y_n=x^2\cdot z_n$ to give the equivalent
differential system to (<ref>):
\begin{equation}
(x-a)\cdot z_n^\prime +a\cdot z_n^\prime + z_n = z_{n+1}^2.
\label{150423:1255}
\end{equation}
Inserting then the general Ansatz solution
\begin{equation}
z_n(x)=\sum_{k=0}^\infty \lambda_{n,k}\, (x-a)^k,\;\; n\geq 1,
\end{equation}
will turn this system of equations (<ref>) into
\begin{align}
0 \; = & \;\; (x-a)\sum_{k=0}^\infty k\cdot\lambda_{n,k}\,
(x-a)^{k-1} +a
\sum_{k=0}^\infty k\cdot\lambda_{n,k}\, (x-a)^{k-1}\nonumber\\
&\;\; +\, \sum_{k=0}^\infty \lambda_{n,k}\, (x-a)^k-
\left(\;\sum_{k=0}^\infty \lambda_{n+1,k}\, (x-a)^k\right)\cdot
\left(\;\sum_{k=0}^\infty \lambda_{n+1,k}\,
= &\;\; \sum_{k=0}^\infty k\cdot\lambda_{n,k}\, (x-a)^{k} +a
\sum_{k=1}^\infty k\cdot\lambda_{n,k}\, (x-a)^{k-1}\nonumber\\
&\;\; +\, \sum_{k=0}^\infty \lambda_{n,k}\, (x-a)^k-
\sum_{k=0}^\infty\sum_{l=0}^k \lambda_{n+1,l}\lambda_{n+1,k-l}\,
= &\;\; \sum_{k=0}^\infty (x-a)^k\left[k\cdot \lambda_{n,k}+
a\cdot (k+1)\cdot \lambda_{n,k+1}+\lambda_{n,k}-\sum_{l=0}^k
\lambda_{n+1,l}\cdot \lambda_{n+1,k-l}\right],
\end{align}
which, termwise equated, gives the following recurrence relation
for the expansion coefficients
\begin{equation}
k\cdot \lambda_{n,k}+ a\cdot (k+1)\cdot
\lambda_{n,k+1}+\lambda_{n,k}=\sum_{l=0}^k \lambda_{n+1,k-l}\cdot
\lambda_{n+1,l}\, ,\;\text{for all}\;\; n\geq 1,\, k\geq 0.
\end{equation}
For every arbitrary but fixed order $n$, the above relation
represents a 1-dimensional recurrence relation of first order
relative to index $k$, which can be uniquely solved by imposing
for all $n\geq 1$ at $k=0$ an initial condition
$\lambda_{n,0}=c_n$, where $c_n$ is some arbitrary constant. Note
that for the singular case $a=0$ the solution for the expansion
coefficients $\lambda_{n,k}$ will be different to those for all
$a\neq 0$.
§ PROOF THAT THE GENERAL SOLUTION CAN BE PARTIALLY MATCHED
TO A SPECIAL SOLUTION
The proposition is that for $a\neq 0$ the general solution $y_n$
(<ref>) can only be matched to the special solution
$y_n^{(3)}$ (<ref>) in the domain $|x-a|<|a|$. This
can be straightforwardly seen when performing the following two
steps: Firstly, equating these two solutions relative to $x^2$
\begin{equation}
\frac{y_n^{(3)}(x)}{x^2}=\frac{y_n(x)}{x^2},\;\;\text{for all}\;\:
n\geq 1,\label{150424:1819}
\end{equation}
will give the matching relation
\begin{equation}
\frac{1}{5^{2^{-n}}}\left[\,\prod_{k=0}^{n-1}
\left(1+\frac{1}{2^{k-2}}\right)^{2^{k-n}}\,\right]\cdot
x^{\frac{1}{2^{n-2}}}=\sum_{k=0}^\infty \lambda_{n,k}\cdot
(x-a)^k,\;\;\text{for all}\;\: n\geq 1, \label{150424:0907}
\end{equation}
which then, secondly, will undergo the transformation
$x\mapsto\hat{x}=x-a$ to finally give the equivalent matching
relation[Note that the coordinate transformation
$x\mapsto\hat{x}=x-a$ is a permissible transformation within the
determination process for the expansion coefficients
$\lambda_{n,k}$ according to (<ref>), simply because
the process itself is not affected by this transformation.]
\begin{equation}
\frac{1}{5^{2^{-n}}}\left[\,\prod_{k=0}^{n-1}
\left(1+\frac{1}{2^{k-2}}\right)^{2^{k-n}}\,\right]\cdot
\lambda_{n,k}\cdot \hat{x}^k,\;\;\text{for all}\;\: n\geq 1,
\label{150424:1046}
\end{equation}
which, in contrast to (<ref>), is easier to match. In
order to explicitly determine the coefficients $\lambda_{n,k}$
such that equality (<ref>) is satisfied for all orders
$n$, it is necessary to expand the power term on the left-hand
\begin{equation}
\frac{a^{\beta-k}}{k!}\cdot\frac{\Gamma(\beta+1)}{\Gamma(\beta-k+1)}\cdot
\hat{x}^k,\;\;\text{with}\;\; \beta=\frac{1}{2^{n-2}}\geq
0,\;\text{for all}\; n\geq 1. \label{150424:1824}
\end{equation}
Now, since this (transformed) expansion (<ref>) only
converges for $|\hat{x}|<|a|$, the original(non-transformed) matching relation (<ref>) will
therefore only be valid for $|x-a|<|a|$, with the corresponding
matched coefficients
\begin{equation}
\lambda_{n,k}=\frac{1}{5^{2^{-n}}}\left[\,\prod_{i=0}^{n-1}
\left(1+\frac{1}{2^{i-2}}\right)^{2^{i-n}}\,\right]\cdot
\frac{a^{\frac{1}{2^{n-2}}-k}}{k!}
\cdot\frac{\Gamma\big(\frac{1}{2^{n-2}}+1\big)}{\Gamma\big(\frac{1}{2^{n-2}}-k+1\big)},
\;\; n\geq 1, \;\; k\geq 0.\qquad\square \label{150424:1908}
\end{equation}
Note that since the general solution $y_n$ (<ref>) was
matched to a genuine solution of (<ref>), namely to
the special solution $y_n^{(3)}$ (<ref>), and not to
some arbitrary function, the matched coefficients
(<ref>) will thus automatically satisfy the
corresponding solved relations (<ref>) for $a\neq 0$,
i.e. at least one set of constants $c_n$ for all $n\geq 1$ can be
found which then, according to (<ref>), uniquely
represent the expansion coefficients $\lambda_{n,k}$
§ THE EXISTENCE OF ONLY TWO UNCOUPLED LIE POINT GROUP
Performing a systematic Lie point group invariance analysis on the
infinite system of first order Riccati-ODEs (<ref>),
and looking out only for uncoupled solutions in the overdetermined
system for the generating infinitesimals, which themselves can
then only take the consistent form
\begin{equation}
\xi(x,y_1,y_2,\dotsc)=\phi(x),\;\;\text{and}\;\;\;
\eta_n(x,y_1,y_2,\dotsc)=\psi_n(x,y_n),\;\;\text{for all}\; n\geq
\end{equation}
one obtains the following infinite recursive set of constraint
\begin{equation}
\left[\frac{\partial\psi_n}{\partial x}x^3+
\left(\frac{\partial\psi_n}{\partial
x^2\right]+\left[\left(\frac{\partial \psi_n}{\partial
\right)y_{n+1}^2\right]=0.
\end{equation}
This equation can only be fulfilled if the terms in each of the
two square brackets vanish separately, because, due to that the
first square bracket only depends on $y_n$ and the second one on
$y_{n+1}$, both square brackets are independent of each other. The
above equation thus breaks apart into the following two equations
\begin{align}
\frac{\partial\psi_n}{\partial x}x+
\left(\frac{\partial\psi_n}{\partial
\frac{\partial \psi_n}{\partial
\end{align}
The last equation (<ref>), however, is only
consistent if $\psi_n$ is restricted to be a non-shifted linear
function of $y_n$, i.e. if
\begin{equation}
\psi_n(x,y_n)=\alpha_n(x)\cdot y_n,
\end{equation}
which then reduces the system
(<ref>)-(<ref>) respectively to
\begin{align}
\frac{d\alpha_n}{dx}x-\frac{d\phi}{dx}+\frac{\phi}{x}=0,\label{150519:1927}\\[0.75em]
\alpha_n-2\alpha_{n+1}-\frac{d\phi}{dx}+\frac{3\phi}{x}=0.\label{150519:1930}
\end{align}
Since (<ref>) leads to the result that
$\alpha_n=\alpha_{n+1}$, equation (<ref>) gives the
solution for $\alpha_n$ in terms of $\phi$
\begin{equation}
\alpha_n = -\frac{d\phi}{dx}+\frac{3\phi}{x}.\label{150519:2000}
\end{equation}
Inserting this result back into (<ref>) leads to the
following differential equation for $\phi$
\begin{equation}
\end{equation}
which has the general solution
\begin{equation}
\phi(x)=c_1\cdot x + c_2\cdot x^2,
\end{equation}
which finally, according to (<ref>), implies that
\begin{equation}
\alpha_n(x)=2c_1+c_2\cdot x.
\end{equation}
Hence, the only possible combination in the infinitesimals which
lead to uncoupled Lie point group invariances in the infinite
system (<ref>) is given by the 2-dimensional Lie
\begin{equation}
\xi(x,y_1,y_2,\dotsc)=c_1\cdot x + c_2\cdot x^2,\quad
\eta_n(x,y_1,y_2,\dotsc) = \left(2c_1+c_2\cdot x\right) y_n,\;\;
\text{for all $n\geq 1$}, \label{150522:0049}
\end{equation}
with $[X_1^\infty,X^\infty_2]=X^\infty_2$, where the to
(<ref>) corresponding scalar operators $X^\infty_1$
and $X^\infty_2$ are given by (<ref>).
|
1511.00033
|
Ram-Pressure Stripping in Massive Clusters]Jellyfish: The origin and distribution of extreme ram-pressure stripping events in massive galaxy clusters
C. McPartland et al.]Conor McPartland$^1$, Harald Ebeling$^1$, Elke Roediger$^2$ & Kelly Blumenthal$^1$
$^1$ Institute for Astronomy, University of Hawai'i at Manoa, 2680 Woodlawn Drive, Honolulu, HI, 96822, USA
$^2$ E.A. Milne Centre for Astrophysics, Department of Physics & Mathematics, University of Hull, Cottinton Road, Hull, HU6 7RX, United Kingdom
We investigate the observational signatures and physical origin of ram-pressure stripping (RPS) in 63 massive galaxy clusters at $z=0.3-0.7$, based on images obtained with the Hubble Space Telescope. Using a training set of a dozen “jellyfish" galaxies identified earlier in the same imaging data, we define morphological criteria to select 211 additional, less obvious cases of RPS. Spectroscopic follow-up observations of 124 candidates so far confirmed 53 as cluster members. For the brightest and most favourably aligned systems we visually derive estimates of the projected direction of motion based on the orientation of apparent compression shocks and debris trails.
Our findings suggest that the onset of these events occurs primarily at large distances from the cluster core ($>400$ kpc), and that the trajectories of the affected galaxies feature high impact parameters. Simple models show that such trajectories are highly improbable for galaxy infall along filaments but common for infall at high velocities, even after observational biases are accounted for, provided the duration of the resulting RPS events is $\lesssim$500 Myr. We thus tentatively conclude that extreme RPS events are preferentially triggered by cluster mergers, an interpretation that is supported by the disturbed dynamical state of many of the host clusters. This hypothesis implies that extreme RPS might occur also near the cores of merging poor clusters or even merging groups of galaxies.
Finally, we present nine additional “jellyfish" galaxies at z$>$0.3 discovered by us, thereby doubling the number of such systems known at intermediate redshift.
galaxies: evolution - galaxies: clusters: intracluster medium - galaxies: structure
§ INTRODUCTION
Evidence of accelerated galaxy evolution in galaxy clusters has been presented as early as 1980, the most well known examples being the increased occurrence of ellipticals in dense environments <cit.> and the higher fraction of blue galaxies in clusters at higher redshift <cit.>. The physical mechanisms responsible for these effects are, however, still very much debated. A variety of processes have been proposed in the literature, ranging form slow-acting gravitational interactions such as galaxy-galaxy harassment <cit.> to potentially extremely rapid galaxy transformations brought about by interactions with the gaseous intracluster medium (ICM).
The latter process, ram-pressure stripping (RPS) is expected to be especially efficient in massive galaxy clusters, as the pressure imparted on a galaxy is directly proportional to the local gas density of the ICM and to the square of the galaxy's velocity with respect to the ICM <cit.>. The resulting removal of the galaxy's interstellar medium (ISM) occurs in the direction of motion of the galaxy relative to the ICM, generating a trail of star-forming regions in the galaxy's wake. For fortuitous viewing angles, this trail, or at least the associated deformation of the galactic disk, is accessible to observation, thus creating a rare opportunity to constrain the motion of galaxies in the plane of the sky. Observations of RPS events thus constitute a valuable complement to spectroscopic radial-velocity surveys and permit a detailed investigation of the kinematics and spatial evolution of galaxies in the dense cluster environment.
The physics and observational signature of RPS have been the subject of extensive numerical simulations which predict that gradual stripping should be pervasive even in low-mass clusters <cit.>. Indeed RPS events have been studied in great detail in the Virgo <cit.> and Coma clusters <cit.>, as well as in other nearby systems, such as the Shapley Concentration <cit.> or Abell 3627 <cit.>. As expected, these events are relatively modest though, with observations showing atomic hydrogen to be displaced and only partially removed <cit.>, while the denser, more centrally located molecular gas is found to be essentially unperturbed <cit.>.
By contrast, in the most massive clusters the environment encountered by infalling galaxies can lead to their entire gas reservoir being stripped in a single pass through the cluster core <cit.>.
Observational evidence of extreme ram-pressure stripping is, however, sparse, due to
their reliance on favourable circumstances, such as suitable infall trajectory, gas mass, galaxy orientation, and high ICM density. Considering the small number and relatively low masses of nearby clusters (except for Coma), these conditions are unlikely to be met in the local Universe.
The extreme environment that is a prerequisite for extreme RPS is, however, routinely encountered by galaxies falling into massive clusters where galaxy peculiar velocities in excess of 1000 km s-1 are common and the ICM particle density easily exceeds 10-3 cm-3. Since massive clusters are rare, larger volumes have to be searched to efficiently probe such truly high-density environments. Although their numbers are still small, striking examples of extreme RPS events have been discovered in Hubble Space Telescope (HST) images of moderately distant ($z\gtrsim0.2$) massive clusters <cit.> and, most recently, in X-ray selected massive clusters at $z{>}0.3$ <cit.>. Importantly, these clusters are not only intrinsically more massive, they are also dynamically less evolved and more likely to be undergoing mergers than systems in the local Universe <cit.>, a critical requirement if extreme RPS events are triggered by merger-driven shocks, as suggested by <cit.>. Increasing the size of the still small sample of RPS examples clearly constitutes a crucial step toward a meaningful statistical investigation of the physics of accelerated galaxy evolution.
In this paper, we aim to compile a statistically significant sample of galaxies that might be undergoing RPS in very massive clusters. We then use this sample to establish which galaxy trajectories are most conducive to creating extreme RPS, and thereby elucidate whether the most dramatic RPS events are triggered by massive cluster mergers <cit.>, rather than during regular infall of galaxies from the field or along filaments. In order to compile the required sample, we develop morphological criteria to select RPS candidates from archival HST imaging data for a well defined sample of massive clusters at $z>0.3$, and compare the spatial and dynamical distribution of the selected objects with expectations from numerical and theoretical models.
This paper is structured as follows:
in Section <ref> we introduce the cluster sample and present an overview of the observations and data-reduction procedures;
in Section <ref> we discuss our morphological criteria for the identification of galaxies experiencing ram-pressure stripping and present the sample of RPS candidates;
in Section <ref> we present the a simple model of clustre infall which we use to interpret our data;
in Section 5 we present our results for the spatial distribution and dynamical properties of RPS events in massive clusters; and in Section 6 we draw conclusions about the origin, trajectories, and physics of extreme RPS. We present a summary of our work in Section 7.
Throughout this paper, we assume a concordance $\Lambda$CDM cosmology with $\Omega_M$ = 0.3, $\Omega_\Lambda$ = 0.7, H$_0$ = 70 km s-1 Mpc-1.
As the clusters in our sample span a range of redshifts of $0.3<z<0.7$, the metric scale of our images varies from 4.45 to 7.15 kpc arcsec-1.
[innerleftmargin=5pt,innerrightmargin=5pt, linecolor=red]
The 12 galaxies deemed textbook examples of ram-pressure stripping and thus used as our training set; six of these (top two rows) were published previously by <cit.>. Three members of our training set were recently found not to be cluster members (see Section <ref>) and are highlighted in the bottom row.
The distribution of all galaxies in our target fields in various parameter spaces. Left: Concentration–Asymmetry; centre: Gini–$M_{20}$; right: $Sk_{0-1}$–$Sk_{1-2}$. Our final sample of RPS candidates is marked by filled blue circles; the morphologically most compelling examples are shown as yellow asterisks. Members of our training set (see Fig. <ref>) are shown with open symbols. Squares show the six systems published in <cit.>, and circles show the six additional galaxies from their extended sample. Three members of our training set, all part of the extended ESE sample, were recently found not to be cluster members (see Section <ref>) and are shown in red. The cuts defining our final morphological selection criteria are indicated by red dashed lines.
§ DATA USED IN THIS STUDY
§.§ The MACS sample
Our cluster sample is drawn from a master list of clusters identified in the course of the Massive Cluster Survey <cit.>, designed to provide a large, statistically complete sample of X-ray luminous ($L_X \gtrsim 5\times 10^{44}$ erg s-1, 0.1-2.4 keV) and moderately distant ($z \gtrsim 0.3$) galaxy clusters. Covering over 22,000 sq.deg., the MACS sample comprises the majority of massive galaxy clusters in the observable Universe, making it ideally suited for our investigation. At redshifts $z\gtrsim0.3$, the sub-kiloparsec angular resolution needed to identify the characteristic morphological traits of RPS events can only be achieved with the Advanced Camera for Surveys (ACS) aboard HST. We thus limit our sample to MACS clusters with archival HST/ACS images as described in more detail in the following section.
§.§ Imaging data
As our primary observational diagnostics revolve around morphological features traced by star-forming regions, we limit our study to MACS clusters that have been observed in the HST/ACS F606W band. The F606W filter is well suited as it corresponds roughly to the B band in the cluster rest frame and has been used in a large number of HST observations of MACS clusters. We further require clusters in our sample to also have imaging data in the ACS F814W passband, as the resulting F814W–F606W colours provide a straightforward means to discriminate against the population of passively evolving cluster ellipticals.
Of the entire MACS sample, 44 clusters were successfully observed in both the ACS F606W and F814W passbands as part of the HST SNAPshot programmes GO-10491, -10875, -12166, and -12884 (PI: Ebeling). These programmes use short exposures (1200 seconds for F606W and 1440 seconds for F814W) designed to reveal bright strong-lensing features and provide constraints on the physical nature of galaxy-galaxy and galaxy-gas interactions in cluster cores. Fundamental properties of this subset of the MACS cluster sample are presented and discussed by Ebeling & Repp (in preparation). Supplementing these SNAPshots, we also include data from observations of 17 additional MACS clusters obtained by the Cluster Lensing and Supernova Survey with Hubble <cit.>, an HST Multi-Cycle Treasury Program employing 16 filters from the UV to the NIR, including F606W and F814W. Exposure times for the CLASH observations are nominally one and two orbits for all ACS filters, but vary substantially between cluster fields around median exposure times of 4060 and 8480 seconds for the F606W and F814W passbands, respectively (see Table <ref> & <ref> for a summary of the observations).
In total, our sample thus comprises 63 MACS clusters. At the redshifts relevant to our study, the field of view of the ACS Wide Field Channel ($202^{\prime\prime}\times202^{\prime\prime}$) covers an inscribed circle of radius between 450 and 720 kpc and thus samples primarily the cluster core region. Charge-transfer-efficiency corrected images in the two passbands were registered using the astrometric solution of the F606W image as a reference, and source catalogs were created using SExtractor <cit.> in dual-image mode, with F606W chosen as the detection band. We removed stars as well as cosmic rays and other artefacts as objects falling on or below the star lines in both magnitude-$\mu_\mathrm{max}$ and magnitude-$r_{20\%, \mathrm{ell}}$ space[Here, $\mu_\mathrm{max}$ and $r_{20\%, \mathrm{ell}}$ are the peak surface brightness and the elliptical radius encircling 20% of the total flux, respectively.]. After removing spurious detections, we have a 5$\sigma$ 90% completeness limiting magnitude of 24.9 in F606W (here and in the following the magnitudes quoted are measured within the Petrosian radius).
As the quantitative morphological indicators we employ to identify RPS candidates (see Section <ref>) require signal-to-noise ratios of $\langle$S/N$\rangle{>}5$ per pixel, we limit our galaxy sample to objects with $m_{\rm F606W}{<}24$, which leaves a total of 15,875 galaxies (11,550 in the SNAPshot data and 4,325 in the CLASH data). We note that, due to the high density of objects in cluster cores and the presence of objects of complex morphology, some of the objects in our master catalogue may in fact be blends of several objects, whereas others have suffered fragmentation, i.e., were broken up into multiple sources.
To mitigate the effect of fragmentation in our master catalog, we enforce strict deblending criteria (DEBLEND_NTHRESH=16, DEBLEND_MINCONT=0.2). Due to the relatively shallow depth ($\sim$1200 s) of the imaging data, the faint extraplanar tails that characterize jellyfish galaxies often fall below our detection limit. For the quantitative selection criteria (see Section 3.1), we, therefore, focus on identifying robust morphological features (disturbances) in the high signal to noise regions of galaxies. However, note that the presence of optical tails is a requirement for an object to be classified as a compelling jellyfish candidate during our visual screeening process.
As for the completeness of the sample of candidates presented here, it is almost certain that modest cases of RPS (in particular when occuring in low mass galaxies) will have been missed due the lack of pronounced morphological features, whereas essentially all the brightest objects would have been easily identified by eye. We note however that regarldless of brighness, objects moving close to our line of sight are likely to be missed as the tell-tale debris trails will be obscured by the the much brighter disks of the galaxies. We discuss this bias in detail in Section <ref> and <ref>.
§.§ Spectroscopic data
The sample of RPS candidates compiled in this work using morphological selection is expected to be heavily contaminated by galaxies that are in fact not members of the respective MACS cluster and / or whose morphology is irregular for reasons other than RPS (see Section <ref> for details). In order to eliminate interlopers, we have embarked on a comprehensive spectroscopic survey of our RPS candidate sample, aimed at (a) excluding fore- and background galaxies from our sample of RPS candidates, and (b) obtaining peculiar radial velocities of those systems that are cluster members. We refer to a forthcoming paper (Blumenthal et al., in preparation) for a more extensive report on these efforts, including a description of the data-reduction procedure. We note here though that all spectroscopic observations were conducted with the DEIMOS spectrograph on the Keck-II 10m-telescope on Maunakea, using multi-object spectroscopy with slits of 1mm width, the 600 l/mm Zerodur grating set to a central wavelength of 6300Å, the GG455 blocking filter, and exposure times ranging from 3$\times$10 to 3$\times$15 minutes. For almost all targeted galaxies, redshifts were measured from emission lines detected in these spectra, yielding a precision of approximately 0.0002 in redshift or 60 km s$^{-1}$ in radial velocity.
§ GALAXY MORPHOLOGY
A recent study by <cit.> presented six textbook examples of “jellyfish" galaxies (thought to be extreme RPS events[Although the observed morphology of these objects does not prove the occurrence of RPS, in-depth follow-up studies of galaxies sharing the same striking features unambiguously confirmed RPS to be at work (; ; ).]) discovered in HST imaging data for 36 of the 63 clusters used in this work. These objects were visually identified, having to meet the following criteria: (1) a strongly disturbed morphology in optical images indicative of unilateral external forces; (2) a pronounced brightness and colour gradient suggesting extensive triggered star formation; (3) compelling evidence of a debris trail. Furthermore, the direction of motion implied by each of these features had to be consistent. We expand the ESE sample by six additional, unpublished, jellyfish candidates, identified by the same authors, that satisfy at least two of these criteria[Note that the inferred direction of motion for two candidates (leftmost two in the bottom row of Fig. <ref>) is largely aligned with our line of sight.], and use the resulting superset of 12 objects (shown in Fig <ref>) as a training set for the identification of additional, less obvious candidate objects.
For each of the galaxies in our catalogue we compute several non-parametric galaxy morphology statistics defined previously in the literature: concentration ($C$) and asymmetry ($A$) <cit.>, Gini coefficient ($G$) and $M_{20}$ <cit.>. While these statistics were originally designed to identify the morphological features of galaxy mergers, we find that they can be applied more widely to characterise and select objects featuring disturbed morphologies. In addition to the aforementioned four statistics, we introduce two “skeletal decomposition" parameters ($Sk_{0-1}$ and $Sk_{1-2}$; see Appendix <ref>).
We compute values for each of these indicators using the ellipticities, position angles, and locations provided by SExtractor but note that the precise location of the centre of each object is iteratively refined through minimisation procedures, as described in <cit.>. Acknowledging the difficulty of cleanly separating galaxies in crowded cluster cores, we resort to using SExtractor's segmentation maps to identify the pixels belonging to a given galaxy rather than relying on an isophotal definition of a galaxy's extent. We stress that, as a result, the morphological quantities measured here should not be directly compared to those from other work.
Venn diagram of the sets of galaxies selected by each of the morphological criteria shown in the three panels of Fig. <ref>. Although each type of cut selects a similar number of galaxies (represented by the area of each circle), the modest overlap between these sets makes the final selection, achieved by requiring all criteria to be met, much more restrictive.
§.§ Selection criteria and visual screening
The fact that the extended ESE sample (Fig. <ref>) contains some of the most extreme examples of jellyfish galaxies known to date (i.e., the brightest and most morphologically disturbed) makes it well suited as a training set for an iterative, semi-automated search for additional RPS candidates. To this end, we examine the location of the training-set members in $C$–$A$, Gini–$M_{\rm 20}$, and $Sk_{0-1}$–$Sk_{1-2}$ space, and define cuts in these parameter spaces that preserve the training set but eliminate the vast majority of other galaxies. The physical rationale behind these cuts is to discard extremely diffuse objects (achieved by a cut in $C$), almost perfectly symmetric sources (cut in $A$), morphologically undisturbed disk and elliptical galaxies (cut in $G$-$M_{20}$), and, finally, objects with little substructure (cuts in $Sk_{0-1}$ and $Sk_{1-2}$).
We apply an initial set of morphological criteria (cuts in $C$–$A$ and Gini–$M_{20}$) to galaxies detected in the 10 cluster fields from which the extended ESE sample originates. The $\sim$650 candidate objects thus selected are then visually scrutinised independently by two of us (CM and HE) and classified according to their plausibility as RPS events. We attempt to reduce the subjectivity of this procedure by reviewing jointly, in a second iteration, all objects classified either as compelling jellyfish galaxies or as plausible candidates by one of the inspectors and assigning a consensus classification. From the resulting set of potential RPS events we select the most compelling candidates, add them to our original training set, and re-evaluate our initial morphological constraints. Cuts in colour-magnitude space were considered too during this process but ultimately dismissed as largely redundant with the aforementioned morphological cuts, which already remove the majority cluster ellipticals and faint blue objects. The full set of morphological criteria (now also including cuts in $Sk_{0-1}$–$Sk_{1-2}$) are then applied to the remaining clusters, and the resulting subset is once again visually screened. Fig. <ref> shows the distribution of all galaxies in various projections of our multi-dimensional morphology parameter space, as well as the applied selection criteria. Members of the extended training set and of our final sample of RPS candidates are highlighted. Although the three sets of selection criteria shown in Fig. <ref> all select approximately the same fraction of galaxies (30-40%), their doing so largely non-redundantly leads to a much more restrictive selection of merely 8% (1263 galaxies) when all criteria are combined (Fig. <ref>).
It is evident from Fig. <ref> that the adopted selection criteria, although highly efficient in eliminating regular disk galaxies and ellipticals, still select mostly galaxies that, although morphologically disturbed, are not necessarily undergoing RPS. In fact less than 20% of the automatically selected systems are classified as RPS candidates in our visual screening process. The disturbed sources rejected after visual inspection can largely be assigned to one of the following classes: strong-gravitational-lensing features (including both cluster-galaxy and galaxy-galaxy lensing events), foreground irregular galaxies, close pairs of ellipticals, unclassifiable clumpy emission in low signal-to-noise areas, and artefacts due to source confusion in crowded regions. We also note that, while colour information was not directly included in our selection procedure, the availability of images in both the F606W and F814W passbands proved essential in our visual classification to distinguish between the morphological disturbances caused by RPS and irregular extinction due to dust (see Fig. <ref>).
Importance of colour information for our visual inspections. Viewed solely in the F606W passband (left) this object could be considered a (remotely) plausible RPS candidate. A false-colour image including data in the F814W filter (right) strongly suggests a slightly disturbed dusty disk galaxy.
§.§ RPS-candidate sample
Properties of the morphologically most compelling "jellyfish" galaxies that constitute our training set. The projected radius $r_{\rm BCG}$ is the projected distance to the (nearest) BCG; the listed angle of incidence is the mean of the values assigned by the three reviewers (see arrows in Fig. <ref>. The first six galaxies form the jellyfish sample of ESE.).
Name $\alpha$ [J2000] $\delta$ [J2000] $m_{\rm F606W}$ $m_{\rm F814W}$ $r_{\rm BCG}$ [kpc] Incidence [deg.] $z$
MACSJ0257-JFG1 02 57 41.4 $-$22 09 53 18.75 18.22 166 10 0.3241
MACSJ0451-JFG1 04 51 57.3 $+$00 06 53 19.66 19.29 298 50 0.4362
MACSJ0712-JFG1 07 12 18.9 $+$59 32 06 19.10 18.39 87 107 0.3430
MACSJ0947-JFG1 09 47 23.1 $+$76 22 52 19.81 19.69 210 34 0.3417
MACSJ1258-JFG1 12 57 59.6 $+$47 02 46 19.10 18.70 133 45 0.3424
MACSJ1752-JFG1 17 51 56.1 $+$44 40 20 20.13 19.61 370 120 0.3739
MACSJ0035-JFG1 00 35 27.3 $-$20 16 18 19.49 19.02 182 103 0.3597
MACSJ0257-JFG2 02 57 43.5 $-$22 08 38 19.92 19.44 243 130 0.3297
MACSJ0429-JFG1 04 29 33.3 $-$02 53 02 20.97 20.64 203 113 0.4000
MACSJ0429-JFG1 04 29 40.4 $-$02 53 18 20.75 20.36 334 40 0.4049
MACSJ0916-JFG1 09 16 12.9 $-$00 25 01 20.43 19.97 334 81 0.3300
MACSJ1142-JFG1 11 42 37.0 $+$58 31 48 20.25 19.62 549 87 0.3267
MACSJ1720-JFG1 17 20 13.6 $+$35 37 17 20.05 19.52 309 30 0.3832
MACSJ1752-JFG1 17 52 06.3 $+$44 40 05 20.25 20.06 747 86 0.3527
RXJ2248-JFG1 22 48 40.2 $-$44 30 50 20.66 20.18 335 64 0.3515
The process described in the previous section yielded 223 possible ram-pressure stripping events (including the training set). We consider 15 of these to be classical jellyfish galaxies (yellow symbols in Fig. <ref>); an additional 115 objects show characteristic features of RPS (albeit less extreme), and 93 are at least plausible candidates. While we cannot rule out that physical processes other than RPS (e.g., minor mergers or tidal interactions) contribute to, or in fact cause, the observed morphology of our candidates, such alternative scenarios are likely to be relevant mainly for the fainter galaxies in our sample for which the most compelling sign of RPS (evidence of a debris trail) cannot be discerned in the shallow imaging data in hand.
As a complement to the first six "jellyfish" galaxies discovered in MACS clusters by <cit.>, we show in Fig. <ref> a second sample of nine compelling jellyfish galaxies; fundamental properties of these systems are further described in Section <ref> and listed in Table <ref>.
Nine additional textbook examples of ram-pressure stripping discovered in this work; the first three of these were previously identified but not published by <cit.> (see also Fig. <ref>). The blue, green, and red arrows indicate the direction of motion assigned to the respective galaxy by the three reviewers; the yellow arrow and metric separation denote the direction and distance to the cluster centre (unknown to the reviewers).
§.§.§ Observational biases
Impressive as the list of 223 RPS candidates may appear, we caution again that most of these galaxies may not even be cluster members, and that, for those that are, the cause of the observed morphological features need not be RPS. In addition, our list is almost certainly incomplete. Two primary observational biases are to blame: (a) our inability to reliably discriminate against non-RPS events solely from morphological data (leading to contamination by non-cluster galaxies) and (b) our inability to identify RPS events in galaxies moving close to our line of sight (leading to incompleteness regarding true RPS events in our target clusters).
First results from a comprehensive spectroscopic survey of all candidates (Blumenthal et al., in preparation) indeed indicate that more than half of the objects we selected are in fact fore- or background galaxies. The hazards of morphological selection alone are underlined not just by this high percentage of projection effects, but also by the elimination of three members of our extended training set (see bottom row Fig. <ref>): the edge-on disk with a stellar tail in MACSJ1236.9+6311 is in the foreground of the cluster, while the dramatically distorted face-on spiral galaxy near the core of MACSJ1652.3+5534 was found to be a background object gravitationally lensed by the massive MACS cluster. The bright blue face-on spiral in MACSJ1731.6+2252, finally, turned out to be a member of a foreground group of galaxies. Although the removal of these three objects from our training set has no effect on our selection criteria, as can be seen from Fig. <ref> in which these galaxies are marked by red circles, the misidentification of galaxies we considered "textbook" cases of RPS serves as a warning about the robustness of morphological selection and underlines the need for spectroscopic follow-up observations.
The impact of the second observational bias cannot trivially be quantified by means of additional observations. Galaxies moving close to our line of sight lack the tell-tale debris trail and bow-shock morphology readily apparent for RPS proceeding in the plane of the sky (see Fig. <ref>) and are thus likely to be missed. We attempt to account for the resulting systematic incompleteness when modelling galaxy trajectories in Section <ref>.
§.§ Direction of motion and location within the cluster
Since one of the goals of our study is to distinguish between the different geometric and kinematic scenarios associated with
"stream-fed" infall along filaments, and cluster mergers, we focus on two key properties of cluster galaxies: the angle of incidence of their trajectory with respect to the gravitational centre of the cluster and the distance from the cluster centre. To observationally constrain the former, we consult the results of hydrodynamical modeling of RPS <cit.> for insights regarding the correlation between the morphological disturbances caused by RPS and the galaxy's direction of motion.
Figure <ref> shows model predictions for the distribution of gas and newly formed stars in galaxies undergoing RPS while moving face-on through the ICM. As expected, identifying the direction of motion becomes challenging when a galaxy moves through the ICM along our line of sight or is observed early in the stripping process.
Distribution of gas (white) and newly formed stars (turquoise) for a simulated RPS event involving a spiral galaxy moving face-on through the ICM. A comparison with Fig. <ref> shows that our morphological selection is, unsurprisingly, most sensitive to features typical of mature RPS events in galaxies viewed edge-on. <cit.>.
We attempt to assign projected directions of motion visually according to the following prescriptions: (1) if tails are discernible, the velocity vector is assumed to be parallel to the tail; (2) edge-on disks showing significant curvature are assigned velocity vectors oriented perpendicular to said curvature and originating at its apex; (3) if extended regions of star formation appear to be present, the velocity vector is placed perpendicular to the dominant elongation of said regions; (4) if none of the previously mentioned indicators are present (or if they are contradictory), we attempt to make the best physically motivated estimate. To avoid systematic biases, galaxies are inspected using small thumbnail images covering only the region immediately surrounding the galaxy with no indication of the direction to the cluster centre. In recognition of the subjective nature of our visual measurements <cit.>, the process is performed independently by three reviewers to derive an approximate grade for the robustness of each estimated direction of motion. Figure <ref> shows examples of objects falling into each of our quality grades with uncertainty increasing top to bottom and left to right. We then define the angle of incidence as the angle between the apparent velocity vector and the position vector with respect to the cluster centre (taken to be the location of the brightest cluster galaxy, BCG), i.e., the angular deviation from a purely radial infall trajectory (note again that all of these quantities are defined and observed in projection).
A second galaxy property that is critical to our efforts to deduce trajectories is location within the cluster. For RPS candidates lacking radial-velocity measurements, we are unable to assess whether an object is located in front or behind the cluster centre (defined by the redshift of the BCG), let alone further constrain its physical distance to the latter along our line of sight. Projected distances, however, measured in the plane of the sky and relative to the location of the BCG, are trivially obtained for comparison with the distribution expected for different geometries of galaxy infall.
Examples of RPS candidate events illustrating our process to estimate direction of motion and the associated error. The arrows are the same as in Fig. <ref>.
§ A SIMPLE MODEL OF GALAXY TRAJECTORIES
In order to understand which kind of galaxy trajectories are most compatible with the observed distributions of (projected) incidence angle and cluster-centric distance, we compare our observations with the results of a simple theoretical model. To this end, we calculate orbits in a canonical cluster representative of the MACS clusters in our sample and use simple prescriptions, described below, to predict the projected radii and incidence angles at which extreme RPS events are most likely to occur.
As an infalling galaxy approaches the dense cluster core, the ICM exerts an increasing ram-pressure, $p_\mathrm{ram} = \rho_\mathrm{ICM}v_\mathrm{gal}^2$, where $\rho_\mathrm{ICM}$ is the ICM mass density and $v_\mathrm{gal}$ is the relative velocity between the galaxy and ICM <cit.>. By comparing $p_\mathrm{ram}$ to the gravitational restoring force per unit area on the gas within the galaxy,
\begin{equation}
f_\mathrm{grav}(R) = \Sigma_\mathrm{gas}(R)\frac{\partial\Phi}{\partial Z}(R),
\end{equation}
we find the critical radius where $p_\mathrm{ram} = f_\mathrm{grav}(R_\mathrm{strip})$ <cit.>. Here $\Sigma_\mathrm{gas}$, $\Phi$, and $Z$ are the ISM mass surface density, the gravitational potential of the galaxy, and its scale height, respectively. Beyond $R_\mathrm{strip}$, the galaxy potential is not strong enough to retain the gas and stripping sets in. <cit.> give an analytic estimate for the GG criterion which determines the stripping radius:
\begin{equation}
\Sigma_\mathrm{gas} v_\mathrm{rot}^2 R_\mathrm{strip}^{-1} = p_\mathrm{ram},
\end{equation}
where $v_\mathrm{rot}$ is the rotation speed of the galaxy. Although, in reality, the onset of RPS is likely to be a highly non-linear process, the simple GG criterion has proven suitable for global characterisations of RPS in in-depth numerical simulations <cit.>.
§.§ Galaxy properties
Since our simple model aims only to predict the distribution of RPS events along galaxy orbits, but not the detailed properties of such events, we model all galaxies in our simulation as thin disks with radius $R_\mathrm{gal} = 15$ kpc and gas surface density $\Sigma_\mathrm{gas}$ = $10^{21}$ atoms per cm$^{2}$ moving face-on through the ICM.
To account for galaxy-to-galaxy variation in $f_\mathrm{grav}$, we also run all models for a range of galaxy masses, parametrized by the rotational velocity $v_\mathrm{rot}$ (see Eq. 2). The explored range of $v_\mathrm{rot}$ from 150 to 350 km s$^{-1}$ corresponds to dynamical masses, within 15 kpc, of $8\times10^{10}$, $2\times10^{11}$, and $4\times10^{11}$ M$_\odot$. The adopted range of rotational velocities covers a spectrum of masses from sub- to super-Milky Way sized objects.
§.§ Cluster properties
We describe the gas and total mass distribution within the cluster using a spherical $\beta$-model <cit.>
\begin{equation}
\rho = \rho_0\left[1 + \left(\frac{r}{r_0}\right)^2\right]^{-\frac{3}{2}\beta },
\end{equation}
where $\rho_0$ is the central mass (or gas) density, $\beta$ and $r_0$ are the power-law index and core radius, respectively, and $r$ is the cluster-centric radius.
We adopt a total mass of $1.3\times10^{15} M_\odot$ <cit.>. As the majority ($\sim$2/3) of the clusters in our sample do not show dramatic large-scale substructure, we assume that our model cluster is largely relaxed, featuring gas and total mass distributions that share a common centre, core radius $r_0$ and power-law slope $\beta$. We adopt $r_0$ = 180 kpc and $\beta$ = 0.59, the median of the values from the spatial X-ray analysis of <cit.>. Assuming a gas fraction $f_{\rm gas}=0.074$ <cit.> and the model parameters above, our canonical cluster has a central particle density $n_0$ of $2.29\times10^{-3}$ cm-3.
§.§ Galaxy trajectories
The orbits of test particles falling into our model cluster are computed for a wide range of initial orbital parameters that encompass expectations for infall along connected filaments and from cluster mergers. Orbit calculations begin at the end of a filament which is assumed to be at a distance of 2.5 Mpc from the cluster core ($\approx R_\mathrm{vir}$). In Fig. <ref>, we show a schematic of the quantities that characterise orbits in our model: the speed of a galaxy in the direction of the filament axis $v_\parallel$, the transverse velocity perpendicular to the filament flow $v_\perp$, and the impact parameter $b$.
Radial profiles of filaments in cosmological simulations show a well defined edge at a radius of 1.0–2.0 $h^{-1}$ Mpc ($\sim$1.4–2.8 Mpc in our assumed cosmology) beyond which the matter density essentially vanishes <cit.>. We therefore model filaments as cylinders of constant density with radius $b_\mathrm{max}$. We populate these filaments with $3\times10^4$ galaxies using Monte Carlo sampling designed to provide constant density within $b_\mathrm{max}$ and a normal distribution in $v_\perp$ to account for the velocity dispersion of galaxies within the filament.
In the following, we consider three infall scenarios that differ primarily in the approach velocity of galaxies at the cluster's virial radius: 1) stream-fed infall along filaments; 2) a slow merger; and 3) a fast merger. Table <ref> lists the model parameters that characterise each of these scenarios. For each infall scenario, we fix the initial velocity $v_\parallel$ at one value for all orbits. For the stream-fed model, we choose $b_{\rm max}$=1.5 Mpc and $v_\parallel$ = 200 km s$^{-1}$, the average filament radius and the average velocity of matter at the cluster-filament interface, respectively <cit.>, as well a velocity dispersion characteristic of group environments ($\sim$100 km s$^{-1}$). The slow and fast merger models are characterised by initial velocities of 1000 and 3000 km s$^{-1}$, respectively, and a velocity dispersion of 1000 km s$^{-1}$ and $b_\mathrm{max}$ = 2.5 Mpc for either merger scenario. Since we know neither the number and orientation of connected filaments for our cluster sample, nor the orientation of a putative merger axis, we place filaments/merging clusters at $10^3$ positions, sampled isotropically on a 2.5 Mpc sphere. In total, this results in $3\times10^6$ orbits per scenario which are each followed for 5 Gyr ($\sim t_\mathrm{cross}$) in time steps of 5 Myr.
Schematic diagram of the quantities that characterise the initial conditions and orbits of galaxies in our infall models: the maximal impact parameter $b_\mathrm{max}$, the initial velocities $v_\parallel$ and $v_\perp$, the cluster-centric radius $\hat{r}$, and the inclination angle $i$.
Model Parameters
Model $v_\parallel$ [km s$^{-1}$] $\sigma_v$ [km s$^{-1}$] $b_\mathrm{max}$ [Mpc]
Stream-fed 200 100 1.5
Slow Merger 1000 1000 2.5
Fast Merger 3000 1000 2.5
* See Fig. <ref> for a schematic illustration of $v_\parallel$ and $b_\mathrm{max}$; $\sigma_v$ indicates the velocity dispersion of infalling galaxies.
Defining the start of the RPS event as the time step in which the GG criterion is first satisfied, we explore a range of RPS event durations, from 50 Myr to 1 Gyr, during which the resulting event is assumed to remain observationally detectable. This choice is motivated by numerical simulations: <cit.> find that the signature RPS morphology should be observable in galaxies overrun by an ICM shock for between $\sim$several 10 Myr to a few 100 Myr. Slightly longer durations are quoted by <cit.> for a scenario similar to our stream-fed infall model (see also Fig. <ref>).
For comparison with our observational results, segments of the orbits corresponding to an RPS event (under our definition) are projected onto the plane of the sky, thus providing the projected angle of incidence (the projected angle between the galaxy's velocity and position vectors), $i$, and the projected radius from the cluster centre. We then tabulate the amount of time spent in bins of projected radius and inclination angle to construct simulated probability distributions for each scenario.
§.§ Accounting for observational bias
As mentioned in Section <ref> and illustrated in Fig. <ref>, RPS events in galaxies moving along or close to our line of sight are likely to be missed, as, for this particular geometry, the pronounced morphological features that our selection process is build upon are obscured by the galaxy being stripped. We examine the importance of this observational bias by imposing on our modeling results that all RPS events are undetectable that occur in galaxies moving along an axis that is inclined to our line of sight by 0, 15, 30, or 45 degrees. As detailed in the following section, even the most severe implementation of this line-of-sight bias results in only modest changes in the model predictions, suggesting that the effect does not significantly affect the conclusions drawn from our comparison with the data.
Cumulative distribution of the incidence angles of the jellyfish and cluster-member samples (asterisks and squares, respectively). The left, center, and right panels show predictions for the stream-fed, slow-merger, and fast-merger models (see Table <ref>), respectively. The dotted, dashed, and solid lines correspond to event durations of 50 Myr, 300 Myr, and 1 Gyr, respectively. Colors denote the mass of the infalling galaxy: blue (thin), green (medium), and red (thick) correspond to dynamical masses of $8\times10^{10}$, $2\times10^{11}$, and $4\times10^{11} M_\odot$, respectively. Model predictions shown in the top row assume that RPS events are identifiable as such regardless of the inclination of the galaxy's direction of motion with respect to our line of sight; results shown in the bottom row mimic the observational bias discussed in Sections <ref> and <ref> by excluding all events triggered in galaxies with velocity vectors within 30 degrees of our line of sight.
As Fig. <ref> but for the projected radius, $r_\mathrm{proj}$.
KS model probabilities for the projected incidence angle $i$ (top row) and the projected radius $r_{\rm pro}$ (bottom row) for the sample of cluster members (green squares in Figs. <ref> and <ref>), shown as a function of the duration of the RPS event, $\tau_\mathrm{event}$, and the mass of the respective galaxy (see legend). Infall along filaments (leftmost panels) is clearly disfavoured.
§ RESULTS
In order to reduce contamination by interlopers (fore- or background galaxies), we restrict our analysis to the subset of candidate RPS events with measured redshifts within $\pm$4000 km s$^{-1}$ of the redshift of the host cluster; the 53 objects (of 124 with measured redshifts) meeting this criterion are hereafter referred to as the “spectroscopic sample". Of these, we select a subset of the 15 galaxies exhibiting the most compelling “jellyfish" morphology comprised of the six systems presented by ESE and the nine shown in Fig. <ref> (“jellyfish sample"). We further restrict the comparison between data and model predictions to a projected radius of 415 kpc from the cluster core, which leaves 23 and 11 galaxies in the spectroscopic and jellyfish samples respectively. This radial cutoff minimises systematic incompleteness introduced at larger cluster-centric radii, which are covered only by images of the most distant clusters in our sample.
Fig. <ref> shows the cumulative distributions of the incidence angle for our two RPS subsamples plotted against predictions from our infall model, with (bottom row) and without (top row) correction for the bias discussed in Section <ref> and <ref>. The bottom The left, center, and right columns of Fig. <ref> show predictions for the stream-fed, slow-merger, and fast-merger models, respectively (see Table <ref> for the parameters characterising these models).
Visual comparison suggests that the observations are best matched by the model predictions for the slow-merger scenario, provided that the duration of the stripping process is less than a Gyr[Note that, in the top panel of Fig. <ref>, all of the solid lines, as well as the red dashed line, fall on top of each other and are thus indistinguishable by eye.]. Contrary to the traditional picture of RPS being driven purely by infall from the low-density field, preferably along filaments, we find poor agreement between the data and stream-fed models which over-predict events at extreme incidence angles (at $\lesssim$40$^\circ$ for almost all combinations of model parameters explored by us, and at $\gtrsim$140$^\circ$ for low-mass galaxies experiencing long RPS events). In this scenario, the motion of galaxies is dominated by the cluster potential, which leads to a preferential alignment of trajectories toward the cluster core (at least in our projected view) and thus a highly anisotropic distribution of incidence angles.
Fig. <ref> shows the cumulative distributions of the number of RPS events within a given projected cluster-centric radius. To provide more natural, equal-area sampling, we bin the data in equal steps of $r_\mathrm{proj}^2$; a uniform areal distribution thus appears as a straight line from zero to one. We find that both of the cluster merger models predict a nearly uniform areal distribution of events in agreement with our observations. Stream-fed models with the most massive galaxies and/or the longest event timescales predict an excess at small projected radii which is not supported by our data. Note, that this comparison also effectively rules out the stream-fed model with a Milky-Way sized galaxy and 300 Myr timescale that at least marginally matched the observed distribution of incidence angles and is shown as the green dashed line in Fig. <ref>.
A more quantitative assessment of the significance of the discrepancies between the observed and predicted distributions can be obtained with Kolmogorov-Smirnov (KS) tests. In Fig. <ref>, we show KS probabilities for the null hypothesis that the observed distributions are drawn from the same parent population as the predictions of a given model. Correcting all models for the aforementioned line-of-sight bias (Sections <ref> and <ref>) does not change our conclusions significantly. For simplicity, we therefore ignore the bias due to motion along the line of sight in the KS tests. To maximize the number of objects in the comparison, we show results for the spectroscopic sample only. However, considering the smaller jellyfish subsample does not significantly alter our conclusions.
Consistent with our qualitative assessments above, we find no agreement with the observed distribution of incidence angles for any model assuming infall along filaments, although the distribution of projected radii does not rule out such models (at least not for low-mass galaxies, see bottom panel of Fig. <ref>). By contrast, practically all of the models for the two merger scenarios provide an acceptable (or good) description of the data, with the exception of those involving the most massive galaxies, for which models assuming long RPS durations of $\tau_\mathrm{event}\gtrsim$ 300 Myr are ruled out at more than $2\sigma$ confidence.
§ CONCLUSIONS
Since our models are intrinsically three dimensional, the comparisons presented above, although involving solely parameters measured in projection, allow us to distinguish between distinctly different three-dimensional scenarios.
In the merger scenarios, RPS events are triggered in fast-moving galaxies near the outskirts of the cluster and, due to the relatively short duration of $\sim$500 Myr required by our incidence angle data (see top row of Fig. <ref>), remain confined to a shell well outside a (three-dimensional) cluster-centric radius of 400 kpc. On the other hand, the projected radius data favour event durations longer than $\sim$100 Myr to explain the uniform areal distribution (Fig. <ref>). The RPS candidates detected by us are thus the projection of the essentially uniform distribution of much more distant RPS events in the fore- and background segments of this shell. In principle, galaxies of all masses may contribute to the observed RPS distribution; however, the majority are likely to be systems of low to intermediate mass, since models for extremely massive galaxies generally require finely tuned, short RPS lifetimes of about 100 Myr approximately to match the observations (red lines in Fig. <ref>).
By contrast, galaxies falling into the cluster along filaments do so at much lower peculiar velocities and thus require higher ICM densities for the GG criterion to be met; as a result, RPS events are triggered only much closer to the cluster core. To match the observed, broad distribution of incidence angles, these galaxies need time to enter our field of view from all sides, which mandates that the associated RPS events remain observable for 300 Myr or longer (Fig. <ref>). Such long life-times, however, lead in turn to an excess in the number of events close to the cluster core that is not observed (Fig. <ref>).
We therefore tentatively conclude that extreme RPS events in massive clusters are generally short-lived ($\lesssim$500 Myr) and triggered far from the cluster core, likely driven by cluster mergers. Interestingly this preference of our analysis for RPS events being most readily observed in galaxies moving at high speed through an only modestly dense ICM suggests that textbook cases of “jellyfish galaxies" might also be observed near the cores of less massive clusters (or even groups of galaxies, see also ) provided a cluster or group merger event ensures sufficiently high peculiar initial velocity. Note also that, while our data disfavour infall along filaments as the primary trigger, they do not rule out a contribution from such a scenario. Wide-field imaging surveys that are able to detect RPS events out to the virial radius are needed to determine the relative contributions of stream-fed infall and cluster mergers.
§ SUMMARY
We have conducted a systematic search for galaxies experiencing ram-pressure stripping (RPS) in 63 MACS clusters at $z{=}$0.3–0.7. Using quantitative morphological parameters for $\sim$16,000 galaxies detected in Hubble Space Telescope images of these systems we identify 211 potential cases of RPS that complement a training set of 12 “jellyfish" galaxies used to define our selection criteria. Where possible, the direction of motion in the plane of the sky is estimated for these systems based on morphological indicators such as the curvature and orientation of the apparent galaxy-ICM interface region or a visible debris trail. Several systematic biases are inherent to our approach: (a) the classification of galaxies according to their likelihood of undergoing RPS is partly based on visual inspection and thus to some extent subjective, (b) the small field of field of view our observations prevents us from sampling the galaxy population in the outer regions of our cluster targets (except in projection) where RPS events might be initially triggered, and (c) our selection process is fundamentally unable to robustly identify RPS events in galaxies moving along, or close to, our line of sight.
We attempt to address the first of these biases by obtaining spectroscopic redshifts of all our RPS candidates. While the resulting spectra do not immediately confirm or refute an RPS event, they allow us to establish whether or not a morphologically selected candidate is in fact a cluster member and whether its spectral characteristics are consistent with ongoing or recent star formation. So far, 53 of 124 systems targeted in spectroscopic follow-up observations were confirmed as cluster members. A detailed analysis of these galaxies' spectral properties will be presented in a forthcoming paper (Blumenthal et al., in preparation).
The remaining two observational biases mentioned above can be accounted for by three-dimensional modelling of the trajectories and environment of galaxies falling into a massive cluster. Specifically, we
compare the distributions of the observed projected incidence angle and distance from the BCG with predictions from simple models of galaxy orbits in a MACS-like cluster. We investigate two scenarios: accretion of galaxies from an attached filament, and a cluster merger event.
We find significantly better agreement for the merger scenario, provided the duration of RPS events is $\lesssim$500 Myr. We thus tentative conclude that extreme ram-pressure stripping events is primarily triggered in massive cluster mergers (rather than by infall alone) where relative velocities between galaxies and the ICM are large enough to initiate RPS far from the cluster core ($\gg$ 400 kpc). Although our study is, by design, limited to relatively massive clusters, we note that this result implies that extreme RPS events may also occur in mergers of poorer clusters and even groups of galaxies, where the required ingredients (high peculiar velocity and moderately high ICM density) are both met by galaxies close to core passage.
We also find that galaxies of mass similar to, or less than, our Milky Way are likely to dominate the set of observable RPS events in massive clusters, although more massive galaxies may contribute too at a lower level. Although models assuming infall along a filament were found to yield predictions that are largely in conflict with our data, both processes (accretion along filaments and via cluster mergers) can be expected to contribute. The extent to which the two mechanisms are responsible for the observed population of RPS events in our sample is difficult to quantify but could be tested by imaging surveys that probe the distribution of RPS events to larger cluster-centric radii.
In-depth studies of the X-ray properties of RPS host clusters along with spectroscopic investigations of the star-formation rates and histories of the candidates identified in this study will be critical to test our conclusions and allow a quantitative comparison of observational diagnostics with predictions of numerical models of ram-pressure stripping.
§ ACKNOWLEDGEMENTS
We thank the anonymous referee for their helpful comments, questions, and suggestions on revising the manuscript. CM thanks J. Lotz for providing the galaxy morphology source code which was adapted for this work. HE gratefully acknowledges financial support from STScI grants GO-10495, -10875, -12166, and -12884. This research made use of Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration, 2013).
§ SKELETAL DECOMPOSITION PARAMETERS
The morphological indicators discussed in Section 3 were generally defined to identify characteristic morphological traits of galaxy mergers <cit.>. We introduce a new metric based on the concept of the morphological skeleton <cit.> to both quantify the amount of substructure in a galaxy while concurrently identifying arm/tail-like structures. Conceived in the context of mathematical morphology <cit.> and originally introduced as a means for binary image compression, the morphological skeleton (or medial axis transform) reduces a shape to a line that maintains the topological structure of the full image, thus allowing exact reconstruction.
We here generalise the definition of the morphological skeleton to images with non-binary, continuous greyscale pixel values. However, we must be cautious as noise in relatively short exposures used in this survey ($\sim$1200 sec.) can manifest as small scale substructure in the skeleton if applied naively. To reduce this erroneous signal from noise, we smooth the image using a Gaussian kernel before determining the skeleton and then prune the result to remove any disconnected segments. We define the result of this process as $Sk_i$. We perform skeletal decompositions under three smoothing scales corresponding to the Petrosian radius $r_p$, the half light radius $r_{50\%}$, and the 10% light radius $r_{10\%}$ which define $Sk_0$, $Sk_1$, and $Sk_2$. Note that due to the cleaning process we apply here, exact reconstruction of the original image is not possible.
To further reduce erroneous signal due to residual noise, we define $Sk_{x+y}$ (where $y=x+1$) as comprising all pixels in the higher-order skeleton (i.e. under a smaller smoothing kernel) connected to that of the lower-order skeleton (larger smoothing kernel). To generate a common reference point and to avoid bias due to image size and lower order structure, we then subtract the length of the lower-order skeleton from $Sk_{x+y}$ and normalise by the length of the lower-order skeleton (e.g. $[|Sk_{0+1}|-|Sk_0|]/|Sk_0|$) defining a final numerical measure $Sk_{x-y}$ which quantifies the excess in substructure under smoothing scale $y$ with respect to $x$ (see Fig. <ref>).
A simple way to understand this qualitatively is to consider a case where $Sk_{0-1}$ or $Sk_{0-1}$ is equal to zero. This would imply that image smoothed on a finer scale (smaller kernel) does not reveal any more substructure or that the galaxy's light profile is essentially smooth below the upper smoothing scale. However, as a full interpretation of the meaning and reliability of these indicators is beyond of the scope of this paper, we here characterise $Sk_{0-1}$ only to be a measure of bending in the galaxy or the deviation from a symmetric object (somewhat correlated with asymmetry), while $Sk_{1-2}$ quantifies the amount of clumpy substructure connected to the brighter regions of the galaxy.
An example of the results of the greyscale skeletonization process which we use to define the skeletal decomposition parameters $Sk_{0-1}$ and $Sk_{1-2}$.
SNAPS Observations
Name $\alpha$ [J2000] $\delta$ [J2000] $t_{exp}$ [s] GO Prop. ID
EMACSJ1057.5+5759 10:57:31.680 +57:59:33.72 1200 12884
MACSJ0032.1+1808 00:32:11.344 +18:07:49.37 1200 12166
MACSJ0035.4-2015 00:35:26.957 -20:15:50.66 1200 10491
MACSJ0140.0-0555 01:40:01.626 -05:55:06.71 1200 10491
MACSJ0152.5-2852 01:52:35.361 -28:53:39.88 1200 10491
MACSJ0257.6-2209 02:57:40.596 -22:09:27.80 1200 10875
MACSJ0308.9+2645 03:08:56.839 +26:45:43.91 1200 12166
MACSJ0451.9+0006 04:51:55.443 +00:06:11.66 1200 10491
MACSJ0521.4-2754 05:21:25.808 -27:55:06.91 1200 10491
MACSJ0547.0-3904 05:47:01.796 -39:04:13.24 1200 12166
MACSJ0553.4-3342 05:53:23.850 -33:42:42.21 2092 12362
MACSJ0712.3+5931 07:12:21.985 +59:32:24.82 1200 10491
MACSJ0845.4+0327 08:45:28.224 +03:27:28.46 1200 10491
MACSJ0916.1-0023 09:16:12.344 -00:23:47.00 1200 10491
MACSJ0947.2+7623 09:47:10.744 +76:23:21.62 1200 10491
MACSJ0949.8+1708 09:49:52.655 +17:07:06.38 1200 10491
MACSJ1006.9+3200 10:06:55.632 +32:01:33.91 1200 10491
MACSJ1115.2+5320 11:15:15.968 +53:19:47.47 1200 10491
MACSJ1124.5+4351 11:24:29.365 +43:51:32.97 1200 12166
MACSJ1133.2+5008 11:33:14.109 +50:08:29.50 1200 10491
MACSJ1142.4+5831 11:42:26.434 +58:32:01.30 1200 12166
MACSJ1226.8+2153C 12:26:41.421 +21:53:07.58 1200 12166
MACSJ1236.9+6311 12:36:59.868 +63:11:02.26 1200 10491
MACSJ1258.0+4702 12:58:02.708 +47:02:42.87 1200 10491
MACSJ1319.9+7003 13:20:09.685 +70:04:28.16 1200 10491
MACSJ1354.6+7715 13:54:31.253 +77:15:08.71 1200 10491
MACSJ1447.4+0827 14:47:26.289 +08:28:37.08 1200 12166
MACSJ1452.9+5802 14:52:57.957 +58:02:43.28 1200 12166
MACSJ1526.7+1647 15:26:42.342 +16:47:48.83 1200 12166
MACSJ1621.3+3810 16:21:23.928 +38:10:16.28 1200 12166
MACSJ1644.9+0139 16:45:01.729 +01:40:09.83 1200 12166
MACSJ1652.3+5534 16:52:19.726 +55:34:46.63 1200 10491
MACSJ1731.6+2252 17:31:39.268 +22:52:05.09 1200 12166
MACSJ1738.1+6006 17:38:05.383 +60:06:14.92 1200 12166
MACSJ1752.0+4440 17:51:57.961 +44:39:45.45 1200 12166
MACSJ1806.8+2931 18:06:51.898 +29:30:23.03 1200 12166
MACSJ2050.7+0123 20:50:42.381 +01:23:24.69 1200 12166
MACSJ2051.1+0215 20:51:10.058 +02:16:00.72 1200 12166
MACSJ2135.2-0102 21:35:12.822 -01:02:51.52 1200 10491
MACSJ2241.8+1732 22:41:56.386 +17:32:47.33 1200 12166
SMACSJ0234.7-5831 02:34:43.512 -58:31:16.51 1200 12166
SMACSJ0549.3-6205 05:49:18.358 -62:05:07.88 1200 12166
SMACSJ0600.2-4353 06:00:12.915 -43:53:19.33 1200 12166
SMACSJ0723.3-7327 07:23:18.709 -73:27:06.01 1200 12166
SMACSJ2031.8-4036 20:31:46.993 -40:37:03.68 1200 12166
SMACSJ2131.1-4019 21:31:05.693 -40:19:12.22 1200 12166
CLASH Observations
Name $\alpha$ [J2000] $\delta$ [J2000] $t_{exp}$ [s] GO Prop. ID
MACSJ0329-0211 03:29:41.560 -02:11:46.10 4104 12452
MACSJ0416-2403 04:16:08.380 -24:04:20.79 4036 12459
MACSJ0429-0253 04:29:36.049 -02:53:06.10 3938 12788
MACSJ0647+7015 06:47:50.269 +70:14:54.99 4128 12101
MACSJ0717.5+3745-POS5 07:17:32.629 +37:44:59.70 7920 10420
MACSJ0744+3927 07:44:52.819 +39:27:26.89 4128 12067
MACSJ1115+0129 11:15:51.900 +01:29:55.10 3870 12453
MACSJ1149+2223 11:49:34.704 +22:24:04.75 4128 12068
MACSJJ1206.2-0847 12:06:12.055 -08:47:59.44 6608 10491
MACSJ1311-0310 13:11:01.800 -03:10:39.79 4158 12789
RXJ1347-1145 13:47:32.110 -11:45:11.36 3878 12104
MACS1423+2404 14:23:47.88 +24:04:42.49 4240 12790
RXJ1532+3021 15:32:53.779 +30:20:59.39 4060 12454
MACSJ1720+3536 17:20:16.780 +35:36:26.49 4040 12455
MACSJ1931-2635 19:31:49.62 -26:34:32.90 3850 12456
MACSJ2129-0741 21:29:26.059 -07:41:28.79 3728 12100
RXJ2248-4431 22:48:43.960 -44:31:51.30 3976 12458
|
1511.00138
|
Hypermagnetic gyrotropy, inflation
and the baryon asymmetry of the Universe
Massimo Giovannini
[Electronic address: massimo.giovannini@cern.ch]
Theory Division, CERN, 1211 Geneva 23, Switzerland
INFN, Section of Milan-Bicocca, 20126 Milan, Italy
We investigate the production of the hypermagnetic gyrotropy
when the electric and magnetic gauge couplings evolve at different rates, as it happens
in the the relativistic theory of the Van der Waals forces. If a pseudo-scalar interaction breaks the duality symmetry of the corresponding equations, the gyrotropic configurations of the hypermagnetic fields can be amplified from the vacuum during an inflationary stage of expansion. After charting the parameter space of the model in terms of the rates of evolution of the magnetic and electric gauge couplings, we identify the regions where the gyrotropy is sufficiently intense to seed the baryon asymmetry of the Universe at the electroweak epoch while the backreaction constraints, the strong coupling bounds and the other astrophysical limits are concurrently satisfied.
Since the seminal work of Sakharov <cit.> various attempts have been made to account for the baryon asymmetry
of the Universe (BAU in what follows). The standard lore of baryogenesis (see e.g. the first article of Ref. <cit.>) stipulates that during a strongly
first-order electroweak phase transition the expanding bubbles are nucleated while the baryon number
is violated by sphaleron processes. Given the current value of the Higgs mass, to produce a sufficiently strong (first-order) phase transition and to get enough $CP$ violation at the
bubble wall, the standard electroweak theory must be appropriately extended. The second complementary lore for the generation of the BAU is leptogenesis (see e.g. the second article of Ref. <cit.>) which can be conventionally realized thanks to heavy Majorana neutrinos decaying out of equilibrium and producing an excess of lepton
number ($L$ in what follows). The excess in $L$ can lead to the observed baryon number thanks to sphaleron interactions violating $(B+ L)$.
An admittedly less conventional perspective stipulates that the BAU could be the result of the decay of maximally helical configurations of the hypercharge field (sometimes dubbed hypermagnetic knots) <cit.>. Indeed, while the $SU_{L}(2)$ anomaly is typically responsible for $B$ and $L$ non-conservation via instantons and sphalerons, the $U_{Y}(1)$ anomaly might lead to the transformation
of the infra-red modes of the hypercharge field into fermions <cit.>. As previously suggested (see third and fourth paper of <cit.>) the production of the BAU demands, in this
context, the dynamical generation of the hypermagnetic gyrotropy[The magnetic and kinetic gyrotropies
play a crucial role in the mean-field dynamo <cit.> and the same notion occurs, with the due differences, in the present context.] ${\mathcal G}^{(B)}(\vec{x}, \tau) = \vec{B}_{Y}\cdot \vec{\nabla} \times \vec{B}_{Y}$ (where $\vec{B}_{Y}$ denotes the hypermagnetic field).
This hypothesis has been subsequently scrutinized, within diverse frameworks, by several authors (see e.g. <cit.> for an incomplete list of references).
In this investigation we propose a specific scenario accounting at once for the formation
of the hypermagnetic knots and for the existence of large-scale magnetic.
We shall therefore posit a derivative interaction of the hypercharge field with one or more scalar fields
during a quasi-de Sitter stage of expansion. Let us consider, fo the sake of concreteness the following four-dimensional action:
\begin{equation}
S = \int d^4 x \, \sqrt{- g} \biggl[ {\mathcal M}_{\sigma}^{\rho}(\varphi,\psi)
Y_{\rho\alpha}\, Y^{\sigma\alpha} - {\mathcal N}_{\sigma}^{\rho}(\varphi,\psi) \widetilde{Y}_{\rho\alpha}\, \widetilde{Y}^{\sigma\alpha}
+ {\mathcal Q} _{\sigma}^{\rho}(\varphi,\psi) Y_{\rho\alpha}\, \widetilde{Y}^{\sigma\alpha}\biggr],
\label{action}
\end{equation}
where $Y_{\alpha\beta}$ and $\widetilde{Y}^{\alpha\beta}$ are, respectively, the hypercharge field strength and its dual. The symmetric tensors ${\mathcal M}_{\sigma}^{\rho}(\varphi,\psi)$, ${\mathcal N}_{\sigma}^{\rho}(\varphi,\psi)$ and
${\mathcal Q} _{\sigma}^{\rho}(\varphi,\psi)$ contain the couplings of the hypercharge either to the inflaton field
itself (be it for instance $\varphi$) or to some other spectator field (be it for instance $\psi$). The typical derivative coupling arising in the relativistic theory of Casimir-Polder and Van der Waals interactions <cit.> is implicitly contained in Eq. (<ref>) when ${\mathcal Q} _{\sigma}^{\rho}(\varphi,\psi)=0$: in this case Eq. (<ref>) offers a viable framework for inflationary magnetogenesis <cit.> characterized by unequal electric and magnetic susceptibilities. For immediate convenience the three symmetric tensors appearing in Eq. (<ref>) can be parametrized as follows:
\begin{eqnarray}
{\mathcal M}^{\lambda}_{\,\, \rho} &=& - \frac{\lambda}{16\pi} \delta^{\lambda}_{\,\rho} - \frac{\lambda_{E}(\varphi, \psi)}{16 \pi}\, \, u^{\lambda} \, u_{\rho}, \qquad {\mathcal N}^{\lambda}_{\,\, \rho} = - \frac{\lambda_{B}(\varphi, \psi)}{16 \pi}\, \overline{u}^{\lambda} \, \overline{u}_{\rho},
\nonumber\\
{\mathcal Q}^{\lambda}_{\,\, \rho} &=& \frac{1}{16 \pi}\, [\lambda_{1}(\varphi,\psi) \delta^{\lambda}_{\rho} + \lambda_{2}(\varphi,\psi) \, \overline{u}^{\lambda} \, \overline{u}_{\rho}],
\label{POS}
\end{eqnarray}
where $u_{\rho} = \partial_{\rho}\varphi/\sqrt{g^{\alpha\beta} \partial_{\alpha} \varphi \partial_{\beta} \varphi}$ and
$\overline{u}_{\rho} = \partial_{\rho}\psi/\sqrt{g^{\alpha\beta} \partial_{\alpha} \psi \partial_{\beta} \psi}$ are the normalized
gradients of the scalar fields. When all the coupling functions of Eq. (<ref>) vanish except for $\lambda$, we recover the standard situation where the scalar fields are only coupled to the gauge kinetic term of the hypercharge without Van der Waals interactions. In a conformally flat background geometry[We focus now on the case where the background metric is conformally flat, i.e. $g_{\mu\nu}(\tau) = a^{2}(\tau) \eta_{\mu\nu}$ where $\eta_{\mu\nu}$ is the Minkowski metric and $\tau$ is the conformal time coordinate.] the gauge field strength can be expressed in terms of the electric and magnetic fields as
$Y^{i0} = e^{i}/a^2$ and $Y^{ij} = - \epsilon^{ijk} b_{k}/a^2$, the explicit components of ${\mathcal M}_{\rho}^{\sigma}$ and ${\mathcal N}_{\rho}^{\sigma}$ can be directly obtained from Eqs. (<ref>).
The (comoving) hyperelectric and hypermagnetic are defined, respectively, as
$\vec{B}_{Y} = a^2 \, \sqrt{\Lambda_{B}}\, \vec{b}$ and $\vec{E}_{Y} = a^2 \, \sqrt{\Lambda_{E}}\, \vec{e}$; note that $\Lambda_{B} = (\lambda + \lambda_{B}/2)$ and $\Lambda_{E} = (\lambda + \lambda_{E}/2)$ can be physically interpreted as the squares of the hypermagnetic and hyperelectric susceptibilities. The evolution equations of $\vec{E}_{Y}$ and $\vec{B}_{Y}$
follow directly from Eqs. (<ref>)–(<ref>) and they are:
\begin{eqnarray}
&& \vec{\nabla} \times \biggl( \sqrt{\Lambda_{B}} \vec{B}_{Y} \biggr) = \partial_{\tau} \biggl( \sqrt{\Lambda_{E}} \vec{E}_{Y} \biggr) + 4 \pi \vec{J}_{Y} + \partial_{\tau} \biggl[\frac{\Lambda_{BE}}{\sqrt{\Lambda_{B}} } \vec{B}_{Y}\biggr] + \vec{\nabla}\times \biggl[ \frac{\Lambda_{BE}}{\sqrt{\Lambda_{E}}} \vec{E}_{Y} \biggr],
\label{one}\\
&& \vec{\nabla} \times \biggl(\frac{\vec{E}_{Y}}{\sqrt{\Lambda_{E}}}\biggr) + \partial_{\tau} \biggl(\frac{\vec{B}_{Y}}{\sqrt{\Lambda_{B}}}\biggr) =0,
\label{two}\\
&& \vec{\nabla} \cdot \biggl(\frac{\vec{B}_{Y}}{\sqrt{\Lambda_{B}}}\biggr)=0,\qquad \vec{\nabla}\cdot ( \sqrt{\Lambda_{E}}\, \vec{E}_{Y} ) + \vec{\nabla}\cdot \biggl[ \frac{\Lambda_{BE}}{\sqrt{\Lambda_{B}}} \vec{B}_{Y}\biggr]= 4 \pi \rho_{Y},
\label{three}
\end{eqnarray}
where $\Lambda_{BE} = (2 \lambda_{1} + \lambda_{2}/2)$ and $\partial_{\tau}$ denotes the derivative with respect to the conformal time coordinate. The hyperelectric and hypermagnetic couplings are, respectively, $g_{E} = (4\pi/\Lambda_{E})^{1/2}$ and as $g_{B} = (4\pi/\Lambda_{B})^{1/2}$.
When $\Lambda_{BE} \to 0$ and in the absence of sources (i.e. $\vec{J}_{Y} \to 0$ and $\rho_{Y} \to 0$), Eqs. (<ref>), (<ref>) and (<ref>) are invariant under a symmetry[More specifically
the generalized duality transformation stipulates that under the exchange and inversion of the susceptibilities ($\sqrt{\Lambda}_{E} \to 1/\sqrt{\Lambda_{B}}$ and $\sqrt{\Lambda_{B}} \to 1/\sqrt{\Lambda_{E}}$) or of the corresponding couplings (i.e. $g_{E} \to 1/g_{B}$ and $g_{B} \to 1/g_{E}$) Eqs. (<ref>), (<ref>) and (<ref>) maintain the same form provided the electric and magnetic fields are also exchanged as $\vec{E} \to - \vec{B}$ and $\vec{B} \to \vec{E}$.] that generalizes the conventional duality transformation <cit.>.
While the expression of ${\mathcal M}_{\rho}^{\sigma}$ and ${\mathcal N}_{\rho}^{\sigma}$ may contain
supplementary terms, a general analysis shows that these terms will simply modify the relation of $\Lambda_{B}$ and $\Lambda_{E}$ to the parameters appearing in the Lagrangian without altering the form of Eqs. (<ref>), (<ref>) and (<ref>).
The duality symmetry is explicitly broken when $\Lambda_{BE} \neq 0$. In this case the hypermagnetic gyrotropy is amplified from the vacuum fluctuations of the hyperelectric and hypermagnetic fields. To estimate this effect let us consider $\Lambda_{E}$, $\Lambda_{B}$ and $\Lambda_{BE}$ as time dependent but otherwise homogeneous. The hyperelectric and hypermagnetic field operators can be represented in Fourier space as[We use the circular polarization
basis where $\epsilon^{(\pm)}_{i}(\hat{k}) = [ e^{(1)}_{i} \pm i e^{(2)}_{2}]/\sqrt{2}$, and
$\vec{k} \times \vec{\epsilon}^{\,\,(\pm)}(\hat{k}) = \mp i \, k \, \vec{\epsilon}^{\,\,(\pm)}(\hat{k})$; $\hat{e}_{1}$, $\hat{e}_{2}$ and $\hat{k}$ are a set of mutually orthgonal unit vectors. In Eqs. (<ref>) and (<ref>) the sums run over the circular polarizations, i.e. $\alpha=\pm$.]:
\begin{eqnarray}
\hat{B}_{i}^{(Y)}(\vec{p},\eta) &=& - \frac{i}{\sqrt[4]{f}} \sum_{\alpha} \biggl\{ [ \vec{p} \times \vec{\epsilon}^{\,\,(\alpha)}]_{i} \,\, \overline{F}_{p,\,\alpha}(\eta) \, \hat{a}_{\vec{p},\alpha} - [ \vec{p} \times \vec{\epsilon}^{\,\,(\alpha)\, *}]_{i} \,\,\hat{a}^{\dagger}_{-\vec{p},\alpha} \overline{F}^{*}_{p,\,\alpha}(\eta)\biggr\},
\label{test3}\\
\hat{E}_{i}^{(Y)}(\vec{p},\eta) &=& - \frac{1}{\sqrt[4]{f}} \sum_{\alpha} \biggl\{ \epsilon^{\,\,(\alpha)}_{i} \,\, \overline{G}_{p,\,\alpha}(\eta) \, \hat{a}_{\vec{p},\alpha} + \epsilon^{\,\,(\alpha)\, *}_{i} \,\,\hat{a}^{\dagger}_{-\vec{p},\alpha} \overline{G}^{*}_{p,\,\alpha}(\eta)\biggr\};
\label{test3a}
\end{eqnarray}
$\eta$ is an auxiliary time variable defined as $ \sqrt{f} \, d\tau = d\eta$ where $f = \Lambda_{E}/\Lambda_{B}$. In the limit $f\to 1$ the conformal time variable coincides with $\eta$. In Eqs. (<ref>) and (<ref>) $\overline{F}_{p,\,\alpha}(\eta)$ and $\overline{G}_{p,\,\alpha}(\eta)$ (with $\alpha = \pm$) are the mode functions obeying the following evolution equations:
\begin{eqnarray}
\overline{F}_{k,\,\pm}^{\,\,\prime\prime} + \biggl(k^2 - \frac{\sqrt[4]{\Lambda_{E}\, \Lambda_{B}}^{\,\,\prime\prime} }{\sqrt[4]{\Lambda_{E}\, \Lambda_{B}}}\biggr) \overline{F}_{\pm} \pm k\,\frac{\Lambda_{BE}^{\prime}}{\sqrt{\Lambda_{B} \Lambda_{E}}} \overline{F}_{\pm} =0, \qquad \overline{G}_{k,\,\pm} = \overline{F}_{k,\,\pm}^{\,\,\prime} - \frac{\sqrt{\Lambda_{B} \Lambda_{E}}^{\,\,\prime}}{\sqrt{\Lambda_{E}\, \Lambda_{B}}}\, \overline{F}_{k,\,\pm},
\label{modef}
\end{eqnarray}
where the prime shall denote hereunder a derivation with respect to $\eta$.
After computing the averages of pairs of field operators at equal times (but different comoving three-momenta) over the initial vacuum state, the corresponding two-point functions can be readily obtained:
\begin{eqnarray}
\langle \hat{B}^{(Y)}_{i}(\vec{k}, \eta) \, \hat{B}^{(Y)}_{j}(\vec{p},\eta) \rangle &=& \frac{2 \pi^2}{k^3} \delta^{(3)}(\vec{k} + \vec{p}) \biggl[ P_{ij}(\hat{k}) \, P_{B}(k, \eta)
+ i \, \epsilon_{i j \ell}\, \hat{k}^{\ell} \,P^{(B)}_{{\mathcal G}}(k,\eta) \biggr],
\label{test4}\\
\langle \hat{E}^{(Y)}_{i}(\vec{k},\eta) \, \hat{E}^{(Y)}_{j}(\vec{p},\eta) \rangle &=& \frac{2 \pi^2}{k^3} \delta^{(3)}(\vec{k} + \vec{p}) \biggl[ P_{ij}(\hat{k}) \, P_{E}(k, \eta)
+ i \, \epsilon_{i j \ell}\, \hat{k}^{\ell} \,P^{(E)}_{{\mathcal G}}(k,\eta) \biggr],
\label{test4a}
\end{eqnarray}
where $P_{ij}(\hat{k}) = (\delta_{ij} - \hat{k}_{i} \hat{k}_{j})$ and $\epsilon_{i j \ell}$ is the three-dimensional Levi-Civita symbol. The hypermagnetic and hyperelectric power spectra appearing in Eqs. (<ref>) and (<ref>) are:
\begin{equation}
P_{B}(k,\eta) = \frac{k^{5}}{4 \pi^2 \sqrt{f}}\biggl[ | \overline{F}_{k, \, +}|^2 + | \overline{F}_{k, \, -}|^2\biggr],\quad
P_{E}(k,\eta) = \frac{k^{3}}{4 \pi^2 \sqrt{f}}\biggl[ | \overline{G}_{k, \, +}|^2 + | \overline{G}_{k, \, -}|^2\biggr],
\label{test5a}
\end{equation}
while $P^{(B)}_{{\mathcal G}}(k,\eta)$ and $P^{(E)}_{{\mathcal G}}(k,\eta)$ are the corresponding gyrotropic contributions:
\begin{equation}
P^{(B)}_{{\mathcal G}}(k,\eta) = \frac{k^{5}}{4 \pi^2 \sqrt{f}}\biggl[ | \overline{F}_{k, \, -}|^2 - | \overline{F}_{k, \, +}|^2\biggr],\quad
P^{(E)}_{{\mathcal G}}(k,\eta) = \frac{k^{3}}{4 \pi^2 \sqrt{f}}\biggl[ | \overline{G}_{k, \, -}|^2 - | \overline{G}_{k, \, +}|^2\biggr].
\label{test6a}
\end{equation}
The quasi-de Sitter evolution affects not only
the energy densities but also the gyrotropies
(i.e. ${\mathcal G}^{(B)}= \vec{B}_{Y}\cdot \vec{\nabla} \times \vec{B}_{Y}$ and
${\mathcal G}^{(E)}= \vec{E}_{Y}\cdot \vec{\nabla} \times \vec{E}_{Y}$). In the approximation
of sudden reheating the quasi-de Sitter phase is replaced by the radiation epoch at $\tau_1$ and the value of the comoving conductivity $\sigma_{c}$
depends on the temperature of the plasma so that $T/\sigma_{c} \simeq {\mathcal O}(2 \alpha^{\prime}) \sim 1/70$ in the limit of high temperatures <cit.>; note that $\alpha^{\prime} = g'^2/(4\pi)$ and $g'\simeq 0.3 $ is the $U(1)_{Y}$ coupling after inflation. Under these conditions the hyperelectric gyrotropy is washed out by finite conductivity effects while the contribution of the hypermagnetic gyrotropy determines the comoving baryon to entropy ratio $\eta_{B} = n_{B}/\varsigma$ <cit.>:
\begin{equation}
\eta_{B}(\vec{x},\tau) = \frac{ 3 \alpha' n_{f}}{8\pi \, H} \biggl(\frac{T}{\sigma_{c}}\biggl) \frac{{\mathcal G}^{(B)}(\vec{x}, \tau)}{a^4 \rho_{crit}},
\label{BAU}
\end{equation}
where $\varsigma= 2 \pi^2 T^3 N_{eff}/45$ is the entropy density of the plasma;
and $n_{f}$ is the number of fermionic generations. In what follows $N_{eff}$ shall be fixed to its standard
model value (i.e. $106.75$). Equation (<ref>) holds when the rate of the slowest reactions in the plasma (associated with the right-electrons) is larger than the dilution rate caused by the hypermagnetic field itself <cit.>: at the phase transition
the hypermagnetic gyrotropy is converted back into fermions since the ordinary magnetic fields does not couple to fermions.
The expectation value of $\eta_{B}(\vec{x}, \tau_{ew})$, i.e. $\langle \eta_{B}\rangle = \langle \eta_{B}(\vec{x}, \tau_{ew})\rangle$
depends on the average gyrotropy $\langle {\mathcal G}^{(B)}(\vec{x},\tau_{ew}) \rangle $ and thanks to Eq. (<ref>) the result is:
\begin{equation}
\langle \eta_{B}\rangle = \frac{ 3 \,\alpha^{\prime}\, n_{f}}{4 \pi H_{ew} a^4 \rho_{crit}} \biggl(\frac{T}{\sigma_{c}}\biggr) \int_{0}^{q_{\sigma}} P_{\mathcal G}^{(B)}(q,\tau_{1})\, d q.
\label{BAU2}
\end{equation}
The integral appearing in Eq. (<ref>) extends from $q_{ew} \simeq H_{ew} a_{ew}$
to $q_{\sigma} \simeq \sqrt{ a_{ew} H_{ew} \sigma_{c}}$ where $q_{\sigma}$ is the hypermagnetic diffusivity scale. If the
Hubble radius at the electroweak epoch is around $H_{ew}^{-1} \simeq 3$ cm, the diffusivity scale is roughly $10^{-7}\, H_{ew}^{-1}$
for a typical electroweak temperature ${\mathcal O}(100)$ GeV. The lower extremum of integration can be
even extrapolated to $0$ since the integrand converges in this limit.
It is convenient to express the integrand of Eq. (<ref>) in terms of the critical fraction of the hypermagnetic energy density multiplied by $R(q,\tau)$ measuring the asymmetry of the mode functions in the circular basis:
\begin{equation}
\langle \eta_{B}\rangle = \frac{3 \,\alpha^{\prime}}{4 \pi} n_{f} \biggl(\frac{T}{\sigma_{c}}\biggr) \int_{0}^{q_{\sigma}} \frac{d q}{q} R(q, \tau_{1}) \biggl(\frac{q}{q_{ew}}\biggr) \Omega_{B}(q,\tau_{1}), \qquad R(q, \tau) = \frac{|\overline{F}_{-}|^2 - |\overline{F}_{+}|^2}{|\overline{F}_{-}|^2 + |\overline{F}_{+}|^2},
\label{BAU3}
\end{equation}
where we introduced the notation already used in <cit.> for $\Omega_{B}(q,\tau) =3 P_{B}(q,\tau)/[4 \pi H^2 a^4 M_{P}^2] $; the analog quantity in the hyperelectric case
is $\Omega_{E}(q,\tau) = 3 P_{E}(q,\tau)/[4 \pi H^2 a^4 M_{P}^2]$.
In a mode-independent approach the gauge couplings can always be parametrized as $g_{E}(a) =\overline{g}_{E} (a/a_{1})^{F_{E}}$ and $g_{B}(a) =\overline{g}_{B} (a/a_{1})^{F_{B}}$ for $a \leq a_{1}$ where $F_{E}$ and $F_{B}$ denote the rates of variation in units of the Hubble rate and $a_{1}$ is the scale factor at the end of inflation. In the first quadrant of the $(F_{B},\, F_{E})$ plane where the gauge coupling are both increasing Eq. (<ref>) becomes:
\begin{equation}
\overline{F}_{k\,\pm}^{\prime\prime} + \biggl[ k^2 - \frac{\sigma^2 - 1/4}{\eta^2} \pm \frac{k \Lambda_{BE}^{\prime}}{\sqrt{\Lambda_{B} \Lambda_{E}}} \biggr] \overline{F}_{k\,\pm}=0,\qquad \sigma =\frac{1 - 2 F_{E}}{2( 1 + F_{B} - F_{E})}.
\label{F1}
\end{equation}
We now remark that the contribution of $\Lambda_{BE}$ is suppressed in the limit $k \to 0$. Introducing then two numerical factors of order $1$ (i.e. $\gamma$ and $\beta$ in what follows), $\Lambda_{BE}$ can be expressed
as the sum of of two complementary contributions, namely $\Lambda_{BE} = \gamma \sqrt{\Lambda_{E} \Lambda_{B}} + \beta \eta_{1} \sqrt{\Lambda_{E} \Lambda_{B}}^{\,\,\prime}$. Recalling that $\eta_{1} =\tau_{1}$, if either $\gamma$ or $\beta$ vanish
separately the leading contribution to $R(q, \tau_{1})$ goes as $q\tau_{1}$ up to logarithmic corrections. If both $\beta$ and $\gamma$ are
present the contribution of the term containing $\beta$ is always the leading one. Note, incidentally, that
Eq. (<ref>) can be written in one of the well known forms of the Whittaker's equation (see e.g. <cit.> for the first applications of this equation to the evolution of the cosmological inhomogeneities). Recalling that the amplified modes satisfy $k \tau_{1} < 1$, we also have that
$R(q,\tau_{1}) = 2\beta (2 |\sigma| - 1) \,k\tau_{1} \,\ln{k\tau_{1}}$.
We can then estimate the integral over the modes and obtain, from Eq. (<ref>), the following rather general result:
\begin{eqnarray}
\langle \eta_{B}\rangle &=& {\mathcal C}_{B}(F_{B}, F_{E}, \beta) \,\epsilon \,{\mathcal A}_{{\mathcal R}} \biggl(\frac{q_{\sigma}}{q_{ew}}\biggr)^{7 - 2 |\sigma|}
\biggl(\frac{q_{ew}}{q_1}\biggr)^{6 - 2 |\sigma|} [ (7 - 2 |\sigma|) \ln{(q_{\sigma}/q_{1})} -1],
\nonumber\\
{\mathcal C}_{B}(F_{B}, F_{E}, \beta) &=&\beta\,\, \frac{2^{2 (|\sigma|+1)} \Gamma^2(|\sigma|) ( 2 |\sigma|-1)}{3 \pi ( 7 - 2 |\sigma|)^2} \, |1 +F_{B} - F_{E}|^{ 2 |\sigma| -1},
\label{F6}
\end{eqnarray}
where, for $T_{ew} \simeq 10^{2}$ GeV we have that the two previous dimensionless ratios are given, respectively, by $(q_{\sigma}/q_{ew}) =8.43\times 10^{7} (\sigma_{c}/T_{ew})^{1/2} $ and by $(q_{ew}/q_{1}) = 3.79\times 10^{-17}(\epsilon {\mathcal A}_{{\mathcal R}})^{-1/4}$.
Note that ${\mathcal A}_{{\mathcal R}}$ is the amplitude of the scalar power spectrum at the pivot wavenumber $0.002\,\mathrm{Mpc}^{-1}$
and $\epsilon$ is the slow-roll parameter.
The functional dependence of the magnetic spectral index upon $F_{B}$ and $F_{E}$ in the first quadrant of the $(F_{E},\, F_{B})$ plane is given by $n = (5 + 6 F_{B} - 4 F_{E})/(1 + F_{B} - F_{E})$ (the scale-invariant limit corresponds to $n\to 1$, see also <cit.>).
For illustration it is interesting to report the magnetic power spectrum at the present time and the BAU at the electroweak epoch in the nearly scale-invariant
limit for the fiducial choice of the parameters[Note that the hypercharge field projects onto the electromagnetic field
through the cosine of the Weinberg angle. Recall also that $\mathrm{nG} = 10^{-9} \mathrm{G}$.] mentioned above:
\begin{eqnarray}
\frac{P_{B}}{\mathrm{nG}^2} &=& 10^{-2.8} \cos^2{\theta_{W}} \biggl(\frac{h_{0}^2 \Omega_{R0}}{4.15\times 10^{-5}}\biggr) \biggl(\frac{{\mathcal A}_{{\mathcal R}}}{2.41\times 10^{-9}}\biggr) \biggl(\frac{\epsilon}{0.01}\biggr) \biggl(\frac{1 + 2 F_{B}}{5} \biggr)^4,
\label{pp}\\
|\langle \eta_{B} \rangle| &=& 1.35 \,\beta \,( \epsilon {\mathcal A}_{{\mathcal R}})^{3/4}\, | 1 - 2 F_{B}|^{4} \biggl[ 18.33 + 0.24 \ln{(\epsilon {\mathcal A}_{{\mathcal R}})}\biggr],
\label{ee}
\end{eqnarray}
implying that $|\langle \eta_{B} \rangle|$ can be as large as $10^{-7}$. As we shall see from Fig. <ref>
the values of $F_{E}$ and $F_{B}$ (or $\beta$) can be tuned to obtain a value ${\mathcal O}(10^{-10})$ we stress that larger values are
phenomenologically safer since, after the electroweak epoch, different physical processes can release entropy
and further reduce the generated BAU. The scale-invariant limit corresponds, in both plots of Fig. <ref>
to the two lower boundaries with equation $F_{E} = 5 F_{B}/3 + 4/3$.
[a]In the plot on the left we illustrate the common logarithm of the magnetic power spectrum in the $(F_{B}, F_{E})$ plane; in the plot on the right
we report the common logarithm of the BAU. In both plots the boundaries of the allowed regions coincide with $1.56 + 2.13 F_{B}$ (upper boundaries) and with
$5 F_{B}/3 + 4/3$ (lower boundaries). In both plots we took $\beta =1$ since we do not want to tune the pseudo-scalar coupling to a particularly
small (or a particularly large) value.
The physical region of Fig. <ref> is therefore located between the scale-invariant limit and the upper boundary determined by the magnetogenesis requirement, i.e. $5 F_{B}/3 + 4/3 \leq F_{E} \leq 1.46 + 1.91 F_{B}$. In this region we have that
$P_{B}(q_{*}, \tau_{g}) \geq 10^{-22} \mathrm{nG}^2$ where $\tau_{g}$ denotes the time of the protogalactic collapse and $q_{*}$
a typical comoving wavenumber ranging between $0.01\, \mathrm{Mpc}^{-1}$ and few $\mathrm{Mpc}^{-1}$ <cit.>. This constraint
can be relaxed down to $10^{-32} \, \mathrm{nG}^2$ (depending on the subsequent protrogalactic evolution) and therefore the allowed region may become
even wider, i.e. $5 F_{B}/3 + 4/3 \leq F_{E} \leq 1.56 + 2.13 F_{B}$.
The backreaction constraints stipulate that $\Omega_{B}(q,\tau)$ and $\Omega_{E}(q,\tau)$ integrated
over $d \ln{q}$ must be, at most, $10^{-3}$ and this requirement excludes the other regions in the $(F_{B}, F_{E})$ plane, as it can be
appreciated from Fig. <ref>. Note that in Fig. <ref> the total number of efolds $N_{t}$ coincides with $N_{\mathrm{max}}= 63.25 + 0.25 \ln{\epsilon}$ which is the maximal number of efolds which are today accessible to our observations.
In summary the hypermanetic gyrotropy can be amplified from vacuum fluctuations during a quasi-de Sitter stage of expansion when the hyperelectric and the hypermagnetic susceptibilities evolve at different rates. If the gauge couplings unify at the end of inflation there is only one region in the $(F_{E},\, F_{B})$ plane where all the physical requirements (i.e. the backreaction constraints, the magnetogenesis bounds and the naturalness of the initial conditions of the scenario) are jointly satisfied. Since the coupling of hypermagnetic fields to fermions is chiral the produced hypermagnetic gyrotropy may seed the BAU with typical values ranging between $10^{-7}$ and $10^{-10}$ at the electroweak epoch.
While the present proposal is certainly less conventional than the standard realizations of baryogenesis and leptogenesis, the scope of this investigation has just been to suggest and partially scrutinize an explicit model where the hypermagnetic gyrotropy seeds the BAU in the framework of a consistent magnetogenesis scenario.
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1511.00351
|
Cosmic Ray Induced EM Showers in the NO$\nu$A Detectors
Hongyue Duyang
For the NO$\nu$A Collaboration
The NO$\nu$A experiment is an electron neutrino appearance neutrino oscillation experiment at Fermilab. Electron neutrino events are identified by the electromagnetic (EM) showers induced by electrons in the final state of neutrino interactions. EM showers induced by cosmic muons or rock muons, are abundant in NO$\nu$A detectors. We use a Muon-Removal Technique to get pure EM shower samples from cosmic and rock muon data. Those samples can be used to characterize the EM signature and provide valuable checks of the MC simulation, reconstruction, PID algorithms, and calibration across the NO$\nu$A detectors.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
NO$\nu$A (NuMI Off-Axis $ \nu_{e} $ Appearance) is a long-baseline $\nu_{\mu}$ to $\nu_e$ neutrino oscillation experiment designed to determine the neutrino mass hierarchy and constrain CP-violation <cit.>. It uses neutrino beam generated by the Main Injector (NuMI) at Fermilab <cit.>, and measures the neutrino flavors by the near detector (ND, 0.3 kiloton) located at Fermilab, and the far detector (FD, 14 kiloton) at Ash River, MN.
The NO$\nu$A detectors are constructed from liquid scintillator contained inside extruded PVC modules.
The extrusions are assembled in alternating layers of vertical and horizontal planes, which are 0.15 radiation lengths in width, optimized for electron reconstruction.
NO$\nu$A finds signal $ \nu_{e} $ events by detecting electrons in the final state of charged current electron neutrino interaction ($ \nu_{e} $-CC) <cit.>. This requires correct modeling, reconstruction and particle identification (PID) algorithms for the electromagnetic (EM) showers.
Calibration effects such as attenuation and alignment should also be under control.
Cosmic ray muons are abundant (148 kHz) in the NO$\nu$A far detector which is on the surface. In the near detector, rock muons are abundant.
They induce EM showers by three different means: energetic muons undergoing bremsstrahlung radiation (Brem), muons decaying into electrons in flight (DiF), and muons stoping in the detectors and decaying into Michel electrons. Michel electrons' energy is small compared to $ \nu_{e} $ events (0.5GeV$ \sim$ 4GeV), and have been used for calibration. Brem and DiF, on the other hand, provide abundant EM showers at few-GeV energy region. This sample makes possible s data-driven method to benchmark EM shower modeling, PID, as well as the detector calibration.
For this purpose, a cosmic muon-removal (MR) algorithm is developed to identify the Brem and DiF showers, remove the muons, and save the remaining EM showers at raw digit level. Figure <ref> show examples in the FD for EM shower events before and after MR.
The shower digits can then be put into standard $\nu_e$ reconstruction and PID algorithms. Data and MC comparison is performed with reconstructed shower variables and PID outputs to validate EM shower modeling and PID. Calibration effects can be checked by comparing PID efficiencies as function of vertex position.
Top: event display cosmic muon candidate with Electromagnetic (EM) Bremsstrahlung (Brem) Shower from NO$\nu$A simulation. Bottom: event display of hits of the EM shower after the removal of hits associated with the muon track from NO$\nu$A simulation. What left are the hits of Brem Showers.
§ PROCEDURE
We take cosmic data and MC simulation in NO$\nu$A FD. Both are subject to standard cosmic reconstructions. The following procedure breaks into 4 steps.
1. Track Selection: Apply selection cuts to search for cosmic track candidates in the far detector. Basically We want the tracks to be through-going muons which are not too steep compared to the beam direction (cos$\theta>0.5$) and long enough to have a good chance of generating Brem showers (number of track planes $>$ 30). Then to compare with beam $\nu_e$ events we want the shower to have energy close to the $\nu_e$ energy region ($0.5\sim5$ GeV).
2. Shower Finding: Taking candidate tracks as input, an EM shower finding algorithm is used to identify the muons with possible EM shower dE/dx information. Muons deposit energy as a minimum ionization particle (MIP) in the detector cells. The additional EM shower hits can deposit much more energy in a small detector region and overlaps with muon energy. It is then possible to take the energy deposition of reconstructed cosmic tracks and look for the region where the energy is significantly greater than a MIP.
More specifically,
If we find 5 consecutive planes with energy greater than 2 MIP, call it the shower start point;
If we find 5 consecutive plane with energy in the range of 0.5 MIP to 1.5 MIP, call it the shower end point;
If we find both the shower start and end point, then the shower is identified as Brem; If we find only the shower start point, and reach the end of track without an end point, the shower is identified as an electron from DiF.
Showers with energy deposition passing the energy cut are saved as raw digit objects.
3. Muon Removal (MR): A Muon-Remove algorithm for EM showers is modified from the Muon-Removal algorithm for charged current events.<cit.>.
It first looks at the slice where a cosmic EM shower is found.
In the case of DiF muon, with a shower region defined by the shower-finding algorithm, it just removes all hits outside that region and what left will be pure electron hits.
In the case of bremsstrahlung showers one additional problem is that we have a muon track inside the EM shower region. Therefore the muon-removal algorithm should remove hits that belong to muon track corresponding to the energy of a MIP in the shower region.
All other hits in those slices where no shower is found are removed. Those hits not associated with any slices are also removed.
§ EM SHOWER RECONSTRUCTION AND IDENTIFICATION
The saved shower raw digits with muon removed are put into the standard $\nu_e$ reconstruction.
The reconstruction starts with clustering together hits from a single neutrino interaction in to a slice clustering hits by space-time coincidence[ref].
A modified Hough transform is used to identify prominent straight-line features in a slice. The slice vertex is defined by tuning the lines in an iterative procedure and finding the converge point.
Prongs are then reconstructed as groups of hits based on their distances to the lines associated with the reconstructed vertex [ref].
Reconstructed energy and angle are compared between data and MC, and to $\nu_e$ MC events (figure <ref>).
Although different in shape, cosmic EM showers cover the same range as $\nu_e$ events.
Other reconstructed variables such as Shower length, shower width, number of cells, number of planes are also compared between data and MC (figures <ref>). Good data and MC agreement is seen.
The reconstructed prongs are then subject to the particle identification (PID) algorithms. NO$\nu$A has two PID algorithms to distinguish $\nu_e$ signal from background, dominated by $\nu_{\mu}$-NC. The primary PID, Likelihood Based $\nu_e$ Identifier (LID), uses the dE/dx information of a particle to compute the likelihoods that the candidate particle is an electron <cit.><cit.>. The alternative algorithm, Library Event Matching (LEM), identifies event types by comparing an unknown trial event to a library of known event from MC <cit.>. The distribution shows consistent result between data and MC for both PIDs. While LID identify most of the cosmic EM showers as signal-like (LID$>$0.7), LEM shows most of showers in its background region (LEM$<$0.6). This is due to the fact that LEM is more sensitive to the angle of the showers with respect to the beam direction. A re-weight of cosmic EM showers according to $\nu_e$ events in energy and angle is able to fix this problem.
Efficiencies are calculated as number of showers passing PID cuts (LID$>$0.7) divided by all showers selected as function o vertex position in x and y to check the calibration effects (Figure <ref>). Overall we see a relatively flat efficiency distribution across the detector, with good data and MC agreement. The extent to which the efficiencies from data and MC do not agree motivates a systematic error on the predicted electron neutrino signal efficiency for the first electron neutrino appearance analysis.
Left: reconstructed shower energy after the cosmic Muon Removal(MR) in the far detector.
Right: cosine of the reconstructed shower angle with respect to the beam direction. Distribution of $\nu_e$ events are drawn on the same plots. Cosmic EM showers show good data/MC agreement, and cover the range of $\nu_e$ events.
The reconstructed shower variables of cosmic EM showers in data and MC, including the reconstructed shower length (top left), width (top right), number of planes (bottom left), and number of cell hits (bottom right).
LID (left) and LEM (right) distribuion of cosmic EM showers in data and MC.
Selection efficiency as a function of reconstructed vertex position x (left) and y (right) with LID $>$ 0.7 cut. The bottom plots show the ratio between data and MC efficiencies. The agreement between data and MC is well with in 5 %.
§ SUMMARY
We use Cosmic Muon-Removal algorithm for finding and isolating EM showers from data and MC in the NO$\nu$A far detector. The shower digits are put into standard NO$\nu$A $\nu_e$ reconstruction and PID algorithms. A comparison using first analysis data and MC dataset shows consistent distributions, validating the EM shower modeling and reconstruction.
The PID efficiencies as functions of vertex position are calculated with good data/MC agreement. It shows the calibration effects are well controlled.
D. S. Ayres et al. (NO$\nu$A), FERMILAB-DESIGN-2007-01 (2007); R. B. Patterson,
for NO$\nu$A, Nucl. Phys. Proc. Suppl. 235-236, 151 (2013)
NuMI Technical Design Handbook, NuMI Technical Design Handbook,
http://www-numi.fnal.gov/numwork/tdh/tdh index.html
P. Adamson, et al. [The NO$\nu$A Collaboration], “First measurement of electronneutrino
appearance in NO$\nu$A”, FERMILAB-PUB-15-262-ND, to be submitted to Phys. Rev. Lett.
K. Sachdev, “A Data-Driven Method of Background Prediction at NO$\nu$A” [arXiv:1501.00968].
J. Bian, “First Results of $\nu_e$ Appearance Analysis and Electron Neutrino Identification at NO$\nu$A”, [arXiv:1510.05708]
E. D. Niner, “Observation of Electron Neutrino Appearance in the NuMI Beam
with the NO$\nu$A Experiment”, Ph.D. Thesis, FERMILAB-THESIS-2015-16.
C. Backhouse and R. B. Patterson, “Library Event Matching event classification
algorithm for electron neutrino interactions in the NO$\nu$A detectors”, Nucl.
Instrum. Meth. A 778, 31 (2015) [arXiv:1501.00968 [physics.ins-det]].
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1511.00066
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University of North Alabama,
Florence, Alabama 35632, USA
Racah Institute of Physics, Hebrew University, 91904 Jerusalem, Israel
A. F. Ioffe Physical-Technical Institute, 194021 St. Petersburg, Russia
The present paper explores possible features of electron elastic scattering off endohedral fullerenes $A$@C$_{60}$. It focuses on how dynamical polarization of the encapsulated atom $A$ by an incident electron might
alter scattering off $A$@C$_{60}$ compared
to the static-atom-$A$ case, as well as how the C$_{60}$ confinement modifies the impact of atomic polarization on electron scattering compared to the free-atom case. The aim is to provide researchers with a “relative frame of reference” for understanding which part of the scattering processes could be due to electron scattering off the encapsulated atom and which due to scattering off the C$_{60}$ cage.
To meet the goal, the C$_{60}$ cage is modeled by an attractive spherical potential of a certain inner radius, thickness, and depth which is a model used frequently in a great variety of fullerene studies to date. Then, the Dyson equation for the self-energy part of the Green's function
of an incident electron moving in the combined field of an encapsulated atom $A$ and C$_{60}$ is solved in order to account for the impact of dynamical polarization of the encaged atom upon $e + A@{\rm C_{60}}$ scattering. The Ba@C$_{60}$ endohedral is chosen as the case study.
The impact is found to be significant, and its utterly different role compared to that in $e + Ba$ scattering is unraveled.
31.15.ap, 31.15.V-, 34.80.Dp, 34.80.Bm
§ INTRODUCTION
Electron elastic scattering off quantum targets is an important fundamental phenomenon of
nature. It has significance to both the basic and applied sciences
and technologies. Yet, to date, the knowledge on the process of
electron collision with such important quantum
targets as endohedral fullerenes $A$@C$_{60}$ is far from complete.
Endohedral fullerenes are
nanostructure formations where an atom $A$ is encapsulated
inside the hollow interior of a C$_{60}$ molecule. The authors are aware of only a handful of works on this
subject. These are the theoretical studies of fast charged-particle ionization of $A$@C$_{60}$ <cit.> and
low-energy electron scattering off $A$@C$_{60}$ <cit.>
calculated in the framework of two different model approximations. Namely, in Refs. <cit.> a static Hartree-Fock (HF) approximation was employed. There, both the atom $A$ and C$_{60}$ were considered as
nonpolarizable targets and the C$_{60}$ cage was modeled by an attractive spherical potential of a certain inner radius, thickness, and depth. In Ref. <cit.>,
the authors kept the encaged atom “frozen”, modeled the C$_{60}$ cage by the potential similar to that used in Ref. <cit.>, but accounted for polarization of the C$_{60}$ cage by incident electrons. The latter was evaluated
in a simplified manner by adding a static polarization potential $-\alpha/r^{4}$ ($\alpha$ being the static polarizability of C$_{60}$) to the model C$_{60}$-potential. Note, a meager amount of research on
$e +A@{\rm C_{60}}$ collision is in contrast to the study of photon-$A$@C$_{60}$ collision, different aspect of which has been intensely
scrutinized in a great variety of theoretical works to date (see, e.g., Refs. <cit.> and references therein),
including experimental studies <cit.> (and references therein). Such disbalance in favor of the number and quality of
studies of $A$@C$_{60}$ photoionization versus research on electron-$A$@C$_{60}$ collision is not accidental. Electronic collision with
a multielectron target is a more complicated multifaceted process compared to photonic collision with the same target.
Therefore, the comprehensive description
of electron scattering by a multielectron target is too challenging
for theorists even with regard to a free atom, not to mention $A$@C$_{60}$ targets.
The present study does not aim at solving the difficult problem of electron-$A$@C$_{60}$ scattering in its entirety. Instead, it focuses on the contribution of electron scattering only off the encapsulated atom $A$ to the entire collision process. The significance of the present study is that it provides an important frame of reference for (future) understanding of which part of electron-$A$@C$_{60}$ scattering could be due to scattering off the encapsulated atom $A$ (unraveled in the present work) and which is due to other “facets” of the entire $A$@C$_{60}$ system. Research results, thus, provide a relative rather than absolute knowledge. To meet the goal, the C$_{60}$ cage is modeled, as in Refs. <cit.>, by a spherical potential of a certain inner radius, thickness, and depth. Polarization of the C$_{60}$ cage by incident electrons will, thus, be ignored (being not the subject of the focused study). This is in contrast to accounting for polarization of the encapsulated atom in the present study. The neglect by polarizability of C$_{60}$ by incident electrons should not be over-dramatized. The effect of polarizability is electron-energy-dependent and may either enhance or decrease scattering cross section, at certain electron energies. Therefore, when scattering off C$_{60}$ is dramatically decreased, or where scattering of the encapsulated atom $A$ is dramatically increased, a relative role of scattering off the atom $A$ will (might) be significant. Furthermore, $A$@C$_{60}$ has the hollow interior which is not totally occupied by the atom (i.e., not totally filled in with charge density). As such, it acts as a resonator relative to incident electronic waves. Therefore, at wave frequencies, matching resonance frequencies of the $A$@C$_{60}$-resonator, there will be a significant incident-electron-density build-up in the hollow interior of $A$@C$_{60}$. Obviously, this build-up of electron density will be positioned near the encapsulated atom $A$. Therefore, the effect of atomic polarization on electron scattering might become comparable or even more important than the C$_{60}$ polarization effect, at resonance frequencies. As such, the impact of atomic polarizability on $e +A@{\rm C_{60}}$ scattering it cannot be dropped out of the consideration at all. It is, therefore, indisputably needed (and interesting, and does make sense) to study how polarization of the encapsulated atom can affect the scattering process even in the neglect by polarization of the fullerene cage by incident electrons. To account for atomic polarization under confinement, the authors employ the Dyson formalism
for the self-energy part of the Green's function of a scattered electron <cit.>, adapt it to the case of the electron motion in a combined field of the encapsulated multielectron atom $A$ and the model static C$_{60}$ cage, solve
the generalized Dyson equation, and, thus, calculate the electron elastic-scattering phase shifts and corresponding cross sections for the $e + A@{\rm C_{60}}$ scattering reaction. The study is restricted to electron elastic scattering at low electron energies $\epsilon \alt 3$ eV where the most interesting effects occur.
Finally, the present study also has a significance which is independent of its direct applicability to electron-$A$@C$_{60}$ scattering. This is because it falls into a mainstream of intense modern studies where numerous aspects of the structure and spectra of atoms under various kinds of confinements (impenetrable spherical, spheroidal, dihedral, Debye-like potentials, etc.) are being attacked from many different angles by
research teams world-wide (see, e.g., numerous review articles in Refs. <cit.>). Such studies are interesting from the view point of basic science.
Results of the present study add new knowledge to the collection of atomic properties under confinement as well, particularly revealing the impact of atomic polarization under confinement on electron-atom scattering.
Atomic units are used throughout the paper unless specified otherwise.
§ THEORY
§.§ $e + A@{\rm C}_{60}$ scattering in the framework of static C$_{60}$
§.§.§ Model static HF approximation
In the present work, the C$_{60}$ cage is modeled by a spherical potential $U_{\rm c}(r)$ defined as follows:
\begin{eqnarray}
U_{\rm c}(r)=\left\{\matrix {
-U_{0}, & \mbox{if $r_{0} \le r \le r_{0}+\Delta$} \nonumber \\
0 & \mbox{otherwise.} } \right.
\label{SWP}
\end{eqnarray}
Here, $r_{0}$, $\Delta$, and $U_{0}$ are the inner radius, thickness, and depth of the potential well, respectively.
Next, the wavefunctions $\psi_{n \ell m_{\ell} m_{s}}({\bm r}, \sigma)=r^{-1}P_{nl}(r)Y_{l m_{\ell}}(\theta, \phi) \chi_{m_{s}}(\sigma)$
and binding energies $\epsilon_{n l}$ of atomic electrons
($n$, $\ell$, $m_{\ell}$ and $m_{s}$ is the standard set of quantum numbers of an electron in a central field, $\sigma$ is the electron spin variable) are the solutions of a system of the “endohedral”
HF equations:
\begin{eqnarray}
&&\left[ -\frac{\Delta}{2} - \frac{Z}{r} +U_{\rm c}(r) \right]\psi_{i}
({\bm x}) + \sum_{j=1}^{Z} \int{\frac{\psi^{*}_{j}({\bm x'})}{|{\bm
x}-{\bm x'}|}} \nonumber \\
&& \times[\psi_{j}({\bm x'})\psi_{i}({\bm x})
- \psi_{i}({\bm x'})\psi_{j}({\bm x})]d {\bm x'} =
\epsilon_{i}\psi_{i}({\bm x}).
\label{eqHF}
\end{eqnarray}
Here, $Z$ is the nuclear charge of the atom, ${\bm x} \equiv ({\bm r}, \sigma)$, and the integration over ${\bm x}$ implies both the integration over ${\bm r}$ and summation over
$\sigma$. Eq. (<ref>) differs from the ordinary HF equation for a free atom by the presence of the $U_{\rm c}(r)$ potential in there. This equation is first solved in order to calculate the electronic ground-state wavefunctions of the encapsulated atom. Once the electronic ground-state wavefunctions are determined, they are plugged back into
Eq. (<ref>) in place of $\psi_{j}({\bm x'})$ and $\psi_{j}({\bm x})$ in order to calculate the electronic wavefunctions of scattering-states $\psi_{i}({\bm x})$ and their radial parts $P_{\epsilon_{i}\ell_{i}}(r)$.
Corresponding electron elastic-scattering phase shifts $\delta_{\ell}(k)$
are then determined by
referring to $P_{k\ell}(r)$ at large $r$:
\begin{eqnarray}
P_{k\ell}(r) \rightarrow \sqrt{\frac{2}{\pi}}\sin\left(k r -\frac{\pi\ell}{2}+\delta_{\ell}(k)\right).
\label{P(r)}
\end{eqnarray}
Here, $k$ and $k'$ are the wavenumbers of the incident and scattered electrons, respectively, and
$P_{k\ell}(r)$ is normalized to $\delta(k-k')$.
The total electron elastic-scattering cross section $\sigma_{\rm el}(\epsilon)$ is then found in accordance with the standard
formula for electron scattering by a central-potential field:
\begin{eqnarray}
\sigma_{\rm el}(k)= \frac{4\pi}{k^2}\sum^{\infty}_{\ell=0}(2\ell+1)\sin^{2}\delta_{\ell}(k).
\label{sigma}
\end{eqnarray}
This approach solves the problem of $e + $A$@{\rm C_{60}}$ in a static approximation, i.e.,
without accounting for polarization of the $A$@C$_{60}$ system by incident electrons.
In the literature, some inconsistency is present in
choosing the magnitudes of $\Delta$, $U_{0}$ and $r_{0}$ of the model potential $U_{\rm c}(r)$ for C$_{60}$:
$r_{0}=5.8$, $\Delta=1.9$, $U_{0}=0.302$ <cit.> (and references therein),
or $r_{0}= 6.01$, $\Delta=1.25$ and $U_{0}=0.422$ <cit.>, or $\Delta = 2.9102$, $r_{0} = 5.262$, $U_{0} = 0.2599$
<cit.>. In order to evaluate which of these sets of parameters is best suited for studying $e + A@{\rm C_{60}}$ scattering,
we performed the corresponding calculations of $e + {\rm C_{60}}$ scattering. Calculated results are plotted in Fig. <ref> against
calculated data obtained in the framework of the sophisticated ab initio static-exchange molecular-Hartree-Fock (MHF) approximation <cit.>.
(Color online) $e + {\rm C_{60}}$ elastic-scattering cross section (in units of $a_{0}^{2}$, $a_{0}$ being the first Bohr radius of the hydrogen atom)
calculated both with the use of different values of the parameters
$r_{0}$, $\Delta$, and $U_{0}$ of the
spherical potential $U_{\rm c}(r)$ (present work) and in the framework of ab initio MHF <cit.>, as marked.
One can see that it is the set of parameters proposed in Ref. <cit.> which leads <cit.> to the overall qualitative and
semi-quantitative agreement between some of the most prominent features of
$e + {\rm C_{60}}$ elastic scattering predicted by the model spherical-potential approximation and ab initio MHF.
Correspondingly, in the present work, as in Ref. <cit.>, the $U_{\rm c}(r)$ potential is defined by
$\Delta = 2.9102$, $r_{0} = 5.262$, and $U_{0} = 0.2599$.
§.§.§ Multielectron approximation: a polarizable atom $A$
In order to account for the impact of polarization
of an encapsulated atom $A$ by incident electrons on $e + A@{\rm C}_{60}$
elastic scattering, let us utilize the concept of the self-energy part of the
Green's function of an incident electron <cit.>.
In the simplest second-order perturbation theory in the Coulomb
interelectron interaction $V$ between the incident and atomic electrons,
the irreducible self-energy part of the
Green's function $\Sigma(\epsilon)$ of the incident electron
is depicted with the help of Feynman diagrams in Fig. <ref>.
The irreducible self-energy part $\Sigma(\epsilon)$
of the Green function of a
scattering electron in the second-order perturbation theory in the Coulomb interaction, referred to as the SHIFT approximation (see text). Here,
a line with a right arrow denotes an electron, whether a scattered electron
(states $|\epsilon_{\ell}\rangle$ and $|\epsilon_{\ell^{\prime}}^{\prime}\rangle$) or an atomic excited electron (a state $|m \rangle$), a line with a
left arrow denotes a vacancy (hole) in the atom (states $\langle j|$ and $\langle i|$), a
wavy line denotes the Coulomb interelectron interaction $V$.
The diagrams of Fig. <ref> illustrate how a scattered electron “$\epsilon _{\ell }$” perturbs (read: polarizes) a $j$-subshell
of the atom by causing $j$ $\rightarrow $ $m$ excitations from the subshell and then gets coupled with
these excited states itself via both the Coulomb direct [diagrams (a) and (b)] and exchange [diagrams (c) and (d)] interactions.
Numerical calculation of electron elastic-scattering phase shifts in the framework of this approximation is addressed by the computer code from Ref. <cit.> labeled as the “SHIFT” code.
Correspondingly, the authors refer to this approximation as the “SHIFT” approximation everywhere in the present paper.
A fuller account of electron correlation (read: polarization) in $e + A@{\rm C}_{60}$ elastic scattering is determined by the reducible
$\tilde{\Sigma}(\epsilon)$ part of the self-energy part of
the electron's Green function <cit.>. The matrix element of the latter are represented diagrammatically in Fig. <ref>.
The matrix element of the reducible self-energy part $\tilde{\Sigma}(\epsilon)$
of the Green's function of a scattering electron, where $\Sigma$ is the irreducible self-energy part of the Green's function depicted in Fig. <ref>. This approximation is
referred to as the SCAT approximation (see text). Note, when calculating
$\langle \epsilon_{\ell}|\tilde{\Sigma}|\epsilon_{\ell}\rangle$ analytically, the summation over unoccupied discreet and integration over continuum excited states (marked as $\epsilon_{\ell ''}''$)
along with the summation over the occupied states in the atom marked as $E_{n\ell '}$ must be performed.
The above diagrammatic equation for $\tilde{\Sigma}$ can be written in an operator form as follows:
\begin{eqnarray}
\hat{\tilde{\Sigma}}=\hat{\Sigma}-\hat{\Sigma}\hat{G}^{(0)}\hat{\tilde{\Sigma}}
\label{EqGreen}.
\end{eqnarray}
Here, $\hat{\Sigma}$
is the operator of the irreducible
self-energy part of the Green's function calculated in the
framework of SHIFT (Fig. <ref>), $\hat{G}^{(0)}=(\hat{H}^{(0)}-\epsilon)^{-1}$ is the HF operator of the
electron Green's function, and $\hat{H}^{(0)}$ is the HF
Hamiltonian operator of the $electron + A@C_{60}$ system. Clearly, the equation for the matrix elements of $\tilde{\Sigma}$
accounts for an infinite series of diagrams by coupling the diagrams of Fig. <ref> in various combinations.
Numerical calculation of electron elastic-scattering phase shifts in the framework of this approximation is addressed by the computer code from Ref. <cit.> labeled as the “SCAT” code.
Correspondingly, the authors refer to this approximation as the “SCAT” approximation everywhere in the present paper.
SCAT works well for the case of electron scattering off free atoms <cit.>.
This gives us confidence in that SCAT is a sufficient approximation for pinpointing the impact of correlation/polarization
on $e + A@{\rm C_{60}}$ scattering as well.
In the framework of SHIFT or SCAT, the electron elastic-scattering
phase shifts $\zeta_{\ell}$ are determined as follows <cit.>:
\begin{eqnarray}
\zeta_{\ell }=\delta_{\ell }^{\rm HF}+\Delta\delta_{\ell }.
\end{eqnarray}
Here, $\Delta\delta_{\ell}$ is the correlation/polarization
correction term to the calculated HF phase shift $\delta_{\ell}^{\rm HF}$:
\begin{eqnarray}
\Delta\delta_{\ell}=\tan^{-1}\left(-\pi
\left\langle\epsilon\ell|\tilde{\Sigma}|\epsilon\ell\right\rangle \right).
\end{eqnarray}
The mathematical expression for
is cumbersome. The
interested reader is referred to <cit.> for details. The matrix element
$\left\langle\epsilon\ell|\tilde{\Sigma}|\epsilon\ell\right\rangle$ becomes
complex for electron energies exceeding the
ionization potential of the atom-target, and so does the correlation term
$\Delta\delta_{\ell}$ and, thus, the phase shift
$\zeta_{\ell}$ as well. Correspondingly,
\begin{eqnarray}
\zeta_{\ell }=\delta_{\ell}+i\mu_{\ell},
\end{eqnarray}
\begin{eqnarray}
\delta_{\ell}=\delta_{\ell }^{\rm HF}+
\rm Re\Delta\delta_{\ell},\quad \mu_{\ell} =
Im\Delta\delta_{\ell }.
\end{eqnarray}
The total electron elastic-scattering cross section $\sigma_{\rm el}$ is then given by the expression
\begin{eqnarray}
\sigma_{\rm el}=\sum_{\ell =0}^{\infty }\sigma_{\ell},
\end{eqnarray}
where $\sigma_{\ell}$ is the electron elastic-scattering partial cross section:
\begin{eqnarray}
\sigma_{\ell}=\frac{2\pi }{k^2}(2\ell+1)\frac{\cosh{2\mu_{\ell}}-
\cos{2\delta_{\ell}}}{{\rm e}^{2\mu_{\ell}}}.
\label{EqSgmRPAE}
\end{eqnarray}
§ RESULTS AND DISCUSSION: $E + {\RM BA@C_{60}}$ SCATTERING
§.§.§ Preliminary notes
In the aims of the present paper, the authors focus on electron scattering off Ba@C$_{60}$, as the case study.
This is because the Ba atom is a highly polarizable atom. It is expected to retain its high polarizability under the
C$_{60}$ confinement as well. This provides one with a better opportunity to learn how atomic polarization can alter electron scattering off $A$@C$_{60}$ compared to scattering off the static target.
Furthermore, the study focuses on the electron energy region of up to $\epsilon \approx 3$ eV.
First, at such energies,
the electron wavelength $\lambda > 6$ $\AA$ exceeds greatly
the bond length $D \approx 1.44$ $\AA$ between the carbon atoms in
C$_{60}$. Correspondingly, the incoming electrons will “see” the
C$_{60}$ cage as a homogeneous rather than “granular” cage. This makes it
appropriate to model the C$_{60}$ cage by a non-granular, i.e., “smooth” potential. Moreover, as was shown in Refs. <cit.>, a low-energy electron motion in the field of C$_{60}$ is insensitive to details of the
“smooth” potential, i.e., to whether the potential is the potential with round borders and unparallel walls or simply a square-well potential, as long as $\lambda \gg D$. This additionally justifies the use of the square-well potential, Eq. (<ref>), in the present study.
Second, correlation/polarization effects, which are of the primary concern of this paper, are expected, as usually, to be most strong primarily at low-energy
electron collisions with multielectron targets.
Third, at these low energies, earlier, there were predicted spectacular
confinement-induced resonances in $e + A@{\rm C_{60}}$ scattering <cit.>, similar to those predicted in $e + {\rm C_{60}}$ scattering <cit.>.
The presence of the confining C$_{60}$ cage in this case, as well as in the case of scattering off empty C$_{60}$, results in the emergence of positive quasi-discrete states in the field of $A$@C$_{60}$ or C$_{60}$. When
quasi-discrete states are present, then, in accordance with the known fact, “... resonance scattering occurs because a positive level with $\ell \neq 0$, though not a true discrete level, is quasi-discrete: owing to the presence of the centrifugal potential barrier, the probability that a particle of low energy will escape from this state to infinity is small, so that the lifetime of the state is long” <cit.>.
It is interesting to explore how the predicted resonances
in $e + A@{\rm C_{60}}$ scattering can be altered by the effects of polarization of the encapsulated atom $A$ by incident electrons.
Next, the calculations of electron scattering off Ba@${\rm C_{60}}$ and free Ba, performed in the present work in the framework of both SHIFT and SCAT, accounted for
the monopole, dipole, quadrupole, and octupole perturbations of the valence $6s^{2}$ and $5p^{6}$ subshells of free and encaged Ba by incident electrons.
Finally, in view of low values of the electron energies, only the $s$-, $p$-, $d$-, $f$-, and $g$-partial electronic waves have been accounted for in the calculations. The contribution of other partial electronic
waves is negligible, at given energies.
§.§.§ Results and discussion
Corresponding calculated HF, SHIFT, and SCAT data for the real parts of phase shifts $\delta_{\ell}(\epsilon)$, partial $\sigma_{\ell}(\epsilon)$ and total $\sigma_{\rm el}(\epsilon)$ cross sections for
electron elastic scattering off Ba@C$_{60}$ are displayed
in Fig. <ref> (the imaginary parts $\mu_{\ell}$ of the phase shifts, when present, are small at the chosen energies and present little interest for discussion).
(Color online) Main panels: Calculated partial $\sigma_{\ell}(\epsilon)$ and total $\sigma_{\rm el}(\epsilon)$ cross sections (in units of $a_{0}^2$)
for electron elastic scattering off Ba@C$_{60}$, obtained in the frameworks of the model static HF (dashed line), multielectron SHIFT (dash-dotted line) and SCAT (solid line) approximations.
Insets: Real parts $\delta_{\ell}(\epsilon)$ of the phase shifts $\zeta_{\ell}(\epsilon)$ (in units of radian) calculated in HF (dashed line), multielectron SHIFT (dash-dotted line) and SCAT (solid line) approximations.
As an important finding, one learns from Fig. <ref> that the C$_{60}$ confinement does not at all “extinguish” the possibility for the encapsulated Ba atom to be strongly polarized by incident electrons.
On the contrary, the polarization impact is found to affect dramatically both the electron scattering phase shifts and partial $\sigma_{\ell}$ as well as total $\sigma_{\rm el}$ cross sections.
All this is obvious from the comparison of calculated results obtained in the framework of HF, on the one hand, and SHIFT and SCAT, on the other hand, approximations. As another important result, one finds that accounting for only the second-order (with respect to the Coulomb interaction) correlation effects, as in SHIFT, is important but far insufficient for the calculation of low-energy electron scattering off Ba@C$_{60}$. Indeed, the correlation impact
beyond the second-order approximation, i.e., accounted for in the SCAT approximation, is utterly significant – the lower the energy, the stronger the impact.
For the next, it is both interesting and necessary to bring to the attention of the reader that the empty static C$_{60}$ cage can only support the $s$-, $p$-, and $d$-bound states <cit.>. In contrast, the “stuffed” C$_{60}$, i.e., Ba@C$_{60}$, was found to lose a $s$-bound state, but acquire, instead, a $f$-bound state,
in the static HF approximation <cit.>. Now, it follows from the present study that by “unfreezing” the encapsulated Ba atom, i.e., by making it polarizable, the lost $s$-bound state is returned to the Ba@C$_{60}$ system, and the latter keeps the former $p$-, $d$- and $f$-bound states. Moreover, the Ba@C$_{60}$ system is found to start supporting a second $p$-bound state as well. All this becomes clear by noting the calculated SCAT values of the $s$-, $p$-, $d$-, and $f$-phase shifts at $\epsilon \rightarrow 0$: $\delta_{s}^{\rm SCAT} \rightarrow 7\pi$, $\delta_{p}^{\rm SCAT} \rightarrow 6\pi$, $\delta_{d}^{\rm SCAT} \rightarrow 3\pi$, and $\delta_{f}^{\rm SCAT} \rightarrow \pi$. The zero-energy values of these phase shifts satisfy the generalized version of Levinson theorem for scattering in a potential field <cit.> which (the generalized theorem) could be derived by the direct numerical calculation and written as follows:
\begin{eqnarray}
\left. \delta_{\ell}(\epsilon)\right\vert_{\epsilon \rightarrow 0}\rightarrow (N_{n_{\ell} }+ q_{\ell })\pi.
\label{Levinson}
\end{eqnarray}
Here, $N_{n_{\ell}}$ is the number of closed subshells with given $\ell$ in
the ground-state configuration of an atom, whereas $q_{\ell }$ is the
number of additional empty bound states with the same $\ell$ in the field of $A$@C$_{60}$. Given that, for the ground-state of the encapsulated Ba atom,
$N_{n_{s}}=6$, $N_{n_{p}} = 4$, $N_{n_{d}} = 2$, and $N_{n_{f}}= 0$, one finds that, in accordance with the SCAT values of the phase shifts, $q_{n_{s}}=1$, $q_{n_{p}} = 2$, $q_{n_{d}} = 1$, and $q_{n_{f}}= 1$.
This translates into one $s$-, two $p$-, one $d$-, and one $f$-bound states supported (one at a time) by Ba@C$_{60}$. Note that calculated SHIFT phase shifts, in contrast to SCAT data, point to the existence of
neither $s$- nor second $p$-bound state in the field of Ba@C$_{60}$. This discrepancy between the calculated SHIFT and SCAT data speaks, in general, to the importance of a fuller account of electron correlation in the
calculation of $e + $A$@{\rm C_{60}}$ scattering.
The discovered in the framework of SCAT emergence of a $s$-bound state and a second $p$-bound state in the field of Ba@C$_{60}$ has profound consequences for both the corresponding partial and total electron-scattering cross sections. Namely, because the phase shift $\delta_{s}^{\rm SCAT}$, on the way to its value of $7\pi$ at $\epsilon=0$, passes through the value of modulo $\pi/2$ (at $\epsilon \approx 0.23$ eV),
$\sigma_{\rm s}^{\rm SCAT}$ becomes large, at lower energies, in contrast to the predictions obtained with the help of inferior HF and SHIFT. Similarly, the increase of $\delta_{\rm p}^{\rm SCAT}$ to $6\pi$ at $\epsilon =0$ results in large
$\sigma_{\rm p}^{\rm SCAT}$ as well, at low energies, again, in contrast to the predictions by HF and SHIFT.
Talking about the $d$- and $g$-partial cross sections, one can see that each of them is dominated by the strong resonance. Its emergence clearly follows from the behavior of the $d$- and $g$-phase shifts. Indeed, the phase shifts first tend to increases towards modulo $\pi$ with decreasing energy, but, because the field turns out to be not strong enough, before that value is reached, they
sharply decrease, passing through the value of modulo $\pi/2$. This is a typical behavior of phase shifts upon electron scattering on quasibound states <cit.>.
Next, note how the resonances in the $d$- and $g$-partial cross sections become higher, narrower, and shift towards lower energy as more correlation is accounted for in the calculation (compare
calculated HF versus SHIFT versus SCAT results). We thus find that the inclusion of more correlation into the calculation of $e + {\rm Ba@C_{60}}$ scattering increases the strength of the
field of the Ba@C$_{60}$-scatterer, since the above noted changes in the resonances are typical for electron scattering in a field of increasing strength <cit.>.
Next, note that no $f$-resonance emerges in $e + {\rm Ba@C_{60}}$ scattering calculated in either of the three approximations utilized in this paper. This is in contrast to electron elastic-scattering off empty static
C$_{60}$. There, the sharp $f$-resonance was predicted to emerge at low energies <cit.> (this is the extremely narrow resonance near zero plotted in Fig. <ref>). The reason for the absence of the $f$-resonance in $e + {\rm Ba@C_{60}}$ scattering is interesting. It is directly associated with that a noticeable part of the valence electronic charge-density of encapsulated Ba is drawn into the C$_{60}$ shell <cit.>.
Therefore, the field inside the “stuffed” C$_{60}$ becomes more attractive than in empty C$_{60}$. It turns the $f$-state into a bound state, thereby eliminating the emergence of a $f$-resonance
in $e + {\rm Ba@C_{60}}$ scattering. Now, as it has been shown in the paragraph above, the inclusion of correlation into the calculation of $e + {\rm Ba@C_{60}}$ scattering increases the field of Ba@C$_{60}$. Therefore, the $f$-state remains bound. This is why the $f$-resonance does not take place in $e + {\rm Ba@C_{60}}$ scattering even if polarization of the encapsulated Ba atom by incident electrons is accounted for in the calculation. Our general prediction is that there will be no $f$-resonances on quasibound states in electron scattering off any $A$@C$_{60}$ system in case where there is a noticeable transfer of electronic charge-density from the encapsulated atom to the C$_{60}$ shell.
Furthermore, it is interesting to compare how much differently polarization of the Ba atom by incident electrons affects electron elastic scattering off free Ba versus
Ba@C$_{60}$. The corresponding calculated HF and SCAT total electron elastic-scattering cross sections are depicted in Fig. <ref>.
(Color online) Calculated total electron elastic-scattering cross sections $\sigma_{\rm el}(\epsilon)$ (in units of $a_{0}^2$)
for electron scattering off Ba@C$_{60}$, obtained in the frameworks of the model static HF (dashed line) and multielectron SCAT (solid line) approximations,
as well as off free Ba (HF, dash-dot-dot; SCAT, dash-dot), as marked.
The calculated data reveal a spectacular difference between the role of polarization in electron scattering off Ba and Ba@C$_{60}$. Namely,
it appears that the effects of polarization in $e + {\rm Ba@C_{60}}$ scattering act oppositely to the effects in $e + {\rm Ba}$ scattering. Thus, whereas
$\sigma_{\rm el}^{\rm SCAT}(e +{\rm Ba}) \ll \sigma_{\rm el}^{\rm HF}(e +{\rm Ba})$ at $\epsilon \alt 1$ eV, the situation
is exact opposite for $e + {\rm Ba@C_{60}}$ scattering in about the same energy region:
$\sigma_{\rm el}^{\rm SCAT}(e +{\rm Ba@C_{60}}) \gg \sigma_{\rm el}^{\rm HF}(e +{\rm Ba@C_{60}})$.
Alternatively, whereas $\sigma_{\rm el}^{\rm SCAT}(e + {\rm Ba}) \gg \sigma_{\rm el}^{\rm HF}(e + {\rm Ba})$
at $\epsilon \agt 1.4$ eV, one observes that $\sigma_{\rm el}^{\rm SCAT}(e + {\rm Ba@C_{60}}) \ll \sigma_{\rm el}^{\rm HF}(e + {\rm Ba@C_{60}})$ in there.
It is, thus, found in the present study that the effects of atomic polarization in electron scattering off the free and encapsulated inside C$_{60}$ atoms
may follow opposite routes. This is an interesting observation.
Lastly, note that there are energy regions, specifically,
$0.8 \alt \epsilon \alt 1.1$ eV and $\epsilon \agt 1.2$ eV, where
$\sigma_{\rm el}^{\rm SCAT}(e + {\rm Ba@C_{60}}) \ll \sigma_{\rm el}^{\rm SCAT}(e + {\rm Ba})$. This means that the gas-medium of big-sized $A@{\rm C_{60}}$s
can be more transparent to incident electrons than the gas-medium of smaller-sized isolated atoms $A$ themselves.
This counter-intuitive effect was earlier unveiled in Ref. <cit.> in the framework of the static HF approximation, but appears to
retain its place even if the encapsulated atom is polarizable, as is shown in the present paper.
§ CONCLUSION
The present work has provided a deeper insight into
possible features of low-energy electron elastic scattering off
$A$@C$_{60}$ fullerenes. This has been
achieved by studying the dependence of $e + {\rm Ba@C_{60}}$ elastic scattering with account for polarization
of encapsulated Ba by incident electrons. It has been demonstrated that the polarization effect
results in dramatic differences between electron scattering off Ba@C$_{60}$ evaluated with and without inclusion of polarization into the calculation.
It has been found that a fuller account for correlation effects in $e + A@C_{60}$ scattering is utterly important. Furthermore, it has been unraveled in the present study
that the impact of polarization on electron scattering off $A$@C$_{60}$ may be both qualitatively and quantitatively different than that
in the case of
electron scattering by the free atom $A$. For instance, it has been demonstrated that where polarization significantly enhances the $e + {\rm Ba@C_{60}}$
scattering cross section, it significantly diminishes the $e + {\rm Ba}$ scattering cross section and vice verse. This leads to the possibility for electron scattering
off $A$@C$_{60}$ to become significantly weaker than in the case of electron scattering by the isolated atom $A$, in certain energy regions. This counter-intuitive effect
has been found to be stronger and occur in a broader energy region than when polarization is ignored.
Lastly, the present study provides researchers with
background information which is useful for future studies of electron scattering by $A$@C$_{60}$, particularly aimed
at elucidating of a possible significance of a simultaneous polarization of both the C$_{60}$ cage and encaged atom by incident electrons. This
will make the $A$@C$_{60}$ more attractive, so that predicted in the present study features of $e + A@{\rm C_{60}}$ may appear at different energies, or disappear
at all, some actual bound states may be converted to resonances, etc. Such effects, however, are subject to an
independent study.
§ ACKNOWLEDGEMENTS
V.K.D. acknowledges the support by NSF Grant No. PHY-1305085.
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1511.00016
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Parton Dynamics at PHENIX
Joe Osborn
Investigating partonic interactions is one of the primary goals of the PHENIX experiment at the Relativistic Heavy Ion Collider (RHIC). RHIC is specially tailored for studying intrinsic partonic spin-momentum correlations due to its unique ability to collide polarized proton beams. Transverse single-spin asymmetries of order 10% have been measured in PHENIX at center of mass energies from 62.4 GeV to 200 GeV, similar to previous measurements. These results indicate that there exist partonic transverse momentum effects within the proton and/or within the process of hadronization. The MPC-EX, a new silicon-tungsten preshower detector at PHENIX, has taken data for the first time this year with the intent of shedding further light on the origins of these asymmetries. A review of the status of the detector and of future planned measurements will be presented. An overview of ongoing work by PHENIX aimed at measuring intrinsic partonic transverse momentum will be discussed.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
In transversely polarized proton-proton collisions, the transverse single spin asymmetry is an optimal observable to probe for partonic dynamics. The asymmetry is defined as follows:
\begin{equation}
A_N = \frac{\sigma^\uparrow(\phi)-\sigma^\downarrow(\phi)}{\sigma^\uparrow(\phi)+\sigma^\downarrow(\phi)}
\end{equation}
In the forward direction of the polarized proton, asymmetries that are orders of magnitude larger than the perturbative QCD (pQCD) prediction have been experimentally observed for a number of decades at energies well into the pQCD regime <cit.>. Figure <ref> shows an example of the $\pion$ asymmetry measured by the STAR collaboration, where the asymmetry is seen to be nonzero up to $Q^2\approx 49$ $[GeV/c]^2$ at small $x_F=\frac{2p_L}{\sqrt{s}}\approx 0.2$. At the center of mass energies that RHIC operates ($\sqrt{s}$=62.4-510 GeV), the PHENIX detector has an optimal opportunity to study transverse single spin asymmetries in a regime where pQCD is applicable yet these non-perturbative effects are still observable. In both experiment and theory we now have the tools to understand the origins of the asymmetries in terms of partonic dynamics since we have facilities that can observe the non-perturbative effects in a regime where theorists can interpret them with pQCD.
The STAR collaboration has measured the transverse single spin asymmetry for $\pi^0$s at a center of mass energy $\sqrt{s}=500$ GeV. The asymmetry is non-zero up to $p_T\approx 7$ GeV/c in the forward direction of the polarized proton, at $x_F\approx 0.2$. The asymmetry has been measured up to nearly 40% from other collaborations at high $x_F\approx 0.8$ at a large range of center of mass energies <cit.>.
In order to theoretically understand these non-perturbative effects, Dennis Sivers introduced the idea of a Transverse Momentum Dependent Parton Distribution Function (TMD PDF)<cit.>. Transverse momentum dependent PDFs are written as a function of both x and the transverse momentum of the interacting partons $k_\perp^2$. Similarly, TMD fragmentation functions can also be written as a function of both z and $k_\perp^2$. Taking into account both the spin of the nucleon and the interacting quarks, there are several TMD PDF (and TMD FF) that quantify spin-momentum and spin-spin correlations between the nucleon spin and partonic transverse momentum (or spin). As an example, the Sivers TMD PDF quantifies a correlation between the initial-state proton spin and quark orbital angular momentum. On the other side of the hard scattering, the Collins TMD FF quantifies a correlation between the final-state quark spin and hadron orbital angular momentum.
Understanding these non-perturbative dynamics can be difficult in proton-proton collisions because of possible contributions from both the initial and final states. Separating contributions from initial-state effects and final-state effects is inconvenient and problematic because they are mixed up in hadron production. Direct photons, defined as photons coming directly from the hard scattering, are the optimal observable to ameliorate this problem. In $qg\rightarrow q\gamma$ and $q\bar{q}\rightarrow g\gamma$ the photon emitted does not suffer from hadronization or fragmentation effects, therefore it contains partonic information directly from the hard scattering. Understanding non-perturbative intitial-state effects with direct photons is one of the goals of the PHENIX experiment and the new MPC-EX detector recently installed in PHENIX.
§ EXPERIMENT
The PHENIX detector at the Relativistic Heavy Ion Collider (RHIC) is well suited to study these non-perturbative initial-state effects. RHIC is a versatile accelerator that can collide protons as well as many different species of heavy ions, such as Au, Al, and U. RHIC is also unique due to its ability to polarize one or both of the proton beams. Because of this RHIC is the only collider based facility in the world where studies of spin-momentum correlations are possible. In 2015, for the first time, RHIC also collided polarized proton beams on nuclei. This unique accelerator facility makes it possible to investigate spin-momentum correlations in many different species of collisions.
PHENIX is an optimal detector to search for non-perturbative initial-state effects due to its electromagnetic calorimetry at both midrapidity and forward rapidity. At midrapidity ($|\eta|<0.35$), PHENIX has two arms that each span $\Delta\phi=\pi/2$ radians with the capability for photon detection with an electromagnetic calorimeter as well as tracking of charged hadrons with a drift chamber, shown in the top of figure <ref>. The EMCal has 8 sectors, 6 that are lead plastic scintillating towers, and 2 that are lead glass cherenkov detectors. The drift chamber allows for unidentified charged hadron tracking and has a 150 $\mu$m spatial resolution. The combination of the the EMCal and drift chamber allows for the analysis of a two-particle angular correlation measurement between direct photons and charged hadrons, which will be discussed further in section 3.
PHENIX also has electromagnetic calorimetry in the forward directions, at $3.1<|\eta|<3.8$, shown in the bottom of figure <ref>. The Muon Piston Calorimeter, (MPC, named for its location in between the PHENIX muon magnets), is a lead tungstate calorimeter that was constructed in a region where spin-momentum correlations are known to be large. The MPC-EX (MPC-EXtension) is a new preshower detector that was installed at the end of 2014 to enhance the ability of the MPC. On its own, the MPC can only resolve $\pion\rightarrow\gamma\gamma$ decays up to $p^{\pion}=20$ GeV; at higher momenta the two photons merge into one cluster and cannot be resolved. The MPC-EX is a silicon tungsten preshower detector that will extend the momentum resolution of neutral pions in the forward region covered by the MPC up to 80 GeV. This increased resolution to $\pion$ decays will allow for measurements of $\pion$ asymmetries up to larger $x_F$, as well as larger $\pion$ background rejection for identification of forward direct photons.
The PHENIX detector configuration, shown both along the beam pipe (top) as well as from the side with the beam pipe going across the page from left to right (bottom).
§ INITIAL-STATE NON-PERTURBATIVE PHYSICS
The intrinsic partonic transverse momentum is one fundamental quantity that characterizes the initial-state of partonic dynamics. One way to measure this quantity is through two-particle angular correlations, for example between a direct photon and away side charged hadron. Figure <ref> shows the vector kinematics of a direct photon and away side charged hadron, where the direct photon is represented by the red vector on the left and the away side hadron fragments from the recoiled quark or gluon on the away side, shown by the black vector on the right of the nominal interaction point. The intrinsic transverse momentum can be seen from this vector diagram, and quantifies the vector sum of the two initial partons intrinsic transverse momentum.
Direct photon and away side charged hadron angular correlation kinematic diagram.
Drell Yan Z Boson production cross section from CDF <cit.>.
The effects of intrinsic transverse momentum can be seen in distributions as simple as a cross section, such as the CDF Drell-Yan Z boson cross section. For Drell-Yan Z boson production at small $p_T$, the $p_T$ of the Z boson will be generated solely by the intrinsic partonic transverse momentum of the $q\bar{q}$ pair that annihilated. At large $p_T$ figure <ref> shows that the cross section exhibits a power law shape which is described by pQCD. At small $p_T$, where the transverse momentum is generated by intrinsic $k_T$, the cross section turns over and exhibits a gaussian like structure. This transition from pQCD power law to gaussian shape is characteristic of where non-perturbative effects become evident. In the case of angular correlations, similar behavior is expected from the vector $p_{out}$ in figure <ref>. This vector characterizes the out of jet plane transverse momentum of the fragmented hadron, and at small $p_{out}$ is generated mainly from the intrinsic transverse momentum of the colliding partons. Therefore one would expect similar behavior in the $p_{out}$ distribution from angular correlations to the CDF Z boson cross section plot at small $p_T$. This will help characterize the intrinsic transverse momentum of the colliding partons. At PHENIX, there is an ongoing two particle correlations analysis in the high statistics Run 13 $\sqrt{s}=510$ GeV $p+p$ data set that will investigate these initial-state non-perturbative effects due to intrinsic transverse momentum.
In the forward direction, investigations of non-perturbative initial-state physics will be greatly benefited by the addition of the MPC-EX. Because of the increased resolution to $\pion$ production via their 2 photon decay in the forward direction, measurements of the transverse single spin asymmetry for $\pion$s will be possible up to large $x_F\approx 0.8$ at $\sqrt{s}$=200 GeV. More importantly the increased resolution to $\pion$ production will also allow for increased decay photon rejection, and thus an increased signal to background ratio for direct photons in the forward direction. Measuring the direct photon single spin asymmetry is an important measurement for the purpose of understanding what initial state effects are generating the large asymmetries observed. Understanding if the asymmetries are generated by a Sivers type effect or Collins type effect (or both, and by how much) would be an important step to learning what underlying processes could be generating the large transverse single spin asymmetries that have been observed in many different types of collisions and center of mass energies. In 2015 the MPC-EX finished taking data for the first time, collecting an integrated luminosity of approximately 60 pb$^{-1}$ of $p^\uparrow+p$, 205 nb$^{-1}$ of $p^\uparrow+Au$, and 450 nb$^{-1}$ of $p^\uparrow+Al$ collisions. Analysis efforts are currently ongoing and should produce important and interesting physics results that help unveil how initial-state non-perturbative effects are generating the large asymmetries observed.
L. Adamczyk et al. (STAR Collaboration), PRD 89,12001 (2014).
A. Adare et al. (PHENIX Collaboration), PRD 90,072008 (2014).
D. Sivers, PRD 41, 83 (1990).
T. Aaltonen et al. (CDF Collaboration), PRD 86, 052010 (2012).
PRL 36,929 (1976)
C.E. Allgower et al. (E925 Collaboration) PRD 65,092008 (2002)
PLB 261, 201 (1991)
I. Arsene et al. (BRAHMS Collaboration) PRL 101, 042001 (2008)
|
1511.00471
|
In this note we show that conforming Galerkin approximations for
$p$-harmonic functions tend to $\infty$-harmonic functions in the
limit $p\to \infty$ and $h\to 0$, where $h$ denotes the Galerkin discretisation parameter.
§ INTRODUCTION AND THE $\INFTY$-LAPLACIAN
Let $\W \subset \reals^d$ be an open and bounded set. For a given
function $u : \W \to \reals$ we denote the gradient of $u$ as $\D u :
\W \to \reals^d$ and its Hessian $\Hess u:\W \to \reals^{d\times
d}$. The $\infty$-Laplacian is the partial differential equation
\begin{equation}
\label{eq:inflap}
\Delta_\infty u
\frob{\qp{\D u \otimes \D u}}{\Hess u}
\sum_{i,j=1}^d \partial_i u \ \partial_j u \ \partial^2_{ij} u = 0,
\end{equation}
where “$\otimes$” is the tensor product between $d$-vectors and
“$:$” the Frobenius inner product between matrices.
This problem is the prototypical example of a PDE from Calculus of
Variations in $\leb{\infty}$, arising as the analogue of the Euler–Lagrange
equation of the functional
\begin{equation}
\cJ[u;\infty] := \Norm{\D u}_{\leb{\infty}(\W)}
\end{equation}
and as the (weighted) formal limit of the variational $p$-Laplacian
\begin{equation}
\label{eq:introplap}
\Delta_p u := \div{\qp{\norm{\D u}^{p-2} \D u}} = 0.
\end{equation}
The $p$-Laplacian is a divergence form problem and appropriate weak
solutions to this problem are defined in terms of duality, or integration by parts. In passing
to the limit ($p\to\infty$) the problem loses its divergence
structure. In the nondivergence setting we do not have access to the
same stability framework as in the variational case and a different
class of “weak” solution must be sought. The correct concept to use
is that of viscosity solutions
The main idea behind this solution concept is to pass derivatives to test functions through the maximum principle, that is, without using duality.
The design of numerical schemes to approximate this solution concept
is limited, particularly in the finite element context, where the only
provably convergent scheme is given in <cit.>
(although it is inapplicable to the problem at hand). In the finite
difference setting techniques have been developed
<cit.> and applied to this problem and also
the associated eigenvalue problem <cit.>. In fact both
in the finite difference and finite element setting the methods of
convergence are based on the discrete monotonicity arguments of
<cit.> which is an extremely versatile
framework. Other methods exist for the problem, for example in
<cit.>, the authors propose a biharmonic
regularisation which yields convergence in the case (<ref>)
admits a strong solution. In <cit.> the author proposed an
$h$-adaptive finite element scheme based on a residual type error
indicator. The underlying scheme was based on the method derived in
<cit.> for fully nonlinear PDEs.
In this note we examine a different route. We will review and use the
known theory used in the derivation of the $\infty$-Laplacian
<cit.> where a
$p$-limiting process is employed to derive
(<ref>). We study how well Galerkin approximations of
(<ref>) approximate the solutions of (<ref>)
and show that by forming an appropriate limit we are able to select
candidates for numerical approximation along a “good” sequence of
solutions. This is due to the equivalence of weak and viscosity
solutions to (<ref>)
<cit.>. To be very clear about where
the novelty lies in this presentation, the techniques we use are not
new. We are summarising existing tools from two fields, one set from
PDE theory and the other from numerical analysis. While both sets of
results are relatively standard in their own field, to the authors'
knowledge, they have yet to be combined in this fashion.
We use this exposition to conduct some numerical experiments which
demonstrate the rate of convergence both in terms of $p$ approximation
[The terminology $p$ approximation we use here should not be
confused with $p$-adaptivity which is local polynomial enrichment of
the underlying discrete function space. ] and
$h$ approximation. These results illustrate that for practical
purposes, as one would expect, the approximation of $p$-harmonic functions for large $p$
gives good resolution of $\infty$-harmonic functions. The numerical
approximation of $p$-harmonic functions is by now quite standard in
finite element literature, see for example <cit.>. There has been a lot of activity in the area since
then however. In particular, the quasi-norm introduced in
<cit.> gave significant insight in the numerical
analysis of this problem and spawned much subsequent research for
which <cit.> form an
inexhaustive list.
While it is not the focus of this work, we are interested in this
approach as it allows us to extend quite simply and reliably into the
vectorial case. When moving from scalar to vectorial calculus of
variations in $\leb{\infty}$ viscosity solutions are no longer
applicable. One notion of solution currently being investigated is
$\mathcal D$-solutions <cit.> which is based on
concepts of Young measures. The ultimate goal of this line of research
is the construction of reliable numerical schemes which allow for
various conjectures to be made as to the nature of solutions and even
what the correct solution concepts in the vectorial case are
The rest of the paper is set out as follows: In <ref> we
formalise notation and begin exploring some of the properties of the
$p$-Laplacian. In particular, we recall that the notion of weak and
viscosity solutions to this problem coincide, allowing the passage to
the limit $p\to\infty$. In <ref> we describe a conforming
discretisation of the $p$-Laplacian and its properties. We show that
the method converges to the weak solution for fixed $p$.
experiments are given in <ref> illustrating the
behaviour of numerical approximations to this problem.
§ APPROXIMATION VIA THE $P$-LAPLACIAN
In this section we describe how $\infty$-harmonic functions
can be approximated using $p$-harmonic functions. We give a brief
introduction to the $p$–Laplacian problem, beginning by introducing the Sobolev spaces
\begin{gather}
\leb{p}(\W)
\ensemble{\phi}
{\int_\W \norm{\phi}^p < \infty} \text{ for } p\in[1,\infty)
\text{ and }
\leb{\infty}(\W)
\ensemble{\phi}
{\esssup_\W \norm{\phi} < \infty},
\\
\sob{l}{p}(\W)
\ensemble{\phi\in\leb{p}(\W)}
{\D^{\vec\alpha}\phi\in\leb{p}(\W), \text{ for } \norm{\geovec\alpha}\leq l}
\text{ and }
\sobh{l}(\W)
\sob{l}{2}(\W),
\end{gather}
which are equipped with the following norms and semi-norms:
\begin{gather}
\Norm{v}_{\leb{p}(\W)}^p
{\int_\W \norm{v}^p} \text{ for } p \in [1,\infty)
\text{ and }
\Norm{v}_{\leb{\infty}(\W)}
\esssup_\W |v|
\\
\Norm{v}_{l,p}^p
\Norm{v}_{\sob{l}{p}(\W)}^p
\sum_{\norm{\vec \alpha}\leq l}\Norm{\D^{\vec \alpha} v}_{\leb{p}(\W)}^p
\\
\norm{v}_{l,p}^p
\norm{v}_{\sob{l}{p}(\W)}^p
\sum_{\norm{\vec \alpha} = l}\Norm{\D^{\vec \alpha} v}_{\leb{p}(\W)}^p
\end{gather}
where $\vec\alpha = \{ \alpha_1,\dots,\alpha_d\}$ is a
multi-index, $\norm{\vec\alpha} = \sum_{i=1}^d\alpha_i$ and
derivatives $\D^{\vec\alpha}$ are understood in a weak sense. We pay
particular attention to the case $l = 1$ and
\begin{gather}
\sobg{1}{p}(\W)
\ensemble{\phi\in\sob{1}{p}(\W)}{\phi\vert_{\partial \W} = g},
\end{gather}
for a prescribed function $g\in\sob{1}{\infty}(\W)$.
Let $L = L\qp{\geovec x, u, \D u}$ be the Lagrangian. We
will let
\begin{equation}
\label{eq:action-functional}
\dfunkmapsto[]
{\cJ[\ \cdot \ ; p]}
\phi
{\cJ[\phi; p] := \int_\W L(\geovec x, \phi, \D \phi) \d \geovec x}
\end{equation}
be known as the action functional. For the $p$–Laplacian the
action functional is given as
\begin{equation}
\cJ[u;p]
\int_\W L(\geovec x, u, \D u)
\int_\W \norm{\D u}^p. \quad \footnote{Typically $L(\geovec x, u , \D u) = \tfrac{1}{p} \norm{\D u}^p$. Note here the rescaling of $L$ has no effect on the resultant Euler--Lagrange equations as to $L$ is independent of $u$.}
\end{equation}
We then look to find a
minimiser over the space $\sobg{1}{p}(\W)$, that is, to find
$u\in\sobg{1}{p}(\W)$ such that
\begin{equation}
\cJ[u;p] = \min_{v\in\sobg{1}{p}(\W)} \cJ[v;p].
\end{equation}
If we assume temporarily that we have access to a smooth minimiser,
$u\in\cont{2}(\W)$, then, given that the Lagrangian is of first order,
we have that the Euler–Lagrange equations are (in general) second
The Euler–Lagrange equations for this problem are
\begin{equation}
\label{eq:p-biharm}
\cL[u; p] :=
\div\qp{\norm{\D u}^{p-2}\D u} = 0.
\end{equation}
Note that, for $p=2$, the problem coincides with the Poisson
problem $\Delta u = 0$. In general, the $p$-Laplace problem is to find $u$ such that
\begin{equation}
\label{eq:plap}
\begin{split}
\Delta_p u:= \div\qp{\norm{\D u}^{p-2} \D u} &= 0 \text{ in } \W
\\
u &= g \text{ on } \partial\W.
\end{split}
\end{equation}
[weak solution]
The problem (<ref>) is associated to a weak formulation, set
\begin{gather}
\bi{u}{v} = \int_\W \qp{\norm{\D u}^{p-2}\D u} \cdot \D v
\end{gather}
to be a semilinear form, then $u \in
\sobg{1}{p}(\W)$ is a weak solution of (<ref>) if it satisfies
\begin{equation}
\bi{u}{v} = 0 \Foreach v \in \sobz{1}{p}(\W).
\end{equation}
[existence and uniqueness of weak solutions to (<ref>)]
There exists a unique weak solution to (<ref>).
The proof is standard and can be found in <cit.> for example. It is based on the strict
convexity of $\cJ[\cdot; p]$, yielding uniqueness, together with
appropriate growth conditions for existence.
[viscosity super and sub-solutions]
A function $u\in\cont{0}(\W)$ is a viscosity sub-solution of
a general second order PDE
\begin{equation}
\label{eq:fnl}
F(u,\D u,\D^2 u) = 0
\end{equation}
at a point $\geovec x \in \W$ if for any $\phi\in\cont{2}(\W)$
satisfying $u(\geovec x) = \phi(\geovec x)$, and touching $u$ from
above, that is $u - \phi \leq 0$ in a neighbourhood of $\geovec x$,
we have
\begin{equation}
F(\phi,\D \phi,\D^2 \phi) \leq 0.
\end{equation}
Similarly, a function $u\in\cont{0}(\W)$ is a viscosity
super-solution of (<ref>) at a point $\geovec x\in\W$ if for
any $\phi\in\cont{2}(\W)$ satisfying $u(\geovec x) = \phi(\geovec
x)$ and touches $u$ from below we have
\begin{equation}
F(\phi,\D \phi,\D^2 \phi) \geq 0.
\end{equation}
[viscosity solution]
The function $u$ is a viscosity solution of (<ref>) in $\W$
if it is a viscosity super and sub-solution at any $\geovec x
\in\W$.
[weak solutions of the $p$-Laplacian are viscosity solutions]
Let $g\in\sob{1}{\infty}(\W)$ and suppose $p > d \geq 2$ is
fixed. Then weak solutions of
\begin{equation}
\label{eq:p-lap}
\begin{split}
\Delta_p u &= 0 \text{ in } \W
\\
u &= g \text{ on } \partial\W
\end{split}
\end{equation}
are viscosity solutions of
\begin{equation}
\label{eq:normalised}
\begin{split}
\Delta_\infty u + \frac{\norm{\D u}^2}{p-2} \Delta u &= 0 \text{ in } \W
\\
u &= g \text{ on } \partial\W.
\end{split}
\end{equation}
We begin by noting that by expanding the derivatives
\begin{equation}
\label{eq:norm}
\begin{split}
\Delta_p u &= \div\qp{\norm{\D u}^{p-2} \D u}
\\
\norm{\D u}^{p-2} \Delta u
\qp{p-2}\norm{\D u}^{p-4} \frob{\D u \otimes \D u}{\Hess u}
\\
\norm{\D u}^{p-2} \Delta u
\qp{p-2}\norm{\D u}^{p-4} \Delta_\infty u,
\end{split}
\end{equation}
is a renormalisation of (<ref>). The two formulations (<ref>) and (<ref>)
of the $p$-Laplacian are equivalent in the viscosity sense, see for
example <cit.>.
It remains to show that weak solutions of (<ref>) are viscosity solutions of (<ref>). As $g\in\sob{1}{\infty}(\W)$ we have
\begin{equation}
\Norm{\D g}_{\leb{p}(\W)}^p
\int_\W \norm{\D g}^p
\cJ[g;p]
< \infty.
\end{equation}
Since $u$ solves (<ref>) weakly, it minimises the functional $\cJ[\cdot;p]$ and hence the minimiser must be of finite energy. In view of the existence and uniqueness of the minimisation problem from Proposition <ref> and Morrey's inequality, we may infer $u\in\cont{0,\alpha}(\W)$ and hence $u\in\cont{0}(\W)$.
Now assume by contradiction that $u$ is not a viscosity subsolution of
\begin{equation}
\norm{\D u}^{p-2} \Delta u
\qp{p-2}\norm{\D u}^{p-4} \Delta_\infty u = 0
\end{equation}
then by Definition <ref> we can find an $\geovec x\in\W$, a $\psi\in\cont{2}(\W)$ and an $r>0$ such that $u - \psi <0$ on $B(\geovec x,r)$, $\qp{u - \psi}(\geovec x) = 0$ and
\begin{equation}
\Delta_p \psi
\norm{\D \psi}^{p-2} \Delta \psi
\qp{p-2}\norm{\D \psi}^{p-4} \Delta_\infty \psi \leq C < 0 \text{ in } B(\geovec x, r),
\end{equation}
for some $C>0$. Since $u - \psi$ has a strict maximum at $\geovec x$ we may find an $\epsilon > 0$ such that
\begin{equation}
\w := \ensemble{\geovec x}{u(\geovec x) - \qp{\psi(\geovec x) - \epsilon} > 0}
\subset
B(\geovec x,r).
\end{equation}
\begin{equation}
\begin{split}
C \int_\w \qp{u - \qp{\psi - \epsilon}}
\int_\w - \Delta_p \psi \qp{u - \qp{\psi - \epsilon}}
\\
\int_\w \qp{\D \psi}^{p-2} \D \psi \cdot \D \qp{u - \psi},
\end{split}
\end{equation}
as $u = \psi - \epsilon$ on $\partial \w$. Now by the convexity of the Lagrangian $L(\geovec x, u, \D u) = \norm{\D u}^p$ we have
\begin{equation}
\norm{\D v}^{p-2} \D v \cdot \D \qp{w - v} \leq \norm{\D w}^p - \norm{\D v}^p,
\end{equation}
\begin{equation}
0 \leq C \int_\w \qp{u - \qp{\psi - \epsilon}}
\leq
\norm{\D u}^p - \norm{\D \qp{\psi - \epsilon}}^p
\cJ[u;p] - \cJ[\psi-\epsilon ;p]
\end{equation}
and we see
\begin{equation}
\cJ[u;p] \leq \cJ[\psi - \epsilon ;p].
\end{equation}
This means that $\w = \emptyset$ and we have a contradiction. The complete proof in full generality for convex minimisation problems can be found in <cit.>. See also <cit.> where the author extends the arguments of <cit.> to singular PDEs.
[viscosity solutions of the $p$-Laplacian are weak solutions]
The converse to Theorem <ref> is also true, thus weak and viscosity solutions are an equivalent concept for the $p$-Laplacian and its evolutionary relative. This has been shown in <cit.>.
[the limit as $p\to\infty$]
Let $u_p \in\sobg{1}{p}(\W)$ denote a sequence of weak/viscosity
solutions to the $p$-Laplacian then there exists a subsequence such
that as $p\to\infty$ that sequence converges to a candidate
$\infty$-harmonic function $u_\infty\in\sob{1}{\infty}(\W)$, that is,
\begin{equation}
u_{p_j} \to u_\infty \text{ in } C^{0}.
\end{equation}
We denote $u_p \in \sobg{1}{p}(\W)$ as the weak solution of (<ref>). In view of Proposition <ref> we know that $u_p$ minimises the energy functional
\begin{equation}
\cJ[u_p;p] = \int_\W \norm{\D u_p}^p.
\end{equation}
Hence in particular
\begin{equation}
\cJ[u_p;p] \leq \cJ[g;p],
\end{equation}
where $g$ is the associated boundary data to (<ref>). Using this fact
\begin{equation}
\Norm{\D u_p}_{\leb{p}(\W)}^p
\cJ[u_p;p]
\leq
\cJ[g;p]
\Norm{\D g}_{\leb{p}(\W)}^p,
\end{equation}
and we may infer that
\begin{equation}
\label{eq:limitpf1}
\Norm{\D u_p}_{\leb{p}(\W)}
\leq
\Norm{\D g}_{\leb{p}(\W)}.
\end{equation}
Now fix a $k > d$ and take $p \geq k$, then using Hölders inequality
\begin{equation}
\label{eq:limitpf3}
\Norm{\D u_p}_{\leb{k}(\W)}^k
\int_\W \norm{\D u_p}^k
\leq
\qp{\int_\W 1^q}^{1/q} \qp{\int_\W \norm{\D u_p}^p}^{1/r},
\end{equation}
with $r = \tfrac{p}{k}$ and $q = \tfrac{r-1}{r}$ such that $\tfrac{1}{r} + \tfrac{1}{q} = 1$. Hence
\begin{equation}
\Norm{\D u_p}_{\leb{k}(\W)}^k
\leq
\norm{\W}^{\tfrac{r}{r-1}} \Norm{\D u_p}^{k}_{\leb{p}(\W)}
\norm{\W}^{1-\tfrac{k}{p}} \Norm{\D u_p}^{k}_{\leb{p}(\W)}
\end{equation}
and we see
\begin{equation}
\label{eq:limitpf2}
\Norm{\D u_p}_{\leb{k}(\W)}
\leq
\norm{\W}^{\tfrac{1}{k}-\tfrac{1}{p}} \Norm{\D u_p}_{\leb{p}(\W)}.
\end{equation}
Using the triangle inequality
\begin{equation}
\begin{split}
\Norm{u_p}_{\leb{k}(\W)}
\Norm{u_p - g}_{\leb{k}(\W)}
\Norm{g}_{\leb{k}(\W)}
\\
C_P \Norm{\D u_p - \D g}_{\leb{k}(\W)}
\Norm{g}_{\leb{k}(\W)},
\end{split}
\end{equation}
in view of the Poincaré inequality. Using the triangle inequality again we have
\begin{equation}
\begin{split}
\Norm{u_p}_{\leb{k}(\W)}
C_P \Norm{\D u_p}_{\leb{k}(\W)}
\Norm{g}_{\sob{1}{k}(\W)}
\\
C_P \norm{\W}^{\tfrac{1}{k}-\tfrac{1}{p}} \Norm{\D u_p}_{\leb{k}(\W)}
\Norm{g}_{\sob{1}{k}(\W)},
\end{split}
\end{equation}
by (<ref>). Hence using (<ref>)
\begin{equation}
\label{eq:limitpf4}
\begin{split}
\Norm{u_p}_{\sob{1}{k}(\W)}
C \Norm{g}_{\sob{1}{k}(\W)}.
\end{split}
\end{equation}
This means that for any $k > d$ we have uniformly that
\begin{equation}
\sup_{p > k} \Norm{u_p}_{\sob{1}{k}(\W)} \leq C.
\end{equation}
Hence, in view of weak compactness, we may extract a subsequence $\{
u_{p_j} \}_{j=1}^\infty \subset \{ u_{p} \}_{p=1}^\infty$ and a
function $u_\infty\in\sob{1}{k}(\W)$ such that for any $k > n$
\begin{equation}
u_{p_j} \rightharpoonup u_\infty \text{ weakly in } \sob{1}{k}(\W)
\end{equation}
\begin{equation}
\begin{split}
\Norm{u_\infty}_{\sob{1}{k}(\W)} &\leq \liminf_{j\to\infty} \Norm{u_{p_j}}_{\sob{1}{k}(\W)}
\\
\liminf_{j\to\infty}
\end{split}
\end{equation}
Taking the limit $k\to\infty$ we have
\begin{equation}
\begin{split}
\Norm{u_\infty}_{\sob{1}{\infty}(\W)} &\leq C \Norm{g}_{\sob{1}{\infty}(\W)}
\end{split}
\end{equation}
and thus
$u_\infty \in \sob{1}{\infty}(\W)$. The result follows from Morrey's
inequality, concluding the proof.
[An alternative to the $p$-Dirichlet functional]
We note that an alternative sequence of solutions is given in <cit.>, where rather than studying the limit of the $p$-Dirichlet functional, the authors propose
\begin{equation}
\widetilde{\cJ}[u;p]
\int_\W \exp\qp{{p}\norm{\D u}^2}.
\end{equation}
This functional may have some merit over the $p$-Dirichlet functional since the Euler–Lagrange equations
\begin{equation}
\begin{split}
0 &= \div\qp{\exp\qp{p\norm{\D u}^2}\D u}
\\
&= \exp\qp{p\norm{\D u}^2} \Delta u + p \exp\qp{p\norm{\D u}^2} \Delta_\infty u
\end{split}
\end{equation}
yield a clearer relation between $\Delta u$ and $\Delta_\infty
u$. We will not explore this issue further in this work.
[existence and uniqueness of viscosity solutions to the
$\infty$-Laplacian <cit.>]
The “candidate” $\infty$-harmonic function $u_\infty$ is the
unique viscosity solution to the $\infty$-Laplacian
The proof is detailed in <cit.>. Roughly, existence of
$u_\infty$ has been shown in Theorem <ref>. For
uniqueness one must prove and make use of the maximum principle for (<ref>). Note also the result of <cit.> where the authors use difference equations to prove the same result in a simpler fashion.
§ DISCRETISATION OF THE $P$-LAPLACIAN
In this section we describe a conforming finite element discretisation of the $p$-Laplacian.
Let $\T{}$ be a conforming triangulation of $\W$,
namely, $\T{}$ is a finite family of sets such that
* $K\in\T{}$ implies $K$ is an open simplex (segment for $d=1$,
triangle for $d=2$, tetrahedron for $d=3$),
* for any $K,J\in\T{}$ we have that $\closure K\meet\closure J$ is
a full lower-dimensional simplex (i.e., it is either $\emptyset$, a vertex, an
edge, a face, or the whole of $\closure K$ and $\closure J$) of both
$\closure K$ and $\closure J$ and
* $\union{K\in\T{}}\closure K=\closure\W$.
The shape regularity constant of $\T{}$ is defined as the number
\begin{equation}
\label{eqn:def:shape-regularity}
\mu(\T{}) := \inf_{K\in\T{}} \frac{\rho_K}{h_K},
\end{equation}
where $\rho_K$ is the radius of the largest ball contained inside
$K$ and $h_K$ is the diameter of $K$. An indexed family of
triangulations $\setof{\T n}_n$ is called shape regular if
\begin{equation}
\label{eqn:def:family-shape-regularity}
\mu:=\inf_n\mu(\T n)>0.
\end{equation}
Further, we define $\funk h\W\reals$ to be the piecewise
constant meshsize function of $\T{}$ given by
\begin{equation}
h\equiv h(\vec{x}):=\max_{\closure K\ni \vec{x}}h_K.
\end{equation}
A mesh is called quasiuniform when there exists a positive constant
$C$ such that $\max_{x\in\Omega} h \le C \min_{x\in\Omega} h$. In what
follows we shall assume that all triangulations are shape-regular and
quasiuniform although the results may be extendable even in the
non-quasiuniform case using techniques developed in
We let $\E{}$ be the skeleton (set of common interfaces) of the
triangulation $\T{}$ and say $e\in\E$ if $e$ is on the interior of
$\W$ and $e\in\partial\W$ if $e$ lies on the boundary $\partial\W$
and set $h_e$ to be the diameter of $e$.
Further, we define the broken gradient $\D_h$, Laplacian
$\Delta_h$ and Hessian $\Hess_h$ to be defined element-wise by
$\D_h w|_K = \D w$, $\Delta_h w|_K = \Delta w$, $\Hess_h w|_K
= \Hess w$ for all $K\in \T{}$, respectively, for respectively smooth
functions on the interior of $K$,
We let $\poly k(\T{})$ denote the space of piecewise polynomials of
degree $k$ over the triangulation $\T{}$,
\begin{equation}
\poly k (\T{}) = \{ \phi \text{ such that } \phi|_K \in \poly k (K) \}
\end{equation}
and introduce the finite element space
\begin{gather}
\label{eqn:def:finite-element-space}
\fes := \poly k(\T{}) \cap \cont{0}(\W)
\end{gather}
to be the usual space of continuous piecewise polynomial
functions of degree $k$.
[finite element sequence]
A finite element sequence $\{ V, \fes \}$ is a sequence of
discrete objects indexed by the mesh parameter, $h$, and
individually represented on a particular finite element space
$\fes$, which itself has a discretisation parameter $h$, that is
$\fes = \fes(h)$.
[$\leb{2}(\W)$ projection operator]
The $\leb{2}(\W)$ projection operator, $P_k : \leb{2}(\W) \to \fes$ is
defined for $v\in\leb{2}(\W)$ such that
\begin{equation}
\int_\W P_k v \Phi = \int_\W v \Phi \Foreach \phi\in\fes.
\end{equation}
It is well known that this operator satisfies the following
approximation properties for $v \in \sob{1}{p}(\W)$
\begin{gather}
\lim_{h\to 0}\Norm{v - P_k v}_{\leb{p}(\W)} = 0
\\
\lim_{h\to 0}\Norm{\D v - \D \qp{ P_k v}}_{\leb{p}(\W)} = 0.
\end{gather}
§.§ Galerkin discretisation
We consider the Galerkin discretisation of (<ref>), to find $U \in \fes$ with $U\vert_{\partial \W} = P_k g$ such that
\begin{equation}
\label{eq:plapdis}
\bi{U}{\Phi} = 0 \Foreach \Phi \in \fes.
\end{equation}
[existence and uniqueness of solution to (<ref>)]
There exists a unique solution of (<ref>).
The proof is standard and, in fact, equivalent to that of the smooth case, as in <cit.>.
[convergence of the discrete scheme to weak solutions]
Let $\{ U_p, \fes\}$ be the finite element sequence generated by
solving (<ref>) and $u_p$, the weak solution of (<ref>), then for fixed $p$ we have that
\begin{equation}
U_p \to u_p \text{ in } \cont{0}(\W).
\end{equation}
We begin by noting the discrete weak formulation (<ref>)
is equivalent to the minimisation problem: Find $U\in\fes$ such that
\begin{equation}
\label{eq:dismin}
\cJ[U; p] = \min_{V\in\fes} \cJ[V; p].
\end{equation}
Using this, we immediately have
\begin{equation}
\Norm{\D U}_{\leb{p}(\W)}^p \leq \cJ[U; p] \leq \cJ[P g; p] \leq \Norm{\D \qp{P g}}_{\leb{p}(\W)}^p.
\end{equation}
In view of the stability of the $\leb{2}$ projection in
$\sob{1}{p}(\W)$ <cit.> we have
\begin{equation}
\Norm{\D U}_{\leb{p}(\W)} \leq C,
\end{equation}
uniformly in $h$.
Hence by weak compactness there exists a (weak) limit to the finite
element sequence, which we will call $u^*$. Due to the weak
semicontinuity of $\cJ[\cdot; p]$ we have
\begin{equation}
\cJ[u^*; p] \leq \cJ[U; p].
\end{equation}
In addition, in view of the approximation properties of $P_k$ given in
<ref> we have for any $v\in\cont{\infty}$ that
\begin{equation}
\cJ[v; p] = \liminf_{h\to 0} \cJ[P_k v; p].
\end{equation}
Using the fact that $U$ is a discrete minimiser of (<ref>)
we have
\begin{equation}
\cJ[u^*; p] \leq \cJ[U; p] \leq \cJ[P_k v; p],
\end{equation}
whence sending $h\to 0$ we see
\begin{equation}
\cJ[u^*;p] \leq \cJ[v;p].
\end{equation}
Now, as $v$ was generic we may use density arguments and that $u_p$ was the unique minimiser to conclude $u^* = u_p$, concluding the proof.
[convergence of the discrete scheme to viscosity solutions]
In view of Theorem <ref> the discrete scheme converges
to viscosity solutions of the $p$-Laplacian.
[convergence in the limit $p\to\infty$]
Let $U_p$ solve the discrete problem (<ref>), then for fixed $h$ along
a subsequence we have $U_p \to U_\infty$.
The proof follows similarly to Theorem <ref>. Since $U_p$ is the Galerkin solution to (<ref>), it minimises $\cJ[\cdot,p]$ over $\fes$. Hence we know
\begin{equation}
\Norm{\D U_p}_{\leb{p}(\W)}
\leq
\Norm{\D \qp{P_k g}}_{\leb{p}(\W)}
\leq
C \Norm{\D g}_{\leb{p}(\W)},
\end{equation}
in view of the stability of $P_k$ in $\sob{1}{p}(\W)$. In addition, analogously to (<ref>)–(<ref>) we may find a constant such that
\begin{equation}
\Norm{U_p}_{\sob{1}{k}} \leq C,
\end{equation}
allowing the extraction of a subsequence $\{ U_{p_j}\}_{j=1}^\infty$ and a limit $U_\infty$ such that for $k > d$
\begin{equation}
U_{p_j} \rightharpoonup U_\infty \text{ weakly in }\sob{1}{k}(\W).
\end{equation}
The rest of the proof parallels that of Theorem <ref>.
[Summarising the results thus far]
Up to this point we have shown the green (solid) lines on the following diagram
VertexStyle/.append style = shape=rectangle,inner sep=0pt
[every node/.style=midway,>=latex']
[->] (4)–(3) node [above] Thm <ref>;
[->] (4)–(3) node [below] $h \to 0$;
[->,green] (4)–(3);
[->] (3)–(2) node [right] $p \to \infty$;
[->] (3)–(2) node [left] Thm <ref>;
[->,green] (3)–(2);
[->] (4)–(1) node [left] $p \to \infty$;
[->] (4)–(1) node [right] Lem <ref>;
[->,green] (4)–(1);
hold. We would like to select a route for which we can pass the limits together, that is, we want to select an appropriate route for which the red (dashed) line is true.
Let $U_p$ be the Galerkin solution of (<ref>) and $u_\infty$ the unique viscosity solution of (<ref>) then along a subsequence
\begin{equation}
U_{p_j} \to u_\infty \text{ in } \cont{0} \text{ as } p \to \infty \text{ and } h \to 0
\end{equation}
The proof is a consequence of Theorems <ref> and <ref> noting that along the same subsequence used in Theorem <ref> we have that
\begin{equation}
\Norm{U_{p_j} - u_\infty}_{\cont{0}(\W)}
\leq
\Norm{U_{p_j} - u_{p_j}}_{\cont{0}(\W)}
\Norm{u_{p_j} - u_\infty}_{\cont{0}(\W)}
\end{equation}
and hence $\Norm{U_{p_j} - u_\infty}_{\cont{0}(\W)} \to 0$ as $p\to \infty$ and $h\to 0$.
[consequences of Theorem <ref>]
An immediate consequence of Theorem <ref> and the
previous arguments are that for $H : \W \times \reals \times
\reals^d$ with appropriate conditions (convexity for example) ,
finite element approximations to the $p$-functional
\begin{equation}
\cJ[u;p] = \Norm{H(\cdot, u, \D u)}_{\leb{p}(\W)}
\end{equation}
can be used as approximations to
\begin{equation}
\cJ[u;\infty] = \Norm{H(\cdot, u, \D u)}_{\leb{\infty}(\W)}.
\end{equation}
[discontinuous Galerkin approximations]
All the above results can be extended into the discontinuous Galerkin framework. This is based on the discrete action functional
\begin{equation}
\label{eq:action-dg}
\cJ_h[U; p] := \int_\W \norm{G(U)}^p + \int_\E h_e^{1-p} \norm{\jump{U}}^p,
\end{equation}
\begin{equation}
\int_\W G(U) \phi = \int_\W \D_h U \phi - \int_\E \jump{U} \avg{\phi} \Foreach \phi \in \poly{k}(\T{}),
\end{equation}
where $\jump{U} = U|_{K^+} - U|_{K^-}$ denotes the jump over an edge $e$ shared by neighbouring elements $K^+$ and ${K^-}$ and $\avg{\phi} = \tfrac{1}{2} \qp{\phi|_{K^+} + \phi|_{K^-}}$, the average of a quantity over an edge.
Using the results of <cit.> discrete minimisers to (<ref>) satisfy the equivalent weak convergence results to the conforming finite elements.
§ NUMERICAL EXPERIMENTS
In this section we summarise numerical experiments validating the analysis done in previous sections.
[practical computation of (<ref>) for large $p$]
The computation of $p$-harmonic functions is an extremely challenging problem in its own right. The class of nonlinearity in the problem results in the algebraic system, which ultimately yields the finite element solution, being extremely badly conditioned. One method to tackle this class of problem is the use preconditioners based on descent algorithms <cit.>. For extremely large $p$, say $p \geq 1000$ this may be required, however for our purposes we restrict our attention to $p \sim 100$. This yields sufficient accuracy for the results we want to illustrate.
Even tackling the case $p\sim 100$ is computationally tough. Our numerical approximation is based on a Newton solver. As is well known, Newton solvers require a sufficiently close initial guess to converge. For large $p$ a reasonable initial guess is given by numerically approximating the $q$-Laplacian for $q < p$ sufficiently close to $p$. This leads to an iterative process in the generation of the initial guess.
§.§ Test 1 : Approximation of the Aronsson viscosity solution
We begin by approximating the viscosity solution derived by Aronsson using separation of variables <cit.>. The function
\begin{equation}
\label{eq:aronson}
u(x,y) = \frac{3}{8}\qp{\norm{x}^{4/3} - \norm{y}^{4/3}} \in \cont{1,1/3}(\W)
\end{equation}
is a viscosity solution of the $\infty$-Laplacian. Notice that this is a weighted version of the Aronsson solution. We have chosen this as $\norm{\D u} \leq 1$ on the domains we consider to try to overcome the severe restrictions in computing $p$-harmonic functions with large $p$. In this test we take $\W = [-1.0001, 0.9999]^2$ and triangulate with a criss-cross mesh. This is so the singularity will not be aligned with the mesh. We approximate the solution of the $p$-Laplacian with boundary data given by (<ref>) for a variety of increasing $p$. Examples of solutions are given in Figure <ref>. In Figure <ref> we plot the error against $p$ for a various levels of mesh refinement. In Table <ref> we demonstrate the convergence of the finite element approximations as $h\to 0$.
[Numerical Results for Problem (<ref>) with
$\poly1$ elements]
Finite element approximations to the $\infty$-harmonic Aronsson function (<ref>) using $p$-harmonic functions for various $p$. Notice as $p$ increases the approximation better catches the singularity on the coordinate axis.
The finite element approximation to the $5$-Laplacian.
The finite element approximation to the $15$-Laplacian.
The finite element approximation to the $50$-Laplacian.
The finite element approximation to the $100$-Laplacian.
The error of the finite element approximation to the $p$-Laplacian compared to the viscosity solution to the $\infty$-Laplacian for various $p$. The colours represent different mesh refinement levels. The darker the colour, the more refined the mesh. In this experiment the meshsize ranges from $h\sim 0.7$ to $h \sim 0.005$ Notice as the mesh is refined, best approximation is achieved for higher and higher $p$.
In this Table we show the convergence of the finite element approximation $U_p$ to $u_\infty$, a viscosity solution of (<ref>), as the meshsize is decreased. We study the $\leb{\infty}$ error of the approximation, the associated convergence rate and give $p^*$, the smallest such $p$ for which $\inf_p \Norm{u_\infty - U_p}_{\leb{\infty}(\W)}$ is attained. Notice that as the mesh is refined, the critical value increases.
$\dim{\fes}$ $\inf_p \Norm{u_\infty - U_p}_{\leb{\infty}(\W)}$ EOC $p^*$
$25$ $0.0162$ $0.00$ $5$
$81$ $0.00836$ $0.95$ $5$
$289$ $0.00390$ $1.10$ $10$
$1089$ $0.00278$ $0.49$ $15$
$4225$ $0.00166$ $0.74$ $20$
$16641$ $0.00130$ $0.35$ $30$
$66049$ $0.00104$ $0.33$ $45$
$263169$ $0.000805$ $0.37$ $60$
§.§ Test 2 : Approximation of a smooth solution
To test the approximation of a known smooth solution of (<ref>) we look at (<ref>) away from the coordinate axis. In this test we take $\W = [0.5, 1.5]^2$ and triangulate with a criss-cross mesh. As in Test 1, we approximate the solution of the $p$-Laplacian with boundary data given by (<ref>) for a variety of increasing $p$. In Figure <ref> we plot the error against $p$ for a various levels of mesh refinement. In Table <ref> we demonstrate the convergence of the finite element approximations as $h\to 0$.
The error of the finite element approximation to the $p$-Laplacian compared to the viscosity solution to the $\infty$-Laplacian for various $p$. The colours represent different mesh refinement levels. The darker the colour, the more refined the mesh. In this experiment the meshsize ranges from $h\sim 0.7$ to $h \sim 0.005$ Notice as the mesh is refined, best approximation is achieved for higher and higher $p$.
In this Table we show the convergence of the finite element approximation $U_p$ to $u_\infty$, a smooth solution of (<ref>), as the meshsize is decreased. We study the $\leb{\infty}$ error of the approximation, the associated convergence rate and give $p^*$, the smallest such $p$ for which $\inf_p \Norm{u_\infty - U_p}_{\leb{\infty}(\W)}$ is attained. Notice that as the mesh is refined, the critical value increases much quicker than in the nonsmooth case of Table <ref>.
$\dim{\fes}$ $\inf_p \Norm{u_\infty - U_p}$ EOC $p^*$
$25$ $0.00301$ $0.00$ $5$
$81$ $0.000883$ $1.77$ $10$
$289$ $0.000244$ $1.86$ $20$
$1089$ $0.0000946$ $1.37$ $30$
$4225$ $0.0000448$ $1.08$ $50$
$16641$ $0.0000218$ $1.04$ $75$
$66049$ $0.0000105$ $1.06$ $125$
$263169$ $0.0000052$ $1.01$ $185$
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|
1511.00151
|
In recent work, Rosenbaum and Wagner
showed that isomorphism of explicitly listed $p$-groups of order $n$ could
be tested in $n^{\frac{1}{2}\log_p n + O(p)}$ time, roughly
a square root of the classical bound. The $O(p)$ term is entirely due
to an $n^{O(p)}$ cost of testing
for isomorphisms that match fixed composition series
in the two groups.
We focus here on the fixed-composition-series
subproblem and exhibit a polynomial-time
algorithm that is valid for general groups.
A subsequent paper
will construct canonical forms within the same time bound.
§ INTRODUCTION
The complexity of testing isomorphism of groups of order $n$ (input via Cayley tables) has long been
cited as $\,n^{\log_p n + O(1)}$,
where $p$ is the smallest prime divisor of $n$; this follows
immediately from the fact that the group requires no more than $\,\log_p n\,$ generators
<cit.>. Wagner <cit.> suggested that this might
be improved by a careful consideration of the isomorphisms that
match fixed composition series.
While composition series have long been a staple for such computational problems
(e.g., <cit.>), Wagner's
insight was that this could lead to an analyzable advantage
in terms of a provable worst-case bound.
Indeed, using that approach,
Rosenbaum and Wagner <cit.> were able to improve the
bound for $p$-groups to $n^{(1/2)\log_p n + O(p)}$. The $O(p)$
term is contributed by their
complexity analysis of
fixed-composition-series isomorphism in $p$-groups.
Their main result then uses the fact that there are $n^{(1/2)\log_p n + O(1)}$
composition series to consider.
Rosenbaum <cit.> extended the result to
solvable groups, achieving an $n^{(1/2)\log_p n + O(\log n/\log\log n)}$ time
for isomorphism, the fixed-composition series subproblem contributing to the
$\,\log n/\log\log n\,$ term.
Making use of a canonical-form version of the
fixed-composition-series subproblem,
Rosenbaum <cit.> subsequently showed that a “collision” method
yields a square-root improvement. That result is striking and there
is much to be appreciated in the methods. However,
the composition-series-isomorphism subproblem remains appealing in its own right
and it appears to be susceptible to established algebraic and computational
Thus, we focus on the following problem.
Given: Groups $G_1,G_2$ given by Cayley tables;
composition series
$G_1=G_{1,0} \vartriangleright G_{1,1} \vartriangleright G_{1,2} \vartriangleright \cdots \vartriangleright G_{1,m} =\1$,
$G_2=G_{2,0} \vartriangleright G_{2,1} \vartriangleright G_{2,2} \vartriangleright \cdots \vartriangleright G_{2,m} =\1$.
Is there an isomorphism $~f: G_1 \rightarrow G_2~$ such that
$f(G_{1,i})= G_{2,i}$, for $0\le i\le m$?
The treatments of for nilpotent and solvable groups
in <cit.>, <cit.>, <cit.>, put great effort into reductions to instances of bounded-valence graph isomorphism so as to
plug in the main result of <cit.>. However, underlying the latter was a method for set-stabilizers in permutation groups
which we can apply directly in a natural approach to . This results in both a better bound
and an extension to general groups.
is in polynomial time.
If one separates the elements needed for just the solvable-group case, the proof
can be compressed to a few lines.
It should be no surprise that the method actually returns all isomorphisms
in the form of a coset of the analogous automorphism group. In fact,
our discussion concentrates mostly on finding automorphism groups.
Given: A group $G$ given by Cayley table;
a composition series
$G=G_0 \vartriangleright G_1 \vartriangleright G_2 \vartriangleright \cdots \vartriangleright G_m =\1.
(Generators for) the group $\{f\in \Aut(G) \mid f(G_{i})= G_{i} \mbox{ for } 0\le i\le m\}$.
We prove
is in polynomial time.
The application
to isomorphism then follows a quick and standard path.
An immediate consequence of Theorem <ref> is the time bound
$\,n^{(1/2)\log_p n + O(1)}\,$ for testing isomorphism
of groups of order $n$,
where $p$ is the smallest prime divisor of $n$;
this is due to the upper bound of
$\, n^{(1+\log_p n)/2}\,$
on the number of composition series (credited to Babai in <cit.>).
Rosenbaum <cit.>
has already realized that timing for isomorphism
using “bidirectional collision”, though his method comes
at a substantial cost in space.
Collision and other innovative methods in
<cit.> also mesh well with our results, with
further implications for group isomorphism.
We will be pleased to borrow from those methods
in a follow-up paper which will first extend our algorithm
to the computation of canonical forms.
We remark that Babai <cit.> has found a holomorph approach
to which also improves the earlier bounds and is in polynomial time for solvable groups.
Our method for involves repeated consideration
of a classic issue in automorphism-group computation.
Specifically, for $H\normaleq G$, we are given $\calA \le \Aut(G/H)$ and $\calB\le \Aut(H)$,
and we want to determine the pairs $(\alpha,\beta)\in \Aut(G/H)\times \Aut(H)$
that “lift” to automorphisms of $G$, if any such exist. The obstruction to such lifting
is easy to formulate algebraically and, in general, it poses a difficult
computational problem. However, for the $\calA$ and $\calB$
that arise herein, we are able to view the obstruction in stages,
each of which is resolvable using methods that are in polynomial time
for solvable permutation groups.
We give two procedures for this. An elementary method, described in <ref>,
is all one needs for the resolution of
for solvable groups, but it is not effective for
general groups. The method is then strengthened in <ref> to one that
generally applicable.
In <ref>, we recall the divide-and-conquer method for set-stabilizer
in permutation groups and show that it is in polynomial time for
the groups we encounter since they are shown to have solvable subgroups
of “small” index.
There are two demonstrations of Theorem <ref>
in <ref>.
They exhibit two ways of breaking the problem down into instances of the
“lift” scenario that are amenable to a set-stabilizer approach
(assuming they are not already in polynomial time via brute-force enumeration).
Theorem <ref> is succinctly resolved in
<ref> by viewing it
as an extended application of the methods of <ref>.
Suppose $G$ is a group acting on the set $\Omega$.
For $g \in G$, we denote the image of $\alpha \in \Omega$ under the
action of $g$ by $\alpha^g$. For $\Delta\subseteq\Omega$, let
{G}_{\Delta} = \{g\in G \mid \Delta^g = \Delta\}
The automorphism group of $G$ is denoted by $\Aut(G)$, which we view as
a subgroup of $\Sym(G)$. Thus, for $H\le G$, $\Aut(G)_H$ denotes the group
of automorphisms stabilizing (or normalizing) $H$.
Implicit in the statement that $G$ is given by a Cayley table is the assumption that
the elements of $G$ can be listed in polynomial time.
Permutation groups
are assumed to be input or output via generating sets. A coset of
a permutation group $J$ is input or output via generators for $J$ and
a single representative.
For other concepts and notation, we
refer the reader to <cit.>. For background on
polynomial-time computability in
permutation groups, see <cit.>, <cit.>, <cit.>, <cit.>.
§ THE KEY SUBPROBLEM
Throughout this section, we assume $G$ is given by a Cayley table and $H\normaleq G$.
Then $\Aut(G)_H$ can be viewed as a subgroup of the
wreath product $\Sym(H) \wr \Sym(G/H)$, and especially a subgroup of
$\Sym(H) \wr \Sym(G/H)_{\{H/H\}}$.
(The subscript $\{H/H\}$ signifies that we fix this single “point”
in the permutation domain $G/H$.)
For the reader's convenience
we give a more explicit indication of
the latter group, namely,
$$\Sym(G)_{H,G/H} := \{ \gamma\in\Sym(G)_H \mid
\gamma \mbox{ permutes the cosets of }H\}.$$
There is a natural homomorphism
\Theta_{G,H}: \Sym(G)_{H,G/H} \,\rightarrow\, \Sym(G/H)\times\Sym(H).
Inasmuch as $H\normaleq G$ will always be clear in context, we let $\Theta:=\Theta_{G,H}$. For $x\in G$,
we let $\inn{x}$ denote the restriction to $H$ of the inner automorphism due on $x$, i.e., for $h\in H$
$h^{\inn{x}} = x^{-1} h x$; this is extended to $X\subseteq G$ by $\InnH{X} = \{ \InnH{x} \mid x\in X\}$.
Because of repeated usage, it is useful to
set $\C :=\{g\in G \mid \forall h\in H, gh=hg\}$, i.e., the centralizer of
$H$ in $G$.
§.§ Lifting automorphisms from $G/H$ and $H$
Noting that
$\Theta(\Aut(G)_H)\,\le \Aut(G/H)\times\Aut(H)
$, we are concerned with the following problem.
Given: $H\normaleq G$;
$\calA\le \Aut(G/H)$; $\calB \le \Aut(H)$.
$\,\Aut(G)_H \cap \Theta^{-1}(\calA\times \calB).\,$
There are a couple of natural ways to reduce Theorem <ref> to
polynomial-time instances of AutLifting. We will describe these in <ref>,<ref>.
The reader may already recognize AutLifting
as a frequent issue in studies of group automorphism/isomorphism, theoretical or computational
(see, e.g., <cit.>).
Special cases are attacked with varied machinery and success.
Our method is guided by properties of the relevant groups that enable polynomial-time steps.
Our approaches to AutLifting each involve defining a group $\calL$ such that
* $\Aut(G)_H \le \calL \le \Sym(G)_{H,G/H}$.
* It is “easy” to find $\,\Theta(\calL )\cap (\calA\times \calB).$
* It is “easy” then to lift to $\,\widehat{\calL}=\calL \cap \Theta^{-1}(\calA\times \calB).$
* It is “easy” to find $\,\widehat{\calL} \cap \Aut(G)$.
§.§ First choice of $\calL$
We consider the following supergroup of $\Aut(G)_H$.
\calL_1~=~
\{ \gamma\in \Sym(G)_{H,G/H} \,\mid\,
\forall g\in G, h\in H:~ (hg)^\gamma = h^\gamma g^\gamma\}.
§.§.§ Theory
* The kernel of $\,\restr{\Theta}{\calL_1}\,$ is isomorphic to the direct product
of $~|G/H| -1~$ copies of $\,H$.
* (Generators for) $\,\Ker(\restr{\Theta}{\calL_1})\,$ can be found in polynomial time.
Let $\gamma\in\Ker(\restr{\Theta}{\calL_1})$. Then $\forall h\in H, \, h^\gamma = h$ and
$\forall x\in G,\, (Hx)^\gamma = Hx$.
Consider the action induced by $\gamma$ on $Hg \ne H$.
We have $g^\gamma = gk$ for some $k\in H$ (since $gH=Hg$). Then for any $hg\in Hg$,
$(hg)^\gamma = h^\gamma g^\gamma = hgk$, that is, $\forall x\in gH$, $x^\gamma = xk$.
Thus, the action of $\Ker(\restr{\Theta}{\calL_1})$ on each $Hg\ne H$
is the group of right multiplications by $H$, a group isomorphic to $H$. Since the actions are independent
across the cosets, (i) follows.
For (ii), we deal with each $Hg\ne H$ independently. Focussing on a coset $X$,
we form for each $h\in H$, the permutation $\phi_{X,h}$ of $G$ that fixes all $x\not\in X$
and maps $x\mapsto xh$ for $x\in X$. Then $\Ker(\restr{\Theta}{\calL_1})$
is generated by $\{\phi_{X,h} \mid h\in H, X\in G/H \mbox{ with } X\ne H \}$.
Take the union of all these permutations for all $Hg\ne H$.
(A bit more economically, only use $h$ in a generating set for $H$.)
$\Aut(G/H)\times \Aut(H) \le \Theta(\calL_1)$.
Furthermore, given $(\alpha,\beta)\in\calA\times\calB$,
$\calL_1 \cap \Theta^{-1}(\alpha,\beta)$ can be constructed
in polynomial time.
Proof: It suffices to show,
for any $\alpha\in \Aut(G/H),\,\beta\in\Aut(H)$, we can construct a single
for which $\Theta(\gamma)=(\alpha,\beta)$ since, by Lemma <ref> the coset
$\,\gamma \,\Ker(\restr{\Theta}{\calL_1})$ then comprises the set
of preimages of $(\alpha,\beta)$ in $\calL_1$.
We construct $\gamma\in\calL_1$ as follows. For $h\in H$,
define $h^\gamma := h^\beta$. For each coset of $X$ of $H$ in $G$ with $X\ne H$,
define $\gamma$ on $X$ as follows:
* Fix any $a\in X$ (so $X=Ha$).
* Fix any $~b\in X^\alpha$.
* For all $h\in H$, define $(ha)^\gamma := h^\beta b$.
Using Lemmas <ref> and <ref>, we conclude
Given $\,\calA\le \Aut(G/H)$ and $\calB \le \Aut(H)$,
$\widehat{\calL}_1:=\calL_1 \cap \Theta^{-1}(\calA\times\calB)$ can be constructed in polynomial
$\widehat{\calL}_1$ is an extension of $\,\calA\times\calB\,$ by a direct product of
copies of $H$.
Proof: For (i), $\widehat{\calL}_1$ is generated by the lifts of the generators of $\,\calA \times \calB\,$
together with generators of $\,\Ker(\restr{\Theta}{\calL_1})$.
§.§.§ Algorithm
Step 1. $\widehat{\calL}_1:=\calL_1 \cap \Theta^{-1}(\calA\times\calB)$
By Lemma <ref>(i), this is in polynomial time for any $\calA, \calB$.
Step 2.
Find $~\{\gamma \in \widehat{\calL}_1 \mid \gamma\in\Aut(G)\}$.
Method: This can be expressed as a set-stabilizer problem for the natural extension of
$\widehat{\calL}_1 \le \Sym(G)$
to an action on $G\times G\times G$. The set to stabilize
is $\{(a,b,ab) \mid a,b\in G\}$.
In an inductive approach to for solvable groups the calls to result
in a solvable $\widehat{\calL}_1$, thus putting Step 2 in polynomial time.
(We provide this forecast in the expectation
that some readers would like to finish the solvable case
as an exercise.)
A more restrictive $\calL$ will yield our main results for general groups,
thus making this section superfluous. Nevertheless,
we retain this discussion of $\calL_1$. By switching back to $\calL_1$ in <ref>
whenever $H$ is solvable, we limit the requisite machinery to the early paper <cit.>.
§.§ Second choice of $\calL$
Consider now a more restricted supergroup of $\Aut(G)_H$.
\calL_2~=~
\{ \gamma\in \Sym(G)_{H,G/H} \,\mid\,
\forall g\in G, h\in H,\, (hg)^\gamma = h^\gamma g^\gamma~\mbox{and}~ (gh)^\gamma = g^\gamma h^\gamma \}.
§.§.§ Theory
The advantage of cutting down from $\calL_1$ to $\calL_2$ is that
$\Ker(\restr{\Theta}{{\calL}_2})$ is abelian,
which plays a role in guaranteeing polynomial-time
set-stabilizers for
the instances of $\widehat{\calL}_2$ that arise. (Actually, we only need
that kernel to be solvable.)
However, since
$\calA \times \calB$ may not be contained in
$\Theta(\calL_2)$, we will first have to determine the
liftable subgroup.
We are guided in this by some properties of $\Theta(\calL_2)$.
Let $\gamma\in \calL_2$. Then for $g\in G$, $h\in H$,
$(g^{-1}hg)^\gamma = (g^\gamma)^{-1}h^\gamma g^\gamma$.
For $g\in G$, $h\in H$, $\gamma\in\calL_2$,
g^\gamma(g^{-1}hg)^\gamma = (gg^{-1}hg)^\gamma
= (hg)^\gamma
= h^\gamma g^\gamma,
the first and third equalities using properties of $\calL_2$.
* The kernel of $\restr{\Theta}{\calL_2}$ is isomorphic to the direct product
of $~|G/H| -1~$ copies of $\Z(H)$ (the center of $H$).
* (Generators for) $\Ker(\restr{\Theta}{\calL_2})$ can be found in polynomial time.
Let $\gamma\in\Ker(\restr{\Theta}{\calL_2})$.
In particular, $h^\gamma = h$ for $h\in H$.
Consider $Hg\ne H$.
As in the proof of Lemma <ref>, there is some $k\in H$
such that the action of $\gamma$ on $Hg=gH$ is right-multiplication by $k$.
It suffices then to show $k\in \Z(H)$. Using Lemma <ref>,
for any $h\in H$,
$$g^{-1}hg = (g^{-1}hg)^\gamma = (g^\gamma)^{-1}h^\gamma g^\gamma
= (gk)^{-1}h gk = k^{-1}(g^{-1}hg)k.$$
The normality of $\Z(H)$ in $G$ then implies $k\in \Z(H)$.
For (ii), we proceed as in Lemma <ref>(ii) but restrict the choice of maps $x\mapsto xh$ to
$h\in \Z(H)$ (or just to a generating set of $\Z(H)$).
The next two lemmas give necessary
conditions on $(\alpha,\beta) \in \Aut(G/H) \times \Aut(H)$ for it
to be liftable to
Recall that $\C$ denotes
the centralizer of
$H$ in $G$, and $\inn{x}$ denotes the restriction to $H$ of the inner automorphism
corresponding to $x$.
Let $\gamma\in \calL_2$. Suppose that
$\Theta(\gamma) = (\alpha,\beta) \in \Aut(G/H) \times \Aut(H)$. Then
* $\alpha$ normalizes ${\CH}/{H}$, and therefore induces an automorphism of $G/\CH$.
$\forall g\in G:~\beta^{-1} \,\InnH{\CH g} \, \beta =\InnH{(\CH g)^\alpha}.$
Lemma <ref> implies that $\C^\gamma = \C$, so that
$(H\C)^\gamma = H^\gamma \C^\gamma$, proving (i).
For (ii), first note that $\forall g\in G, h\in H$,
$$ h^{\beta^{-1} \InnH{g} \beta} = (g^{-1} h^{\beta^{-1}} g)^\beta
= (g^\gamma)^{-1} h g^\gamma = h^{\InnH{g^\gamma}},
the second equality following from Lemma <ref>.
In other words,
$$ \forall g\in G,~
\beta^{-1}\InnH{g} \beta = \InnH{g^\gamma}.$$
Suppose $(\alpha,\beta) \in \Aut(G/H) \times \Aut(H)$ satisfies (i),(ii) in
Lemma <ref>.
\begin{equation}
\label{cosetconj}
\forall g\in G:~\beta^{-1} \left(\InnH{H g} \right) \beta =\InnH{(H g)^\alpha}.
\end{equation}
For any $A\subseteq G$,
$\InnH{A \C}= \InnH{A}$. Also, by (i),
$\alpha$ permutes the cosets of $H\C/H$ in $G/H$ so that
$(H\C g)^\alpha = \C(Hg)^\alpha$.
This organization may seem convoluted seeing that
equation (<ref>)
could also be viewed as a direct consequence of
Lemma <ref>. Our motive is that, in the process of cutting down to
our algorithmic
route runs through (<ref>) after first forcing (i),(ii) of Lemma <ref>.
Suppose $(\alpha,\beta) \in \Aut(G/H) \times \Aut(H)$ satisfies property (<ref>).
$\,\Theta^{-1}(\alpha,\beta)\cap\calL_2\,$ is nonempty and can be found in polynomial
Construct $\gamma\in\Theta^{-1}(\alpha,\beta)$ as follows.
For $h\in H$, $h^\gamma := h^\beta$.
To define $\gamma$ on a coset $X$ of $H$ with $X\ne H$. Fix $a \in X$.
Then, by (1), there is some $b\in X^\alpha$ such that
\beta^{-1}\InnH{a}\beta = \InnH{b}.
Fix any such $b$. For any $g\in X$, say $g=ka$ with $k\in H$,
set $g^\gamma := k^\beta b$.
It follows easily that
\begin{equation}
\forall h\in H, ~ (hg)^\gamma = h^\gamma g^\gamma
\end{equation}
\begin{equation}
\beta^{-1}\InnH{g}\beta = \InnH{g^\gamma}
\end{equation}
$\beta^{-1} \InnH{ka} \beta =
\beta^{-1} \InnH{k} \beta\, \beta^{-1} \InnH{a} \beta =
\InnH{k^\beta} \InnH{b} = \InnH{k^\beta b}.
Having established (2) and (3) for $g$ in each coset $X$, they hold for
all $g\in G$.
Then, for all $h\in H, g\in G$,
$$(gh)^\gamma = (h^{\inn{g}^{-1}}g)^\gamma = h^{{\inn{g}^{-1}}\beta}g^\gamma
= h^{\beta (\beta^{-1} \inn{g} \beta)^{-1}}g^\gamma =
h^{\beta \, \inn{g^{\gamma}}^{-1} }g^\gamma = g^\gamma h^\gamma.
We conclude that $\gamma\in \calL_2$.
The complete set of preimages of
$(\alpha,\beta)$ is
$\gamma\, \Ker(\restr{\Theta}{\calL_2})$, for which we apply
Lemma <ref>(ii).
To summarize the main points of this subsection, we have the following
consequence of
Lemmas <ref>, <ref>, <ref>.
Suppose $\calM\le \Aut(G/H) \times \Aut(H)$ such that
property (<ref>) is satisfied
by all $(\alpha,\beta)\in \calM$. Then
* $\calM \le \Theta({\calL}_2)$
* $\calL_2\cap\Theta^{-1}(\calM)$ is an extension of $\calM$ by an abelian group.
* Given $\calM$, $\calL_2\cap\Theta^{-1}(\calM)$ can be constructed in polynomial time.
Proof: For (iii), $\calL_2\cap\Theta^{-1}(\calM)$ is generated by the lifts of the generators of $\,\calM\,$
together with generators of $\,\Ker(\restr{\Theta}{\calL_2})$.
§.§.§ Algorithm
Steps 1-3 cut $\calA\times\calB$ to the subgroup $\calM$ consisting of elements that
can be lifted to $\calL_2$.
Step 1.
$\calA\,:=\, \nrm{\calA}{{\CH}/{H}}$.
Viewing $\calA\le \Sym(G/H)$,
this can be viewed as stabilizing the subset $\CH/{H}$
of the polynomial-size domain $G/H$.
By Lemma <ref>(i), Step 1 does not affect
$~\Theta(\calL_2) \cap (\calA\times\calB)$.
Step 2. $\calB \,:=\, \nrm{\calB}{\InnH{G}}$.
Here we consider $\calB \subseteq \Sym(H)$ acting on
$\Aut(H)\subseteq\Sym(H)$ via conjugation. So, at first glance, this
appears to be a normalizer problem for permutation groups.
However, the specific reductions of
our main problems to will reveal that required
instances of Step 2 can be handled via set-stabilizers.
By Lemma <ref>(ii), Step 2 does not affect
$~\Theta(\calL_2) \cap (\calA\times\calB)$.
Step 2 does not yet accomplish the
compatibility condition on $(\alpha,\beta)$ expressed in property (1)
and required in Lemma <ref>,
but it establishes a structure for getting there.
We now have that $\calA \le {\Aut(G/H)}_{\CH/H}$, so there is an induced action
$$\fra: \calA \rightarrow \Aut\left(\frac{G}{\CH}\right).
We also now have $\calB$ acting on $\InnH{G}$ and, since $\calB\le\Aut(H)$
also normalizes $\InnH{H}$, there is an induced action
$$\frb: \calB \rightarrow \Aut\left( \frac{\InnH{G}}{\InnH{H}} \right)
But there is a natural identification
\frac{G}{\CH} \,\,\,{\simeq}\,\,\,
\frac{\InnH{G}}{\InnH{H}}.$$
(For all $g\in G$, $g\CH \leftrightarrow\InnH{g}\InnH{H}$.)$~~$
Via this identification, property (1)
is expressible in the form
\fra(\alpha)=\frb(\beta).
This motivates
Step 3. $\calM \,:=\, \{(\alpha,\beta)\in\calA\times \calB \mid \fra(\alpha)=\frb(\beta)\}$.
By the above, this is a permutation-group intersection problem, solvable for example as a
set-stabilizer problem: consider $\fra(\calA)\times \frb(\calB)$ acting on $\,\Omega\times \Omega\,$
where $\,\Omega = G/\CH$; the object then is
to stabilize the diagonal $\{(\omega,\omega) \mid \omega\in \Omega\}$.
By Lemmas <ref>,<ref> and Proposition <ref>(i),
$~\Theta(\calL_2) \cap (\calA\times\calB)\,=\, \calM$.
Hence, $\calL_2\cap\Theta^{-1}{(\calA\times\calB)} \,=\, \calL_2\cap\Theta^{-1}(\calM)$.
So the next step is
Step 4.
$\widehat{\calL}_2 \,:=\ \calL_2\cap\Theta^{-1}(\calM)$.
This is in polynomial time by Proposition <ref>(iii).
The final step is
Step 5.
Find $~\{\gamma \in \widehat{\calL}_2 \mid \gamma\in\Aut(G)\}$.
As in <ref>,
this can be expressed as a set-stabilizer problem for the action of
$\widehat{\calL}_2$ on $G\times G\times G$.
Thus the computational complexity of our use of
rests on properties of $\calA$ and $\calB$ that will enable efficient
routines for Steps 1,2,3,5. The properties are
described in <ref>.
§ NICE GROUPS
§.§ A reminder on polynomial-time set stabilizers
In <cit.>, the author proposed
an algorithm for finding $G_\Delta$ where $G\le \Sym(\Omega)$
and $\Delta\subseteq \Omega$. For the reader's convenience in what follows, we offer a
brief description.[See also <cit.> for an extended discussion of the
divide-and-conquer paradigm for this and other applications.]
To accommodate a recursion,
the problem is generalized to cosets.
For a $G$-stable subset
$\,\Pi \subseteq \Omega\,$ and a coset $\,X=Ga\,$ of $\,G$, define
$$X_{\Delta \mid \Pi} = \{ x\in X \mid (\Delta\cap\Pi)^x = \Delta\cap\Pi^a\}.
So $\,X_{\Delta \mid \Pi}\,$ is either $\emptyset$ or a right coset of $\,G_{\Delta\cap\Pi}$.
If $\,\Pi = \Pi_1\dot\cup\Pi_2\,$ for $G$-stable $\,\Pi_i$
$$X_{\Delta \mid \Pi} ~:=~ (X_{\Delta \mid \Pi_1})_{\Delta\mid \Pi_2}
If $G$ acts transitively on $\Pi$ and $|\Pi| >1$, find a minimal
decomposition $\,\Pi = \Pi_1 \dot\cup \cdots \dot\cup \Pi_m\,$ into blocks
of imprimitivity and decompose $\,G= \bigcup_{1\le i\le |G/H|} Ht_i\,$ where $H$
is the kernel of the action of $G$ on $\,\{\Pi_i\}_{1\le i\le m}$.
$$X_{\Delta \mid \Pi} ~:=~ \bigcup_{1\le i\le |G/H|} (Ht_ia)_{\Delta \mid \Pi}.
If $\,\Pi =\{\pi\}\,$ then
if $\,|\Delta \cap \{\pi,\pi^a\}|=1\,$ then $\,X_{\Delta \mid \Pi} := \emptyset\,$
else $\,X_{\Delta \mid \Pi} := X$.
The computation of $\,G_\Delta\,$ starts with $\,\Pi:=\Omega\,$ and $\,a:=1$.
The key to the complexity of the method lies in the sizes of the primitive
groups arising in the action on the $m$ blocks. The algorithm will run in polynomial time for an hereditary
class of groups if such induced subgroups of $S_m$
have order $O(m^{\rm constant})$.
Notice that the “set-stabilizer” algorithm actually dealt with cosets. Thus, we can find set-stabilizers for a
group $A$ that has a small-index subgroup $B$ in a good class: we start by
breaking $A$ into cosets of $B$.
We will find ourselves in exactly
that situation in <ref>.
If we were only given a black-box for set-stabilizers in groups with bounded non-cyclic composition factors, i.e.,
without knowing the algorithm, we could still use that for the coset problem. It is useful
for another purpose in <ref> to express that as a “set-transporter” problem;
namely, given $\Delta,\Lambda \subseteq \Omega$, find
$$G_{\Delta \mapsto \Lambda } = \{g\in G \mid \Delta^g = \Lambda\}.$$
For this, consider
the natural action of $\widetilde{G}= G\wr C_2$
on $\Omega\,\dot\cup\, \Omega'$, where $\Omega'$ is a copy of $\Omega$.
The desired transporter can be deduced from
$\widetilde{G}_{\Delta \,\dot\cup \,\Lambda'}$ where $\Lambda'$ is the corresponding
copy of $\Lambda$. In particular, $(Ga)_\Delta =
G_{\Delta\mapsto \Delta^{a^{\nega}} }\,a$.
§.§ The groups that turn up
In effect, we will only rely on the polynomial boundedness
of primitive solvable groups. However, we deal with groups
that are not quite solvable.
Notation. For a group $X$, $\Sol(X)$ is the solvable radical of $X$, i.e., the maximum normal
solvable subgroup.
Definition. Let $X\le \Sym(\Omega)$. We call $X$ $\,$[
The author regrets using a term that has had other meanings. However, the usage
of “” is already inconsistent in the literature and he could not think of an
unused term with
similar connotation.]$\,$ if
$\,|X:\Sol(X)| \le |\Omega|^2$.
groups still enable the
polynomial-time divide-and-conquer paradigm. For example,
Given an $G\le\Sym(\Omega)$ and $\Delta\subseteq\Omega$,
$G_\Delta$ can be found in polynomial time.
By decomposing $\,G\,$ into cosets of $\,\Sol(G)\,$, we reduce to
$|\Omega|^2$ problems involving
solvable groups.
Our breakdown to cosets of a nice group is reminiscent of the first use of the set-stabilizer method.
When <cit.> was written,
primitive groups in the relevant class, specifically the class of groups with bounded composition factors, were not known to
be polynomially bounded.
This difficulty was overcome by partitioning the primitive
group into cosets of a small-index $p$-subgroup. That additional
complication soon became unnecessary as polynomial bounds were
first for primitive solvable
groups (independently by Pálfy <cit.> and Wolf <cit.>), and ultimately for
primitive groups in a
class that even includes groups with bounded non-cyclic composition factors
(Babai et al. <cit.>). We
point, however, to
a subtle difference between what happens in our current passage
to cosets of a nice group and the earlier use of that trick.
In <cit.>, after passing to cosets of a $p$-group acting on the blocks, the
divide-and-conquer narrows our window to a single, stabilized block. Since the group acting within the block is not necessarily
a $p$-group, we may again arrive at a primitive group that requires the cosets-of-$p$-group
By contrast, in the present
case there is a single such decomposition in the lifetime of the set-stabilizer process, i.e.,
we only visit solvable groups thereafter.
Lemma <ref> is a weak consequence of the
discussion in <ref>.
The good subgroup can be of any polynomially-bounded index,
need not be normal,
and need only be in the broader class
described in <cit.>.
However, “” is a good fit for our situation because we next see that the
property arises so conveniently.
Furthermore, in our present application, one can keep track of small-index normal
solvable subgroups
as we cut down the group or take preimages, so there
is no need even to implement a radical-finder. Nevertheless,
we do note that radicals of
permutation groups can be found in polynomial-time
(<cit.>, <cit.>).
The relevance to our problem is seen in
With reference to the groups in 2,
suppose $\calA\le \Sym(G/H)$ and $\calB\le \Sym(H)$ are .
* If $H$ is solvable, then $\widehat{\calL}_1\le\Sym(G)$ is .
* $\widehat{\calL}_2\le\Sym(G)$ is .
We have $|\calA\times\calB:\Sol(\calA\times\calB)| =
| \calA:\Sol(\calA)| \cdot |\calB:\Sol(\calB) | \le |G/H|^2|H|^2 = |G|^2$.
Thus, (i) follows from Proposition <ref>(ii).
Using $\calM$ as in <ref> (Steps 3,4), $\,\calM\le \calA\times\calB$ implies
$|\calM : \Sol(\calM)| \le |\calA\times\calB:\Sol(\calA\times\calB)| \le |G|^2$.
Thus (ii) follows from Proposition <ref>(ii).
Note, as the methods for are to be used repeatedly, the of
$\,\widehat{\calL}_i\,$ implies that of the subgroup $\,\widehat{\calL}_i \cap \Aut(G)$.
For a base case, we need a consequence of the Classification of Finite Simple Groups.
Using the fact that any simple group $T$ can be generated by two elements (<cit.>),
we immediately have $|\Aut(T)| \le |T|^2$. Hence,
If $T$ is simple then $\Aut(T)$, viewed as a subgroup of $\Sym(T)$, is
In fact, it is well known that the Classification
yields a stronger bound of the form $|\Aut(T)| = O( |T| \log |T|)$ (e.g., see <cit.>).
However, the square bound in “” is easier to maintain
through our process.
§ AUTOMORPHISMS STABILIZING A COMPOSITION SERIES
Applying the machinery of <ref>-<ref>, we offer two
polynomial-time Turing reductions of to polynomial-time instances of .
Moreover, we show that the
deepest tool needed in either implementation is
set-stabilizer for solvable permutation groups.
Aside from reducing to the most basic tool, this extra effort will be useful
in a subsequent development of canonical forms. (See also <cit.>
for an indication of how the divide-and-conquer method for set-stabilizer
translates to canonical set placement.)
Given: A group $G$ given by Cayley table;
a composition series
$G=G_0 \vartriangleright G_1 \vartriangleright G_2 \vartriangleright \cdots \vartriangleright G_m =\1.
$\Aut(G)_{\{G_1,G_2,\ldots \}}\,$.
§.§ Bottom-up on given series
We go from a solution for $G_i$ to a solution for $G_{i-1}$.
The group on top, $G_{i-1}/G_i$, is simple and so
one can enumerate $\Aut(G_{i-1}/G_i)$.
Even for a simple permutation group $T$ given
by generators, one can produce
generators for $\Aut(T)$ (which is all that is necessary for some of our subproblems).
This follows from Kantor's demonstration <cit.>
that one can obtain the “natural” representation
of $T$.
Before proceeding further,
we can claim to have already established a polynomial-time solution for this use of
. That is, by virtue of almost-solvability (<ref>), there are citable polynomial-time methods for
Steps 1,2,3,5 in <ref>. Thus,
a message of this subsection is that we do not need the
full power of available polynomial-time tools. That leads to speculation
that there are more general problems to solve.
Let us consider the steps of the algorithm in <ref>.
Since there are more interesting things to say about Step 2, we will postpone that discussion and
dispense with the other steps.
Step 1:
Since we always arrive at with $G/H$ simple,
either $\,\CH=G\,$ or $\,\CH= H$. In either case, $\calA$ already normalizes $\CH/H$ so there is nothing
more to do.
Step 3:
This is a group intersection. So it would be in polynomial time if just one group is
<cit.> and, as indicated,
just a polynomial-time set-stabilizer if both groups are , as is the case here.
However, in this situation,
none of the machinery is even needed because
$\calA$ is listable. Note also that this is trivial when $\,\CH = G\,$ (which means $\calM = \calA\times\calB$).
Step 5:
As indicated, this is a set-stabilizer for a group that we now know to be .
Now for Step 2:
Three approaches will be indicated, winding up with just set-stabilizers.
We state these for a
broader class than groups.
For an integer constant $d > 0$,
let $\,\Gamma_d\,$ denote the class of finite
groups all of whose nonabelian composition factors lie in
$\,S_d$. Also, bear in mind that the polynomial timing
for $\,\Gamma_d\,$ groups immediately extends to situations
where we are in possession of a $\,\Gamma_d\,$
subgroup of polynomial index (see <ref>.)
In particular, these methods apply to
Problem (I)
Given: $X,Y\le \Sym(\Omega)$ with $X\in \Gamma_d$.
$X_Y$. (Where $X$ is acting on $\Sym(\Omega)$ via conjugation.)
This was shown to be in polynomial-time by Luks and Miyazaki <cit.>.
[While $\Gamma_d$ is slightly smaller
than the class available just for set-stabilization <cit.>, the restriction
is still needed for this method.]
In our situation, we are trying to normalize a polynomial-size $Y$ and there
is a more elementary approach to this special situation.
Problem (II)
Given: $X,Y\le \Sym(\Omega)$ with $X\in \Gamma_d$ and
$|Y|= {\rm O}(|\Omega^{\rm const}|)$.
Let $\Sym(\Omega)$ act on
$\Omega\times\Omega$ diagonally. Also, for $s\in\Sym(\Omega)$, let
$$\Delta_s \,:=\, \{(\omega,\omega^s) \mid \omega\in\Omega\}.
Then for $y,x\in\Sym(\Omega)$,
$$\Delta_y^x\,=\, \Delta_{x^{-1} y x}.
Thus, $x$ normalizes $Y$ iff $x$ stabilizes the collection
$\{\Delta_y\}_{y\in Y}$. So, finding $X_Y$ is a matter of finding the
subgroup of $X$ inducing automorphisms of a hypergraph. Miller <cit.>
has shown that to be in polynomial time for $X\in \Gamma_d$.
That's fine but if we want to avoid hypergraphs when we get to the canonization analogue,[
We do know that hypergraph canonization can be done. However, our version has not been
published and Miller gave no details of his own.]
it is worth showing that Step 2 requires no more than set-stabilizer.
There is just one place where this requires more scrutiny. This occurs in the
“bottom-up” approach in Oct3Notes (3.2).[
The “top-down” approach has another alternative.] Consider then a still
more special situation.
Our situation is even more special.
Problem (III)
Given: $X,Y\le \Sym(\Omega)$ with $X\in \Gamma_d$ and
$|Y|= {\rm O}(|\Omega|^{\rm const})$;
$K< Y$ with $K\normal \langle X,Y\rangle$ and
we are able to list $\Aut(Y/K)$.
Note that $X_Y=X_{Y/K}$.
For each $\sigma\in\Aut(Y/K)$, we find those $x\in X$ such that conjugation
by $x$ induces $\sigma$. This will be case iff
$\forall y\in Y: x^{-1}y x \,\in \, (yK)^\sigma $
which is the case iff
$\forall y\in Y, \exists z \in(yK)^\sigma: \Delta_y^x = \Delta_{z}$
(See Remark <ref> for viewing these
“set-transporters” as set-stabilizers.) It is feasible to run through all $y$ and then all $z$.
To apply the Problem (III) method in
our situation, $Y=\InnH{G}$, $K=\InnH{H}$.
Furthermore, we are only concerned with the case $\CH=H$, else
which is already
normalized by $\calB$. Thus $Y/K$ is simple.
So we can not only list
$\Aut(Y/K)$ but we can even shorten the process by checking that
$x$ conjugates correctly on just $y_1, y_2\in Y$, where $y_1K, y_2K$ generate
A further savings on the number of set-stabilizer calls can be realized by using
a quotient-group method of Kantor and Luks <cit.>.
Finding the $x\in X$ such that $(yK)^x = (yK)^\sigma$ can be accomplished
with a single set-stabilizer.
§.§ Top-down on a refinement of a characteristic series
Recall that a subgroup $H$ of a group $G$ is called characteristic if it is invariant
under $\Aut(G)$. A group with no proper characteristic subgroups is called
characteristically simple and is necessarily a product of isomorphic simple groups.
A characteristic series in $G$ is a chain
$G=K_0 \vartriangleright K_1 \vartriangleright K_2 \vartriangleright \cdots \vartriangleright K_r =\1$
for which each $K_i$ is characteristic in $G$.
A characteristic series can be constructed
with characteristically simple quotients${K_i/K_{i+1}}\,$
even if $G$ is a permutation group given by generators; see, e.g., <cit.>. For groups
given by a Cayley table, the construction is elementary: find the minimal normal subgroups
of $G$ by considering the normal closures of all elements; these will each be
characteristically simple, so select those whose simple factors are of designated type
and let them generate the subgroup $K$. Continue the process with $G/K$, etc.
Without loss of generality, we may assume that the series $G_0,G_1,G_2,\ldots,G_m$
in has a characteristic subseries
$$G=K_0 \vartriangleright K_1 \vartriangleright K_2 \vartriangleright \cdots \vartriangleright K_r =\1$$
where each $K_i/K_{i+1}$ is characteristically simple.[
The Wagner-Rosenbaum composition series are already of the special type.]
We are given a composition series
$$G=G_0 \vartriangleright G_1 \vartriangleright G_2 \vartriangleright \cdots \vartriangleright G_m =\1.
As indicated above, we construct a characteristic series
$G=K_0 , K_1 , K_2 , \ldots, K_r =\1$
with characteristically simple quotients $K_i/K_{i+1}$.
Refine the series between each $K_i$ and $K_{i+1}$ by inserting
$$K_i =(K_i\cap G_0)K_{i+1} \trianglerighteq (K_i\cap G_1)K_{i+1}
\trianglerighteq (K_i\cap G_2)K_{i+1} \trianglerighteq \cdots \trianglerighteq (K_i\cap G_m)K_{i+1} =K_{i+1}$$
and eliminate duplicates.
We again have a composition series and
automorphisms stabilizing the original series will stabilize this new one. Having computed
the automorphisms stabilizing the new series, we have an group, so
cutting down the result to
stabilize the original series is done with set stabilizers.
Method for Theorem <ref>.
Having reset the series as in Lemma <ref>,
successively compute the appropriate subgroup of $\Aut(K_0/K_i)$, where $(K_i)_i$ is
the embedded normal series as in Lemma <ref>.
In the base case $K_0/K_0$ is trivial.
For the inductive step, the call to
AutLifting involves $K_0/K_{i+1}\,$ and its normal subgroup $K_i/K_{i+1}$.
We arrive with the inductive input $\calA \le \Aut(K_0/K_{i})$.
The group $\calB$ should consist of
the automorphisms of the semisimple group $H := K_i/K_{i+1}$ that fix
the composition
series induced on this section.
If $H$ is nonabelian, $\calB$ is the direct product of the
automorphism groups of the simple factors. If $H$ is a product
of cyclic groups of prime order $p$, $\calB$
can be viewed as the upper triangular matrices over ${\rm GF}(p)$.
Using the $\calL_2$ method at each stage,
the procedure is in polynomial time for all groups, but that seemed
to require hypergraph stabilizer in Step 2.
So let us revisit that step.
Suppose $H$ is nonabelian. Then $\calB$ (which now fixes
the simple factors) is of polynomial size, e.g. $|\calB|\le |H|^2$ in which
case Step 2 can be carried out
by testing each element of $\calB$. This is not the case if $H$ is abelian, but
then we simply revert to the
$\calL_1$ method of <ref> for this round.
§ ISOMORPHISM MATCHING FIXED COMPOSITION SERIES
We prove Theorem <ref> via
a familiar technique
in isomorphism studies, applying the automorphism-group result to finding isomorphisms.
For example, an algorithm for finding automorphism groups of graphs will find the isomorphisms
between two connected graphs $X_1,X_2$ by finding the automorphism group of the disjoint
union $X_1\dot\cup X_2$. An analogous construction works here.
We are given composition series
$$G_1=G_{1,0} \vartriangleright G_{1,1} \vartriangleright G_{1,2} \vartriangleright \cdots \vartriangleright G_{1,m} =\1$$
$$G_2=G_{2,0} \vartriangleright G_{2,1} \vartriangleright G_{2,2} \vartriangleright \cdots \vartriangleright G_{2,m}=\1$$
Form the single subnormal series
$$G_1\!\times\!G_2=G_{1,0}\!\times\! G_{2,0}\vartriangleright G_{1,1}\!\times\! G_{2,1} \vartriangleright
G_{1,2} \!\times\! G_{2,2} \vartriangleright \cdots \vartriangleright G_{1,m}\!\times\! G_{2,m} =\1\times\1.$$
We directly accommodate the setup of <ref> to this situation. (The
method of <ref> could be adapted as well.)
We now want the automorphisms of $G_1\!\times\!G_2$ that not only fix the terms in this series but, in doing so,
fix or switch the factors. If, for any $i$, we find that there are no relevant automorphisms of
$G_{1,i}\!\times\! G_{2,i}$ that switch factors, we exit with a negative
response to .
Calls to will now
have $G/H$ as the product of two isomorphic simple groups. For $\calA$ we take the automorphisms
that fix or switch the factors. (That would always be the case if the simple groups are nonabelian.)
The rest of the discussion, including the various methods for Step 2, proceed as before.
A trivial technicality.
Our weak version
of Lemma <ref> does not quite say
$\Aut({T\times{}T})\allowbreak\le\Sym (T\times T)$ is for
simple nonabelian $T$ since $|\Aut(T\times T)| = 2 |\Aut(T)|^2$.
As already noted, Lemma <ref> could be strengthened,
but it is clear that the theory is unaffected by an extra factor of 2.
The author is delighted to acknowledge communications with Laci Babai and
James Wilson that stimulated the development of these ideas
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|
1511.00428
|
First]Sneha Gajbhiye
First]Ravi N. Banavar
[First]Systems and Control Engineering, Indian Institute of Technology Bombay,
India, 400076. (e-mail: sneha@sc.iitb.ac.in, banavar@iitb.ac.in)
This paper presents tracking control laws for two different objectives of a nonholonomic system - a spherical robot - using a geometric approach. The first control law addresses orientation tracking using a modified trace potential function. The second law addresses contact position tracking using a $right$ transport map for the angular velocity error. A special case of this is position and reduced orientation stabilization. Both control laws are coordinate free. The performance of the feedback control laws are demonstrated through simulations.
Differential geometry, feedback stabilization, spherical robot.
§ INTRODUCTION
The problem of tracking of nonholonomic systems is a challenging one in control theory. Applications include robotics, rolling and locomotive mechanisms. A better understanding of the system's intrinsic properties simplify the control synthesis. Geometric control theory plays an important role in accomplishing such design strategies, see <cit.>, <cit.>, <cit.>. In this paper we study the tracking problem of one such nonholonomic system - a spherical robot.
A spherical mobile robot is a spherical shell actuated by a driving mechanism mounted inside to make the shell roll. In this paper we consider the driving actuators as three rotors. So the robot has three input degrees of freedom (rotors) which are used to control two translation and three rotational degrees of freedom (shell). Several modeling approaches and motion planning algorithms have been proposed for the spherical robot to achieve desired orientation and position,
see <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, that are solely based on coordinate dependent approach like quaternions and Euler parametrizations. Geometric control addressed the development of control laws for systems evolving on manifolds in a coordinate free setting. Recently, <cit.> has derived the dynamic model of the Chaplygin's sphere using geometric mechanics and presented orientation stabilization of a Chaplygin's sphere with a rotor by the controlled Lagrangian matching condition. In <cit.>, <cit.>, the authors address the control methods based on quaternions and stereographic projection respectively. In <cit.>, the authors applied backstepping to achieve position stabilization and tracking by expressing attitude in quaternion representation. In <cit.>, the authors propose motion planning algorithms using symmetric products on manifold (Lie group) to achieve position convergence with arbitrary orientation and vice versa. As spherical robot is a nonholonomic system, it fails to satisfy a necessary condition for asymptotic stabilization on $SO(3) \times \mathbb{R}^{2}$ by a continuous feedback law, see Brockett <cit.>. Due to this negative result, point-to-point stabilization of position and orientation of spherical robot through continuous state-feedback is not possible. In <cit.>, the authors consider stereographic projection map and design smooth kinematic control law to achieve position tracking and position with reduced attitude stabilization.
The contribution of this paper is to present two geometric control laws to achieve two different objectives: 1) tracking of a desired orientation trajectory; 2) contact position trajectory tracking asymptotically. The intermediate result of contact position tracking law is position and reduced attitude stabilization. The use of the transport map for velocity error on $SO(3)$ gives a better and complete understanding of the nonholonomic constraint in case of position tracking. For orientation tracking, a potential function which is the trace of the relative orientation and desired orientation, is constructed. The stability result is derived using Lyapunov direct method <cit.> which is recently restated in <cit.> to achieve asymptotic stability. While the notion of transport map in velocity error has been considered (see, <cit.> for tracking of fully actuated and <cit.> for underactuated systems), this is the first instance where such a treatment is considered in the presence of a nonholonomic constraint and with underactuation.
The model is derived using Lagrangian reduction defined on a symmetry group. The well developed theory on geometric nonholonomic mechanics is presented in <cit.>, <cit.>, <cit.>, <cit.>, <cit.>. By symmetry we can study the dynamics of a mechanical system on a reduced space and the reduced equations are in the Euler-Poincaré form. Due to nonholonomic constraints the system may or may not have full symmetry as in the case of the rigid body with gravitational field, for example, a heavy top; and the Euler-Poincaré equation will depend on an advection term <cit.>. In this paper we follow this modelling tool and derive the reduced equations of motion. The paper is organized as follows: In Section 2 we present the description and modelling of the spherical robot using Lagrangian reduction theory. In section 3 we formulate the control problem for orientation tracking and then position tracking and axis stabilization. We identify this stabilization as position and axis stabilization. Section 4 follows with the concluding remarks on the above control strategies.
§ DESCRIPTION OF A SPHERICAL ROBOT
Consider a spherical mobile robot with internal rotors which can roll without slipping on a flat surface under a uniform gravitational field. All the three rotors are placed along three mutually orthogonal axes of the sphere-body frame, as shown in Fig. (<ref>). To balance the mass symmetrically, the rotor is placed on one side and a dead weight is placed on the diametrically opposite side. All the rotors and dead weights are placed such that the center of mass of the robot coincides with the geometric center of the sphere.
Spherical robot on horizontal plane
Let the sphere body coordinate frame be located with its origin at the center of the sphere. Let $\mathbf{x} \in R^{3}$ be the position of the center of the sphere in an inertial frame, and let $R_{s}\in SO(3)$ be the rotation matrix which maps from the sphere body coordinate frame to the inertial coordinate frame. The relative motion of three rotors with respect to the sphere body frame is given by generalized shape coordinates $\theta_{i} \in S^{1}$, where $i = 1,2,3$. Hence, the configuration space is $Q={\mathbb R}^{2} \times SO(3)\times Q_{s}$, where $Q_{s} = S^{1} \times S^{1} \times S^{1}$. The following notation is adopted here:
* $(\hat{e}_{1},\hat{e}_{2},\hat{e}_{3})$ - Unit vectors in inertial frame,
* $\omega_{s}^{I}$, $\omega_{s}^{s}$ - Angular velocity of the sphere in inertial frame and sphere frame respectively; $\dot{\theta}_{i}$ - Angular velocity of the $i^{th}$ rotor,
* $m_{s}, m_{i}$ - Mass of the sphere and $i^{th}$ rotors; $\mathbb{I}^{s} = diag(I_{s},I_{s},I_{s})$ - Inertia matrix of the sphere without rotors about its center of mass in sphere frame and $J^{i} $ - Moment of inertia of the rotors about the three principal axes, i.e. $J^{1}=diag(J_{a},J_{b},J_{b});J^{2}=diag(J_{b},J_{a},J_{b});J^{3}=diag(J_{b},J_{b},J_{a})$ with $J_{a}=2J_{b}$.
The Lagrangian of the system consists only of kinetic energy and is given as
\begin{align*}
L = \frac{1}{2} \left( m_{T} \|\dot{\mathbf{x}}\|^{2} + \sum_{i=1}^{3} (I_{i} \omega_{i}^{2} + J_{a}(\omega_{i} + \dot{\theta}_{i})^{2}) \right)
\end{align*}
where $\omega_{s}^{s} \triangleq [\omega_{1},\omega_{2},\omega_{3}]$, $m_{T}= (m_{s} + \sum_{i = 1}^{3} m_{i})$ and $I_{i} = I_{s} + 2 J_{b}$. The Lagrangian is now expressed as
\begin{equation}\label{lagrange}
L = \frac{1}{2} m_{T} \|\dot{\mathbf{x}}\|^{2} + \frac{1}{2} \omega_{s}^{T}I^{s} \omega_{s}^{s} + \frac{1}{2}\left( \dot{\Theta} + \omega_{s}^{s}\right)^{T}\mathbb{J}\left(\dot{\Theta} + \omega_{s}^{s}\right)
\end{equation}
where $\mathbb{J} = diag(J_{a},J_{a},J_{a})$, $I^{s}=dia(I_{1},I_{2},I_{3})$, $\&$ $\dot{\Theta}=(\dot{\theta}_{1},\dot{\theta}_{2},\dot{\theta}_{3})$. The rolling without slipping assumption on the robot yields a nonholonomic constraint given as
\begin{equation}\label{constr1}
\dot{\mathbf{x}}- \omega_{s}^{I}\times r\hat{e}_{3} =0.
\end{equation}
§.§ Dynamics of the spherical robot
The configuration space $Q$ is a smooth manifold, $TQ$ is the velocity space called the tangent bundle and a smooth distribution $\mathcal{D} \subset TQ$ defines the constraints, the set of admissible velocities.
With the Lagrangian $L$ defined in (<ref>) and distribution $\mathcal{D}\subset TQ$ satisfying the constraint (<ref>), let $G =SO(3) \times \mathbb{R}^{3}$ be a group with its Lie algebra $\mathfrak{g} = \mathfrak{so}(3) \times \mathbb{R}^{3}$, where $\mathfrak{so}(3)$ is the Lie algebra of $SO(3)$. The group action of $G$ on $Q$ is given by $\Phi: G \times Q \mapsto Q$; $\Phi_{(\bar{R}_{s},\bar{b})}(R_{s},b,\Theta) = (\bar{R}_{s}R_{s},\bar{R}_{s}b + \bar{b}, \Theta)$. It is seen that the Lagrangian $L$ and distribution $\mathcal{D}$ is invariant with respect to the subgroup $G_{\hat{e}_{3}}$ of $G$ given as
$ G_{\hat{e}_{3}}=\lbrace (R_{s},b)\in G \vert R_{s}^{T} \hat{e}_{3}=\hat{e}_{3} \rbrace = SO(2)\times \mathbb{R}^{2}$. When the Lagrangian $L$ and the distribution $\mathcal{D}$ are invariant under the action of the subgroup group $G_{\hat{e}_{3}}$, the system is reduced to the space $TQ/G_{\hat{e}_{3}}$ and the Lagrangian is termed as the reduced Lagrangian $l$. Define
\begin{equation}\label{advection}
\bar{Y}= R_{s}^{T} \dot{\mathbf{x}} \quad \quad \Gamma\triangleq R_{s}^{T}\hat{e}_{3},
\end{equation}
where $\bar{Y}$ is the velocity of the contact point in the sphere frame and $\Gamma$ is called an advected variable <cit.>. The reduced Lagrangian $l: TQ/G_{\hat{e}_{3}} \longrightarrow \mathbb{R} $
\begin{equation}\label{equ2}
l = \frac{1}{2} m_{T} \|\bar{Y}\|^{2} + \frac{1}{2} \omega_{s}^{s}\cdot (I^{s} + \mathbb{J}) \omega_{s}^{s} + \frac{1}{2}\left( \dot{\Theta} \cdot \mathbb{J}\dot{\Theta} + 2 \omega_{s}^{s} \cdot \mathbb{J}\dot{\Theta}\right)
\end{equation}
and the rolling constraint is now expressed in the sphere body coordinate frame as $\bar{Y} = r \widehat{\omega}_{s}^{s}\Gamma$,
where $\bar{Y}$, $\Gamma$ $\in \mathbb{R}^{3}$ and $\widehat{\omega}_{s}^{s} = R_{s}^{T}\dot{R}_{s} \in \mathfrak{so}(3)$ is the (left-invariant) sphere-body angular velocity. Substituting $\bar{Y}$ in $l$, the system is reduced to the quotient space $\mathcal{D}/G_{\hat{e}_{3}}$ given by the reduced-constraint Lagrangian $l_{c}$ as
$$l_{c} = + \frac{1}{2} \omega_{s}^{s}\cdot (- m_{T}r^{2}\widehat{\Gamma}\widehat{\Gamma} + I^{s} + \mathbb{J}) \omega_{s}^{s} + \frac{1}{2}\left( \dot{\Theta} \cdot \mathbb{J}\dot{\Theta} + 2 \omega_{s}^{s}\cdot \mathbb{J}\dot{\Theta}\right)$$
Due to subgroup symmetry, there is an advection dynamic and differentiating (<ref>) it is calculated as
\begin{equation}\label{final_adv}
\dot{\Gamma} = - \omega_{s}^{s} \times \Gamma.
\end{equation}
The dynamics is calculated using the intermediate theorem given by <cit.>. The equations of motion is given by the Euler-Poincaré equation for the group variable $R_{s}$ and the Euler-Lagrange equation for the shape variable $\Theta$.
Let $\Pi_{s} = \frac{\partial l_{c}}{\partial \omega_{s}^{s}} $ be the angular momentum of the sphere(momentum conjugate to $\omega_{s}^{s}$) and $\Pi_{i}$ be the angular momentum of the $i^{th}$ rotor,the dynamics is given as,
\begin{align}\label{equ3}
\dot{\Pi}_{s} = \Pi_{s} \times \omega_{s}^{s}; \quad \quad \dot{\Pi}_{i} = u,
\end{align}
Recasting the dynamic equation (<ref>) as,
\begin{align}\label{reduced_dyanamic}
\begin{split}
& M(\Gamma) \dot{\omega}_{s}^{s} = (I^{s} \omega_{s}^{s} + \mathbb{J}\dot{\Theta}) \times \omega_{s}^{s} - u,\\
& \mathbb{J}\dot{\omega}_{s}^{s} + \mathbb{J}\ddot{\Theta} = u.
\end{split}
\end{align}
where $M(\Gamma)= I^{s} - m_{T}r\widehat{\Gamma}^{T}\widehat{\Gamma}$ and using the solution $\omega_{s}^{s}$ of the equation (<ref>), we can find the curve $R_{s}(t)$ by solving the reconstruction equation
\begin{equation}\label{rec}
\dot{R}_{s}(t) = R_{s}(t)\widehat{\omega}_{s}^{s} \text{ with }~~R_{s}(0)=R_{s_{0}}.
\end{equation}
Hence, equations (<ref>), (<ref>) and (<ref>), together with the reconstruction equation (<ref>), give the complete dynamics of the spherical robot. If $u^{i}=0$, it is easily seen that any configuration is an equilibrium and hence the equilibrium manifold is the whole configuration manifold $Q$. By expressing the control in terms of the gradient of a potential function (or error function), the equilibrium can be changed to any desired point. A similar procedure is followed in the next two sections to achieve tracking.
§.§ Remarks on controllability
The controllability for the three rotor case has been analysed by <cit.>, <cit.> in the literature. One can use fiber configuration controllability definition to check the controllability. This controllability has been studied for the Chaplygin's sphere with rotors in <cit.>, <cit.>. We will mention this result and then design stabilization/tracking control laws for our system. To check the controllability, equation (<ref>) is cast in an affine-control form as
\begin{equation}\nonumber
\dot{q} = f(q) + g(q)u
\end{equation}
where, $q = (\omega_{s}^{s}, \dot{\Theta})$, the drift vector field $f =0$ and the control vector fields $ g = [g_{1}~g_{2}~g_{3}]$, are expressed as
\begin{equation}
g_{i} = \begin{bmatrix}
- (I^{s} + \mathbb{J} - m_{T}r\widehat{\Gamma}^{T}\widehat{\Gamma})^{-1} \mathbb{J} \Delta^{-1}\\
\Delta^{-1}
\end{bmatrix} \hat{e}_{i}
\end{equation}
where $\Delta = \mathbb{J} - \mathbb{J}(I^{s} + \mathbb{J} - m_{T}r\widehat{\Gamma}^{T}\widehat{\Gamma})^{-1} \mathbb{J}$. If $v = \Delta~u$ is an equivalent control input, where $v = (v_{1},v_{2},v_{3})$ is a transformed control input; then the control vector fields on $SO(3) \times Q_{s}$ are written as
\begin{equation}
g_{i} \simeq \begin{bmatrix}
- A_{i}(\Gamma) \\
\hat{e}_{i}
\end{bmatrix}
\end{equation}
where $A_{i}$ is the $i^{th}$ column of $A = (I^{s} - m_{T}r\widehat{\Gamma}^{T}\widehat{\Gamma})^{-1} \mathbb{J}$. Let $\pi_{s} = R_{s} \Pi_{s}$, then from (<ref>), which is the Euler-Poincaré form, we see
\begin{equation}\label{conserve1}
\frac{d}{dt}\left( R_{s} \Pi_{s} \right) = 0.
\end{equation}
that is, the inertial momentum $\pi_{s}$ is conserved. Suppose that the system is initially at equilibrium, then $\Pi_{s} = 0$ and therefore $\omega_{s} = - (I^{s} + \mathbb{J} - m_{T} r \widehat{\Gamma}\widehat{\Gamma})^{-1}\mathbb{J}\dot{\Theta} = - A(\Gamma) \dot{\Theta}$, where $A(\Gamma)$ is called as mechanical connection <cit.>. From (<ref>), $\bar{Y} = r A(\Gamma)\dot{\Theta} \times \Gamma$, then the control vector fields for the complete configuration $SO(3) \times Q_{s} \times \mathbb{R}^{2}$ are expressed as
\begin{equation}
\bar{g}_{i} \simeq \begin{bmatrix}
- A(\Gamma)\hat{e}_{i} \\
\hat{e}_{i} \\
r A(\Gamma)\hat{e}_{i} \times \Gamma
\end{bmatrix}
\end{equation}
The iterative Lie brackets $\{\bar{g}_{1}, \bar{g}_{2}, \bar{g}_{3}, [\bar{g}_{1},\bar{g}_{2}], [\bar{g}_{1},\bar{g}_{3}] \}$ span the tangent space of $SO(3) \times \mathbb{R}^{2}$ (termed as the fiber configuration) at any configuration. Hence, the system is fiber configuration accessible at any configuration and therefore fiber configuration controllable.
§ ORIENTATION TRAJECTORY TRACKING
The control objective here is to design a feedback control law which tracks a desired orientation trajectory $R_{d}(t)$. The rotational system dynamics described by (<ref>) and (<ref>) can be expressed in the standard control form with $q = (R_{s},\omega_{s}^{s})$ as
\begin{equation}\label{control_equation2}
\dot{q} = f(q) + g(q)u
\end{equation}
\begin{align*}
f = \begin{bmatrix}
R_{s}\widehat{\omega}_{s}^{s} \\
M^{-1}\left( (I^{s}\omega_{s}^{s} + \mathbb{J}\dot{\Theta}) \times \omega_{s}^{s}\right)
\end{bmatrix}; \quad
g = [g_{1}~ g_{2}~ g_{3}] = \begin{bmatrix}
\mathbf{0} \\
\end{bmatrix}.
\end{align*}
where for notational simplification we write $M(\Gamma)=M$ and $b_{i}'s$ are the columns of $M^{-1}$.
We now define a scalar valued potential function to achieve this objective and then prove the stability of the system. Subsequently, we add a damping term to get asymptotic convergence to the equilibrium. Let $V: Q \longrightarrow \mathbb{R}$ be an error function about $R_{d}(t)\in SO(3)$ constructed by a modified trace function as
\begin{equation}
V(R_{s}) = \mbox{trace}(K_{p}(I_{3\times 3} - R_{d}^{T}R_{s})).
\end{equation}
where $K_{p} = diag(\lambda_{1},\lambda_{2},\lambda_{3})$ with $\lambda_{1},\lambda_{2},\lambda_{3}>0$ and $\lambda_{1} \neq \lambda_{2} \neq \lambda_{3}$. The modified trace function was first employ by <cit.> for the purpose of feedback stabilization. Define $\widehat{\omega}_{d}^{s} = R_{d}^{T}\dot{R}_{d}$ and set $R_{e} = R_{d}^{T}R_{s}$ and the error in angular velocity as $e_{\omega} \triangleq \omega_{s}^{s} - R_{e}^{T}\omega_{d}^{s}$. Taking time derivative of $V$,
\begin{align}\nonumber
\begin{split}
& \frac{d V}{dt} =\mbox{trace}( K_{p}(R_{d}^{T}\dot{R}_{d}R_{d}^{T}R_{s} - R_{d}^{T} \dot{R}_{s})) \\
& = \mbox{trace}( K_{p} R_{e}(R_{e}^{T} \widehat{\omega}_{d}^{s}R_{e} - \widehat{\omega}_{s}^{s})) = - \mbox{trace}(k_{p}R_{e}\widehat{e}_{\omega})\\
& = - \frac{1}{2} \mbox{trace}([\mbox{skew}(K_{p}R_{d}^{T}R_{s}) + \mbox{sym}(K_{p}R_{d}^{T}R_{s})] \widehat{e}_{\omega})\\
& = - \frac{1}{2} \mbox{trace}(\mbox{skew}(K_{p}R_{d}^{T}R_{s}) (\widehat{e}_{\omega}))
\end{split}
\end{align}
and from the equality $\mbox{trace}(\widehat{x}\widehat{y}) = - 2 x \cdot y $, where $\widehat{\cdot} : \mathbb{R}^{3} \longrightarrow \mathfrak{so}(3)$ is a hat map and $(\cdot)^{\vee} : \mathfrak{so}(3)\longrightarrow \mathbb{R}^{3}$ is a breve map (inverse of hat map), it follows that
\begin{align}\nonumber
\begin{split}
& \dot{V} = \mbox{skew}(K_{p}R_{d}^{T}R_{s})^{\vee} \cdot e_{\omega} = (\sum_{i = 1}^{3} \lambda_{i} R_{s}^{T}\hat{e}_{i} \times R_{d}^{T}\hat{e}_{i})\cdot e_{\omega} = dV \cdot e_{\omega},
\end{split}
\end{align}
where $dV$ can also be termed as differential of $V$ with respect to $R_{s}$.
We compute the feedforward (FF) control term which tracks the desired velocity and add the proportional-derivative (PD) term to stabilize/track the orientation asymptotically. The velocity error has geometric interpretation since $\widehat{\omega}$ is Lie algebraic element. Since $\dot{R}_{s}$ and $\dot{R}_{d}$ are the two velocities taking values in different tangent spaces, to define the error velocity we need to compare tangent vectors in the same tangent space. This can be achieved by the transport map $\tau$ as defined in [<cit.>, 11]. If $\dot{R}_{s} \in T_{R_{s}}SO(3)$ and $\dot{R}_{d} \in T_{R_{d}}SO(3)$ are the two vectors at the points $R_{s}$ and $R_{d}$ respectively, then a right transport map $\tau(R_{s},R_{d})$ transforms $\dot{R}_{d}$ into a vector at $T_{R_{s}}SO(3)$ and the error is expressed as
\begin{align}\nonumber
\begin{split}
& \dot{R}_{s} - \tau(R_{s},R_{d}) (\dot{R}_{d}) = \dot{R}_{s} - \dot{R}_{d}(R_{d}^{T}R_{s}),\\
& = R_{s}R_{s}^{T} \dot{R}_{s} - (R_{s}R_{s}^{T})(R_{d}R_{d}^{T})\dot{R}_{d}(R_{d}^{T}R_{s}),\\
& = R_{s}\widehat{\omega}_{s}^{s} - R_{s}(R_{s}^{T}R_{d})\widehat{\omega}_{d}^{s}(R_{d}^{T}R_{s})= R_{s}[\widehat{\omega}_{s}^{s} - (R_{e}^{T}\omega_{d}^{s})^{\wedge}] = R_{s}\widehat{e}_{\omega}
\end{split}
\end{align}
Now, the feedforward control term is calculated by taking the covariant derivative of the transport map along $\omega_{s}^{s}$. Associated with a Riemannian manifold is the notion of the affine connection $\nabla$ that defines the covariant derivative. For details on Riemannain manifolds and affine differential geometry one can refer to <cit.>, <cit.>. For a given affine connection $\nabla$, two vector fields $X_{\xi},X_{\eta}$ with $\xi, \eta \in \mathfrak{g}$ its Lie algebra, the covariant derivative is defined as $\nabla_{X_{\xi}}X_{\eta}$. If $X_{\xi},X_{\eta}$ are left-invariant vector fields on $Q$, then the covariant derivative is $\nabla_{X_{\xi}}X_{\eta} = \frac{d}{dt}X_{\eta} + \overset{\mathfrak{g}}{\nabla}_{\xi}\eta$,
where $\overset{\mathfrak{g}}{\nabla}_{\xi}\eta$ is a bilinear map defined as
\begin{equation}\label{covariant}
\overset{\mathfrak{g}}{\nabla}_{\xi}\eta = M^{-1}\left( \frac{1}{2}(\xi \times M\eta) + \frac{1}{2}(\eta \times M\xi) \right).
\end{equation}
In our case $X_{\xi} = \dot{R}_{s}$, $X_{\eta} = \tau(R_{s},R_{d})\dot{R}_{d}$ and $\mathfrak{g} = \mathfrak{so}(3)$ with $\xi=\widehat{\omega}_{s}^{s}$ and $\eta = (R_{d}^{T}\omega_{d}^{s})^{\wedge}$. With this the $f_{FF}$ is calculated as $f_{FF} = M \left( \nabla_{\dot{R}_{s}}\tau(R_{s},R_{d})\dot{R}_{d} \right), $
\begin{align}
& = M \left( \frac{d R_{e}^{T}}{dt} \omega_{d}^{s} + \overset{\mathfrak{so}(3)}{\nabla}_{\widehat{\omega}_{s}^{s}}(R_{e}^{T}\omega_{d}^{s})^{\wedge} \right), \nonumber \\
& = M \left( \frac{d R_{e}^{T}}{dt}\omega_{d}^{s} + R_{e}^{T} \frac{d \omega_{d}^{s}}{dt} + \overset{\mathfrak{so}(3)}{\nabla}_{\widehat{\omega}_{s}^{s}}(R_{e}^{T}\omega_{d}^{s})^{\wedge} \right), \nonumber \\
& = M \left( (\omega_{s}^{s} \times R_{e}^{T}\omega_{d}^{s}) + R_{e}^{T}\dot{\omega}_{d}^{s} +\overset{\mathfrak{so}(3)}{\nabla}_{\widehat{\omega}_{s}^{s}}(R_{e}^{T}\omega_{d}^{s})^{\wedge} \right).\nonumber
\end{align}
From (<ref>) calculating the bilinear map and therefore
\begin{align}
f_{FF} &= M \left(\frac{1}{2} M^{-1}(\omega_{s}^{s} \times M R_{e}^{T}\omega_{d}^{s}) - \frac{1}{2} M^{-1}(M\omega_{s}^{s} \times R_{e}^{T}\omega_{d}^{s})\right) \nonumber \\
& \quad + M (\omega_{s}^{s} \times R_{e}^{T}\omega_{d}^{s}) + M R_{e}^{T}\dot{\omega}_{d}^{s}. \label{feedfrwd}
\end{align}
Under the feedback torque $u(R_{s}) = dV(R_{s}) - f_{FF}$ the closed loop system (<ref>) is Lyapunov stable about $(R_{d}, \omega_{d}^{s})$.
Proof: Define the function $H : TQ \longrightarrow \mathbb{R}$
\begin{equation}\label{lyapunov_function}
H(R_{s},\omega_{s}^{s}) = V(R_{e}) + \frac{1}{2}\| e_{\omega}\|_{M}^{2} = V(R_{e}) + \frac{1}{2}\mathbb{G}(I)(e_{\omega},e_{\omega}),
\end{equation}
where $\mathbb{G}(I) = M$ is the Riemanian metric on $Q$. Since, $V$ is an error function and $M>0$, it follows that the function $H$ is locally positive definite around $(R_{d}, \omega_{d}^{s})$. It follows
\begin{align}
& \frac{d}{dt}H(\mathbf{x},\omega_{s}^{s}) = \frac{d}{dt}V + \mathbb{G}(I)(e_{\omega},\nabla_{\omega_{s}^{s}}e_{\omega}), \nonumber \\
& = \dot{V} + \mathbb{G}(I)(e_{\omega},\nabla_{\omega_{s}^{s}}\omega_{s}^{s} - \nabla_{\omega_{s}^{s}}R_{e}^{T} \omega_{d}^{s}), \nonumber \\
& = \dot{V} + \langle e_{\omega}, M\left( \frac{d}{dt}\omega_{s}^{s} + \overset{\mathfrak{so}(3)}{\nabla}_{\widehat{\omega}_{s}^{s}}\widehat{\omega}_{s}^{s} \right) \rangle - \langle e_{\omega}, M \nabla_{\omega_{s}^{s}}R_{e}^{T} \omega_{d}^{s} \rangle, \nonumber \\
& = \dot{V} + \langle e_{\omega}, M\dot{\omega}_{s}^{s} + ( \omega_{s}^{s} \times (I^{s}\omega_{s}^{s} + \mathbb{J} \dot{\Theta})) - M \nabla_{\omega_{s}^{s}}R_{e}^{T} \omega_{d}^{s} \rangle, \nonumber \\
& = \dot{V} + \langle e_{\omega}, - u \rangle - \langle e_{\omega}, M \nabla_{\omega_{s}^{s}}R_{e}^{T} \omega_{d}^{s} \rangle, \nonumber \\
& = dV \cdot e_{\omega} + \langle e_{\omega}, -dV + f_{FF} \rangle - \langle e_{\omega}, f_{FF}\rangle = 0. \label{lyap:2}
\end{align}
Thus, $H$ is a Lyapunov function about $(R_{d}, \omega_{d}^{s})$ and therefore $(R_{d},\omega_{d}^{s})$ is stable in the sense of Lyapunov for system (<ref>). $\blacksquare$
The next step is to introduce damping or the dissipative term $u_{diss}$ to achieve asymptotic stability. Introducing damping to the control by defining $u = (dV + f_{FF}) + u_{diss}$
where $u_{diss} = [u^{1}_{diss}~ u^{2}_{diss}~ u^{3}_{diss}]^{T}$. Then the closed loop control system becomes
\begin{equation}\label{control_equation3}
\dot{q} = F_{cl}(q) + g(q)u_{diss}
\end{equation}
where $g = [g_{1}~g_{2}~g_{3}]$ and
\begin{align*}
F_{cl} & = \begin{bmatrix}
R_{s}\widehat{\omega}_{s}^{s} \\
M^{-1}\left( I^{s}\omega_{s}^{s} \times \omega_{s}^{s} + \mathbb{J}\dot{\Theta} \times \omega_{s}^{s} + (dV + f_{FF})\right)
\end{bmatrix},~ g_{i} = \begin{bmatrix}
0 \\
\end{bmatrix}.
\end{align*}
Lemma 1: The control system (<ref>) is locally controllable on $SO(3) \times \mathbb{R}^{3}$.
Proof: The proof is given in Appendix. $\blacksquare$
We now prove asymptotic stability of our system about the desired equilibrium. To prove this we use the stability result stated in [<cit.>,theorem 1].
Consider the system (<ref>) with input torque $u_{diss}$. Let $H$ be described in (<ref>). If $\mathcal{L}_{F_{cl}}H = 0$ and $u_{diss} = - \mathcal{L}_{g}H$ is the dissipative input, then the closed loop system asymptotically stabilize $(R_{d}, \omega_{d}^{s})$.
Proof: Consider the Lyapunov function $H$ as defined in (<ref>), Computing the rate of $H$ we get
\begin{align}\nonumber
\begin{split}
\frac{dH}{dt}& = \frac{\partial H}{\partial q} \dot{q} = \mathcal{L}_{F_{cl}}H + \mathcal{L}_{g}H u_{diss},
\end{split}
\end{align}
From (<ref>), we see that, $\mathcal{L}_{F_{cl}}H = 0$ which implies
\begin{equation}\label{asymptotic_H}
\dot{H} = \mathcal{L}_{g}H u_{diss}.
\end{equation}
Defining $u_{diss} = - \mathcal{L}_{g}H$ yield $\dot{H} = - (\mathcal{L}_{g}H)^{2}$. We know that $u_{diss}=\begin{bmatrix}
u_{diss}^{1} & u_{diss}^{2} & u_{diss}^{2}
\end{bmatrix}^{T}$ then calculating $u_{diss}^{i}$ as
\begin{align*}
u_{diss}^{i} = - \left(\frac{\partial H}{\partial q}\right)^{T}g_{i} = -\begin{bmatrix}
\left( \frac{\partial H}{\partial R_{s}}\right)^{T} & \left( \frac{\partial H}{\partial \omega_{s}^{s}}\right)^{T}
\end{bmatrix}
\begin{bmatrix}
\textbf{0} \\
\end{bmatrix} = M e_{\omega} \cdot b_{i}
\end{align*}
where $i = {1,2,3}$. From this the dissipative control is calculated as
\begin{align*}
u_{diss} & = - \begin{bmatrix}
\mathcal{L}_{g_{1}}H & \mathcal{L}_{g_{2}}H & \mathcal{L}_{g_{3}}H
\end{bmatrix}^{T} = M e_{\omega} \cdot (M^{-1})^{T}= K_{v} e_{\omega},
\end{align*}
where $M=M^{T}$ is a symmetric positive-definite matrix and $K_{v}$ is a positive constant. Substituting the value of $\mathcal{L}_{g}H$ in (<ref>) we get $\dot{H} = - K_{v} (e_{\omega})^{2} \leq 0.$
Since, system is locally controllable from Lemma 1 and $\dot{H}$ is negative semidefinite we conclude from the Theorem 1 of <cit.> that the point $(R_{d}, \omega_{d}^{s})$ is local asymptotically stable. $\blacksquare$
If $\omega_{d}^{s} = 0$ implies $f_{FF} =0$, then the system (<ref>) with control $u = dV + K_{v}\omega_{s}^{s}$ is local asymptotically stable about $(R_{d},0)$.
Infact the system is local exponential stable about
$(R_{d},0)$. To check exponential stability, we compute the second variation of $H$. If the second variation is positive definite about the equilibrium point we say that the equilibrium is exponentially stable. From (<ref>) $H = T + V$ where $T= (1/2)\omega_{s}^{s}\cdot M \omega_{s}^{s}$ and for $T$ being a kinetic energy, yields $\partial^{2}T(q) > 0~ \forall q$. The second variation of the error functions $V$ is calculated as follows; let $\widehat{\eta} = R_{s}^{T}\delta R_{s} \in \mathfrak{so}(3)$, then
\begin{align}
\delta V(R_{s}) & = \delta (\mbox{trace}( K_{p}( I_{3\times 3} - R_{s}R_{d}^{T})))=\mbox{trace}(\widehat{\eta} K_{p} R_{s}R_{d}^{T}), \nonumber \\
\delta^{2} V(R_{s}) &= \delta \mbox{trace}(\widehat{\eta} K_{p} R_{s}R_{d}^{T})= \mbox{trace}(- \widehat{\eta} K_{p} \delta R_{s}R_{d}^{T}) \nonumber \\
& = \langle \widehat{\eta}, K_{p}R_{s}R_{d}^{T} \widehat{\eta} \rangle \quad \quad \forall \widehat{\eta}\neq 0. \label{control_equation5}
\end{align}
At $R_{d}$ we have $\partial^{2}V>0$ and $\partial^{2}H(q_{0})= \partial^{2}T(q_{0}) + \partial^{2}V(q_{0}) > 0$, where $q_{0} = R_{d}$. Since, the second variation of $H$ is positive definite at equilibrium one can conclude the system achieves the desired orientation exponentially.
§.§.§ Simulation:
We choose the model parameters as: $m_{s} = 1kg;~ m_{i}= 0.672kg;~ r_s = 0.176m;~I^{s} = \hbox{diag}(0.0153,0.0153,$ $0.0153)kg-m^{2}$; $\mathbb{J}= \hbox{diag}(0.672,0.672,0.672)kg-cm^{2}$;
and control parameters as $K_{p} = \mbox{diag}(2,8,1)$ and $K_{v} = 0.5$. Choosing rotation matrix $R_{s}=\mbox{exp}(\alpha \hat{e}_{1})\mbox{exp}(\beta \hat{e}_{3})\mbox{exp}(\gamma \hat{e}_{1})$, the simulations are carried out for both attitude tracking and stabilization by three rotors with the following control law:
$$u = -(\sum_{i=1}^{3} \lambda_{i}R_{s}^{T}\hat{e}_{i} \times R_{d}^{T}\hat{e}_{i}) + K_{v}e_{\omega} + f_{FF}.$$
Keeping the desired orientation trajectory as $R_{d}(t) = exp(2 \pi (1 - \cos \pi t)\hat{e}_{2})$, then Fig. (<ref>(a)) and (<ref>(b)) shows the error in angular velocity ($e_{\omega}$) of the sphere and the error norm of $R_{s}$, indicating asymptotic convergence to the desired trajectory. The error norm is calculated as
$$E_{R} = (3 - \mbox{trace}(K_{p}R_{d}^{T}R_{s}))^{1/2}.$$
For stabilization, setting the desired orientation as $R_{d}=R_{x}(\pi/9)R_{y}(\pi/18)R_{z}(\pi/3)$ and initial angular velocity as $\omega_{s}^{s}(0) = (12.5,7,1)$.
Then Fig. (<ref>(a)) shows the torque applied at the internal rotors. The angular velocity of the three rotors is shown in Fig (<ref>(b)) converges to the initial momentum.
(a)Angular velocity of spherical robot. (b) Error norm of orientation
(a) Torque to the internal rotors. (b) Angular position of the rotors.
§ CONTACT POINT TRACKING AND AXIS STABILIZATION
In this section we derive a control law based on a configuration error function which ensures contact position tracking by tracking the angular velocity. The control objective is to design a control law which aligns $\omega_{s}^{s}$ to a desired angular velocity and stabilizes/tracks the contact position on the plane asymptotically. Suppose $\dot{\mathbf{x}}_{d} = R_{d}\omega_{d}^{s} \times r e_{3}$ is the desired contact point velocity, where $R_{d}$ is a desired orientation. Note that given the nonholonomic constraint, $\dot{\mathbf{x}}_{d}$ gets determined by $\omega_{d}^{I} = R_{d} \omega_{d}^{s}$ which eventually determine $\mathbf{x}_{d}$ . Let $V_{1}$ be a potential function given by $V_{1} = \frac{1}{2}\| \mathbf{x} - \mathbf{x}_{d} \|^{2}.$
Taking the time derivative of $V$ along the system's trajectory,
\begin{align}\label{track_eq1}
\begin{split}
\dot{V}_{1} & = (\mathbf{x} - \mathbf{x}_{d}) \cdot (\dot{\mathbf{x}} - \dot{\mathbf{x}}_{d}) \\
& = (\mathbf{x} - \mathbf{x}_{d}) \cdot [(R_{s}\omega_{s}^{s} \times r \hat{e}_{3}) - (R_{d}\omega_{d}^{s} \times r \hat{e}_{3})],\\
& = - r \hat{e}_{3} \times (\mathbf{x} - \mathbf{x}_{d}) \cdot R_{s}(\omega_{s}^{s} - R_{s}^{T}R_{d}\omega_{d}^{s}), \\
& = - r R_{s}^{T}[\hat{e}_{3} \times (\mathbf{x} - \mathbf{x}_{d})] \cdot (\omega_{s}^{s} - R_{s}^{T}R_{d}\omega_{d}^{s}),
\end{split}
\end{align}
Set $R_{e} = R_{d}^{T}R_{s}$ and define the error in angular velocity as $e_{\omega} \triangleq \omega_{s}^{s} - R_{e}^{T}\omega_{d}^{s}$. The proportional-derivative (PD) control term is given as $f_{PD} = k_{p}r R_{s}^{T}[\hat{e}_{3} \times (\mathbf{x} - \mathbf{x}_{d})] - k_{d}e_{\omega}$,
where $k_{p}$ and $k_{d}$ are positive definite matrices. With this PD control and (<ref>), the nonlinear controller is given as
\begin{equation}\label{control_track}
u = -(f_{PD} + f_{FF}).
\end{equation}
The system (<ref>) with control input (<ref>), given by
\begin{equation}\nonumber
\dot{\omega}_{s}^{s} = M^{-1}(I^{s} \omega_{s}^{s} + \mathbb{J}\dot{\Theta}) \times \omega_{s}^{s} + M^{-1}(f_{PD} + f_{FF}) ,
\end{equation}
is local asymptotically stable at $(\mathbf{x}_{d},\omega_{d}^{s})$.
Proof: Define a candidate error function
\begin{equation}
H(\mathbf{x},\omega_{s}^{s}) = V_{1} + \frac{1}{2}\| e_{\omega}\|_{M}^{2} = V_{1} + \frac{1}{2}\mathbb{G}(I)(e_{\omega},e_{\omega}),
\end{equation}
The time derivative of the Lyapunov function is
\begin{align}
& \frac{d}{dt}H(\mathbf{x},\omega_{s}^{s}) = \frac{d}{dt}V_{1} + \mathbb{G}(I)(e_{\omega},\nabla_{\omega_{s}^{s}}e_{\omega}), \nonumber \\
& = \dot{V}_{1} + \mathbb{G}(I)(e_{\omega},\nabla_{\omega_{s}^{s}}\omega_{s}^{s} - \nabla_{\omega_{s}^{s}}R_{e}^{T} \omega_{d}^{s}), \nonumber \\
& = - r R_{s}^{T} [\hat{e}_{3} \times (\mathbf{x} - \mathbf{x}_{d})]\cdot e_{\omega} + \langle e_{\omega}, f_{PD} + f_{FF} \rangle - \langle e_{\omega}, f_{FF}\rangle, \nonumber \\
& = - k_{d} e_{\omega}\cdot e_{\omega} \leq 0. \label{lyap:1}
\end{align}
Let $\Omega_{c} = \{ (\mathbf{x}, \omega_{s}^{s}) | H(\mathbf{x},\omega_{s}^{s}) \leq c \}$,where $c>0$ and since $\dot{H}(\mathbf{x},\omega_{s}^{s}) \leq 0$ all the trajectories are bounded and contained within $\Omega_{c}$. Define $N$ to be the set of all points of $\Omega_{c}$ satisfying $\dot{H}=0$. From (<ref>), we have $N = \{(\mathbf{x},\omega_{s}^{s})\in \Omega_{c} | e_{\omega}=0 \}$. As $e_{\omega}=0$ implies $\omega_{s}^{s} = R_{e}^{T}\omega_{d}^{s}$ which yields the dynamics, $\dot{\mathbf{x}} = \dot{\mathbf{x}}_{d}$ and $\dot{\omega}_{s}^{s} = - r M^{-1}R_{s}^{T}[\hat{e}_{3} \times (\mathbf{x} - \mathbf{x}_{d})] + \frac{d}{dt}(R_{e}^{T}\omega_{d}^{s})$. Since the robot rolls on a horizontal plane, at any point $(\mathbf{x} - \mathbf{x}_{d}) \neq \hat{e}_{3}$. So the only possibility of $\dot{e}_{\omega} = 0$ to happen is when $x = x_{d}$. Hence, the largest invariant set will be the set $N_{1} = \{(\mathbf{x}, e_{\omega})| \omega_{s}^{s}=R_{e}^{T}\omega_{d}^{s}, \mathbf{x}=\mathbf{x}_{d}\}$ in $\Omega_{c}$. And from LaSalle's invariance principle, the trajectories in $\Omega_{c}$ converge to $N_{1}$ as $t \rightarrow \infty $, i.e, to the equilibrium $(\mathbf{x}_{d},\omega_{d}^{s})$. $\blacksquare$
§.§.§ Position and reduced attitude stabilization:
For position stabilization there are two cases: 1) when $x_{d}=0$ $\& $ $R_{d}\omega_{d}^{s} = 0$ $\Rightarrow$ $\omega_{d}^{s}=0$; and 2) when $x_{d}=0$ $\& $ $R_{d}\omega_{d}^{s} = \alpha \hat{e}_{3}$, where $\alpha$ is any scalar. Case 1 is immediate. When $\omega_{d}^{s}=0$ the control law $u = f_{PD}$ and the robot converges to the origin asymptotically. In case 2, $R_{d}\omega_{d}^{s} = \alpha \hat{e}_{3}$ implies that the robot's final contact position is the origin and the angular velocity is about the $z-$axis. The control law in this case is expressed as
\begin{align}\nonumber
\begin{split}
u & = - k_{p}r R_{s}^{T}(\hat{e}_{3} \times \mathbf{x}) + k_{d} (\omega_{s}^{s} - \alpha R_{s}^{T}\hat{e}_{3}) - \frac{1}{2} (\omega_{s}^{s} \times \alpha M R_{s}^{T}\hat{e}_{3}) \\
& + \frac{1}{2} (M\omega_{s} \times \alpha R_{s}^{T}\hat{e}_{3}) - M (\omega_{s}^{s} \times \alpha R_{s}^{T}\hat{e}_{3}).
\end{split}
\end{align}
The first term in the control law is responsible for the contact position stabilization and remaining terms will orient the sphere such that the angular velocity tracks $\hat{e}_{3}$. Such is a case of reduced attitude stabilization where stabilizing $R_{s}$ upto a rotation about $\hat{e}_{3}$ is equivalent to stabilizing the angular velocity direction of the axis $R_{s}^{T}\hat{e}_{3}$ <cit.>. Thus, we can restate the attitude as $R_{s} \in \mathbb{S}^{2}$ and conclude that the control law (<ref>) gives the contact point and reduced attitude stabilization in terms of the points in $\mathbb{R}^{2} \times \mathbb{S}^{2}$.
§.§.§ Simulation:
We take the model parameters as in section (<ref>) with initial orientation $R_{s_{0}} = exp(\frac{\pi }{6} \hat{e}_{1}) $ and starting point on the horizontal plane as $(x_{0},y_{0}) = (4,2)\mbox{units}$. Setting the desired orientation $R_{d} = exp(\frac{\pi}{4} \hat{e}_{3})$ and $\omega_{d}^{s} = \hat{e}_{3}$ which satisfy $R_{d}^{T}\omega_{d}^{s}=\hat{e}_{3}$, Fig. <ref>(a)) and Fig. (<ref>(a)) shows that as the angular velocity achieve $\omega_{d}^{s}$ asymptotically, the sphere attains the desired Line-of-sight that is $R_{s}^{T}\hat{e}_{3}= (\Gamma_{1},\Gamma_{2},\Gamma_{3})$ converges to $\hat{e}_{3}=(0,0,1)$. The initial oscillations in $R_{s}^{T}\hat{e}_{3}$ plot are due to the sphere rotating in the spiral type motion on plane and then asymptotic converges to $(0,0,1)$. The position on $xy$ plane is illustrated in Fig. (<ref>(b)).
a)Angular velocity of spherical robot. b) Position on the plane.
$\Gamma$ plot and torques on rotors.
To illustrate contact point trajectory tracking, we choose $\mathbf{x}_{d}$ to track line and circle, as shown in Fig. (<ref>), (<ref>) and (<ref>). Keeping $\omega_{d}^{s}=0$, then Fig. (<ref>)(a) and (b) shows the angular velocity of the sphere and the phase plane of position. Setting $\mathbf{x}_{d} = (r_{s}\sin (t),r_s \cos (t))$ which yields $\omega_{d}^{s}=(-\sin(t),-\cos(t))$. Fig. (<ref>) shows the spherical robot follows the desired circular trajectory. Setting $\omega_{d}^{s}= (0.2,0.3)$ and $\mathbf{x}_{d} = (0.2t + 0.4,0.3t+0.6)$ then the sphere will rotate at constant speed shown in Fig. (<ref>) and follows the line given by $\mathbf{x}_{d}$.
Phase plane of $xy$ position.
Angular velocity and torques on rotors.
Angular velocity and phase plane of $xy$ position tracks the circular trajectory.
Angular velocity and phase plane of $xy$ position tracks the line trajectory.
§ DISCUSSION
In conclusion, we say that both the control strategies derived using the geometric approach, without parametrization, illustrate a more general philosophy on the control design, preserving the mechanical notions of the system. To the best of our knowledge this is the first instance where such a strategy has been employed to a nonholonomic system. Both the control strategies are derived using the notions of the affine connection, the error functions and a transport map on tangent spaces. The first feedback strategy results in a continuous feedback law which tracks the desired orientation trajectory. In position tracking strategy, an intermediate result while proving the stability is $\nabla_{\omega_{s}^{s}}e_{\omega} = f_{PD}$, which provides an interpretation about feedforward control $f_{FF}$. The closed-loop system with $f_{FF}$ has the property that $\nabla_{\omega_{s}^{s}}e_{\omega}$ vanishes along the trajectory. That is, if $\e_{\omega} =0 \Rightarrow \dot{e}_{x}=(\dot{\mathbf{x}} - \dot{\mathbf{x}}_{d})$ is zero at initial time, it will remain zero at final time. And if we keep $\omega_{d}^{s}=\hat{e}_{3}$ we get the $\dot{e}_{x}(0)=\dot{e}_{x}(T)=0$ for all time, and the result is contact position and a reduced attitude stabilization.
§ APPENDIX
§.§.§ Proof of Lemma 1:
In this section we will compute the Lie brackets of $F_{cl}$ and $g_{i}$, where $i=1,2,3$. Given two vector field $X,Y \in TQ$ the Lie derivative (bracket) of $Y$ along $X$ is $[X,Y] \equiv \frac{d}{dt}|_{t=0}\Phi_{t}^{*}(Y)$,
where $\Phi$ is the flow of $X$ and $\Phi_{t}^{*}(Y)$ is pull-back of a vector field $Y$. From system (<ref>), the Lie bracket of $F_{cl}$ and $g_{i}$ will be
\begin{equation}\nonumber
[F_{cl},g_{i}](q) = -[g_{i},F_{cl}](q) = - \frac{d}{dt}|_{t=0} (D\Phi_{t}^{g_{i}}(q))^{-1}\cdot F_{cl}(\Phi_{t}^{g_{i}}(q))
\end{equation}
where $\Phi_{t}^{g_{i}}$ is the flow of $g_{i}$. The control vector field $g_{i}(q) = (\mathbf{0}, M^{-1}\hat{e}_{i})$
then flow of $g_{i}$ is given as $\Phi_{t}^{g_{i}}(q) = \left(R_{s},\omega_{s}^{s} + t M^{-1}\hat{e}_{i} \right)$
and $(D\Phi_{t}^{g_{i}}(q))^{-1}$ is the identity map on the manifold $Q$.
\begin{align*}
& [g_{1},F_{cl}](q) = \frac{d}{dt}|_{t=0} (D\Phi_{t}^{g_{i}}(q))^{-1}\cdot F_{cl}(\Phi_{t}^{g_{i}}(q)) \nonumber \\
& = \frac{d}{dt}|_{t=0} (D\Phi_{t}^{g_{i}}(q))^{-1}\cdot \begin{bmatrix}
R_{s}(\omega_{s} + t M^{-1}\hat{e}_{1})^{\bigwedge} \\
\ast
\end{bmatrix} = \begin{bmatrix}
\ast
\end{bmatrix}.
\end{align*}
Similarly, $[g_{2},F_{cl}](q)$ and $[g_{3},F_{cl}](q)$ are calculated and given as
\begin{equation*}
[g_{2},F_{cl}] = \begin{bmatrix}
\ast
\end{bmatrix}\mbox{ and }
[g_{3},F_{cl}] = \begin{bmatrix}
\ast
\end{bmatrix}.
\end{equation*}
where $\ast$ denotes some functions we are not interested in. The vectors $g_{1}$,$g_{2}$,$g_{3}\in T_{q}Q$ are linearly independent since $\{\hat{e}_{1},\hat{e}_{2},\hat{e}_{3}\}$ are linearly independent. To see the linear independence, we write all the six vectors as
\begin{align}
& \alpha_{i}M^{-1}\hat{e}_{i} + \beta_{i}R_{s}(M^{-1}\hat{e}_{i})^{\wedge}=0. \label{lin_indep2}
\end{align}
Now, for these vectors to be linearly independent, all the scalars $\alpha_{i}$'s and $\beta_{i}$'s equal to zero. From the values of $g_{i}$ and $[g_{i},F_{cl}]$, for (<ref>) to hold that, it follows that
\begin{align}\label{lin_indep1}
\alpha_{i}M^{-1}\hat{e}_{i} = 0 \quad \beta_{i}R_{s}(M^{-1}\hat{e}_{i})^{\wedge}=0.
\end{align}
for all $i$. Since $\{g_{1},g_{2},g_{3}\}$ is linearly independent, (<ref>) will hold only when $\alpha_{i}=0$. And $\{R_{s}(M^{-1}\hat{e}_{1})^{\wedge},R_{s}(M^{-1}\hat{e}_{2})^{\wedge},$ $R_{s}(M^{-1}\hat{e}_{3})^{\wedge}\}$ is linear independent, then $\beta_{i}=0$ to satisfy (<ref>). Hence, the set
$\{ g_{1},g_{2},g_{3},[f,g_{1}],[f,g_{2}],[f,g_{3}]\}$
are linearly independent on $Q =SO(3)\times \mathbb{R}^{3}$ of dimensional six and spans the tangent space of the configuration space at any configuration. Therefore, the system is locally controllable.
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|
1511.00277
|
Department of Physics and Institute of Theoretical Physics and Astrophysics,
Xiamen University, Xiamen 361005, Fujian, China
Heat and energy are conceptually different, but often are assumed to
be the same without justification. An effective method for investigating
diffusion properties in equilibrium systems is discussed. With this method, we demonstrate that for one-dimensional systems, using the indices of particles as the space variable , which has been accepted as a convention, may lead to misleading conclusions. We then show that though in one-dimensional systems there is no general connection between energy diffusion and heat conduction, however, a general connection between heat diffusion and heat conduction may exist. Relaxation behavior of local energy current fluctuations and that of local heat current fluctuations are also studied. We find that they are significantly different, though the global energy current equals the globe heat current.
05.60.Cd, 89.40.-a, 44.10.+i, 51.20.+d
§ INTRODUCTION
By definition, it is clear that heat and internal energy are conceptually
different. Internal energy is referred to as the total kinetic and
potential energy of a system, which is a function of the system state,
while heat is a quantity that characterizes a process. For one-dimensional
(1D) systems, combining the continuous equations of energy and mass,
i.e., $\frac{\partial e(x,t)}{\partial t}+\frac{\partial}{\partial x}{j}^{e}(x,t)=0$
and $\frac{\partial\rho(x,t)}{\partial t}+\frac{\partial}{\partial x}{p}(x,t)=0$,
one can obtain
\begin{equation}
\frac{\partial}{\partial t}[e(x,t)-\frac{(e+P)\rho(x,t)}{\rho}]+\frac{\partial}{\partial x}{j}^{q}(x,t)=0,
\end{equation}
and thus introduce the heat density function as <cit.>
\begin{equation}
\end{equation}
Here $e(x,t)$, $\rho(x,t)$, ${p}(x,t)$, ${j}^{e}(x,t)$, and ${j}^{q}(x,t)$
represent, respectively, the density of energy, mass, momentum, energy
current, and heat current; $e$ ($\rho$) and $P$ represent, respectively,
the spatially averaged energy (mass) density and the internal pressure
of the system at the equilibrium state. The local heat current is
related to the local energy current as
\begin{equation}
\end{equation}
Therefore, the physical meaning of the change rate of $q(x,t)$ is
definite: it represents the divergence of the local heat current.
But, on the contrary, the value of $q(x,t)$ itself lacks definite
meanings. Usually $q(x,t)$ is negative, because $P>0$ and $\langle\rho(x,t)/\rho\rangle=1$.
Indeed, following Eq. (1), the heat density can be defined as $q(x,t)+c$
with an arbitrary constant $c$. Energy contains two parts, corresponding
to regular and irregular motions, respectively, but heat is always
related to thermal processes that feature random motions.
In spite of this fact, in some previous studies energy and heat are
not distinguished. An example is in the study of the relation between
anomalous diffusion and transport properties in low-dimensional (one-
and two-dimensional) systems. It is well known that in bulk (three-dimensional)
materials the thermal conductivity $\kappa$ and the heat diffusion
coefficient $D$ can be generally related by $\kappa=\rho c_{P}D$,
where $c_{P}$ is the constant pressure specific heat. But this relation
does not hold in low-dimensional systems. In recent decades, stimulated
by the rapid progress in nanoscience <cit.>, transport properties
of low-dimensional systems has attracted intensive investigations
It has been found that in general diffusion and transport are abnormal
in low-dimensional systems. In particular, in 1D momentum conserving
systems, the heat conductivity diverges with the system size $L$
as $\kappa\sim L^{\alpha}$ and the heat diffusion coefficient diverges
with time as $D\sim t^{\beta-1}$. ($\alpha$ and $\beta$ are constants.)
In some studies <cit.>
the heat diffusion behavior has been assumed, implicitly, to be the
same as the energy diffusion behavior, and the exponent $\beta$ is
calculated by tracing energy diffusion instead.
It has been conjectured that there exists a general relation between
exponents $\alpha$ and $\beta$. Two formulae, $\alpha=2-2/\beta$
by Li and Wang <cit.>
and $\alpha=\beta-1$ by Denisov et al. <cit.>,
have been proposed. It is worth noting that in these studies <cit.>
the authors did not distinguish heat diffusion and energy diffusion
and treated them identically. Though this is legal for the specific
example systems they studied where energy and heat are coincidentally
the same, we emphasize that in principle $\beta$ involved in these
two formulae should be the exponent that characterizes heat
diffusion, rather than that of energy diffusion. This is particularly
important for clarifying which one of these two formula is correct,
which has been a focused issue recently. Indeed, in our very recent
study <cit.>
it has been shown that the diffusion behaviors of energy and heat
can be qualitatively different, suggesting that there should be no
general relation between energy diffusion and heat conduction, but
a general relation between heat diffusion and heat
conduction may exist and can be established.
This paper is an effort to verify this conjecture. We shall first
ascertain a correct way for calculating the exponent $\beta$, so
as to put our studies on a solid basis. So far there are two classes
of methods for probing diffusion processes, i.e., the equilibrium
and non-equilibrium <cit.>
methods. With both methods, the probability density function (PDF)
of local fluctuations of interested physical quantities are calculated.
The space variable of the PDF should correctly give the positions
of local fluctuations, but in practice the indices of particles are
often taken as the space variable instead to facilitate the simulations
In doing so, the underlying philosophy is that the index of a particle
is equivalent to its position in 1D systems, as the particle simply
moves around its equilibrium position. But as we shall demonstrate
in the following, this is not the case: Using the index variable may
result in not only quantitative but also qualitative deviations, which
may be responsible for the confusing results of $\beta$ reported
previously in the literatures. By taking the correct space variable
and the equilibrium method (which has been shown to be more efficient
and accurate), we calculate the exponent $\beta$ of heat diffusion
with high precision in a 1D hard-point gas model <cit.>
By comparing it with the values of the exponent $\alpha$ obtained
in previous studies <cit.>
we shall show that the anomalous heat diffusion and heat conduction
can be accurately connected by the formula $\alpha=2-2/\beta$.
In addition, we shall also discuss the behaviors of the local heat
current and the local energy current. By properly setting the coordinate
system to guarantee the system has a vanishing total momentum, we
find although the total heat current is always equal to the total
energy current, the relaxation behaviors of local currents of energy
and heat can be remarkably different.
The rest of this paper is organized as follows: The model to be studied
will be described in the next section, and the methods for probing
energy and heat diffusion will be detailed in Sec. 3. The main results
will be provided and discussed in Sec. 4, followed by a brief summary
in the last section.
§ MODELS
We consider two paradigmatic 1D models extensively employed for studying
transport properties of low-dimensional systems. Each model is composed
of $N$ point particles arranged in order. We denote by $m_{k}$,
$x_{k}$, $v_{k}$, and $p_{k}$, respectively, the mass, the position,
the velocity, and the momentum of the $k$ particle.
The first model is a 1D hard-point gas <cit.>
with alternative mass $m_{o}$ for odd-numbered particles and $m_{e}$
for even-numbered particles. We set $m_{o}=1$ and $m_{e}=3$, the
same as in Ref. <cit.>,
for the sake of comparison. The particles travel freely except for
elastic collisions with their nearest neighbors. After a collision
between the $k$th particle and the $(k+1)$th particle, their velocities
change to
\begin{equation}
\end{equation}
\begin{equation}
\end{equation}
Another model is a 1D lattice; i.e., the well known Fermi-Pasta-Ulam
(FPU) model defined by the Hamiltonian
\begin{equation}
\end{equation}
where the masses of all particles are set to be unity.
In our simulations the periodic boundary condition is applied and
the system size $L$ is set to be the same as the particle number
$N$, so that the averaged particle number density is unity. The local
temperature is defined as $T_{k}\equiv\frac{\langle p_{k}^{2}\rangle}{k_{B}m_{k}}$,
where $k_{B}$ (set to be unity) is the Boltzmann constant and $\langle\cdot\rangle$
stands for the ensemble average. For both models the average energy
per particle is fixed to be unity, corresponding to a system temperature
$T=2$ in the gas model and $T\approx1.2$ in the FPU model.
§ METHODS
In principle, one can probe the diffusion behavior directly by adding
an external perturbation to the equilibrium system and observing its
ensuing relaxation process <cit.>.
This method requires demanding computing resource, so that a satisfactory
precision is usually hard to reach <cit.>.
A more effective method <cit.>
is instead to study the properly rescaled spatiotemporal correlation
functions of fluctuations in the equilibrium state. The basic idea
of this method is detailed in the following.
Let $A(x,t)$ represents the density distribution function of a physical
quantity ${\cal A}$. In numerical simulations, in order to calculate
the spatiotemporal correlation function of fluctuations of ${\cal A}$,
we have to discretize the space variable first. For this aim, we divide
the space occupied by the system into $N_{b}=L/b$ bins of equal size
of $b$. The total quantity of ${\cal A}$ in the $k$th bin, denoted
by $A_{k}(t)$, is defined as $A_{k}(t)\equiv\int_{x\in k\mathrm{th\, bin}}A(x,t)dx$.
As such $A_{k}(t)/b$ gives the coarse-grained density of ${\cal A}$
in the $k$th bin. The fluctuations of ${\cal A}$ are therefore captured
by $\Delta A_{k}(t)\equiv A_{k}(t)-\langle A\rangle$, where $\langle A\rangle$
represents the ensemble average of $A_{k}(t)$. The positions of the
bin centers can then be used as the coarse-grained space variable.
-.8cm The probability density distribution function of energy obtained by
using our correlation function method and the perturbation method
with the space variable being, respectively, the coarse-grained space
variable and the particle indices. (a)-(d) are for the gas model at
$t=400$, obtained by (a)-(b) the correlation function method and
(c)-(d) the perturbation method. (e)-(f) are for the FPU model at
$t=600$ by using the correlation function method.
For a conserved physical quantity ${\cal A}$, it has been derived
in Ref. <cit.>
that the PDF corresponding to a local fluctuation initially located
in the $k$th bin, which is specified by $\Delta A_{k}(0)$, can be
calculated as
\begin{equation}
\rho_{A}(\Delta x_{k,l},t)=\frac{\langle\Delta A_{l}(t)\Delta A_{k}(0)\rangle}{\langle\Delta A_{k}(0)\Delta A_{k}(0)\rangle}+\frac{1}{N_{b}-1}
\end{equation}
if the microcanonical ensemble is considered, and
\begin{equation}
\rho_{A}(\Delta x_{k,l},t)=\frac{\langle\Delta A_{l}(t)\Delta A_{k}(0)\rangle}{\langle\Delta A_{k}(0)\Delta A_{k}(0)\rangle}
\end{equation}
if the canonical ensemble is considered. Here $\Delta x_{k,l}$ denotes
the displacement from the $k$th bin to the $l$th bin, i.e., $\Delta x_{k,l}\equiv(l-k)b$.
For the sake of convenience, in the following we shall use $x$ to
denote $\Delta x_{k,l}$ without confusion. The spatiotemporal correlation
function defined above gives the causal relation between a local fluctuation
and the effects it induces at another position and at a later time,
thus it is in essence equivalent to the probability density function
that describes the diffusion process of the fluctuation. In order
to facilitate numerical simulations, we suggest considering the microcanonical
ensemble where all systems are isolated from the environment. Hence
one does not have to simulate the environment, which saves greatly
the simulation time.
In previous studies, e.g., <cit.>,
the authors constantly use the indices of particles to represent the
space variable. In particular, the value of ${\cal A}$ of the $k$th
particle, denoted by $A_{k}^{{\rm ind}}(t)$, is adopted to represent
the density distribution of ${\cal A}$ at the position of $kL/N$,
and $\langle\Delta A_{l}^{{\rm ind}}(t)\Delta A_{k}^{\text{i}nd}(0)\rangle$
is assumed to represent the correlation between two positions with
a distance of $x=(l-k)L/N$ and a time delay of $t$. In the following,
we shall refer to this “coordinate as the index variable.
Although the index represents the mean position of a particle
in the equilibrium state, it by no means gives the position of the
particle at instant times that is crucial for correctly calculating
the spatiotemporal correlation functions. For this reason, Dhar once
questioned the effectiveness of the index variable because it may
result in large position fluctuations <cit.>.
We find that it is even worse: The deviations caused by using the
index variable is not only quantitative, but also qualitative.
-.4cm -.3cm -.8cm
The simulation results of the spatiotemporal correlation function
of heat fluctuations, $\rho_{q}(x,t)$, for the 1D gas model. (a)
presents a snapshot of $\rho_{q}(x,t)$ at time $t=400$, and (b)
shows the time dependence of the height of the center peak of $\rho_{q}(x,t)$.
The red solid line in (b) is the best linear fitting of the data for
revealing their asymptotic characteristics, suggesting that $\rho_{q}(0,t)\sim t^{-0.60}$.
In (c) $\rho_{q}(x,t)t^{\lambda}$ versus $x/t^{\lambda}$ at three
different times are compared with the rescaling factor $\lambda=0.60$
obtained via best linear fitting in (b). The fact that three curves
overlap perfectly verifies the scaling property of $\rho_{q}(x,t)$.
Taking the energy fluctuations as an example, we show that indeed
the index variable may lead to qualitatively wrong results. We denote
the spatiotemporal correlation function obtained by using the coarse-grained
space variable and the index variable as $\rho_{e}(x,t)$ and $\rho_{e}^{{\rm ind}}(x,t)$,
respectively. The 1D gas model is considered first. To prepare an
equilibrium gas, the system is efficiently simulated for a sufficient
long time by using the event-driven algorithm that employs the heap
data structure to identify the collision times <cit.>.
Then $\rho_{e}(x,t)$ and $\rho_{e}^{{\rm ind}}(x,t)$ are calculated
with $N=4000$ and $b=1$. Figure 1(a)-(b) show the results. One can
see that they are remarkably different: With the coarse-grained space
variable, the spatiotemporal correlation function has two peaks, while
with the index variable it has three peaks. The two peaks of $\rho_{e}(x,t)$
move outwards with a constant speed $v=1.75$, which can be shown
easily to be the sound speed <cit.>.
The two side peaks of $\rho_{e}^{{\rm ind}}(x,t)$ move outwards with
the same speed. The center peak of $\rho_{e}^{{\rm ind}}(x,t)$ does
not move but broadens as $\Delta w\sim t^{0.67}$, where $\Delta w$
represents its half-height width. It is tempting to think that the
two side peaks represent the sound mode and the center peak represents
the heat mode, as in the case of the mass fluctuations that gives
the dynamic structure factor of the system <cit.>.
But we find the ratio of the area of the center peak to that of the
two side peaks equals $1/2$, while it should equal $2$, i.e., the
Landau-Placzek ratio <cit.>
of an ideal gas, if it characterizes the dynamical structural factor
As will be discussed in the next section, the decaying behavior of
the center peak is also different from the heat mode. Therefore, $\rho_{e}^{{\rm ind}}(x,t)$
fails to capture the properties of the heat mode. It is worth noting
that the three peak structure has also been reported in other studies
of the gas model by using the index variable <cit.>.
As mentioned above, diffusion properties can also be investigated
directly by observing the evolution of an added perturbation to the
system. We find that by applying this method, one can obtain the same
results. [See Fig. 1(c)-(d).] In addition, we find that in the
1D FPU model [see Fig. 1(e)-(f)], the qualitative difference between
$\rho_{e}(x,t)$ and $\rho_{e}^{{\rm ind}}(x,t)$ is also obvious
<cit.>. These results suggest clearly that the position of a particle
at instant times cannot be approximated by its equilibrium position
in order to correctly calculate spatiotemporal correlation functions.
§ RESULTS AND DISCUSSIONS
In Fig. 1(a), it is clearly shown that the energy fluctuations transport
ballistically in the 1D gas model, implying that the mean square displacement
of the transported energy increases in time as $\langle x^{2}(t)\rangle\sim t^{2}$.
On the other hand, the heat conduction properties of this model have
been extensively studied in the literatures <cit.>.
It has been found that the heat conductivity $\kappa$ diverges with
the system size $L$ as $\kappa\sim L^{\alpha}$ with $\alpha=1/3$
suggesting that heat diffuses in a supperdiffusive manner rather than
ballistically. Therefore, energy diffusion and heat conduction do
not fall into the same anomalous class. We then calculate the PDF
of heat fluctuations, i.e., $\rho_{q}(x,t)$, following Eq. (2) and
present the results in Fig. 2. It can be seen that the profile of
$\rho_{q}(x,t)$ is completely different from that of energy fluctuations,
$\rho_{e}(x,t)$ [see Fig. 1(a)]; the former has only one single
peak. This outstanding feature has been reported in ref. <cit.>,
but here we perform the simulation with a larger system size of $N=4096$
which allows us to measure the decaying rate of $\rho_{q}(x,t)$ more
accurate. As presented in Fig. 2(b), we obtain that the height of
the peak of $\rho_{q}(x,t)$ goes as $h=\rho_{q}(0,t)\sim t^{-\lambda}$
with $\lambda=0.60$. As heat is a conserved quantity, the peak of
$\rho_{q}(x,t)$ must keep its area unchange, and as a consequence
its half-height width should broaden as $\Delta w\sim t^{-0.60}$.
As such $\rho_{q}(x,t)$ at different time may be rescaled onto a
common function. This is confirmed by our study presented in Fig.
2(c): $\rho_{q}(x,t)$ is indeed invariant upon rescaling $x\rightarrow t^{\lambda}x$
so that $t^{\lambda}\rho_{q}(x,t)=t_{0}^{\lambda}\rho_{q}(x_{0},t_{0})$
for $x=({t}/{t_{0}})^{\lambda}x_{0}$ with scaling factor $\lambda=0.60$.
-.1cm -.8cm The spatiotemporal correlation function of local energy current $C_{e}(x,t)$
(a) and of local heat current $C_{q}(x,t)$ (b) at time $t=400$ for
the 1D gas model $N$=3000. In (c), the autocorrelation function of
the globe energy current $\langle J_{e}(t)J_{e}(0)\rangle$ (black
solid line) is compared with $\int C_{e}(x,t)dx$ (blue open squares)
and $\int C_{q}(x,t)dx$ (red solid triangles). The three sets of
data fall onto the same curve upon proper shifts in vertical direction.
This result confirms for the first time that the relaxation of the
heat mode follows the scaling law with $\lambda=3/5$ as predicted
by the hydrodynamic mode-coupling theory <cit.>.
This scaling property implies that $\rho_{q}(x,t)$ relaxes as $\langle x^{2}(t)\rangle=\langle x_{0}^{2}(t_{0})\rangle(\frac{t}{t_{0}})^{2\lambda}$;
i.e., heat fluctuations diffuse in a power law $\langle x^{2}(t)\rangle\sim t^{\beta}$
with the diffusion exponent $\beta=2\lambda$ <cit.>.
We obtain $\beta=1.20$ accordingly, suggesting that heat diffusion
is superdiffusive, in clear contrast with energy diffusion which is
ballistic. More important and interesting, by combining $\beta=1.20$
obtained here and $\alpha=1/3$ obtained in previous analytical and
numerical studies <cit.>,
we find they follow perfectly the general formula proposed by Li and
Wang <cit.>
that trying to connect energy diffusion (should be heat diffusion
instead) and heat conduction in 1D systems.
We would like to point out that the center peak of $\rho_{e}^{{\rm ind}}(x,t)$
[see Fig. 1(a)] is also invariant upon the rescaling but with
$\lambda=0.67$ instead, which implies $\beta=1.34$. One may notice
that the formula $\alpha=\beta-1$ correctly describes the relation
of energy diffusion and heat conduction in this case, as pointed in
Ref. <cit.>.
We can therefore conclude that this is a consequence by improperly
replacing the space variable with the particle indices.
Finally, we study the relaxation behavior of local energy current
and local heat current. We set the total momentum of the system to
be zero. As having been pointed out in Ref. <cit.>,
the global energy current always equals the globe heat current, but
this fact does not imply the properties of local heat current and
local energy current are also identical. Figure 3(a)-(b) show the
spatiotemporal correlation functions of local energy current $C_{e}(x,t)=\langle j_{l}^{e}(t)j_{k}^{e}(0)\rangle$
and that of local heat current $C_{q}(x,t)=\langle j_{l}^{q}(t)j_{k}^{q}(0)\rangle$.
Here $x=(l-k)L/N$, ( the size of a bin is $b=L/N$ ), the local energy
current is defined as $j_{l}^{e}(t)=\sum_{k:x_{k}\in l{\rm th~bin}}m_{k}v_{k}^{3}/2$,
and the local heat current is obtained by substituting $j_{l}^{e}(t)$
into Eq. (3). It can be seen that $C_{e}(x,t)$ and $C_{q}(x,t)$
have remarkably different features: The former has a global negative
bias, and its two peaks moving oppositely at the sound speed. The
latter is more complicated. There are two pulses which look like a
negative Mexican hat wavelet and move outwards at the sound speed,
but there is no global bias. Instead, there is a dip at the origin.
Therefore, the properties of $C_{q}(x,t)$ can not be probed by studying
$C_{e}(x,t)$, and vice versa.
The globe energy current $J_{e}(t)=\sum_{k}j_{k}^{e}(t)$. One has
$\langle J_{e}(t)J_{e}(0)\rangle=\langle\sum_{k}j_{k}^{e}(0)\sum_{l}j_{l}^{e}(t)\rangle\propto\langle j_{k}^{e}(0)\sum_{l}j_{l}^{e}(t)\rangle$
for any index $k$ because of the homogeneity of the system. In other
words, $\langle J_{e}(t)J_{e}(0)\rangle\propto\int C_{e}(x,t)dx$.
Because $J_{e}(t)=J_{q}(t)$ as a result of the null total momentum,
we have $\int C_{e}(x,t)dx\propto\int C_{q}(x,t)dx$. This result
implies that though the relaxation behaviors of local heat current
and local energy current are different, their integrals decay in the
same way, which is also confirmed by direct simulation results shown
in Fig. 3(c).
§ SUMMARY
We demonstrate that in 1D systems, using the particle indices as the
space variable may result in qualitative deviations in probing the
diffusion properties, and hence should be abandoned. Instead, the
coarse-grained space variable is a correct and practical choice. By
taking advantage of it, we have verified that in the 1D gas model,
heat diffusion and heat conduction, rather than energy diffusion and
heat conduction, can be connected by the formula $\alpha=2-2/\beta$
accurately, rather than by that proposed in <cit.>.
Our analysis has also shown that
energy diffusion and heat conduction follows $\alpha=\beta-1$ as
observed in Ref. <cit.>
may be a misunderstanding caused by misusing the particle indices
as the space variable. We emphasize that the position of a particle
at instant times cannot be approximated by its equilibrium position
in probing spatiotemporal correlation functions of a system.
In addition, in the case of null total momentum, the globe energy
current and the globe heat current are found to be the same. But local
energy currents are different from local heat currents. The relaxation
behavior of the former is significantly different from that of the
latter as well. We conclude that in general, the relaxation and transport
properties of heat can not be identified with those of energy.
This work is supported by the NNSF (Grants No. 10925525, No. 11275159,
and No. 10805036) and SRFDP (Grant No. 20100121110021) of China.
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School of Information Science, JAIST, Japan.
Suppose that we are given two independent sets $\bfI_b$ and $\bfI_r$ of
a graph such that $\msize{\bfI_b}=\msize{\bfI_r}$,
and imagine that a token is placed on each vertex in $\bfI_b$.
Then, the sliding token problem is to determine whether
there exists a sequence of independent sets which transforms $\bfI_b$
into $\bfI_r$ so that each independent set in the sequence results from
the previous one by sliding exactly one token along an edge in the graph.
The sliding token problem is one of the reconfiguration problems
that attract the attention from the viewpoint of theoretical computer science.
The reconfiguration problems tend to be PSPACE-complete in general,
and some polynomial time algorithms are shown in restricted cases.
Recently, the problems that aim at finding a shortest reconfiguration sequence are investigated.
For the 3SAT problem, a trichotomy for the complexity of finding the shortest sequence has been shown;
that is, it is in P, NP-complete, or PSPACE-complete in certain conditions.
In general, even if it is polynomial time solvable to decide whether two instances are reconfigured with each other,
it can be NP-complete to find a shortest sequence between them.
Namely, finding a shortest sequence between two independent sets
can be more difficult than the decision problem of reconfigurability between them.
In this paper, we show that the problem for finding a shortest sequence between two independent sets
is polynomial time solvable for some graph classes which are subclasses of the class of interval graphs.
More precisely, we can find a shortest sequence between two independent sets on a graph $G$
in polynomial time if either $G$ is a proper interval graph, a trivially perfect graph, or a caterpillar.
As far as the authors know, this is the first polynomial time algorithm for the shortest sliding token
problem for a graph class that requires detours.
§ INTRODUCTION
Recently, the reconfiguration problems attract the attention
from the viewpoint of theoretical computer science.
The problem arises when we wish to find a step-by-step transformation between
two feasible solutions of a problem such that
all intermediate results are also feasible and
each step abides by a fixed reconfiguration rule, that is,
an adjacency relation defined on feasible solutions of the original problem.
The reconfiguration problems have been studied extensively for several well-known problems, including
independent set <cit.>,
satisfiability <cit.>,
set cover, clique, matching <cit.>,
vertex-coloring <cit.>, shortest path <cit.>, and so on.
The 15 puzzle, Dad's puzzle, and its Chinese variant.
The reconfiguration problem can be seen as a natural “puzzle” from the viewpoint of
recreational mathematics. The 15 puzzle is one of the most famous classic puzzles,
that had the greatest impact on American and European society of any mechanical puzzle
the word has ever known in 1880 (see <cit.> for its rich history).
It is well known that the 15 puzzle has a parity;
for any two placements, we can decide whether two placements are reconfigurable or not by checking the parity.
Therefore, we can solve the reconfiguration problem in linear time just by
checking whether the parity of one placement coincides with the other or not.
Moreover, we can say that the distance between any two reconfigurable placements is $O(n^3)$,
that is, we can reconfigure from one to the other in $O(n^3)$ sliding pieces when the size of the board is $n\times n$.
However, surprisingly, for these two reconfigurable placements,
finding a shortest path is NP-complete in general <cit.>.
Namely, although we know that it is $O(n^3)$, finding a shortest one is NP-complete.
Another interesting property of the 15 puzzle is in another case of generalization.
In the 15 puzzle, every peace has the same unit size of $1\times 1$.
We have the other famous classic puzzles that can be seen as a generalization of this viewpoint.
That is, when we allow to have rectangles, we have the other classic puzzles,
called “Dad puzzle” and its variants (see <ref>).
Gardner said that “These puzzles are very much in want of a theory” in 1964 <cit.>,
and Hearn and Demaine have gave the theory after 40 years <cit.>;
they prove that these puzzles are PSPACE-complete in general
using their nondeterministic constraint logic model <cit.>.
That is, the reconfiguration of the sliding block puzzle is PSPACE-complete in general decision problem,
and linear time solvable if every block is the unit square.
However, finding a shortest reconfiguration for the latter easy case is NP-complete.
In other words, we can characterize these three complexity classes using the model of sliding block puzzle.
From the viewpoint of theoretical computer science, one of the most important problems is the 3SAT problem.
For this 3SAT problem, a similar trichotomy for the complexity of finding a shortest sequence has been shown
recently; that is, for the reconfiguration problem of 3SAT, finding a shortest sequence between two satisfiable
assignments is in P, NP-complete, or PSPACE-complete in certain conditions <cit.>.
In general, the reconfiguration problems tend to be PSPACE-complete,
and some polynomial time algorithms are shown in restricted cases.
In the reconfiguration problems, finding a shortest sequence can be a new trend in theoretical computer
science because it has a great potential to characterize the class NP from a new viewpoint.
Beside the 3SAT problem, one of the most important problems in theoretical computer science is the
independent set problem.
Recall that an independent set of a graph $G$ is a vertex-subset of $G$ in which
no two vertices are adjacent.
(See <ref> which depicts five different independent sets in the same graph.)
For this notion, the natural reconfiguration problem is called the sliding token problem introduced by
Hearn and Demaine <cit.>:
Suppose that we are given two independent sets $\bfI_b$ and $\bfI_r$ of a graph
$G = (V,E)$ such that $\msize{\bfI_b}=\msize{\bfI_r}$,
and imagine that a token (coin) is placed on each vertex in $\bfI_b$.
Then, the sliding token problem is to determine whether there exists a sequence
$\langle \bfI_1, \bfI_2, \ldots, \bfI_{\ell} \rangle$ of independent sets of $G$ such that
(a) $\bfI_1=\bfI_b$, $\bfI_{\ell}=\bfI_r$,
and $\msize{\bfI_i} = \msize{\bfI_b}=\msize{\bfI_r}$ for all $i$, $1 \le i \le \ell$; and
(b) for each $i$, $2 \le i \le \ell$,
there is an edge $\{u,v\}$ in $G$ such that $\bfI_{i-1} \setminus\bfI_{i}=\{u\}$
and $\bfI_{i}\setminus\bfI_{i-1}=\{v\}$,
that is, $\bfI_{i}$ can be obtained from $\bfI_{i-1}$ by sliding exactly
one token on a vertex $u \in \bfI_{i-1}$ to its adjacent vertex $v$ along $\{u,v\} \in E$.
Figure <ref> illustrates a sequence
$\langle \bfI_1, \bfI_2, \ldots, \bfI_5 \rangle$ of independent sets
which transforms $\bfI_b = \bfI_1$ into $\bfI_r = \bfI_5$.
Hearn and Demaine proved that the sliding token problem is PSPACE-complete for planar graphs.
A sequence $\langle \bfI_1, \bfI_2, \ldots, \bfI_5 \rangle$ of independent sets of the same graph,
where the vertices in independent sets are depicted by small black circles (tokens).
(We note that the reconfiguration problem for independent set have some variants.
In <cit.>,
the reconfiguration problem for independent set is studied
under three reconfiguration rules called
“token sliding,” “token jumping,” and “token addition and removal.”
In this paper, we only consider token sliding model, and
see <cit.> for the other models.)
For the sliding token problem, some polynomial time algorithms are
investigated as follows: Linear time algorithms have been shown for
cographs (also known as $P_4$-free graphs) <cit.>
and trees <cit.>.
Polynomial time algorithms are shown
for bipartite permutation graphs <cit.>,
and claw-free graphs <cit.>.
On the other hand, PSPACE-completeness is also shown for
graphs of bounded tree-width <cit.>, and planar graphs <cit.>.
In this context, we investigate for finding a shortest sequence of the sliding token problem,
which is called the shortest sliding token problem. That is, our problem is formalized as follows:
Input: A graph $G=(V,E)$ and two independent sets $\bfI_b,\bfI_r$ with $\msize{\bfI_b}=\msize{\bfI_r}$.
Output: A shortest reconfiguration sequence $\bfI_b=\bfI_1$, $\bfI_2$, $\ldots$, $\bfI_{\ell}=\bfI_r$ such that
$\bfI_{i}$ can be obtained from $\bfI_{i-1}$ by sliding exactly
one token on a vertex $u \in \bfI_{i-1}$ to its adjacent vertex $v$ along $\{u,v\} \in E$ for
each $i$, $2 \le i \le \ell$.
We note that $\ell$ is not necessarily in polynomial of $\msize{V}$;
this is an issue how we formalize the problem, and if we do not know that $\ell$ is in polynomial or not.
If the length $k$ is given as a part of input, we may be able to decide whether $\ell \le k$ in polynomial time
even if $\ell$ itself is not in polynomial. However, if we have to output the sequence itself,
it cannot be solved in polynomial time if $\ell$ is not in polynomial.
In this paper, we will show that the shortest sliding token problem is solvable in polynomial time
for the following graph classes:
Proper interval graphs:
We first prove that any two independent sets of
a proper interval graph can be transformed into each other.
In other words, every proper interval graph with two independent sets
$\bfI_b$ and $\bfI_r$ is a yes-instance of the problem if $\msize{\bfI_b}=\msize{\bfI_r}$.
Furthermore, we can find the ordering of tokens to be slid in a minimum-length sequence in $O(n)$ time (implicitly),
even though there exists an infinite family of independent sets on paths
(and hence on proper interval graphs) for which any sequence requires $\Omega(n^2)$ length.
Trivially perfect graphs:
We then give an $O(n)$-time algorithm for trivially perfect graphs
which actually finds a shortest sequence if such a sequence exists.
In contrast to proper interval graphs,
any shortest sequence is of length $O(n)$ for trivially perfect graphs.
Note that trivially perfect graphs form a subclass of cographs,
and hence its polynomial time solvability has been known <cit.>.
We finally give an $O(n^2)$-time algorithm for caterpillars for the shortest sliding token problem.
To make self-contained, we first show a linear time algorithm for decision problem that
asks whether two independent sets can be transformed into each other.
(We note that this problem can be solved in linear time for a tree <cit.>.)
For a yes-instance, we next show an algorithm that finds
a shortest sequence of token sliding between two independent sets.
We here remark that, since the problem is PSPACE-complete in general,
an instance of the sliding token problem may require
the exponential number of independent sets to transform.
In such a case, tokens should make detours to avoid violating to be independent (as shown in <ref>).
As we will see, caterpillars certainly require to make detours to transform.
Therefore, it is remarkable that any yes-instance on
a caterpillar requires a sequence of token-slides of polynomial length.
This is still open even for a tree.
That is, in a tree, we can determine if two independent sets are reconfigurable in linear time due to
<cit.>, however, we do not know if the length of the sequence is in polynomial.
As far as the authors know, this is the first polynomial time algorithm for the shortest sliding token
problem for a graph class that requires detours of tokens.
§ PRELIMINARIES
In this section, we introduce some basic terms and notations.
In the sliding token problem,
we may assume without loss of generality that graphs are simple and connected.
For a graph $G=(V,E)$, we let $n=\msize{V}$ and $m=\msize{E}$.
§.§ Sliding token
For two independent sets $\bfI_i$ and $\bfI_j$ of the same cardinality in a graph $G=(V,E)$,
if there exists exactly one edge $\{u,v\}$ in $G$ such that $\bfI_{i} \setminus\bfI_{j}=\{u\}$
and $\bfI_{j}\setminus\bfI_{i}=\{v\}$,
then we say that $\bfI_{j}$ can be obtained from $\bfI_{i}$ by sliding a token on
the vertex $u \in \bfI_{i}$ to its adjacent vertex $v$ along the edge $\{u,v\}$,
and denote it by $\bfI_{i} \vdash \bfI_{j}$.
We remark that the tokens are unlabeled, while the vertices in a graph are labeled.
A reconfiguration sequence between two independent sets $\bfI_1$ and $\bfI_{\ell}$ of $G$
is a sequence $\langle \bfI_1, \bfI_2, \ldots, \bfI_{\ell} \rangle$ of
independent sets of $G$ such that $\bfI_{i-1} \vdash \bfI_i$ for $i=2, 3, \ldots, \ell$.
We denote by $\bfI_{1} \vdash^* \bfI_{\ell}$ if there exists
a reconfiguration sequence between $\bfI_1$ and $\bfI_{\ell}$.
We note that a reconfiguration sequence is reversible, that is,
we have $\bfI_{1} \vdash^* \bfI_{\ell}$ if and only if $\bfI_{\ell} \vdash^* \bfI_{1}$.
Thus we say that two independent sets $\bfI_1$ and $\bfI_{\ell}$ are
reconfigurable into each other if $\bfI_{1} \vdash^* \bfI_{\ell}$.
The length of a reconfiguration sequence $\calS$ is defined as
the number of independent sets contained in $\calS$.
For example, the length of the reconfiguration sequence in <ref> is $5$.
The sliding token problem is to determine whether two given independent sets
$\bfI_b$ and $\bfI_r$ of a graph $G$ are reconfigurable into each other.
We may assume without loss of generality that $\msize{\bfI_b} = \msize{\bfI_r}$;
otherwise the answer is clearly “no.”
Note that the sliding token problem is a decision problem asking for the existence of a reconfiguration sequence
between $\bfI_b$ and $\bfI_r$, and hence it does not ask an actual reconfiguration sequence.
In this paper, we will consider the shortest sliding token problem that
computes a shortest reconfiguration sequence between two independent sets.
Note that the length of a reconfiguration sequence may not be in polynomial of the size of the graph
since the sequence may contain detours of tokens.
We always denote by $\bfI_b$ and $\bfI_r$ the initial and target independent sets of $G$,
respectively, as an instance of the (shortest) sliding token problem;
we wish to slide tokens on the vertices in $\bfI_b$ to the vertices in $\bfI_r$.
We sometimes call the vertices in $\bfI_b$ blue,
and the vertices in $\bfI_r$ red;
each vertex in $\bfI_b\cap\bfI_r$ is blue and red.
§.§ Target-assignment
We here give another notation of the sliding token problem,
which is useful to explain our algorithm.
Let $\bfI_b=\{b_1,b_2,\ldots,b_k\}$ be an initial independent set of a graph $G$.
For the sake of convenience, we label the tokens on the vertices in $\bfI_b$;
let $t_i$ be the token placed on $b_i$ for each $i$, $1 \le i \le k$.
Let $\calS$ be a reconfiguration sequence between $\bfI_b$ and
an independent set $\bfI$ of $G$, and hence $\bfI_b \vdash^* \bfI$.
Then, for each token $t_i$, $1 \le i \le k$,
we denote by $\fmap{\calS}{t_i}$ the vertex in $\bfI$ on which
the token $t_i$ is placed via the reconfiguration sequence $\calS$.
Notice that $\{ \fmap{\calS}{t_i} \mid 1 \le i \le k\} = \bfI$.
Let $\bfI_r$ be a target independent set of $G$,
which is not necessarily reconfigurable from $\bfI_b$.
Then, we call a mapping $\fallmap: \bfI_b \to \bfI_r$
a target-assignment between $\bfI_b$ and $\bfI_r$.
The target-assignment $\fallmap$ is said to be proper
if there exists a reconfiguration sequence $\calS$ such that
$\fmap{\calS}{t_i} = \fallmap(b_i)$ for all $i$, $1 \le i \le k$.
Note that there is no proper target-assignment between $\bfI_b$ and $\bfI_r$
if $\bfI_b \not\vdash^* \bfI_r$.
Therefore, the sliding token problem can be seen as the problem of
determining whether there exists at least
one proper target-assignment between $\bfI_b$ and $\bfI_r$.
§.§ Interval graphs and subclasses
The neighborhood of a vertex $v$ in a graph $G=(V,E)$ is
the set of all vertices adjacent to $v$,
and we denote it by $N(v) = \{u\in V \mid \{u,v\}\in E\}$.
Let $N[v] = N(v)\cup\{v\}$.
For any graph $G=(V,E)$, two vertices $u$ and $v$ are called strong twins if $N[u]=N[v]$,
and weak twins if $N(u)=N(v)$.
In our problem, strong twins have no meaning:
when $u$ and $v$ are strong twins, only one of them can be used by a token.
Therefore, in this paper, we only consider the graphs without strong twins.
That is, for any pair of vertices $u$ and $v$, we have $N[u]\neq N[v]$.
(We have to take care about weak twins; see Section <ref> for the details.)
A graph $G = (V,E)$ with $V = \{v_1,v_2,\ldots,v_n\}$ is an interval graph
if there exists a set $\calI$ of (closed) intervals $I_1,I_2,\ldots,I_n$
such that $\{v_i,v_j\}\in E$ if and only if $I_i\cap I_j\neq\emptyset$
for each $i$ and $j$ with $1\le i,j\le n$.[In this paper,
a bold $\bfI$ denotes an “independent set,” an italic $I$ denotes an
“interval,” and calligraphy $\calI$ denotes “a set of intervals.”]
We call the set $\calI$ of intervals an interval representation of the graph,
and sometimes identify a vertex $v_i \in V$ with its corresponding interval $I_i \in \calI$.
We denote by $L(I)$ and $R(I)$ the left and right endpoints of
an interval $I \in \calI$, respectively.
That is, we always have $L(I)\le R(I)$ for any interval $I=[L(I),R(I)]$.
To specify the bottleneck of the running time of our algorithms,
we suppose that an interval graph $G=(V,E)$ is given as an input by
its interval representation using $O(n)$ space.
(If necessary, an interval representation of $G$ can be found in $O(n+m)$ time <cit.>.)
More precisely, $G$ is given by a string of length $2n$ over alphabets
$\{L(I_1), L(I_2), \ldots, L(I_n), R(I_1), R(I_2), \ldots, \allowbreak R(I_n)\}$.
For example, a complete graph $K_3$ with three vertices can be given by
an interval representation $L(I_1) L(I_2) L(I_3) \allowbreak R(I_1) R(I_2) R(I_3)$,
and a path of length two is given by an interval representation $L(I_1) L(I_2) R(I_1) L(I_3) R(I_2) R(I_3)$.
An interval graph is proper if it has an interval representation
such that no interval properly contains another.
The class of proper interval graphs is also known as
the class of unit interval graphs <cit.>:
an interval graph is unit if it has an interval representation
such that every interval has unit length.
Hereafter, we assume that each proper interval graph is given in
the interval representation of intervals of unit length.
In the context of the interval representation,
an interval graph is proper if and only if $L(I_i)<L(I_j)$ if and only if $R(I_i)<R(I_j)$.
An interval graph is trivially perfect if
it has an interval representation such that the relationship between
any two intervals is either disjoint or inclusion.
That is, for any two intervals $I_i$ and $I_j$ with $L(I_i)<L(I_j)$, we have either
$L(I_i)<L_(I_j)<R(I_j)<R(I_i)$ or $L(I_i)<R(I_i)<L(I_j)<R(I_j)$.
A caterpillar $G=(V,E)$ is a tree (i.e., a connected acyclic graph)
that consists of two subsets $S$ and $L$ of $V$ as follows.
The vertex set $S$ induces a path $(s_1,\ldots,s_{n'})$ in $G$,
and each vertex $v$ in $L$ has degree 1, and its unique neighbor is in $S$.
We call the path $(s_1,\ldots,s_{n'})$ spine, and each vertex in $L$ leaf.
In this paper, without loss of generality, we assume that $n'\ge 2$,
$\deg(s_1)\ge 2$, and $\deg(s_{n'})\ge 2$.
That is, the endpoints $s_1$ and $s_{n'}$ of
the spine $(s_1,\ldots,s_{n'})$ should have at least one leaf.
It is easy to see that the class of caterpillars is a proper subset of the class of interval graphs,
and these three subclasses are incomparable with each other.
§ PROPER INTERVAL GRAPHS
We show the main theorem in this section for proper interval graphs,
which first says that the answer of sliding token is always “yes” for connected proper interval graphs.
We give a constructive proof of the claim, and it certainly finds a shortest sequence in linear time.
For a connected proper interval graph $G=(V,E)$,
any two independent sets $\bfI_{b}$ and $\bfI_{r}$ with $\msize{\bfI_b} = \msize{\bfI_r}$
are reconfigurable into each other, that is, $\bfI_{b}\vdash^* \bfI_{r}$.
Moreover, the shortest reconfiguration sequence can be found in polynomial time.
We give a constructive proof for Theorem <ref>, that is,
we give an algorithm which actually finds a shortest reconfiguration sequence between
any two independent sets $\bfI_b$ and $\bfI_r$ of a connected proper interval graph $G$.
A connected proper interval graph $G=(V,E)$ has a unique interval representation (up to reversal), and
we can assume that each interval is of unit length in the representation <cit.>.
Therefore, by renumbering the vertices, we can fix an interval representation
$\calI =\{I_1,I_2,\ldots,I_n\}$ of $G$ so that $L(I_i) < L(I_{i+1})$ (and $R(I_i) < R(I_{i+1})$)
for each $i$, $1 \le i \le n-1$, and each interval $I_i \in \calI$ corresponds to the vertex $v_i \in V$.
Let $\bfI_b = \{b_1, b_2, \ldots, b_k\}$ and $\bfI_r=\{r_1, r_2, \ldots, r_k\}$ be
any given initial and target independent sets of $G$, respectively.
Without loss of generality, we assume that the blue vertices $b_1, b_2, \ldots, b_k$
are labeled from left to right (according to the interval representation $\calI$ of $G$),
that is, $L(b_i) < L(b_j)$ if $i < j$; similarly,
we assume that the red vertices $r_1, r_2, \ldots, r_k$ are labeled from left to right.
Then, we define a target-assignment $\fallmap: \bfI_b \to \bfI_r$, as follows:
for each blue vertex $b_i \in \bfI_b$
\begin{equation}
\label{eq:map_proper}
\fallmap(b_i) = r_i.
\end{equation}
To prove Theorem <ref>, it suffices to show that $\fallmap$ is proper,
and each token takes no detours.
§.§ String representation
By traversing the interval representation $\calI$ of a connected proper interval graph $G$
from left to right, we can obtain a string $S =s_1s_2\cdots s_{2k}$ which is a superstring of
both $b_1b_2 \cdots b_k$ and $r_1r_2 \cdots r_k$, that is,
each letter $s_i$ in $S$ is one of the vertices in $\bfI_b \cup \bfI_r$ and
$s_i$ appears in $S$ before $s_j$ if $L(s_i) < L(s_j)$.
We may assume without loss of generality that $s_1=b_1$
since the reconfiguration rule is symmetric in sliding token.
If a vertex is contained in both $\bfI_b$ and $\bfI_r$,
as $b_i$ and $r_j$, then we assume that it appears as $b_i r_j$ in $S$,
that is, the blue vertex $b_i$ appears in $S$ before the red vertex $r_j$.
Then, for each $i$, $1 \le i \le 2k$, we define the height $h(i)$ at $i$ by
the number of blue vertices appeared in the substring $s_1 s_2 \cdots s_i$ minus
the number of red vertices appeared in $s_1 s_2 \cdots s_i$.
For the sake of notational convenience, we define $h(0) = 0$.
Then $h(i)$ can be recursively computed as follows:
\begin{equation}
\label{eq:height}
h(i) = \left\{
\begin{array}{ll}
0 & ~~~\mbox{if $i=0$}; \\
h(i-1) + 1 & ~~~\mbox{if $s_i$ is blue}; \\
h(i-1) - 1 & ~~~\mbox{if $s_i$ is red}.
\end{array} \right.
\end{equation}
Note that $h(2k) = 0$ for any string $S$ since $\msize{\bfI_b} = \msize{\bfI_r}$.
Using the notion of height, we split the string $S$ into substrings $S_1, S_2, \ldots, S_h$
at every point of height $0$, that is, in each substring $S_j = s_{2p+1} s_{2p+2} \cdots s_{2q}$,
we have $h(2q) = 0$ and $h(i) \neq 0$ for all $i$, $2p+1 \le i \le 2q-1$.
For example, a string $S=b_1b_2r_1r_2 b_3r_3 r_4r_5b_4r_6b_5r_7b_6r_8b_7b_8 b_9r_9$
can be split into four substrings
$S_1= b_1b_2r_1r_2$, $S_2=b_3r_3$, $S_3=r_4r_5b_4r_6b_5r_7b_6r_8b_7b_8$ and $S_4=b_9r_9$.
Then, the substrings $S_1, S_2, \ldots, S_h$ form a partition of $S$,
and each substring $S_j$ contains the same number of blue and red tokens.
We call such a partition the partition of $S$ at height $0$.
Let $S_j = s_{2p+1} s_{2p+2} \cdots s_{2q}$ be a substring in the partition of the string $S$ at height $0$. Then,
(a) the blue vertices $b_{p+1}, b_{p+2}, \ldots, b_{q}$ appear in $S_j$,
and their corresponding red vertices $r_{p+1}, r_{p+2}, \ldots, r_{q}$ appear in $S_j$;
(b) if $S_j$ starts with the blue vertex $b_{p+1}$,
then each blue vertex $b_i$, $p+1 \le i \le q$,
appears in $S_j$ before its corresponding red vertex $r_i$; and
(c) if $S_j$ starts with the red vertex $r_{p+1}$,
then each blue vertex $b_i$, $p+1 \le i \le q$, appears in $S_j$ after its corresponding red vertex $r_i$.
By the definitions, the claim (a) clearly holds.
We thus show that the claim (b) holds. (The proof for the claim (c) is symmetric.)
Since $h(2p) = 0$ and $S_j$ starts with a blue vertex, we have $h(2p+1) = 1 > 0$.
We now suppose for a contradiction that there exists
a blue vertex $s_x = b_{i'}$ which appears in $S_j$ after its corresponding red vertex $s_y = r_{i'}$.
Then, $y < x$.
We assume that $y$ is the minimum index among such blue vertices in $S_j$.
Then, in the substring $s_1 s_2 \cdots s_y$ of $S$, there are exactly $i'$ red vertices.
On the other hand, since $y < x$, the substring $s_1 s_2 \cdots s_y$ contains at most $i'-1$ blue vertices.
Therefore, by the definition of height, we have $h(y) < 0$.
Since $h(2p+1) = 1 > 0$ and $h(y) < 0$, by Eq. (<ref>) there must exist an index $z$
such that $h(z) = 0$ and $2p < z < y$.
This contradicts the fact that $S_j$ is a substring in the partition of $S$ at height $0$.
§.§ Algorithm
Recall that we have fixed the unique interval representation $\calI =\{I_1,I_2,\ldots,I_n\}$ of
a connected proper interval graph $G$ so that $L(I_i) < L(I_{i+1})$ for each $i$,
$1 \le i \le n-1$, and each interval $I_i \in \calI$ corresponds to the vertex $v_i \in V$.
Since all intervals in $\calI$ have unit length, the following proposition clearly holds.
For two vertices $v_i$ and $v_j$ in $G$ such that $i < j$,
there is a path $P$ in $G$ which passes through only intervals (vertices)
contained in $[L(I_{i}), R(I_{j})]$.
Furthermore, if $I_{i'} \cap I_i = \emptyset$ for some index $i'$ with $i'<i$,
no vertex in $v_1, v_2, \ldots, v_{i'}$ is adjacent to any vertex in $P$.
If $I_{j} \cap I_{j'} = \emptyset$ for some index $j'$ with $j<j'$,
no vertex in $v_{j'}, v_{j'+1}, \ldots, v_{n}$ is adjacent to any vertex in $P$.
Let $S$ be the string of length $2k$ obtained from two given independent sets $\bfI_b$ and $\bfI_r$
of a connected proper interval graph $G$, where $k = \msize{\bfI_b} = \msize{\bfI_r}$.
Let $S_1, S_2, \ldots, S_h$ be the partition of $S$ at height $0$.
The following lemma shows that the tokens in each substring $S_j$
can always reach their corresponding red vertices.
(Note that we sometimes denote simply by $S_j$ the set of all
vertices appeared in the substring $S_j$, $1 \le j \le h$.)
Let $S_j = s_{2p+1} s_{2p+2} \cdots s_{2q}$ be a substring in the partition of $S$ at height $0$.
Then, there exists a reconfiguration sequence between $\bfI_b \cap S_j$ and $\bfI_r \cap S_j$
such that tokens are slid along edges only in the subgraph of $G$
induced by the vertices contained in $[L(s_{2p+1}), R(s_{2q})]$.
We first consider the case where $S_j$ starts with the blue vertex $b_{p+1}$, that is, $s_{2p+1} = b_{p+1}$.
Then, by Lemma <ref>(b) each blue vertex $b_i$, $p+1 \le i \le q$,
appears in $S_j$ before the corresponding red vertex $r_i$.
Therefore, we know that $s_{2q} = r_q$, and hence it is red.
Suppose that $s_{x} = b_{q}$, then all vertices appeared in $s_{x+1} s_{x+2} \cdots s_{2q}$ are red.
Roughly speaking, we slide the tokens $t_q, t_{q-1}, \ldots, t_{p+1}$ from left to right in this order.
We first claim that the token $t_q$ can be slid from $b_q$ $(= s_x)$ to $r_q$ $(= s_{2q})$.
By Proposition <ref>
there is a path $P$ between $b_q$ and $r_q$ which passes through only intervals contained in $[L(b_q), R(r_q)]$.
Since $\bfI_b$ is an independent set of $G$,
the vertex $b_q$ is not adjacent to any other vertices $b_{p+1}, b_{p+2}, \ldots, b_{q-1}$ in $\bfI_b \cap S_j$.
Since $L(b_{p+1}) < L(b_{p+2}) < \cdots < L(b_{q-1}) < L(b_q)$,
by Proposition <ref> all vertices in $P$ are not adjacent to
any of tokens $t_{p+1}, t_{p+2}, \ldots, t_{q-1}$ that are now placed on
$b_{p+1}, b_{p+2}, \ldots, b_{q-1}$, respectively.
Therefore, we can slide the token $t_q$ from $b_q$ to $r_q$.
We fix the token $t_q$ on $r_q=s_{2q}$, and will not slide it anymore.
We then slide the next token $t_{q-1}$ on $b_{q-1}$ to $r_{q-1}$ along
a path $P'$ which passes through only intervals contained in $[L(b_{q-1}), R(r_{q-1})]$.
Since $\bfI_r$ is an independent set of $G$,
the corresponding red vertex $r_{q-1}$ is not adjacent to $r_q$ on which the token $t_q$ is now placed.
Recall that $L(r_{q-1}) < L(r_q)$, and hence by Proposition <ref>,
$r_q$ is not adjacent to any vertex in $P'$.
Similarly as above, the tokens $t_{p+1}, t_{p+2}, \ldots, t_{q-2}$ are not adjacent to any vertex in $P'$.
Therefore, we can slide the token $t_{q-1}$ from $b_{q-1}$ to $r_{q-1}$.
Repeat this process until the token $t_{p+1}$ on $b_{p+1}$ is slid to $r_{p+1}$.
In this way, there is a reconfiguration sequence between $\bfI_b \cap S_j$ and $\bfI_r \cap S_j$
such that tokens are slid along edges only in the subgraph of $G$ induced by
the vertices contained in $[L(b_{p+1}), R(r_{q})]$.
The symmetric arguments prove the case where $S_j$ starts with the red vertex $r_{p+1}$.
Note that, in this case, we slide the tokens $t_{p+1}, t_{p+2}, \ldots, t_q$ from right to left in this order.
Proof of Theorem <ref>.
We now give an algorithm which slides all tokens on the vertices in $\bfI_{b}$ to the vertices in $\bfI_r$.
Recall that $S_1, S_2, \ldots, S_h$ are the substrings in the partition of $S$ at height $0$.
Intuitively, the algorithm repeatedly picks up one substring $S_j$,
and slides all tokens in $\bfI_b \cap S_j$ to $\bfI_r \cap S_j$.
By Lemma <ref> it works locally in each substring $S_j$,
but it should be noted that a token in $S_j$ may be adjacent to another token
in $S_{j-1}$ or $S_{j+1}$ at the boundary of the substrings.
To avoid this, we define a partial order over the substrings $S_1, S_2, \ldots, S_h$, as follows.
Consider any two consecutive substrings $S_j$ and $S_{j+1}$,
and let $S_j = s_{2p+1} s_{2p+2} \allowbreak \cdots s_{2q}$.
Then, the first letter of $S_{j+1}$ is $s_{2q+1}$.
We first consider the case where both $s_{2q}$ and $s_{2q+1}$ are the same color.
Then, since $s_{2q}$ and $s_{2q+1}$ are both in the same independent set of $G$,
they are not adjacent.
Therefore, by Proposition <ref> and Lemma <ref>,
we can deal with $S_j$ and $S_{j+1}$ independently.
In this case, we thus do not define the ordering between $S_j$ and $S_{j+1}$.
We then consider the case where $s_{2q}$ and $s_{2q+1}$ have different colors;
in this case, we have to define their ordering.
Suppose that $s_{2q}$ is blue and $s_{2q+1}$ is red;
then we have $s_{2q} = b_q$ and $s_{2q+1} = r_{q+1}$.
By Lemma <ref> the token $t_q$ on $s_{2q}$ is slid to left,
and the token $t_{q+1}$ will reach $r_{q+1}$ from right.
Therefore, the algorithm has to deal with $S_j$ before $S_{j+1}$.
Note that, after sliding all tokens $t_{p+1}, t_{p+2}, \ldots, t_q$ in $S_j$,
they are on the red vertices $r_{p+1}, r_{p+2}, \ldots, r_q$, respectively,
and hence the tokens in $S_{j+1}$ are not adjacent to any of them.
By the symmetric argument, if $s_{2q}$ is red and $s_{2q+1}$ is blue,
$S_{j+1}$ should be dealt with before $S_j$.
Notice that such an ordering is defined only for two consecutive substrings $S_j$ and $S_{j+1}$, $1 \le j \le h-1$.
Therefore, the partial order over the substrings $S_1, S_2, \ldots, S_h$ is acyclic,
and hence there exists a total order which is consistent with the partial order defined above.
The algorithm certainly slides all tokens from $\bfI_b$ to $\bfI_r$ according to the total order.
Therefore, the target-assignment $\fallmap$ defined in Eq. (<ref>) is proper,
and hence $\bfI_{b}\vdash^* \bfI_{r}$.
Therefore, there always exists a reconfiguration sequence
between two independent sets $\bfI_b$ and $\bfI_r$ of a connected proper interval graph $G$.
We now discuss the length of reconfiguration sequences between $\bfI_b$ and $\bfI_r$,
together with the running time of our algorithm.
For two given independent sets $\bfI_b$ and $\bfI_r$ of
a connected proper interval graph $G$ with $n$ vertices,
$(1)$ the ordering of tokens to be slid in a shortest reconfiguration sequence between
them can be computed in $O(n)$ time and $O(n)$ space; and
$(2)$ a shortest reconfiguration sequence between them can be output
in $O(n^2)$ time and $O(n)$ space.
We first modify our algorithm
so that it finds a shortest reconfiguration sequence between $\bfI_b$ and $\bfI_r$.
To do that, it suffices to slide each token $t_i$, $1 \le i \le k$,
from the blue vertex $b_i$ to its corresponding red vertex $r_i$ along the shortest path between $b_i$ and $r_j$.
We may assume without loss of generality that $L(b_i) < L(r_i)$,
that is, the token $t_i$ will be slid from left to right.
Then, for the interval $b_i$, we choose an interval $I_{j} \in \calI$
such that $b_i \cap I_{j} \neq \emptyset$ and $L(I_j)$ is the maximum among all $I_{j'} \in \calI$.
If $L(r_i)\le L(I_j)$, we can slide $t_i$ from $b_i$ to $r_i$ directly;
otherwise we slide $t_i$ to the vertex $I_j$, and repeat.
We then prove the claim (1).
If we simply want to compute the ordering of tokens to be slid in a shortest reconfiguration sequence,
it suffices to compute the partial order over the substrings $S_1, S_2, \ldots, S_h$
in the partition of the string $S$ at height $0$.
It is not difficult to implement our algorithm in Section <ref>
to run in $O(n)$ time and $O(n)$ space.
Therefore, the claim (1) holds.
We finally prove the claim (2).
Remember that each token $t_i$, $1 \le i \le k$, is slid along the shortest path from $b_i$ to $r_i$.
Furthermore, once the token $t_i$ reaches $r_i$, it is not slid anymore.
Therefore, the length of a shortest reconfiguration sequence
between $\bfI_{b}$ and $\bfI_{r}$ is given by
the sum of all lengths of the shortest paths between $b_i$ and $r_i$.
It is clear that this sum is $O(kn)=O(n^2)$.
We output only the shortest paths between $b_i$ and $r_i$,
together with the ordering of the tokens to be slid.
Therefore, the claim (2) holds.
This proposition also completes the proof of Theorem <ref>.
It is remarkable that there exists an infinite family of instances
for which any reconfiguration sequence requires $\Omega(n^2)$ length.
To show this, we give an instance such that each shortest path between $b_i$ and $r_i$ is $\Theta(n)$.
Simple example is: $G$ is a path $(v_1,v_2,\ldots,v_{8k})$ of length $n=8k$ for any positive integer $k$,
$\bfI_b=\{v_1,v_3,v_5,\ldots,v_{2k-1}\}$, and $\bfI_r=\{v_{6k+2},v_{6k+4},\ldots,v_{8k}\}$.
In this instance, each token $t_i$ must be slid $\Theta(n)$ times,
and hence it requires $\Theta(n^2)$ time to output all of them.
We note that a path is not only a proper interval graph, but also a caterpillar.
Thus this simple example also works as a caterpillar.
§ TRIVIALLY PERFECT GRAPHS
The main result of this section is the following theorem.
The sliding token problem for
a trivially perfect graph $G = (V,E)$ can be solved in $O(n)$ time and $O(n)$ space.
Furthermore, one can find a shortest reconfiguration sequence between
two given independent sets $\bfI_b$ and $\bfI_r$ in $O(n)$ time and $O(n)$ space if there exists.
In this section, we explicitly give such an algorithm as a proof of Theorem <ref>.
Note that there are no-instances for trivially perfect graphs.
However, for trivially perfect graphs, we construct
a proper target-assignment between $\bfI_b$ and $\bfI_r$ efficiently if it exists.
§.§ $\MPQ$-tree for trivially perfect graphs
The $\MPQ$-tree of an interval graph $G$ is a kind of decomposition tree,
developed by Korte and Möhring <cit.>,
which represents the set of all feasible interval representations of $G$.
For an interval graph $G$,
although there are exponentially many interval representations for $G$,
its corresponding $\MPQ$-tree is unique up to isomorphism.
For the notion of $\MPQ$-trees, the following theorem is known:
For any interval graph $G=(V,E)$, its corresponding $\MPQ$-tree can be constructed in $O(n+m)$ time.
Since it is involved to define $\MPQ$-tree for general interval graphs,
we here give a simplified definition of $\MPQ$-tree only for the class of trivially perfect graphs.
(See <cit.> for the detailed definition of the $\MPQ$-tree for a general interval graph.)
Let $G=(V,E)$ be a trivially perfect graph. Recall that a trivially perfect graph has
an interval representation such that the relationship between any two intervals is either disjoint or inclusion.
Then, the $\MPQ$-tree $\calT$ of $G$ is a rooted tree such that each node,
called a $\PP$-node, in $\calT$ is associated with a non-empty set of vertices in $G$ such that
(a) each vertex $v\in V$ appears in exactly one $\PP$-node in $\calT$, and
(b) if a vertex $v_i \in V$ is in an ancestor node of another node that contains $v_j \in V$,
then $L(I_i)\le L(I_j)<R(I_j)\le R(I_i)$ in any interval representation of $G$,
where $v_i$ and $v_j$ correspond to the intervals $I_i$ and $I_j$, respectively (see <ref> as an example).
By the property (b), the ancestor/descendant relationship on $\calT$ corresponds to
the inclusion relationship in the interval representation of $G$.
Thus, $N[v_j]\subseteq N[v_i]$ if $v_i$ is in an ancestor node of another node that contains $v_j$ in the $\MPQ$-tree.
Let $\calT$ be the (unique) $\MPQ$-tree of a connected trivially perfect graph $G = (V,E)$.
For two vertices $u$ and $w$ in $G$,
we denote by $\LCA(u,w)$ the least common ancestor in $\calT$ for the nodes containing $u$ and $w$.
By the property (a) the node $\LCA(u,w)$ can be uniquely defined.
(a) A trivially perfect graph in an interval representation, and (b) its $\MPQ$-tree.
§.§ Basic properties and key lemma
Let $\calT$ be the (unique) $\MPQ$-tree of a connected trivially perfect graph $G = (V,E)$.
Recall that the interval representation of a trivially perfect graph has just disjoint or inclusion relationship.
This fact implies the following observation.
Every pair of vertices $u$ and $w$ in a connected trivially perfect graph $G$ has
a path of length at most two via a vertex in $\LCA(u,w)$.
We first observe that every $\PP$-node of $\calT$ is non-empty.
If the root is empty, the graph is disconnected, which is a contradiction.
If some non-root $\PP$-node $P$ is empty, joining all children of $P$ to the parent of $P$,
we obtain a simpler $\MPQ$-tree than $\calT$, which contradicts the construction of
the unique $\MPQ$-tree in <cit.>. Thus every $\PP$-node is non-empty.
Therefore, there is at least one vertex $v$ in $\LCA(u,w)$.
Then, the property (b) implies that $N[u]\subseteq N[v]$ and $N[w]\subseteq N[v]$,
and hence $\{u,v\}$ and $\{w,v\}$ are both in $E$.
Therefore, there is a path $(u,v,w)$ of length at most two between $u$ and $w$ via $v$.
When we have $v=u$ or $v = w$, the path degenerates to the edge $\{u,w\} \in E$.
Let $\LCA^*(u,w)$ be the set of vertices in $V$ appearing in
the $\PP$-nodes on the (unique) path from $\LCA(u,w)$ to the root of the $\MPQ$-tree.
By the definition of $\MPQ$-tree, we clearly have the following observation.
(Recall also that each token must be slid along an edge of $G$.)
Consider an arbitrary reconfiguration sequence $\calS$
which slides a token $t_i$ from $b_i \in \bfI_b$ to some vertex $r_i$.
Then, $t_i$ must pass through at least one vertex in $\LCA^*(b_i, r_i)$,
that is, there exists at least one independent set $\bfI'$ in $\calS$
such that $\bfI' \cap \LCA^*(b_i, r_i) \neq \emptyset$.
We are now ready to give the key lemma for trivially perfect graphs.
Let $\fallmap: \bfI_b \to \bfI_r$ be a target-assignment between $\bfI_b$ and $\bfI_r$.
Then, $\fallmap$ is proper if and only if
the nodes $\LCA(b_i, \fallmap(b_i))$ and $\LCA(b_j, \fallmap(b_j))$ are not in
the ancestor/descendant relationship on $\calT$ for every pair of vertices $b_i, b_j \in \bfI_b$.
We first show the sufficiency.
For a target-assignment $\fallmap$ between $\bfI_b$ and $\bfI_r$,
suppose that the nodes $\LCA(b_i, \fallmap(b_i))$ and $\LCA(b_j, \fallmap(b_j))$
are not in the ancestor/descendant relationship on $\calT$ for every pair of vertices $b_i, b_j \in \bfI_b$.
Then, we can simply slide the tokens one by one in an arbitrary order;
by Observation <ref> each token $t_i$, $1 \le i \le k$,
can be slid along a path from $b_i$ to $\fallmap(b_i)$ via a vertex $v_{i'}$ in $\LCA(b_i, \fallmap(b_i))$.
Note that there is no token $t_j$ adjacent to $v_{i'}$,
because the nodes $\LCA(b_i, \fallmap(b_i))$ and $\LCA(b_j, \fallmap(b_j))$ are not in
the ancestor/descendant relationship on $\calT$.
Thus, there is a reconfiguration sequence between $\bfI_b$ and $\bfI_r$ according to $\fallmap$,
and hence $\fallmap$ is proper.
We then show the necessity.
Suppose that $\fallmap$ is proper,
and suppose for a contradiction that
there exists a pair of vertices $b_i, b_j \in \bfI_b$ such that
the nodes $\LCA(b_i, \fallmap(b_i))$ and $\LCA(b_j, \fallmap(b_j))$ are in
the ancestor/descendant relationship on $\calT$;
without loss of generality, we assume that $\LCA(b_i, \fallmap(b_i))$ is
an ancestor of $\LCA(b_j, \fallmap(b_j))$.
Since $\fallmap$ is proper,
there exists a reconfiguration sequence $\calS$ between $\bfI_b$ and $\bfI_r$
which slides the token $t_i$ from $b_i$ to $\fallmap(b_i)$ and also slides
the token $t_j$ from $b_j$ to $\fallmap(b_j)$.
By Observation <ref>
there is at least one vertex $v_{i'}$ in $\LCA^*(b_i, \fallmap(b_i))$ which is passed through by $t_i$.
Similarly, there is at least one vertex $v_{j'}$ in $\LCA^*(b_j, \fallmap(b_j))$
which is passed through by $t_j$.
Let $P_i$ and $P_j$ be the $\PP$-nodes that contains $v_{i'}$ and $v_{j'}$, respectively.
Since $\LCA(b_i, \fallmap(b_i))$ and $\LCA(b_j, \fallmap(b_j))$ are in
the ancestor/descendant relationship on $\calT$, so are $P_i$ and $P_j$.
First suppose $P_j$ is an ancestor of $P_i$.
Then we have
$N[b_{j}] \subseteq N[v_{i'}]$, $N[\fallmap(b_j)]\subseteq N[v_{i'}]$ and $N[v_{j'}]\subseteq N[v_{i'}]$.
Therefore, if we slide $t_i$ via $v_{i'}$, then $t_i$ would be adjacent to
the other token $t_j$ which is on one of the three vertices $b_{j}$, $\fallmap(b_j)$ and $v_{j'}$.
Thus, the token $t_j$ should “escape” from $b_j$ before sliding $t_i$.
However, we can establish the same argument for any descendant of $P_i$,
and hence $t_j$ must escape to some vertex $u$ that is contained in an ancestor of $P_i$ at first.
However, the vertex $u$ is adjacent to all of $b_{i}, \fallmap(b_i), v_{i'}$,
and hence $t_j$ cannot escape before sliding $t_i$.
This contradicts the assumption that $\calS$ slides the token $t_i$
from $b_i$ to $\fallmap(b_i)$ and also slides the token $t_j$ from $b_j$ to $\fallmap(b_j)$.
The other case, $P_i$ is an ancestor of $P_i$, is symmetric.
§.§ Algorithm and its correctness
We now describe our linear-time algorithm for a trivially perfect graph.
Let $\calT$ be the $\MPQ$-tree of a connected trivially perfect graph $G = (V,E)$.
Let $\bfI_b=\{b_1,b_2,\ldots,b_k\}$ and $\bfI_r=\{r_1,r_2,\ldots,r_k\}$ be given
initial and target independent sets of $G$, respectively.
Then, we determine whether $\bfI_b \vdash^* \bfI_r$ as follows:
(A) construct some particular target-assignment $\fallmap^*$
between $\bfI_b$ and $\bfI_r$; and
(B) check whether $\fallmap^*$ is proper or not, using Lemma <ref>.
We will show later in Lemma <ref> that
it suffices to check only $\fallmap^*$ in order to determine
whether $\bfI_b \vdash^* \bfI_r$ or not.
Indeed, our linear-time algorithm executes (A) and (B) above at the same time,
in the bottom-up manner based on $\calT$.
Description of the algorithm
Remember that the vertex-set associated to each $\PP$-node in $\calT$ induces a clique in $G$.
Therefore, for any independent set $\bfI$ of $G$,
each $\PP$-node contains at most one vertex in $\bfI$,
and hence contains at most one token.
We put a “blue token” for each $\PP$-node containing a blue vertex in $\bfI_b$,
and also put a “red token” for each $\PP$-node containing a red vertex in $\bfI_r$.
Note that a $\PP$-node may contain a pair of blue and red tokens.
Our algorithm lifts up the tokens from the leaves to the root of $\calT$,
and if a blue token $b$ meets a red token $r$ at their least common ancestor
$\LCA(b,r)$ in $\calT$, then we replace them by a single “green token.”
This corresponds to setting $\fallmap^*(b) = r$.
More precisely, at initialization step,
the algorithm first collects all leaves of $\calT$ in a queue, which is called frontier.
The algorithm marks the nodes in the frontier,
and lifts up each token to its parent $\PP$-node.
Each $\PP$-node $P$ is put into the frontier if its all children are marked,
and then, all children of $P$ are removed from the frontier after
the following procedure at $P$:
Case (1) $P$ contains at most one token: the algorithm has nothing to do.
Case (2) $P$ contains only one pair of blue token $b$ and red token $r$:
the algorithm replaces them by a single green token, and let $\fallmap^*(b) = r$.
Case (3) $P$ contains only green tokens:
the algorithm replaces them by a single green token.
Case (4) $P$ contains two or more blue tokens, or two or more red tokens:
the algorithm outputs “no” and halts
(that is, $\bfI_b\not\vdash^* \bfI_r$ in this case).
Case (5) $P$ contains at least one green token and at least one blue or red token:
the algorithm outputs “no” and halts
(that is, $\bfI_b\not\vdash^* \bfI_r$ in this case).
Repeating this process, and the algorithm outputs “yes”
if and only when the frontier contains only
the root $\PP$-node $r$ of $\calT$ which is in one of Cases (1)–(3) above.
Correctness of the algorithm
It is not difficult to implement our algorithm to run in $O(n)$ time and $O(n)$ space.
Therefore, we here prove the correctness of the algorithm.
We first show that $\bfI_b \vdash^* \bfI_r$ if the algorithm outputs “yes.”
In this case, the algorithm is in Cases (1), (2), or (3)
at each $\PP$-nodes in $\calT$ (including the root $r$).
Then, the target-assignment $\fallmap^*$ has been (completely) constructed:
for each blue vertex $b_i \in \bfI_b$,
$\fallmap^*(b_i)$ is the red vertex in $\bfI_r$ such that
$\LCA(b_i, v_{i'})$ has the minimum height in $\calT$ among
all vertices $v_{i'} \in \bfI_r$.
Then, we have the following lemma.
If the algorithm outputs “yes,” then $\bfI_b \vdash^* \bfI_r$.
By Lemma <ref> it suffices to show that
the target-assignment $\fallmap^*$ constructed by
the algorithm satisfies that the nodes $\LCA(b_i, \fallmap^*(b_i))$ and
$\LCA(b_j, \fallmap^*(b_j))$ are not in the ancestor/descendant relationship
on $\calT$ for every pair of vertices $b_i, b_j \in \bfI_b$.
We first consider the case where a $\PP$-node $P$ is in Case (2).
Then, there is exactly one pair of a blue token $b$ and
a red token $r = \fallmap^*(b)$, and $P=\LCA(b,r)$.
Since $b$ and $r$ did not meet any other tokens before $P$,
the subtree $\calT_P$ of $\calT$ contains only the two tokens $b$ and $r$.
Therefore, the lemma clearly holds for $\calT_P$.
We then consider the case where a $\PP$-node $P$ is in Case (3).
Then, two or more least common ancestors of pairs of blue and red tokens meet at this node $P$.
Notice that the green tokens were placed on children's node of $P$ in
the previous step of the algorithm, and hence they were sibling in $\calT$.
Therefore, their corresponding least common ancestors are not in
the ancestor/descendant relationship on $\calT$.
The following lemma completes the correctness proof of our algorithm.
If the algorithm outputs “no,” then $\bfI_b \not\vdash^* \bfI_r$.
We assume that the algorithm outputs “no.”
Then, by Lemma <ref>, it suffices to show that
there is no target-assignment $\fallmap$ between
$\bfI_b$ and $\bfI_r$ such that $\LCA(b_i, \fallmap(b_i))$ and
$\LCA(b_j, \fallmap(b_j))$ are not in the ancestor/descendant
relationship on $\calT$ for every pair of vertices $b_i, b_j \in \bfI_b$.
Suppose for a contradiction that
the algorithm outputs “no,” but there exists
a target-assignment $\fallmap'$ between $\bfI_b$ and $\bfI_r$
such that $\LCA(b_i, \fallmap'(b_i))$ and $\LCA(b_j, \fallmap'(b_j))$
are not in the ancestor/descendant relationship on $\calT$
for every pair of vertices $b_i, b_j \in \bfI_b$.
Then, by Lemma <ref>, $\fallmap'$ is proper and hence $\bfI_b\vdash^* \bfI_r$.
Since the algorithm outputs “no,”
there is a $\PP$-node $P$ which is in either Case (4) or (5).
We first assume that the $\PP$-node $P$ is in Case (4).
Without loss of generality,
at least two blue tokens $b_1$ and $b_2$ meet at this node $P$.
Then, the $\MPQ$-tree $\calT$ contains two red tokens $r_1$ and $r_2$
placed on $\fallmap'(b_1)$ and $\fallmap'(b_2)$, respectively.
Notice that, since $b_1$ and $b_2$ did not meet any red token before
at the node $P$,
both $r_1$ and $r_2$ must be placed on either $P$ or nodes in $\calT \setminus \calT_P$.
Then, the least common ancestor $\LCA(b_1, \fallmap'(b_1))$ must be an ancestor of $P$,
and so is the least common ancestor $\LCA(b_2, \fallmap'(b_2))$.
Therefore, the nodes $\LCA(b_1, \fallmap'(b_1))$ and $\LCA(b_2, \fallmap'(b_2))$
are in the ancestor/descendant relationship, a contradiction.
Thus, the algorithm outputs “no” because the $\PP$-node $P$ is in Case (5).
In this case, without loss of generality,
at least one blue token $b_1$ and at least one green token $c_1$ meet at this node $P$.
Then, the red token $r_1$ corresponding to $\fallmap'(b_1)$ must be placed on
either $P$ or some node in $\calT \setminus \calT_P$.
Therefore, the least common ancestor $\LCA(b_1, \fallmap'(b_1))$ is an ancestor of $c_1$.
Note that $c_1$ corresponds to the least common ancestor of some pair of blue and red tokens,
say $b_j$ and $\fallmap'(b_j)$, and $p$ is an ancestor of it.
Therefore, the nodes $\LCA(b_1, \fallmap'(b_1))$ and $\LCA(b_j, \fallmap'(b_j))$ are
in the ancestor/descendant relationship on $\calT$, a contradiction.
§.§ Shortest reconfiguration sequence
To complete the proof of Theorem <ref>,
we finally show that our algorithm in Section <ref>
can be modified so that it actually finds a shortest reconfiguration sequence between $\bfI_b$ and $\bfI_r$.
Once we know that $\bfI_b \vdash^* \bfI_r$ holds by the $O(n)$-time algorithm
in Section <ref>,
we run it again with modification that “green” tokens are left at
the corresponding least common ancestors.
As in the proof of Lemma <ref>, we can now obtain a reconfiguration sequence
$\calS = \langle \bfI_1, \bfI_2, \ldots, \bfI_{\ell} \rangle$
between $\bfI_b = \bfI_1$ and $\bfI_r = \bfI_{\ell}$ such that
each token $t_i$, $1 \le i \le k$, is slid at most twice.
It is sufficient to output $\bfI_{i+1}\setminus \bfI_i$ and $\bfI_{i}\setminus \bfI_{i+1}$,
and hence the running time of the modified algorithm is proportional to $\ell$,
the number of independent sets in $\calS$.
Since $k=\msize{\bfI_b}=O(n)$ and each token $t_i$, $1 \le i \le k$,
is slid at most twice in $\calS$,
we have $\ell = O(n)$, that is, the length $\ell$ of $\calS$ is $O(n)$.
Therefore, the modified algorithm also runs in $O(n)$ time and $O(n)$ space.
Notice that each token $t_i$ is slid to its target vertex
$\fallmap^*(b_i)$ along a shortest path (of length at most two)
between $b_i$ and $\fallmap^*(b_i)$ without detour, and hence $\calS$ has the minimum length.
This completes the proof of Theorem <ref>.
§ CATERPILLARS
The main result of this section is the following theorem.
The sliding token problem for a connected caterpillar $G = (V,E)$
and two independent sets $\bfI_b$ and $\bfI_r$ of $G$ can be solved in $O(n)$ time and $O(n)$ space.
Moreover, for a yes-instance,
a shortest reconfiguration sequence between them can be output in $O(n^2)$ time and $O(n)$ space.
Let $G=(S\cup L,E)$ be a caterpillar with spine $S$ which induces
the path $(s_1,\ldots,s_{n'})$, and leaf set $L$.
We assume that ${n'}\ge 2$, $\deg(s_1)\ge 2$, and $\deg(s_{n'})\ge 2$.
First we show that we can assume that each spine vertex has at most one leaf without loss of generality.
For any given caterpillar $G=(S\cup L,E)$ and two independent sets $\bfI_b$ and $\bfI_r$ on $G$,
there is a linear time reduction from them to another caterpillar
$G'=(S'\cup L',E')$ and two independent sets $\bfI_b'$ and $\bfI_r'$ such that
(1) $G$, $\bfI_b$, and $\bfI_r$ are a yes-instance of the sliding token problem
if and only if $G'$, $\bfI_b'$, and $\bfI_r$ are a yes-instance of the sliding token problem,
(2) the maximum degree of $G'$ is at most 3, and
(3) $\deg(s_1)=\deg(s_{n'})=2$, where $n'=\msize{S'}$.
In other words, the sliding token problem on a caterpillar
is sufficient to consider only caterpillars of maximum degree 3.
On $G$, let $s_i$ be any vertex in $S$ with $\deg(s_i)>3$.
Then there exist at least two leaves $\ell_i$ and $\ell_i'$ attached to $s_i$
(note that they are weak twins).
Now we consider the case that two tokens in $\bfI_b$ are on $\ell_i$ and $\ell_i'$.
Then, we cannot slide these two tokens at all, and any other token cannot pass through $s_i$ since
it is blocked by them.
If $\bfI_r$ contains these two tokens also,
we can split the problem into two subproblems by removing $s_i$ and
its leaves from $G$, and solve it separately.
Otherwise, the answer is “no” (remind that the problem is reversible;
that is, if tokens cannot be slid, there are no other tokens which slide into the situation).
Therefore, if at least two tokens are placed on the leaves of
a vertex of the original graph, we can reduce the case in linear time.
Thus we assume that every spine vertex with its leaves contains
at most one token in $\bfI_b$ and $\bfI_r$, respectively.
Then, by the same reason, we can remove all leaves but one of each spine vertex.
More precisely, regardless whether $\bfI_b \vdash^* \bfI_r$ or $\bfI_b \not\vdash^* \bfI_r$,
at most one leaf for each spine vertex is used for the transitions.
Therefore, we can remove all other useless leaves but one from each spine vertex.
Especially, removing all useless leaves, we have $\deg(s_1)=\deg(s_{n'})=2$.
Hereafter, we only consider the caterpillars stated in Lemma <ref>.
That is, for any given caterpillar $G=(S\cup L,E)$ with spine $(s_1,\ldots,s_{n'})$,
we assume that $\deg(s_1)=\deg(s_{n'})=2$ and $2\le \deg(s_i)\le 3$ for each $1<i<n'$.
Then, we denote the unique leaf of $s_i$ by $\ell_i$ if it exists.
We here introduce a key notion of the problem on these caterpillars that is named locked path.
Let $G$ and $\bfI$ be a caterpillar and an independent set of $G$, respectively.
A path $P=(p_1,p_2,\ldots,p_k)$ on $G$ is locked by $\bfI$ if and only if
(a) $k$ is odd and greater than 2,
(b) $\bfI \cap P=\{p_1,p_3,p_5,\ldots,p_k\}$,
(c) $\deg(p_1)=\deg(p_k)=1$ (in other words, they are leaves), and
$\deg(p_3)=\deg(p_5)=\cdots =\deg(p_{k-2})=2$.
This notion is simplified version of a locked tree used in <cit.>.
Using the discussion in <cit.>,
we obtain the condition for the immovable independent set on a caterpillar:
Let $G$ and $\bfI$ be a caterpillar and an independent set of $G$, respectively.
Then we cannot slide any token in $\bfI$ on $G$ at all
if and only if there exist a set of locked paths $P_1,\ldots,P_{h}$ for some $h$
such that $\bfI$ is a union of them.
The proof can be found in <cit.>, and omitted here.
Intuitively, for any caterpillar $G$ and its independent set $\bfI$,
if $\bfI$ contains a locked path $P$, we cannot slide any token through the vertices in $P$.
Therefore, $P$ splits $G$ into two subgraphs, and we obtain two completely separated subproblems.
(We note that the endpoints of $P$ are leaves with tokens, and their neighbors are spine vertices without
tokens. This property admits us to cut the graph at the spine vertices on the locked path.)
Therefore, we obtain the following lemma:
For any given caterpillar $G=(S\cup L,E)$ and two independent sets $\bfI_b$ and $\bfI_r$ on $G$,
there is a linear time reduction from them to another caterpillar
$G'=(S'\cup L',E')$ and two independent sets $\bfI_b'$ and $\bfI_r'$ such that
(1) $G$, $\bfI_b$, and $\bfI_r$ are a yes-instance of the sliding token problem
if and only if $G'$, $\bfI_b'$, and $\bfI_r$ are a yes-instance of the sliding token problem, and
(2) both of $\bfI_b'$ and $\bfI_r'$ contain no locked path.
In $G$, when $\bfI_b$ contains a locked path $P$,
it should be appear in $\bfI_r$; otherwise, the answer is no.
Therefore, we can remove all vertices in $P$ and obtain the new graph
$G''$ with two independent sets $\bfI_b'':=\bfI_b\setminus P$ and $\bfI_r:=\bfI_r\setminus P$ such that
$G$ with $\bfI_b$ and $\bfI_r$ is a yes-instance if and only if
$G''$ with $\bfI_b''$ and $\bfI_r''$ is a yes-instance.
Repeating this process, we obtain disconnected caterpillar $\hat{G}$ and two independent sets $\hat{\bfI_b}$ and
$\hat{\bfI_r}$ such that both of $\hat{\bfI_b}$ and $\hat{\bfI_r}$ contain no locked paths.
On a disconnected graph, we can solve the problem separately for each connected component.
Therefore, we can assume that the graph is connected, which completes the proof.
Hereafter, without loss of generality,
we assume that the caterpillar $G$ with two independent sets $\bfI_b$ and $\bfI_r$ satisfies the conditions
in Lemmas <ref> and <ref>.
That is, each spine vertex $s_i$ has at most one leaf $\ell_i$,
$s_1$ and $s_{n'}$ have one leaf $\ell_1$ and $\ell_{n'}$, respectively,
both of $\bfI_b$ and $\bfI_r$ contain no locked path, and $\msize{\bfI_b}=\msize{\bfI_r}$.
By the result in <cit.>, this is a yes-instance.
Thus, it is sufficient to show an $O(n^2)$ time algorithm that
computes a shortest reconfiguration sequence between $\bfI_b$ and $\bfI_r$.
Each pair $(s_i,\ell_i)$ can have at most one token.
Therefore, without loss of generality, we can assume that the blue vertices $b_1, b_2, \ldots, b_k$ in $\bfI_b$
are labeled from left to right (according to the order $(s_1,\ell_1)$, $(s_2,\ell_2)$, $\ldots$, $(s_{n'},\ell_{n'})$ of $G$),
that is, $L(b_i) < L(b_j)$ if $i < j$; similarly,
the red vertices $r_1, r_2, \ldots, r_k$ are also labeled from left to right.
Then, we define a target-assignment $\fallmap: \bfI_b \to \bfI_r$, as follows:
for each blue vertex $b_i \in \bfI_b$
\begin{equation}
\label{eq:map_caterpillar}
\fallmap(b_i) = r_i.
\end{equation}
To prove Theorem <ref>, it suffices to show that $\fallmap$ is proper,
and we can slide tokens with fewest detours.
Here, any token cannot bypass the other token since each token is on a leaf or spine vertex.
Thus, by the results in <cit.>, it has been shown that $\fallmap$ is proper.
We show that we can compute a shortest reconfiguration in case analysis.
Now we introduce direction of a token $t$ denoted by $dir(t)$ as follows:
when $t$ slides from $v_i\in \{s_i,\ell_i\}$ in $\bfI_b$ to $v_j\in \{s_j,\ell_j\}$ in $\bfI_r$ with $i<j$,
the direction of $t$ is said to be R and denoted by $dir(t)=R$.
If $i>j$, it is said to be L and denoted by $dir(t)=L$.
If $i=j$, the direction of $t$ is said to be C and denoted by $dir(t)=C$.
The most right R token $a$ has to precede the most left L token $c$.
We first consider a simple case: all directions are either R or L.
In this case, we can use the same idea appearing in the algorithm for
a proper interval graph in Section <ref>.
We can introduce a partial order over the tokens,
and slide them straightforwardly using the same idea in
Section <ref>.
Intuitively, a sequence of R tokens are slid from left to right,
and a sequence of L tokens are slid from right to left,
and we can define a partial order over the sequences of different directions.
The only additional considerable case is shown in <ref>.
That is, when the token $a$ slides to $\ell_i$ from left and
the other token $c$ slides to $s_{i+1}$ from right, $a$ should precede $c$.
It is not difficult to see that this (and its symmetric case) is the only exception
than the algorithm in Section <ref>
when all tokens slide to right or left.
In other words, in this case, detour is required, and unavoidable.
We next suppose that $\bfI_b$ (and hence $\bfI_r$) contains some token $t$ with $dir(t)=C$.
In other words, $t$ is put on $s_i$ or $\ell_i$ for some $i$ in both of $\bfI_b$ and $\bfI_r$.
We have five cases.
Case (1):
$t$ is put on $\ell_i$ in $\bfI_b$ and $\bfI_r$.
In this case, we have nothing to do; $t$ does not need to be slid.
Case (2):
$t$ is put on $s_i$ in $\bfI_b$ and slid to $\ell_i$ in $\bfI_r$.
In this case, we first slide it from $s_i$ to $\ell_i$, and do nothing any more.
Then no detour is needed for $t$.
Case (3):
$t$ is put on $\ell_i$ in $\bfI_b$ and slid to $s_i$ in $\bfI_r$.
In this case, we lastly slide it from $\ell_i$ to $s_i$, and
no detour is needed for $t$ again.
Case (4):
$t$ is put on $s_i$ in $\bfI_b$ and $\bfI_r$, and $\ell_i$ exists.
Using a simple induction by the number of tokens, we can determine if $t$ should make a detour or not in linear time.
If not, we never slide $t$. Otherwise, we first slide $t$ to $\ell_i$, and lastly slide back from $\ell_i$ to $s_i$.
It is clear that the length of detour with respect to $t$ is as few as possible.
Case (5):
$t$ is put on $s_i$ in $\bfI_b$ and $\bfI_r$, and $\ell_i$ does not exist.
By assumption, $1<s<n'$ (since $\ell_1$ and $\ell_{n'}$ exist).
Without loss of generality, we suppose $t$ is the leftmost spine vertex having the condition.
We first observe that $\msize{\bfI_b\cap\{s_{i-1},\ell_{i-1},s_{i+1},\ell_{i+1}\}}$ is at most 1.
Clearly, we have no token on $s_{i-1}$ and $s_{i+1}$.
When we have two tokens on $\ell_{i-1}$ and $\ell_{i+1}$, the path
$(\ell_{i-1},s_{i-1},s_i,s_{i+1},\ell_{i+1})$ is a locked path, which contradicts the assumption.
We also have $\msize{\bfI_r\cap\{s_{i-1},\ell_{i-1},s_{i+1},\ell_{i+1}\}}\le 1$ by the same argument.
Now we consider the most serious case since the other cases are simpler and easier than this case.
The most serious case is that a blue token on $\ell_{i-1}$ and a red token on $\ell_{i+1}$.
Since any token cannot bypass the other, $\bfI_b$ contains an L token on $\ell_{i-1}$, and
$\bfI_r$ contains an L token on $\ell_{i+1}$.
In this case, by the L token on $\ell_{i-1}$, first, $t$ should make a detour to right,
and by the L token in $\bfI_r$, $t$ next should make a detour to left twice after the first detour.
It is clear that this three slides should not be avoided, and this ordering of three slides cannot be violated.
Therefore, $t$ itself should slide at least four times to return to the original position,
and $t$ can done it in four slides.
During this slides, since $t$ is the leftmost spine with this condition,
the tokens on $s_1,\ell_1,s_2,\ell_2,\ldots,s_{i-1},\ell_{i-1}$ do not make any detours.
Thus we focus on the tokens on $s_{i+1},\ell_{i+1},\ldots$.
Let $t'$ be the token that should be on $\ell_{i+1}$ in $\bfI_r$.
Since $t$ is on $s_i$, $t'$ is not on $\{s_{i+1},\ell_{i+1}\}$.
If $t'$ is on one of $\ell_{i+2},s_{i+3},\ell_{i+3},s_{i+4},\ldots$ in $\bfI_b$, we have nothing to do;
just make a detour for only $t$.
The problem occurs when $t'$ is on $s_{i+2}$ in $\bfI_b$.
If there exists $\ell_{i+2}$, we first slide $t'$ to it, and this detour for $t'$ is unavoidable.
If $\ell_{i+2}$ does not exist, we have to slide $t'$ to $s_{i+3}$ before slide of $t$.
This can be done immediately except the only considerable case;
when we have another L or S token $t''$ on $s_{i+3}$.
We can repeat this process recursively and confirm that each detour is unavoidable.
Since $G$ with $\bfI_b$ and $\bfI_r$ contains no locked path, this process will halts.
(More precisely, this process will be stuck if and only if this sequence of tokens forms a locked path on $G$,
which contradicts the assumption.)
Therefore, traversing this process, we can construct the shortest reconfiguration sequence.
Proof of Theorem <ref>.
For a given independent set $\bfI_b$ on a caterpillar $G=(V,E)$,
we can check if each vertex is a part of locked path as follows in $O(n)$ time
(which is much simpler than the algorithm in <cit.>):
(0) Initialize a state $S$ by “not locked path”.
(1) For $i=1,2,\ldots,n'$, check $s_i$ and $\ell_i$.
We here denote their states by $(s_i,\ell_i)=(x,y)$,
where $x\in \{0,1\}$, $y\in \{0,1,-\}$ such that $1$ means “token is placed on the vertex”,
$0$ means “no token is placed on the vertex”, and $-$ means “the leaf does not exist.”
In each case, update the state $S$ as follows:
Case $(0,1)$: If $S$ is “not locked path,” set $S$ by “locked path?,”
and remember $i$ as a potential left endpoint of a locked path.
If $S$ is “locked path”, $(s_i,\ell_i)$ is a part of locked path.
Therefore, mark all vertices between the previously remembered
left endpoint to this endpoint as “locked path”.
After that, set $S$ by “locked path?” again, and remember $i$ as
a potential left endpoint of the next locked path.
Case $(0,-)$ and $(0,0)$: If there is a token on $s_{i-1}$, nothing to do.
If there is no token on $s_{i-1}$,
reset $S$ by “not locked path” (regardless of the previous state of $S$).
Case $(1,0)$: reset $S$ by “not locked path” (regardless of the previous state of $S$).
Case $(1,-)$: nothing to do.
Simple case analysis shows that after the procedure above, every vertex in
a locked path is marked in $O(n)$ time.
Thus, we first run this procedure twice for $(G,\bfI_b)$ and $(G,\bfI_r)$ in $O(n)$ time,
and check whether the marked vertices coincide with each other.
If not, the algorithm outputs “no”.
Otherwise, the algorithm splits the caterpillar $G$ into
subgraphs $G_1,G_2,\ldots,G_h$ induced by only unmarked vertices.
Then we can solve the problem for each subgraph; we note that
two endpoints of which tokens are placed of a locked path $P$ are leaves.
That is, for example, when a locked path $P=(p_1,p_2,\ldots,p_k)$ splits $G$ into $G_1$ and $G_2$,
the neighbors of $G_1$ and $G_2$ in $P$ are $p_2$ and $p_{k-1}$, and there are no token on them.
Thus, in the case, we can solve the problem on $G_1$ and $G_2$ separately, and
we do not need to consider their neighbors.
For each subgraph $G_1,\ldots,G_h$, the algorithm next checks whether
each subgraph contains the same number of blue and red tokens.
If they do not coincide with each other, the algorithm output “no.”
Otherwise, we have a yes-instance.
The correctness of the algorithm so far follows from Theorem <ref>
with results in <cit.> immediately.
It is also easy to implement the algorithm to run in $O(n)$ time and space.
It is not difficult to modify the algorithm to output the sequence itself based on the previous
case analysis. For each token, the number of detours made by the token is bounded above by $O(n)$,
the number of slides of the token itself is also bounded above by $O(n)$,
and the computation for the token can be done in $O(n)$ time.
Therefore, the algorithm runs in $O(n^2)$ time, and the length of the sequence is $O(n^2)$.
(As shownn in the last paragraph in Section <ref>,
there exist instances that require a shortest sequence of length $\Theta(n^2)$.)
§ CONCLUDING REMARKS
In this paper, we showed that the shortest sliding token problem can be solved
in polynomial time for three subclasses of interval graphs.
The computational complexity of the problem for chordal graphs, interval graphs, and trees are still open.
Especially, tree seems to be the next target.
We can decide if two independent sets are reconfigurable in linear time <cit.>,
then can we find a shortest sequence for a yes-instance in polynomial time?
As in the 15-puzzle, finding a shortest sequence can be NP-hard.
For a tree, we do not know that the length can be bounded by any polynomial or not.
It is an interesting open question whether there is
any instance on some graph classes whose reconfiguration sequence requires super-polynomial length.
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|
1511.00285
|
$^1$Instituto de Astrofísica e Ciências do Espaço, Universidade de Lisboa, OAL, Tapada da Ajuda, PT1349-018 Lisboa, Portugal
$^2$Inter-University Centre for Astronomy and Astrophysics, Post Bag 4,
Ganeshkhind, Pune University Campus, Pune 411007, India
The thermodynamical evolution of gas during the collapse of the primordial
star-forming cloud depends significantly on the initial degree of rotation.
However, there is no clear understanding of how the initial rotation can
affect the heating and cooling process and hence the temperature that leads
to the fragmentation of the gas during Population III star formation.
We report the results from three-dimensional, smoothed-particle
hydrodynamics (SPH) simulations of a rotating self-gravitating primordial
gas cloud with a modified version of the Gadget-2 code, in which the initial
ratio of the rotational to the gravitational energy ($\beta_0$) is varied
over two orders of magnitude.
We find that despite the lack of any initial turbulence and magnetic
fields in the clouds, the angular momentum distribution leads to the
formation and build-up of a disk that fragments into several clumps.
We further examine the behavior of the protostars that form in both
idealized as well as more realistic minihalos from the cosmological
simulations. The thermodynamical evolution and the
fragmentation behavior of the cosmological minihalos are similar to
that of the artificial cases, especially in those with a similar
$\beta_0$-parameter. Protostars with a higher rotation support exhibit
spiral-arm-like structures on several scales, and have lower accretion
rates. These type of clouds tend to fragment more, while some of the
protostars escape from the cluster with the possibility of surviving until
the present day. They also take much longer to form compared to their
slowly rotating counterparts.
We conclude that the use of appropriate initial conditions of the gas in
minihalos is a pivotal and decisive quantity to study the evolution and
final fate of the primordial stars.
Rotational effect on primordial minihalos
§ INTRODUCTION
Since the late 1960s, the theoretical study of the Universe at high
redshift has persuaded a number of groups to work persistently on
the dynamics of collapsing gas clouds
<cit.>, leading to the formation
of the very first stars in the Universe, the so-called Population III
(or Pop III) stars. Based on these leading$\hbox{-}$edge studies,
the first sources of light are believed to have formed only a few
hundred million years after the Big Bang <cit.>. This
era marks a crucial transition from the simple to the complex Universe
<cit.>. Subsequent pioneering three-dimensional
numerical simulations using adaptive mesh refinement <cit.>,
smoothed-particle hydrodynamics <cit.> or more recently
using the hydrodynamic moving mesh code Arepo
<cit.> have led to the development of a widely
accepted standard model of primordial star formation. In this star
formation model, the first protostars form within dark matter (DM)
halos with a virial temperature of $\sim 1000$ K and masses of
$\sim 10^5\hbox{--}10^6$ $M_\odot$, which had collapsed at
redshift $z \geq$ 20.
However, other simulations show that Pop III stars may still form
well beyond this, in waves that delay their formation <cit.>. These studies suggest that Pop III star
can continue to form down to the redshift $z = 2.5$ with a low peak
rate of 10$^{-5}$ $M_{\odot}$ yr$^{-1}$ Mpc$^{-3}$, which occurred
at $z = 6$. Observational evidence of Pop III stars and the possibility
of Pop III waves reaching lower redshifts has received a significant
boost with the discovery of a luminous Lyman-$\alpha$ (Ly$\alpha$)
emitter with high EW He ii and Ly$\alpha$ emission and no
metal lines <cit.>.
The hydrogen atoms combine with the free electron (present from the
epoch of recombination at $z \sim$ 1100) to produce small fractional
abundances of $\rm H_2$, $x_{\rm H_2} \sim 10 ^{-3}$ <cit.>.
As the collapse proceeds, the gas is cooled via $\rm H_2$ rotational
and vibrational line emission (also termed as gas-phase reaction),
with $\rm H^-$ ion as an intermediate state (first discussed in the
context of the local ISM by )
\begin{equation}
\rm H + e^- \rightarrow H^- + \gamma,
\end{equation}
\begin{equation}
\rm H^- + H \rightarrow H_2 + e^-,
\end{equation}
where the free electrons act as catalysts <cit.>. At this
point, gas attains a temperature $\sim$ 200 K, and is in local
thermodynamical equilibrium (LTE) with the kinetic temperature of
the gas. However, the limited $\rm H_2$ abundance is not sufficient
to cool the gas further, and the gas begins to heat up with increasing
density. This results in the free-fall time being shorter
than the cooling time. The transition from the cooling (low density)
to heating (high density) with increasing density occurs near a
critical density $n_{\rm cr} \approx 10^4$ cm$^{-3}$, and sets
a characteristic Jeans length, allowing the gas to fragment with
the Jeans mass of $M_{\rm J}$ (200 K, $10^4\,$cm$^{-3}$) $\sim$
1000 $M_\odot$ <cit.>. At higher density ($\sim 10^8$ cm$^{-3}$)
hydrogen molecules are formed by the three-body reactions <cit.>
\begin{equation}
\rm H + H + H \rightarrow H_2 + H
\begin{equation}
\rm H + H + H_2 \rightarrow H_2 + H_2.
\end{equation}
There is however a significant uncertainty in the rate coefficients
<cit.> for the above set of reactions that can cool the gas
rapidly by converting almost all the atomic hydrogen into molecules,
making the gas chemothermally unstable <cit.>.
However, the amount of hydrogen molecules produced is a strong function
of temperature. At high temperature, $\rm H_2$ molecules are destroyed
by collisions with atomic $\rm H$ and $\rm H_2$ molecules <cit.>,
\begin{equation}
\rm H_2 + H \rightarrow H + H + H,
\end{equation}
\begin{equation}
\rm H_2 + H_2 \rightarrow H + H + H_2.
\end{equation}
The collision dissociation of $\rm H_2$ prevents the fractional
abundance ($x_{\rm H_2}$) from becoming large.
The study by <cit.> has discussed the equilibrium condition
between the abundances of atomic and molecular hydrogen, known as the
principle of microscopic reversibility. Therefore the rate at which
$\rm H_2$ is produced via the three-body reaction must be
compensated by the destruction rate to bring the system
to a chemical and thermal equilibrium, the abundances of which are
related through the following well-known Saha equation:
\begin{equation}
\frac{n_{\rm H_{2}}}{n_{\rm H}^{2}} = \frac{z_{\rm H_{2}}}{z_{\rm H}^{2}}
\left(\frac{h^{2}}{\pi m_{\rm H} k T} \right)^{3/2} \exp \left(\frac{E_{\rm diss}}{kT}
\right),
\end{equation}
where $n_{\rm H_{2}}$ and $n_{\rm H}$ are the number densities of molecular
and atomic hydrogen, respectively, $z_{\rm H_{2}}$ and $z_{\rm H}$ are the
partition functions of molecular and atomic hydrogen, $E_{\rm diss}$ is the
dissociation energy of the hydrogen molecule, $k$ is the Boltzmann
constant, $h$ is Planck's constant, and $T$ is the temperature.
In LTE, the adopted values of the rate coefficient for the $\rm H_2$
collisional dissociation ($k_{\rm diss}$) must be consistent with the
three-body formation rate coefficient ($k_{\rm form}$) in the sense that
each pair of rate coefficients satisfies the chemical equilibrium condition
$k_{\rm form}/k_{\rm diss} = K$, where $K$ is the equilibrium constant.
Once the gas density reaches a density $\sim 10^{10}$ cm$^{-3}$,
the cloud becomes optically thick to the strongest of $\rm H_2$
lines. Using the Sobolev approximation (as described in ),
the $\rm H_2$ cooling rate in this regime can be expressed as
\begin{equation}
\Lambda_{\rm H_2, \rm thick} = \sum\limits_{u,l} h\nu_{ul} \beta_{\rm esc, ul} A_{\rm ul} n_{\rm u} \, ,
\end{equation}
where $n_{\rm u}$ is the number density of the hydrogen molecules in
upper energy level $u$; $A_{\rm ul}$ is the spontaneous radiative
transition rate, also known as Einstein coefficient, for a
transition between $u$ and $l$; $h\nu_{ul}$ is the energy difference
between $u$ and $l$; and $\beta_{\rm esc, ul}$ is the escape probability
associated with this transition, i.e., the probability that the emitted
photon can escape from the region of interest. In the high-density
regime ($\sim 10^{14}$ cm$^{-3}$), the gas goes through a phase of
cooling instability due to a strong increase in the cooling rate by
$\rm H_2$ collisional induced emission (CIE) <cit.>.
Above the central density $\sim 10^{16}$ cm$^{-3}$,
the gas becomes completely optically thick to the continuum radiation
<cit.>. At this point the remaining $\rm H_2$ dissociates
via reactions (5,6), and hence cools the gas to collapse further.
Once all the $\rm H_2$ dissociates, the gas becomes fully adiabatic
with core mass $\sim 0.01 M_\odot$ surrounded by a massive, dense
envelope that accretes matter rapidly <cit.>.
The vanguard numerical simulations <cit.>
propose the formation of massive (typically $\sim 20\hbox{--}50
M_\odot$) primordial protostars. This calculation result, however,
contrasts with the present day star formation in which protostars with
masses less than 1$M_\odot$ are formed <cit.>.
The recent improved and high resolution numerical simulations have
inferred that the disk around the primordial core is unstable and
fragments to form a small N system with low-mass stars, instead of
a single protostar <cit.>.
In the literature, there are appreciable indications that the collapsing
cloud from which the protostar forms could have strong rotational
support <cit.>. The cloud's rotation can affect the
dynamical as well as the thermal evolution of gas and consequenty determine
the ensuing properties of the Pop III stars <cit.>.
The consequences of the cloud's rotation on the chemical signature of
the zero-metallicity primordial stars have been studied using stellar
models <cit.>. With the use of the sink
particle technique (discussed in <ref>) in the smoothed
particle hydrodynamics (SPH) simulations, <cit.> have discussed
the rotation velocity of the first stars, angular momentum transfer and
the internal structure of the new-born protostars <cit.>. The central protostar rotates
with a significant fraction of the Keplerian velocity. There is however
scatter in the radial velocity, temperature, and accretion rate. In a
recent study, <cit.> performed radiation hydrodynamical
simulations to follow the evolution of 100 primordial protostars.
Although their simulations were in 2D, they nicely compared the angular
momentum of the cloud with the Pop III accretion rate. More recently,
simulations and analytic models have shown the formation of the massive
primordial stars in rapidly rotating disks in the presence of turbulence
and UV backgrounds <cit.>. However,
the extent to which the thermal and dynamical evolution of gas
depends on the initial degree of rotation of the cloud has never been
systematically tested. In addition, there is so far no clear
understanding of how the cloud's rotation can regulate the concurrent
heating and cooling process during the collapse.
The dependence of the resultant fragmentation on the
cloud's initial rotation, however, has been shown in detail in previous
studies <cit.>. After performing a number of
idealized numerical experiments, these parameterized studies have
concluded that the formation of either binary or multiple systems
depend highly on the initial rotation of the cloud.
Nevertheless, these studies could not point out the influence on
the thermal evolution of the primordial gas during collapse because
their calculations adopted the model of equation of state (EOSs). In
this work, we candidly scrutinize the role of initial rotation of the
collapsing gas on the heating and cooling process that controls the
chemothermal evolution of gas inside minihalos. In addition, we
perform rigorous calculations in both the idealized as well as more
realistic cosmological minihalos to thoroughly analyze the evolution
of gas particles and their physical properties. This unique approach
thus enables us to investigate in detail the thermal, chemical, and
dynamical evolution of the baryonic matter in a full 3D simulation
of Pop III collapse to understand better the physical process and
the resulting fragmentation behavior that occurs once the first
object is formed. Finally, we address important issues, such as the
relation of the physical property and accretion phenomenon of the
protostars on the initial rotation of the collapsing core. We also
discuss the possibility of survival of Pop III stars until today.
The paper is organized in the following manner. In <ref>
we describe the numerical setup of the simulations and the initial
conditions. In <ref> we briefly discuss the relevant
physical concepts of the problem with an emphasis on the heating and
cooling process that determines the temperature evolution. The details
of the velocity structure are outlined in <ref>. We
discuss the accretion phenomenon comprehensively, followed by the
implication of this study for the fragmentation of primordial gas in
<ref>. The long-term evolution of protostars
are contoured in <ref>. We summarize the main points and
draw our conclusions in <ref>.
§ SIMULATIONS
In order to follow the gravitational collapse, one needs to ensure
that the gas evolution in the simulations should not depend on the choice
of minihalos. We therefore use two completely different numerical
setups: minihalos from the cosmological simulations of <cit.>
obtained from the hydrodynamic moving mesh code Arepo
<cit.> and the artificial minihalos with an initially
uniform density distribution of gas particles. In the following, we
describe the initial condition and setup of the simulations.
To investigate the cosmological minihalos, we use snapshots at the
epoch when the central number density is just below $10^6$ cm$^{-3}$,
the onset of the crucial three-body $\rm H_2$ formation reaction. The
mesh-generating points of Arepo can be interpreted as the
Lagrangian fluid particles, which is the basic characteristic of the
Gadget-2 SPH code <cit.>. At this time, the
minihalos contain masses of 1030 $M_{\odot}$ and 1093 $M_{\odot}$
with maximum central temperature of 469 K and 436 K, respectively. The
complete physical nature of these minihalos (e.g., number density
($n$), initial rotation ($\beta_0$), maximum and minimum temperature,
etc) is summarized in Table 1. We use the Arepo output of these
minihalos <cit.> as the initial
conditions for our Gadget-2 implementation. Because of the conversion
from the moving mesh to the SPH formalism <cit.>, we denote the minihalos as CH1 and CH2 for the Gadget-2
execution. The numerical resolution for CH1 and CH2 in SPH Gadget-2
simulations (for 100 SPH particles) is $\sim 10^{-2}$ $M_{\odot}$.
For the artificial setup, we use the initial conditions that permit
us to carry out a set of methodical numerical experiments. The randomly
distributed gas particles have been settled using the periodic boundary
conditions in a box. During this initial phase, we keep the gas
temperature fixed and do not follow the primordial chemical network.
Once the gas particles are settled, we are in a position to
perform the simulations for the collapse of gas due to self-gravity.
The gas particles are initially uniformly distributed in a spherical
cloud of size $R_0 \sim 2.7$ pc and a total mass of $M = 2982$ $M_\odot$.
The number density is $n = 10^{3}$ cm$^{-3}$ and the temperature
$T = 200$ K. These initial conditions are equivalent to the primordial
gas clumps that collapse <cit.>. All clouds
are modeled with 5 million SPH particles, and the mass of a single SPH
particle is $m_{\rm SPH} = 5.9 \times10^{-4}$ $M_\odot$. Therefore,
the numerical resolution (roughly 100 SPH particles) is $0.06$ $M_\odot$.
The free-fall time for the uniform density distribution is $t_{\rm ff}
= \sqrt{3/32 \pi G \rho} = 1.37$ Myr and the sound crossing time is
$t_{\rm sc} \approx $ 5 Myr. As the free-fall timescale is shorter
than the sound-crossing timescale, the clouds immediately start to
collapse under their self-gravity. The clouds are then given different
degrees of solid body rotation and are not subject to the internal
turbulent motions. The strength of rotational support can be described
by the $\beta_0$-parameter <cit.>,
\begin{equation}
\beta_0 = \frac{E_{\rm rot}}{E_{\rm grav}} = \frac{R_0^3\Omega^2}{3GM} ,
\end{equation}
where $\Omega$ is the angular velocity, and $E_{\rm rot}$ and
$E_{\rm grav}$ are the magnitudes of the rotational and the gravitational
energies, respectively. We perform ten different numerical experiments with
$\beta_0$ = 0.0, 0.005, 0.007, 0.01, 0.02, 0.04, 0.05, 0.07, 0.1, 0.2.
Physical properties of the cosmological minihalos are summarized:
Halo CH1 CH2
$n$ (cm$^{-3}$) 10$^{6}$ (max) 71 (min) 10$^{6}$ (max) 85 (min)
$T$ (K) 469 (max) 59 (min) 436 (max) 54 (min)
mass ($M_{\odot}$) 1030 1093
$n$-SPH 690855 628773
resolution ($M_{\odot}$) 1.3 $\times 10^{-2}$ 1.4 $\times 10^{-2}$
$\beta_0$ 0.035 (max) 0.025 (min) 0.042 (max) 0.03 (min)
notes: $n$ denotes the number density, $T$ the temperature, $n$-SPH
the number of SPH particles and $\beta_0$ the rotation parameter,
respectively. The numerical resolution is calculated for the 100 SPH particles.
As the collapse progresses in the central region of the cloud,
it is extremely difficult to simulate the higher density regime
because of shorter timescale. To overcome this problem, we use the
sink particle technique in which the high-density region is replaced
by a single sink particle with appropriate boundary conditions
<cit.>. The sink particle can then
be assumed to be (or at least approximated by) a growing protostar.
The density threshold for the sink particles to form is set as the
number density of 5 $\times 10^{13}$ cm$^{-3}$, at which point the
gas has a temperature of $\sim$ 1000 K. The sink particle can
accrete gas particles within its accretion radius $r_{\rm acc}$, which
we fix at 6 AU, the Jeans radius at the density
threshold for sink creation. The corresponding Jeans mass for
both the cosmological as well as the artificial clumps is 0.06
$M_{\odot}$, so we can resolve both the cosmological and artificial
minihalos. The softening parameter of the sinks is 1.2 AU. In
order to avoid spurious formation of new sink particles out of
the gas, the sink particles is prevented further from forming
within $2\, r_{\rm acc}$ of one another.
We follow <cit.> for the implementation of the external
pressure term and a time-dependent primordial chemical network.
To model a constant pressure boundary, we used a modified
version of the Gadget 2 momentum equation,
\begin{equation}
\frac{d v_{i}}{d t} = - \sum_{j} m_{j} \left[
f_{i}\frac{P_{i}}{\rho_{i}^{2}} \nabla_{i} W_{ij}(h_{i})
+ f_{j}\frac{P_{j}}{\rho_{j}^{2}}\nabla_{i} W_{ij}(h_{j}) \right] ,
\end{equation}
where the contribution from the external pressure
($P_{\rm ext}$) is subtracted from both $P_{i}$ and $P_{j}$ (i.e.,
$P_{i}$ and $P_{j}$ are replaced by $P_{i} - P_{\rm ext}$ and
$P_{j} - P_{\rm ext}$, respectively). All quantities have their
usual meaning. The chemical network includes primordial hydrogen,
helium, and deuterium to model the chemical and thermal evolution
of the metal-free gas inside minihalos. The details of all the
chemical reactions are given in <cit.> and references therein.
We adopt the intermediate three-body $\rm H_2$ rate coefficient
$7.7 \times 10^{-31} T^{-0.464}$ cm$^6$ s$^{-1}$ proposed by <cit.>.
§ HEATING AND COOLING RATE
Radial logarithmic binned, mass-weighted averages
of the $\rm H_2$ fraction (A), temperature (B), and various heating and
cooling rates (C to F) are plotted as a function of density for different
degrees of initial rotation $\beta_0$, just before the formation of the
first sink.
In this section, we investigate the cooling and heating mechanism
associated with the emission, chemical reactions and gas contraction
during the collapse of the cloud under its own gravity. The differences
in the cooling and heating rates can force the gas to choose different
paths for its thermodynamic evolution.
If we assume that the gas density ($\rho$) evolves with the free-fall
time ($t_{\rm ff}$) of the gas, i.e., $d\rho/dt = \rho/t_{\rm ff}$
<cit.>, the thermal evolution can
then be followed by solving the energy equation
\begin{equation}
\frac{d\epsilon}{dt} = \frac{p}{\rho}\frac{d\rho}{dt} - \Lambda + \Gamma ,
\end{equation}
where $\epsilon$ is the energy per unit volume in the gas, and $\Lambda$
and $\Gamma$ are the cooling and heating rates, respectively, in units of
erg s$^{-1}$ cm$^{-3}$. Figure <ref> shows the
physical conditions in the gas once the central region has collapsed
to a density of $\sim 5 \times 10^{13}$ cm$^{-3}$, i.e., just before
the first sink formation. The panels show mass-weighted averages of the
properties of individual SPH particles within the radial logarithmic bins.
The three-body $\rm H_2$ formation heating rate provides chemical
heating associated with the release of 4.48 eV each time a $\rm H_2$
molecule forms. Because the $\rm H_2$-fractions are almost similar
(Fig. <ref>A), there is no consequential effect of the
three-body $\rm H_2$ heating on the cloud's initial rotation
(Fig. <ref>C). However, we find that there are substantial
temperature differences between the clouds (Fig. <ref>B).
For example, the temperature of the cloud with $\beta$ = 0.2 is almost
40-60% lower than that with $\beta_0$ = 0.005. This is because
a higher degree of the rotational support slows down the contraction
and reduces the amount of compressional heating (Figure <ref>D).
Thus, the cloud with $\beta$ = 0.2 has a temperature of roughly
$T \leq$ 200 K due to a slower collapse and, hence, less efficient
compressional heating, whereas the temperature is nearly 1100 K in
case of $\beta$ = 0.0 or $\beta$ = 0.005.
The rapid conversion of atomic to molecular hydrogen during the
three-body reaction cools the gas less than the free-fall time and
hence causes chemothermal instability <cit.>. At high
densities the heating rate is as high as the line-cooling rate
(Fig. <ref>E), however, again with slight differences over
the range of $\beta_0$ modeled. At equilibrium, the formation rate
is balanced by the dissociation, and hence the dissociation cooling
rate behaves in the same way with density as the heating rate. The
dissociation cooling rate varies between clouds, with that for
$\beta_0$ = 0.005 nearly 10 orders of magnitude higher than
for the $\beta_0$ = 0.2 model (Fig. <ref>F) in the
density range where the three-body reaction dominates. We therefore
conclude that it is indeed the compressional heating ($pdV$)
that determines the thermal evolution of gas, which strongly depends
on the initial degree of rotation ($\beta_0$).
At this point, it is worth pointing out that the temperature
evolution of the cosmological minihalos are similar to that of the
idealized cases. As expected, both CH1 (with $\beta_0 \approx$ 0.035)
and CH2 (with $\beta_0 \approx$ 0.042) have the temperature variation
that falls in between the highest and lowest $\beta_0$ modeled throughout
the density space. This also confirms that our varied parameter study
with idealized clumps actually represents the cosmological initial
conditions of the minihalos.
§ VELOCITY STRUCTURE
Radial logarithmic binned, mass-weighted
averages of the radial velocity (A), rotational velocity (B), radial
velocity over sound speed (C), rotational velocity over sound speed
(D), rotational velocity over Keplerian speed (E), and radial velocity
over rotational velocity are plotted as a function of radius, just
before the first protostar forms. The initial strength of rotation
introduces a scatter in the velocity.
In this section, we study the dynamics of the gas particles that
can arise as a result of the chemical and thermal evolution of the clouds.
We therefore take a closer look at the velocity of the gas associated
with the cloud collapse. The radial profiles of the gas show the
mass-weighted averages within logarithmic bins and are taken just
before the formation of the first sink.
We find that there are considerable differences in radial velocities
between the clouds (Fig. <ref>A). The radial velocity of the
cloud with $\beta_0$ = 0.005 is almost 30-40% higher than the cloud
with $\beta_0$ = 0.2. This is consistent with the fact that a lower
rate of compressional heating for the gas of the collapsing core
implies a lower radial velocity, which is nearly comparable with the
sound speed (Fig. <ref>C).
In order to quantify the degree of rotational support, we plot the
rotational velocity (Fig. <ref>B) and the ratio of the rotational
velocity ($v_{\rm rot}$) to the Keplerian velocity ($v_{\rm Kep}$),
defined as $v_{\rm K} = \sqrt{GM_{\rm enc}(r)/r}$, where $M_{\rm enc}(r)$
is the mass enclosed within the radius $r$ (Fig. <ref>F). We
find that the rotational speed for all $\beta_0$ modeled is well below
that required for rotational support by a factor of 5 to 7. This is
consistent with previous cosmological simulations <cit.>, which argued that the collapsing gas cloud has the ratio
$v_{\rm rot}/v_{\rm Kep} \sim $ 0.5
For the $\beta_0 = 0.2$ case, the rotation
speed varies between 0.3 and 0.9 times the Keplerian velocity,
indicating that this cloud has gone through an efficient phase of
angular momentum redistribution <cit.>. However, for
$\beta_0 >$ 0.05 the collapsing cloud is almost completely
rotationally supported throughout and the gas at higher densities
is relatively cold. All clouds are consequently sub-Keplerian,
and the radial velocities for all clouds are comparable to the
rotational velocities within 100 AU (Figure <ref>E). From
this discussion, we can infer that the cloud with higher rotation
transfers the angular momentum more efficiently and hence becomes
Keplerian in the outer regime of the collapsing core (for instance,
$v_{\rm rot} \sim v_{\rm Kep}$ for $r \geq 10^4$ AU for $\beta_0 = 0.2$).
Thus the outer regime, which is likely to form a Keplerian disk due to
rotation, becomes unstable enough from accreting mass and, consequently,
has higher chance to fragment (as we see in the next section). We
conclude that the cloud's initial rotation plays a pivotal
role in the dynamical evolution of the gas by affecting the amount
of rotational support even at later stages of the collapse.
§ MASS ACCRETION AND FRAGMENTATION
During collapse, the angular momentum is transported to smaller
scales, resulting in the formation of rotationally supported
disk-like structures <cit.>. However, the
rotation is not sufficient to hold the collapse of the disk, which
then fragments into multiple objects <cit.>. We
thus gauge the accretion and gas instability carefully, as
predicted by the properties of the gas, to check whether there
is any hint of fragmentation already present before the
formation of the first sink.
We follow <cit.> to study the mass accretion rate, estimated
as $\dot{M}(r) = 4 \, \pi \, r^2 \, \rho(r) \, v_{\rm rad}(r)$, as a
function of radius for different degrees of initial
rotation (Fig. <ref>A). We also define the accretion time,
$t_{\rm acc} = M_{\rm enc}(r)/4 \, \pi \, \rho \, v_{\rm rad} \, r^2$.
For all simulations, $\dot{M}$ has a maximum at $\sim$
20-40 AU. Given that $\rm H_2$ line cooling becomes optically
thick at the corresponding densities, $\dot{M}$ for all simulations
converges in this range, as the gas looses its ability to cool
efficiently. We would, however, like to point out that the scale,
where $dM/dt$ becomes maximum, may slightly change with the choice
of density threshold for the sink formation. For example, if we
choose $n \sim 10^{15}$ cm$^{-3}$ for a sink particle to form, the
collapse of the inner structure proceeds and hence the mass accretion
rate reaches a maximum at $\sim$ 15-30 AU, depending on the initial
degree of rotation. However, this is not a substantial issue
compared to the overall thermal nature of the collapsing cloud. For
slowly rotating clouds, the accretion rate is substantially higher
($\sim 0.1$ yr$^{-1}$) and the accretion time is on the order of
free-fall time (Fig. <ref>C).
Since the collapsing core becomes Jeans unstable by accreting more
and more mass, we check the strength of the gravitational instability
by measuring the number of the Bonnor-Ebert masses ($M_{\rm {BE}}$)
contained in the central dense volume <cit.>.
We compute the Bonnor-Ebert masses as
$M_{\rm{BE}} = 1.18 (c_s^4/G^{3/2})P_{\rm{ext}}^{-1/2} \approx
20 M_\odot T^{3/2} n^{-1/2} \mu^{-2} \gamma^2$ <cit.>, where
$c_s$ is the sound speed, $P_{\rm{ext}}$ is the external pressure that
we assume to be equal to the local gas pressure, $\mu$ is the mean
mass per particle, and $\gamma = 5/3$ is the adiabatic index, respectively.
Within the central 10$^4$ AU regime, the enclosed gas mass for all
values of $\beta_0$ contains a roughly equal number of Bonnor-Ebert
masses, although with a factor of two between the highest and lowest
values of $\beta_0$ (Fig. <ref>B). However, in the outer
regime (i.e., $r \geq$ 10$^4$ AU), the clouds with $\beta_0=0.1$ and
$\beta_0=0.2$ contain roughly 7 and 10 Bonnor-Ebert masses respectively.
This is because the higher rotating clouds become close to Keplerian
in the outer regime (as seen in the previous section) and tend to
form a disk-like structure. As the higher rotation obstructs the
infalling gas particles, the disk gradually accumulates enough
Jeans masses through accretion and becomes gravitationally unstable to
Alongside the Bonnor-Ebert analysis, we also compare all relevant
timescales associated with the cloud collapse. Figure <ref>C
represents the accretion timescale ($t_{\rm acc}$) over free-fall
timescale ($t_{\rm ff}$). For a higher rotating cloud, the accretion
time is much longer than the free-fall time. This is consistent with
the above discussion and an alternate way to explain the mass
accretion features of clouds with different $\beta_0$-models shown in
Fig. <ref>A. In Figure <ref>D, we consider the fragmentation
timescale, defined as $t_{\rm frag} \equiv M_{\rm BE}/\dot{M}$
<cit.>, and compare it with the free-fall timescale.
We know that fragmentation generally
occurs when the dynamical timescale of the central collapse becomes
larger than the collapse timescale of individual density fluctuations.
Here the free-fall timescale represents the dynamical timescale.
In the outer regime ($r \geq 2 \times 10^4$ AU), which is likely to
form a disk, the fragmentation time becomes shorter than the free-fall
time, especially for higher rotating clouds. We conclude here that
the clouds with larger $\beta_0$ collapse slowly from high rotation
and tend to form a disk-like structure that becomes Keplerian and
gravitationally unstable by accreting mass, heralding fragmentation.
§ PROTOSTELLAR SYSTEM
Radial logarithmic binned, mass-weighted
averages of the mass accretion rate (A), the number of Bonnor-Ebert
masses (B), the accretion time over free-fall time (C), the
fragmentation time over free-fall time (D), and the free-fall time
over sound crossing time (E) are plotted as a function of radius,
just before the formation of the first sink.
The column density and column-weighted
temperature distribution in a region of 2000 AU centered around the
first protostar for different strengths of the initial rotation of
the cloud are shown when a total of $\sim 30\,M_{\odot}$ have been
converted into, or accreted onto, sink particles.
Time evolution of the protostellar system:
total mass of all the sinks particles and the most massive sink
particle are plotted as a function of time (top left). Total mass
accretion rate by all sink particles are shown as a function of time
(for others rotation parameter $\beta_0$; same as Fig. <ref>A).
The cloud with zero rotation attains the 30 $M_{\odot}$ within a few
hundred years after the formation of the first sink particle. The
rotationally supported cloud takes $\sim$ 100 $\hbox{--}$ 1000 years,
depending on the initial strength of rotation ($\beta_0$). The mass
accretion rate decreases with time until further sink particles form.
The study by <cit.> shows that any fragmentation only
takes place after the disk formation, and that clouds with higher
$\beta_0$ tend to fragment at lower densities. We examine how the
circumstellar accretion disk that formed in the idealized, as well as
more realistic cosmological minihalos, becomes locally unstable and
evolves for different degrees of initial rotation.
Figure <ref> shows the column density and column-weighted
temperature distribution in the inner 2000 AU at the end of the
simulations. These images clearly reflect that all simulations fragment
to form a small N-body system, comprised of secondary protostars,
within the time in which $\sim$ 30 $M_{\odot}$ of material are
accreted onto the sink particles. It is not surprising that clouds
with a higher level of rotational support exhibit the disk-like
structure on several scales within the central region, whereas
slowly rotating clouds are more likely to be centrally condensed.
The top left panel of Fig. <ref> shows the time evolution
of the total mass accreted onto all the sinks ($\Sigma M_*$) and the
maximum sink mass ($M_{*\rm max}$) in the period over which the sink
formation occurs. We find that the cloud with zero rotation attains
the 30 $M_{\odot}$ within a few hundred years after the formation of
the first sink particle, whereas rotationally supported clouds take
$\sim$ 100 $\hbox{--}$ 1000 years, depending on the initial
strength of rotation. Figure <ref> also shows the time
evolution of the total mass accretion rate ($dM_*/dt$) onto all the
sinks (same as Figure <ref>A) for different values of $\beta_0$.
As expected, the mass accretion rate is larger for the slowly rotating
clouds. In each case, $dM_*/dt$ decreases with time until further
sink particles form, and then the total accretion rate increases
again, however, this time with large temporal variations. The mass
accretion rate for the idealized cloud with $\beta_0=0.04$
is similar to that of the cosmological minihalo CH2, which has a
rotation parameter $\sim \beta_0=0.042$. This again confirms that
our idealized clumps with varying $\beta_0$ can actually be the
representatives of cosmological initial conditions to investigate
the thermodynamical evolution of gas and the resulting fragmentation
behavior of the sinks.
Figure <ref>A shows the number of sinks for different values
of $\beta_0$, with rotationally supported clouds fragmenting the most.
This is consistent with the recent study by <cit.>, who have
used Arepo simulations to investigate the dependence of the
high accretion rate and efficient cooling of the gas on the
fragmentation of the disk. However, Fig. <ref>A shows
some scatter in the numbers, indicating the presence of statistical
fluctuations that can be removed by pursuing more realizations to
achieve a desired degree of accuracy. Figure <ref>B shows
the time taken for all the clouds to accumulate $\sim$ 30 $M_{\odot}$.
As expected, higher rotating clouds take longer compared
to their slowly rotating counterparts. Figure <ref>C shows
the distance at which the primordial protostars form from the center
of the cloud. The red line represents the mean distances of all
protostars ($R_{\rm dist}$), which follows a power-law relationship
with the cloud's initial rotation, $R_{\rm dist} \propto \beta_0^{3/4}$.
The protostars of the slowly rotating clouds form near the center
($\leq$ 300 AU), while the others spread over larger distances of
5000 AU, as the conservation of angular momentum acts to move
protostars to larger radii.
Another trend we find in our simulations is that a number of
protostars are ejected from the central gas cloud, as seen in
cosmological simulations <cit.>. Although the resolution
used in those studies was higher than our resolution, we still
find ejection from the cluster. Figure <ref> shows the
radial velocity and the ratio of the radial velocity to the escape
velocity for all protostars. The position of the sinks is measured
from the center of mass of all the sinks. The escape velocity of
the sinks is defined as
$v_{\rm esc} = \sqrt{2GM_{\rm enc}(r)/r}$, where $M_{\rm enc}(r)$
is the total mass (gas + sinks) that is enclosed within the radius $r$.
The radial velocities of the sinks formed with lower $\beta_0$ are
below that required to be kicked out of the cluster. They tend to
remain within the cluster and continue to accrete. For faster-rotating
clouds, some protostars move from the cluster with radial velocities
exceeding the escape velocity. There is, therefore, a greater chance that
some protostars will be ejected, opening up the possibility that
they could survive until the present day.
At this point we would like to point out that we have not mentioned
the behavior of sinks for the $\beta_0=$ 0.2 case. The thermodynamical
evolution of this cloud is considerably different from other cases, so
it is important to compare the fragmentation behavior of $\beta_0=$ 0.2
cloud with others. However, because of very fast rotation it has been
extremely tough to run the simulation up to the point where the total
mass of all sinks reaches $\sim$ 30 $M_{\odot}$. The simulation with
$\beta_0=$ 0.2 stops much before compared to other simulations. For
instance, simulation with $\beta_0=$ 0.2 stops when the total mass of
sinks is only around 3$\hbox{--}$4 $M_{\odot}$ (and to reach up to that
epoch of time; it takes around one month to run on a supercomputer
that is based on graphical processing units$\hbox{,}$
HPC$\hbox{--}$GPU Cluster Kolob). This is usually expected as
the fast rotation can impede the collapse. We find a limiting value of
$\beta_0=$ 0.1 for the simulation to run up to the epoch when the mass
of sinks attains a value of 30 $M_{\odot}$.
However, Figs. (4$\hbox{--}$7) still allow us to extrapolate and
predict the expected results.
§ SUMMARY AND DISCUSSION
The fragmentation behavior is plotted for
different degrees of initial rotation at the epoch when stellar system
accretes $\sim$ 30 $M_\odot$. The number of sinks (A), time taken to
accrete $\sim$ 30 $M_\odot$ (B) and the position of the sinks
($R_{\rm dist}$) from the center of mass is shown in (C) as a
function of the rotation parameter $\beta_0$.
Radial velocity (A) and the ratio of the
radial velocity to the escape velocity (B) of sink particles for
different values of $\beta_0$ are plotted as a function of radius,
at the epoch when the stellar system has accreted $\approx$ 30
$M_\odot$. Some protostars with higher $\beta_0$ move away from the
cluster, with the radial velocity exceeding the escape velocity.
We have minutely investigated the dependence of the thermodynamical
evolution of primordial star-forming gas on the initial degree of
rotation of the cloud, and analyzed its influence on the resulting
fragmentation of the circumstellar accretion disk. For this purpose,
we performed a set of three-dimensional hydrodynamical simulations
of Pop III gas collapse to pursue a systematic parameter study,
specified by $\beta_0$, which spans two orders of magnitude for the
amount of initial rotation, including two simulations runs with
realistic cosmological initial conditions.
The cloud's
initial strength of rotation introduces a significant impact on
the intricate combinations of the heating and cooling process, leading
to a scatter in the temperature evolution of the collapsing gas. Clouds
with slower rotation collapse faster and get heated as a result of the
compressional heating. We also find that the dynamical evolution of the
gas strongly depends on the initial strength rotation. Clouds with higher
rotation form a Keplerian disk that becomes gravitationally unstable by
accreting the infalling mass. Therefore, any change in the thermodynamical
evolution introduces substantial difference in the number of Jeans mass,
which determines the susceptibility to the fragmentation of the gas
between the clouds with highest and lowest initial rotation.
In summary, a higher degree of rotation can hinder the infall,
lead to a cooler gas, and result in more fragmentation.
In addition, we find that the protostars with higher rotational
support have larger spiral arms with lower accretion rates. We
also point out that the newborn protostars are distributed in
such a way to conserve the angular momentum, and some of them
could have survived until today if they were of sufficiently low mass.
We conclude that the initial conditions of the primordial gas in
the minihalos should be chosen scrupulously so as to simulate
the long-term evolution and final fate of the primordial stars.
Despite considerable computational efforts involved, we emphasize
that we cannot accurately predict the final mass of the
primordial protostars.
We have neglected the effect of the magnetic fields, which can
be important in minihalos <cit.>. Recent
cosmological simulations show the importance of the amplification
of even small seed fields <cit.>. In addition,
the radiative feedback can significantly affect the thermal as
well as chemical evolution of the gas <cit.>,
which is not included in our simulations. Notwithstanding,
our approach to the problem enables us to provide good estimates
for the overall trend of the accretion rate, thermodynamical
evolution, and fragmentation behavior of gas in the rotating clouds
in which we are particularly interested. Recent radiation
hydrodynamic simulations <cit.> demonstrated
the effect of UV radiative feedback on the mass accretion, thus
constraining the mass spectrum of the first stars <cit.>.
Moreover, there is still a discrepancy regarding the three-body
$\rm H_2$ rate coefficient <cit.>. However,
the recent study by <cit.> provides the currently best
available rate that is in good agreement with <cit.> at high
temperatures and <cit.> at lower temperatures. It is
therefore of strategic interest to accurately simulate the formation
of the first stars in the Universe with the best available rates,
inclusion of the magnetic fields, and radiative feedback in the
next-generation avant-garde SPH codes.
The author wishes to thank Prateek Sharma, Biman Nath, David Sobral,
and Dominik Schleicher for thoroughly checking the manuscript and
for enormous constructive suggestions. The author also acknowledges
Kazu Omukai, Athena Stacy,
and the referee for helpful and worthwhile comments. The present
work is supported by the Indian Space Research Organization (ISRO) grant
(No. ISRO/RES/2/367/10-11) and Department of Science and Technology (DST)
grant (Sr/S2/HEP-048/2012). The author is grateful to the Centre for
Theoretical Studies (CTS) at the Indian Institutes of Technology
$\hbox{--}$ Kharagpur and Raman Research Institute for the financial
support and hospitality. The author would also like to thank the
Department of Physics, Indian Institute of Technology (Banaras Hindu
University) at Varanasi and the Inter-University Center for Astronomy
and Astrophysics (IUCAA) at Pune for local hospitality.
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|
1511.00201
|
In this paper, we consider an initial-boundary problem for plane magnetohydrodynamics
flows under the general condition on the heat conductivity $\kappa$
that may depend on both the density $\rho$ and the temperature $\theta$ and satisfies
\kappa(\rho,\theta)\geq\kappa_1(1+\theta^{q}) \quad \hbox{\rm with
constants}~ \kappa_1>0 ~\hbox{\rm and}~ q>0.
We prove the global existence of strong solutions for large initial
data and justify the passage to the limit as the shear viscosity
$\mu$ goes to zero. Furthermore, the value $\mu^\alpha$ with any $0<\alpha<1/2$ is established for the boundary layer thickness.
Keywords. plane magnetohydrodynamics
flows; global existence; vanishing shear viscosity; boundary layer.
2010 MSC. 35B40; 35B45; 76N10; 76N20; 76W05; 76X05.
§ INTRODUCTION
Magnetohydrodynamics (MHD) concerns the motion of conducting fluids in
an electromagnetic field and has a very broad range of applications.
The dynamic motion of the fluid and the magnetic field interact
strongly on each other, so the hydrodynamic and electrodynamic
effects are coupled, which make the problem considerably
The plane MHD flows with constant longitudinal magnetic field, which are three-dimensional MHD flows
uniform in the transverse
direction, are governed by the following equations:
\begin{equation}\label{e1}
\begin{split}
&\rho_t+(\rho u)_x=0,\\
&(\rho u)_t+\left(\rho u^2+p+\frac12 |\mathbf{b}|^2\right)_x
=(\lambda u_{x})_x,\\[1mm]
&(\rho \mathbf{w})_t+(\rho u \mathbf{w}-\mathbf{b})_x=(\mu\mathbf{w}_{x})_x,\\[1mm]
& (\rho e)_t+(\rho u e)_x-(\kappa\theta_x)_x+pu_x={\cal Q},\\[1mm]
&{\cal Q}:=\lambda u_x^2+\mu
|\mathbf{w}_x|^2+\nu |\mathbf{b}_x|^2,
\end{split}
\end{equation}
where $\rho$ denotes the density of the flow, $\theta$ the temperature, $u\in \mathbb{R}$
the longitudinal velocity, $\mathbf{w}=(w_1,w_2)\in \mathbb{R}^2$ transverse velocity,
$\mathbf{b}=(b_1,b_2)\in \mathbb{R}^2$ transverse magnetic field, $p=p(\rho,\theta)$ the pressure,
$e=e(\rho,\theta)$ the internal energy, and $\kappa=\kappa(\rho,\theta)$ the heat conductivity.
The coefficients $\lambda$, $\mu$ and $\nu$ standing for the bulk viscosity, shear viscosity and
the magnetic diffusivity, respectively, are assumed to be positive constants in this paper.
We focus on the perfect gas
with the equations of state:
\begin{equation}\label{e2}
p=\gamma\rho \theta, \qquad e=c_v\theta,
\end{equation}
where the constants $\gamma>0$ and $c_v>0$. Without loss of
generality, we set $c_v=1$.
We consider system (<ref>) in the bounded domain $Q_T=\Omega\times (0, T)$ with $\Omega=(0, 1)$
subject to the following initial and boundary conditions:
\begin{equation}\label{e4}
\left\{
\begin{array}{lllllll}
(\rho,u,\theta, \mathbf{w},\mathbf{b})(x,0)=(\rho_0,u_0,\theta_0,\mathbf{w}_0,\mathbf{b}_0)(x),\\[1mm]
(u, \mathbf{b},\theta_x)|_{x=0, 1}=\mathbf{0},\\[1mm]
\mathbf{w}(0,t)=\mathbf{w}^-(t),\quad\mathbf{w}(1,t)=\mathbf{w}^+(t).
\end{array}
\right.
\end{equation}
Because of physical importance, complexity, rich phenomenon, and
mathematical challenges, the MHD problem has been extensively
studied in many papers, see
<cit.> and the references
therein. In particular, if there is no magnetic effect, MHD
reduces to the compressible Navier-Stokes equations, see for example
<cit.> and references therein for some mathematical
studies. However, many fundamental problems for MHD are still open.
For example, even though for the one dimensional
compressible Navier-Stokes equations, there is a pioneer work by
Kazhikhov and Shelukhin <cit.> on the global existence of strong
solutions with large initial data, the corresponding problem for the
MHD system with constant viscosity, heat conductivity and
diffusivity coefficients remains unsolved. The reason is that the
presence of the magnetic field and complex interaction with the
hydrodynamic motion in the MHD flow of large oscillation cause
serious difficulties.
The initial-boundary value problem (<ref>)-(<ref>) has
fundamental importance in the studies on the MHD problem. In this
paper, we investigate the global existence, zero shear viscosity
limit, convergence rate and boundary layer effect
of strong solutions for problem (<ref>)-(<ref>) with large initial data, where $\kappa$ may depend both density
and temperature such that $\kappa=\kappa(\rho,\theta)$ is twice
continuous differential in $\mathbb{R}^+\times\mathbb{R}^+$ and
\begin{equation}\label{kappa}
\kappa(\rho,\theta)\geq \kappa_1(1+\theta^q)\quad\hbox{\rm with
constants}~ \kappa_1>0 ~\hbox{\rm and}~ q>0.
\end{equation}
In kinetic theory of gas, the heat conductivity $\kappa$
is a function of temperature $\theta$ and increases with $\theta$ in
general (cf.<cit.>). From experimental results for gases at
very high temperatures (see <cit.>), the condition
(<ref>) seems reasonable when one considers a gas model that
incorporates real gas effects that occur in high temperature (cf.<cit.>). In <cit.>, one of the assumptions on $\kappa$ is that
there are constants $C_1, C_2>0$ such that the heat conductivity
$\kappa$ satisfies
\begin{equation}\label{assumption30}
\begin{split}
C_1(1+\theta^q)\leq \kappa(\rho,\theta)\leq C_2(1+\theta^q), \quad\forall \rho, \theta>0,\\
\end{split}
\end{equation}
where $q\geq 2$, which implies that $\kappa$ has a positive lower
bound. This type of temperature dependence is also motivated by the
physical fact that $\kappa$ grows like $\theta^q $ with $q=2.5$ for important physical regimes and $q \in [4.5,
5.5]$ for molecular diffusion in gas (see <cit.>). The
assumption (<ref>) with $q> 0$ was also made in many papers (see for example <cit.> and references therein). Clearly, here we
remove these assumptions on $\kappa$.
The well-posedness theory has been studied in many papers, some of which will be mentioned below.
It was Vol'pert and Hudjaev <cit.> who first proved the existence and uniqueness of local smooth solutions.
The global existence of smooth solutions with small initial data was established by Kawashima and Okada <cit.>.
Under the technical condition on $\kappa$:
\begin{equation}\label{assumption4}
\begin{split}
C^{-1}(1+\theta^q)\leq \kappa(\rho,\theta)\equiv\kappa(\theta)\leq C (1+\theta^q),
\end{split}
\end{equation}
for $q\geq 2$, Chen and Wang <cit.> proved the existence, uniqueness and the Lipschitz
continuous dependence of global strong solutions with large $H^1$ initial
data. Similar results can be found in <cit.> under the same technical condition as (<ref>).
The existence of global weak solutions was proved by Fan, Jiang and Nakamura <cit.>
under the condition (<ref>) for $q\geq 1$ or the condition $\kappa\equiv\kappa(\rho)\geq C/\rho$,
while the uniqueness and the Lipschitz continuous
dependence on the initial data of global weak solutions with the initial data in Lebesgue spaces were
obtained by them in <cit.>. Very recently, the case $q>0$ of condition
(<ref>) was treated by Fan, Huang and Li <cit.> where the existence and
uniqueness of global solutions with large initial data and vaccum were shown. A similar result can be found in <cit.> by Hu and Ju. The uniqueness and continuous dependence of weak
solutions for the Cauchy problem have been proved recently by Hoff and Tsyganov in
<cit.>. In this paper, we show the global existence of strong solutions for problem (<ref>)-(<ref>)
under the general condition (<ref>), which extends some global existence results mentioned above.
The problem of small viscosity finds many applications, for example, in the boundary layer theory (cf. <cit.>).
In this direction, some results on the Navier-Stokes equations can be referred to <cit.>
and references therein. The vanishing shear viscosity limit of the weak solution for problem (<ref>)-(<ref>)
was studied by Fan, Jiang and Nakamura <cit.>
under the condition (<ref>) for $q\geq 1$ or $\kappa\equiv \kappa(\rho)\geq C/\rho$.
As pointed out in <cit.>, the result of <cit.> can be transplanted to the case $q>0$ of
the condition (<ref>). In this paper, we justify the passage to the limit
with more strong convergence of $\mathbf{w}$ and $\mathbf{b}$ under the general condition (<ref>).
Thus, we extend and improve some results mentioned above.
The boundary layer theory has been one of the fundamental
and important issues in fluid dynamics since it was proposed by
Prandtl in 1904. Frid and Shelukhin in <cit.> investigated the
boundary layer effect of the compressible isentropic Navier-Stokes
equations with cylindrical symmetry, and proved the existence of
boundary layers of thickness $O(\mu^{\alpha}) (0<\alpha<1/2)$.
Under the assumption on $\kappa$:
\begin{equation}\label{assumption3}
\begin{split}
C^{-1}(1+\theta^q)\leq \kappa(\rho,\theta)\leq C (1+\theta^q),\quad
|\kappa_\rho(\rho,\theta)|\leq C (1+\theta^q), ~~q>1, \\
\end{split}
\end{equation}
Jiang and Zhang <cit.> studied the compressible nonisentropic Navier-Stokes equations with cylindrical symmetry,
and proved that the thickness of boundary layer is of the order $O(\mu^\alpha)
(0<\alpha<1/2)$. Recently, Jiang and Zhang's result is extended to the case of
constant heat conductivity, see <cit.>. A similar result can be found in <cit.> by Yao,
Zhang and Zhu.
To the best of our knowledge, however, there is no corresponding results for the initial
boundary problem (<ref>)–(<ref>). In this paper, the value
$\mu^\alpha (0<\alpha<1/2)$ is established for the boundary layer
thickness of problem (<ref>)-(<ref>).
We introduce some
notations. Let $k\geq 0$ be an integer, $\mathcal{O}$ a domain of $\mathbb{R}^n (n\geq 1)$ and $p\geq 1$. $W^{k,p}(\mathcal{O})$ and $W_0^{k,p}(\mathcal{O})$ denote the usual Sobolev spaces, $W^{0,p}(\mathcal{O})=L^p(\mathcal{O})$.
$C^{k}(\mathcal{O})$ and $C^{k}(\overline{\mathcal{O}})$ denote the spaces consisting of continuous
derivatives up to order $k$ in $\mathcal{O}$. For $0<\alpha<1$, $C^{k+\alpha}(\mathcal{O})$ (resp. $C^{\alpha}(\overline{\mathcal{O}})$) and $C^{k+\alpha,k+\alpha/2}(\mathcal{O})$ (resp. $C^{k+\alpha,k+\alpha/2}(\overline{\mathcal{O}})$) denote the Hölder spaces with the exponent $\alpha$. $L^p(I,B)$ is the
space of all strong measurable, $p^{th}$-power
integrable (essentially bounded if $p=\infty$) functions from $I$ to
$B$, where $I\subset \mathbb{R}$ and $B$ is a Banach space. For
simplicity, we also use the notation $\|(f, g,
\cdots)\|^2_{B}=\|f\|^2_{B}+\|g\|_{B}^2+\cdots$ for functions $f,
g,\cdots$ belonging to
$B$ equipped with a norm $\|\cdot\|_{B}$.
In what follows, we assume that the initial and boundary functions satisfy
\begin{equation}\label{assumption1}
\begin{split}
&\rho_0>0,\,\,\theta_0>0,\,\,\, \|(\rho_0^{-1},\theta_0^{-1})\|_{C(\overline{\Omega})}<\infty,
\,\,\|(\mathbf{w}^-,\mathbf{w}^+)\|_{C^1([0,T])}<\infty, \\[1mm]
&(\rho_0,\mathbf{w}_0,\theta_0)\in W^{1,2}(\Omega),\,\,\mathbf{b}_0\in W_0^{1,2}(\Omega),
\,\, u_0\in W_0^{1,2}(\Omega)\cap W^{2,m}(\Omega)~\hbox{\rm with}~m\in (1, +\infty),\\[1mm]
\end{split}
\end{equation}
Now the results on the global existence, vanishing shear viscosity limit and convergence rate
of strong solutions can be stated as
Let (<ref>) and (<ref>) hold. Then
(i) For any fixed $\mu>0$, there exists a unique strong solution
$(\rho,u,\mathbf{w},\mathbf{b},\theta)$ for problem
(<ref>)–(<ref>). Moreover, there exist some positive
constants $C$ independent of $\mu$ such that
\begin{equation}\label{ve}
\begin{split}
&C^{-1}\leq \rho, \theta \leq C,\quad \|(u,\mathbf{w},\mathbf{b})\|_{L^\infty(Q_T)}\leq C,\\[2mm]
&\|(\rho_t,\rho_x, u_x,\mathbf{b}_x,\theta_x)\|_{L^\infty(0, T;L^2(\Omega))}
+\|(u_t, \mathbf{b}_t,\theta_t, u_{xx}, \theta_{xx})\|_{L^2(Q_T)}\leq C,\\[2mm]
T;L^1(\Omega))}+\|\mathbf{w}_t\|_{L^2(Q_T)}\leq C,\\
&\mu^{1/4}\|\mathbf{w}_x\|_{L^{\infty}(0,T;L^2(\Omega))} +\mu^{3/4}\|\mathbf{w}_{xx}\|_{L^2(Q_T)} \leq C,\\
\sqrt{\omega}\mathbf{b}_{xx}\big)\|_{L^2(Q_T)} \leq C,\\
\end{split}
\end{equation}
where $\omega: [0, 1]\rightarrow [0, 1]$ is defined by
\begin{equation*}
\begin{aligned}
\omega(x)=\left\{\begin{aligned} &x,&0\leq x\leq 1/2,\\ &1-x,&
1/2\leq x\leq 1.
\end{aligned}\right.
\end{aligned}
\end{equation*}
(ii) There exist some functions $\overline\rho, \overline u,
\overline{\mathbf{w}}, \overline{\mathbf{b}}$ and $\overline\theta$ in the class:
\begin{equation*}\label{base0}
\mathbb{F}:\left\{\begin{split}
& \overline\rho, \overline\theta>0,\quad(\overline u, \overline{\mathbf{b}})|_{x=0, 1}=0,\\
&(\overline\rho, 1/\overline\rho,\overline u, \overline{\mathbf{w}}, \overline{\mathbf{b}},
\overline\theta,1/\overline\theta)\in L^\infty(Q_T),\quad
\overline{\mathbf{w}} \in L^\infty(0, T;W^{1,1}(\Omega))\cap BV(Q_T),\\
&(\overline\rho_t,\overline\rho_x,\overline u_x,\overline{\mathbf{b}}_x,\overline\theta_x)
\in L^\infty(0, T;L^2(\Omega)),\quad (\overline u_x,\overline{\mathbf{b}}_x,\overline \theta_x) \in L^2(0, T;L^\infty(\Omega)),\\
&\big(\overline u_t, \overline{\mathbf{w}}_t, \overline{\mathbf{b}}_t,\overline\theta_t,
\overline u_{xx}, \overline\theta_{xx}\big) \in L^2(Q_T),\\
&\sqrt{\omega}\overline{\mathbf{w}}_x\in L^\infty(0, T;L^2(\Omega)),\quad\big(\sqrt{|\overline u|}\overline{\mathbf{w}}_x,\sqrt{\omega}\overline{\mathbf{b}}_{xx} \big) \in L^2(Q_T),\\
\end{split}\right.
\end{equation*}
such that, as $\mu\rightarrow 0$, $(\rho,u,\mathbf{w},\mathbf{b}, \theta)$ converges in the following sense
\begin{equation*}\label{rate}
\begin{split}
&(\rho,u,\mathbf{b}, \theta)\rightarrow (\overline\rho,\overline u,\overline{\mathbf{b}},
\overline\theta)~~ \hbox{\rm strongly in}~~C^\alpha(\overline Q_T),~\forall \alpha\in(0, 1/4),\\
&(u_x, \theta_x)\rightarrow (\overline u_x, \overline\theta_x)~~\hbox{\rm strongly in}
~~L^{s_1}(Q_T),~\forall s_1 \in [1, 6),\\
& \mathbf{b}_x \rightarrow \overline{\mathbf{b}}_x ~~\hbox{\rm strongly in}~~L^{s_2}(Q_T),~\forall s_2 \in [1, 4),\\
& (\rho_t,\rho_x) \rightharpoonup (\overline\rho_t, \overline\rho_x)
~~\hbox{\rm weakly}-*~\hbox{\rm in}~ L^\infty(0, T; L^2(\Omega)),\\
&(u_t,\mathbf{b}_t,\theta_t,u_{xx}, \theta_{xx})
\rightharpoonup(\overline u_t,\overline{\mathbf{b}}_t, \overline\theta_t,\overline u_{xx},
\overline\theta_{xx})~~ \hbox{\rm weakly in}~~L^2(Q_T),\\
& \mathbf{b}_{xx} \rightharpoonup \overline{\mathbf{b}}_{xx}
\quad \hbox{\rm weakly in}~~L^2\big((a+\delta,b-\delta)\times(0, T)\big),~\forall \delta\in\big(0, (b-a)/2\big),\\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
& \mathbf{w} \rightarrow \overline{\mathbf{w}}\quad\hbox{\rm strongly in}~~C^\alpha([a+\delta,b-\delta]\times[0, T]),~\forall \delta\in\big(0, (b-a)/2\big),~\alpha\in(0, 1/4)\\
& \mathbf{w}_t \rightharpoonup \overline{\mathbf{w}}_t ~~\hbox{\rm weakly in}~~L^2(Q_T),\\
& \mathbf{w}_x \rightharpoonup \overline{\mathbf{w}}_x\quad\hbox{\rm weakly}-*~\hbox{\rm in}~ L^\infty(0, T; L^2(a+\delta,b-\delta)),~\forall \delta\in\big(0, (b-a)/2\big),\\
&\mathbf{w}\rightarrow \overline{\mathbf{w}}~~ \hbox{\rm strongly in}~~L^r(Q_T),\quad\forall r \in [1, +\infty),\\
&\sqrt{\mu}\|\mathbf{w}_x\|_{L^4(Q_T)} \rightarrow 0.
\end{split}
\end{equation*}
Moreover, $(\overline{\rho},\overline{u},\overline{\mathbf{w}},\overline{\mathbf{b}},\overline{\theta})$
solves problem (<ref>)–(<ref>) with $\mu=0$ in the sense:
\begin{equation}\label{equations}
\begin{split}&\left.\begin{split}
&\overline\rho_t+(\overline\rho ~\overline u)_x=0, \\[1mm]
&(\bar\rho\bar u)_t+\big(\bar\rho\bar u^2+\gamma\overline\rho\overline\theta+|\overline{\mathbf{b}}|^2/2\big)_x
=\lambda \overline u_{xx},\\[1mm]
&(\bar\rho\overline{\mathbf{w}})_t+ (\bar\rho\bar u\overline{\mathbf{w}}-\overline{\mathbf{b}})_x=0,\\[1mm]
& \overline{\mathbf{b}}_t+ ( \bar u\overline{\mathbf{b}}-\overline{\mathbf{w}})_x=\nu \overline{\mathbf{b}}_{xx},\\[1mm]
& (\overline\rho \overline\theta)_t+(\bar\rho \bar u\overline\theta)_x +\gamma\overline\rho
\overline\theta \overline u_x
= \lambda \overline u_x^2+\nu|\overline{\mathbf{b}}_x|^2,
\end{split}\right\}~\hbox{\rm a.e. in }~Q_T,\\[1mm]
&\iint_{Q_T} \Big\{\big[(\overline\rho \overline\theta)_t+(\bar\rho\bar u\overline\theta)_x
+\gamma\overline\rho \overline\theta \overline u_x-\lambda \overline u_x^2-\nu|\overline{\mathbf{b}}_x|^2\big]\varphi
= 0,
\end{split}\end{equation}
for all $\varphi \in L^2(0, T;W^{1,2}(\Omega))$.
(iii) Assume that
\in \mathbb{F}$ is a solution for the limit problem
(<ref>). Then
\begin{equation*}\label{0u3}
\begin{aligned}
&\|(\rho-\overline\rho, u-\overline u, \mathbf{w}-\overline{\mathbf{w}},\mathbf{b}-\overline{\mathbf{b}},
\theta-\overline\theta )\|_{L^\infty(0, T;L^2(\Omega))}\\
&\quad\quad\quad+\|(u_x-\overline u_x, \mathbf{b}_x-\overline{\mathbf{b}}_x, \theta_x-\overline\theta_x)\|_{L^2(Q_T)}=
\end{aligned}
\end{equation*}
With the estimates appearing in the above
theorem, and following the argument given in <cit.>(cf. <cit.>), if the initial data is in Hölder space,
\begin{equation*}
\rho_0\in C^{1+\alpha}(\Omega),\,\,\quad (u_0,\mathbf{w}_0,\mathbf{b}_0,\theta_0)\in C^{2+\alpha}(\Omega)
\end{equation*}
for some $\alpha\in (0,1)$, then there exists a unique classical solution
\begin{equation*}
\rho\in C^{1+\alpha,1+\alpha/2}(Q_T),\,\,\quad (u,\mathbf{w},\mathbf{b},\theta)\in C^{2+\alpha,1+\alpha/2}(Q_T),
\end{equation*}
and it satisfies (<ref>).
It should be pointed out that if we only consider the global existence with fixed $\mu$, then
the condition $u_0\in W^{2,m} (m>1)$ in (<ref>) can be removed. In fact, this can be done
in a more easy way, but we do not pursue it in the paper.
Compared to <cit.> and some related references, the generality of the condition (<ref>) causes
some other technical difficulties since all the
estimates in (<ref>) must be uniform in $\mu$. Firstly, we must
overcome the difficulty coming from the dissipative estimate on the
temperature. For example, Fan-Jiang-Nakamura <cit.> only
established the $\mu$-uniform estimate of $\theta_x$ in $L^\beta(Q_T)$ with any $ \beta\in(1, 3/2)$
by means of the technique used by Frid and Shelukhin <cit.>. Secondly, to
obtain the stronger convergence of $\mathbf{w}$ and $\mathbf{b}$
(see Theorem 1.1(ii)), we must establish some new uniform estimates
on the derivatives of $\mathbf{w}$ and $\mathbf{b}$. Thirdly, we
must seek a new method to obtain a uniform upper bound of the
To overcome the difficulties, some techniques are developed
here. One of two ingredients in the proof is the boundary estimates
of derivatives of the transverse velocity and the magnetic field,
and the other is that we deduce a uniform upper bound of $\theta$ by
a simple, direct method.
Below we present a sketch of the proof to (<ref>). Firstly, the
uniform upper and lower bounds of the density can be obtained in a
standard way. Next, a key observation is that
we can establish the uniform bound of
$\|u_{xx}\|_{L^{m_0}(Q_T)} (m_0>1)$ by $L^p$-theory of
linear parabolic equations (see Lemma <ref>), which plays an
important role in this paper. It should be pointed out that it is in
this step we ask the condition $u_0\in W^{2,m}(\Omega)$ for some
$m>1$. By virtue of the estimate and a delicate analysis, we then
deduce the difficult bounds of $\|\omega\mathbf{w}_x\|_{L^\infty(0,
T;L^2(\Omega))}$ and
u_{xx},\theta_{x},\omega\mathbf{b}_{xx})\|_{L^{2}(Q_T)}$(see Lemma <ref>).
In this step, the main idea is to use the norm $\|u_{xx}\|_{L^{2}(Q_T)}$ to control the qualities
$\|(\omega\mathbf{w}_x, \mathbf{b}_x)\|_{L^\infty(0, T;L^2(\Omega))}$ and
$\|\mathbf{w}_t\|_{L^2(Q_T)}$ (see Lemmas <ref>-<ref>) and then,
from the equations of $u$ and $\theta$ it follows the uniform bound of $\|u_{xx}\|_{L^{2}(Q_T)}$
by Gronwall's inequality.
With the uniform bound of $\|\mathbf{b}_t\|_{L^{2}(Q_T)}$,
we deduce the uniform bounds of $\|\mathbf{w}_x\|_{L^\infty(0, T;L^1(\Omega))}$
and $\|\mathbf{b}_x\|_{L^2(0, T;L^\infty(\Omega)}$ (see Lemmas <ref> and <ref>),
by which we further obtain the uniform bounds of $\|\sqrt{\omega}\mathbf{w}_x\|_{L^\infty(0, T;L^2(\Omega))}$
and $\big(\mu^{1/4}\|\mathbf{w}_x\|_{L^\infty(0,T;L^2(\Omega))}+\mu^{3/4}\|\mathbf{w}_{xx}\|_{L^2(Q_T)}\big)$
(see Lemma <ref>), which are essential to study both $L^2$ convergence rate and boundary layer thickness.
Due to the above estimates, we finally get an upper bound of
$\theta$ in a direct way (see Lemma <ref>). As a
consequence, the uniform bound of $\|(\theta_t,
\theta_{xx})\|_{L^{2}(Q_T)}$ can be obtained by a brief argument
(see Lemma <ref>). Consequently, the passage to limit is
justified in the more strong sense.
Next, we investigate the thickness of boundary layer. At first, we
give the definition of a BL-thickness defined as in
<cit.> (cf. <cit.>).
A function $\delta(\mu)$ is called a BL-thickness for problem
(<ref>)-(<ref>) with vanishing $\mu$ if
$\delta(\mu)\downarrow 0$ as $ \mu \downarrow 0$, and
\begin{equation*}\label{12}
\begin{aligned}
&\lim\limits_{\mu\rightarrow 0}\|(\rho-\overline\rho, u-\overline u,
\mathbf{w}-\overline{\mathbf{w}}, \mathbf{b}-\overline{\mathbf{b}},
\theta-\overline\theta)\|_{L^\infty(0,T;L^\infty(\Omega_{\delta(\mu)}))}=0,\\
&\mathop{ \inf\lim}\limits_{\mu\rightarrow 0}\|(\rho-\overline\rho,
u-\overline u, \mathbf{w}-\overline{\mathbf{w}},
\mathbf{b}-\overline{\mathbf{b}},\theta-\overline\theta)\|_{L^\infty(0,T;L^\infty(\Omega))}>0,
\end{aligned}
\end{equation*}
where $\Omega_\delta=(\delta, 1-\delta)$ for $\delta \in (0, 1/2)$,
and $(\rho, u, \mathbf{w}, \mathbf{b}, \theta)$ and
$(\overline\rho,\overline u, \overline{\mathbf{w}},
\overline{\mathbf{b}}, \overline\theta)$ are the solutions to
problem (<ref>)-(<ref>) and problem (<ref>)-(<ref>)
with $\mu=0$, respectively.
We shall prove that for any $\alpha \in (0, 1/2)$, the function
is a BL-thickness, which is almost optimal since it is close to the
classical value $O(\sqrt{\mu})$ (see e.g. <cit.>). One can see from
the proof in Section 3 that our method is a bit different from that
used in <cit.>, which is based on an iteration inequality (<ref>). To indicate the
idea clearly, we further assume that
\begin{equation}\label{vw}
\begin{aligned}
\mathbf{w}_0=\mathbf{b}_0\equiv\mathbf{0}.
\end{aligned}
\end{equation}
Let (<ref>),
(<ref>) and (<ref>) hold. Assume that $(\mathbf{w}^-, \mathbf{w}^+)$ is not identically equal to
$\mathbf{0}$. Then the limit problem (<ref>) has a unique solution $(\overline \rho,\overline u, \mathbf{0},
\mathbf{0}, \overline\theta)$ in $\mathbb{F}$, and the function $\delta(\mu)=\mu^\alpha$
for any $\alpha\in(0, 1/2)$
is a BL-thickness for problem (<ref>)-(<ref>) such that
\begin{equation*}
\begin{split}
&\lim\limits_{\mu\rightarrow 0} \|(\rho-\overline\rho,u-\overline u,\mathbf{b},\theta-\overline\theta)\|_{C^\alpha(\overline Q_T)}=0,\quad \forall \alpha\in (0,1/4),\\
&\lim\limits_{\mu\rightarrow 0} \|\mathbf{w}\|_{L^\infty(0, T;L^\infty(\delta(\mu),
1-\delta(\mu)))}=0,\quad \mathop{\inf\lim}\limits_{\mu\rightarrow 0} \|\mathbf{w}\|_{L^\infty(0, T;L^\infty(\Omega))}>0.\\
\end{split}
\end{equation*}
Moreover, $\mathbf{w}$ has the asymptotic property:
\begin{equation*}
\begin{split}
\|\mathbf{w}_x\|^2_{L^\infty(0, T;L^2(\delta, 1-\delta))}\leq \left\{
\begin{split}
& C_n\big(\tau+\tau^3+\cdots+\tau^{n-2}\big)+C_n\mu^{(n-1)/2}/\delta^n~(n=\hbox{\rm odd}),\\
& C_n\big(\tau+\tau^3+\cdots+\tau^{n-1}\big)+C_n\mu^{(n-1)/2}/\delta^n~(n=\hbox{\rm even}),\\
\end{split}\right.
\end{split}
\end{equation*}
where $\delta\in (0, 1/2), \tau=\sqrt{\mu}/\delta$, and the constants $C_n$ are independent of $\mu$ and $\delta$.
The remainder of this paper shall be arranged as follows. In Section
2, we will prove Theorem <ref>. For this, a lot of a
priori estimates independent of $\mu$ are derived in Section 2.1,
which are sufficient to prove this theorem. The second and third
parts of this theorem can be shown in Sections 2.2 and 2.3,
respectively. Finally, we will give the proof of Theorem <ref> in
Section 3.
§ THE PROOF OF THEOREM 1.1
The existence and uniqueness of local solutions can be obtained by
using the Banach theorem and the contractivity of the operator
defined by the linearization of the problem on a small time interval
(cf.<cit.>). The existence of global solutions is proved
by extending the local solutions globally in time based on the
global a priori estimates of solutions. The uniqueness of the global
solution follows from the uniqueness of the local solution. Thus,
the next subsection will focus on deriving required a priori
estimates of the solution $(\rho,u,\mathbf{w}, \mathbf{b},\theta)$. Moreover, all a
priori estimates which will be established are uniform in $\mu$.
Throughout this section, we shall denote by $C$ the various positive
constants dependent on $T$, but independent of $\mu$.
§.§ A priori estimates independent of $\mu$
Rewrite (<ref>) as
\begin{equation}\label{e20}
\begin{split}
&\mathcal{E}_t+\Big[u\big(\mathcal{E}+p+\frac12|\mathbf{b}|^2\big)-\mathbf{w}\cdot\mathbf{b}\Big]_x=\big(\lambda uu_x+\mu\mathbf{w}\cdot\mathbf{w}_x+\nu\mathbf{b}\cdot\mathbf{b}_x+\kappa\theta_x\big)_x,\\[2mm]
&(\rho \mathcal{S})_t+(\rho u \mathcal{S})_x-\left(\frac{\kappa\theta_x}{\theta}\right)_x=\frac{\lambda u_x^2+\mu|\mathbf{w}_x|^2+\nu|\mathbf{b}_x|^2}{\theta}
\end{split}
\end{equation}
where $\mathcal{E}$ and $\mathcal{S}$ are the total energy and the
entropy, respectively,
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
Let (<ref>) and (<ref>) hold. Then
\begin{equation}\label{ba1}
\begin{split}
&\int_\Omega \rho(x,t)dx=\int_\Omega \rho_0(x)dx,\quad\forall t \in (0, T),\\
&\sup\limits_{0<t<T}\int_\Omega\big[\rho(\theta+ u^2+|\mathbf{w}|^2)+ |\mathbf{b}|^2\big] dx\leq
& \iint_{Q_T}
\left(\frac{\lambda u_x^2+\mu|\mathbf{w}_x|^2+\nu|\mathbf{b}_x|^2}{\theta}+\frac{\kappa\theta_x^2}{\theta^2}\right)dxdt
\leq C.
\end{split}
\end{equation}
Integrating $\eqref{e20}_1$ over $Q_t=\Omega\times(0, t)$ with $t\in (0, T)$ yields
\begin{equation}\label{total}
\begin{split}
&\int_\Omega \mathcal{E}dx=\int_\Omega \mathcal{E}|_{t=0}dx+\mu\int_0^t\mathbf{w}\cdot\mathbf{w}_x|_{x=0}^{x=1}ds.
\end{split}
\end{equation}
Let $a=0$ or $1$. We first integrate (<ref>)$_3$ from $x=a$ to $x$, and then integrate the resulting equation over $\Omega$, to obtain
\begin{equation*}\label{w8}
\begin{split}
\mu\mathbf{w}_x(a,t)=\mu\big(\mathbf{w}^+- \mathbf{w}^-\big)-\int_\Omega(\rho u\mathbf{w}-\mathbf{b})dx-\frac{\partial}{\partial t}\int_\Omega\int_a^x\rho\mathbf{w}dydx.
\end{split}
\end{equation*}
Multiplying it by $\mathbf{w}(a,t)$ and integrating over $(0, t)$, we have
\begin{equation*}
\begin{split}
\mu\int_0^t(\mathbf{w}\cdot\mathbf{w}_x)(a,s)ds=&\mu\int_0^t\big(\mathbf{w}^+- \mathbf{w}^-\big)\cdot\mathbf{w}(a,s)ds -\int_0^t\mathbf{w}(a,s)\cdot\left(\int_\Omega(\rho u\mathbf{w}-\mathbf{b})dx\right)ds\\[1mm]
\rho\mathbf{w}dydx\right)+\mathbf{w}(a,0)\cdot\left(\int_\Omega\int_a^x
\rho_0\mathbf{w}_0dydx\right)\\[1mm]
\end{split}
\end{equation*}
hence, by Young's inequality and (<ref>)$_1$,
\begin{equation*}\label{w10}
\begin{split}
\left|\mu\int_0^t(\mathbf{w}\cdot\mathbf{w}_x)(a,s)ds\right|\leq&C+C\int_\Omega\rho|\mathbf{w}|dx+C \iint_{Q_t}\big(\rho |u| |\mathbf{w}|+|\mathbf{b}|+\rho |\mathbf{w}|\big)dxds\\
\leq&C+\frac12\int_\Omega\mathcal{E}dx+C\iint_{Q_t}\mathcal{E}dxds.
\end{split}
\end{equation*}
Substituting it into (<ref>) yields
\begin{equation*}
\begin{split}
&\int_\Omega \mathcal{E}dx\leq C+C\iint_{Q_t}\mathcal{E}dxds,
\end{split}
\end{equation*}
and so, (<ref>)$_2$ follows from Gronwall's inequality.
(<ref>)$_3$ can be proved by integrating (<ref>)$_2$ and using (<ref>)$_2$. The proof is complete.
From Lemma 2.1, the following estimates can be proved.
Let (<ref>) and (<ref>) hold. Then
\begin{equation}\label{rho11}
\begin{split}
& C^{-1}\leq \rho \leq C, \\
& \theta\geq C,\\
&\iint_{Q_T} \frac{\kappa\theta_x^2}{\theta^{1+\alpha}} dxdt
\leq C,\quad\forall \alpha \in (0, \min\{1,q\}),\\
\leq C,\quad\forall \alpha \in (0, \min\{1,q\}),\\
&\iint_{Q_T} \left(\lambda
u_x^2+\mu|\mathbf{w}_x|^2+\nu|\mathbf{b}_x|^2 \right)dxdt
\leq C,\\
\leq C,\\
&\iint_{Q_T} |\theta_x|^{3/2} dxdt\leq C.
\end{split}
\end{equation}
The proofs to the estimates $\rho \leq C$ and
(<ref>)$_3$-(<ref>)$_5$ can be found in <cit.> where
the vacuum is permitted. (<ref>)$_6$ is an immediate
consequence of (<ref>)$_5$, so the estimate $\rho\geq
C^{-1}$ can be proved in a standard way (see <cit.>). We
omit their proofs for brevity.
Now we turn to (<ref>)$_2$, whose proof depends only on the estimate $\rho \leq C$. It follows from (<ref>)$_5$
\begin{equation*}
\begin{split}
\theta_t+u\theta_x-\frac{1}{\rho} (\kappa \theta_x)_x
\geq & \frac{\lambda}{\rho} \left(u_x^2 -\frac{p}{\lambda} u_x \right)
= \frac{\lambda}{\rho} \left(u_x
\end{split}
\end{equation*}
By $ \rho \leq C$, we have
\begin{equation*}\begin{split}
\theta_t+u\theta_x-\frac{1}{\rho} (\kappa
\theta_x)_x +K \theta^2\geq 0,\\
\end{split}\end{equation*}
where $K$ is a positive constant independent of $\mu$. Let $z=\theta-\underline\theta$, where
with $C=K\min_{\overline\Omega}\theta_0$. Then
$ z_x|_{x=0, 1}=0, ~z|_{t=0}\geq0, $ and
\begin{equation*}\begin{split}
&z_t+u z_x-\frac{1}{\rho} (\kappa z_x)_x +K(\theta+\underline\theta)z\\
&=\theta_t+C\frac{\min_{\overline\Omega}\theta_0}{(Ct+1)^2}+u\theta_x-\frac{1}{\rho} (\kappa
\theta_x)_x +K \theta^2- K\left(\frac{\min_{\overline\Omega}\theta_0}{Ct+1}\right)^2\\
&\geq C\frac{\min_{\overline\Omega}\theta_0}{(Ct+1)^2} -K\left(\frac{\min_{\overline\Omega}\theta_0}{Ct+1}\right)^2 = 0,
\end{split}\end{equation*}
and then, $z\geq 0$ on
$\overline Q_T$ by the comparison theorem, so (<ref>)$_2$.
It remains to show (<ref>)$_7$. By (<ref>)$_2$ and (<ref>)$_3$, we have
\begin{equation}\label{theta01}
\begin{split}
\iint_{Q_T}\frac{\theta_x^2}{\theta} dxdt\leq C.
\end{split}
\end{equation}
Then, we have by the mean value theorem, Lemma <ref>, (<ref>)$_1$ and Hölder's inequality
\begin{equation*}
\begin{split}
\theta \leq & \int_\Omega \theta dx
+ \int_\Omega |\theta_x|dx\\
\leq &C+C\left(\int_\Omega\frac{\theta_x^2}{\theta}dx\right)^{1/2}\left(\int_\Omega
\theta dx\right)^{1/2}\\
\leq & C+C\left(\int_\Omega\frac{\theta_x^2}{\theta}dx\right)^{1/2},
\end{split}
\end{equation*}
which together with (<ref>) gives
\begin{equation}\label{theta00}
\begin{split}
\int_0^T\|\theta\|_{L^\infty(\Omega)}^2dt \leq C.\\
\end{split}
\end{equation}
Thus, it follows from Hölder's inequality, Lemma 2.1 and (<ref>) that
\begin{equation*}
\begin{split}
\iint_{Q_T}|\theta_x|^{3/2}dxdt\leq&\left(\iint_{Q_T}\frac{\theta_x^2}{\theta} dxdt\right)^{3/4}
\left(\iint_{Q_T}\theta^3 dxdt\right)^{1/4}\\
\leq& C\left(\int_0^T\|\theta^{2}\|_{L^\infty(\Omega)}\int_\Omega \theta dxdt\right)^{1/4}\leq C.\\
\end{split}
\end{equation*}
The proof is complete.
About the magnetic field $\mathbf{b}$, we have
Let (<ref>) and (<ref>) hold. Then
\begin{equation*}\label{b00}
\begin{split}
& \sup\limits_{0<t<T}\int_\Omega|\mathbf{b}|^4dx+\iint_{Q_T}|\mathbf{b}|^2|\mathbf{b}_x|^2dxdt\leq C.
\end{split}
\end{equation*}
Multiplying (<ref>)$_4$ by $4|\mathbf{b}|^2\mathbf{b}$ and integrating over $Q_t$, we obtain
\begin{equation}\label{b0}
\begin{split}
&\int_\Omega|\mathbf{b}|^4dx+4\nu\iint_{Q_t}|\mathbf{b}|^2|\mathbf{b}_x|^2dxds+8\nu\iint_{Q_t}|\mathbf{b}\cdot \mathbf{b}_x|^2dxds\\
&=\int_\Omega|\mathbf{b}_0|^4dx+4\iint_{Q_t} \mathbf{w}_x \cdot(|\mathbf{b}|^2\mathbf{b})dxds-4\iint_{Q_t} (u \mathbf{b})_x\cdot(|\mathbf{b}|^2\mathbf{b})dxds.\\
\end{split}
\end{equation}
Integrating by parts and using Young's inequality, we have
\begin{equation}\label{b1}
\begin{split}
\iint_{Q_t} \mathbf{w}_x \cdot(\mathbf{b} |\mathbf{b}|^2) dxds
&=-\iint_{Q_t} \mathbf{w} \cdot(\mathbf{b}_x |\mathbf{b}|^2)dxds-2\iint_{Q_t} (\mathbf{w} \cdot\mathbf{b} )(\mathbf{b}\cdot \mathbf{b}_x)dxds\\
&\leq 3\iint_{Q_t} |\mathbf{w}| |\mathbf{b}|^2 |\mathbf{b}_x| dxds\\
&\leq \frac{\nu}{4}\iint_{Q_t} |\mathbf{b}|^2 |\mathbf{b}_x|^2dxds+C\iint_{Q_t} |\mathbf{w}|^2 |\mathbf{b}|^2 dxds\\
&\leq \frac{\nu}{4}\iint_{Q_t} |\mathbf{b}|^2 |\mathbf{b}_x|^2dxds+C\int_0^t\|\mathbf{b}\|_{L^\infty(\Omega)}^2 \int_\Omega |\mathbf{w}|^2dxds\\
&\leq\frac{\nu}{4}\iint_{Q_t} |\mathbf{b}|^2 |\mathbf{b}_x|^2dxds+C,
\end{split}
\end{equation}
where we used (<ref>)$_2$ and (<ref>)$_6$.
On the other hand, we have
\begin{equation}\label{b2}
\begin{split}
&-\iint_{Q_t} (u \mathbf{b})_x\cdot|\mathbf{b}|^2\mathbf{b}dxds=3\iint_{Q_t} u (\mathbf{b}_x \cdot\mathbf{b})|\mathbf{b}|^2 dxds\\
&\leq \frac{\nu}{4}\iint_{Q_t} |\mathbf{b}|^2 |\mathbf{b}_x|^2 dxds+C\iint_{Q_t} u^2 |\mathbf{b}|^4 dxds\\
&\leq\frac{\nu}{4}\iint_{Q_t} |\mathbf{b}|^2 |\mathbf{b}_x|^2 dxds+C\int_0^t\|u^2\|_{L^\infty(\Omega)}\int_\Omega |\mathbf{b}|^4 dxds.
\end{split}
\end{equation}
Plugging (<ref>) and (<ref>) into (<ref>) and using
Gronwall's inequality, we finish the proof by noticing
$\int_0^T\|u^2\|_{L^\infty(\Omega)} dt\leq C\iint_{Q_T}u_x^2dxdt\leq
Let (<ref>) and (<ref>) hold. Then
\begin{equation}\label{rho}
\begin{split}
&\sup\limits_{0<t<T}\int_\Omega\rho_x^2dx+\iint_{Q_T} \big(\rho_t^2+\theta
\rho_x^2\big) dxdt\leq C,\\[1mm]
&\left|\rho(x,t)-\rho(y,s)\right|\leq C\left(|x-y|^{1/2}
+|s-t|^{1/4}\right),\quad\forall (x, t), (y, s) \in \overline Q_T.
\end{split}
\end{equation}
Set $\eta=1/\rho$. It follows from the equation (<ref>)$_1$ that
Substituting it into (<ref>)$_2$ yields
\begin{equation*}
\left[\rho(u-\lambda\eta_x)\right]_t+\left[\rho
\end{equation*}
Multiplying it by $(u-\lambda\eta_x)$ and integrating over $Q_t$, we
\begin{equation*}
\begin{split}
\theta\rho^2\eta_x^{2}dxds\\
&=\frac12\int_\Omega\rho_0(u_0+ \lambda\rho_0^{-2}\rho_{0x})^2dx+\gamma\iint_{Q_t}\rho^2\theta u \eta_x dxds
\\
&\quad-\gamma\iint_{Q_t}\rho^2\eta\theta_x (u-\lambda\eta_x)dxds-\iint_{Q_t}\mathbf{b}\cdot\mathbf{b}_x(u-\lambda\eta_x)dxds.
\end{split}
\end{equation*}
To estimate the second integral on right-hand side, we use Young's
inequality, Lemmas 2.1 and 2.2 to obtain
\begin{equation*}
\begin{split}
\gamma\iint_{Q_t}\rho^2\theta u \eta_x dxds
\frac{\gamma\lambda}{2}\iint_{Q_t}\theta\rho^{2}\eta_x^2dxds+C\iint_{Q_t} \theta u^2 dxds\\
&\leq \frac{\gamma\lambda}{2}\iint_{Q_t}\theta\rho^{2}\eta_x^2dxds+C\int_0^t\|\theta\|_{L^\infty(\Omega)}
\int_{\Omega} u^2 dxds\\
\end{split}
\end{equation*}
On the other hand, we have by Cauchy-Schwarz's inequality, (<ref>) and Lemma <ref>
\begin{equation*}
\begin{split}
&-\gamma\iint_{Q_t}\rho^2\eta\theta_x (u-\lambda\eta_x)dxds-\iint_{Q_t}\mathbf{b}\cdot\mathbf{b}_x(u-\lambda\eta_x)dxds\\
&\leq C+C\iint_{Q_t}\theta \rho
(u-\lambda\eta_x)^2dxds+C\iint_{Q_t}\frac{\theta_x^2}{\theta}dxds+C\iint_{Q_t} \rho(u-\lambda\eta_x)^2dxds\\
&\leq C+C\int_0^t\big(1+\|\theta\|_{L^\infty(\Omega)}\big)\int_\Omega\rho
\end{split}
\end{equation*}
Combining the above results yields
\begin{equation*}
\begin{split}
& \int_\Omega\rho
(u-\lambda\eta_x)^2dx+ \iint_{Q_t} \theta\rho^2\eta_x^{2}dxds\\
&\leq C
\end{split}
\end{equation*}
which together with Gronwall's inequality gives
\begin{equation*}\label{rho22}
\begin{split}
&\sup\limits_{0<t<T}\int_\Omega\rho_x^2dx+\iint_{Q_T} \theta \rho_x^2 dxds\leq C .
\end{split}
\end{equation*}
By this estimate and Lemma 2.2, we derive from the equation (<ref>)$_1$ that
\begin{equation*}
\begin{split}
\iint_{Q_T} \rho_t^2dxdt \leq & C\int_0^T\|u^2\|_{L^\infty(\Omega)}\int_\Omega\rho_x^2dxdt+C\iint_{Q_T}u_x^2dxdt \leq C.\\
\end{split}
\end{equation*}
Thus (<ref>)$_1$ holds.
Now we prove the second estimate. Let $\beta(x)=\rho(x,t)-\rho(x,s)$ for any $x \in [0, 1]$
and $s, t \in [0, T]$ with $s\neq t$. Then for any $x\in [0, 1]$ and $\delta \in (0, 1/2]$, there exist some
$y\in [0, 1]$ and $\xi$ between $x$ and $y$ such that $\delta=|y-x|$ and
$\beta(\xi)=\frac{1}{x-y}\int^x_y\beta(z)dz$, and hence
\begin{equation*}
\begin{split}
\beta(x)=\frac{1}{x-y}\int^x_y\beta(z)dz+\int_\xi^x\beta'(z)dz,
\end{split}
\end{equation*}
therefore, by Hölder's inequality and (<ref>),
\begin{equation*}
\begin{split}
|\beta(x)|\leq& \frac{1}{\delta}\left|\int^x_y\beta(z)dz\right|+\left|\int_\xi^x\beta'(z)dz\right|\\[1mm]
\leq &\frac{1}{\delta}\left|\int^x_y\hspace{-2mm}\int_s^t\rho_\tau d\tau dz\right|+\left|\int_\xi^x\left[\rho_z(z,t)-\rho_z(z,s)\right]dz\right|\\[1mm]
\leq &\frac{1}{\delta}\left(\iint_{Q_T}\rho_\tau^2 d\tau dz\right)^{1/2}|x-y|^{1/2}|s-t|^{1/2}\\[1mm]
&+ \left(\int_0^1(|\rho_z(z,s)|^2+|\rho_z(z,t)|^2)dz\right)^{1/2}|x-\xi|^{1/2}\\[1mm]
\leq & C\delta^{-1/2}|s-t|^{1/2}+C\delta^{1/2}.
\end{split}
\end{equation*}
If $0<|s-t|^{1/2}<1/2$, taking $\delta=|s-t|^{1/2}$ yields
\begin{equation}\label{rho6}
\begin{split}
|\rho(x,s)-\rho(x,t)|\leq C|s-t|^{1/4}.
\end{split}
\end{equation}
If $ |s-t|^{1/2}\geq 1/2$, then (<ref>) holds since $\rho$ is uniformly bounded in $\mu$.
On the other hand, we have by (<ref>)$_1$
\begin{equation}
\begin{split}
|\rho(x,t)-\rho(y,t)|=\left|\int_y^x\rho_zdz\right|\leq C|x-y|^{1/2}.
\end{split}
\end{equation}
Thus, (<ref>)$_2$ is a consequence of the triangle inequality. The proof is complete.
To deduce other required $\mu$-uniform estimates, we need the
following lemma which plays an important role in this paper.
Let (<ref>) and (<ref>) hold. Then
\begin{equation}\label{uxx}
\begin{split}
\iint_{Q_T}|u_{xx}|^{m_0} dxdt\leq C,\quad m_0=\min\{m, 4/3\}.
\end{split}
\end{equation}
In particular,
\begin{equation}\label{u0}
\begin{split}
\int_0^T\|u_{x}\|_{L^\infty(\Omega)}^{m_0}dt \leq C.
\end{split}
\end{equation}
Note that the estimate (<ref>) is an immediate consequence of (<ref>). Thus, it is enough to prove (<ref>). To this end, we rewrite the equation (<ref>)$_2$ as
\begin{equation}\label{f2}
\begin{split}
\gamma\theta_x-\frac{\gamma}{\rho}\rho_x\theta-\frac{1}{\rho}\mathbf{b}\cdot\mathbf{b}_x=:f.
\end{split}
\end{equation}
We will apply $L^p$ estimates of linear parabolic equations (cf.<cit.>) to show (<ref>). From (<ref>)$_2$,
the coefficient $a(x,t):=\lambda/\rho$ satisfies
$$\left|a(x,t)-a(y,s)\right|\leq C\left(|x-y|^{1/2}
+|s-t|^{1/4}\right),\quad\forall (x, t), (y, s) \in \overline Q_T.
Due to the condition on $u_0$ in (<ref>), we only need
to give a uniform bound of $f$ in $L^{4/3}(Q_T)$.
From Lemmas 2.2 and 2.3, the second term and the forth term on right-hand side of (<ref>) are
uniformly bounded in $L^{3/2}(Q_T)$ and $L^{2}(Q_T)$, respectively.
To deal with the first term on right-hand side of (<ref>), we observe by Hölder inequality and Lemma 2.1
\begin{equation*}
\begin{split}
u^2&\leq2\int_\Omega|uu_x|dx \leq 2\left(\int_\Omega u^2dx\right)^{1/2}\left(\int_\Omega u_x^2dx\right)^{1/2} \leq C\left(\int_\Omega u_x^2dx\right)^{1/2},\\
\end{split}
\end{equation*}
therefore, we have by Lemma 2.2
\begin{equation*}
\begin{split}
\int_0^T\|u\|_{L^\infty}^4 dt
\leq C \iint_{Q_T} u_x^2dx\leq C,
\end{split}
\end{equation*}
which together with Young's inequality yields
\begin{equation*}
\begin{split}
\iint_{Q_T}|uu_x|^{3/2}dxdt &\leq C\iint_{Q_T}u_x^2dxdt+C\iint_{Q_T}u^6dxdt\\
&\leq C+ C \int_0^T\|u\|_{L^\infty(\Omega)}^4 \int_{\Omega}u^2dxdt \leq C.
\end{split}
\end{equation*}
As to the third term on right-hand side of (<ref>), we have by (<ref>)$_1$ and (<ref>)
\begin{equation*}
\begin{split}
\iint_{Q_T}|\rho_x \theta|^{4/3}dxdt
&\leq C\iint_{Q_T}\rho_x^2\theta dxdt+C\iint_{Q_T}\theta^{2}dxdt
\leq C.
\end{split}
\end{equation*}
Combining the above results gives $\|f\|_{L^{4/3}(Q_T)}\leq C.$ The
proof is then completed.
By a direct application of the above lemma, we obtain
Let (<ref>) and (<ref>) hold. Then
\begin{equation*}\label{we11}
\begin{split}
& \mu\sup\limits_{0<t<T}\int_\Omega|\mathbf{w}_x|^2dx+\mu^2 \iint_{Q_T}
|\mathbf{w}_{xx}|^2 dxds \leq C.
\end{split}
\end{equation*}
We rewrite (<ref>)$_3$ in the form
\begin{equation}\label{w12}
\begin{split}
\mathbf{w}_t-\frac{\mu}{\rho}\mathbf{w}_{xx}=\frac{1}{\rho}\mathbf{b}_x-u\mathbf{w}_x,
\end{split}
\end{equation}
and multiply it by $\mu\mathbf{w}_{xx}$ and integrating over $Q_t$
to obtain
\begin{equation}\label{0v100}
\begin{split}
& \frac\mu2\int_\Omega|\mathbf{w}_x|^2dx
&\quad -\frac{\mu}{2}\iint_{Q_t}u_x |\mathbf{w}_x|^2dxds+\mu\int_0^t \mathbf{w}_{t}\cdot\mathbf{w}_{x} \Big|_{x=0}^{x=1} ds\\
& \leq C\mu+\frac{\mu^2}{4}\iint_{Q_t}\frac{1}{\rho}|\mathbf{w}_{xx}|^2dxds+C\iint_{Q_t}|\mathbf{b}_x|^2dxds\\
&\quad+ C\int_0^t\|u_x\|_{L^\infty(\Omega)}\left(\mu\int_\Omega|\mathbf{w}_x|^2dx\right)ds+C\mu\int_0^t\|\mathbf{w}_x\|_{L^\infty(\Omega)} ds.
\end{split}
\end{equation}
From the mean value theorem and H$\ddot{o}$lder inequality, we obtain
\begin{equation}\label{wx2}
\begin{split}
|\mathbf{w}_x|^2\leq & \Big|\frac{\mathbf{w}(b,t)-\mathbf{w}(a,t)}{b-a}\Big|^2+2\int_\Omega|\mathbf{w}_{x}||\mathbf{w}_{xx}|dx\\
\leq& C+C\left(\int_\Omega|\mathbf{w}_{x}|^2dx\right)^{1/2}\left(\int_\Omega|\mathbf{w}_{xx}|^2dx\right)^{1/2},
\end{split}
\end{equation}
and so, Young's inequality yields
\begin{equation*}\label{w5}
\begin{split}
\mu\int_0^t\|\mathbf{w}_x\|_{L^\infty(\Omega)} ds
\leq& C\mu+C\int_0^t\mu^{1/4}\left(\mu\int_\Omega|\mathbf{w}_{x}|^2dx\right)^{1/4}
\left(\mu^2\int_\Omega|\mathbf{w}_{xx}|^2dx\right)^{1/4}ds\\[1mm]
\leq &C\sqrt{\mu}+\frac{C \mu}{\epsilon}\iint_{Q_t}
|\mathbf{w}_{x}|^2dxds+\epsilon \mu^2
\iint_{Q_t}\frac{1}{\rho}|\mathbf{w}_{xx}|^2dxds, \forall \epsilon \in (0, 1).
\end{split}
\end{equation*}
Inserting it into (<ref>) and taking a small $\epsilon>0$, we find that
\begin{equation*}
\begin{split}
& \mu\int_\Omega|\mathbf{w}_x|^2dx
+\mu^2\iint_{Q_t}\frac{1}{\rho}|\mathbf{w}_{xx}|^2dxds \\
&\leq C +C\int_0^t\big(1+\|u_x\|_{L^\infty}\big)\left(\mu\int_{Q_t}|\mathbf{w}_x|^2dx\right)ds.
\end{split}
\end{equation*}
Thus, the lemma follows from Gronwall's inequality and (<ref>). This proof is complete.
Our next main task is to show the other estimates appearing in
Theorem 1.1. To this end, we need three preliminary lemmas. The
first one reads as
Let (<ref>) and (<ref>) hold. Then
\begin{equation*}
\begin{split}
& \int_\Omega|\mathbf{b}_x|^2\omega^2dx+ \iint_{Q_t}|\mathbf{b}_{xx}|^2\omega^2dxds
\leq C\iint_{Q_t} |\mathbf{w}_{x}|^2\omega^2 dxds+C\left(\iint_{Q_t} u_{xx}^2
\end{split}
\end{equation*}
where $\omega$ is the same as that in Theorem 1.1.
Multiplying (<ref>)$_4$ by $\mathbf{b}_{xx}\omega^2(x)$ and
integrating over $Q_t$, we have
\begin{equation}\label{b111}
\begin{split}
&-\iint_{Q_t} \mathbf{b}_t\cdot \mathbf{b}_{xx}\omega^2 dxdt+\nu\iint_{Q_t} |\mathbf{b}_{xx}|^2\omega^2 dxds\\
&=\iint_{Q_t} (u\mathbf{b})_x\cdot \mathbf{b}_{xx}\omega^2 dxds-\iint_{Q_t} \mathbf{w}_x\cdot \mathbf{b}_{xx}\omega^2 dxds.
\end{split}
\end{equation}
To estimate the first integral on left-hand side of (<ref>), we
integrate by parts and use (<ref>)$_4$ to obtain
\begin{equation}\label{be1}
\begin{split}
\iint_{Q_t} \mathbf{b}_t\cdot \mathbf{b}_{xx}\omega^2 dxdt
&=-\frac{1}{2}\int_\Omega|\mathbf{b}_x|^2\omega^2 dx
+\frac{1}{2}\int_\Omega|\mathbf{b}_{0x}|^2\omega^2 dx-2\iint_{Q_t} \mathbf{b}_t\cdot \mathbf{b}_{x}\omega\omega' dxdt\\
&\quad-2\iint_{Q_t} \left(\nu\mathbf{b}_{xx}+\mathbf{w}_x-u\mathbf{b}_x-u_x\mathbf{b}\right)\cdot \mathbf{b}_{x}\omega\omega' dxds.
\end{split}
\end{equation}
Below we deal with the third term on right-hand side of
(<ref>). By Cauchy-Schwarz's inequality and Lemmas <ref> and <ref>, we have
\begin{equation*}
\begin{split}
&-2\iint_{Q_t} \left(\nu\mathbf{b}_{xx}+\mathbf{w}_x-u_x\mathbf{b}\right)\cdot \mathbf{b}_{x}\omega\omega' dxds\\
&\leq \frac{\nu}{4}\iint_{Q_t}|\mathbf{b}_{xx}|^2\omega^2dxds +C\iint_{Q_t}|\mathbf{b}_{x}|^2dxds+C\iint_{Q_t} |\mathbf{w}_{x}|^2\omega^2 dxds\\
& \quad+ C\iint_{Q_t} u_x^2dxds+C\iint_{Q_t} |\mathbf{b}\cdot\mathbf{b}_x|^2 dxds\\
&\leq C+ \frac{\nu}{4}\iint_{Q_t}|\mathbf{b}_{xx}|^2\omega^2dxds +C\iint_{Q_t} |\mathbf{w}_{x}|^2\omega^2 dxds.\\
\end{split}
\end{equation*}
Observe that since from the mean value theorem and $u(1,t)=u(0,t)=0$, we have
\begin{equation}\label{u3}
\begin{split}
|u(x,t)|\leq \|u_x\|_{L^\infty(\Omega)}\omega(x),
\end{split}
\end{equation}
\begin{equation*}
\begin{split}
&2\iint_{Q_t} u |\mathbf{b}_{x}|^2\omega\omega'dxds
\leq C\int_{0}^t \|u_x\|_{L^\infty(\Omega)} \int_\Omega|\mathbf{b}_{x}|^2\omega^2dxds.\\
\end{split}
\end{equation*}
Substituting them into (<ref>) yields
\begin{equation}\label{be3}
\begin{split}
\iint_{Q_t} \mathbf{b}_t\cdot \mathbf{b}_{xx}\omega^2 dxdt \leq & C-\frac{1}{2}\int_\Omega|\mathbf{b}_x|^2\omega^2dx+\frac{\nu}{4}\iint_{Q_t}|\mathbf{b}_{xx}|^2\omega^2dxds\\
& +C\int_{0}^t\|u_x\|_{L^\infty(\Omega)}\int_\Omega|\mathbf{b}_{x}|^2\omega^2dxds
+C\iint_{Q_t} |\mathbf{w}_{x}|^2\omega^2 dxds.
\end{split}
\end{equation}
As to the two terms on right-hand side of (<ref>), we have by Young's inequality
\begin{equation}\label{be2}
\begin{split}
& \iint_{Q_t} (u\mathbf{b})_x\cdot \mathbf{b}_{xx}\omega^2 dxds-\iint_{Q_t} \mathbf{w}_x\cdot \mathbf{b}_{xx}\omega^2 dxds\\
&\leq \frac{\nu}{4}\iint_{Q_t} |\mathbf{b}_{xx}|^2\omega^2 dxds+C\iint_{Q_t} |(u\mathbf{b})_x|^2\omega^2 dxds
+C\iint_{Q_t} |\mathbf{w}_{x}|^2\omega^2 dxds.
\end{split}
\end{equation}
It remains to treat the second term on right-hand side of (<ref>). We observe by Lemma <ref>
\begin{equation}\label{ux}
\begin{split}
&\int_0^t\|u_x\|_{L^\infty(\Omega)}^2 ds\leq C\iint_{Q_t}|u_xu_{xx}|dxds \leq C\left(\iint_{Q_t}u_{xx}^2 dxds\right)^{1/2},
\end{split}
\end{equation}
which together with Lemma 2.1 gives
\begin{equation*}
\begin{split}
\iint_{Q_t} |(u\mathbf{b})_x|^2\omega^2 dxds \leq &C\iint_{Q_t} u^2|\mathbf{b}_{x}|^2\omega^2 dxds+C\iint_{Q_t} u_x^2|\mathbf{b} |^2\omega^2 dxds\\
\leq & C\int_0^t\|u^2\|_{L^\infty(\Omega)}\int_\Omega|\mathbf{b}_{x}|^2\omega^2 dxds+C\left(\iint_{Q_t} u_{xx}^2 dxds\right)^{1/2}.
\end{split}
\end{equation*}
Substituting it into (<ref>) and then, substituting the resulting inequality and (<ref>) into (<ref>) and using Gronwall's inequality, we finish the proof.
Let (<ref>) and (<ref>) hold. Then
\begin{equation}\label{w1}
\begin{split}
& \int_\Omega|\mathbf{w}_x|^2\omega^2 dx+\mu \iint_{Q_t} |\mathbf{w}_{xx}|^2\omega^2 dxds\leq C+C\left(\iint_{Q_t} u_{xx}^2 dxds\right)^{1/2},\\
&\iint_{Q_t}\big(|\mathbf{w}_t|^2+u^2|\mathbf{w}_x|^2\big)dxdt\leq C+C \iint_{Q_t}u_{xx}^2 dxds.
\end{split}
\end{equation}
Multiplying (<ref>) by $\mathbf{w}_{xx}\omega^2(x)$ and integrating over $Q_t$, we have
\begin{equation}\label{0v10}
\begin{split}
&-\iint_{Q_t} \mathbf{w}_t\cdot \mathbf{w}_{xx}\omega^2 dxdt+\mu\iint_{Q_t}|\mathbf{w}_{xx}|^2\frac{\omega^2}{\rho}dxds\\
&=\iint_{Q_t} u\mathbf{w}_x\cdot \mathbf{w}_{xx}\omega^2 dxds-\iint_{Q_t} \mathbf{b}_x\cdot \mathbf{w}_{xx}\frac{\omega^2}{\rho} dxds.
\end{split}
\end{equation}
Integrating by parts and using (<ref>), we have
\begin{equation*}
\begin{split}
& \iint_{Q_t} \mathbf{w}_t\cdot \mathbf{w}_{xx}\omega^2 dxdt\\[1mm]
&=-\frac{1}{2}\int_\Omega|\mathbf{w}_x|^2\omega^2 dx
+\frac{1}{2}\int_\Omega|\mathbf{w}_{0x}|^2\omega^2 dx-2\iint_{Q_t} \mathbf{w}_t\cdot \mathbf{w}_{x}\omega\omega' dxdt\\[1mm]
&=-\frac{1}{2}\int_\Omega|\mathbf{w}_x|^2\omega^2 dx
+\frac{1}{2}\int_\Omega|\mathbf{w}_{0x}|^2\omega^2 dx\\[1mm]
&\quad-2\iint_{Q_t} \left(\frac{\mu}{\rho}\mathbf{w}_{xx}-u\mathbf{w}_x
+\frac{\mathbf{b}_x}{\rho}\right)\cdot \mathbf{w}_{x}\omega\omega' dxds\\[1mm]
&\leq C-\frac{1}{2}\int_\Omega|\mathbf{w}_x|^2\omega^2 dx+C\mu^2\iint_{Q_t}|\mathbf{w}_{xx}|^2dxds +C\iint_{Q_t}|\mathbf{w}_x|^2\omega^2 dxds\\[1mm]
&\quad+C\iint_{Q_t}|\mathbf{b}_{x}|^2dxds+C\iint_{Q_t}|u||\mathbf{w}_x|^2\omega dxds.
\end{split}
\end{equation*}
From (<ref>) it follows that
\begin{equation*}
\begin{split}
\iint_{Q_t}|u||\mathbf{w}_x|^2\omega dxds\leq \int_0^t\|u_x\|_{L^\infty(\Omega)} \int_\Omega|\mathbf{w}_x|^2\omega^2dxds,\\
\end{split}
\end{equation*}
which together with Lemmas <ref> and <ref> gives
\begin{equation*}
\begin{split}
& \iint_{Q_t} \mathbf{w}_t\cdot \mathbf{w}_{xx}\omega^2 dxdt\\
& \leq C-\frac{1}{2}\int_\Omega|\mathbf{w}_x|^2\omega^2 dx
+C\int_0^t\big(1+\|u_x\|_{L^\infty(\Omega)} \big)\int_\Omega|\mathbf{w}_x|^2\omega^2dxds.\\
\end{split}
\end{equation*}
To estimate the right-hand side of (<ref>), we have by integrating
by parts and by (<ref>)
\begin{equation*}
\begin{split}
\iint_{Q_t} u\mathbf{w}_x\cdot \mathbf{w}_{xx}\omega^2 dxds
=&-\frac{1}{2}\iint_{Q_t} |\mathbf{w}_x|^2[u_x\omega^2+2u\omega\omega']
\leq&C\int_0^t \|u_x\|_{L^\infty(\Omega)} \int_\Omega|\mathbf{w}_x|^2\omega^2dxds,\\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&-\iint_{Q_t} \mathbf{b}_x\cdot \mathbf{w}_{xx}\frac{\omega^2}{\rho} dxds \\[1mm]
&=\iint_{Q_t}\mathbf{w}_x\cdot \mathbf{b}_{xx} \frac{\omega^2}{\rho}dxds+2\iint_{Q_t} \mathbf{w}_x\cdot \mathbf{b}_{x} \frac{\omega\omega'}{\rho} dxds-\iint_{Q_t} \mathbf{w}_x\cdot \mathbf{b}_{x}\frac{\omega^2\rho_x}{\rho^2} dxds\\[1mm]
& \leq C\iint_{Q_t} |\mathbf{b}_{xx}|^2\omega^2 dxds+C\iint_{Q_t} |\mathbf{w}_x|^2\omega^2 dxds+C\iint_{Q_t} |\mathbf{b}_{x}|^2dxds\\
&\quad+C\iint_{Q_t} |\mathbf{b}_x|^2\omega^2 \rho_x^2dxds\\[1mm]
&\leq C+C\iint_{Q_t} |\mathbf{w}_x|^2\omega^2 dxds+C\iint_{Q_t} |\mathbf{b}_{xx}|^2\omega^2 dxds,
\end{split}
\end{equation*}
where we used the fact by Lemmas <ref> and <ref>
\begin{equation*}
\begin{split}
\iint_{Q_t} |\mathbf{b}_x|^2\omega^2 \rho_x^2dxds\leq &C\int_0^t \left\||\mathbf{b}_x|^2\omega^2\right\|_{L^\infty(\Omega)} ds\\[1mm]
\leq & C\iint_{Q_t} \left|(|\mathbf{b}_x|^2\omega^2)_x\right| dxds\\[1mm]
\leq & C\iint_{Q_t} |\mathbf{b}_{x}|^2|\omega\omega'|dxds+C \iint_{Q_t} |\mathbf{b}_x\cdot\mathbf{b}_{xx}| \omega^2 dxds\\[1mm]
\leq &C+C\iint_{Q_t} |\mathbf{b}_{xx}|^2 \omega^2 dxds.
\end{split}
\end{equation*}
Substituting the above results into (<ref>) and using Lemma <ref>, we have
\begin{equation*}
\begin{split}
&\int_\Omega|\mathbf{w}_x|^2\omega^2 dx+\mu \iint_{Q_t} |\mathbf{w}_{xx}|^2\omega^2 dxds\\
& \leq C + C \int_0^t\big(1+\|u_x\|_{L^\infty(\Omega)}\big)\int_\Omega|\mathbf{w}_x|^2\omega^2 dxds+C\left(\iint_{Q_t} u_{xx}^2 dxds\right)^{1/2}.
\end{split}
\end{equation*}
Thus, the first estimate of this lemma follows from Gronwall's inequality and
Consequently, we have by (<ref>), the first estimate of this lemma and (<ref>)
\begin{equation*}\label{v11}
\begin{split}
\iint_{Q_T} u^2|\mathbf{w}_x|^2 dxdt&\leq \int_0^T\|u_x\|_{L^\infty(\Omega)}^2\int_\Omega |\mathbf{w}_x|^2\omega^2 dxdt \\ &\leq C\int_0^T\|u_x\|_{L^\infty(\Omega)}^2 dt\left[1+\left(\iint_{Q_T}u_{xx}^2 dxdt\right)^{1/2}\right]\\
&\leq C+C \iint_{Q_T}u_{xx}^2 dxdt.
\end{split}
\end{equation*}
Furthermore, by Lemmas <ref> and <ref>, we derive from
(<ref>) that
\begin{equation*}
\begin{split}
&\iint_{Q_T} |\mathbf{w}_t|^2dxdt\leq C +C\iint_{Q_T}u^2|\mathbf{w}_x|^2dxdt\leq C+C \iint_{Q_T}u_{xx}^2 dxdt.
\end{split}
\end{equation*}
The proof is complete.
Let (<ref>) and (<ref>) hold. Then
\begin{equation}\label{b99}
\begin{split}
& \int_\Omega|\mathbf{b}_{x}|^2dx+\iint_{Q_t} |\mathbf{b}_t|^2
\leq C+ C\left(\iint_{Q_t}u_{xx}^2 dxds\right)^{1/2}. \\
\end{split}
\end{equation}
Multiplying (<ref>)$_4$ by $\mathbf{b}_t$ and
integrating over $Q_t$ yield
\begin{equation}\label{b9}
\begin{split}
&\frac{\nu}{2}\int_\Omega|\mathbf{b}_{x}|^2dx+ \iint_{Q_t} |\mathbf{b}_t|^2
=\frac{\nu}{2}\int_\Omega|\mathbf{b}_{0x}|^2dx+\iint_{Q_t} \big[\mathbf{w}_x
- (u\mathbf{b})_{x}\big]\cdot\mathbf{b}_t dxdt. \\
\end{split}
\end{equation}
Integrating by parts yields
\begin{equation*}
\begin{split}
&\iint_{Q_t} \mathbf{w}_x \cdot\mathbf{b}_t dxdt= \int_\Omega\mathbf{w}_x \cdot\mathbf{b}dx
-\int_\Omega\mathbf{w}_{0x} \cdot\mathbf{b}_0dx- \iint_{Q_t} (\mathbf{w}_t)_x \cdot\mathbf{b} dxdt\\
&=-\int_\Omega\mathbf{w}_{0x}\cdot\mathbf{b}_0dx -\int_\Omega\mathbf{w}\cdot\mathbf{b}_xdx
+\iint_{Q_t} \mathbf{w}_t\cdot\mathbf{b}_x dxdt,
\end{split}
\end{equation*}
which together with Lemmas <ref> and <ref> and
(<ref>)$_2$ gives
\begin{equation}\label{b10}
\begin{split}
\iint_{Q_t} \mathbf{w}_x \cdot\mathbf{b}_t dxdt
& \leq C+\frac{\nu}{4}\int_\Omega|\mathbf{b}_x|^2dx+\left(\iint_{Q_t}|\mathbf{b}_x|^2dxdt\right)^{1/2}
\left(\iint_{Q_t}|\mathbf{w}_t|^2dxdt\right)^{1/2}\\
&\leq C +\frac{\nu}{4}\int_\Omega|\mathbf{b}_x|^2dx+C\left(\iint_{Q_t}|\mathbf{w}_t|^2dxdt\right)^{1/2}\\
&\leq C +\frac{\nu}{4}\int_\Omega|\mathbf{b}_x|^2dx+C\left(\iint_{Q_t}u_{xx}^2dxds\right)^{1/2}.
\end{split}
\end{equation}
On the other hand, we have by Cauchy-Schwarz's inequality, Lemma 2.1 and (<ref>)
\begin{equation}\label{b11}
\begin{split}
&-\iint_{Q_t} (u\mathbf{b})_{x} \cdot\mathbf{b}_t dxdt\\
& \leq \frac12 \iint_{Q_t} |\mathbf{b}_t|^2 dxdt
+\frac12\iint_{Q_t} |(u\mathbf{b})_{x}|^2 dxds\\
&\leq \frac12 \iint_{Q_t} |\mathbf{b}_t|^2 dxdt+C\int_0^t\|u^2\|_{L^\infty(\Omega)} \int_{\Omega} |\mathbf{b}_x|^2 dxds+C\int_0^t\|u_x\|_{L^\infty(\Omega)}^2\int_{\Omega} |\mathbf{b}|^2 dxds\\
&\leq \frac12 \iint_{Q_t} |\mathbf{b}_t|^2 dxdt+C\int_0^t\|u^2\|_{L^\infty(\Omega)} \int_{\Omega} |\mathbf{b}_x|^2 dxds+C\left(\iint_{Q_t}u_{xx}^2 dxds\right)^{1/2}.
\end{split}\end{equation}
Substituting them into (<ref>) and using Gronwall's inequality, we complete the proof.
Now we can prove the following desired results.
Let (<ref>) and (<ref>) hold. Then $\|(u,\mathbf{b})\|_{L^\infty(Q_T)}\leq C$, and
\begin{equation}\label{w1112}
\begin{split}
&\sup\limits_{0<t<T}\int_\Omega(\rho_t^2+u_x^2+\theta^2)dx +\iint_{Q_T} \big( u_t^2 +u_{xx}^2+\kappa\theta_x^2\big)dxdt\leq C,\\
& \sup\limits_{0<t<T}\int_\Omega|\mathbf{w}_x|^2\omega^2 dx+\iint_{Q_T}\big(|u|^2|\mathbf{w}_x|^2+ |\mathbf{w}_{t}|^2 \big) dxdt \leq C,\\
&\sup\limits_{0<t<T}\int_\Omega|\mathbf{b}_{x}|^2dx+\iint_{Q_T}\big(|\mathbf{b}_t|^2 + |\mathbf{b}_{xx}|^2\omega^2\big) dxdt\leq C.
\end{split}
\end{equation}
Rewrite the equation (<ref>)$_2$ in the form
\begin{equation*}\label{0u0}
\begin{split}
\gamma\sqrt{\rho}\theta_x-\frac{\gamma}{\sqrt{\rho}}\rho_x\theta-\frac{1}{\sqrt{\rho}}\mathbf{b}\cdot\mathbf{b}_x.
\end{split}
\end{equation*}
Using Cauchy-Schwarz's inequality, we obtain
\begin{equation}\label{0u1}
\begin{split}
& \frac{\lambda}2\int_\Omega u_x^2dx+\iint_{Q_t}\big(\rho u_t^2+\lambda^2\rho^{-1}u_{xx}^2\big)dxdt\\
&\leq \frac{\lambda}2\int_\Omega u_{0x}^2dx+C\iint_{Q_t} \big(u^2u_x^2+ \theta_x^2+\rho_x^2 \theta^2 + |\mathbf{b}\cdot\mathbf{b}_x|^2\big)dxds\\
&\leq C+C\int_0^t\|u^2\|_{L^\infty(\Omega)}\int_\Omega u_x^2dxds+C\iint_{Q_t} \theta_x^2dxds+C\int_0^t\|\theta^2\|_{L^\infty(\Omega)}\int_{\Omega} \rho_x^2 dxds\\
&\leq C+C\int_0^t\|u^2\|_{L^\infty(\Omega)}\int_\Omega u_x^2dxds+C\iint_{Q_t} \big(\theta_x^2+ \theta^2 \big) dxds,
\end{split}
\end{equation}
where we used (<ref>) and $\int_0^t\|\theta^2\|_{L^\infty(\Omega)}ds\leq C \iint_{Q_t} (\theta^2+\theta_x^2)dxds$.
Our next step is to multiply (<ref>)$_5$ by $ \theta $ and
integrate over $Q_t$. We have
\begin{equation}\label{theta0}
\begin{split}
&\frac12\int_\Omega \rho\theta^2dx+ \iint_{Q_t} \kappa \theta_x^2dxds = -\iint_{Q_t}pu_x\theta dxds+\iint_{Q_t} \theta {\cal Q} dxds.
\end{split}
\end{equation}
By Cauchy-Schwarz's inequality and (<ref>), we obtain
\begin{equation*}
\begin{split}
-\gamma\iint_{Q_t} \rho\theta^2 u_xdxds
&\leq C \iint_{Q_t} \theta^2dxds+C\iint_{Q_t}\theta^{2} u_x^2 dxds\\
&\leq C+C\int_0^t\|\theta\|_{L^\infty(\Omega)}^{2} \int_{\Omega}u_x^2
\end{split}
\end{equation*}
On the other hand, we have by Lemmas <ref>, <ref> and <ref>
\begin{equation*}
\begin{split}
\iint_{Q_t} \theta {\cal Q} dxds
&\leq C\int_0^t\|\theta\|_{L^\infty(\Omega)} \left
\{\int_{\Omega}\big(u_x^2 +\mu|\mathbf{w}_x|^2dx+|\mathbf{b}_x|^2\big)dx\right\}ds \\
& \leq C+C\int_0^t\|\theta\|_{L^\infty(\Omega)}\int_{\Omega}u_x^2
+C\left(\iint_{Q_t}u_{xx}^2 dxds\right)^{1/2}.
\end{split}
\end{equation*}
Inserting them into (<ref>) yields
\begin{equation}\label{theta33}
\begin{aligned}
&\int_\Omega \theta^2dx+ \iint_{Q_t} \kappa \theta_x^2dxds\\
&\leq C +C\int_0^t\left[\|\theta\|_{L^\infty(\Omega)}+\|\theta\|_{L^\infty(\Omega)}^{2}\right]\int_{\Omega}u_x^2 dxds +C\left(\iint_{Q_t}u_{xx}^2 dxds\right)^{1/2}.
\end{aligned}
\end{equation}
Plugging it into (<ref>) gives
\begin{equation*}\label{0u2}
\begin{split}
& \int_\Omega u_x^2dx+\iint_{Q_t} \big(u_t^2+ u_{xx}^2\big)dxdt\\
&\leq C +C\int_0^t\left[\|u^2\|_{L^\infty(\Omega)}+\|\theta\|_{L^\infty(\Omega)}+\|\theta\|_{L^\infty(\Omega)}^{2}\right]
\int_{\Omega}u_x^2 dxds +C\left(\iint_{Q_t}u_{xx}^2 dxds\right)^{1/2}\\
&\leq C +C\int_0^t\left[\|u^2\|_{L^\infty(\Omega)}+\|\theta\|_{L^\infty(\Omega)}+\|\theta\|_{L^\infty(\Omega)}^{2}\right]
\int_{\Omega}u_x^2 dxds+\frac12\iint_{Q_t}u_{xx}^2 dxds.\\
\end{split}
\end{equation*}
By Gronwall's inequality and noticing (<ref>), we have
\begin{equation*}
\begin{split}
\int_\Omega u_x^2dx+\iint_{Q_t} \big(u_t^2 + u_{xx}^2\big)dxdt \leq C.
\end{split}
\end{equation*}
Consequently, (<ref>) follows from (<ref>) and Lemmas 2.7-2.9 .
The proof is complete.
As a consequence of Lemma 2.10, we have
\begin{equation}\label{ux4}
\begin{split}
\iint_{Q_T} u_x^6dxdt\leq& C\int_0^T\|u_x\|_{L^\infty(\Omega)}^4\int_\Omega u_x^2dx dt\leq C\int_0^T\|u_x\|_{L^\infty(\Omega)}^4 dt\\
\leq& C\int_0^T\left(\int_\Omega|u_x||u_{xx}|dx\right)^2 dt\leq
C\int_0^T\left(\int_\Omega u_x^2dx\right)\left(\int_\Omega u_{xx}^2dx\right)dt\\
\leq& C.
\end{split}
\end{equation}
Let (<ref>) and (<ref>) hold. Then $\|\mathbf{w}\|_{L^\infty(Q_T)}\leq C$. Moreover, it holds
\begin{equation*}\begin{split}
\sup\limits_{0<t<T}\int_\Omega |\mathbf{w}_x | dx \leq C.
\end{split}\end{equation*}
Set $\mathbf{z}=\mathbf{w}_x$. Differentiating (<ref>) in $x$ gives
\begin{equation}\label{zx}
\begin{split}
\mathbf{z}_t=\left(\frac{\mu}{\rho}\mathbf{z}_x\right)_x-(u\mathbf{z})_x+\left(\frac{\mathbf{b}_x}{\rho}\right)_x.
\end{split}\end{equation}
Denote $\Phi_\epsilon(\cdot):\mathbb{R}^2\rightarrow \mathbb{R}^+ $ for $\epsilon \in (0, 1)$ by
\begin{equation*}\label{phiepsilon}
\Phi_\epsilon(\xi)=\sqrt{\epsilon^2+|\xi|^2},\quad\forall\xi\in\mathbb{R}^2.
\end{equation*}
Observe that $\Phi_\epsilon$ has the properties
\begin{equation}\label{phii}
\left\{\begin{split}
&|\xi|\leq |\Phi_\epsilon(\xi)|\leq |\xi|+\epsilon,\quad\forall\xi\in\mathbb{R}^2,\\
&|\nabla_\xi\Phi_\epsilon(\xi)|\leq 1,\quad\forall\xi\in\mathbb{R}^2,\\
& 0\leq \xi\cdot\nabla_\xi\Phi_\epsilon(\xi)\leq \Phi_\epsilon(\xi),\quad\forall\xi\in\mathbb{R}^2,\\
&\eta D_\xi^2\Phi_\epsilon(\xi)\eta^{\top}\geq 0,\quad \forall\xi, \eta\in\mathbb{R}^2,\\
&\lim\limits_{\epsilon\rightarrow 0^+}\Phi_\epsilon(\xi)=|\xi|,\quad\forall\xi\in\mathbb{R}^2,
\end{split}
\right.
\end{equation}
where $\xi^{\top}$ stands for the transpose of the vector
$\xi=(\xi_1,\xi_2) \in \mathbb{R}^2$, and $D_\xi^2 g$ is the
Hessian matrix of the function $g :\mathbb{R}^2\rightarrow\mathbb{R}$ which is defined by
\begin{equation*}
D^2_\xi g(\xi)=\left(\begin{split}
\end{split}
\right).
\end{equation*}
Multiplying (<ref>) by $\nabla_\xi\Phi_\epsilon(\mathbf{z})$ and integrating over $Q_t$, we have
\begin{equation}\label{wx1}
\begin{split}
&\int_\Omega \Phi_\epsilon(\mathbf{z})dx-\int_\Omega \Phi_\epsilon(\mathbf{w}_{0x})dx\\
&=-\mu\iint_{Q_t}\frac{1}{\rho} \mathbf{z}_x D_\xi^2\Phi_\epsilon(\mathbf{z})(\mathbf{z}_x)^\perp dxds
-\iint_{Q_t}(u\mathbf{z})_x\cdot\nabla_\xi\Phi_\epsilon(\mathbf{z}) dxds\\
&\quad+\iint_{Q_t}\left(\frac{\mathbf{b}_x}{\rho}\right)_x\cdot\nabla_\xi\Phi_\epsilon(\mathbf{z}) dxds
+\mu\int_0^t\frac{\mathbf{z}_x\cdot\nabla_\xi\Phi_\epsilon(\mathbf{z}) }{\rho}\bigg|_{x=0}^{x=1}ds =:\sum_{j=1}^4E_j.
\end{split}
\end{equation}
From (<ref>)$_4$ it follows that
E_1\leq 0.
To estimate $E_2$, we observe by (<ref>)$_3$
\begin{equation*}
\begin{split}
E_2=& -\iint_{Q_t}\big(u\mathbf{z}_x+u_x \mathbf{z}\big)\cdot\nabla_\xi\Phi_\epsilon(\mathbf{z}) dxds\\
=& \iint_{Q_t}\big(u_x\Phi_\epsilon(\mathbf{z})-u_x \mathbf{z}\cdot\nabla_\xi\Phi_\epsilon(\mathbf{z}) \big) dxds\\
\leq& C\int_0^t\|u_x\|_{L^\infty(\Omega)} \int_\Omega\Phi_\epsilon(\mathbf{z})dxds.
\end{split}
\end{equation*}
As to $E_3$, utilizing the equation (<ref>)$_4$ yields
\begin{equation*}
\begin{split}
E_3 =&\iint_{Q_t}\frac{\mathbf{b}_{xx}\cdot\nabla_\xi\Phi_\epsilon(\mathbf{z})}{\rho}dxds
=&\frac{1}{\nu}\iint_{Q_t} \frac{\big[\mathbf{b}_{t}+(u\mathbf{b})_x-\mathbf{z}\big]\cdot\nabla_\xi\Phi_\epsilon(\mathbf{z})}{\rho} dxds -\iint_{Q_t}\frac{\mathbf{b}_{x}\cdot\nabla_\xi\Phi_\epsilon(\mathbf{z}) }{\rho^2}\rho_xdxds\\[1mm]
\leq&C \iint_{Q_t} \big[|\mathbf{b}_t|+|(u\mathbf{b})_x|+|\rho_x||\mathbf{b}_x|\big]dxds
\leq C,
\end{split}
\end{equation*}
where we used (<ref>)$_2$-(<ref>)$_3$ and Lemmas <ref> and <ref>.
It remains to estimate $E_4$. From (<ref>), we have
\begin{equation}\label{wt}
\begin{split}
\left|\frac{\mu}{\rho(a,t)}\mathbf{z}_x(a,t)\right|
\leq C+C |\mathbf{b}_x(a,t)|,\quad \hbox{\rm where}~a=0~\hbox{\rm
\end{split}
\end{equation}
On the other hand, we first integrate (<ref>)$_4$ from $a$ to $y
\in [0, 1]$ in $x$, and then integrate the resulting equation over
$(0, 1)$ in $y$, so that
\begin{equation*}\begin{split}
\mathbf{b}_x(a,t)=&-\frac{1}{\nu}\left\{\int_0^1\hspace{-2mm}\int_a^y
\mathbf{b}_t(x,t)
\end{split}\end{equation*}
so it follows from Lemmas <ref> and <ref> that
\begin{equation*}\begin{split}
\int_0^T|\mathbf{b}_x(a,t)|^2dt \leq C.
\end{split}\end{equation*}
Thus one derives from (<ref>) that
\begin{equation*}
\begin{split}
\int_0^T\left|\frac{\mu}{\rho(a,t)}\mathbf{z}_x(a,t)\right|dt \leq C+C\int_0^T|\mathbf{b}_x(a,t)| dt\leq C,
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
E_4\leq C\int_0^T\left\{\left|\frac{\mu}{\rho(1,t)}\mathbf{z}_x(1,t)\right|
+\left|\frac{\mu}{\rho(0,t)}\mathbf{z}_x(0,t)\right|\right\}dt \leq C.
\end{split}
\end{equation*}
Substituting the above results in (<ref>) and utilizing
Gronwall's inequality, we get
$$\int_\Omega \Phi_\epsilon(\mathbf{z})dx\leq C+\int_\Omega \Phi_\epsilon(\mathbf{w}_{0x})dx.
Passing to the limit as $\epsilon\rightarrow 0$ yields
$$\int_\Omega |\mathbf{w}_x|dx\leq C.
This and $\int_\Omega |\mathbf{w}|^2dx\leq C$ imply that
$|\mathbf{w}|\leq C$. The proof is complete.
Let (<ref>) and (<ref>) hold. Then
\begin{equation*}\begin{split}
\int_0^T\|\mathbf{b}_x\|_{L^\infty(\Omega)}^2dt \leq C.
\end{split}\end{equation*}
For any fixed $ z \in [0, 1]$, we first integrate (<ref>)$_4$ from $z$ to $y \in [0, 1]$ in $x$, and then integrate the resulting equation over $(0, 1)$ in $y$, so that
\begin{equation*}\begin{split}
\mathbf{b}_x(z,t)=&-\frac{1}{\nu}\left\{\int_0^1\hspace{-2mm}\int_z^y
\mathbf{b}_t(x,t)
\end{split}\end{equation*}
which together with Lemmas <ref> and <ref> implies the desired result. The proof is complete.
Combining Lemmas <ref>-<ref>, we have
\begin{equation}\label{bxxbound}\begin{split}
\iint_{Q_T}|\mathbf{b}_x||\mathbf{b}_{xx}|dxdt
&\leq C+C\int_0^T\|\mathbf{b}_x\|_{L^\infty(\Omega)}\int_\Omega|\mathbf{w}_x|dxdt \leq
\end{split}\end{equation}
\begin{equation}\label{bxbound}
\begin{split}
\iint_{Q_T}|\mathbf{b}_x |^4dxdt\leq C\int_0^T\|\mathbf{b}_x\|_{L^\infty(\Omega)}^2\int_\Omega|\mathbf{b}_x|^2dxdt\leq C.
\end{split}\end{equation}
Now some results in Lemmas <ref> and <ref> can be improved
as follows.
Let (<ref>) and (<ref>) hold. Then
\begin{equation*}\label{we1}
\begin{split}
\sqrt{\mu}\sup\limits_{0<t<T}\int_\Omega|\mathbf{w}_x|^2dx+\mu^{3/2}
\iint_{Q_T}
|\mathbf{w}_{xx}|^2 dxdt \leq C,\\
& \sup\limits_{0<t<T}\int_\Omega|\mathbf{w}_x|^2\omega dx+\iint_{Q_T} \big(\mu |\mathbf{w}_{xx}|^2+|\mathbf{b}_{xx}|^2\big)
\omega dxdt\leq C.
\end{split}
\end{equation*}
For the first estimate, we can use an argument similar to Lemma 2.6
to finish the proof. The key is to deal with the term
$-\mu\iint_{Q_t}\frac{1}{\rho}\mathbf{b}_x\cdot\mathbf{w}_{xx}dxds $
in (<ref>).
By integrating by parts and using Cauchy-Schwarz's inequality, we
\begin{equation}\label{w0}
\begin{split}
& -\mu\iint_{Q_t}\frac{1}{\rho}\mathbf{b}_x\cdot\mathbf{w}_{xx}dxds\\[1mm]
-\mu\int_0^T \frac{\mathbf{b}_x\cdot\mathbf{w}_{x}}{\rho}\bigg|_{x=0}^{x=1} ds\\[1mm]
&\leq C\mu\iint_{Q_t}| \mathbf{b}_{xx}|^2 dxds+C\mu\iint_{Q_t}|\mathbf{w}_{x}|^2dxds+\mu\iint_{Q_t}|\mathbf{b}_{x}|^2 \rho_x^2dxds\\[1mm]
&\quad+C\mu\left(\int_0^t\|\mathbf{b}_{x}\|_{L^\infty(\Omega)}^2 ds\right)^{1/2}\left(\int_0^t\|\mathbf{w}_{x}\|_{L^\infty(\Omega)}^2 ds\right)^{1/2}\\[1mm]
&\leq C\mu + C\mu\iint_{Q_t}| \mathbf{b}_{xx}|^2 dxds+C\mu\iint_{Q_t}|\mathbf{w}_{x}|^2dxds
+C\mu \left(\int_0^t\|\mathbf{w}_{x}\|_{L^\infty(\Omega)}^2 ds\right)^{1/2},\\
\end{split}
\end{equation}
where we used the fact by Lemmas <ref> and <ref>
\begin{equation*}
\begin{split}
\iint_{Q_t}|\mathbf{b}_{x}|^2|\rho_x|^2dxds \leq
\int_0^t\|\mathbf{b}_{x}\|_{L^\infty(\Omega)}^2\int_\Omega\rho_x^2dxds\leq
\end{split}
\end{equation*}
By (<ref>), we obtain
\begin{equation}\label{w01}
\begin{split}
&\mu \left(\int_0^t\|\mathbf{w}_{x}\|_{L^\infty(\Omega)}^2 ds\right)^{1/2}\\
&\leq C\mu+C \mu^{1/4}\left(\mu\iint_{Q_t}|\mathbf{w}_{x}|^2dxds\right)^{1/4}
\left(\mu^2\iint_{Q_t}|\mathbf{w}_{xx}|^2dxds\right)^{1/4}\\[1mm]
&\leq C\sqrt{\mu}+\frac{C \mu}{\epsilon}\iint_{Q_t}
|\mathbf{w}_{x}|^2dxds+\epsilon \mu^2
\iint_{Q_t}\frac{1}{\rho}|\mathbf{w}_{xx}|^2dxds,~ \forall
\epsilon>0.
\end{split}
\end{equation}
It remains to show the estimate
\begin{equation}\label{bxx}
\begin{split}
& \iint_{Q_t}|\mathbf{b}_{xx}|^2dxds \leq C+ C\iint_{Q_t} |\mathbf{w}_x|^2dxds.
\end{split}
\end{equation}
Multiplying (<ref>)$_4$ by $\mathbf{b}_{xx}$ and integrating over
$Q_t$, we have
\begin{equation*}
\begin{split}
&=\frac12\int_\Omega|\mathbf{b}_{0x}|^2dx+\iint_{Q_t}u_x\mathbf{b}\cdot\mathbf{b}_{xx} dxds-\frac12\iint_{Q_t}u_x|\mathbf{b}_x|^2 dxds-\iint_{Q_t} \mathbf{w}_x\cdot\mathbf{b}_{xx}dxds \\
&\leq C+\frac{\nu}{2}\iint_{Q_t} |\mathbf{b}_{xx}|^2 dxds+C\iint_{Q_t} |\mathbf{w}_x|^2 dxds +C\int_0^t\|u_x\|_{L^\infty(\Omega)}\int_{\Omega}|\mathbf{b}_x|^2 dxds,
\end{split}
\end{equation*}
where we used Lemma <ref>. Thus, (<ref>) follows from Gronwall's inequality.
Inserting the above estimates into (<ref>) and taking a small
$\epsilon>0$, we have
\begin{equation*}
\begin{split}
-\mu\iint_{Q_t}\frac{1}{\rho}\mathbf{b}_x\cdot\mathbf{w}_{xx}dxdt \leq C\sqrt{\mu} +C\mu\iint_{Q_t}|\mathbf{w}_{x}|^2dxdt+\frac{\mu^2}{4}
\iint_{Q_t}\frac{1}{\rho}|\mathbf{w}_{xx}|^2dxds.
\end{split}
\end{equation*}
Then, an argument similar to Lemma <ref> leads to
\begin{equation*}
\begin{split}
+\mu^2\iint_{Q_t} |\mathbf{w}_{xx}|^2dxds \\
&\leq C\sqrt{\mu}
\end{split}
\end{equation*}
So the first estimate of this lemma follows from Gronwall's inequality and
The second estimate can proved by the arguments similar to Lemma <ref> and
(<ref>)$_1$ and in terms of the first estimate and Lemmas
<ref>-<ref>. In fact, this can be done by using $\omega$ instead of
$\omega^2$ in (<ref>) and (<ref>) and noticing the following facts:
\begin{equation*} \begin{split}
& \mu\iint_{Q_T}|\mathbf{w}_x\cdot\mathbf{w}_{xx}|dxdt \leq C\sqrt{\mu}\iint_{Q_T}|\mathbf{w}_x|^2dxdt+C\mu^{3/2}\iint_{Q_T}|\mathbf{w}_{xx}|^2dxdt\leq C,\\[1mm]
& \iint_{Q_T}|\mathbf{b}_x\cdot\mathbf{w}_{x}|dxdt \leq C\int_0^T\|\mathbf{b}_x\|_{L^\infty(\Omega)}\int_\Omega|\mathbf{w}_x|dxdt \leq
\end{split}\end{equation*}
The proof is complete.
As a consequence of Lemma <ref> and (<ref>), we also have
\begin{equation}\label{wx4}
\begin{split}
\mu^{3/2}\iint_{Q_T} |\mathbf{w}_{x}|^4dxdt \leq C \mu \int_0^T \|\mathbf{w}_{x}\|_{L^\infty(\Omega)}^2 \left(\sqrt{ \mu}\int_{Q_T}|\mathbf{w}_x|^2dx\right)dt\leq C.
\end{split}
\end{equation}
Based on the above lemmas, we can bound the temperature $\theta$ in a direct way.
Let (<ref>) and (<ref>) hold. Then $\theta \leq C.$
Rewrite the equation (<ref>)$_4$ in the form
\begin{equation}\label{theta9}
\begin{split}
\theta_t= a(x,t)\theta_{xx}+b(x,t)\theta_x+c(x,t) \theta+f(x,t),
\end{split}
\end{equation}
\begin{equation*}
\begin{split}
a =\rho^{-1}\kappa,\quad b =\rho^{-1}\kappa_x-u,\quad c =-\gamma
u_x,\quad f =\rho^{-1} (\lambda
\end{split}
\end{equation*}
Set $z=\theta_x$. Differentiating the equation (<ref>) in $x$ yields
\begin{equation}\label{z1}
\begin{split}
z_t=(a z_x)_x+(b z)_x+c z+c_x\theta+f_x.
\end{split}
\end{equation}
For $\epsilon \in (0, 1)$, denote $\varphi_\epsilon:\mathbb{R}\rightarrow \mathbb{R}^+$ by
\varphi_\epsilon(s)=\sqrt{s^2+\epsilon^2}.
Simple calculations show that
\begin{equation*}\label{phi}
\left\{\begin{split}
&\varphi_\epsilon'(0)=0,\quad|\varphi_\epsilon'(s)|\leq 1, \quad\varphi_\epsilon''(s)\geq 0, \quad |s\varphi_\epsilon''(s)|\leq 1,\\
&\lim\limits_{\epsilon\rightarrow 0} \varphi_\epsilon(s)=|s|, \quad\lim\limits_{\epsilon\rightarrow 0}s\varphi_\epsilon''(s)=0.
\end{split}\right.
\end{equation*}
Multiplying (<ref>) by $\varphi_\epsilon'(z)$, integrating over
$Q_t$, and noticing
we have
\begin{equation*}
\begin{split}
\int_\Omega \varphi_\epsilon(z)dx-\int_\Omega \varphi_\epsilon(\theta_{0x})dxds&=-\iint_{Q_t} a \varphi_\epsilon''(z)z_x^2dxds-\iint_{Q_t} b z z_x\varphi_\epsilon''(z)dxds\\
&\quad+\iint_{Q_t} (c z+c_x \theta+f_x)\varphi_\epsilon'(z)dxds,\\
\end{split}
\end{equation*}
and then, we obtain by $\varphi_\epsilon''(s)\geq 0$ and $|\varphi_\epsilon'(s)|\leq 1$
\begin{equation}\label{z2}
\begin{split}
&\int_\Omega \varphi_\epsilon(z)dx-\int_\Omega \varphi_\epsilon(\theta_{0x})dx \\
&\leq \iint_{Q_t}|b z_x||z\varphi_\epsilon''(z)|dxds + \iint_{Q_t}(|c z|+|c_x \theta|+|f_x|)dxds.
\end{split}
\end{equation}
Recalling $|s\varphi_\epsilon''(s)|\leq 1$ and
$s\varphi_\epsilon''(s)\rightarrow 0$ as $\epsilon\rightarrow 0$ and
using Lebesgue's dominated convergence theorem, we obtain
\begin{equation*}
\begin{split}
\lim\limits_{\epsilon\rightarrow 0}\iint_{Q_T} |b z_x||z\varphi_\epsilon''(z)|dxdt=0.
\end{split}
\end{equation*}
Thus, passing to the limit as $\epsilon\rightarrow 0$ in (<ref>) and using $\lim\limits_{\epsilon\rightarrow 0} \varphi_\epsilon(s)=|s|$, we have
\begin{equation}\label{theta10}
\begin{split}
\int_\Omega |\theta_x|dx\leq \int_\Omega|\theta_{0x}|dx + \iint_{Q_t}(|c \theta_x|+|c_x \theta|+|f_x|)dxds.
\end{split}
\end{equation}
By Lemma <ref>, we have
\begin{equation*}
\begin{split}
&\iint_{Q_T}|c_x\theta| dxdt\leq C\left(\iint_{Q_T}u_{xx}^2dxdt\right)^{1/2}\left(\iint_{Q_T}\theta^2dxdt\right)^{1/2}\leq C,\\
&\iint_{Q_T}|c \theta_x| dxdt\leq C\left(\iint_{Q_T}u_{x}^2dxdt\right)^{1/2}\left(\iint_{Q_T}\theta_x^2dxdt\right)^{1/2}\leq C.
\end{split}
\end{equation*}
By Cauchy-Schwarz's inequality, (<ref>), Lemmas <ref>
and <ref>, (<ref>), (<ref>) and (<ref>), we
\begin{equation*}\label{theta6}
\begin{split}
\iint_{Q_T}|f_x| dxdt
&\leq C\iint_{Q_T}\big(|u_x||u_{xx}|+\mu|\mathbf{w}_x\cdot\mathbf{w}_{xx}|+ |\mathbf{b}_x\cdot\mathbf{b}_{xx}|\big)dxdt\\
&\quad+C\iint_{Q_T}\big(u_x^2+\mu|\mathbf{w}_x|^2+ |\mathbf{b}_x|^2\big)|\rho_x|dxdt\\
&\leq C+C\iint_{Q_T}\big(u_x^2+u_{xx}^2 + \sqrt{\mu} |\mathbf{w}_x|^2+\mu^{3/2} |\mathbf{w}_{xx}|^2\big)dxdt \\
&\quad+C\iint_{Q_T}\big(u_x^4+\mu^2|\mathbf{w}_x|^4+|\mathbf{b}_x|^4\big)dxdt+C\iint_{Q_T}\rho_x^2dxdt \leq C.
\end{split}
\end{equation*}
Substituting the above estimates into (<ref>) yields
\int_\Omega|\theta_x|dx\leq C,
which together with $\int_\Omega\theta dx\leq C$ implies the desired result. The proof is complete.
By means of the bounds of $\theta$, we can obtain easily the
following estimates.
Let (<ref>) and (<ref>) hold. Then
\begin{equation*}\begin{split}
\sup\limits_{0<t<T}\int_\Omega \theta_x^2dx+\iint_{Q_T} \big(\theta_t^2+\theta_{xx}^2\big)dxdt\leq C.
\end{split}\end{equation*}
Rewrite the equation (<ref>)$_5$ in the form
\begin{equation}\label{theta1}
\begin{split}
\rho\theta_t-(\kappa\theta_x)_x=\mathcal{Q}-\rho
u\theta_x-\gamma\rho\theta u_x:=f.
\end{split}\end{equation}
We first estimate $\|f\|_{L^2(Q_T)}$. By (<ref>),
(<ref>), (<ref>) and Lemmas <ref> and <ref>,
we have
\begin{equation}\label{f}
\begin{split}
\iint_{Q_T}f^2dxdt\leq &
C\iint_{Q_T}(u_x^4+\mu^2|\mathbf{w}_x|^4+\nu^2|\mathbf{b}_x|^4+\rho^2u^2\theta_x^2+\rho^2u_x^2\theta^2)dxdt \leq C.
\end{split}\end{equation}
Multiplying (<ref>) by $\kappa\theta_t$ and integrating over $Q_t$, we have
\begin{equation}\label{theta2}
\begin{split}
&\iint_{Q_t} \rho\kappa\theta_t^2 dxdt+\iint_{Q_t} \kappa\theta_x (\kappa\theta_t)_x dxdt
\end{split}\end{equation}
Observe that
(\kappa\theta_t)_x=(\kappa\theta_x)_t+\kappa_\rho\rho_x\theta_t+\kappa_\rho\theta_x(\rho_xu+\rho u_x),
so that
\begin{equation*}\label{theta3}
\begin{split}
\iint_{Q_t} \kappa\theta_x (\kappa\theta_t)_x dxdt=& \frac12\int_\Omega \kappa^2\theta_x^2dx-\frac12\int_\Omega \kappa^2(\rho_0,\theta_0)\theta_{0x}^2dx\\
\end{split}\end{equation*}
and substitute it into (<ref>) to yield
\begin{equation}\label{theta8}
\begin{split}
&\iint_{Q_t} \rho\kappa\theta_t^2 dxdt+\int_\Omega
\kappa^2\theta_x^2dx\\
& \leq C-2\iint_{Q_t}\Big[\kappa\kappa_\rho\rho_x\theta_x\theta_t
+\kappa\kappa_\rho\theta_x^2(\rho_xu+\rho u_x)-f\kappa\theta_t\Big]dxdt.\\
\end{split}\end{equation}
By the estimates $C^{-1}\leq \rho, \theta \leq C$ and (<ref>),
we have $ \kappa_1\leq \kappa \leq C, |\kappa_\rho| \leq C. $ By
Young's inequality, (<ref>), (<ref>) and Lemma <ref>, we
\begin{equation}\label{theta4}
\begin{split}
&-2\iint_{Q_t}\Big[\kappa\kappa_\rho\rho_x\theta_x\theta_t+\kappa\kappa_\rho\theta_x^2(\rho_xu+\rho u_x)
& \leq C+\frac14 \iint_{Q_t} \rho\kappa\theta_t^2 dxdt+C\iint_{Q_t} (\kappa\theta_x)^2(\rho_x^2+|\rho_x|+|u_x|) dxds\\
& \leq C+\frac14 \iint_{Q_t} \rho\kappa\theta_t^2 dxdt+C\int_0^t\|\kappa\theta_x\|_{L^\infty(\Omega)}^2 ds.
\end{split}\end{equation}
Now we are ready to deal with the second integral on right-hand side
of (<ref>). By the embedding $W^{1,1}(\Omega)\hookrightarrow
L^\infty(\Omega)$ and Young's inequality, we have
\begin{equation*}\begin{split}
\int_0^t\|\kappa\theta_x\|_{L^\infty(\Omega)}^2 ds \leq& \iint_{Q_t} |\kappa\theta_x|^2 dxds+2\iint_{Q_t} | \kappa\theta_x ||(\kappa\theta_x)_x| dxds\\
\leq & \frac{C}{\epsilon}+\frac{\epsilon}{2}\iint_{Q_t} \big|(\kappa\theta_x)_x\big|^2 dxds, \quad \forall \epsilon>0,
\end{split}\end{equation*}
which together with (<ref>) gives
\begin{equation*}\begin{split}
\int_0^t\|\kappa\theta_x\|_{L^\infty(\Omega)}^2 ds \leq \frac{C}{\epsilon}
+ \epsilon\iint_{Q_t} (\rho^2\theta_t^2+f^2)dxdt.
\end{split}\end{equation*}
Plugging it into (<ref>), taking a small $\epsilon>0$ and
using (<ref>), we obtain
\begin{equation*}
\begin{split}
-2\iint_{Q_t}\Big[\kappa\kappa_\rho\rho_x\theta_x\theta_t+\kappa\kappa_\rho\theta_x^2(\rho_xu+\rho u_x)
\leq C+\frac12 \iint_{Q_t} \rho\kappa\theta_t^2 dxdt,
\end{split}\end{equation*}
from which, (<ref>) and (<ref>) it follows that
\begin{equation}\label{theta}
\begin{split}
\sup\limits_{0<t<T}\int_\Omega \theta_x^2dx+\iint_{Q_T} \theta_t^2
dxdt\leq C.
\end{split}\end{equation}
By (<ref>) and Lemma <ref>, one can derive easily from (<ref>)$_5$ that $\|\theta_{xx}\|_{L^2(Q_T)} \leq C$. The proof is complete.
Due to Lemma <ref>, an argument similar to (<ref>) yields
\begin{equation}\label{theta5}
\begin{split}
\iint_{Q_T}\theta_x^6dxdt\leq C.
\end{split}\end{equation}
Thus, all the estimates appearing in Theorem 1.1 are proved.
§.§ Proof of Theorem 1.1(ii)
By an argument similar to (<ref>)$_2$, one has
\begin{equation}\label{base5}
\begin{aligned}
&\|(u, \mathbf{b}, \theta)\|_{C^{1/2, 1/4}(\overline Q_T)}\leq C,\\
&\|\mathbf{w}\|_{C^{1/2, 1/4}([\delta, 1-\delta]\times[0, T])}\leq C,~\forall\delta\in\big(0, (b-a)/2\big) .
\end{aligned}
\end{equation}
From (<ref>)$_2$, (<ref>), (<ref>),
(<ref>), (<ref>) and Lemmas <ref>, <ref>,
<ref>-<ref> it follows that there exist a subsequence
$\mu_j \rightarrow 0$ and $(\overline\rho, \overline u, \overline{\mathbf{w}},
\overline{\mathbf{b}}, \overline\theta) \in \mathbb{F}$
such that the corresponding solution
for problem (<ref>)-(<ref>) with $\mu=\mu_j $, still denoted by $(\rho,u,\mathbf{w},\mathbf{b}, \theta)$, converges in the sense:
\begin{equation*}\label{rate}
\begin{split}
&(\rho,u,\mathbf{b}, \theta)\rightarrow (\overline\rho,\overline u,\overline{\mathbf{b}}, \overline\theta)~~
\hbox{\rm strongly in}~~C^\alpha(\overline Q_T),~\forall\alpha\in(0, 1/4),\\[1mm]
&(\rho_t,\rho_x,u_x,\mathbf{b}_x,\theta_x)\rightharpoonup(\overline\rho_t, \overline\rho_x,\overline u_x,
\overline{\mathbf{b}}_x,\overline\theta_x)~~ \hbox{\rm weakly}-*~\hbox{\rm in}~ L^\infty(0, T; L^2(\Omega)),\\[1mm]
&(u_t,\mathbf{b}_t,\theta_t,u_{xx}, \theta_{xx})\rightharpoonup(\overline u_t,\overline{\mathbf{b}}_t,
\overline\theta_t,\overline u_{xx}, \overline\theta_{xx})~~ \hbox{\rm weakly in}~~L^2(Q_T),\\[1mm]
& \mathbf{b}_{xx} \rightharpoonup \overline{\mathbf{b}}_{xx}~~ \hbox{\rm weakly in}~~L^2((a+\delta,b-\delta)\times(0, T)),~\forall \delta\in(0, (b-a)/2),\\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
& \mathbf{w} \rightarrow \overline{\mathbf{w}}\quad\hbox{\rm strongly in}~~C^\alpha([a+\delta,b-\delta]\times[0, T]),~\forall \delta\in\big(0, (b-a)/2\big),~\alpha\in(0, 1/4),\\
& \mathbf{w}_t \rightharpoonup \overline{\mathbf{w}}_t \quad\hbox{\rm weakly in}~~L^2(Q_T),\\
& \mathbf{w}_x \rightharpoonup \overline{\mathbf{w}}_x\quad\hbox{\rm weakly}-*~ \hbox{\rm in}~ L^\infty(0, T; L^2(a+\delta,b-\delta)),~\forall \delta\in(0, (b-a)/2),\\
&\mathbf{w}\rightarrow \overline{\mathbf{w}}~~ \hbox{\rm strongly in}~~L^r(Q_T),\quad\forall r \in [1, +\infty),\\
&\sqrt{\mu}\|\mathbf{w}_x\|_{L^4(Q_T)} \rightarrow 0.
\end{split}
\end{equation*}
Next we show the strong convergence of $(u_x,\mathbf{b}_x,\theta_x)$ in $L^2(Q_T)$. Multiplying (<ref>)$_2$ with $\mu=\mu_j$ by $(u-\overline u)$ and integrating over $Q_T$, we have
\begin{equation*}
\begin{split}
&\lambda\iint_{Q_T} \big(u_{x}-\overline u_x\big)^2dxdt+\lambda\iint_{Q_T}\overline u_x\big(u_{x}-\overline u_x\big)dxdt\\
&=-\iint_{Q_T}\left[(\rho u)_{t}
+\left(\rho u^2+\gamma\rho\theta +\frac12|\mathbf{b}|^2\right)_x\right](u-\overline u) dxdt,
\end{split}
\end{equation*}
which together with Lemmas <ref>, <ref> and <ref> implies that
u_{x}\rightarrow\overline u_x~\hbox{\rm strongly in}~ L^2(Q_T) ~\hbox{\rm as}~\mu_j\rightarrow 0.
Similarly, one has
\begin{equation*}
\begin{split}
&(\mathbf{b}_{x},\theta_{x})\rightarrow (\overline{\mathbf{b}}_x,\overline \theta_x)~\hbox{\rm strongly in}~ L^2(Q_T) ~\hbox{\rm as}~\mu_j\rightarrow 0.
\end{split}
\end{equation*}
Furthermore, since from (<ref>), (<ref>) and
(<ref>), we have
\begin{equation*}
\begin{split}
&(u_x, \theta_x)\rightarrow(\overline u_x,\overline\theta_x)~~\hbox{\rm strongly in}~ L^{s_1}(Q_T)
~\hbox{\rm as}~\mu_j\rightarrow 0,~\forall s_1 \in [1, 6),\\
&\mathbf{b}_{x}\rightarrow \overline{\mathbf{b}}_x ~~\hbox{\rm strongly in}~ L^{s_2}(Q_T)
~\hbox{\rm as}~\mu_j\rightarrow 0,~\forall s_2 \in [1, 4).\\
\end{split}
\end{equation*}
Then, it is easy to check that $(\overline\rho,\overline u,
\overline{\mathbf{w}}, \overline{\mathbf{b}}, \overline\theta)$
satisfies (<ref>).
On the other hand, one can see from Theorem 1.1(iii) that the limit problem
(<ref>) admits at most one solution in $\mathbb{F}$. Thus, the above convergence relations hold for any $\mu_j\rightarrow 0$. The proof of Theorem 1.1(ii) is then completed.
§.§ Proof of Theorem 1.1(iii)
The proof is divided into several steps among which the fourth step is the key that can be proved in terms of the boundary estimates of $\mathbf{w}_x$. For convenience, we set
\begin{equation*}
\begin{split}
&\widetilde{\rho}=\rho-\overline\rho,\quad \widetilde{u}= u-\overline
\quad\widetilde{\mathbf{b}}=\mathbf{b}-\overline{\mathbf{b}},\quad
\widetilde{\theta}=\theta-\overline\theta,\\[1mm]
&\mathbb{H}(t)=\|(\widetilde{\rho}, \widetilde{u},\widetilde{\mathbf{w}}, \widetilde{\mathbf{b}}, \widetilde{\theta})\|_{L^2(\Omega)}^2,\\[1mm]
&D(t)=1+ \|(u_x,\mathbf{b}_x, \overline u_x, \overline{\mathbf{b}}_x,\overline{\theta}_x)\|_{L^\infty(\Omega)}^2+\|(\overline u_t, \overline\theta_t, \overline u_x, \overline{\mathbf{b}}_x,\overline{\theta}_x)\|_{L^2(\Omega)}^2.
\end{split}
\end{equation*}
Clearly, $D(t)\in L^1(0, T)$.
Step 1 We claim that
\begin{equation}\label{rho4}
\begin{split}
\int_\Omega \widetilde{\rho}^2dx\leq \epsilon\iint_{Q_t} \widetilde{u}_x^2dxds+\frac{C}{\epsilon}\int_0^tD(s)\mathbb{H}(s)ds,~~ \forall \epsilon \in (0, 1).
\end{split}
\end{equation}
From (<ref>)$_1$ and (<ref>)$_1$ it follows that
\widetilde{\rho}_t=-\big(\rho \widetilde{u} +\overline
Multiplying it by $\widetilde{\rho}$ and integrating over $Q_t$, we
have by Young's inequality
\begin{equation*}\label{rho1}
\begin{split}
\frac12\int_\Omega \widetilde{\rho}^2dx=&-\iint_{Q_t}\big(\rho \widetilde{u}_x\widetilde{\rho}+\rho_x\widetilde{u}\widetilde{\rho} \big)dxds -\frac12\iint_{Q_t}\overline
\leq&\frac{\epsilon}{4}\iint_{Q_t} \widetilde{u}_x^2dxds+C\iint_{Q_t} \widetilde{u}^2\rho_x^2dxds\\
& +\frac{C}{\epsilon}
\int_0^t(1+\|\overline u_x\|_{L^\infty(\Omega)})\int_\Omega\widetilde{\rho}^2dxds, \forall \epsilon \in (0,
\end{split}
\end{equation*}
Since from (<ref>), we have
\begin{equation*}\label{rho3}
\begin{split}
C\iint_{Q_t} \widetilde{u}^2\rho_x^2dxds
\leq C\int_0^t\|\widetilde{u}\|_{L^\infty(\Omega)}^2 ds\leq \frac{\epsilon}{4}\iint_{Q_t} \widetilde{u}_x^2dxds+\frac{C}{\epsilon}\iint_{Q_t} \widetilde{u}^2dxds.
\end{split}
\end{equation*}
Thus, the claim (<ref>) is proved.
Step 2 We claim that
\begin{equation}\label{u14}
\begin{split}
\int_\Omega \widetilde{u}^2dx+ \iint_{Q_t} \widetilde{u}_x ^2dxds\leq C \int_0^tD(s)\mathbb{H}(s)ds.
\end{split}
\end{equation}
(<ref>)$_1$ and (<ref>)$_1$, we derive from (<ref>)$_2$ and (<ref>)$_2$ that
\begin{equation*}
\begin{split}
&\big(\rho \widetilde{u} \big)_t+\big(\rho u\widetilde{u}\big)_x+\widetilde{\rho}\overline u_t
+(\rho u-\overline\rho~\overline u)\overline u_x +\gamma(\rho\theta-\overline\rho\overline\theta)_x
+\frac12(|\mathbf{b}|^2 -|\overline{\mathbf{b}}|^2 )_x =\lambda \widetilde{u}_{xx}.
\end{split}
\end{equation*}
Multiplying it by $\widetilde{u}$ and integrating over $Q_t$, we have
\begin{equation}\label{uu}
\begin{split}
&\frac12\int_\Omega\rho \widetilde{u}^2dx+\lambda\iint_{Q_t} \widetilde{u}_x ^2dxds\\
&=-\iint_{Q_t}\widetilde{\rho} \overline u_t \widetilde{u}dxds
-\iint_{Q_t}(\rho u-\overline\rho~\overline u)\overline u_x\widetilde{u} dxds
+\gamma\iint_{Q_t} (\rho\theta-\overline\rho\overline\theta) \widetilde{u}_x dxds
\\
&\quad+\frac12\iint_{Q_t} (|\mathbf{b}|^2 -|\overline{\mathbf{b}}|^2 ) \widetilde{u}_x dxds =:\sum_{i=1}^4 I_i.
\end{split}
\end{equation}
Observe that $\rho u-\overline\rho~\overline u=\rho \widetilde{u} +\overline u \widetilde{\rho} $ and
$\rho \theta-\overline\rho\overline\theta =\rho\widetilde{\theta}+\overline\theta \widetilde{\rho} $. We have
\begin{equation}\label{u1}
\begin{split}
&I_1+I_2 \\
&\leq C \int_0^t\|\widetilde{u}\|_{L^\infty(\Omega)} \int_\Omega|\widetilde{\rho}|(|\overline u_t|+|\overline u_x|) dt
+C\iint_{Q_t} |\overline u_x| \widetilde{u}^2dxds\\
& \leq C \int_0^t\left(\int_\Omega\widetilde{\rho}^2 dx\right)^{1/2}
\left(\int_\Omega(\overline u_t^2+\overline u_x^2)dx\right)^{1/2} \|\widetilde{u}\|_{L^\infty(\Omega)} dt +C\int_0^t\|\overline u_x\|_{L^\infty(\Omega)}\int_\Omega \widetilde{u}^2dxds\\
&\leq C \int_0^t \left(\int_\Omega(\overline u_t^2+\overline u_x^2)dx\right)\left(\int_\Omega\widetilde{\rho}^2 dx\right)dt
+C\int_0^t \|\widetilde{u}\|_{L^\infty(\Omega)}^2 ds\\
&\quad +C\int_0^t\|\overline u_x\|_{L^\infty(\Omega)}\int_\Omega \widetilde{u}^2dxds \leq C \int_0^tD(s)\mathbb{H}(s)ds
+ \frac\lambda4\iint_{Q_t} \widetilde{u}_x ^2 dxds,
\end{split}
\end{equation}
\begin{equation}\label{u2}
\begin{split}
I_3&\leq \frac\lambda4\iint_{Q_t} \widetilde{u}_x ^2 dxds+C\iint_{Q_t} (\widetilde{\theta}^2+\widetilde{\rho}^2) dxds.
\end{split}
\end{equation}
Utilizing the estimates
$\|(\mathbf{b},\overline{\mathbf{b}})\|_{L^\infty(Q_T)}\leq C$, we
\begin{equation}\label{b01}
\begin{split}
I_4&\leq \frac\lambda4\iint_{Q_t} \widetilde{u}_x ^2 dxds+C\iint_{Q_t} |\widetilde{\mathbf{b}}|^2 dxds.
\end{split}
\end{equation}
Substituting (<ref>)-(<ref>) into (<ref>) completes the proof to (<ref>).
Step 3 We claim that
\begin{equation}\label{Theta}
\begin{split}
&\int_\Omega\widetilde{\theta}^2dx+ \iint_{Q_t} \widetilde{\theta}_x^2dxds\\
&\leq C\sqrt{\mu}
+\frac{C}{\epsilon}\int_0^tD(s)\mathbb{H}(s)ds,~~\forall\epsilon\in(0, 1).
\end{split}
\end{equation}
From (<ref>)$_5$ and
(<ref>)$_5$ it follows that
\begin{equation*}\label{ll}
\begin{split}
& \big(\rho\widetilde{\theta}\big)_t+(\rho u\widetilde{\theta})_x+\widetilde{\rho}\overline\theta_t+(\rho \widetilde{u}+\overline u \widetilde{\rho})\overline\theta_x+\gamma\rho\theta \widetilde{u}_x
+\gamma\big(\rho\widetilde{\theta}+\widetilde{\rho}\overline\theta\big)\overline u_x =\big[\kappa(\rho,\theta)\widetilde{\theta}_x\big]_x\\
&+\big[(\kappa(\rho,\theta)-\kappa(\overline\rho,\overline\theta))\overline\theta_x\big]_x +\lambda(u_x^2-\overline u_x^2) +\mu|\mathbf{w}_x|^2+\nu(|\mathbf{b}_x|^2 -|\overline{\mathbf{b}}_x|^2).
\end{split}
\end{equation*}
Multiplying it by $\widetilde{\theta}$ and integrating over $Q_t$, we obtain
\begin{equation}\label{21}
\begin{split}
& \frac12\int_\Omega\rho\widetilde{\theta}^2dx+ \iint_{Q_t}\kappa\widetilde{\theta}_x^2dxds \\
&=-\iint_{Q_t}\widetilde{\rho}\widetilde{\theta}\overline\theta_tdxdt-\iint_{Q_t}(\rho \widetilde{u}+\overline u \widetilde{\rho})\widetilde{\theta}\overline\theta_xdxds
-\gamma\iint_{Q_t} \rho\theta \widetilde{u}_x \widetilde{\theta} dxds\\
&\quad-\gamma\iint_{Q_t}\rho \overline u_x\widetilde{\theta}^2dxds -\gamma\iint_{Q_t}\overline\theta\overline u_x\widetilde{\rho}\widetilde{\theta} dxds
-\iint_{Q_t}\overline\theta_x[\kappa(\rho,\theta)-\kappa(\overline\rho,\overline\theta)]\widetilde{\theta}_xdxds \\
&\quad+\lambda\iint_{Q_t} (u_x+\overline u_x)\widetilde{u}_x \widetilde{\theta} dxds+\mu\iint_{Q_t} |\mathbf{w}_x|^2 \widetilde{\theta} dxds\\
&\quad+\nu\iint_{Q_t}(|\mathbf{b}_x|^2 -|\overline{\mathbf{b}}_x|^2)\widetilde{\theta} dxds=:\sum_{i=1}^9 E_i.
\end{split}
\end{equation}
By Hölder's inequality and Young's inequality, we have
\begin{equation*}\label{u8}
\begin{split}
&E_1+E_2+E_5 \leq C\int_0^t\left(\int_\Omega\big(\widetilde{\rho}^2+ \widetilde{u}^2\big)dx\right)^{1/2}\left(\int_\Omega\widetilde{\theta}^2(\overline\theta_t^2+\overline\theta_x^2+\overline u_x^2)dx\right)^{1/2}dt\\
&\leq C\int_0^t \left(\int_\Omega(\overline\theta_t^2+\overline\theta_x^2+\overline u_x^2)dx\right)^{1/2} \left(\int_\Omega(\widetilde{\rho}^2+ \widetilde{u}^2)dx\right)^{1/2}\|\widetilde{\theta}\|_{L^\infty(\Omega)}dt\\
&\leq C\int_0^t\left(\int_\Omega(\overline\theta_t^2+\overline\theta_x^2+\overline u_x^2)dx\right)\left(\int_\Omega\big(\widetilde{\rho}^2+ \widetilde{u}^2\big)dx\right)dt
&\leq C\int_0^tD(s)\mathbb{H}(s)ds
&\leq C\int_0^tD(s)\mathbb{H}(s)ds +\frac{\kappa_1}{4}\iint_{Q_t}\widetilde{\theta}_x^2dxds.\\
\end{split}
\end{equation*}
By Young's inequality, we have
\begin{equation*}\label{u111}
\begin{split}
& E_3+E_4 +E_7\\
&\leq \epsilon\iint_{Q_t} \widetilde{u}_x ^2dxds + \frac{C}{\epsilon}\int_0^t\big(1+\|u_x\|_{L^\infty(\Omega)}^2+\|\overline u_x\|_{L^\infty(\Omega)}^2\big)\int_{\Omega} \widetilde{\theta}^2dxds\\
&\leq \epsilon\iint_{Q_t} \widetilde{u}_x ^2dxds + \frac{C}{\epsilon}\int_0^tD(s)\mathbb{H}(s)ds,\quad \forall\epsilon \in (0, 1).
\end{split}
\end{equation*}
By the mean value theorem and $C^{-1}\leq \rho, \overline\rho, \theta, \overline\theta \leq C$, we obtain
|\kappa(\rho,\theta)-\kappa(\overline\rho,\overline\theta)|\leq C(|\widetilde{\rho} |+|\widetilde{\theta}|),
\begin{equation*}\label{u9}
\begin{split}
E_6 \leq & \frac{\kappa_1}{4}\iint_{Q_t} \widetilde{\theta}_x^2dxds+C\iint_{Q_t}|\overline\theta_x|^2\big(\widetilde{\rho}^2+\widetilde{\theta}^2\big)dxds \\ \leq & \frac{\kappa_1}{4} \iint_{Q_t} \widetilde{\theta}_x^2dxds+C\int_0^t\|\overline\theta_x\|^2_{L^\infty(\Omega)}\int_{\Omega}\big(\widetilde{\rho}^2+\widetilde{\theta}^2\big)dxds\\
\leq & \frac{\kappa_1}{4} \iint_{Q_t} \widetilde{\theta}_x^2dxds+C\int_0^tD(s)\mathbb{H}(s)ds.
\end{split}
\end{equation*}
By (<ref>), we have
\begin{equation*}\label{u10}
\begin{split}
E_8\leq C\iint_{Q_t}\widetilde{\theta}^2dxds
+C\mu^2\iint_{Q_t} |\mathbf{w}_x|^4 dxds
\leq C\sqrt{\mu} + C\int_0^tD(s)\mathbb{H}(s)ds.
\end{split}
\end{equation*}
As to $E_9$, we have by the relation:
\begin{equation*}
\begin{split}
E_9\leq & \epsilon\iint_{Q_t}|\widetilde{\mathbf{b}}_x|^2dxds
\int_{\Omega}\widetilde{\theta}^2dxds\\
\leq & \epsilon\iint_{Q_t}|\widetilde{\mathbf{b}}_x|^2dxds
+\frac{C}{\epsilon}\int_0^tD(s)\mathbb{H}(s)ds,~~\forall\epsilon\in(0, 1).
\end{split}
\end{equation*}
Substituting the results into (<ref>) completes the proof to
Step 4 We claim that
\begin{equation}\label{0v3}
\begin{split}
& \int_\Omega |\widetilde{\mathbf{w}}|^2dx \leq C\sqrt{\mu} +\epsilon\iint_{Q_t} |\widetilde{\mathbf{b}}_x|^2 dxds + \frac{C}{\epsilon}\int_0^tD(s)\mathbb{H}(s)ds,\quad\forall \epsilon \in (0, 1).
\end{split}
\end{equation}
From (<ref>)$_3$ and
(<ref>)$_3$, we have
\begin{equation*}
\begin{split}
& \rho\widetilde{\mathbf{w}}_t+ \rho u \widetilde{\mathbf{w}}_x+ \rho \widetilde{u} \overline{\mathbf{w}}_x-\widetilde{\mathbf{b}}_x+\frac{\widetilde{\rho}}{\overline\rho}\overline{\mathbf{b}}_x= \mu \mathbf{w}_{xx}.
\end{split}
\end{equation*}
Multiplying it by $\widetilde{\mathbf{w}}$ and integrating over $Q_t$, we have
\begin{equation}\label{0v33}
\begin{split}
\frac12\int_\Omega \rho|\widetilde{\mathbf{w}}|^2dx
& =\mu\iint_{Q_t} \mathbf{w}_{xx}\cdot\widetilde{\mathbf{w}} dxds
-\iint_{Q_t} \rho\widetilde{u}\overline{\mathbf{w}}_x\cdot\widetilde{\mathbf{w}} \\
&\quad+ \iint_{Q_t} \widetilde{\mathbf{b}}_x\cdot\widetilde{\mathbf{w}}dxds
\overline{\mathbf{b}}_x \cdot\widetilde{\mathbf{w}}dxds\\
&\leq C\mu^2\iint_{Q_t} |\mathbf{w}_{xx}|^2dxds+C\iint_{Q_t} \widetilde{u}^2|\overline{\mathbf{w}}_x|^2dxds
\\
&\quad +\frac{C}{\epsilon}\int_0^tD(s)\mathbb{H}(s)ds +\epsilon\iint_{Q_t} |\widetilde{\mathbf{b}}_x|^2 dxds,\quad\forall \epsilon\in(0, 1).
\end{split}
\end{equation}
Observe that
\begin{equation*}
\begin{split}
& |\widetilde{u}(x,t)|= |u(x,t)-\overline u(x,t)|=\left|\int_0^x\widetilde{u}_x dx\right|\leq \left(\int_0^1\widetilde{u}_x^2dx\right)^{1/2}\omega^{1/2}(x),
\forall x \in [0, 1/2],\\
&|\widetilde{u}(x,t)|= |u(x,t)-\overline u(x,t)|=\left|\int_x^1\widetilde{u}_x dx\right|\leq \left(\int_0^1\widetilde{u}_x^2dx\right)^{1/2}\omega^{1/2}(x),
\forall x \in [1/2, 1].\\
\end{split}
\end{equation*}
We have
\begin{equation*}
\begin{split}
|\widetilde{u}(x,t)|^2 \leq \left(\int_0^1\widetilde{u}_x^2dx\right) \omega(x),\quad \forall (x,t) \in \overline Q_T,
\end{split}
\end{equation*}
which together with Lemma <ref> and (<ref>) gives
\begin{equation*}
\begin{split}
\iint_{Q_t} \widetilde{u}^2|\overline{\mathbf{w}}_x|^2dxds
\leq & \int_0^t\left(\int_0^1\widetilde{u}_x^2dx\right) \left(\int_\Omega|\overline{\mathbf{w}}_x|^2\omega dx\right)ds
\leq C\iint_{Q_t}\widetilde{u}_x^2dxds\\
\leq & C \int_0^tD(s)\mathbb{H}(s)ds.
\end{split}
\end{equation*}
Substituting it into (<ref>) completes the proof to (<ref>).
Step 5 We claim that
\begin{equation}\label{B}
\begin{split}
& \int_\Omega |\widetilde{\mathbf{b}}|^2dx +
\iint_{Q_t} |\widetilde{\mathbf{b}}_x|^2dxds\leq C\int_0^t D(s)\mathbb{H}(s)ds.
\end{split}
\end{equation}
From (<ref>)$_4$ and (<ref>)$_4$, we have
\begin{equation*}
\begin{split}
& \widetilde{\mathbf{b}}_t+ \big(u \widetilde{\mathbf{b}}\big)_x
+ \big(\widetilde{u}\overline{\mathbf{b}}\big)_x- \widetilde{\mathbf{w}}_x-\nu\widetilde{\mathbf{b}}_{xx}=0.
\end{split}
\end{equation*}
Multiplying it by $\widetilde{\mathbf{b}}$ and integrating over $Q_t$ yield
\begin{equation*}
\begin{split}
&\frac12\int_\Omega |\widetilde{\mathbf{b}}|^2dx +
\nu\iint_{Q_t} |\widetilde{\mathbf{b}}_x|^2dxds \\
& = -\frac12\iint_{Q_t}u_x|\widetilde{\mathbf{b}}|^2dxds+\iint_{Q_t} \widetilde{u} \overline{\mathbf{b}}\cdot\widetilde{\mathbf{b}}_xdxds- \iint_{Q_t} \widetilde{\mathbf{b}}_x \cdot\widetilde{\mathbf{w}}dxds\\
&\leq \frac{\nu}{2}\iint_{Q_t} |\widetilde{\mathbf{b}}_x|^2 dxds +C\int_0^t D(s)\mathbb{H}(s)ds.
\end{split}
\end{equation*}
Thus, the claim (<ref>) is proved.
Adding the above five inequalities and taking a small $\epsilon>0$, we complete the proof of Theorem 1.1(iii)
by Gronwall's inequality.
Thus, the proof to Theorem 1.1 is complete.
§ PROOF OF THEOREM 1.3
Let (<ref>), (<ref>) and (<ref>) hold. Then $\overline{\mathbf{b}}=\overline{\mathbf{w}}=0$. Moreover,
\begin{equation*}
\begin{split}
&\sup\limits_{0<t<T}\int_\Omega \big(|\mathbf{b}|^2+|\mathbf{w}|^2\big) dx+\iint_{Q_T}|\mathbf{b}_x|^2dxdt \leq C\sqrt{\mu}.
\end{split}
\end{equation*}
From Theorem 1.1(iii), it suffices to show that $\overline{\mathbf{b}}=\overline{\mathbf{w}}=0$.
To this end, multiplying the equations (<ref>)$_3$ and
(<ref>)$_4$ by $\overline{\mathbf{w}}$ and
$\overline{\mathbf{b}}$, respectively, and integrating over $Q_t$,
we have
\begin{equation*}
\begin{split}
&\frac12\int_\Omega\overline\rho |\overline{\mathbf{w}}|^2dx-\iint_{Q_t} \overline{\mathbf{b}}_x\cdot\overline{\mathbf{w}}dxds=0,\\
&\frac12\int_\Omega |\overline{\mathbf{b}}|^2 dx+\nu\iint_{Q_t}|\overline{\mathbf{b}}_x|^2dxds+\iint_{Q_t} \overline{\mathbf{b}}_x\cdot\overline{\mathbf{w}}dxds+\frac12\iint_{Q_t}\overline u_x |\overline{\mathbf{b}}|^2dxds=0.
\end{split}
\end{equation*}
Adding the two equations yields
\begin{equation*}
\begin{split}
\big(\overline\rho|\overline{\mathbf{w}}|^2+|\overline{\mathbf{b}}|^2
\big)dx +\nu\iint_{Q_t}|\overline{\mathbf{b}}_x|^2dxdt\leq
\frac12\int_0^t\|\overline
\end{split}
\end{equation*}
which together with Gronwall's inequality completes the proof.
Let (<ref>), (<ref>) and (<ref>) hold. Then
\begin{equation*}
\begin{split}
\sup\limits_{0<t<T}\int_{\Omega}|\mathbf{w}_{x}|^2\omega^2dx \leq C\sqrt{\mu}.\\
\end{split}
\end{equation*}
Note that $\mathbf{w}_0=\mathbf{b}_0\equiv0$. By means of Lemmas <ref> and 3.1, similar arguments as in Lemmas <ref> and <ref> give
\begin{equation*}
\begin{split}
& \iint_{Q_t}|\mathbf{b}_{xx}|^2\omega^2dxds
\leq C\sqrt{\mu}+C\iint_{Q_t} |\mathbf{w}_{x}|^2\omega^2 dxds,\\[1mm]
\leq C\sqrt{\mu} +C\iint_{Q_t}|\mathbf{b}_{xx}|^2\omega^2dxds,
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\int_{\Omega}|\mathbf{w}_{x}|^2\omega^2dx
\leq C\sqrt{\mu} +C\iint_{Q_t}|\mathbf{w}_{x}|^2\omega^2dxds,
\end{split}
\end{equation*}
and then, Gronwall's inequality yields the desired result.
The proof is complete.
Let (<ref>), (<ref>) and (<ref>) hold. Then
\begin{equation*}
\begin{split}
\iint_{Q_T}|\mathbf{w}_{t}|^2dxdt \leq C\sqrt{\mu}.\\
\end{split}
\end{equation*}
Using Lemmas <ref>, <ref>, <ref> and <ref> and noticing (<ref>), we derive from (<ref>) that
\begin{equation*}
\begin{split}
\iint_{Q_T}|\mathbf{w}_{t}|^2dxdt &\leq C\mu^2\iint_{Q_T}|\mathbf{w}_{xx}|^2dxdt+C\iint_{Q_T}|\mathbf{b}_{x}|^2dxdt+C\iint_{Q_T}u^2|\mathbf{w}_{x}|^2dxdt\\
&\leq C\sqrt{\mu}+\int_0^T\|u_x\|_{L^\infty(\Omega)}^2\int_\Omega|\mathbf{w}_{x}|^2\omega^2dxdt\leq C\sqrt{\mu}.\\
\end{split}
\end{equation*}
The proof is complete.
Let (<ref>), (<ref>) and (<ref>) hold. Then
\begin{equation*}
\begin{split}
\sup\limits_{0<t<T}\int_{\Omega}|\mathbf{b}_{x}|^2 dx +\iint_{Q_T}|\mathbf{b}_{t}|^2dxdt \leq C\sqrt{\mu}.\\
\end{split}
\end{equation*}
Multiplying (<ref>)$_4$ by $\mathbf{b}_t$, integrating over
$Q_t$ and noticing $\mathbf{w}_0=0$, we have
\begin{equation}\label{b999}
\begin{split}
&\frac{\nu}{2}\int_\Omega|\mathbf{b}_{x}|^2dx+ \iint_{Q_t} |\mathbf{b}_t|^2
= \iint_{Q_t} \mathbf{w}_x\cdot\mathbf{b}_t dxdt - \iint_{Q_t}(u\mathbf{b})_{x}\cdot\mathbf{b}_t dxdt. \\
\end{split}
\end{equation}
Using Lemmas <ref>, <ref> and <ref>, we obtain by the similar
arguments as in (<ref>) and (<ref>)
\begin{equation*}
\begin{split}
\iint_{Q_t} \mathbf{w}_x \cdot\mathbf{b}_t dxdt
& \leq C\sqrt{\mu}+\frac{\nu}{4}\int_\Omega|\mathbf{b}_x|^2dx,
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&-\iint_{Q_t} (u\mathbf{b})_{x} \cdot\mathbf{b}_t dxdt \leq C\sqrt{\mu}+ \frac12 \iint_{Q_t} |\mathbf{b}_t|^2
\end{split}\end{equation*}
Substituting them into (<ref>), we complete the proof.
Now we can prove Theorem 1.3.
Proof of Theorem 1.3 By Theorem 1.1 (ii)-(iii) and Lemma <ref>, one sees that there exists a unique solution
$(\overline \rho,\overline u, \mathbf{0}, \mathbf{0}, \overline\theta)$ for
the limit problem (<ref>) in $\mathbb{F}$.
Next, we are ready to show the second part of this theorem. Denote $\omega_\delta : [0,
1]\rightarrow [0, 1]$ for $\delta\in (0, 1/2)$ by
\begin{equation*}
\omega_\delta (x)=\left\{\begin{split}&x,&&0\leq x\leq \delta,\\
&\delta,&&\delta\leq x\leq 1-\delta,\\
&1-x,&&1-\delta\leq x \leq 1.\\
\end{split}\right.
\end{equation*}
Multiplying (<ref>) by $\mathbf{w}_{xx}\omega_\delta
^n(x)~(n=1,2,\cdots)$ and integrating over $Q_t$, we have
\begin{equation}\label{0v101}
\begin{split}
\mu\iint_{Q_t}|\mathbf{w}_{xx}|^2\frac{ \omega_\delta ^n}{\rho}dxds
&=\iint_{Q_t} \mathbf{w}_t\cdot \mathbf{w}_{xx} \omega_\delta ^n dxdt+\iint_{Q_t} u\mathbf{w}_x\cdot \mathbf{w}_{xx} \omega_\delta ^n dxds\\
&\quad-\iint_{Q_t} \mathbf{b}_x\cdot \mathbf{w}_{xx} \frac{ \omega_\delta ^n}{\rho} dxds.
\end{split}
\end{equation}
Integrating by parts, using (<ref>) and noticing $\mathbf{w}_0=0$, we have
\begin{equation}\label{wtxx}
\begin{split}
&\iint_{Q_t} \mathbf{w}_t\cdot \mathbf{w}_{xx} \omega_\delta ^n dxdt\\
&=-\frac{1}{2}\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^n dx
-n\iint_{Q_t} \mathbf{w}_t\cdot \mathbf{w}_{x} \omega_\delta ^{n-1} \omega_\delta ' dxdt\\
&=-\frac{1}{2}\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^n dx
-n\iint_{Q_t} \left(\frac{\mu}{\rho}\mathbf{w}_{xx}-u\mathbf{w}_x
+\frac{\mathbf{b}_x}{\rho}\right)\cdot \mathbf{w}_{x} \omega_\delta ^{n-1} \omega_\delta ' dxds\\
&\leq -\frac{1}{2}\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^n dx+ \frac{\mu}{2}\iint_{Q_t}|\mathbf{w}_{xx}|^2\frac{ \omega_\delta ^n}{\rho}dxds +C_n\mu\iint_{Q_t}|\mathbf{w}_x|^2 \omega_\delta ^{n-2}
&\quad+C_n\iint_{Q_t}|u||\mathbf{w}_x|^2 \omega_\delta ^{n-1}|\omega_\delta '| dxds -n\iint_{Q_t}
\mathbf{b}_x\cdot \mathbf{w}_{x} \frac{\omega_\delta ^{n-1} \omega_\delta '}{\rho} dxds,\quad n=2,3,\cdots.
\end{split}
\end{equation}
Here and in what follows, $C$ and $C_n$ are positive constants
independent of $\mu$ and $\delta$.
By the mean value theorem and $u(1,t)=u(0,t)=0$, we have
\begin{equation}\label{u33}
\begin{split}
|u(x,t)|\leq \|u_x\|_{L^\infty(\Omega)}\omega_\delta (x),\quad\forall x\in [0, \delta]\cup[1-\delta,1],\\
\end{split}
\end{equation}
which together with the definition of $\omega_\delta $ gives
\begin{equation*}
\begin{split}
\iint_{Q_t}|u||\mathbf{w}_x|^2 \omega_\delta ^{n-1}|\omega_\delta '| dxds=&\int_0^t\hspace{-2mm}\int_0^\delta|u||\mathbf{w}_x|^2 \omega_\delta ^{n-1}dxds+\int_0^t\hspace{-2mm}\int_{1-\delta}^1|u||\mathbf{w}_x|^2 \omega_\delta ^{n-1}dxds\\
\leq& \int_0^t\|u_x\|_{L^\infty(\Omega)} \int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^ndxds,\\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
& \iint_{Q_t} \mathbf{w}_t\cdot \mathbf{w}_{xx} \omega_\delta ^n dxdt\\
&\leq -\frac{1}{2}\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^n dx
+ \frac{\mu}{2}\iint_{Q_t}|\mathbf{w}_{xx}|^2\frac{\omega_\delta ^{n}}{\rho}dxds +C_n\mu\iint_{Q_t}|\mathbf{w}_x|^2 \omega_\delta ^{n-2}
&\quad+C\int_0^t \|u_x\|_{L^\infty(\Omega)} \int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^ndxds-n\iint_{Q_t}
\mathbf{b}_x\cdot \mathbf{w}_{x} \frac{\omega_\delta ^{n-1} \omega_\delta '}{\rho} dxds.\\
\end{split}
\end{equation*}
To estimate the second integral on the right-hand side of
(<ref>), we have by integrating by parts and noticing
\begin{equation*}
\begin{split}
\iint_{Q_t} u\mathbf{w}_x\cdot \mathbf{w}_{xx} \omega_\delta ^n dxds
=&-\frac{1}{2}\iint_{Q_t} |\mathbf{w}_x|^2[u_x \omega_\delta ^n+nu \omega_\delta ^{n-1} \omega_\delta ']
\leq&C_n\int_0^t \|u_x\|_{L^\infty(\Omega)} \int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^ndxds.\\
\end{split}
\end{equation*}
As to the third term on the right-hand side of (<ref>), we
\begin{equation*}
\begin{split}
&-\iint_{Q_t} \mathbf{b}_x\cdot \mathbf{w}_{xx}\frac{ \omega_\delta ^n}{\rho} dxds\\[1mm]
&=\iint_{Q_t} \mathbf{w}_x\cdot \mathbf{b}_{xx} \frac{ \omega_\delta ^n}{\rho}dxds-\iint_{Q_t} \mathbf{w}_x\cdot \mathbf{b}_{x} \frac{\omega_\delta ^n\rho_x}{\rho^2} dxds+n\iint_{Q_t} \mathbf{b}_{x}\cdot\mathbf{w}_x\frac{ \omega_\delta ^{n-1} \omega_\delta '}{\rho} dxds\\[1mm]
& \leq C\iint_{Q_t} |\mathbf{b}_{xx}|^2 \omega_\delta ^n dxds+C\iint_{Q_t} |\mathbf{w}_x|^2 \omega_\delta ^n dxds+C\iint_{Q_t} |\mathbf{b}_x|^2 \omega_\delta ^n \rho_x^2dxds\\[1mm]
&\quad+n\iint_{Q_t} \mathbf{b}_{x}\cdot\mathbf{w}_x\frac{ \omega_\delta ^{n-1} \omega_\delta '}{\rho} dxds\\[1mm]
&\leq C\sqrt{\mu}\delta^{n-1}+C\iint_{Q_t} |\mathbf{b}_{xx}|^2 \omega_\delta ^n dxds + C\iint_{Q_t} |\mathbf{w}_x|^2 \omega_\delta ^n dxds\\[1mm]
&\quad+n\iint_{Q_t} \mathbf{b}_{x}\cdot\mathbf{w}_x\frac{ \omega_\delta ^{n-1} \omega_\delta
'}{\rho} dxds,
\end{split}
\end{equation*}
where we used the fact by (<ref>)$_1$, Lemma <ref> and $0\leq \omega_\delta
(x)\leq \delta$
\begin{equation*}
\begin{split}
\iint_{Q_t} |\mathbf{b}_x|^2 \omega_\delta ^n \rho_x^2dxds\leq&C\int_0^t \left\||\mathbf{b}_x|^2 \omega_\delta ^n\right\|_{L^\infty(\Omega)} ds
\leq C \iint_{Q_t} \left|(|\mathbf{b}_x|^2 \omega_\delta ^n)_x\right| dxds\\[1mm]
\leq & C_n\iint_{Q_t} |\mathbf{b}_{x}|^2| \omega_\delta ^{n-1} \omega_\delta '|dxds+C \iint_{Q_t} |\mathbf{b}_x\cdot\mathbf{b}_{xx}| \omega_\delta ^n dxds\\[1mm]
\leq &C_n\sqrt{\mu}\delta^{n-1} +C\iint_{Q_t} |\mathbf{b}_{xx}|^2 \omega_\delta ^n dxds.
\end{split}
\end{equation*}
Substituting the above results into (<ref>) yields
\begin{equation}\label{w000}
\begin{split}
&\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^n dx+\mu \iint_{Q_t} |\mathbf{w}_{xx}|^2 \omega_\delta ^n dxds\\
& \leq C_n\sqrt{\mu}\delta^{n-1}+C_n\mu\iint_{Q_t}|\mathbf{w}_x|^2 \omega_\delta ^{n-2}
dxds \\
&\quad+ C \int_0^t \big[1+\|u_x\|_{L^\infty(\Omega)}\big] \int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^n dxds+C\iint_{Q_t} |\mathbf{b}_{xx}|^2 \omega_\delta ^n dxds.
\end{split}
\end{equation}
It remains to treat the relation between the terms $\iint_{Q_t}
|\mathbf{b}_{xx}|^2 \omega_\delta ^n dxds$ and $\iint_{Q_t}
|\mathbf{w}_{x}|^2 \omega_\delta ^n dxds$. To this end, we multiply
(<ref>)$_4$ by $\mathbf{b}_{xx} \omega_\delta ^n(x) ~(n=2,3,\cdots)$ and integrate over $Q_t$ to obtain
\begin{equation}\label{b1111}
\begin{split}
\nu\iint_{Q_t} |\mathbf{b}_{xx}|^2 \omega_\delta ^n dxds
&=\iint_{Q_t} \mathbf{b}_t\cdot \mathbf{b}_{xx} \omega_\delta ^n dxdt+\iint_{Q_t} (u\mathbf{b})_x\cdot \mathbf{b}_{xx} \omega_\delta ^n dxds\\
&\quad-\iint_{Q_t} \mathbf{w}_x\cdot \mathbf{b}_{xx} \omega_\delta ^n dxds.
\end{split}
\end{equation}
To estimate the first term on right-hand side of (<ref>), we
use Young's inequality, Lemma <ref> and $0\leq \omega_\delta
(x)\leq \delta$ to obtain
\begin{equation*}
\begin{split}
\iint_{Q_t} \mathbf{b}_t\cdot \mathbf{b}_{xx} \omega_\delta ^n dxdt
\leq C\sqrt{\mu}\delta^{n}+\frac{\nu}{4}\iint_{Q_t}|\mathbf{b}_{xx}|^2 \omega_\delta ^n dxdt.
\end{split}
\end{equation*}
Next we deal with the second term on right-hand side of
(<ref>). By $0\leq \omega_\delta (x)\leq \delta$ and Lemmas
<ref> and <ref>, we have
\begin{equation*}
\begin{split}
\iint_{Q_t} |(u\mathbf{b})_x|^2 \omega_\delta ^n dxds \leq &C\iint_{Q_t} u^2|\mathbf{b}_{x}|^2 \omega_\delta ^n dxds+C\iint_{Q_t} u_x^2|\mathbf{b} |^2 \omega_\delta ^n dxds\\
\leq & C\sqrt{\mu}\delta^n+C\delta^n\int_0^t\|u_x\|_{L^\infty(\Omega)}^2\int_\Omega|\mathbf{b} |^2 dxds\leq C\sqrt{\mu}\delta^n.
\end{split}
\end{equation*}
As to the third term on right-hand side of (<ref>), we have by Young's inequality
\begin{equation*}
\begin{split}
- \iint_{Q_t} \mathbf{w}_x\cdot \mathbf{b}_{xx} \omega_\delta ^n dxds\leq C\iint_{Q_t} |\mathbf{w}_x|\omega_\delta ^n dxds
+\frac{\nu}{4}\iint_{Q_t} |\mathbf{b}_{xx}|\omega_\delta ^n dxds.
\end{split}
\end{equation*}
Substituting them into (<ref>) yields
\begin{equation}\label{bel}
\begin{split}
& \iint_{Q_t} |\mathbf{b}_{xx}|^2 \omega_\delta ^n dxds\leq C\sqrt{\mu}\delta^{n}+C \iint_{Q_t}|\mathbf{w}_x|^2 \omega_\delta ^{n}dxds,
\end{split}
\end{equation}
and inserting it into (<ref>) and using Gronwall's
inequality, we obtain the iteration
\begin{equation}\label{iteration}
\begin{split}
&\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^n dx \leq C_n \sqrt{\mu}\delta^{n-1} + C_n\mu\iint_{Q_t} |\mathbf{w}_x|^2 \omega_\delta ^{n-2} dxds,\quad n=2,3,\cdots.
\end{split}
\end{equation}
Note that the above results still hold for $n=1$. Since the term
$-\iint_{Q_t} \frac{\mu}{\rho}\mathbf{w}_{xx}\cdot \mathbf{w}_{x}
\omega_\delta ' dxds$ in the equality of (<ref>) with $n=1$ can
be dealt with as follows
\begin{equation*}
\begin{split}
&-\iint_{Q_t} \frac{\mu}{\rho}\mathbf{w}_{xx}\cdot \mathbf{w}_{x} \omega_\delta ' dxds\leq C\sqrt{\mu}\iint_{Q_t} |\mathbf{w}_{x}|^2 dxds+C\mu^{3/2}\iint_{Q_t}|\mathbf{w}_{xx}|^2dxds\leq C,
\end{split}
\end{equation*}
where we used Lemma <ref>, a similar argument as above gives,
instead of (<ref>),
\begin{equation}\label{wx3}
\begin{split}
&\int_\Omega|\mathbf{w}_x|^2 \omega_\delta dx \leq C.
\end{split}
\end{equation}
So, we derive from (<ref>) that
\begin{equation*}
\begin{split}
&\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^2 dx \leq C(\sqrt{\mu}\delta+\sqrt{\mu}),\\
&\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^3 dx \leq C\big(\sqrt{\mu}\delta^2+\mu\big).\\
\end{split}
\end{equation*}
Next, taking $n=4,5,6,7$ in
(<ref>), respectively, we get
\begin{equation*}
\begin{split}
&\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^4 dx \leq C\big(\sqrt{\mu}\delta^3+\mu^{3/2}\delta+\mu^{3/2}\big),\\
&\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^5 dx \leq C\big(\sqrt{\mu}\delta^4+\mu^{3/2}\delta^2+\mu^{2}\big),\\
&\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^6 dx \leq C\big(\sqrt{\mu}\delta^5+\mu^{3/2}\delta^3+\mu^{5/2}\delta+\mu^{5/2}\big),\\
&\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^7 dx
\leq C\big(\sqrt{\mu}\delta^6+\mu^{3/2}\delta^4+\mu^{5/2}\delta^2+\mu^3\big),\\
\end{split}
\end{equation*}
thus, an induction gives
\begin{equation}\label{0w1}
\begin{split}
\int_\Omega|\mathbf{w}_x|^2 \omega_\delta ^n dx\leq \left\{
\begin{split}
& C_n\big(\sqrt{\mu}\delta^{n-1}+\mu^{3/2}\delta^{n-3} +\cdots+\mu^{(n-2)/2}\delta^{2}+ \mu^{(n-1)/2}\big)~~(n=\hbox{\rm odd}),\\
& C_n\big(\sqrt{\mu}\delta^{n-1}+\mu^{3/2}\delta^{n-3} +\cdots+\mu^{(n-1)/2}\delta
+\mu^{(n-1)/2}\big)~~(n=\hbox{\rm even}),\\
\end{split}\right.
\end{split}
\end{equation}
where $n=2,3,\cdots,$ which together with the definition of
$\omega_\delta $ gives
\begin{equation}\label{0w0}
\begin{split}
\int_{\delta}^{1-\delta}|\mathbf{w}_x|^2 dx\leq \left\{
\begin{split}
& C_n\big(\tau+\tau^3+\cdots+\tau^{n-2}\big)+C_n\mu^{(n-1)/2}/\delta^n~(n=\hbox{\rm odd}),\\
& C_n\big(\tau+\tau^3+\cdots+\tau^{n-1}\big)+C_n\mu^{(n-1)/2}/\delta^n~(n=\hbox{\rm even}),\\
\end{split}\right.
\end{split}
\end{equation}
where $\delta\in (0, 1/2), \tau=\sqrt{\mu}/\delta.$
On the other hand, we have by the mean value theorem and
Lemma 3.1
\begin{equation*}
\begin{split}
\|\mathbf{w}\|_{L^\infty(\delta, 1-\delta)} \leq & \frac{1}{1-2\delta}\int_\delta^{1-\delta}|\mathbf{w}|dx+\int_\delta^{1-\delta}|\mathbf{w}_x|dx\\
\leq &C\mu^{1/4}+\left(\int_\delta^{1-\delta}|\mathbf{w}_x|^2dx\right)^{1/2},\quad \forall\delta\in(0, 1/4),
\end{split}
\end{equation*}
which together with (<ref>) implies that any function
$\delta(\mu)$ with $\delta(\mu)\downarrow0$ and
$\frac{\mu^{(n-1)/(2n)}}{\delta(\mu)}=\frac{\mu^{1/2-1/(2n)}}{\delta(\mu)}\rightarrow 0$ as $ \mu
\rightarrow0$ satisfies
\begin{equation}\label{v99}
\begin{split}
&\lim\limits_{\mu\rightarrow 0} \|\mathbf{w}\|_{L^\infty(0, T;L^\infty(\delta(\mu), 1-\delta(\mu)))}=0.
\end{split}
\end{equation}
Since $n$ can be arbitrarily large, we see that $\delta(\mu)= \mu^\alpha $ satisfies (<ref>) for any $\alpha\in (0, 1/2)$. The proof is completed.
§ ACKNOWLEDGMENTS
The authors would like to thank Professor Tong Yang at City
University of Hong Kong for valuable discussions on Lemma <ref> during their visit to City University of Hong Kong.
The research was supported in part by the NSFC (grants 11571062,11571380,11401078), Guangdong Natural Science
Foundation (grant 2014A030313161), the Program for Liaoning
Excellent Talents in University (grant LJQ2013124)
and the Fundamental Research Fund for the Central Universities (grant DC201502050202).
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|
1511.00042
|
[I. R. Sellers]sellers@ou.edu
Homer L. Dodge Department of Physics & Astronomy, University of Oklahoma, 440 W. Brooks St., Norman, Oklahoma 73019, USA
School of Chemical, Biological and Materials Engineering, Sarkey’s Energy Center, University of Oklahoma, East Boyd Street-T301, Norman, OK 73019, USA
InAs/AlAs$_{x}$Sb$_{1-x}$ quantum wells are investigated for their potential as hot carrier solar cells. Continuous wave power and temperature dependent photoluminescence indicate a transition in the dominant hot carrier relaxation process from conventional phonon-mediated carrier relaxation below 90 K to a regime where inhibited radiative recombination dominates the hot carrier relaxation at elevated temperatures. At temperatures below 90 K photoluminescence measurements are consistent with type-I quantum wells that exhibit hole localization associated with alloy/interface fluctuations. At elevated temperatures hole delocalization reveals the true type-II band alignment; where it is observed that inhibited radiative recombination due to the spatial separation of the charge carriers dominates hot carrier relaxation. This decoupling of phonon-mediated relaxation results in robust hot carriers at higher temperatures even at lower excitation powers. These results indicate type-II quantum wells offer potential as practical hot carrier systems.
§ INTRODUCTION
Hot carrier solar cells (HCSCs) have been proposed as devices, which can increase the conversion efficiency of a single junction solar cell above the Shockley-Queisser limit <cit.>. Since thermalization of photogenerated carriers is a major loss mechanism in conventional solar cells, HCSCs have the potential to produce higher efficiency devices using simple single gap semiconductor architectures by eliminating the thermal losses associated with electron-phonon interactions <cit.>.
However, before their practical implementation can be realized, HCSCs must circumvent two main challenges <cit.>: 1. Find an absorber material in which hot phonons are longer lived than hot carriers such as to provide the required condition to promote reabsorption of these hot phonons, a phonon bottleneck, which significantly reduces hot carrier relaxation through phonon channels <cit.>; 2. Implement energy selective contacts <cit.> in which only a narrow range of energy (within the hot carrier distribution) can be extracted, restricting the energy distributed through carriers cooling, therefore minimizing the entropy heat transfer loss <cit.>.
Here, InAs/AlAs$_{0.16}$Sb$_{0.84}$ quantum-wells are investigated as a candidate hot carrier absorber. The use of quantum wells also offers the potential to facilitate the development of energy-selective contacts and fast carrier extraction via resonant tunneling from quantum wells (QWs) making them an attractive potential system for HCSCs.
§ EXPERIMENTAL RESULTS AND ANALYSIS
A schematic of the sample used in this investigation is shown in Fig. <ref>(a). The InAs/AlAs$_{0.16}$Sb$_{0.84}$ multi-quantum-well heterostructure was grown by molecular beam epitaxy (MBE) at a substrate temperature of 465C. A 2000 nm InAs buffer layer was grown on a nominally undoped GaAs substrate to reduce the density of crystalline defects arising from the lattice mismatch of the active region (MQWs) and the substrate. The thickness of the InAs QWs is 2.4 nm and the AlAs$_{0.16}$Sb$_{0.84}$ barriers are 10 nm.
As shown in Fig. <ref>(b), there is a lower quantum confinement in the valence band (VB) and much larger confinement in the conduction band (CB). The lower energy barrier layers in the VB results in the rapid transfer of the holes absorbed directly in the InAs QWs to the AlAsSb barriers and their enhanced mobility with increasing temperature. Conversely, due to the large confinement in the CB, electrons remain strongly confined at all temperatures <cit.>. The type-II band alignment shown in Fig. <ref> (b) (magnified in 1(d) for clarity) and thermal diffusion of holes results in an excess of electrons (with respect to holes) in the QWs due to the reduced radiative recombination rate.
In addition, the large energy band offset between the QW and barrier facilitates absorption of a large proportion of the solar spectrum directly in the InAs QWs, without significant losses in the barriers. Finally, the narrow QWs enable a design in which the separation of the energy levels in the CB is large ($\sim$ 0.7 eV) resulting in a hot carrier distribution that predominately occupies the ground-state subband; without significant influence of broadening due to occupation of higher order (or barrier) subbands (Fig. <ref>(c), (d)).
Energy dependent photoluminescence (PL) spectra for a range of power densities at 10 K are shown in Fig.<ref>(a). At lower powers, a shift in the PL peak energy is evident, which reflects the effects of alloy fluctuations that have a significant effect at low power and temperature <cit.>. At intermediate powers, the peak energy stabilizes and a broadening of the high-energy tail becomes evident. Such high-energy broadening is indicative of the presence of hot carriers <cit.> generated by non-equilibrium photogenerated carriers in the CB.
The observation of the shift in peak energy at low power is also evident in Fig.<ref>(b), which shows the dependence of the peak PL energy (at increasing temperature) versus absorbed power $(P_{abs})$. At powers below 1-2 W/cm$^{2}$ a large increase in the peak PL is observed. However, at higher $P_{abs}$ the peak PL energy saturates, particularly at higher temperatures. This behavior has been shown to be due to the presence of alloy fluctuations at the InAs-AlAsSb interface and the resulting spatial localization of carriers, which is quenched or saturated, with increasing temperature and/or excited carrier density. <cit.>
(a) Schematic representation of the InAs/AlAs$_{0.16}$Sb$_{0.84}$ quantum well sample investigated. (b) Simulated energy profiles showing the relative energy of the confinement potential at the Γ point in the conduction (high) and valence band (low). (c) 2D Electron density of states as a function of the energy for this structure. (d) Shows a magnification of the band offsets displaying the type II band alignment with large separation of the energy subbands.
The effect of the alloy fluctuations is also illustrated (somewhat) in Figure 2(c), which shows the power dependent behavior of the temperature difference (the difference between carrier and lattice temperatures: $\Delta T=T_{e}-T_{L}$). The carrier temperature, and therefore, $\Delta T$, can be quantified by fitting the high-energy tail of the PL spectrum, using the generalized Planck relation <cit.>
\begin{equation}
I(E)\propto\varepsilon(E) exp\left(-\frac{E}{k_{B}T_{e}}\right)
\end{equation}
Where I is the energy-dependent PL intensity, $\varepsilon$ is the effective emissivity, which is related to the absorption profile, $k_{B}$ is Boltzmann's constant, and $T_e$ represents the carrier temperature extracted from the slope of the PL at energy greater than the band gap. Although hot carriers have been predominately investigated using ultrafast time-resolved spectroscopy <cit.>, Equation (1) describes a technique to study the behavior of hot carriers in continuous-wave operation, the mode of operation of solar cells, and therefore presents a more realistic method to interpret the hot carrier dynamics in practical photovoltaic systems.
At low powers, a large shift of the carrier temperature is observed (below 1 W/cm$^{2}$). The validity of Equation (1) for the extraction of $T_e$ assumes that the effective emissivity $(\varepsilon)$, therefore absorption, is constant at a fixed energy. That is, it is independent of the excitation power. Since the PL energy changes rapidly in the low power regime, the initial increase in $T_e$ is attributed to an artifact of the increasing absorption rather than the real carrier temperature. However (as described above) at higher powers ($P_{abs} >$ 2 W/cm$^{2}$) the energy shift stabilizes (Fig.<ref>(b)) and as such, reflects the (true) carrier temperature; independent of fluctuation effects, which are saturated under these excitation conditions.
It is important to emphasize, once more, that since there is a large energy difference between the ground-state transition and the higher-order states (Fig.<ref>(d)), the high energy tail represents hot carrier effects related solely to the ground-state of the QW, unperturbed by state-filling effects. It must be noted, however, that although band-to-band recombination in the AlAsSb barriers has little effect on the high-energy tail of QW luminescence, the effects of impurities in the QW <cit.> and/or localized states at the QW/barrier interface cannot be totally dismissed as contributing to the high-energy tail; the latter of which is discussed in more detail below (see Fig.<ref>).
(a) Power dependent PL spectrum at 10 K. (b) Peak PL energy at selected various temperatures. (c) Temperature difference ($\Delta T$) versus absorbed power ($P_{abs}$) at 10 K.
The number of carriers generated $(N_c)$ with increasing intensity is indicated with respect to the absorbed power on the upper axis of Fig.<ref>(b) and (c). The densities absorbed are the order of, or less than, 3$\times$10$^{9}$ cm$^{-2}$ for the excitation levels used. If this is compared to the total 2D density of states calculated for the ground state for the InAs QWs, which is shown in Fig.<ref>(c) (1$\times$10$^{13}$ cm$^{-2}$), then it becomes clear that the increased contribution of the high energy tail is not the result of significant state-filling, or the saturation of the ground-state, and therefore likely has its origin in inhibited hot carrier relaxation via the creation of a phonon bottleneck <cit.>. Similar effects have been observed recently in InAs QDs, where the spatial separation of carriers in impurity states leads to the observation of inhibited carrier relaxation as a result of reduced carrier-carrier scattering <cit.>. The type-II nature of InAs/AlAsSb is expected to lead to similar results here.
Equation (1) shows that the PL spectrum can also give information about the absorption and the effective band gap of the QWs. The pre-exponential term of the Planck distribution describes an effective emissivity term, which is an energy dependent parameter. Fig.<ref>(a) shows the natural logarithm of this effective emissivity $(ln \varepsilon)$ (closed squares) as a function of photon energy $(E)$ at 10 K for low excitation power, prior to significant hot carrier generation. These data are shown with respect to the power dependent PL. As the energy increases towards the peak PL energy, and therefore band gap, the effective emissivity increases rapidly. Once the energy gap is reached, the effective emissivity increases much more slowly, reflecting (somewhat) the lower rate of change of the absorption at higher energy <cit.>.
(a) Natural logarithm of effective emissivity (blue squares) and power dependent PL spectrum as a function of energy. (b) The comparison of the behavior of the highest power PL spectrum and natural logarithm of effective emissivity for several intensities.
Fig.<ref>(b) illustrates the effect of the emissivity term with increasing excitation power obtained by plotting $ln \varepsilon$ $vs.$ $E$ for the PL spectra of Fig.<ref>(a). These data are compared to highest power PL (shown in black). The behavior of the effective emissivity data is constant across the spectra with the shift in absolute value related to the increasing carrier temperature, as extracted from the slope of the PL, which is inserted in the Eqn. (1) transposed for the natural logarithm of the effective emissivity. The consistency of $ln \varepsilon$ confirms that the pre-exponential term in Equation (1) is indeed independent of power (for the conditions used to extract $T_e$) and demonstrates further that a large separation exists between the ground and the first excited state in the QWs. Therefore, since the carrier temperature is extracted in a region with the constant effective emissivity over a large energy range, $T_e$ is determined with relatively low uncertainty.
Fig.<ref>(a) and (c) display the dependence of the temperature difference $(\Delta T)$ for temperatures between 10 K and 90 K, and 90 K and 130 K, respectively. The inset to Fig.<ref>(c) shows the same data at 225 K and 295 K. In Fig.<ref>(a), the carrier temperature, and therefore $\Delta T$, tends to increase with increasing excitation power. The dependence of the hot carriers and their thermalization rate can be evaluated by studying the rate of the thermalized energy (which is the same as absorbed power in the $V_{oc}$ condition) per degree of temperature change <cit.> as described by:
\begin{equation}
\noindent P_{th}=\frac{ntE_{LO}}{\tau_{th}}exp\left(-\frac{E_{LO}}{k_{B}T_{e}}\right)=Q\Delta T exp\left(-\frac{E_{LO}}{k_{B}T_{e}}\right)
\end{equation}
Where $P_{th}$ is the thermalized (absorbed) power, $n$ is carrier density, $t$ is thickness, $\tau_{th}$ is the thermalization time, $E_{LO}$ is the phonon energy for InAs, $k_{B}$ is Boltzmann’s constant, and $T_e$ is the carrier temperature. $\Delta T$ is the difference in temperature between the carriers and the lattice, and $Q$ is the thermalization coefficient <cit.>.
Equation (2) can be used to extract $Q$, an empirical parameter used to assess the contribution of phonon mediated carrier relaxation in QWs <cit.>. A high $Q$ is indicative of efficient phonon-mediated relaxation of hot carriers; therefore, systems with lower $Q$ are desired for practical HCSCs <cit.>. In Fig.<ref>(b) and (d), the slope of each data set is used to extract $Q$ for that particular temperature. In Fig.<ref>(b) it can be seen that increasing the temperature to 90 K results in a $Q$ that is increasing, consistent with the increasing contribution of LO phonon scattering at elevated temperature <cit.>. As the temperature is increased further (90 K - 130 K), as shown in Fig.<ref>(d), $Q$ starts to become less dependent on $T_{L}$; stabilizing between 90 K and 130 K despite increasing phonon densities at elevated temperature. To reveal the mechanism for this unusual behavior the effect of $T_e$ with increasing excitation power at higher temperatures needs to be considered.
Fig.<ref>(a) shows $T_e$ versus power between 10 K and 90 K. As excitation power is increased the carrier temperature also increases, as the ratio of excited carriers to phonon density becomes larger. As the lattice temperature is increased to 90 K, the absolute increase in $T_e$ (with power) begins to slow and reduces. This behavior is expected since the phonon density is larger at elevated temperatures (see Fig.<ref>b), increasing the prevalence of carrier thermalization. This behavior consequently leads to an increased $Q$ as observed in Fig.<ref>(b). However, for higher temperatures ($>$ 130 K), we can see the dependence of $T_e$ with absorbed power is less than the dependence of $T_e$ at lower temperature (Fig.<ref>b). Indeed, although $T_e$ is reduced up to 90 K - stabilizing somewhat through 130 K rather than producing the expected equilibrium carrier distribution via strong LO phonon-relaxation, $T_e$ actually increases; again, despite an increasing phonon density at elevated lattice temperatures.
In addition to this apparent decoupling of the phonon-relaxation channels above 130 K, the effect of excitation temperature, also, becomes less pronounced at higher temperature. The inset to Fig.<ref>(b) shows the power dependence at 225 K (solid squares) and 295 K (solid circles), respectively. What is evident is: that the absolute $T_{e}$ increases relative to $T_L$ above 130 K and from 225 K to 295 K, the carriers are “hot", even at lower excitation levels.
This behavior presents an interesting question with respect to the validity of using analysis of $Q$ in type-II systems. The empirical parameter $Q$ has been used previously to assess, or qualify the contribution of phonon-relaxation channels in type-I QWs and evaluate their potential for applications as the absorber in HCSCs <cit.>. Indeed recently, this analysis has also been extended to determine the absolute efficiency that may be produced if such systems were applied to HCSCs under concentrated illumination <cit.>.
This analysis, however, is based on two principles: 1) that at high temperatures the dominant relaxation channels are related to LO phonon scattering and 2) a constant carrier temperature with respect to excitation power, occurs at (and represents) the equilibrium condition; i.e., the carriers are thermalized at $T_L$. In the case of the type-II QWs investigated here, the behavior of the system is not consistent with these assumptions, particularly at $T > 130$ K. The high (and increasing) $T_{e}$, at $T > 130$ K, along with the relative insensitivity to excitation power, suggests the relaxation of hot carriers in this system is not dominated by electron-phonon interaction.
(a), (c) $\Delta T$ versus power density for several lattice temperatures. (b), (d) Gradients of $P_{abs}$/exp(-E$_{LO}$/(k$_B$T$_{e}$ )) against $\Delta T$ give the thermalization coefficient $(Q)$. The inset graph in (c) displays the independency of $\Delta T$ from power densities at temperatures above 200 K.
This is further illustrated in the inset to Fig.<ref>, which shows the $Q$ analysis at 225 K (closed circles) and 295 K (closed squares). Here, the difficulty in interpreting a thermalization coefficient becomes clear since the independence of $T_e$ with power at these temperatures results in a $Q$ that is large, sometimes infinite, but can also (dependent upon fitting methodology) produce a negative value!
To understand the apparent anomalies in the system under investigation with respect to previous systems presented in the literature <cit.>, the nature of the band alignment should be considered. The type-II nature of the InAs/AlAs$_{x}$Sb$_{1-x}$ QWs introduces important differences in the behavior of the samples at high temperature and under intense illumination. At low excitation and at temperatures below 90 K, the PL measured is dominated by a quasi-type-I transition. This is related to recombination of electrons confined in the QWs and holes localized at the InAs/AlAsSb interface <cit.>.
At T $>$ 90 K the holes localized at the QW/barrier interface are thermally activated and redistribute into the lower energy AlAsSb barrier region. This delocalization of trapped charges reveals the true type-II band alignment of this system, and consequently the excitons will be spatially separated. It should be noted, if the alloy fluctuations were eliminated, or the materials properties improved, the type-II behavior would be observed at all temperatures.
A consequence of the separation of the electrons and holes is a reduced radiative recombination efficiency, and therefore a longer radiative lifetime. This behavior will result in an excess of electrons in the QWs, reduced carrier-carrier scattering <cit.>, and (once more) the development of a phonon bottleneck <cit.>. As such, the dominant relaxation process in the type-II QWs presented appears related predominantly to the radiative recombination lifetime, rather than phonon mediated processes.
Relation between $Q$ and $\Delta T$ against lattice temperature are displayed to $T_e$ = 130 K (blue triangles). The inset shows that $Q$ cannot be determined through for these data sets. Also shown (closed stars) is the temperature dependent hot carrier temperature, $\Delta T$.
Relation between $Q$ and $\Delta T$ against lattice temperature are displayed to $T_{e} = 130$ K (blue triangles). The inset shows that $Q$ cannot be determined through for these data sets. Also shown (closed stars) is the temperature dependent hot carrier temperature, $\Delta T$.
This behavior further illustrates that the analysis of a thermalization coefficient $(Q)$ used for type-I systems <cit.> appears invalid here. Indeed, since the rapid spatial separation of carriers absorbed directly in the QWs is a general feature of type-II systems, the decoupling of LO-phonons via inhibited radiative recombination should be a general feature across other type-II QWs investigated this way.
Fig.<ref> illustrates further the unique difference between the dominant hot carrier relaxation processes in type-I and type-II systems. Specifically, Fig.<ref> shows a comparison of the change of $Q$ (open triangles) and $\Delta T$ (closed stars), versus lattice temperature, $T_{L}$. These data are extracted as in a similar manner to those in Fig.<ref>. At $T < 90$ K, where the sample displays type-I behavior, ΔT decreases with increasing lattice temperature, i.e., the hot carriers are being thermalized by conventional LO-phonon interaction. In this temperature regime (10 K – 90 K) $Q$ is shown to increase with temperature from 0.2 WK$^{-1}$cm$^{-2}$ to 2 WK$^{-1}$cm$^{-2}$, supporting the idea that $Q$-analysis is valid in this regime, when the system behaves as a (quasi)-type-I QW <cit.>.
It must be noted, however, that the $Q$ determined here should be considered an upper limit since the diffusion (and therefore mobility) of the carriers absorbed is temperature dependent. Practically, this will result in a change of the absorbed power density as lateral carrier diffusion (and luminescence density area) increases at higher temperatures, before radiative recombination occurs.
At T $>$ 90 K, the nature of the system changes: as the holes delocalize from alloy fluctuations, the system transitions from (quasi)-type-I to type-II. As such, $T_e$ begins to increase, increasing linearly with increasing lattice temperature, up to 300 K. At temperatures between 130 K and 150 K the behavior, or interpretation, of $Q$ becomes ambiguous as the dependence of the hot carriers with excitation power becomes less pronounced (See Fig.<ref>(c)). In this regime, $T_e$ is dominated by the efficiency of the radiative recombination, which in type-II systems has been shown to extend for 100’s of nanoseconds <cit.>. Therefore, the analysis of $Q$ at T $>$ 130 K, or more generally in type-II systems, becomes moot.
To investigate this hypothesis further, first-principles density-functional-theory (DFT) calculations using the VASP package <cit.> were used to explore the electronic structure of an analogous InAs/AlSb heterostructure in which the InAs layer is 2.4 nm thick and AlSb is about 9.4 nm thick. A PBE-GGA exchange-correlation functional <cit.> was used for the structural relaxation; when calculating the density of states, a hybrid functional was used <cit.>. The heterostructure is very similar to that studied experimentally and allows a qualitative picture of its behavior to be determined. In this theoretical system, AlSb, rather than AlAsSb, is used to simplify the interpretation.
Fig.<ref>(a) shows the 3D density of states (DOS) calculated for the structure, which is magnified about the energy gap. The valence band edges of InAs and AlSb in this heterostructure are almost degenerate, while the conduction bands are well separated between InAs and AlSb. The calculations thus support the type-II band alignment. It should be noticed that the band gap is normally underestimated in DFT-PBE calculations and also hybrid-functional calculations. <cit.>
The first peak above the Fermi level at 0.3 eV is an interfacial state is mainly located at the InAs/AlSb interface, which may account for the carrier localization and the aforementioned transition from type-I to type-II in these QWs. This heterostructure displays two distinct interfaces, i.e. an AlSb-InAs (interface i) and an Sb-Al/As-In (interface ii) with the difference arising from the varied stacking sequence. A close inspection of the (3D) DOS by projecting it to each atom suggests that the interfacial state is more pronounced at the interface i, particularly on the interfacial In, As, and Sb atoms. The origin of these interfacial states is under further investigation but may originate from the interfacial strain effect (Fig.<ref>(a)).
The results indicate that reducing the amount of the Sb (that is, deposit more Al and As) at the interface between the QWs and the barriers may help to reduce these charge trapping levels. The available (3D) DOS of this interfacial state is in the order of 10$^{20}$ cm$^{-3}$ (or 10$^{13}$ cm$^{-2}$ assuming one-nm-thick 2D-interfaces), similar to the InAs conduction band edge. On the other hand, the (3D) phonon density (Fig.<ref>(b)), that is the overall phonon density without distinguishing different types of phonon, is much higher than the electron density and increases rapidly as a function of temperature. The increased phonon density at higher temperature and the experimentally observed reduced thermalization suggests that phonon-mediated carrier relaxation does not dominate at high temperatures, which is consistent with the hypothesis that a phonon bottleneck results in the type-II QWs presented, supporting the conclusion that the relaxation of hot carriers is dominated by the reduced radiative efficiency in these systems.
The demonstration of robust hot carriers at elevated temperatures (and at reasonable excitation densities), coupled with the relaxation of phonon loss channels, indicates that type-II systems offer a viable route to practical hot-carrier solar cells.
(a) Calculated 3D electron density of states (DOS) for an InAs/AlSb heterostructure, shown inset (upper). The DOS of InAs and AlSb are plotted by projecting the total DOS onto each component. (b) Calculated 3D phonon density of InAs as a function of temperature. The atomic stacking is schematically illustrated in (a) to show the two distinct interfaces. The size of the atoms is shown based on their covalent radius.
The InAs/AlAsSb system, specifically, has several attractive features making it a leading candidate: 1) The large QW to barrier energy separation, which is tunable across the solar spectrum, facilitates efficient absorption of the sun’s energy; 2) The degeneracy of the valence band enables efficient hole extraction, while resonant tunneling structures are a reasonable route for fast – energy selective - hot electron extraction in these systems; and 3) Since the photogenerated carriers absorbed directly in the InAs QW are rapidly separated by the type-II band alignment, the loss of photogenerated carriers to photoluminescence is minimized in the QWs. Work is now underway to develop device architectures to further evaluate these systems in practical solar cell devices.
§ ACKNOWLEDGMENTS
The authors would like to acknowledge the contribution of James Dimmock of Sharp Laboratories of Europe Ltd for useful discussions and the critical reading of this manuscript, and Professor Rui Yang (University of Oklahoma) for his insight into the early sample design. The computation was performed at the Extreme Science and Engineering Discovery Environment (XSEDE) and the OU Supercomputing Center for Education & Research (OSCER) at the University of Oklahoma.
§ REFERENCES
|
1511.00131
|
We construct a hyperbolic modular double – an algebra lying in between the Faddeev
modular double for $U_q(sl_2)$ and the elliptic modular double.
The intertwining operator for this algebra leads to an integral operator solution
of the Yang-Baxter equation associated with a generalized Faddeev-Volkov
lattice model introduced by the second author. We describe also the L-operator
and finite-dimensional R-matrices for this model.
§ INTRODUCTION
The representation theory is intimately related to special functions.
The quantum groups and Yang-Baxter equation (YBE) provide a wide class of novel
functions that do not appear in the classical representation theory of Lie groups.
These functions possess a number of peculiar properties and satisfy many intricate
identities which do not have classical counterparts.
The noncompact (or modular) quantum dilogarithm
<cit.> is a remarkable special function significant for a large class
of quantum integrable systems. In particular, it plays a prominent role in the
space-time discretization of the Liouville model and in the construction of
the lattice Virasoro algebra <cit.>, as well as in the investigations
of the XXZ spin chain model in a particular regime <cit.>.
The observation that there exist two mutually commuting Weyl pairs
led Faddeev <cit.> to the notion of a modular double of the quantum algebra $U_q(s\ell_2)$.
It is formed by two copies of $U_q(s\ell_2)$ with different deformation parameters
whose generators mutually (anti)commute with each other.
This doubling enables unambiguous fixing of the representation space of the algebra.
The elliptic modular double introduced by the second author in <cit.> carry over
the idea of doubling to the Sklyanin algebra <cit.>. This doubling is extremely useful.
The symmetry constraints with respect to the extended algebra are much more powerful
as compared to the initial algebra. They enable again unambiguous description of the relevant objects.
The Faddeev-Volkov model <cit.> is a solvable two-dimensional lattice model of statistical
mechanics <cit.>. In contrast to the Ising model,
its spin variables take continuous values. The Boltzmann weights are expressed in terms of
the modular quantum dilogarithm. In <cit.> the free energy per edge
of this model was derived in the thermodynamic limit
using a particular form of the star-triangle relation.
A generalization of the Faddeev-Volkov model has been proposed by the second author in <cit.>.
In this extension the Boltzmann weights are expressed in terms of the modular quantum dilogarithm as well,
but they have more involved form as compared to the original model.
The corresponding star-triangle relation is a degeneration of the elliptic beta integral
evaluation formula <cit.>. The star-triangle relation associated with the
latter integral appeared first in the operator form as main identity
behind the integral Bailey lemma discovered in <cit.> (see also <cit.>
for a detailed discussion) and later it was formulated in the functional form in <cit.>.
In the present work we study an algebraic structure
underlying the generalized Faddeev-Volkov model of <cit.> and related quantum integrable systems.
First we consider a contraction of the Sklyanin algebra described in <cit.>,
which is more general than $U_q(sl_2)$.
Then we show that this symmetry algebra can be enhanced using the doubling construction.
So, we will supplement the algebra with a dual set of generators (anti)commuting with
the initial generators. We baptize the resulting algebra as the hyperbolic modular
double, following the terminology of <cit.> for the modular quantum dilogarithm
considered as a generalization of the Euler gamma function. It lies in between
the elliptic modular double and the modular double of $U_q(s\ell_2)$ in the sense that
the three algebras are arranged in a sequence of contractions.
We will pass naturally from the language of lattice models to the standard
YBE and find the most general solution of YBE having the symmetry of the hyperbolic modular double.
It is an integral operator acting on a pair of infinite-dimensional spaces which are representation
spaces of the latter algebra. An integral operator solution of the YBE
(at the plain non-deformed level)
was constructed for the first time in <cit.>. The factorization property of
the corresponding R-operator was noticed later in <cit.>,
which resulted in a powerful almost purely algebraic techniques of
building general R-operators <cit.>.
Previously in <cit.> we described finite-dimensional solutions of YBE
for the elliptic modular double, modular double of $U_q(s\ell_2)$ and
the Lie group $\mathrm{SL}(2,\mathbb{C})$ in a concise form and elucidated
their factorization property <cit.>.
One of the principal aims of this paper is to find all finite-dimensional solutions of YBE
having the symmetry of the hyperbolic modular double with generic deformation parameter.
However, a detailed construction of the representation theory of the latter algebra
together with the analysis of special functions associated with that, following
the considerations of <cit.> or <cit.> and references therein, is left aside.
We do not discuss modular doubling of quantum
affine algebras as well, this subject was considered recently in <cit.>.
The plan of the paper is as follows. We start in Sect. <ref> with a description
of the hyperbolic gamma function (modular quantum dilogarithm) and state its basic properties.
Then we present the solvable model of statistical mechanics generalizing the Faddeev-Volkov model
and the corresponding hyperbolic star-triangle relation.
In Sect. <ref> we construct an integral R-operator in terms of the Boltzmann weights of this solvable vertex model and show that it satisfies the Yang-Baxter equation. We also rewrite it in the factorized form.
In Sect. <ref> we describe an algebra emerging as a degeneration of the Sklyanin algebra and construct
the corresponding intertwining operator of equivalent representations.
As a natural extension of this quantum algebra we introduce the hyperbolic modular double.
Then we study finite-dimensional irreducible representations of the hyperbolic modular double.
In Sect. <ref> we reduce the integral R-operator to a finite-dimensional
invariant subspace in one of its infinite-dimensional spaces.
In this way we obtain an explicit formula for finite-dimensional solutions of the YBE
with the symmetry of the hyperbolic modular double.
In Sect. <ref> we apply the reduction formula in the
simplest nontrivial setting. We choose the fundamental representation in one of the spaces
and recover the L-operator from the integral R-operator. It automatically takes the factorized form.
Finally, in Sect. <ref> the reduction formula is elaborated further on
by a simplification to finite-dimensional matrix solutions of YBE such that all of
them take a factorized form.
§ A SOLVABLE LATTICE MODEL
Gamma functions are the main building blocks in the construction of special
functions of hypergeometric type. The hierarchy of hyperbolic gamma functions
is formed by particular combinations of two multiple Barnes gamma functions
<cit.> (the standard Jackson's $q$-gamma function <cit.>
is a combination of two Barnes gamma functions of the second order as well,
but we do not consider this function here).
The hyperbolic gamma function of the second order is a homogeneous function
of $u,\omega_1,\omega_2\in{\mathbb{C}}$. For $\text{Re}(\omega_1), \text{Re}(\omega_2)>0$ and
$0<\text{Re}(u)<\text{Re}(\omega_1+\omega_2)$ it has the form
\begin{align}
& \gamma^{(2)}(u;\omega_1,\omega_2):=
\exp\left(-\frac{\pi \textup{i}}{2}B_{2,2}(u;\omega_1,\omega_2)- \right. \notag \\ & \makebox[5em]{} \left.
-\int_{\mathbb{R}+\textup{i} 0}\frac{e^{ux}}
{(1-e^{\omega_1 x})(1-e^{\omega_2 x})}\frac{dx}{x}\right),
\label{hgf}
\end{align}
where $B_{2,2}$ is a multiple Bernoulli polynomial of the second order
\left( \left(u-\frac{\omega_1+\omega_2}{2}\right)^2
Denoting $q=e^{2\pi \textup{i}\omega_1/\omega_2}$ and
$\tilde q=e^{-2\pi \textup{i}\omega_2/\omega_1}$ and
assuming that $|q|<1$, one can write
\exp\left(-\int_{\mathbb{R}+\textup{i} 0}\frac{e^{ux}}
{(1-e^{\omega_1 x})(1-e^{\omega_2 x})}\frac{dx}{x}\right)
=\frac{(e^{2\pi \textup{i} u/\omega_1}\tilde q; \tilde q)_\infty}
{(e^{2\pi \textup{i} u/\omega_2}; q)_\infty},
where $(t;q)_\infty=\prod_{k=0}^\infty(1-tq^k)$.
The modular quantum dilogarithm is usually defined as a function
obtained from (<ref>) by removing the exponential factor involving
$B_{2,2}(u;\omega_1,\omega_2)$, the shift $u\to u+(\omega_1+\omega_2)/2$,
and some renormalization of the variables $u$ and $\omega_{1,2}$.
In particular, in the context of $2d$ conformal field theory it is accepted
to denote $\omega_1=b,\, \omega_2=b^{-1}$.
We shall use the following representation
\begin{align}
& \gamma(z):=\gamma(z;b) :=
\exp \left(- \frac{\textup{i} \pi}{2}\left(z-\frac{b+b^{-1}}{2}\right)^2 +
\right. \notag \\
& \makebox[5em]{} \left. + \frac{\textup{i} \pi}{24}(b^2+b^{-2}) + \int\limits^{+\infty}_{-\infty} \frac{d t}{4t} \frac{e^{t(2z-b-b^{-1})}}{\sin( \textup{i} b t) \sin( \textup{i} b^{-1} t)}
\right),\lb{hypg}
\end{align}
where the integration contour goes above the singularity at $t$ = 0.
One can easily restore the original function
\gamma^{(2)}(z;\omega_1,\omega_2) = \gamma(z/\sqrt{\omega_1\omega_2};\sqrt{\omega_1/\omega_2})\,.
This integral representation is valid for ${\rm Re}(b) > 0$ and $0<{\rm Re}(z)<{\rm Re}(b+b^{-1})$.
The analytic continuation enables one to extend the definition hypg
to a wider range of parameters.
The inverse of this special function is called
also the double sine-function and denoted either $S(u;\omega_1,\omega_2)$
<cit.> or $S_b(z)$ <cit.>. The notation $\gamma^{(2)}(z;\omega_1,\omega_2)$
is taken from <cit.>.
Here we use the terminology suggested in <cit.>.
Interrelations between main known modifications
of function hypg are described in Appendix A of <cit.>.
Various identities for the quantum dilogarithm can be found
in <cit.> and some other papers.
The definition hypg implies that $\gamma(z)$ is invariant under the swap
$b \leftrightarrows b^{-1}$. It satisfies two linear difference equations of
the first order
\begin{align} \lb{period}
\gamma(z+b) = 2 \sin (\pi b z)\, \gamma(z) ,\qquad
\gamma(z+b^{-1}) = 2 \sin (\pi b^{-1} z)\, \gamma(z).
\end{align}
Let us introduce the crossing parameter
\eta := -\frac{b+b^{-1}}{2}.
Then the reflection formula can be written as follows
\begin{align} \label{refl}
\gamma(z)\,\gamma(-2\eta-z) = 1.
\end{align}
The function $\gamma(z)$ is meromorphic. It has a double series of zeros
\begin{align} \lb{zero}
z = b (n+1) + b^{-1} (m+1) ,\;\;\; n,\,m \in \mathbb{Z}_{\geq 0},
\end{align}
and a double series of poles
\begin{align} \lb{pole}
z = - b \,n - b^{-1} m ,\;\;\; n,\,m \in \mathbb{Z}_{\geq 0}.
\end{align}
In the following we deal with multiple products of the hyperbolic gamma function.
In order to avoid lengthy formulae we adopt the convention
\gamma(\pm x + g) := \gamma(x + g) \,\gamma(-x + g)\,,\;\;
\gamma(\pm x \pm y) := \gamma(\pm x + y) \,\gamma(\pm x - y).
In <cit.> a nontrivial generalization of the Faddeev-Volkov model
<cit.> has been proposed.
Such models of statistical mechanics are defined on the square lattice.
The continuous spin variables sit in the lattice vertices.
The rapidity variables are associated with the medial graph built
with the help of pairs of directed lines crossing edges in the middle
with the inclination of 45 or 135 degrees (see, e.g., a picture on Fig. <ref>).
Self-interaction of spins and interactions between the nearest neighboring spins are allowed.
The Boltzmann weight $W(\alpha-\beta; x,y)$ is assigned to a horizontal edge connecting
a pair of vertices with spins $x$, $y$ that is crossed by a pair of
medial graph lines carrying the rapidities $\alpha$ and $\beta$.
Similarly the Boltzmann weight $\overline{W}(\alpha-\beta; x,y)$ is assigned to a
vertical edge connecting a pair of vertices with the spins $x$, $y$ that is crossed by a
pair of medial graph lines carrying rapidities $\alpha$ and $\beta$.
The self-interaction at a vertex $z$ contributes the Boltzmann weight $\rho(z)$ to the partition function.
The edge Boltzmann weights $W$ and $\overline{W}$ of the model <cit.>
are given by fourfold products of hyperbolic gamma functions
and the vertex Boltzmann weight $\rho$ is a product of two hyperbolic gamma functions
\begin{align} \lb{BW}
& W(\alpha;x,y) = \gamma(\alpha-\eta \pm \textup{i} x \pm \textup{i} y) \,, \\ \notag &
\overline{W}(\alpha;x,y) = \gamma(-\alpha \pm \textup{i} x \pm \textup{i} y) \,,\;\;\;
\rho(z) = \frac{1}{2\gamma(\pm 2 \textup{i} z)}\,.
\end{align}
The edge Boltzmann weights depend on the difference of rapidities and they
are symmetric in the spin variables $W(\alpha;x,y) = W(\alpha;y,x)$,
$\overline{W}(\alpha;x,y) = \overline{W}(\alpha;y,x)$.
Let us recall that the Boltzmann weights depend on $b$ which is a temperature-like parameter.
Physical interpretation of $W$ and $\overline{W}$ as true Boltzmann weights
requires their positivity. This is possible in two regimes of the key parameter $b$:
1) $b$ is real and $0<b<1$; 2) $|b|=1,$ Im$(b^2)>0$. In both regimes $\eta<0$
and $\gamma(z)^*=\gamma(z^*)$. As a result one should demand that the spin variables
are real. As to the rapidities, one can set $\eta < \alpha <-\eta$ for $W(\alpha)$
and $0<-\alpha <-2\eta$ for $\overline{W}(\alpha)$.
These constraints should correspond to unitary representations of the
hyperbolic modular double to be described below.
The Boltzmann weights possess a crossing symmetry, i.e. the horizontal and vertical
edge weights are related as follows
\begin{align} \lb{crossing}
\overline{W}(\alpha;x,y) = W(\eta-\alpha;x,y).
\end{align}
We note that in view of the reflection formula refl and quasiperiodicity period
the vertex Boltzmann weight $\rho(z)$ can be rewritten solely in terms of
the trigonometric functions, i.e. its expression in terms of the hyperbolic gamma
functions is overcomplicated. In contrast, the edge Boltzmann weights $W$ and $\overline{W}$
are genuine products of hyperbolic gamma functions and the number of such
functions in the products cannot be reduced.
The formulated model is solvable because of the star-triangle relation depicted in Fig. <ref>.
This relation equates the partition functions of two elementary cells,
\begin{align} \label{HSTR}
\int\limits^{+\infty}_{-\infty} \rho(z) \, \overline{W}(\alpha-\beta;x,z)\, W(\alpha - \gamma;y,z)\, \overline{W}(\beta-\gamma;w,z) \,d z \notag \\ =
\chi(\alpha,\beta,\gamma)\,W(\alpha-\beta;y,w)\,\overline{W}(\alpha-\gamma;x,w) \,
\end{align}
up to a normalization constant $\chi$. Using this example one can see that
the horizontal edge Boltzmann weights $W(\alpha-\beta; x,y)$ depends on the difference $\alpha-\beta$,
where $\alpha$ is the rapidity of the upward directed median line of 45 degrees
and $\beta$ is the rapidity of the upward directed median line of 135 degrees.
For the vertical edge weights $\overline{W}(\alpha-\beta; x,y)$ the situation is similar –
$\alpha$ is the rapidity of the line going to the right of the edge and $\beta$ – of
the line going to the left.
We call identity HSTR with the weights BW the hyperbolic star-triangle relation.
Corresponding normalization constant has the following form
\chi(\alpha,\beta,\gamma) = \gamma\left(2\beta-2\alpha\right)\gamma\left(2\gamma-2\beta\right)
\gamma\left(2\alpha-2\gamma-2\eta\right).
\begin{array}{c}
\includegraphics[width = 4.0 cm]{str-triangLeft.eps}
\end{array} \quad = \quad \chi \;\; \times \quad
\begin{array}{c}\includegraphics[width = 4.0 cm]{str-triangRight.eps}\end{array}
The star-triangle relation.
This local relation enables one to construct a family of commuting
row-to-row transfer matrices and then to
calculate the partition function of the model using the machinery of QISM <cit.>.
The free energy per edge of the model in the thermodynamical limit
has been calculated in <cit.> following the method from <cit.>.
Let us substitute in (<ref>)
\begin{equation}
W(\alpha;x,y)= m(\alpha) W_r(\alpha;x,y), \;\;
\overline{W}(\alpha;x,y)= m(\eta-\alpha)\overline{W}_r(\alpha;x,y)
\label{reno}\end{equation}
and choose the function $m(\alpha)$ in such a way that the normalization constant
$\chi$ on the right-hand side of (<ref>) disappears,
\frac{m(\alpha-\beta)m(\eta-\alpha+\gamma)m(\beta-\gamma)}
{m(\eta-\alpha+\beta)m(\alpha-\gamma)m(\eta-\beta+\gamma)}\, \chi(\alpha,\beta,\gamma) =1.
Ascribe now to the edges the renormalized weights $W_r(\alpha;x,y)$
$\overline{W}_r(\alpha;x,y)$. Then, denoting the total number of edges in the infinitely
growing lattice as $N$, one finds that the free energy per edge
\beta f_{edge} = - \underset{ N\to \infty}{\lim} \frac{\log Z^{(r)}}{N} =0,
where $Z^{(r)}$ is the total partition function for the model with
renormalized Boltzmann weights.
Equivalently, we can keep the original Boltzmann weights intact and
compute the contribution of the renormalizing factors in the asymptotics
of the partition function. Take the finite rectangular lattice with $N$ spins
along the horizontal
line and $M$ spins along the vertical line. Such lattice has $(N-1)M$ horizontal
edges and $N(M-1)$ vertical edges. Therefore the indicated renormalization of
the Boltzmann weights yields a scaling of the partition function
Z_{N,M}= Z_{N,M}^{(r)}\, m(\alpha)^{(N-1)M}\, m(\eta-\alpha)^{N(M-1)},
i.e. the free energy per edge of the original models is
\text{ free energy per edge}= - \underset{ N,M\to \infty}{\lim} \frac{\log Z_{N,M}}{NM}=- \log m(\alpha)m(\eta-\alpha).
It is easy to see that the needed normalization constant $m(\alpha)$ is found
from the equation
As shown in <cit.>, the solution of this equation satisfying
the unitarity relation $m(\alpha)m(-\alpha)=1$
is given by the ratio of two hyperbolic gamma functions of the third
order $\gamma^{(3)}(u;\omega_1,\omega_2,\omega_3)$ for a special choice
of the quasiperiods $\omega_i$. In the current notation it has the following
integral representation
\begin{eqnarray} \nonumber &&
\exp\Big(- \pi \textup{i}\left(\alpha^2+\frac{1}{24}(1-2(b+b^{-1})^2)\right)+
\\ && \makebox[4em]{}
+\frac{1}{8}\int_{\mathbb{R}+\textup{i}0}\frac{e^{4\alpha t}}{\sin(\textup{i}bt) \sin(\textup{i}b^{-1}t)
\cos(\textup{i}(b+b^{-1})t)}
\frac{dt}{t}\Big).
\label{m}\end{eqnarray}
It happens that this result coincides with a similar normalization
factor $m(\alpha)$ for the
original Faddeev-Volkov model derived in <cit.>.
The hyperbolic star-triangle relation HSTR can be written as the integral identity
\begin{align} \label{hypgamma}
\int\limits^{+\infty}_{-\infty} \prod_{k = 1}^{6} \gamma(g_k \pm \textup{i} z) \frac{d z}{2\gamma(\pm 2 \textup{i} z)}
= \prod_{1 \leq j < k \leq 6}\gamma(g_j + g_k),
\end{align}
where parameters $g_k,\, k = 1 , \ldots,6,$ satisfy the
constraints Re$(g_k)>0$ and the balancing condition
\sum_{k = 1}^{6} g_k = -2 \eta.
Note that the condition Re$(g_k)>0$ restricts the
domain of complex values of the rapidities and spins as follows
\text{Re}(\beta-\alpha\pm\textup{i} x), \;
\text{Re}(\alpha-\gamma-\eta\pm\textup{i} y), \;
\text{Re}(\gamma-\beta\pm\textup{i} w) >0.
In principle these restrictions can be relaxed by the analytical continuation
(e.g., it can be reached by a deformation of the integration contour).
The first mathematically rigorous proof of relation (<ref>), and, so,
of the star-triangle relation (<ref>), was obtained in <cit.>.
However, this identity is a special limiting
case of the elliptic beta integral evaluation formula rigorously
established in <cit.>.
Let us note that the exactly computable integral hypgamma is a non-compact
(hyperbolic) analogue of the Rahman (trigonometric) $q$-beta integral <cit.>.
Another important property of the described Boltzmann weights is the unitarity relation.
For real values of the spins $x$ and $y$ one has
\begin{align}
&\int\limits^{+\infty}_{-\infty} \rho(z)\, \overline{W}(\alpha-\beta;x,z)\,
\overline{W}(\beta-\alpha;y,z)\, d z \notag \\
&\kern 35pt = \frac{1}{2\rho(x)} \,\gamma(2\alpha-2\beta)\gamma(2\beta-2\alpha)
\left( \delta(x-y) + \delta(x+y) \right).
\lb{unit}
\end{align}
This identity can be rigorously obtained by taking the limit $\gamma \to \alpha$
in HSTR. In the computation of partition functions of three-dimensional
supersymmetric field theories on the squashed spheres such relations indicate
the chiral symmetry breaking phenomenon <cit.>.
The Boltzmann weights $W$ and $\overline{W}$ satisfy the reflection equation that
is an evident consequence of Eq. refl,
\begin{align} \lb{reflW}
W(\alpha-\beta;x,y) \,W(\beta-\alpha;x,y) = 1.
\end{align}
§ FROM THE LATTICE MODEL TO THE INTEGRAL R-OPERATOR
The star-triangle relations imply integrability of the two-dimensional lattice
models, similar to the case outlined in the previous section. Another wide class
of integrable systems is associated with
the quantum spin chains. Formulation of the latter models in the framework of QISM <cit.>
requires definition of the R-matrix solving the YBE.
In this section we show how to construct such R-matrices associated with the hyperbolic star-triangle
relation HSTR. Since the spin variables sitting in vertices of the two-dimensional lattice
take continuous values (in contrast to the discrete spins of the Ising and chiral Potts models),
on the spin chains side we deal with integral operators instead of the finite-dimensional R-matrices.
In other words, the quantum spaces of the relevant spin chain are infinite-dimensional functional spaces.
Therefore we call solutions of the corresponding YBE the R-operators to emphasize this aspect.
We will indicate in Sect. <ref> that the integral R-operators represent in some sense the most general
YBE solutions, since they embrace all finite-dimensional R-matrices.
Our presentation below follows to some extent the original construction of <cit.>.
Thus we are interested in the integral operator $\mathbb{R}_{12}(u|g_1,g_2)$
defined on the tensor product of two infinite-dimensional function spaces that are representation spaces
with labels (“spins") $g_1$ and $g_2$ (arbitrary complex numbers) of some algebra.
The symmetry algebra underlying the hyperbolic star-triangle relation will be introduced
in Sect. <ref>. The R-operator depends on a complex number $u$ – the spectral parameter and
satisfies the YBE
\begin{align}
&\mathbb{R}_{12}(u-v|g_1,g_2) \,\mathbb{R}_{13}(u|g_1,g_3) \,\mathbb{R}_{23}(v|g_2,g_3) \notag\\ &=
\mathbb{R}_{23}(v|g_2,g_3) \,\mathbb{R}_{13}(u|g_1,g_3)\, \mathbb{R}_{12}(u-v|g_1,g_2).\label{YBE}
\end{align}
Instead of the integral operator $\mathbb{R}_{12}(u|g_1,g_2)$, we can
consider first a more general notation operator $\mathbb{R}_{12}(u_1,u_2|v_1,v_2)$
depending on four complex parameters and satisfying the equation
\begin{align} \lb{YBErap}
&\mathbb{R}_{12}(u_1,u_2|v_1,v_2) \,\mathbb{R}_{13}(u_1,u_2|w_1,w_2) \,\mathbb{R}_{23}(v_1,v_2|w_1,w_2) \notag \\
&=\mathbb{R}_{23}(v_1,v_2|w_1,w_2) \,\mathbb{R}_{13}(u_1,u_2|w_1,w_2)\, \mathbb{R}_{12}(u_1,u_2|v_1,v_2).
\end{align}
This equation is rather similar to YBE. An operator solution of Eq. YBErap can be easily constructed
in terms of the lattice model formulated in the previous section. The parameters $u_1,u_2,v_1,v_2,w_1,w_2$
are identified with the rapidities. The kernel of the integral operator is a product of four edge Boltzmann
weights (two horizontal and two vertical) and two vertex Boltzmann weights,
\begin{align} \label{Rint}
&\bigl[\mathbb{R}_{12}(u_1,u_2|v_1,v_2)\Phi\bigr](z_1,z_2) =
\int\limits^{+\infty}_{-\infty}\int\limits^{+\infty}_{-\infty} \rho(x_1)\rho(x_2) W(u_1-v_2;z_1,z_2)\times \\ &
\overline{W}(u_1-v_1;z_1,x_2)
\overline{W}(u_2-v_2;z_2,x_1)
W(u_2-v_1;x_1,x_2) \Phi(x_1,x_2) d x_1 d x_2. \notag
\end{align}
Thus the kernel of $\mathbb{R}_{12}(u_1,u_2|v_1,v_2)$
is the partition function of an elementary square cell, see Fig. <ref>.
\begin{align*}
&\bigl[\mathbb{R}_{12}(u_1,u_2|v_1,v_2)\, \Phi\bigr](z_1,z_2)= \\[0.2cm] & =
\int\limits^{+\infty}_{-\infty}\int\limits^{+\infty}_{-\infty} \quad \begin{array}{c}\includegraphics[width = 4 cm]{RBW.eps}\end{array}
\Phi(x_1,x_2) \, d x_1 \, d x_2
\end{align*}
The kernel of the integral R-operator is the partition function of an elementary square cell.
Taking into account explicit expressions for the Boltzmann weights we rewrite Eq. Rint
as follows
\begin{align}
= \int\limits^{+\infty}_{-\infty}\int\limits^{+\infty}_{-\infty}
\frac{d x_1 \,d x_2\,\Phi(x_1,x_2)}{4\gamma(\pm 2 \textup{i} x_1)\gamma(\pm 2 \textup{i} x_2)} \times \notag \\
& \makebox[4em]{} \gamma(\pm \textup{i} z_1 \pm \textup{i} z_2 + u_1 -v_2 -\eta)\,
\gamma(\pm \textup{i} x_1 \pm \textup{i} z_2 + v_1-u_1) \times \notag \\
& \makebox[4em]{} \gamma(\pm \textup{i} x_2 \pm \textup{i} z_1 + v_2-u_2)\,\gamma(\pm \textup{i} x_1 \pm \textup{i} x_2 + u_2 - v_1 - \eta).
\lb{explR}
\end{align}
This R-operator corresponds to the generalized Faddeev-Volkov model of <cit.>.
It can be derived from the elliptic hypergeometric R-operator constructed in <cit.> after taking
a particular limit in the parameters, but the rigorous proof of this fact would
require quite intricate techniques. Remind that the latter R-operator
intertwines representations of the Sklyanin algebra.
Let us recall that the vertex Boltzmann weight $\rho$ is in fact a trigonometric function.
The integrand function in the expression explR is a genuine product of 16 hyperbolic gamma functions (modulo a trigonometric multiplier) and their number cannot be reduced.
In <cit.> the R-operator associated with the Faddeev modular double of $U_q(s\ell_2)$
has been constructed in a similar form. It corresponds to the Faddeev-Volkov lattice model itself.
However, the integrand function of the corresponding integral operator solution of YBE is a product
of only 8 hyperbolic gamma functions (up to a trigonometric multiplier).
One can obtain this R-operator rigorously from expression explR by taking appropriate limits
in the parameters such that the $\gamma$-functions depending on the sum $x_1 + x_2$ disappear.
E.g., such a limiting procedure is described in <cit.> for the reduction of
the hyperbolic star-triangle relation down to the original Faddeev-Volkov model case.
Similar to the very first integral R-operator considerations in <cit.>,
it is easy to check that the operator $\mathbb{R}_{12}(u_1,u_2|v_1,v_2)$, Eq. Rint,
solves Eq. YBErap.
In order to demonstrate how it works we resort to graphical representation of
the integral R-operators. In Fig. <ref> at the right top we depicted convolution
of the kernels of the integral operators from
the left-hand side expression in Eq. YBErap
and at the right bottom we depicted convolution of the kernels from the RHS of Eq. YBErap.
External points are marked by numbers $1,2,3$. They denote three quantum spaces.
Integration over internal points is assumed (convolution of the kernels). The dotted lines
are the rapidity lines.
The left-hand and right-hand side expressions in Eq. YBErap are connected to each other by a sequence of moves. Each move is the application of the star-triangle transformation, Eq. HSTR.
Thus the YBE YBErap boils down to a combination of the star-triangle relations HSTR.
Keeping track of the normalization factors $\chi$ arising at each step
one can check that they eventually drop out.
The sequence of star-triangle moves transforming the left-hand side expression in
YBErap to its right-hand form.
However, we still need to understand the algebraic meaning of solution Rint,
which is the primary goal of the present paper. In other words, we have to
give an algebraic interpretation of the rapidities $u_1,u_2,v_1,v_2$. As we will see further
they can be chosen as linear combinations of the spectral parameters $u, v$ and the
representation labels $g_1,g_2$ of a certain algebraic structure which is studied in the next section,
\begin{align} \label{uv}
u_1 =
\frac{u+g_1}{2}\ ,\
u_2 = \frac{u-g_1}{2}\ \ ,\ \
v_1 =
\frac{v+g_2}{2}\ , \
v_2 = \frac{v-g_2}{2}\ .
\end{align}
This relation yields a solution of the YBE in the form YBE,
\mathbb{R}_{12}(u-v|g_1,g_2)= \mathbb{R}_{12}(u_1,u_2|v_1,v_2).
Our considerations below indicate that this is in fact the most general solution of YBE
compatible with certain quantum algebra. The most strong argument follows from the
fact that Eq. Rint embraces all finite-dimensional solutions of the YBE YBE
associated with this algebra.
Factorization of the kernel of the integral operator Rint implies factorization of
the operator itself. Indeed, it can be written as a product of five elementary operators
\begin{align} \lb{R4fact}
\mathbb{R}_{12}(u_1,u_2|v_1,v_2) = \mathrm{P}_{12}\,\mathrm{S}(u_1-v_2)\,\mathrm{M}_2(u_2-v_2)\,
\mathrm{M}_1(u_1-v_1)\,\mathrm{S}(u_2-v_1).
\end{align}
Here $\mathrm{P}_{12}$ is a permutation operator of two tensor factors, i.e. $\mathrm{P}_{12}\Phi(z_1,z_2)\\ = \Phi(z_2,z_1)$.
$\mathrm{S}(u)$ is an operator of multiplication by a particular function,
\begin{align} \lb{S}
\mathrm{S}(u) = W(u;z_1,z_2)= \gamma(\pm \textup{i} z_1 \pm \textup{i} z_2 + u -\eta).
\end{align}
$\mathrm{M}_1$ and $\mathrm{M}_2$ are two copies of the integral operator
\begin{align} \lb{Mbw}
\bigl[\mathrm{M}(g)\,\Phi \bigr](z) = \frac{1}{\gamma(-2g)}\int\limits^{+\infty}_{-\infty} \rho(x) \, \overline{W}(g;z,x) \,\Phi(x)\,d x
\end{align}
acting in the first and second quantum spaces, respectively.
This operator is a degeneration of an elliptic hypergeometric integral operator introduced in <cit.>.
Owing to the chosen definition Mbw, normalizations of the R-operators
R4fact and explR differ by the multiplicative numerical factor
$\gamma(2v_1-2u_1)\gamma(2v_2-2u_2)$. This renormalization of the R-operator
removes certain divergences appearing during the reduction we describe
below and makes this procedure smooth.
At this point we should specify an appropriate function space for the operator $\mathrm{M}$, Eq. Mbw.
Firstly, the kernel of $\mathrm{M}$ is invariant with respect to the reflections
$x \to -x$ and $z \to -z$.
Consequently, $\mathrm{M}$ projects out odd functions and maps onto the space of even functions.
Moreover, calculating the asymptotics of the kernel at $x \to \pm \infty$ we obtain a restriction
on the asymptotic behavior of $\Phi(x)$. Thus we assume that $\Phi$ is an even function, i.e. $\Phi(-x) = \Phi(x)$,
and $e^{4\pi \textup{i} x g}\Phi(x)$ is rapidly decaying at $x \to +\infty$.
In view of the reflection equation reflW and unitarity, Eq. unit, of the Boltzmann weights
the operators $\mathrm{S}$ and $\mathrm{M}$ satisfy (in the space of even functions) the inversion relations
\begin{align} \lb{sq}
\mathrm{S}(u) \, \mathrm{S}(-u) = 1 ,\qquad \mathrm{M}(g)\, \mathrm{M}(-g) = 1.
\end{align}
The second relation is a degeneration of the inversion formula proved in <cit.> for the
elliptic hypergeometric integral operator of <cit.>.
The inversion relation sq for $\mathrm{S}(u)$ is valid for generic values of
$g \in \mathbb{C}$, but for $\mathrm{M}(g)$ it is violated for particular discrete
lattice points on ${\mathbb{C}}$. In the latter case a nontrivial null-space of $\mathrm{M}(g)$
appears which is described in the next section.
Let us note that for generic parameters the operators $\mathrm{S}$, $\mathrm{M}_1$, $\mathrm{M}_2$
provide a twisted representation of generators of the permutation group $S_4$ satisfying the Coxeter
relations. More precisely, the star-triangle relation HSTR can be reformulated as
the cubic Coxeter relations for $\mathrm{S}$, $\mathrm{M}_1$, $\mathrm{M}_2$,
whereas the inversion relations (<ref>) represent quadratic Coxeter relations.
For more details on this interpretation, see the end of Sect. <ref> and
<cit.> where the allied constructions are elaborated in detail.
Finally we note that the identities sq lead to the unitarity-like relation for the R-operator R4fact,
\begin{align}
\mathbb{R}_{12}(u|g_1,g_2)\,\mathbb{R}_{12}(-u|g_2,g_1) = 1.
\end{align}
§ THE INTERTWINING OPERATOR OF A DEGENERATE SKLYANIN ALGEBRA
In this section we study in detail the operator $\mathrm{M}$, Eq. Mbw, in order to infer its algebraic meaning.
We rewrite Eq. Mbw explicitly in terms of the hyperbolic gamma functions
\begin{align} \lb{M}
\bigl[\mathrm{M}(g)\,\Phi \bigr](z) = \frac{1}{2}\int\limits^{+\infty}_{-\infty}
\frac{\gamma(-g \pm \textup{i} z \pm \textup{i} x)}
{\gamma(\pm 2 \textup{i} x)\gamma(-2g)} \Phi(x)\,d x\, .
\end{align}
For certain discrete values of $g$ the operator $\mathrm{M}(g)$ simplifies considerably.
First of all at the origin $g=0$ it is the identity operator, $\mathrm{M}(0) =1$,
which can be seen by taking the limit $g\to 0$ after a simple residue calculus.
Moreover one can easily check that $\mathrm{M}(g)$ respects a pair of the
contiguous relations involving the shifts of $g$ by $\frac{b}{2}$ and $\frac{1}{2b}$,
\begin{eqnarray}\label{rec1}
- \frac{\textup{i}}{\sin(2\pi \textup{i} b z)} \,\sin({\textstyle \frac{b}{2}\dd_z}) \,
\mathrm{M}(g) = \textstyle \mathrm{M}(g+\frac{b}{2}) ,
\\ &&
- \frac{\textup{i}}{\sin(2\pi \textup{i} b^{-1} z)} \,\sin({\textstyle\frac{1}{2b}\dd_z}) \,
\mathrm{M}(g) = \textstyle \mathrm{M}(g+\frac{1}{2b}).
\label{rec2}\end{eqnarray}
Elliptic analogues of these relations can be found in <cit.>.
Applying recurrences (<ref>), (<ref>) for constructing $\mathrm{M}(g)$ in the
discrete quarter-infinite lattice of points $g=\frac{n b}{2}+\frac{m}{2b}$,
$n,\,m \in \mathbb{Z}_{\geq 0}$, we find that the integral operator (<ref>)
is converted to a product of $n+m$ finite-difference operators of the first order
\begin{align} \lb{Mfact}
\mathrm{M}({\textstyle \frac{n b}{2}+\frac{m}{2b}}) =
\left[ \frac{-\textup{i}}{\sin(2\pi \textup{i} b z)} \,\sin({\textstyle \frac{b}{2}\dd_z}) \right]^{n}
\left[ \frac{-\textup{i}}{\sin(2\pi \textup{i} b^{-1} z)} \,\sin({\textstyle \frac{1}{2b}\dd_z}) \right]^{m}.
\end{align}
We have already mentioned above that the R-operator Rint
is related to a certain quantum algebra. Let us now specify it explicitly.
It is a contraction of the Sklyanin algebra <cit.> that has been introduced
in <cit.> and then investigated in <cit.>.
This degenerate Sklyanin algebra is formed by four generators $A,B,C,D$ which
respect the following commutation relations
\begin{align}
& C \,A = e^{\textup{i} \pi b^2} \,A\, C \, ,\;\;
D \,C = e^{\textup{i} \pi b^2} \,C\, D \,, \notag \\
& [\,A\,,\,D\,] = -2 \textup{i} \sin^3 \pi b^2 \, C^2\, ,\;\;
[\,B\,,\,C\,] = \frac{A^2-D^2}{2 \textup{i} \sin \pi b^2} \,, \notag \\
& A\,B - e^{\textup{i} \pi b^2}\, B\,A = e^{\textup{i} \pi b^2} \, D \, B -B\,D
= \frac{\textup{i}}{2} \sin 2 \pi b^2 \, (C\,A-D\,C)\,.\lb{alg}
\end{align}
It has a pair of Casimir operators
\begin{align}
&K_0 = e^{\textup{i} \pi b^2} A\, D - \sin^2 \pi b^2 \, C^2 ,\notag\\
&K_1 = \frac{e^{-\textup{i} \pi b^2} A^2 + e^{\textup{i} \pi b^2} D^2}{4\sin^2 \pi b^2} -B\, C -\frac{1}{2}\cos \pi b^2 \, C^2 \label{Cas}
\end{align}
commuting with all four generators. This algebra is different from the conventional
quantum deformation of the rank 1 Lie algebra, $U_q(s\ell_2)$ <cit.>, in particular,
it does not obey the Hopf algebra structure. As shown in <cit.>,
the spectral problem for a special quadratic combination of the generators
of this algebra reproduces the eigenvalue problem for the Askey-Wilson
polynomials. Therefore this algebra comprises the Zhedanov algebra
as well <cit.>, which was constructed precisely with the aim of
interpreting Askey-Wilson polynomials as a representation space of some
quadratic algebra.
The Sklyanin algebra possesses a representation by finite-difference operators with elliptic
coefficients which depend on an arbitrary complex parameter $g$ labeling representations <cit.>.
Particular linear combinations of this algebra generators in a degeneration limit,
such that the elliptic nome goes to zero and the elliptic functions reduce to trigonometric ones,
take the form (the details of this procedure can be found in <cit.>, see also <cit.>)
\begin{align}
&A(g)= \frac{\textup{i}e^{\frac{\textup{i}\pi}{2} b^2}}{2} \frac{e^{-\pi \textup{i} b g}}{\sin 2\pi \textup{i} b z}
\left[ e^{2 \pi b z} e^{\frac{\textup{i} b}{2}\dd_z} - e^{-2 \pi b z} e^{-\frac{\textup{i} b}{2}\dd_z} \right] ,\lb{rep1} \\
&B(g)= -\frac{1}{2}\cos \pi b^2 \, C(g) -\frac{1}{4\sin \pi b^2} \frac{1}{\sin 2 \pi \textup{i} b z} \times \notag \\
& \kern 32pt
\left[ \cos\pi b(2g + 4 \textup{i} z - b) \,e^{\frac{\textup{i} b}{2}\dd_z}
- \cos\pi b(2g - 4 \textup{i} z - b) \,e^{-\frac{\textup{i} b}{2}\dd_z} \right], \lb{rep2} \\
&C(g)= \frac{1}{2\sin \pi b^2} \frac{1}{\sin 2\pi \textup{i} b z}
\left[ e^{\frac{\textup{i} b}{2}\dd_z} - e^{-\frac{\textup{i} b}{2}\dd_z} \right] , \lb{rep3} \\
&D(g)= - \frac{\textup{i}e^{-\frac{\textup{i}\pi}{2} b^2}}{2} \frac{e^{\pi \textup{i} b g}}{\sin 2\pi \textup{i} b z}
\left[ e^{-2 \pi b z} e^{\frac{\textup{i} b}{2}\dd_z} - e^{2 \pi b z} e^{-\frac{\textup{i} b}{2}\dd_z} \right]. \lb{rep4}
\end{align}
These operators satisfy defining relations (<ref>).
In this representation the Casimir operators Cas take the values
K_0(g) = e^{\textup{i} \pi b^2} ,\qquad K_1(g) = \frac{\cos 2\pi b g}{2\sin^2 \pi b^2}.
We can construct Verma module representations of the algebra alg following an
analogy with the $s\ell_2$ algebra.
We choose $|0 \rangle = 1$ to be a lowest weight vector in the representation annihilated by the
lowering operator $C$, $C |0 \rangle = 0$. In order to obtain the basis of the Verma module
we act by the raising operator $B$ on the lowest weight vector a number of times:
$| k \rangle := B^k \,| 0 \rangle,\, k\in \mathbb{Z}_{\geq 0}$.
Using relations alg one can check that
\begin{align}
& A \, | k \rangle = \sum_{ l = 0}^{[k/2]} a_{k,l}(g) | k - 2 l \rangle \,,\;\;
D \, | k \rangle = \sum_{ l = 0}^{[k/2]} d_{k,l}(g) | k - 2 l \rangle \,, \notag\\
& C \, | k \rangle = \sum_{ l = 0}^{[\frac{k-1}{2}]} c_{k,l}(g) | k - 1 - 2 l \rangle \,, \label{Verma}
\end{align}
where $a_{k,l},\,d_{k,l},\,c_{k,l}$ are some functions of $g$.
Contrary to the familiar situation of $s\ell_2$ the lowering operator
$C$ acting on the vector $| k \rangle$ produces not $| k-1 \rangle$ but a linear combination of
vectors with descending weights $k-1, k-3, k-5,\ldots$. Similarly, the operators $A$ and $D$ are not diagonal
contrary to their counterpart in $s\ell_2$. Acting on the vector $|k \rangle$ they mix it
with the vectors having descending weights $k-2, k-4, k-6, \ldots$.
For the chosen trigonometric polynomial realization, the vector $| k \rangle$ is
a linear combination of $\cos(2\pi \textup{i} j b z)$, where $j = k, k-2 ,k-4,\ldots, 1$ (or 0).
For the generic values of $g$ the representation is infinite-dimensional.
However, if $g = (n+1)\frac{b}{2},\, n \in \mathbb{Z}_{\geq 0},$ the situation drastically changes.
Then $| n+1 \rangle$ is a linear combination of $\{| k \rangle\}_{k=0}^{n}$. Acting
by powers of the raising operator $B$ on the vector $| n \rangle$ we do not get out of the $n$-dimensional space.
In order to avoid misunderstanding we note that $C\,| n+1 \rangle \neq 0$, unlike the $s\ell_2$ case.
Since representations with the labels $g$ and $-g$ have the same values of the Casimir
operators, they are equivalent. Indeed, they are intertwined by
the operator $\mathrm{M}(g)$, Eq. M, as follows from the relations
\begin{align}\notag
&\mathrm{M}(g) \,A(g) = A(-g)\, \mathrm{M}(g),\;\;
\mathrm{M}(g) \,B(g) = B(-g)\, \mathrm{M}(g),\\
&\mathrm{M}(g) \,C(g) = C(-g)\, \mathrm{M}(g),\;\;
\mathrm{M}(g) \,D(g) = D(-g)\, \mathrm{M}(g),
\lb{intw1}\end{align}
which can be checked by an explicit calculation.
The operator $\mathrm{M}(g)$ is invariant under the swap $b \leftrightarrows b^{-1}$.
Therefore it is natural to introduce the second set of generators $\widetilde{A},\widetilde{B},\widetilde{C},\widetilde{D}$
respecting the commutation relations alg with the replacement $b \to b^{-1}$, i.e.
\begin{align}
&\widetilde{C} \,\widetilde{A} = e^{\frac{\textup{i} \pi}{b^2}} \,\widetilde{A}\, \widetilde{C} \,,\;\;
\widetilde{D} \,\widetilde{C} = e^{\frac{\textup{i} \pi}{b^2}} \,\widetilde{C}\, \widetilde{D} \,, \notag \\ &
[\,\widetilde{A}\,,\,\widetilde{D}\,] = -2 \textup{i} \sin^3 \frac{\pi}{b^2} \, \widetilde{C}^2 \,, \;\;
[\,\widetilde{B}\,,\,\widetilde{C}\,] = \frac{\widetilde{A}^2-\widetilde{D}^2}{2 \textup{i} \sin \frac{\pi}{b^2}} \,, \notag\\ &
\widetilde{A}\,\widetilde{B} - e^{\frac{\textup{i} \pi}{b^2}}\, \widetilde{B}\,\widetilde{A}
= e^{\frac{\textup{i} \pi}{b^2}} \, \widetilde{D} \, \widetilde{B} -\widetilde{B}\,\widetilde{D}
= \frac{\textup{i}}{2} \sin \frac{2 \pi}{b^2} \, (\widetilde{C}\,\widetilde{A}-\widetilde{D}\,\widetilde{C}).\lb{algtilde}
\end{align}
We also need to specify the commutation relations for generators from different sets.
The generators $A,D$ anticommute with $\widetilde{B},\widetilde{C}$;
the generators $B,C$ anticommute with $\widetilde{A},\widetilde{D}$;
the generators $A,D$ commute with $\widetilde{A},\widetilde{D}$;
and the generators $B,C$ commute with $\widetilde{B},\widetilde{C}$.
An explicit realization of the generators $\widetilde{A},\widetilde{B},
\widetilde{C},\widetilde{D}$ by finite-difference operators is given by the formulae
rep1–rep4, where $b$ should be replaced by $b^{-1}$.
New Casimir operators have the form
\begin{align}
&\widetilde{K}_0 = e^{\textup{i} \pi/b^2} \widetilde{A}\, \widetilde{D}
- \sin^2 \pi/b^2 \, \widetilde{C}^2 = e^{\textup{i} \pi/b^2} ,
\notag\\
&\widetilde{K}_1 = \frac{e^{-\textup{i} \pi/ b^2} \widetilde{A}^2
+ e^{\textup{i} \pi/b^2}\widetilde{D}^2}{4\sin^2 \pi/ b^2}
-\widetilde{B}\, \widetilde{C} -\frac{1}{2}\cos \pi/b^2 \, \widetilde{C}^2
= \frac{\cos 2\pi g/b}{2\sin^2 \pi/b^2}.
\label{Cas2}
\end{align}
Taken together, two sets of generators $A,B,C,D$
and $\widetilde{A},\widetilde{B},\widetilde{C},\widetilde{D}$ form the
algebra which we call the hyperbolic modular double.
It lies in between the Faddeev's modular double of $U_q(s\ell_2)$ <cit.> and the elliptic modular double <cit.>
in the sense that these algebraic structures are related by a sequence of contractions
\begin{align*}
&\text{Elliptic modular double} \to \\ & \kern 4em \to \text{Hyperbolic modular double} \to \\ & \kern 9em \to \text{Modular double of $U_q(s\ell_2)$}.
\end{align*}
Using particular combinations of the generators of this algebra similar to the one
considered in <cit.>, it is possible to construct
a modular double of the Zhedanov algebra <cit.> as well.
Evidently, $\mathrm{M}(g)$ works as the intertwining operator for the second set of
generators as well,
\begin{align}\notag
\mathrm{M}(g) \,\widetilde{A}(g) = \widetilde{A}(-g)\, \mathrm{M}(g),\;\;
\mathrm{M}(g) \,\widetilde{B}(g) = \widetilde{B}(-g)\, \mathrm{M}(g),\\
\mathrm{M}(g) \,\widetilde{C}(g) = \widetilde{C}(-g)\, \mathrm{M}(g),\;\;
\mathrm{M}(g) \,\widetilde{D}(g) = \widetilde{D}(-g)\, \mathrm{M}(g).
\lb{intw2}\end{align}
Irreducible representations of the hyperbolic modular double are
fixed by one complex number $g$, with the $g$ and $-g$ label representations
being equivalent. Realization of the generators of this algebra in the space
of analytical functions is unique (up to a multiplication by a numerical factor),
because solutions of a system of finite-difference equations with the shifts by $b$
and $b^{-1}$ (which can be taken real and incommensurate) are determined up to
multiplication by a number. Relations (<ref>) and (<ref>) are natural
extensions of the intertwining relations for the $U_q(sl_2)$ algebra and its
modular double derived by Ponsot and Teschner <cit.>. In the limit described
in <cit.>, when the hyperbolic R-matrix is degenerated to that of
Faddeev and Volkov, the integral operator $\mathrm{M}(g)$ passes to
the intertwining operator of <cit.>.
Intertwining operators are quite useful, since they enable one to get insight
to the structure of representations of the corresponding algebra.
Indeed, the null-space of $\mathrm{M}(g)$, $\mathrm{Ker}\,\mathrm{M}(g)$, and the image of $\mathrm{M}(-g)$, $\mathrm{Im}\,\mathrm{M}(-g)$,
are invariant spaces of the representation with the label $g$ that follows from intw1 and intw2.
The inversion formula sq implies that
\mathrm{M}_{z}(g) \gamma(g \pm \textup{i} x \pm \textup{i} z) = 0 ,\; \text{where} \; g = \frac{n b}{2} + \frac{m}{2b}
,\; n, m \in \mathbb{Z}_{\geq 0},\; (n,m) \neq (0,0),
since the normalization factor $1/\gamma(2g)$ of $\mathrm{M}(-g)$ is divergent at
the specified values of $g$ (see Eq. zero). Here the subindex in the operator
$\mathrm{M}_{z}$ indicates that $z$ is used as the integration variable.
Hence, expanding $\gamma(g \pm \textup{i} x \pm \textup{i} z)$ in $x$ we recover the null-space of $\mathrm{M}_{z}(g)$.
Moreover, the function $\gamma(g \pm \textup{i} x \pm \textup{i} z)$ is proportional to the
integrand of the operator $\mathrm{M}(-g)$.
Consequently, its expansion in $x$ lies in the image of $\mathrm{M}(-g)$.
Let us remind that for these values of $g$ the integral operator
$\mathrm{M}(g)$ turns to the finite-difference operator Mfact.
In the following we will be interested in irreducible finite-dimensional representations of the hyperbolic modular double at
\begin{align} \lb{g-lat}
g_{n,m} = \frac{b}{2} (n+1) + \frac{1}{2b} (m+1),\;\; n,m \in\mathbb{Z}_{\geq 0},
\end{align}
which have the dimension $(n+1)(m+1)$. They are realized in the invariant space
$$\mathrm{Ker}\,\mathrm{M}(g_{n,m}) \cap\mathrm{Im}\,\mathrm{M}_{ren}(-g_{n,m}),$$
where $\mathrm{M}_{ren}(-g)=\gamma(2g)\mathrm{M}(-g)$.
All basis vectors of the finite-dimensional irreducible representation are embraced by the generating function
\textstyle \gamma(\pm \textup{i} x \pm \textup{i} z + g_{n,m}),
where $x$ is an auxiliary parameter.
Indeed, owing to Eqs. period and refl, it turns into the finite product of
trigonometric functions
\begin{align} \lb{gen-fun}
&\gamma(\pm \textup{i} x \pm \textup{i} z + g_{n,m})= \\ & \kern 4em
\prod_{r = 0}^{n-1}2\sin \pi b \textstyle (\pm \textup{i} x + \textup{i} z + \frac{b}{2}(n-1-2r)+\frac{1}{2b}(m+1)) \times \notag\\
& \kern 4em \prod_{s = 0}^{m-1}2\sin \textstyle \frac{\pi}{b}(\pm \textup{i} x + \textup{i} z + \frac{1}{2b}(m-1-2s)-\frac{b}{2}(n-1)). \notag
\end{align}
From the latter formula we extract the natural basis of the finite-dimen-sional representation
\cos(2j\pi \textup{i} b z) \cos (2l \pi\textup{i}z/b) ,\;\; j = 0 ,1, \ldots, n ,
\;\; l =0 ,1, \ldots, m .
Note that the generating function coincides with the edge Boltzmann weight BW.
§ REDUCTIONS OF THE INTEGRAL R-OPERATOR
In this section we show that the integral operator solution, Eq. Rint, of the YBE YBE
enables one to recover all finite-dimensional solutions of YBE as well.
In order to do it we apply the operator $\mathbb{R}_{12}(u|g_1,g_2)$ to the function
$\gamma(\pm \textup{i} z_1 \pm \textup{i} z_3 + u_1-u_2)\Phi(z_2)$, where $z_3$ is
an auxiliary parameter and $\Phi(z_2)$ is an arbitrary function from the second space.
For $g_1 = g_{n,m}$ the first factor turns to the generating function, Eq. gen-fun.
Temporarily we assume $g_1$ to be generic.
Computation of the result of this action is pictorially presented in Fig. <ref>,
where we slightly changed the graphical rules.
Now all edges represent the Boltzmann weights $\overline{W}$, Eq. BW.
The black blob corresponds to the vertex Boltzmann weight $\rho$.
We omit rapidity lines and indicate corresponding differences of the rapidities explicitly.
At the first step we apply the star-triangle relation, Eq. HSTR, implementing
integration at the vertex $x_1$. Thus the only integration left is at the vertex
$x_2$. It corresponds to the integral operator $\mathrm{M}_1(u_2-v_2)$
with the kernel $\rho(x_2)\,\overline{W}(u_2-v_2;z_1,x_2)$ which acts on the product
At the second step we just rearrange the factors such that we gain the integral operator $\mathrm{M}_2(u_1-u_2+\eta)$
with the kernel $\rho(x_2)\,\overline{W}(u_1-u_2+\eta;z_2,x_2)$
which acts on the product $\overline{W}(u_1-v_2;z_1,z_2)\,\overline{W}(v_1-u_1+\eta;z_2,z_3)\,\Phi(z_2)$.
The sequence of transformations converting the integral operator
$\mathbb{R}_{12}(u|g_1,g_2)$ at $g_1 = g_{n,m}$ to a finite-dimensional matrix in the first space.
Then we note that the remaining integral operator
$\mathrm{M}_2(u_1-u_2+\eta) = \mathrm{M}_2(g_1+\eta)$ for $g_1 = g_{n,m}$
turns to the finite-difference operator
$\mathrm{M}_2({\textstyle \frac{nb}{2}+\frac{m}{2b}})$, Eq. Mfact.
Thus we have obtained the reduction formula which encompasses all solutions of the YBE YBE
that have the symmetry of the hyperbolic modular double and
are realized on the tensor product of the finite-dimensional representation
with the label $g_{n,m}$, Eq. g-lat, in the first space, and arbitrary infinite-dimensional
representation with the label $g$ in the second one,
\begin{align}\lb{reduct}
&\textstyle \mathbb{R}_{12}(u|g_{n,m},g) \,\gamma(\pm \textup{i} z_1 \pm \textup{i}
z_3 + \frac{b}{2} (n+1)+ \frac{1}{2b}(m+1))\,\Phi(z_2) \\ &
= c \cdot \frac{\gamma(\pm \textup{i} z_2 \pm \textup{i} z_3 + \frac{-u+g_{n,m}+g}{2})}
{\gamma(\pm \textup{i} z_1 \pm \textup{i} z_3 + \frac{-u-g_{n,m}-g-2\eta}{2})}\times \notag \\ & \kern 4em
\mathrm{M}_2({\textstyle \frac{nb}{2}+\frac{m}{2b}})\,\frac{\gamma(\pm \textup{i} z_1 \pm \textup{i} z_2 + \frac{-u+g_{n,m}-g}{2})}
{\gamma(\pm \textup{i} z_2 \pm \textup{i} z_3 + \frac{-u-g_{n,m}+g-2\eta}{2})}\notag
\,\Phi(z_2),
\end{align}
c=\frac{1}{\gamma(u+g_{n,m}\pm g)}.
The same result can be obtained using the fusion following the procedure
described in <cit.>.
Expanding both sides of this formula in the auxiliary parameter $z_3$ we recover
the reduced R-operator that is a matrix whose entries are some
finite-difference operators acting in the second space, i.e. we have the L-operator.
One can straightforwardly reduce further the L-operator (<ref>) to R-matrices
which are finite-dimensional in both spaces. In order to achieve it we just need to
force the representation
label $g$ of the second space to lie on the second copy of the lattice g-lat.
Remarkably, the factorized form of the integrand function of the integral operator
Rint is inherited by the reduced R-operator.
We will see in Sect. <ref> and <ref> that the reduced
solution of YBE reduct can be further arranged to the factorized product
of matrices form.
In <cit.> an analogous reduction formula has been derived for the integral
R-operators in the following three cases: 1) the R-operator with the symmetry group
$\mathrm{SL}(2,\mathbb{C})$; 2) solutions of YBE with the symmetry of the modular
double of $U_q(s\ell_2)$; 3) the most general known R-operator obeying
the symmetry of the elliptic modular double (and of the Sklyanin algebra, of course).
§ THE FUNDAMENTAL REPRESENTATION $\ML$-OPERATOR AND ITS FACTORIZATION
Let us show how formula reduct works in practice.
We consider the simplest nontrivial representation in the first space
$g_1 = g_{1,0} = b+\frac{1}{2b}$ (see Eq. g-lat), i.e. the fundamental
representation of the $(A,B,C,D)$-generated part of the hyperbolic double and trivial
representation for the $(\widetilde{A},\widetilde{B},\widetilde{C},\widetilde{D})$-part.
Corresponding solution of the YBE is known as the (spin $\frac{1}{2}$) L-operator.
The generating function in this case has the following form (see Eq. gen-fun)
\textstyle
\gamma(\pm \textup{i} z_1 \pm \textup{i} z_3 + b + \frac{1}{2b}) = 2 \cos 2\pi \textup{i} b z_1 + 2 \cos 2\pi \textup{i} b z_3 =
\mathbf{e}_1 \, 2 \cos 2\pi \textup{i} b z_3 + \mathbf{e}_2,
where the basis of the 2-dimensional representation in the first space $\mathbb{C}^2$ is formed by
$\mathbf{e}_1 = 1$ and $\mathbf{e}_2=2\cos 2 \pi \textup{i} b z_1$.
The intertwining operator from reduct simplifies to
$\mathrm{M}_2(\frac{b}{2}) = c \cdot \frac{1}{\sin 2 \pi \textup{i} b z_2}
\left( e^{\frac{\textup{i}b}{2}\dd_2} - e^{-\frac{\textup{i}b}{2}\dd_2} \right)$ (see Eq. Mfact).
To simplify the formulae we shift the spectral parameter $u \to u+\frac{b}{2}$.
Now we wish to rewrite formula reduct in a matrix form.
In the formula reduct we pull the hyperbolic gamma functions depending
on $\pm \textup{i} z_1 \pm \textup{i} z_2$ to the left and the hyperbolic gamma functions
depending on $\pm \textup{i} z_2 \pm \textup{i} z_3$ to the right.
Then we simplify them by means of Eqs. period and refl.
Thus the right-hand side expression in reduct takes the form
\begin{align}
&\Bigl[ 2\cos 2 \pi \textup{i} b z_1 - 2\cos\pi b ( 2 \textup{i} z_2 + u + g) \Bigr] \times \notag \\
&\kern 4em e^{\frac{\textup{i}b}{2}\dd_2}
\Bigl[ 2\cos 2 \pi \textup{i} b z_3 - 2\cos\pi b ( 2 \textup{i} z_2 - u + g) \Bigr] \notag \\
&- \Bigl[ 2\cos 2 \pi \textup{i} b z_1 - 2\cos\pi b ( 2 \textup{i} z_2 - u - g) \Bigr] \times \notag \\
&\kern 3.5em e^{-\frac{\textup{i}b}{2}\dd_2}
\Bigl[ 2\cos 2 \pi \textup{i} b z_3 - 2\cos\pi b ( 2 \textup{i} z_2 + u - g) \Bigr].
\end{align}
Now it is straightforward to rewrite the reduced R-operator
$\mathbb{R}_{12}(u+\frac{b}{2}|g_{1,0},g) =: \mathrm{L}(u|g)$
in a matrix form in the basis $\{\mathbf{e}_1,\mathbf{e}_2 \}$ of $\mathbb{C}^2$,
using the definition of matrix elements $\mathrm{L}(u|g)\, \mathbf{e}_k :=
\sum_{i}\mathbf{e}_i \left[\mathrm{L}(u|g)\right]_{i,k}$,
\begin{align}
&\mathrm{L}(u|g) = \frac{1}{\sin 2 \pi \textup{i} b z}
\begin{pmatrix}
-2 \cos 2 \pi b (\textup{i} z + u_1) & -2 \cos 2 \pi b (\textup{i} z - u_1) \notag\\
1 & 1
\end{pmatrix} \times \\ & \kern 70 pt \times
\begin{pmatrix}
e^{\frac{\textup{i}b}{2}\dd} & 0 \\
0 & -e^{-\frac{\textup{i}b}{2}\dd}
\end{pmatrix}
\begin{pmatrix}
1 & -2 \cos 2 \pi b (\textup{i} z - u_2) \\
1 & -2 \cos 2 \pi b (\textup{i} z + u_2)
\end{pmatrix}.\lb{Lax}
\end{align}
Here we substituted $z_2 \to z$.
The rapidities $u_1,u_2$ are defined as $u_1 = \frac{u+g}{2}$ and $u_2 = \frac{u-g}{2}$ (recall Eq. uv).
We stress that the $\mL$-operator is automatically obtained in the factorized form,
Eq. Lax. This might be expected since the initial formula reduct obeys
similar factorization.
Choosing in the $\mL$-operator the second space representation label as
$g =g_{1,0} = b+\frac{1}{2b}$, i.e. restricting it to the fundamental representation
as well, we recover a $4 \times 4$ matrix solution of the YBE YBE.
This solution differs from the standard trigonometric R-matrix with 6 nonzero entries <cit.>.
It is the R-matrix of the 7-vertex model <cit.>.
This factorized representation is analogous to the one found in <cit.>
(compare Eq. Lax with the normal-ordered factorized $\mL$-operator Eq. (2.20)
in <cit.>). There the lateral matrices are identified with the trigonometric
intertwining vectors that provide the vertex-face correspondence between the
7-vertex model and a trigonometric SOS model.
The $\mL$-operator, Eq. Lax, can be written in terms of the
degenerate Sklyanin algebra generators as well,
\begin{align} \lb{genLax}
&\mathrm{L}(u|g) =
- e^{-\textup{i} \pi b u} \, A(g) - e^{\textup{i} \pi b u}\, D(g) \\
\sin \pi b^2 \,C(g)
\end{array}\right. \\ \notag
&\kern 3em \left.\begin{array}{c}
- 4 \sin \pi b^2 \, B(g) - 2 \sin \pi b^2 (\cos 2\pi b u + \cos \pi b^2) \,C(g) \\
e^{\textup{i} \pi b u} \, A(g) + e^{-\textup{i} \pi b u}\, D(g)
\end{array}\right).
\end{align}
In <cit.> this L-operator genLax has been identified
with the quantum L-operator for the 2-particle trigonometric Ruijsenaars model.
Taking $g_1 = g_{0,1} = \frac{b}{2}+\frac{1}{b}$ one recovers in the same way
the second $\mL$-operator $\widetilde{\mL}(u|g)$ whose entries are generators
of the second half of the hyperbolic modular double.
Thus we see that our construction is self-consistent. A particular reduction of
the integral R-operator results in the generators of the hyperbolic modular double.
Consequently the latter quantum algebra is indeed the symmetry algebra of the
integral R-operator explR.
The algebraic interpretation of the rapidities stated above uv is correct.
The infinite-dimensional spaces in the construction of the integral R-operator
are equipped with the structure of representations of the hyperbolic modular double.
Implementing the reduction condition $g_3 = g_{1,0}$ in the YBE YBE we obtain
an $\mathrm{RLL}$-relation,
\begin{align}
& \mathbb{R}_{12}(u-v|g_1,g_2)\,\mathrm{L}_1(u|g_1)\,\mathrm{L}_2(v|g_2) = \notag \\ &
=\mathrm{L}_2(v|g_2)\,\mathrm{L}_1(u|g_1)\,\mathbb{R}_{12}(u-v|g_1,g_2). \label{RLL}
\end{align}
Here the integral R-operator explR acts in a pair of infinite-dimensional spaces with
the representation labels $g_1$ and $g_2$ and intertwines the matrix product of two
L-operators. The lower index (1 or 2) of the L-operator enumerates the
infinite-dimensional spaces where it acts nontrivially, i.e. the entries of
$\mathrm{L}_i$ (see Eq. Lax) are some difference operators in the variable $z_i$.
The RLL-relation RLL can be rewritten in terms of the rapidities as well
in a full analogy with Eq. YBErap,
\begin{align}
&\mathbb{R}_{12}(u_1,u_2|v_1,v_2)\,\mathrm{L}_1(u_1,u_2)\,\mathrm{L}_2(v_1,v_2)= \notag \\
\end{align}
where $\mathrm{L}(u_1,u_2) := \mL(u|g_1)$,
$\mathrm{L}(v_1,v_2) := \mL(v|g_2)$ (recall Eq. uv).
Now we can give a natural interpretation of the operator $\mathrm{S}(u)$, Eq. S,
which is one of the factors of the R-operator R4fact.
It implements the permutation of rapidities $(u_1,u_2,v_1,v_2) \mapsto (u_1,v_1,u_2,v_2)$
in the matrix product of two L-operators, i.e.
\mathrm{S}(u_2-v_1)\,\mL_1(u_1,u_2)\,\mL_2(v_1,v_2) = \mL_1(u_1,v_1)\,\mL_2(u_2,v_2)\,\mathrm{S}(u_2-v_1).
This statement can be checked by a straightforward calculation.
On the other hand, the R-operator itself implements the permutation of
the rapidities $(u_1,u_2,v_1,v_2) \mapsto (v_1,v_2,u_1,u_2)$
in Eq. RLLrap. For more details of such permutation of parameters
in various models, see <cit.>.
§ FACTORIZED FINITE-DIMENSIONAL SOLUTIONS OF THE YBE
In the previous section we have shown that the reduction formula reduct
produces the $\mL$-operator in the factorized form from the fundamental
representation in the first space for the integral R-operator. Now we are
going to demonstrate that the same pattern persists for all finite-dimensional
representations, i.e. we show that the higher-spin solutions of the YBE can be
factorized as well. Finite-dimensional representations of the hyperbolic modular
double naturally factorize to products of finite-dimensional representations of
its two halves. Therefore without loss of generality we can consider
nontrivial representations for only one of its halves.
Thus, we choose $g_1 = g_{n,0} = \frac{b}{2} (n+1) + \frac{1}{2b}$,
$n\in \mathbb{Z}_{\geq 0}$, (recall Eq. g-lat) in the reduction formula reduct.
The generating function of the $(n+1)$-dimensional
representation of interest takes the form (recall Eq. gen-fun)
\begin{align} \lb{gen-fun2}
\gamma( \pm \textup{i} z \pm \textup{i} x + g_{n,0} )
= \prod_{r = 0}^{n-1} \Bigl[ 2\cos 2 \pi \textup{i} b z + 2\cos \pi b( 2 \textup{i} x + b(n-1-2r)) \Bigr] \notag \\
= \sum_{j = 1}^{n+1} \psi_{n+2-j}^{(n)}(x)\, \varphi_{j}^{(n)}(z)
= \sum_{j = 1}^{n+1} \varphi_{n+2-j}^{(n)}(x)\,\psi_{j}^{(n)}(z),
\end{align}
\varphi^{(n)}_j (z) := (2\cos 2 \pi \textup{i} b z)^{j-1},\;\;\; j = 1, 2,\ldots,n+1.
The second equality in (<ref>) is used to define the dual basis
$\psi^{(n)}(x)$, whereas the third equality follows from the invariance of the
generating function under the permutation of $x$ and $z$.
Thus the generating function produces two natural bases
$\{\mathbf{e}_{j}\}_{j=1}^{n+1}$ and $\{\mathbf{f}_j\}_{j=1}^{n+1}$ of $\mathbb{C}^{n+1}$,
\mathbf{e}_{j} = \varphi^{(n)}_j(z)\;\; , \;\;
\mathbf{f}_{j} = \psi^{(n)}_j(z) \;\; , \;\; j = 1 , 2, \ldots , n+1.
Expanding both sides of Eq. reduct as linear combinations of $\varphi^{(n)}(z_3)$,
we obtain a matrix form of the reduced R-operator written in the indicated pair of bases,
\mathbb{R}_{12}(u|g_{n} ,\,g)\,\psi^{(n)}_{j}(z_1) =
\varphi^{(n)}_{l}(z_1) \bigl[\mathbb{R}_{12}(u|g_{n} ,\,g)\bigr]_{lj}.
In the previous section we did not have such a subtlety, since for $n = 1$
(the fundamental representation) both bases coincide.
Similar to the pattern given in the previous section, the direct calculation yields
the following factorization formula for the reduced R-operator
ℝ_12(u|g_n,0 , g) = V(u+g,z) D(z,) 𝐂 V^T(u-g,z)
consisting of the product of five matrices. Here we substituted $z_2 \to z$ for brevity.
$\mathbf{C}$ is a numerical matrix with the unities on the antidiagonal, i.e.
$\left( \mathbf{C} \right)_{lj} = \delta_{n+2-l , j}$.
Entries of the diagonal matrix
\begin{align}
\left[D(z,\dd)\right]_{lj}:= \delta_{lj}\, \beta^{(n)}_{l}(z)\,e^{(n+2-2l) \frac{\textup{i}b}{2}\dd_z}.
\end{align}
are the shift operators determined by the expansion of
$\mathrm{M}({\textstyle\frac{nb}{2}})$ of the form
M(nb/2) = ∑_l = 1^n+1 β^(n)_l(z) e^(n+2-2l)η_z.
Entries of the matrix $V$, $\left[V(u,z)\right]_{jl} = V^{(n)}_{jl}(u,z)$, are
some trigonometric functions. They are defined by the relations
\begin{align}\label{Vdef}
&\sum_{j = 1}^{n+1} \varphi_j^{(n)}(x)\,V_{jl}^{(n)}(u,z) := \\ & \kern 4em
\prod_{r = 0}^{l-2} \Bigl[ 2\cos 2 \pi \textup{i} b x - 2\cos \pi b ( 2\textup{i} z - u - 2\eta - g_{n,0} + 2 b r) \Bigr] \times \notag \\
& \kern 4em \prod_{r = 0}^{n-l} \Bigl[ 2\cos 2 \pi \textup{i} b x - 2\cos \pi b ( -2\textup{i} z - u - 2\eta - g_{n,0} + 2 b r) \Bigr]. \notag
\end{align}
It is easy to see that $V_{jl}^{(n)}(u,-z) = V_{j,n+2-l}^{(n)}(u,z)$, i.e.
$V(u,-z) = V(u,z)\,\mathbf{C}$.
Let us recall that for the factorized $\mL$-operator the lateral matrices are composed out of
the trigonometric intertwining vectors providing the vertex-face correspondence <cit.>.
Then, in the case of the $(n+1)$-dimensional representation, the lateral matrices $V$
are constructed out of the fused trigonometric intertwining vectors (see Eq. Vdef)
providing the vertex-face correspondence for the higher-spin models.
In <cit.> an analogous factorization formula has been derived for
finite-dimensional R-operators with the symmetry algebras $s\ell_2$, $U_q(s\ell_2)$, and
the Sklyanin algebra.
§ ACKNOWLEDGEMENTS
We are indebted to S. E. Derkachov for useful discussions of the results
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1511.00336
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We consider a model for the low-luminosity gamma-ray burst GRB 060218 that plausibly accounts for multiwavelength observations to day 20. The model components are: (1) a long-lived ($t_j \sim 3000$ s) central engine and accompanying low-luminosity ($L_j \sim 10^{47}$ erg s$^{-1}$), semirelativistic ($\gamma \sim 10$) jet; (2) a low-mass ($\sim 4 \times 10^{-3} M_\odot$) envelope surrounding the progenitor star; and (3) a modest amount of dust ($A_V \sim 0.1$ mag) in the interstellar environment. Blackbody emission from the transparency radius in a low-power jet outflow can fit the prompt thermal X-ray emission, and the nonthermal X-rays and $\gamma$-rays may be produced via Compton scattering of thermal photons from hot leptons in the jet interior or the external shocks. The later mildly relativistic phase of this outflow can produce the radio emission via synchrotron radiation from the forward shock. Meanwhile, interaction of the associated SN 2006aj with a circumstellar envelope extending to $\sim 10^{13}$ cm can explain the early optical emission. The X-ray afterglow can be interpreted as a light echo of the prompt emission from dust at $\sim 30$ pc. Our model is a plausible alternative to that of Nakar, who recently proposed shock breakout of a jet smothered by an extended envelope as the source of prompt emission. Both our results and Nakar's suggest that bursts such as GRB 060218 may originate from unusual progenitors with extended circumstellar envelopes, and that a jet is necessary to decouple the prompt emission from the supernova.
supernovae: SN2006aj – gamma-ray burst: GRB 060218 – stars: mass loss – circumstellar matter – shock waves – hydrodynamics
§ INTRODUCTION
Low-luminosity gamma-ray bursts (LLGRBs) are a subclass of long-duration gamma-ray bursts (GRB) that, although rarely detected and not yet well understood, have the potential to shed light on more commonly observed cosmological bursts. Though the uncertainty is large, estimated volumetric rates indicate that LLGRBs occur some $10 - 100$ times more often than typical GRBs <cit.>, making them a compelling population for further study. In addition, LLGRBs take place nearby, so the associated supernovae (SNe) can be detected easily and studied in detail, placing constraints on energetics and circumstellar environment and giving clues about the SN-GRB connection. Phenomena like central engine activity, jet-star and jet-wind interactions, and the transition from beamed to spherical outflow can be probed more thoroughly than is possible at high redshift, and any insight into the radiation mechanisms of LLGRBs can inform our understanding of the GRB population at large.
Among known LLGRB sources, the remarkably similar sources GRB 060218/SN 2006aj <cit.> and GRB 100316D/SN 2010bh <cit.> stand out as unique due to their long time-scale, smooth single-peaked light curve, anomalously soft and bright X-ray afterglow, and the presence of significant thermal X-ray and optical components at early times <cit.>. Several important and compelling questions concerning these two bursts remain open. In a narrow sense, the atypical prompt emission, the origin of the X-ray blackbody component, and the unusual X-ray afterglow are hard to explain in terms of standard GRB theory. In a broader sense, we do not know whether the progenitor system is the same as for typical GRBs: do these ultra-long LLGRBs represent the low-luminosity end of a continuum of collapsar explosions, or a different stellar endpoint altogether? The answer to this question has important implications for high-mass stellar evolution, the connection between SNe and GRBs, and the low-energy limits of GRB physics, especially considering that events similar to GRB 060218 and GRB 100316D are likely more common than cosmological GRBs. The peculiar nature of these bursts, the wealth of timely observations in many wavebands (especially for GRB 060218), and the lack of a consensus picture for their behaviour make these particular LLGRBs prime targets for theory.
Accordingly, a wide variety of models have been proposed to explain the many facets of GRB 060218. <cit.> and <cit.> originally modeled the prompt X-ray emission as shock breakout from a circumstellar shell at $\sim 10^{12}$ cm. The breakout duration in this case, assuming spherical symmetry, is only a few hundred seconds; however, <cit.> suggested asphericity as a means to lengthen the burst time-scale. On the other hand, <cit.> argued against the shock breakout interpretation, showing that fine tuning is required to bring about a large change in breakout time-scale through asymmetrical effects. <cit.> presented an alternative synchrotron self-absorption model for the prompt emission, but the high brightness temperature and small emitting area in their model are at odds with radio observations, which suggest only mildly relativistic speeds <cit.>. <cit.> found that Compton scattering of soft input photons off relativistic external shocks driven by an inner outflow could roughly reproduce the observed prompt light curve. In the same vein, <cit.> showed that a Fermi acceleration mechanism could upscatter breakout thermal photons, creating a high energy power law tail to the thermal distribution. However, it is unlikely that thermal equilibrium is obtained in a relativistic breakout, and photon energies are limited by Compton losses <cit.>. <cit.> and <cit.> investigated the prompt UV/optical emission, and demonstrated that shock breakout could reproduce the optical flux, given a large breakout radius of $5 \times 10^{13}$ cm. (This large radius was initially viewed as a problem; see, however, the discussion in Section <ref> below.) <cit.> also put forth a model for the prompt UV, based on optically thick cyclotron emission. <cit.> showed that an early UV/optical peak could be attributed to cooling emission from an extended low-mass circumstellar envelope shock-heated by the passage of fast SN ejecta. They did not discuss the case of SN 2006aj, although it appears in their Figure 1 as an example of extended envelope interaction.
Another possibility for the prompt emission is that GRB 060218 is an ordinary GRB jet viewed off-axis. However, a solely geometrical effect should result in a frequency-independent, or achromatic, break in the light curve, whereas the break in GRB 060218 is chromatic in nature <cit.>. <cit.> considered a scenario for GRB 060218 in which primary radiation scatters off material radiatively accelerated slightly off-axis from the line of sight; this acceleration can explain the chromatic behaviour of the afterglow. However, as their model still required an unusually soft, long-duration, and low-luminosity primary photon source, it did not give insight into the fundamental factor distinguishing LLGRBs from most bursts.
<cit.> and <cit.> tackled the X-ray and radio afterglow. In each case, the radio could be explained by a synchrotron self-absorption in a wide ($\theta \ga 1$), mildly relativistic ($\Gamma \sim 2$–$3$) outflow, but the high X-ray afterglow flux could not be accounted for in a simple external shock synchrotron model. <cit.> attributed this X-ray excess to a forming magnetar, while <cit.> preferred late-time fallback accretion on to a central compact object. <cit.> suggested that the radio afterglow was better explained by the late non-relativistic phase of an initially collimated jet outflow. They inferred a jet luminosity $10^{45}$ erg s$^{-1}$, an initial jet Lorentz factor $\Gamma \sim 5$, and an initial jet opening angle $\theta \sim 0.3$, and showed that a hot low-luminosity jet could successfully penetrate a WR star and expand upon breakout to achieve these initial conditions. Based on the smooth light curve and long engine duration, they posited a neutron star-powered rather than black hole-powered central engine. <cit.> calculated the synchrotron afterglow light curves from a relativistic shock breakout, and while their model could fit the radio emission of GRB 060218, it predicted too low a flux and too shallow a temporal decay for the X-ray afterglow. <cit.> analysed the X-ray afterglows of 12 nearby GRBs and established that GRB 060218 and GRB 100316D belong to a distinct subgroup marked by long duration, soft-photon index, and high absorption. They proposed the possibility that these afterglows are in fact dust echoes from shells $\sim$tens of parsecs across that form at the interface between the progenitor's stellar wind and the ISM.
Until recently, most existing models have focused on explaining a particular aspect of this burst (e.g., the prompt nonthermal emission, the radio afterglow, or the optical emission), while leaving the other components to speculation. <cit.> recently suggested a model that attempts to unify the prompt X-rays, early optical peak, and radio emission. In his picture, the prompt X-ray and optical emission arise from the interaction of a typical GRB jet with a low-mass envelope surrounding the progenitor star. The short-lived jet is able to tunnel through the progenitor star, but is choked in the envelope, powering a quasi-spherical, mildly relativistic explosion akin to a low-mass SN. The prompt X-rays are produced by the shock breaking out of the optically thick envelope <cit.>, and optical radiation is emitted as the envelope expands and cools <cit.>. Interaction of the breakout ejecta with circumstellar material (CSM) generates the radio via synchrotron radiation <cit.>. Nakar's model does not, however, explain the unusual X-ray afterglow or the presence of thermal X-rays at early times.
In this paper, we present a plausible alternative to Nakar's model for this peculiar burst, building on previous jet models. In Section <ref>, we give an overview of observations of GRB 060218, and discuss the key features that must be reproduced by any model. In Section <ref>, we address some problems with a straightforward shock breakout view for the prompt emission, and provide motivation for adopting a long-lived jet instead. In Section <ref>, we describe how each component of the observed radiation is generated in our engine-driven model for GRB 060218, and check that our picture is self-consistent. Advantages, drawbacks, and predictions of our model, ramifications for GRB classification, and future prospects are discussed in Section <ref>, before we conclude in Section <ref>.
§ OVERVIEW OF OBSERVATIONS
The X-ray evolution of GRB 060218 and GRB 100316D can be divided into a prompt phase, an exponential or steep power-law decline, and an afterglow phase. Remarkably, these two objects share many observational features, perhaps suggesting that they have similar origins. In both objects, we see:
* Prompt nonthermal X-rays and $\gamma$-rays with a Band-like spectrum, but with lower luminosity, lower peak energy, and longer time-scale as compared to cosmological GRBs.
* Thermal X-rays with roughly constant temperature $\sim 0.1$ keV over the first $\sim1000$ s.
* Strong thermal UV/optical emission on a time-scale of hours to days.
* A radio afterglow lasting tens of days and implying mildly relativistic outflow.
* An X-ray afterglow that is brighter and softer than expected in standard synchrotron models.
Any unified model for these bursts must account for each of these components. Here we summarise multiwavelength observations during the prompt and afterglow phases of GRB 060218 and GRB 100316D.
Prompt X-rays/$\gamma$-rays: The nonthermal spectrum of GRB 060218 from $t=200$ s to $t=3000$ s is well fit by a Band function <cit.> with low-energy photon index $\Gamma_1 = -1.0$ and high-energy photon index $\Gamma_2 = -2.5$, implying $F_\nu \propto \nu^0$ at low energies and $F_\nu \propto \nu^{-1.5}$ at high energies. $\Gamma_1$ and $\Gamma_2$ remain roughly constant over the evolution <cit.>. <cit.> found a somewhat different low-energy index, $\Gamma_1 = -1.4$, when fitting the spectrum with a cut-off power law instead of a Band function. These values are typical for long GRBs <cit.>. The peak energy $E_p$ of the best-fitting Band function decreases as $E_p \propto t^{-1.6}$ from $t=600$ s until the end of the prompt phase. At $700$ s, $E_p = 10$ keV <cit.>. Despite its low luminosity, GRB 060218 obeys the Amati correlation between $E_p$ and luminosity <cit.>. In addition to the nonthermal Band function, a significant soft thermal component was detected in the spectrum. <cit.> found that the blackbody temperature remains nearly constant at $0.17$ keV throughout the prompt phase <cit.>. A later analysis by <cit.> determined a slightly lower temperature, $0.14$ keV, for times after several hundred seconds (see their Figure 7).
The prompt XRT ($0.3$–$10$ keV) light curve of GRB 060218 can be decomposed into contributions from the thermal and nonthermal parts; the nonthermal component dominates until approximately 3000 s (; see also Figure 1 in ). The total (nonthermal + thermal) isotropic-equivalent luminosity in the XRT band grows as $L_{XRT} \propto t^{0.6}$ for the first $1000$ s, when it reaches a peak luminosity $\sim 3 \times 10^{46}$ erg s$^{-1}$, then declines as roughly $t^{-1}$ until $\sim 3000$ s, fading exponentially or as a steep power law after that <cit.>. The thermal component initially comprises about $\sim 1/6$ of the total XRT band luminosity, and its light curve evolves similarly: at first it increases as a power law, rising steadily as $L_{th} \propto t^{0.66}$ <cit.> until it peaks at $1 \times 10^{46}$ erg s$^{-1}$ at $t = 3000$ s <cit.>. At that time, the thermal and nonthermal luminosities are about equal, but during the steep decline phase ($3000$–$7000$ s), the thermal component comes to dominate the luminosity, indicating that the nonthermal part must decline more steeply <cit.>. The light curve in the BAT band ($15$–$150$ keV) is initially very similar to the XRT light curve, increasing as about $t^{0.8}$ with roughly the same luminosity. Though its maximum luminosity ($\sim 3 \times 10^{46}$ erg s$^{-1}$) is similar to the peak XRT flux, the BAT flux peaks earlier, at $t=400$ s. Furthermore, it decays faster after the peak, falling off as $L_{BAT} \propto t^{-2}$ from $400$–$3000$ s <cit.>.
Evidence for a blackbody spectral component has also been claimed for GRB 100316D, with a similar constant temperature $kT = 0.14$ keV <cit.>. However, the presence of this thermal component has been called into question based on a large change in its statistical significance with the latest XRT calibration software <cit.>. The nonthermal spectrum of this burst is similar to that of GRB 060218: its peak energy has about the same magnitude and declines in a similar fashion, and its low-energy photon index is also nearly the same over the first $\sim 1000$ seconds <cit.>.
Compared to GRB 060218, GRB 100316D is more luminous in the XRT band, with $L_{XRT} \sim 10^{48}$ erg s$^{-1}$. In this case, the XRT light curve has nearly constant luminosity ($\propto t^{-0.13}$) for the first $800$ s <cit.>. (For this burst, there are no X-ray data available from $800$–$30000$ s.) If the light curve is broken into blackbody and Band function components, the nonthermal flux strongly dominates over the thermal contribution, with $L_{XRT}/L_{th} \sim 30$.
Optical photometry: From the first detection of GRB 060218 at a few hundred seconds, the UV/optical emission slowly rises to a peak, first in the UV at $31$ ks, and then in the optical at $39$ ks <cit.>. The light curves dipped to a minimum at around $\sim 200$ ks, after which a second peak occurred around $800$ ks, which can be attributed to the emergence of light from the supernova SN 2006aj. Like other GRB-supernovae, 2006aj is a broad-lined Type Ic <cit.>, but its kinetic energy $E_k \approx 2 \times 10^{51}$ erg is an order of magnitude smaller than usual <cit.>.
GRB 100316D was not detected with UVOT <cit.>. Its associated supernova, SN 2010bh, peaked at $\sim 10$ days <cit.>. While detailed optical data is not available for the earliest times, SN 2010bh does show an excess in the B-band at $t=0.5$ days <cit.>, which is at least consistent with an early optical peak.
X-ray/radio afterglow: Once the prompt emission of GRB 060218 has faded, another component becomes visible in the XRT band at $10000$ s. This afterglow has luminosity $L_{ag} = 3 \times 10^{43}$ erg s$^{-1}$ when it first appears, and fades in proportion to $t^{-1.2}$ until at least $t=10^6$ s. While this power law decay is typical for GRBs, the time-averaged X-ray spectral index ($\beta$ in $F_\nu \propto \nu^{\beta}$) is unusually steep, $\beta_X = -2.2$ <cit.>. The measured spectral index at late times ($0.5$–$10$ days) is $\beta_X = -4.5$ <cit.>, suggesting that the spectrum softens over time.
Radio observations of GRB 060218 beginning around $\sim 1$ day indicate a power-law decay in the radio light curve with spectral flux $F_\nu \propto t^{-0.85}$ <cit.>, not so different from the X-ray temporal decay and typical for GRBs. At 5 days, the spectrum peaked at the self-absorption frequency $\nu_a \approx 3$ GHz <cit.>. The radio to X-ray spectral index is unusually flat, $\beta_{RX} = -0.5$ <cit.>. No jet break is apparent in the radio data <cit.>, and self-absorption arguments indicate mildly relativistic motion (see Section <ref>).
The X-ray afterglow light curve of GRB 100316D can also be described by a simple power-law decay: $L_{ag} \propto t^{-0.87}$ from $t=0.4$–$10$ days, with X-ray luminosity $ \sim 10^{43}$ erg s$^{-1}$ at $t=0.4$ days. Like GRB 060218, its X-ray spectrum is also very soft, with $\beta_X = -2.5$ over the period $0.5$–$10$ days <cit.>. Because of the gap in coverage, it is unclear precisely when the prompt phase ends and the afterglow phase begins.
GRB 100316D was detected at $5.4$ GHz from $11$–$70$ days, with a peak at that frequency at $t \approx 30$ days <cit.>. This peak comes much later than that of GRB 060218, where the $5$ GHz peak occurred at $2$–$3$ days <cit.>. The late-time radio to X-ray spectral index is $\beta_{RX} < -0.4$, comparable to GRB 060218 <cit.>. No jet break is detected out to $66$ days, and the estimated Lorentz factor is again mildly relativistic, $\Gamma \sim 1.5$–$2$ on day 1 <cit.>.
§ SHOCK BREAKOUT OR CENTRAL ENGINE?
The majority of models for the prompt X-rays of GRB 060218 fall into two categories: shock breakout <cit.> or IC scattering of blackbody radiation by external shocks <cit.>. The latter type requires seed thermal photons for IC upscattering; while <cit.> and <cit.> assumed these photons were produced by a shock breakout event, other thermal sources such as a dissipative jet are also possible. Here, we point out some difficulties with a shock breakout interpretation of the prompt X-ray emission, and suggest some reasons to consider a long-lived central engine scenario instead.
Early models for GRB 060218 <cit.> considered the case where matter and radiation are in thermal equilibrium behind the shock, and the thermal X-rays and thermal UV/optical emission arise from shell interaction and shock breakout, respectively. However, for sufficiently fast shocks the radiation immediately downstream of the shock is out of thermal equilibrium, so the breakout temperature can be higher than when equilibrium is assumed <cit.>. In this scenario the prompt emission peaks in X-rays and the prompt spectrum is a broken power law with $F_\nu \propto \nu^0$ at low energies and $F_\nu \propto \nu^{-1.74}$ at high energies <cit.>. This is similar to the Band function spectrum observed in GRB 060218, motivating consideration of the case where the nonthermal X-rays originate from a relativistic shock breakout while the thermal UV/optical component comes from a later equilibrium phase of the breakout, as described in <cit.>. This interpretation still has possible problems. For one, the origin of a separate thermal X-ray component is unclear in this picture. In addition, the evolution of the prompt peak energy differs from the shock breakout interpretation. In GRB 060218, the peak energy falls off as $t^{-1.6}$, while in the shock breakout model it declines more slowly as $t^{-(0.5-1)}$. Consequently, while the peak energy inferred from relativistic shock breakout, $\sim 40$ keV, is consistent with observations at early times (less than a few hundred seconds), it overestimates $E_p$ for most of the prompt phase. Another problem is that the optical blackbody emission is observed from the earliest time in GRB 060218, and it rises smoothly in all UVOT bands until peak. In the nonequilibrium shock breakout scenario, thermal optical emission would not be expected until later times, when equilibrium has been attained. A final issue with the shock breakout picture of <cit.> is that it involved a stellar mass explosion. Since only a small fraction of the energy goes into relativistic material in a standard SN explosion, the energy required for the breakout to be relativistic was extreme, $E_{SN} \ga 10^{53}$ ergs. This high energy is inconsistent with the unremarkable energy of the observed SN, $2 \times 10^{51}$ ergs.
One can also consider the case where the prompt optical emission is attributed to shock breakout, but the prompt X-rays have a different origin. The large initial radius in this case is incompatible with a bare WR star, and initially seemed to rule out a WR progenitor <cit.>. However, this calculation assumed that much of the stellar mass was located close to the breakout radius. An extended optically thick region containing a relatively small amount of mass could circumvent this difficulty. Such an envelope might be created by pre-explosion mass loss or a binary interaction. There is mounting evidence for the existence of such dense stellar environments around other transients such as SN Type IIn <cit.>, SN Type IIb <cit.>, SN Type Ibn <cit.>, and SN Type Ia-CSM <cit.>.
The model of <cit.> builds on the relativistic shock breakout model of <cit.>, while solving several of its problems. <cit.> introduces a low-mass, optically thick envelope around a compact progenitor. In his model, the explosion powering the breakout is driven not by the SN, but by a jet that tunnels out of the progenitor star and is choked in the envelope, powering a quasi-spherical explosion. Having a large optically thick region preserves the long shock breakout time-scale, but in this case most of the mass is concentrated in a compact core. Since the envelope mass is much smaller than the star's mass, the energy required for a relativistic breakout is reduced as compared to the model of <cit.>. This picture also provides a natural explanation for the optical blackbody component via cooling emission from the shocked envelope. However, since the prompt X-rays still arise from a relativistic shock breakout, Nakar's model inherits some problems from that scenario as well, namely that the predicted peak energy evolution is too shallow, and the thermal X-rays lack a definitive origin. It remains unclear, too, whether Nakar's model can account for the simultaneous observation of optical and X-ray emission at early times.
Given the possible difficulties with shock breakout, a different source for the prompt X-ray radiation should be considered. <cit.> have shown that a central engine origin for certain LLGRBs is unlikely as their duration ($T_{90}$) is short compared to the breakout time. However, due to their relatively long $T_{90}$, engine-driven models are not ruled out for GRB 060218 and GRB 100316D. Furthermore, as discussed in Section <ref>, the prompt X-ray/$\gamma$-ray emission of GRB 060218 shares much in common with typical GRBs. As these similarities would be a peculiar coincidence in the shock breakout view, a collapsar jet origin for GRB 060218 is worth considering. Motivated by this, we consider the case where the early optical emission is powered by interaction of the SN ejecta with a circumstellar envelope, but the prompt X-rays originate from a long-lived jet.
§ A COMPREHENSIVE MODEL FOR GRB 060218
The origin of different components of the prompt and afterglow emission in our model. The figure is not to scale. The progenitor has a core-envelope structure. $M_c \sim 2\,M_\odot$ is confined to a core of $R_c \sim 10^{11}$ cm (blue), while a mass $M_{ext} \ll M_c$ is contained mostly near the edge of an extended envelope with $R_{ext} \gg R_c$ (purple). Upper left: A long-lived, dissipative jet tunnels through the progenitor system. Upon breakout, it emits blackbody radiation from radius $R_{ph}$. Some thermal photons IC scatter from external shocks (orange) or the jet interior (yellow) to create the Band-like nonthermal component. The jet obtains terminal opening angle $\theta_0$ and Lorentz factor $\Gamma_0$ after breakout. Upper right: Fast SN ejecta shock the envelope, heating it. The slower bulk of SN ejecta (red) then lift the hot envelope (pink), which emits in the optical and UV as it expands and cools. Lower left: The prompt X-rays undergo scattering in a dusty region with inner radius $R_d \sim $ tens of pc and X-ray scattering optical depth $\tau_d$. The resulting light echo outshines the synchrotron afterglow, giving rise to a characteristic soft spectrum. Lower right: External shock synchrotron emission from the mildly relativistic phase of the jet generates the radio afterglow.
A schematic of our model is presented in Figure <ref>. The essential physical ingredients are a long-lived jet, an extended low-mass circumstellar envelope, and a modest amount of dust at tens of parsecs, which are responsible for the prompt X-rays/radio afterglow, early optical, and X-ray afterglow, respectively. Below we consider the origin of each observed component in detail, and show that a reasonable match to observations can be obtained for appropriate choices of the progenitor, jet, and CSM properties.
§.§ Prompt thermal emission
The thermal X-ray component is a puzzling aspect of GRB 060218, and it is not unique in this regard. A recent review by <cit.> lists a number of typical GRBs for which a Band $+$ blackbody model improves the spectral fit, which has been claimed as evidence for thermal emission. <cit.> have also found evidence for thermal radiation in several other bursts. In fact, <cit.> have recently suggested that most bursts must contain a broadened thermal component, because in the majority of observed bursts, the full width half maximum of the spectral peak is narrower than is physically possible for synchrotron radiation. Although prompt thermal radiation is observationally indicated, the physical origin of this emission is yet unclear. One possible source of thermal X-rays is a jet-blown cocoon, although the flat early light curve of GRB 060218 and GRB 100316D is hard to explain in this case <cit.>. Another possibility is that the blackbody emission is produced at the transparency radius in a dissipative jet outflow, as discussed in the context of GRB 060218 by <cit.> and <cit.>. Here we consider the latter scenario.
<cit.>, who considered photospheric emission from super-Eddington neutron star winds, showed that the photospheric radius in a spherical mildly relativistic outflow is $R_{ph} = (\dot{M}_{iso} \kappa/4 \pi \tau_{ph} \beta c) (1-\beta)$, where $\dot{M}_{iso}$ is the isotropic mass loss rate, $c$ is the speed of light, $\beta$ is the the wind velocity in units of $c$, $\kappa$ is the opacity, and $\tau_{ph} \approx 1$ is the optical depth at the photosphere. Let $L_{th}$ and $T_0$ be the observer-frame luminosity and temperature, and define the Lorentz factor $\gamma = (1-\beta^2)^{-1/2}$. Then the comoving luminosity and temperature are $\bar{L}_{th} = (1+\beta^2)^{-1} \gamma^{-2} L_{th}$ and $\bar{T}_0 = \gamma^{-1} T_0$. (Here and below, a bar indicates that a quantity is measured in the frame comoving with the engine-driven outflow.) Substituting $\bar{L}_{th}$, $\bar{T}_0$, and $R_{ph}$ into the Stefan-Boltzmann equation, one may derive $\dot{M}_{iso}$ in terms of the observables $L_{th}$ and $T_0$:
\begin{equation}
\label{Mdot}
\dot{M}_{iso} = \frac{c \tau_{ph}}{\kappa}\left(\frac{4 \pi L_{th}}{\sigma_B T_0^4} \right)^{1/2} \frac{\beta \gamma}{(1-\beta)(1+\beta^2)^{1/2}} ,
\end{equation}
where $\sigma_B$ is the Stefan-Boltzmann constant.
Based on observations of the thermal component in GRB 060218, we take the luminosity to evolve as a power law, $L_{th} = L_0(t/t_L)^k$, before some time $t_L$, and to decline exponentially as $L_{th} = L_0 e^{(t_L-t)/t_{fold}}$ after $t_L$. An empirical fit to the data of <cit.> gives $L_0 \approx 10^{46}$ erg s$^{-1}$, $t_L \approx 2800$ s, and $t_{fold} \approx 1140$ s. <cit.> found $k \approx 0.66$ by fitting the thermal component in the $0.3$–$2$ keV band. We set the temperature to a constant, $T_0$, and define $\xi = T_0/(0.17 \text{\,keV})$. For simplicity, we assume the outflow is injected at a constant Lorentz factor; this is supported by the near constant observed temperature, as otherwise the comoving temperature would have to vary in such a way to precisely cancel the change in $\gamma$. Scaling $L_0$ to $10^{46}$ erg s$^{-1}$ and $\kappa$ to $0.2$ cm$^2$ g$^{-1}$, and setting $\tau_{ph} =1$, the mass loss and kinetic luminosity $L_{iso} = (\gamma-1) \dot{M}_{iso} c^2$ prior to $t_L$ are
\begin{equation}
\label{Mdot2}
\dot{M}_{iso}(t) = 1.3 \times 10^{-9} \kappa_{0.2}^{-1} L_{0,46}^{1/2} \xi^{-2} (t/t_L)^{k/2} \beta \gamma^3\,M_\odot \text{\,s}^{-1}
\end{equation}
\begin{equation}
\label{Lj}
L_{iso}(t) = 2.3 \times 10^{45} \kappa_{0.2}^{-1} L_{0,46}^{1/2} \xi^{-2} (t/t_L)^{k/2} \beta \gamma^3 (\gamma-1) \text{\,erg\,s}^{-1}.
\end{equation}
We have assumed the Lorentz factor $\gamma$ of the outflow is large enough that the approximation $(1-\beta)^{-1}(1+\beta^2)^{-1/2} \approx \sqrt{2} \gamma^2$ applies; at worst, this differs from the exact expression by a factor $2^{1/2}$ when $\beta \rightarrow 0$. The isotropic mass and energy of the jet are then
\begin{equation}
\label{Mj}
\begin{aligned}
M_{iso} & = \int_0^{\infty} M_{iso} dt \\
& = 3.9 \times 10^{-6} \kappa_{0.2}^{-1} L_{0,46}^{1/2} \xi^{-2} \left(\dfrac{t_{eng}}{3100 \text{ s}} \right) \beta \gamma^3 \,M_{\odot}
\end{aligned}
\end{equation}
\begin{equation}
\label{Ej}
\begin{aligned}
E_{iso} & = \int_0^{\infty} L_{iso} dt \\
& = 7.1 \times 10^{48} \kappa_{0.2}^{-1} L_{0,46}^{1/2} \xi^{-2} \left(\dfrac{t_{eng}}{3100 \text{ s}}\right) \beta \gamma^3 (\gamma-1) \text{\,ergs},
\end{aligned}
\end{equation}
where $t_{eng} \equiv 2t_L/(2+k) + t_{fold}$. If the outflow is beamed into opposing jets with a small opening angle $\theta_0$, the true mass and energy of the ejected material are $M_j = (\theta_0^2/2) M_{iso}$ and $E_j = (\theta_0^2/2) E_{iso}$. For a mildly relativistic flow ($\gamma \sim$ a few) with $\theta_0 \sim 10 \degree$, we have $M_j \sim 10^{-6}\,M_\odot$ and $E_j \sim 10^{49}$ ergs. The photosphere at $R_{ph} \approx 6.6 \times 10^{11} L_{0,46}^{1/2} \xi^{-2} (t/t_L)^{k/2} \gamma \text{ cm}$ expands subrelativistically with average speed $\sim R_{ph}/t_L \sim 0.01 \gamma c$. Note that the photosphere lies within the radius of the low-mass envelope, $R_{ext} \simeq 9 \times 10^{12}$ cm, that we derive in Section <ref> below, suggesting that dissipation occurs within the envelope. The time $t_L$ corresponds to the time when the central engine shuts off.
In the above calculation, we have assumed for simplicity that the jet outflow is directed into an uncollimated cone. However, as we show in Section <ref>, the jet may be collimated within the envelope, and become uncollimated only after breaking out. The decollimation time-scale can be estimated as the time for the jet's cocoon to expand and become dynamically unimportant after breakout, which is $\sim R_{ext}/c_s \sim 3^{1/2}R_{ext}/c \sim 500$ s, where $c_s$ is the sound speed. This is short compared to the duration of prompt emission; therefore, outside of the envelope, the assumption of an uncollimated outflow is reasonable for most of the prompt phase. However, it appears that the photosphere is within the envelope. The decollimation time-scale there might be longer because it will take the jet some time to excavate the walls of the narrow hole left by its passage. Collimation has the joint effect of decreasing the outflow's opening angle (due to the confining effect of the cocoon) and decreasing its Lorentz factor (due to more of the total jet energy going into internal versus kinetic energy). Both of these effects lead to a smaller $M_j$ and $L_j$, for the same observed thermal luminosity and temperature. Thus, by ignoring collimation we potentially overestimate these quantities; our derived mass loss rate and kinetic luminosity should really be viewed as upper limits.
§.§ Extinction and absorption
The optical/UV extinction and the X-ray absorption to GRB 060218 are crucial for the interpretation of observations of the event, as well as giving information on gas and dust along the line of sight. The early optical/UV emission is strongly weighted to the ultraviolet, which is especially sensitive to absorption. The amount of Galactic absorption is not controversial; extinction maps of the Galaxy yield $E(B - V ) = 0.14$ mag, while the Galactic Na I D lines indicate $E(B - V ) = 0.13$ mag <cit.>. The reddening has been estimated from the narrow Na I D lines in the host galaxy as being $E(B - V ) = 0.042$ mag, or $A_V = 0.13 \pm 0.01$ mag <cit.>. As noted by <cit.>, a larger reddening is possible if there is ionization in the host galaxy. However, the properties of the host galaxy derived from fitting the spectral energy distribution and the observed Balmer line decrement point to a low extinction so <cit.> advocate the low value obtained from the Na I D line. Our model for the late-time X-rays (see Section <ref>) also suggests a similar low extinction.
A higher host galaxy reddening, $E(B - V ) = 0.2$ mag, was advocated by <cit.> and <cit.> because the early ($< 1$ day) emission could be fitted by a Rayleigh-Jeans spectrum, consistent with high temperature emission. This suggestion allowed a shock breakout model for both the thermal X-ray emission and the early optical emission. This value of the reddening was also used by <cit.>, who noted that the implied blackbody temperature is $> 50,000$ K. <cit.> advocates the large reddening based on the slow color evolution leading up to the peak, which is expected in the Rayleigh-Jeans limit. However, his model could in principle accommodate a smaller extinction, if the model is consistent with a constant temperature leading to the peak.
In view of the lack of direct evidence for the larger values of extinction in the host, we take the small value that is directly indicated. <cit.> had derived some of the observed parameters for GRB 060218 based on Galactic extinction only. As expected, the spectrum is then not well approximated by a Rayleigh-Jeans spectrum and a temperature in the range $30,000$–$35,000$ K is deduced over the first half day. A blackbody fit gives the radius at the time of peak luminosity, $10^{14}$ cm, which yields a luminosity of $5 \times 10^{43}$ erg s$^{-1}$. This can be compared to the luminosity $> 3 \times 10^{44}$ erg s$^{-1}$ found by <cit.> in his larger extinction model.
The X-ray absorption column density has been obtained by fitting the observed spectrum to a model with a power-law continuum, a blackbody thermal component and interstellar absorption; <cit.> obtain an absorbing hydrogen column density of $N_H = 6 \times 10^{21}$ cm$^{-2}$ over 10 spectra covering the time of peak luminosity. <cit.> infer the same absorption column from fitting an absorbed power law to the afterglow spectra. There is no evidence for evolution of $N_H$. Using a standard conversion of $N_H$ to $A_V$ for the Galaxy, $N_H = 2 \times 10^{21} A_V$ cm$^{-2}$ <cit.>, the corresponding value of $A_V$ is 3. There is a significant difference between the extinction determined from the Na I line and that from the X-ray absorption.
One way to reconcile the difference is to have the dust be evaporated in the X-ray absorbing region. <cit.> have discussed evaporation of dust by the radiation from a GRB; optical/UV photons with energies $1$–$7$ eV are responsible for the evaporation. A normal burst with an optical/UV luminosity of $L_{opt} = 1 \times 10^{49}$ erg s$^{-1}$ can evaporate dust out to a radius of $R_d \simeq 10$ pc <cit.>. Since $R_d \propto L_{opt}^{1/2}$ and the peak luminosity of GRB 060218 was about $1 \times 10^{43}$ erg s$^{-1}$, we have $R_d \approx 0.01$ pc and the absorbing gas is likely to be circumstellar in origin.
§.§ UV/optical emission
Here we investigate the possibility that the optical emission is from shocked gas, but the X-ray emission is not. We take a supernova energy of $2 \times 10^{51}$ ergs and a core mass of $2\,M_\odot$, as determined from modeling the supernova emission (Mazzali et al. 2006b). The optical emission has a time-scale of $\sim 1$ day, which is characteristic of supernovae thought to show the shock breakout phenomenon <cit.>, but the emission is brighter than that observed in more normal supernovae. As discussed in Section <ref>, there is increasing evidence that massive stars can undergo dense mass loss before a supernova. We thus consider the possibility that an extended, low-mass circumstellar medium is responsible for the high luminosity.
<cit.> have discussed how the shock breakout process is affected by the mass of an extended envelope. When most of the stellar mass is at the radius of the surrounding envelope, a standard shock breakout, as in <cit.>, is expected. This case applies to SN 1987A <cit.>. However, when the envelope mass is much less than the core mass, the early emission is determined by the emission from the envelope that is heated by the expansion of the outer part of the core. One of the distinguishing features of the non-standard case is that the red luminosity can drop with time, which is not the case for standard shock breakout. <cit.> note in their Fig. 1 that the early emission from GRB 060218 shows a drop in the V emission that implies the non-standard, low-mass envelope case. Another difference is that in the standard case, the initially rising light curves turn over because the blackbody peak passes through the wavelength range of interest as the emission region cools <cit.>, while in the non-standard case the turnover is due to all the radiative energy in the envelope being radiated and the temperature remains steady <cit.>. The set of Swift–UVOT light curves in fact show approximately constant colors (and thus temperatures) through the luminosity peak at $\sim 3.5 \times 10^4$ s <cit.>. The UVOT observations of GRB 060218 give the best set of observations of a supernova during this early non-standard phase.
<cit.> has recently discussed the early emission from GRB 060218 in terms of interaction with a low mass envelope. The mass of the envelope was estimated at $0.01\,M_\odot$ based on the time-scale of the optical peak and an estimate of the shell velocity. However, the expansion of the envelope was attributed to an explosion driven by the deposition of energy from an internal jet. In this case, the event is essentially a very low mass supernova. In our model, the expansion is driven by the outer, high velocity gas of the supernova explosion, as in the non-standard expansion case of <cit.>. The input parameters are a supernova explosion energy $E_{SN} = 2 \times 10^{51}$ ergs and core mass $M_c = 2\,M_\odot$ <cit.>, a peak luminosity of $L_{p} = 5 \times 10^{43}$ erg s$^{-1}$ <cit.>, and a time of peak of $t_p = 3.5 \times 10^4$ s <cit.>. Since SN 2006aj was of Type Ic (no Helium or Hydrogen lines), we assumed an opacity $\kappa = 0.2$ cm$^2$ g$^{-1}$, appropriate for an ionized heavy element gas. These parameters can then be used to find the properties of the low mass extended envelope (subscript ext): $M_{ext} \approx 4 \times 10^{-3}\,M_\odot$, shell velocity $v_{ext} \approx 2.9 \times 10^9$ cm s$^{-1}$, and energy $E_{ext} \approx 2.8 \times 10^{49}$ ergs. These results come from the dynamics of the outer supernova layers sweeping up and out the low mass envelope around the star, and the time of the peak luminosity <cit.>. The value of $R_{ext} \approx 9 \times 10^{12}$ cm is proportional to luminosity, because of adiabatic expansion. The radius is related to the luminosity and thus the assumed absorption. These results are not sensitive to the density distribution in the extended envelope provided that most of the envelope mass is near $R_{ext}$. The mass in the envelope derived here is sufficient that the shock wave breaks out in the envelope, as assumed in the model. At the time of maximum luminosity, the radius of the shell is $R_p \approx v_{ext}t_p = 1 \times 10^{14}$ cm. As noted by <cit.>, the minimum luminosity between the two peaks of the light curve can give an upper limit to the initial radius of the core. In the case of GRB 060218, the drop in the luminosity between the peaks is shallow so that only a weak limit on the core radius can be set, $R_c \la 1.6 \times 10^{12}$ cm.
In the choked-jet scenario, the flow must reach a quasi-spherical state prior to breaking out of the envelope semi-relativistically, which could be difficult. Numerical simulations suggest that the bulk of the jet outflow does not become quasi-spherical until long after it becomes nonrelativistic <cit.>. In the lab frame, this occurs on a time-scale $\sim 5 t_{NR}$, where $t_{NR}$ is set by $\frac{4}{3} \pi \rho_{ext} (c t_{NR})^3 = E_{iso} c^{-2}$. For a constant density envelope, we obtain
\begin{equation}
\label{t_NR}
t_{NR} \simeq 1000 \text{ s } L_{iso,51}^{1/3} M_{ext,-2}^{-1/3}
\left(\dfrac{R_{ext}}{3 \times 10^{13} \text{ cm}}\right) \left(\dfrac{t_{jet}}{20 \text{ s}}\right)^{1/3} ,
\end{equation}
where we have scaled to the values of jet isotropic luminosity, jet duration, and envelope mass and radius used by <cit.>. As $t_{NR}$ is comparable to the time-scale $t_{ext} = R_{ext}/c \sim 1000 \left(\frac{R_{ext}}{3 \times 10^{13}\text{\,cm}}\right)$ s for a relativistic jet to break out of the envelope, the jet may only be marginally nonrelativistic at breakout. This suggests that the breakout time $t_{br}$ is not much larger than $t_{ext}$, and it may well be that $t_{br} < 5 t_{NR}$, in which case the breakout will be aspherical. Therefore, it is questionable whether the jet of <cit.> can become approximately spherical in the envelope, which is an assumption in his model. As $t_{NR}/t_{ext}$ is independent of $R_{ext}$, changing the envelope's size does not help with this problem. $t_{jet}$ can not be made much lower since it must remain larger than the time to break out of the star, which is $\sim 10$ s in this case, and $M_{ext}$ cannot be much larger or the envelope kinetic energy would exceed the SN energy <cit.>. Thus, the problem can only be solved via a jet with lower $L_{iso}$.
These considerations show that the overall properties of the early optical/UV emission from GRB 060218 can be accounted for by a model in which there is shock breakout in a low mass, extended envelope. The model makes further predictions that can be tested in the case of GRB 060218. Approximating the observed temperature at the peak as the effective temperature leads to $T_{obs} \approx 3.5 \times 10^4$ K, which is consistent with the observed temperature of GRB 060218 at an age of $0.085$–$0.5$ days <cit.>. The high temperature justifies the neglect of recombination in the model. <cit.> note that the optical depth of $M_{ext}$ becomes unity at $t \approx t_p(c/v_{ext})^{1/2}$, which is day 1.3 for GRB 060218; the photospheric velocity at this time gives an estimate for $v_{ext}$. The earliest spectrum of <cit.> is on day 2.89, when they estimate a photospheric velocity of 26,000 km s$^{-1}$. The photospheric velocity is higher at earlier times, so there is rough agreement of the model with observations.
§.§ X-ray afterglow
After a steep drop, the X-ray emission from GRB 060218 enters an apparent afterglow phase at an age of $0.1$–$10$ days. During this time, the flux spectrum is approximately a power law and the evolution is a power law in time: $F_\nu \propto \nu^{\beta_X} t^{-1.1}$ <cit.>. Continuous spectral softening is observed, with $\beta_X$ decreasing from $-2.2$ at $0.1$ day to $\sim -4.5$ at $\sim 3$ days. The time evolution is typical of a GRB afterglow, but the spectrum is unusually steep and the indices do not obey the standard “closure" relations for GRB afterglows <cit.>. In view of this, other proposals have been made for this emission, e.g., late power from a central magnetar. <cit.> considered a wide, accretion-powered outflow as the afterglow source, but the expected light curve in that case is $F_\nu \propto t^{-5/3}$, which seems too steep to explain the observations.
In standard GRB afterglow emission, there is one population of relativistic particles that gives rise to the emission, from radio to X-ray wavelengths. However, in GRB 060218, it is difficult to join the radio spectrum with the X-rays <cit.>; a flattening of the spectrum above radio frequencies would be necessary, as well as a sharp steepening at X-ray energies. In fact, some young supernova remnants such as RCW 86 show such spectra <cit.>. The steepening would require some loss process for the high energy particles; however, <cit.> find that synchrotron losses set in at a relatively low energy, so the observed spectrum cannot be reproduced. In addition, the X-ray evolution does not show a jet break, as might be expected if the afterglow is produced in the external shocks of a collimated outflow. <cit.> examined a shock breakout afterglow model for the late radio and X-ray emission. They were able to model the radio emission quite adequately, but the predicted X-ray emission was considerably below that observed, decayed too slowly in time, and had the incorrect spectral index. They concluded that the X-ray emission had some other source.
An alternative model for the emission was suggested by <cit.>, that it is a dust echo of emission close to maximum light. The light curve shape expected for an X-ray echo is a plateau followed by evolution to a $t^{-2}$ time dependence. The observed light curve for GRB 060218 is between these cases, which specifies the distance of the scattering dust in front of the source, $\sim 50$ pc <cit.>. <cit.> applied the echo model widely to GRB light curves. However, <cit.> noted two problems with this model for typical bursts. First, the required value of $A_V$ is typically $\sim$10, substantially larger than that deduced by other means. Second, the evolution is generally accompanied by a strong softening of the spectrum that is not observed.
The case of GRB 060218 is different from the standard cases; it had a long initial burst and a large ratio of early flux to late flux. These properties are more favorable for echo emission. The early flux was $F_{pr} \approx 1\times 10^{-8}$ erg cm$^{-2}$ s$^{-1}$ lasting for $t_{pr}\sim 2000$ s, while the late flux of $F_{late} \approx 1\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$ lasted for $t_{late} \sim 20,000$ s. If the late emission is produced as an echo, the optical depth of the dust region is $\tau_0 = F_{late}t_{late}/F_{pr}t_{pr} = 0.01$ <cit.>. The corresponding value of $A_V$ is $0.01$–$0.1$ <cit.>. This value of $A_V$ is roughly consistent with that determined from the Na I D line, giving support to the echo interpretation.
To better understand the spectral softening and determine the dust properties, we numerically investigated the expected dust echo emission from a dust shell at radius $R_d$. We used the theory of <cit.>, with some modifications to specify to GRB 060218. While <cit.> assumed a flat prompt spectrum in the range $0.3$–$10$ keV as is typical for cosmological GRBs, we instead used an empirical model including a blackbody as described in Section <ref> and a Band function with flux and peak energy evolving according to <cit.>. In particular, the inclusion of the thermal component – which dominates at low energies – results in a steeper echo spectrum than predicted by <cit.>.
The parameters of the model are the dust radius $R_d$, the scattering optical depth at 1 keV $\tau_{\text{keV}}$, the minimum and maximum grain sizes $a_-$ and $a_+$, and the power-law indices $s$ and $q$ that set how the scattering optical depth per unit grain size scales with energy and grain radius, i.e. $\tau_a \propto \tau_{\text{\,keV}} E^{-s} a^{4-q}$ with $2 < s < 3$ and $3 < q < 4$ typically. The echo flux is integrated over the range $0.3$–$10$ keV, appropriate for the Swift XRT band. The parameter $a_- \approx 0.005\,\mu\text{m}$ is based on observations of galactic dust grains <cit.>. The prompt photons are approximated as being injected instantaneously at $t=0$.
Our calculated echo light curve is shown in Fig. <ref>. We find a reasonably good fit to the light curve with reduced chi-squared of 2.1 when $\tau_{\text{\,keV}} \approx 0.006$, $R_d \approx 35$ pc, $a_+ = 0.25\,\mu\text{m}$, $s=2$ and $q=4$. The same model can satisfactorily reproduce the spectral evolution at late times, as depicted in Fig. <ref>. The optical depth is well-determined and robust to changes in the other parameters. There is a degeneracy between $R_d$ and $a_+$ because the afterglow flux depends only on the combination $R_d a_+^{-2}$; however, $a_+ = 0.25\,\mu\text{m}$ is roughly consistent with Galactic observations <cit.>. Varying $s$ does not greatly affect the light curve, but $s\approx2$ is preferred to match the spectral index at late times. The model is mostly insensitive to $q$, but a larger $q$ improves the fit slightly by marginally increasing the flux at late times. The scattering depth $\tau_{sca}$ at energy $0.8 \text{ keV} \la E \la 10 \text{ keV}$ can be converted to an optical extinction via the relation $\tau_{sca}/A_V \approx 0.15(E/1\text{\,keV})^{-1.8}$ <cit.>. For $\tau_{sca} = 0.006$ at $1$ keV, $A_V \approx 0.15$, in line with Na I D line observations and the simple estimate above. (We note that it is not necessary for these values to coincide: as the typical scattering angle is $\alpha_{sca} \sim 0.1\degree$–$1\degree$, the line of sight to the afterglow and the prompt source are separated by $\sim \alpha_{sca} R_d \sim 0.1$–$1$ pc. It is possible that the ISM properties could vary on this scale.) We conclude that a moderate amount of dust located $\sim$tens of parsecs from the progenitor can plausibly explain the anomalous X-ray afterglow.
Prompt and afterglow light curves for a dust echo model with $\tau_{\text{\,keV}} \approx 0.006$, $R_d \approx 35$ pc, $a_+ = 0.25 \,\mu$ m, $s=2$ and $q=4$. The prompt data points are fit with a simple exponentially cut-off power-law, shown by the green line. The black line indicates the contribution from dust scattering at $R_d$.
Spectral evolution in GRB 060218. The blue, green, and red points are taken from Table 1 in <cit.>, Figure 1 in <cit.>, and <cit.> respectively. The lower solid black line indicates the typical high-energy spectral index of the Band function, $F_\nu \propto \nu^{-1.5}$. The upper solid black line shows the two-point XRT flux spectral index, $\log(F(10 \text{\,keV})/F(0.3 \text{\,keV}))/\log(10/0.3)$, as a function of time for our best-fitting echo model. The time when the echo flux first exceeds the prompt flux in our model is shown by the vertical dashed line.
Due to the gap in observations from $1000$–$30000$ s, the late X-ray light curve in GRB 100316D is difficult to model in detail. None the less, some simple estimates can be made. The prompt X-ray emission has luminosity $L_{pr} \sim 3 \times 10^{46}$ erg s$^{-1}$ and time-scale $t_{pr} \sim 1000$ s <cit.>. The X-ray afterglow has luminosity $L_{late} \sim 2 \times 10^{43}$ erg s$^{-1}$ at $t_{late} \sim 3 \times 10^4$ s and decays as $ t^{-0.87}$ <cit.>, so $L_{late} t_{late}$ gives a reasonable estimate of the reradiated energy. The above lead to a similar estimate for the optical depth as for GRB 060218, $\tau_d \sim 0.02$, or $A_V \sim 0.2$. One interesting difference between the two bursts is that the spectral index of the late afterglow, $\beta_X = -2.5$, is harder in GRB 100316D than in GRB 060218 where $\beta_X = -4.5$. (Notably, GRB 060218 is the only burst with such a steep afterglow spectrum; GRB 100316D is more typical, as other soft-afterglow bursts such as GRB 090417B and GRB 130925A also show $\beta_X \sim -2.5$ <cit.>.) In the echo interpretation, this discrepancy can be explained partially by a difference in the prompt spectrum. Due to the presence of a strong thermal component at low energies, the time-averaged prompt $0.3 - 10$ keV spectrum of GRB 060218 is steeper than in GRB 100316D, where the thermal component is weak and the spectrum is essentially flat at low energies. However, this effect alone is not sufficient, as it only produces a change in spectral index of $\sim$ 1. $R_d$ and $a_+$ also have a strong effect on $\beta_X$ because they change time-scale for spectral steepening, as does the energy dependence of the scattering cross section. A larger $R_d$, smaller $a_+$, or lower value of $s$ (compared to our values for GRB 060218) may be necessary to obtain the correct $\beta_X$ in GRB 100316D. However, due to a lack of data regarding the time dependence of $\beta_X$ and an insufficient light curve, we cannot say which of these effects is the relevant one.
<cit.> have recently argued that four bursts, including GRB 060218 and GRB 100316D, belong to a distinct subclass of transient taking place in a complicated CSM. They base their claim on the unlikelihood of three unrelated properties – high absorption column, soft afterglow spectrum, and long duration – occurring together by chance. They invoke a wind-swept dusty shell to account for the high X-ray absorption and steep afterglow spectrum (through an echo of the prompt emission), and propose shock breakout in a complex local CSM to explain the long duration of prompt emission, preferring this interpretation to one in which the central engine duration is intrinsically long. Our findings support their suggestion that the very soft spectrum of GRB 060218 arises from a dust echo, but as the amount of dust in our model is not particularly high, an especially dense shell is not necessary; the dust could exist in an ISM of typical density and chemistry. We stress that the absorption column implied by dust extinction in our model is not consistent with the X-ray absorption column inferred from the prompt emission, as the latter is larger by a factor of $\sim 30$. For this reason, dust scattering and X-ray absorption are unlikely to be occurring in the same place in GRB 060218. Rather, the X-ray absorption is likely happening at small radii where dust has been evaporated. Also, while our results do indicate a dense envelope around the progenitor star, we also differ from the <cit.> picture by adopting an intrinsically long-lived central engine.
Our results can be compared to two other objects for which dust echo models have been proposed, GRB 130925A <cit.> and GRB 090417B <cit.>. The optical extinction inferred from modeling the afterglow as a dust echo is $A_V = 7.7$ mag in GRB 130925A, <cit.>, and in GRB 090417B it is $A_V \ga 12$ mag <cit.>. In each case, the amount of dust required to fit the X-ray afterglow via an echo is consistent with the absorbing hydrogen column needed to fit the X-ray spectrum <cit.>. In GRB 090417B, the high extinction can also explain the lack of an optical detection <cit.>. In contrast to GRB 060218 and GRB 100316D, GRB 130925A and GRB 090417B appear to have taken place in an unusually dusty environment, with the dust accounting for both the X-ray scattering afterglow and the large $N_H$.
Interestingly, these bursts also differ in their prompt emission. GRB 130925A appears typical of the ultra-long class of objects described by <cit.>, which also includes GRB 101225A, GRB 111209A, and GRB 121027A. Compared to GRB 060218 and GRB 100316D, these ultra-long bursts are more luminous and longer lived, and they show variability in their light curves on short time-scales, reminiscent of typical GRBs <cit.>. The light curve of GRB 090417B is qualitatively similar to GRB 130925A, and it likewise has a longer time-scale, higher luminosity, and more variability compared to GRB 060218 <cit.>. Thus, while <cit.> have made a strong case that GRB 060218, GRB 100316D, GRB 130925A, and GRB 090417B constitute a population distinct from cosmological LGRBs, upon closer inspection GRB 130925A and GRB 090417B differ strikingly from GRB 060218 and GRB 100316D. It seems, then, that three discrete subclasses are needed to explain their observations: 1) smooth light curve, very low-luminosity ultra-long bursts like GRB 060218/GRB 100316D, with echo-like afterglows implying a modest amount of dust; 2) spiky light curve, somewhat low-luminosity ultra-long bursts like GRB 130925A/GRB 090417B, with echo-like afterglows implying a large amount of dust; and 3) spiky-light curve bursts with typical time-scale and luminosity, and synchrotron afterglows.
The underlying reason why the afterglow is dominated by dust-scattered prompt emission in some cases, and synchrotron emission from external shocks in others, is unclear. One possibility is that kinetic energy is efficiently converted to radiation during the prompt phase, resulting in a lower kinetic energy during the afterglow phase as discussed by <cit.> in the context of GRB 130925A. A second possibility is that the external shocks do not effectively couple energy to postshock electrons and/or magnetic fields. We return to this question at the end of Section <ref>.
§.§ Radio afterglow
An essential feature of the radio afterglow in GRB 060218 is that it shows no evidence for a jet break, but instead decays as a shallow power law in time, with $F_\nu \propto t^{-0.85}$ at $22.5$ GHz <cit.>. This behaviour runs contrary to analytical models of GRB radio afterglows <cit.> which predict that, after a relatively flat decay during the Blandford–McKee phase, the on-axis light curve should break steeply to $t^{-p}$ after a critical time $t_j$. Here, $p$ is the power law index of accelerated postshock electrons, i.e. $N(E) \propto E^{-p}$, which typically takes on values $2 < p < 3$. The steepening is due to a combination of two effects that reduce the observed flux: when the jet decelerates to $\Gamma \sim \theta_0$, the jet edge comes into view, and also the jet begins to expand laterally. The same general behaviour of the radio light curve is also seen in numerical simulations <cit.>. The steep decay lasts until a time $t_s$, which is the time-scale for the flow to become quasi-spherical if sideways expansion is fast, i.e. if the increase in radius during sideways expansion is negligible <cit.>. While detailed simulations have demonstrated that the transition to spherical outflow is much more gradual and that the flow remains collimated and transrelativistic at $t_s$ <cit.>, numerical light curves none the less confirm that analytical estimates of the radio flux that assume sphericity and nonrelativistic flow remain approximately valid for on-axis observers at $t > t_s$ <cit.>. After $t_s$, the light curve gradually flattens as the flow tends toward the Sedov–Taylor solution, eventually becoming fully nonrelativistic on a time-scale $t_{NR}$. Therefore, the smooth and relatively flat light curve of GRB 060218 over the period $2$–$20$ days suggests one of two possibilities: either we observed the relativistic phase of an initially wide outflow that took $t_j \ga 20$ days to enter the steep decay phase, or we observed the late phase of an outflow that became transrelativistic in $t_s \la 2$ days and that may have been beamed originally.
In either scenario, a light curve as shallow as $t^{-0.85}$ is not easily produced in the standard synchrotron afterglow model. One issue is that such a shallow decay suggests that the circumstellar density profile and postshock electron spectrum are both flatter than usual. Throughout the period of radio observations, the characteristic frequency $\nu_m$, the synchrotron self-absorption frequency $\nu_a$, and the cooling frequency $\nu_c$ are related by $\nu_m < \nu_a < \nu_c$ <cit.>. As the $22.5$ GHz band lies between $\nu_a$ and $\nu_c$, the expected light curve slope in the relativistic case is $t^{3(1-p)/4}$ for a constant density circumstellar medium, and $t^{(1-3p)/4}$ for a wind-like medium <cit.>. In the nonrelativistic limit, the slopes are $t^{3(7-5p)/10}$ (constant density) and $t^{(5-7p)/6}$ (wind) <cit.>. In order to fit the observed slope $t^{-0.85}$, we require a constant density medium and $p = 2.1$ (relativistic) or $p=2.0$ (nonrelativistic). However, <cit.> found that the afterglows of several typical GRBs were best explained with a constant density model, and a low $p$-value was indicated for a number of bursts in their sample. Hence, GRB 060218 does not seem so unusual in this regard.
A second point of tension with the shallow light curve is the observed Lorentz factor. <cit.> inferred a mildly relativistic bulk Lorentz factor $\Gamma \simeq 2.4$ from an equipartition analysis. However, they based their analysis on the treatment of <cit.>, which did not include the effects of relativistic expansion. A more accurate calculation that takes relativistic and geometrical effects into account was carried out by <cit.>. From Figure 2 in <cit.>, we estimate that, at day 5, the spectral flux at peak was $F_p \sim 0.3$ mJy and the peak frequency was $\nu_p = \nu_a \sim 3$ GHz. Applying equation (5) in <cit.>, we obtain a bulk Lorentz factor $\Gamma \approx 0.8$. On the other hand, using their equation (19) for the equipartition radius $R_{N}$ in the nonrelativistic limit, we find $\beta \sim R_N/ct \approx 1.3$. These results indicate that the outflow is in the mildly relativistic ($\beta \Gamma \sim 1$) limit, where neither the Blandford–McKee solution (which applies when $\Gamma \gg 2$) nor the Sedov–Taylor solution (which applies when $\beta \ll 1$) is strictly valid. As discussed above, one expects a relatively shallow light curve slope in these limits, but during the transrelativistic transitional regime the slope tends to be steeper. In spite of these caveats, we press on and compare the relativistic and nonrelativistic limits of the standard synchrotron model.
The possibility of a wide, relativistic outflow was first considered by <cit.>. Their spherical relativistic blast wave model predicts an ejecta kinetic energy $E_{k} \sim 2 \times 10^{48}$ ergs and a circumburst density $n \sim 100$ cm$^{-3}$, assuming fractions $\epsilon_e \sim 0.1$ and $\epsilon_B \sim 0.1$ of the postshock energy going into relativistic electrons and magnetic fields, respectively. In order to postpone the jet break, they presumed the initial outflow to be wide, with $\theta_0 \ga 1.4$ <cit.>. Yet, as <cit.> pointed out, given the isotropic equivalent $\gamma$-ray energy $6 \times 10^{49}$ ergs, the parameter set of <cit.> predicts an unreasonably high $\gamma$-ray efficiency, $\eta_\gamma \approx 98\%$. <cit.> refined this analysis and showed that parameters $E_{k} \sim 10^{50}$ ergs, $n \sim 100$ cm$^{-3}$, $\epsilon_e \sim 10^{-2}$, and $\epsilon_B \sim 10^{-3}$ also fit the data while keeping the $\gamma$-ray efficiency within reason, but the origin of the spherical (or very wide) relativistic outflow is still unclear. One possibility is that the some fraction of the SN ejecta is accelerated to relativistic speeds. However, <cit.> have found that, even for a large SN energy $\sim 10^{52}$ ergs, only a fraction $\sim 10^{-4}$ goes into relativistic ejecta. It therefore seems implausible that $\sim 5\%$ of the SN energy $2 \times 10^{51} \text{ ergs}$ could be coupled to relativistic material in GRB 060218. A choked jet in a low-mass envelope, as discussed by <cit.>, provides an alternative way to put significant energy into a quasi-spherical, relativistic flow.
Given the difficulties with the relativistic scenario, <cit.> considered the possibility that the radio emission comes from the late spherical phase of an originally collimated outflow instead. With the same assumption of $\epsilon_e = \epsilon_B = 0.1$, <cit.> infer the same kinetic energy and circumstellar density as <cit.>. The advantage of their view is that it eliminates the efficiency problem, as the isotropic equivalent kinetic energy during the early beamed phase is larger by a factor $2/\theta_0^2$.
<cit.> also looked at a mildly relativistic synchrotron model in the context of SN shock breakout. In this case, the light curve decays more slowly since energy is continuously injected as the outer layers of the SN ejecta catch up to the shocked region. As a result, the radio light curve is better fit by a wind profile than a constant density in the breakout case <cit.>. Their study adopts a fixed $\epsilon_e= 0.2$, and a fixed energy and Lorentz factor for the fast shell dominating breakout emission, $E_f = 2 \times 10^{50}$ ergs and $\gamma_f = 1.3$, which are derived from the relativistic breakout model of <cit.>. They then vary $\epsilon_B$ and the wind density parameter $A_*$, concluding that $\epsilon_B = 1.5 \times 10^{-4}$ and $A_*=10$ give the best fit. Due to degeneracy, however, other parameter sets with different energy and $\epsilon_e$ may fit the radio light curves as well.
Unfortunately, such degeneracies involving the unknown quantities $\epsilon_e$ and $\epsilon_B$ are an unavoidable limitation when deriving $E_k$ and $n$ in the standard synchrotron model. The available observations give only the specific flux $F_\nu$, the self-absorption frequency $\nu_a$, and an upper limit on the cooling frequency $\nu_c$, which is not sufficient to uniquely determine the four model parameters. In practice, this is typically addressed by fixing two of the parameters to obtain the other two. (For example, <cit.> choose $\epsilon_e$ and $\epsilon_B$; <cit.> fix $\epsilon_e$ and $E_k$.) We take a different approach. In this section and Section <ref>, we consider a number of constraints from dynamics, time-scales, and direct radio, optical, and X-ray observations, assuming that the emission is from the late phase of an initially collimated jet. We apply these conditions to constrain the available $(E_k, n , \epsilon_e, \epsilon_B, \theta_0, \gamma)$ parameter space. We then consider whether any reasonable parameter set is consistent with a jet that could produce the observed thermal X-rays through dissipation at early times, as described in Section <ref>.
We begin with the constraints inferred directly from radio observations. We have $\nu_a \sim 4 \times 10^{9}$ Hz at 5 days, $F_\nu(22.5 \text{\,GHz}) \sim 0.25$ mJy at 3 days, and $\nu_c \la 5 \times 10^{15}$ Hz so that the synchrotron flux remains below the observed X-ray afterglow flux throughout observations <cit.>. Lower limits on $E_k$ and $n$ can be deduced by assuming $\epsilon_e < 1/3$ and $\epsilon_B < 1/3$. For a relativistic blast wave with $p=2.1$, we have $\epsilon_{B,-2}^{0.34} \epsilon_{e,-1}^{0.36} C_p^{0.36} E_{k,51}^{0.34} n^{0.33} \sim 0.44$, $\epsilon_{B,-2}^{0.78} \epsilon_{e,-1}^{1.1} C_p^{1.1} E_{k,51}^{1.28} n^{0.5} \sim 0.0032$, and $\epsilon_{B,-2}^{-1.5} E_{k,51}^{-0.5} n^{-1} \la 0.43$ <cit.>, where $C_p = 13(p-2)/3(p-1) \approx 0.39$. In this case, we find $E_k \ga 7 \times 10^{47}$ ergs and $n \ga 3$ cm$^{-3}$. Similarly, in the nonrelativistic limit <cit.> derived $\epsilon_{B,-2}^{1/3} \epsilon_{e,-1}^{1/3} E_{k,51}^{1/3} n^{1/3} \sim 1$, $\epsilon_{B,-2}^{3/4} \epsilon_{e,-1} E_{k,51}^{1.3} n^{0.45} \sim 0.003$, and $\epsilon_{B,-2}^{3/2} E_{k,51}^{3/5} n^{0.9} \ga 0.4$. This leads to the constraints $E_k \ga 1 \times 10^{47}$ ergs and $n \ga 60$ cm$^{-3}$.
The minimum synchrotron energy $E_{min}$ provides a further constraint on burst energetics. In general, calculating $E_{min}$ requires integrating the specific synchrotron luminosity $L_\nu$ over a range of frequencies $\nu_{min}$–$\nu_{max}$. However, when $p \simeq 2.5$, the dependence of $E_{min}$ on $\nu_{min}$, $\nu_{max}$, and $p$ is weak <cit.>. In that case, if $L_\nu$ is measured at frequency $\nu$, one can obtain a rough estimate of $E_{min}$ by setting $\nu = \nu_{min}$: with quantities given in cgs units, $E_{min} \simeq 8.0 \times 10^{13} (1+\eta)^{4/7} V^{3/7} \nu^{2/7} L_\nu^{4/7}$ ergs <cit.>, where $\eta$ is the ratio of proton energy to electron energy, which is not known. <cit.> estimated that the size of the radio-emitting region is $R = 3 \times 10^{16}$ cm at $t=5$ days, so the emitting volume at that time can be approximated by $V \sim \frac{4}{3} \pi R^3 \sim 1.1 \times 10^{50}$ cm$^3$. At the same time, the flux density at $\nu = 4.86$ GHz was $S_\nu = 300 \,\mu\text{Jy}$, implying a specific luminosity $L_\nu = 7.5 \times 10^{27}$ erg s$^{-1}$ Hz$^{-1}$ given the distance $D=145$ Mpc. With these parameters, we find $E_{min} \sim 1.2 \times 10^{47} (1+\eta)^{4/7}$ ergs. Compared to the above estimate, this puts a stricter lower limit on the energy when $\eta$ is large.
A further condition comes from time-scale considerations, since the steep $t^{-p}$ part of the light curve should fall outside of the observational period. For an on-axis observer, a numerically calibrated expression for the jet break time in the observer frame is $t_j = 3.5 E_{iso,53}^{1/3} n^{-1/3} (\theta_0/0.2)^{8/3}$ days <cit.>. In the relativistic case, we need $t_j \ga 20$ days, so $E_{k,51}^{1/3} n^{-1/3} (\theta_0/0.2)^2 \ga 5.7$. On the other hand, the time $t_s$ that roughly marks the end of the steep light curve phase is $t_s \simeq 365 E_{iso,53}^{1/3} n^{-1/3} (\theta_0/0.2)^{2/3}$ days <cit.>. Since $t_s \la 2$ is needed for the nonrelativistic model, we have $E_{k,51}^{1/3} n^{-1/3} \la 0.0055$. Note that $t_j \sim (\theta_0^2/4)t_s$.
For typical burst energies and CSM densities, the relativistic scenario requires a very wide opening angle to make $t_j$ sufficiently large. For example, the parameters of <cit.> require $\theta_0 \ga 80 \degree$. An equally large $\theta_0$ is inferred for GRB 100316D. In that object, the radio afterglow has a similar slow temporal decay, but the time-scale of the $F_\nu$ peak at $8.5$ GHz was $\sim 10$ times longer, occurring at $30$ days <cit.> as compared to $3$ days in GRB 060218 <cit.>, and the radio luminosity is about 10 times higher at $20$ days <cit.>. Assuming the same microphysics, this implies about the same burst energy, but a circumstellar density that is higher by a factor of $100$–$1000$ <cit.>, even for a quasi-spherical outflow. It seems unusual that the progenitors of these similar bursts have such different circumstellar environments. In addition, the higher density leads to a smaller $t_j$ than in GRB 060218, while radio observations show a flat light curve over the period $20$–$70$ days <cit.> implying $t_j \ga 70$ days, larger than GRB 060218. This problem is alleviated by considering a wind-like medium as in <cit.>, but in that case the expected light curve is $\propto t^{-3/2}$ for $p=2$, which seems too steep to fit observations unless one adopts $p < 2$.
One can consider the nonrelativistic case instead, but due to the weak dependence on $E_k$ and $n$, the condition on $t_s$ is also hard to satisfy unless the burst energy is extremely low or the CSM is extremely dense. In addition, because the flow is still highly aspherical at $t \sim t_s$, the model light curve slope will be too steep if $t \ga t_s$ only holds marginally, even if the flux is approximately correct. The slope does not settle to the limiting Sedov–Taylor value until the outflow sphericizes, which according to numerical simulations does not occur until $\sim 5 t_{NR} \simeq 4700 E_{k,51}^{1/3} n^{-1/3} (\theta_0/0.2)^{-2/3} \text{\,days} \gg t_s$ <cit.>.
However, so far we have assumed that the CSM near the progenitor star is the same as the CSM at $\sim$ a few $10^{16}$ cm where the radio is emitted. It is possible that the circumstellar environment is more complicated, and in particular that the CSM density is higher closer to the progenitor star. In fact, there is some evidence that this is the case. The X-ray absorption column, $N_H \approx 6 \times 10^{21}$ cm$^{-2}$, measured during the prompt phase is higher than one would expect for a constant density medium with $n \sim 100$ cm$^{-3}$, even if that medium extended to $\sim 10$ pc scales. Thus, we speculate that the shell emitting the radio could have undergone additional deceleration by sweeping up the material responsible for X-ray absorption at some time $t < 2$ days. While the absorbing column through the expanding outflow is expected to change with time, the measured $N_H$ is constant during the prompt phase. Therefore, the bulk of the absorbing material would have to lie outside the jet throughout the first $10^4$ s. Note that $N_H$ also stays constant throughout the X-ray afterglow from $10^4$–$10^6$ s, but this does not provide any additional constraints on our model, since the afterglow in our picture is a light echo and thus inherits the absorption of the prompt component (see Section <ref>). Hence most of the absorbing mass must be confined to the radial range $R(10^4 \text{\,s}) < R_{abs} < R(2 \text{\,days})$. Taking $R \sim 2 \Gamma^2 c t$, where $\Gamma$ is the Lorentz factor of the forward shock, we find $0.002 \text{\,pc} \la R_{abs} (\Gamma/3)^{-2} \la 0.03 \text{\,pc}$. The total mass of absorbing material, assuming it is distributed isotropically, is $M_{abs} \sim 4\pi R_{abs}^2 N_H m_p$, so $0.002\,M_\odot \la M_{abs} (\Gamma/3)^{-4} \la 0.5\,M_\odot$. Note that less absorbing mass is necessary if it is located closer to the star, or if it is distributed preferentially along the poles. The jet will sweep a mass $M_{sw} \sim (\theta_0^2/2) M_{abs}$, which is sufficient to decelerate it to nonrelativistic speeds if $M_{sw} \gg M_j$. From equation (<ref>), $M_j \sim 10^{-4} (\theta_0^2/2) (\gamma/3)^3 M_\odot$ for the parameters of GRB 060218, thus we find
\begin{equation}
7 \Gamma (\Gamma/\gamma)^3 \la M_{sw}/M_j \la 2000 \Gamma (\Gamma/\gamma)^3.
\end{equation}
$M_{sw}/M_j > 1$ is possible to satisfy for $\Gamma \ga 0.1 \gamma^{4/3}$, and is always satisfied for $\Gamma \ga 0.6 \gamma^{4/3}$. Therefore, for the mildly relativistic case we consider, it seems plausible that the mass responsible for the X-ray absorption could also be responsible for decelerating the jet.
The conditions $\epsilon_e < 1/3$, $\epsilon_B < 1/3$, $E_k > E_{min}$, $\nu_c < 5 \times 10^{15}$ Hz, and the constraint on $t_j$ (or $t_s$) can be conveniently expressed in the $\epsilon_e$–$\epsilon_B$ plane. The result is shown in Figure <ref>. (We do not plot the line $E_k = E_{min}$ as it is largely irrelevant as long as $\eta \sim 1$.) The first two panels show the standard, constant CSM density case, for a relativistic flow with $p=2.1$ and a nonrelativistic flow with $p=2.0$, respectively. In the relativistic case, if the jet is very wide ($\theta_0 \simeq 1.4$), the $t_j$-condition can be marginally satisfied ($t_j \ga 10$ days) for $\epsilon_B \ga \epsilon_e$ as depicted in the top panel. However, because $t_j$ is sensitive to $\theta_0$, the available parameter space rapidly shrinks when $\theta_0$ is reduced: the green region disappears from the plot when $\theta_0 \la 1.0$, and the yellow region when $\theta_0 \la 0.7$. Thus, the relativistic scenario disfavors a tightly collimated outflow for any sensible combination of $\epsilon_e$ and $\epsilon_B$. In the nonrelativistic case, shown in the middle panel, we see that the constraints $t_s < 2$ days and $\nu_c < 5 \times 10^{15}$ Hz can not be jointly satisfied for any choice of parameters. At best, the $t_s$-condition can be met marginally ($t_s \la 4$ days) if $10^{-3} \la \epsilon_B/\epsilon_e \la 10^{-1}$. (Here, we also show the condition $t(\beta=1) < 2$ days, which was used by <cit.>. However, as discussed above and in <cit.>, the radio flux still deviates considerably from the Sedov–Taylor prediction at this time because the outflow is semirelativistic.) The situation changes if some additional mass is swept up by the jet before the first radio observation, in which case the condition $E_k < M_{sw} c^2$ replaces the upper limit on $t_s$. This scenario is shown in the bottom panel, assuming $M_{sw} = 10^{-5}\,M_\odot$ (corresponding to an isotropic mass $10^{-3}\,M_\odot$ for $\theta_0=0.2$). Compared to the standard cases, this case accommodates a larger set of possible parameters. The effect of increasing (decreasing) $M_{sw}$ is to increase (decrease) the size of the green region by moving the critical line $E_k = M_{sw} c^2$ towards the lower left (upper right).
Constraints described in Section <ref>, depicted in the $\epsilon_e$–$\epsilon_B$ plane. In each plot, thin dash-dotted lines of constant $E_k$ ($10^{48}$ ergs and $10^{50}$ ergs) and $n$ ($10^2$ cm$^{-3}$ and $10^4$ cm$^{-3}$) are drawn. The conditions $\epsilon_B = 1/3$, $\epsilon_e = 1/3$, and $\nu_c = 5 \times 10^{15}$ Hz are shown as heavy solid lines, as labelled in the diagram. Regions where all conditions are met are shaded in green, while yellow regions indicate that the conditions are met if the time-scale constraints are relaxed by a factor of 2. Top: The relativistic case with $p=2.1$. The condition $t_j = 20$ days is shown, assuming a wide jet ($\theta_0 \simeq 1.4$). Middle: The nonrelativistic case with $p=2.0$, assuming the same density for all $r > R_{ext}$, with the conditions $t_s = 2$ days, $t_s = 4$ days, and $t(\beta \simeq 1) = 2$ days. Note that $t_s \la 2$ days and $\nu_c <5 \times 10^{15}$ Hz cannot be jointly satisfied. Bottom: The nonrelativistic case, assuming some additional mass $M_{sw} = 10^{-5}\,M_\odot$ is swept up prior to $2$ days. See the text for discussion.
§.§ Jet propagation
We now examine the evolution of the jet as it drills the star and breaks out into the surrounding medium. For our picture so far to be plausible, several conditions must be met. First, the initial kinetic energy of the outflow $E_{iso}$ must exceed the prompt isotropic radiated energy $E_{\gamma,iso} = 6 \times 10^{49}$ ergs, i.e. the radiative efficiency $\eta_\gamma = E_{\gamma,iso}/E_{iso} < 1$. Using Equation <ref>, this implies $\gamma > 2.1$. Note that $E_{iso} = E_{\gamma_iso} + E_{k,iso}$, where $E_{k,iso} = (2/\theta_0^2) E_k$. Second, the total breakout time from the stellar core and extended envelope, $t_b=t_{b,*} + t_{b,ext}$, should be shorter than the duration of prompt X-rays $t_L$. Third, the interaction with the extended envelope should be dominated by the supernova, and not by the jet or cocoon. In other words, the jet/cocoon system should not sweep up or destroy the envelope before the supernova has a chance to interact with it. Finally, we expect that the energy in relativistic ejecta will be less than the SN energy: $E_j < E_{SN}$.
In what follows, we scale the collimation-corrected jet luminosity $L_j$ to $10^{46}$ erg s$^{-1}$, corresponding to a jet energy $E_j \sim L_j t_L \sim 3 \times 10^{49}$ ergs. We assume a constant jet luminosity for simplicity. (A time-varying luminosity does not affect our general conclusions, as long as the average value of $L_j$ remains the same.)
In order for such a low-luminosity jet to penetrate the progenitor star, <cit.> found that it must be hot and have a narrow opening angle $\theta_j \la 0.03$, conditions that are satisfied by a collimated jet. The general theory of jet propagation in the collimated and uncollimated regimes was put forth by <cit.>. Their model is applicable when the jet is injected with a Lorentz factor $\gamma_{inj}$ and opening angle $\theta_{inj}$ that satisfy $\gamma_{inj} \ga \theta_{inj}^{-1}$. They showed that the jet is collimated if $L_j < \pi r_h^2 \rho_a c^3 \theta_{inj}^{2/3}$, where $r_h$ is the radius of jet's head and $\rho_a$ is the density of the ambient medium. For a typical WR star with mass $M_* = 10 M_\odot$ and radius $R_* = 10^{11}$ cm, the jet is collimated for $L_j \la 10^{52} \text{ erg s}^{-1}$ <cit.>, so we are well within this regime. While propagating in the star, collimation by the uniform-pressure cocoon keeps the jet cross section approximately constant, and the Lorentz factor below the jet head is $\gamma_j \sim \theta_{inj}^{-1}$, independent of the injection Lorentz factor $\gamma_{inj}$ <cit.>. Later, once the jet breaks out into a low-density medium, it becomes uncollimated and its opening angle and Lorentz factor tend towards $\sim \theta_{inj}$ and $\sim \gamma_{inj}$, respectively <cit.>. Therefore, the values $\theta_0$ and $\gamma$, which describe the jet post-breakout, provide an estimate of the injection conditions at much smaller radii, i.e. $\theta_{inj} \sim \theta_0 $ and $\gamma_{inj} \sim \gamma$. As a result,
\begin{equation}
\label{gamma_lower}
\gamma \ga \theta_0^{-1}
\end{equation}
will hold after adiabatic expansion.
In the strongly collimated limit the jet head moves nonrelativistically with speed $\beta_h \simeq L_j^{1/5} \rho_a^{-1/5} t^{-2/5} \theta_{inj}^{-4/5} c^{-1}$ <cit.>. Let the stellar density profile be $\rho_a = \rho_0 (r/R_*)^{-\delta_*}$, with $\rho_0 = \frac{(3-\delta_*)M_*}{4\pi R_*^3}$. Typically, $1.5 \la \delta_* \la 3$ for WR stars <cit.>. Equation (B-2) in <cit.> gives the radius of the jet head as a function of time for the case of a nonrelativistic head; substituting $r=R_*$ and $\rho_a = \rho_0$ into that expression, we find the breakout time
\begin{equation}
\label{tb_star}
t_{b,*} = 94 \text{ } C(\delta_*) L_{j,46}^{-1/3} M_{c,1}^{1/3} R_{c,11}^{2/3} \left(\dfrac{\theta_{0}}{0.2}\right)^{4/3} \text{\,s}.
\end{equation}
The order-unity constant $C(\delta_*) = \left[\frac{3}{7}\frac{5-\delta_*}{3-\delta_*}\right]^{-2/3}$ scales the result to $\delta_* = 1.5$, as this gives the most conservative estimate of the breakout time for the typical range of $\delta_*$.
Breakout from the low-mass envelope proceeds similarly to breakout from the stellar core, the main differences being that the jet head is faster and harder to collimate due to the lower ambient density. For an envelope density profile $\rho_{ext} \propto r^{-\delta_{ext}}$, $\rho_{ext}(R_{ext}) = \frac{(3-\delta_{ext}) M_{ext}}{4 \pi R_{ext}^3}$. While $\delta_{ext}$ is not known in general, requiring that the density decreases outwards and that most of the envelope mass is at large radii restricts its value to $0 \leq \delta_{ext} \leq 3$. The collimation condition at the edge of the envelope can be rewritten as
\begin{equation}
\label{envelope_collimation}
L_{j,46} \la 46 (3-\delta_{ext}) M_{ext,-3} R_{ext,13}^{-1} \left(\dfrac{\theta_0}{0.2}\right)^{2/3} \equiv L_{coll,46}.
\end{equation}
For our parameters, we find that the jet remains collimated throughout the envelope, though we note that the high-luminosity jet of a typical GRB would be uncollimated in the same envelope <cit.>. The parameter $\tilde{L} = (L_j/\rho_{ext} t^2 \theta_0^4 c^5)^{2/5}$ then determines if the jet head is relativistic ($\tilde{L} > 1$) or Newtonian ($\tilde{L} < 1$) <cit.>. The condition for a nonrelativistic head at breakout is
\begin{equation}
\label{envelope_nonrelativistic}
L_{j,46} \la 6.9 \times 10^{-2} (3-\delta_{ext}) M_{ext,-3} R_{ext,13}^{-1} \left(\dfrac{\theta_0}{0.2}\right)^{4}.
\end{equation}
For the range of parameters we consider, the jet is usually relativistic at the time of breakout; however, for a low luminosity or a somewhat wide opening angle, it is possible that the jet head breaks out nonrelativistically.
Since we are interested in a lower bound on the jet luminosity, we compute the breakout time in the nonrelativistic limit. In this case, we can reuse equation (<ref>) with $R_*$ and $M_*$replaced by $R_{ext}$ and $M_{ext}$ to calculate the breakout time from the envelope. We have
\begin{equation}
\label{tb_ext}
t_{b,ext} \simeq 120 C(\delta_{ext}) L_{j,46}^{-1/3} M_{ext,-3}^{1/3} R_{ext,13}^{2/3} \left(\dfrac{\theta_0}{0.2}\right)^{4/3} \text{\,s},
\end{equation}
where in this case we have scaled to $\delta_{ext} = 0$ via $C(\delta_{ext}) = \left[\frac{3}{5}\frac{5-\delta_{ext}}{3-\delta_{ext}}\right]^{2/3}$, which maximizes $t_{b,ext}$. Combining equations (<ref>) and (<ref>) with the parameters $M_c \approx 2\,M_\odot$, $M_{ext} \approx 4 \times 10^{-3}\,M_\odot$, and $R_{ext} \approx 9 \times 10^{12}$ cm inferred in Section <ref>, and an assumed core radius $R_c \sim 10^{11}$ cm, we find the total breakout time
\begin{equation}
t_b \simeq 230 L_{j,46}^{-1/3} \left(\dfrac{\theta_0}{0.2}\right)^{4/3} \text{\,s}.
\end{equation}
The condition $t_b < t_L$ is satisfied as long as $L_{j,46} \ga 5 \times 10^{-4} (\theta_0/0.2)^4$, which generally holds in our model so long as the jet is reasonably beamed.
To ensure that the interaction with the envelope is dominated by the supernova ejecta, the supernova energy should exceed the energy of the jet-blown cocoon, so that the former overtakes the latter. The energy deposited into the cocoon up to breakout is $E_c \sim L_j (t_b - R_b/c)$ <cit.>, where $R_b$ is the breakout radius. There are two dynamically distinct cocoons that can potentially disturb the stellar envelope. First, while the jet is within the stellar core, material entering the jet head escapes sideways to form a cocoon of shocked stellar matter. When the jet breaks out of the stellar core and enters the surrounding envelope, this “stellar cocoon" also breaks out and begins to sweep the envelope as it expands outwards. Then, as the jet continues to propagate through the envelope, it blows a second cocoon containing shocked envelope material. This “envelope cocoon" expands laterally as the jet propagates, and then breaks out into the circumstellar medium once the jet reaches the envelope's edge. Here, we show that these cocoons have a negligible effect on the envelope dynamics compared to the supernova, because the stellar cocoon is too slow and the envelope cocoon is too narrow.
Consider first the stellar cocoon. While traversing the star, the jet head is nonrelativistic, so $t_b \gg R_b/c$ and essentially all of the energy goes into the cocoon, i.e. $E_{c,*} \sim L_j t_{b,*}$. From equation (<ref>), we have
\begin{equation}
\label{Ec}
E_{c,*} \sim 3 \times 10^{48} L_{j,46}^{2/3} M_{c,1}^{1/3} R_{c,11}^{2/3} \left(\dfrac{\theta_0}{0.2}\right)^{4/3} \text{\,ergs}.
\end{equation}
Here and for the rest of this section, we ignore order-unity factors that depend on $\delta_*$ or $\delta_{ext}$. When $\beta_h < 1$, the cocoon expands sideways with speed $\beta_c \approx \beta_h \theta_0$, resulting in a cocoon opening angle $\theta_c \sim \beta_c/\beta_h \sim \theta_0$ <cit.>. The mass entrained in the cocoon at the time of breakout is therefore
\begin{equation}
\label{Mc}
M_{c,*} \sim \dfrac{\theta_0^2}{2} M_c \sim 0.2 M_{c,1} \left(\dfrac{\theta_0}{0.2}\right)^{2}\,M_\odot.
\end{equation}
After breakout the cocoon material expands with typical speed $v_{c,*} \sim (2E_{c,*}/M_{c,*})^{1/2}$, so we have
\begin{equation}
v_{c,*} \sim 10^{8} L_{j,46}^{1/3} M_{c,1}^{-1/3} R_{c,11}^{1/3} \left(\dfrac{\theta_0}{0.2}\right)^{-1/3} \text{\,cm\,s}^{-1}.
\end{equation}
This is generally much slower than $v_{ext} \approx 3 \times 10^9$ cm s$^{-1}$ in our model, with $v_{c,*} < v_{ext}$ for $L_{j,46} \la 3 \times 10^4 M_{c,1} R_{c,11}^{-1} (\theta_0/0.2)$. Therefore, the fast supernova ejecta rapidly overtake the stellar cocoon.
As discussed above, the jet stays collimated in the envelope, but the jet head may become relativistic. In this limit, the lateral speed of the cocoon is $\beta_c \sim \tilde{L}^{1/2} \theta_0$ <cit.>, and since $\beta_h \approx 1$, $\theta_c \sim \beta_c$. For a collimated jet, $\tilde{L} \la \theta_0^{-4/3}$ <cit.>, so we have $\theta_c \la \theta^{1/3} < 1$. As the pressure of the envelope cocoon rapidly drops after it breaks out from the envelope's edge and expands freely into the low-density circumstellar medium, little sideways expansion through the envelope is expected after breakout. Thus, as long as $\theta_0$ is small, the passage of the jet and envelope cocoon leaves the envelope relatively intact, and the SN-envelope interaction is quasi-spherical. Note that it is not strictly necessary for the jet to be collimated by the envelope in our model. In principle the jet may be uncollimated, with $\tilde{L}$ somewhat larger than $\theta_0^{-4/3}$, as long as $\theta_c$ remains small, but in practice this regime is not attained in GRB 060218.
Ideally, the jet head should break out of the stellar core before the SN shock. This guarantees that the jet will reach the edge of the envelope before the SN, so the jet will be seen first. Comparing the SN breakout time $t_{SN} \sim R_* (2E_{SN}/M_{SN})^{-1/2}$ to the breakout time in equation (<ref>), one finds $t_{b,*}/t_{SN} \sim 0.6 L_{j,46}^{-1/3} R_{c,11}^{-1/3} \left(\frac{M_c}{2\,M_\odot}\right)^{-1/6} \left(\frac{E_{SN}}{2 \times 10^{51} \text{\,erg\,s}^{-1}}\right)^{1/2} (\theta_0/0.2)^{4/3}$. This condition is satisfied for $L_{j,46} \ga 0.2 \left(\frac{M_c}{2\,M_\odot}\right)^{-1/2} \left(\frac{E_{SN}}{2 \times 10^{51} \text{\,erg\,s}^{-1}}\right)^{3/2} (\theta_0/0.2)^4$, which is only sometimes met for the parameters considered here. However, even if the SN shock reaches the edge of the star first, the jet breaks out soon after. This is because, after the SN crosses the core, the core density drops as $\rho_c \propto (v_{SN} t)^{-3}$, and as $\beta_h$ depends inversely on $\rho_c$, the jet soon accelerates to $c \beta_h > v_{SN}$. Thus, it may be possible for this constraint to be violated, and we do not rule out models for which $v_{SN} > c \beta_h$ initially.
We can get a grasp on the allowed region of $\gamma$–$\theta_0$ parameter space by using equations (<ref>) and (<ref>) to convert the conditions $\eta_\gamma < 1$, $E_j < E_{SN}$, $E_j \ga E_{min}$, $\gamma \ga \theta_0^{-1}$, $t_b \la t_L$, $v_{c,*} < v_{ext}$, and $c \beta_h < v_{SN}$ to relations between $\gamma$ and $\theta_0$. We show the result in Figure <ref>. We see that the available parameter space is bound chiefly by $E_j < E_{SN}$ from above, $\gamma \ga \theta_0^{-1}$ from the left, $\theta_0 \la \pi/2$ from the right, and $\eta_\gamma \la 1$ from below. The other important conditions are always satisfied when these four constraints are met. Note that the conditions related to the jet and cocoon do not necessarily apply when $\theta_0$ is large, because in that limit the explosion is quasi-spherical instead of jet-like, but the conditions on the overall burst energetics are still relevant. The possible values of $E_j$, $\gamma$, and $\theta_0$ lie in the range $7 \la E_{j,48} \la 2 \times 10^3$, $2 \la \gamma \la 25$, and $0.04 \la \theta_0 \la \pi/2$.
Constraints described in Section <ref>, depicted in the $\theta$–$\gamma$ plane. The conditions $E_j > E_{min}$, $E_j > E_{SN}$, $\eta_\gamma < 1$, $\gamma \ga \theta^{-1}$, $t_b < t_L$, and $v_{c,*} < v_{ext}$ are drawn as heavy solid lines. Two other constraints that may be marginally violated ($L_j < L_{coll}$ and $c \beta_h > v_{SN}$) are shown as dashed lines. The regions of parameter space that satisfy all the constraints, and only the strict constraints, are painted green and yellow, respectively. Curves of constant jet energy are shown as thin, dash-dotted lines. See the text for discussion.
Three general classes of solution can satisfy all of the necessary conditions:
* Low kinetic energy, narrow beam, low Lorentz factor:
For low jet energies, e.g. $E_j \simeq 7 \times 10^{48}$ erg s$^{-1}$, the jet is confined to a narrow range around $\theta_0 \simeq 0.5$ and $\gamma \simeq 2.1$. The isotropic jet energy in this case is $E_{iso} \simeq 6 \times 10^{49}$ erg s$^{-1}$, and the radiative efficiency is $\eta_\gamma \simeq 0.5$, implying a mildly hot jet. The kinetic energy during the afterglow phase is $E_k \simeq 3 \times 10^{48}$, which gives $-2.5 \la \log \epsilon_e \la -0.8$, $-3.6 \la \log\epsilon_B \la -0.5$, and $2.5 \la \log n \la 3.9$ using the bottom panel of Fig. <ref>. This solution is similar to that of <cit.>, who also inferred a mildly hot jet.
* High kinetic energy, narrow beam, high Lorentz factor:
For higher kinetic energies, the model is less restrictive: for example, $E_j = 2 \times 10^{50}$ ergs gives $0.1 \la \theta_0 \la 1.5$, allowing for either narrow or wide jets. In the narrow jet case of $\theta_0 \simeq 0.1$, we have $E_{iso} \simeq 4 \times 10^{52}$ erg s$^{-1}$ and $\gamma \simeq 10$. The radiative efficiency in this case is low, $\eta_\gamma \simeq 2 \times 10^{-3}$, and therefore $E_k \approx E_j$. In order to accommodate the higher energy, this model requires lower-than-standard values for $\epsilon_e$ and/or $\epsilon_B$: $-5.3 \la \log \epsilon_e \la -2.6$ and $-5.4 \la \log\epsilon_B \la -0.5$. For this reason, this scenario has not been considered previously. The CSM density in this case is $3.5 \la \log n \la 5.7$.
* High kinetic energy, wide beam, high Lorentz factor:
A high jet energy directed into a wide ($\theta_0 \rightarrow \pi/2$) outflow is also allowed. In this case $E_j \approx E_{iso}$ . For $E_j \simeq 2 \times 10^{50}$ ergs, we find $\gamma \simeq 2.6$ and $\eta_\gamma \simeq 0.3$. Using the top panel of Fig. <ref>, we find $-3.9 \la \log \epsilon_e \la -2.7$, $-3.1 \la \log\epsilon_B \la -0.5$, and $1.7 \la \log n \la 3.0$ This model is similar to the model proposed by <cit.>, and also consistent with the picture in <cit.>, since in that case there is reason to expect a quasi-spherical explosion.
Our picture favors models with a narrow jet, because a wide jet would considerably disrupt the circumstellar envelope, which is problematic for our optical model discussed in Section <ref>. However, we do not have a strong reason to prefer a high jet energy versus a low energy one; each option offers some advantage. The low energy case uses typical values for $\epsilon_e$ and $\epsilon_B$, and more readily transitions to a nonrelativistic outflow without the need for an extreme CSM density. On the other hand, the high energy case is compatible with a wider variety of jets with various $\theta_0$ and $\gamma$, and requires a less efficient dissipation mechanism since $\eta_\gamma \ll 1$.
§.§ Prompt nonthermal emission
Because GRB 060218 adheres to several well-known GRB correlations, it is worth considering whether this object has the same emission mechanism as standard GRBs. In this section, we attempt to glean as much as possible about the emission mechanism in GRB 060218, assuming that the source of the nonthermal X-rays is a mildly relativistic jet. To do so, we first lay out a simple empirical description for the prompt emission that preserves the essential features of the observed spectrum and light curve. We then consider a number of jet scenarios for the prompt emission.
§.§.§ A simple empirical description of GRB 060218
The nonthermal emission observed in the XRT and BAT bands appear to have the same origin <cit.>, and throughout the prompt phase, the joint BAT-XRT spectrum is best fit by a Band function <cit.>. Here, we aim to find the least complex model that fulfills these criteria while also fitting the light curves at various frequencies. Consider a simple Band function model for the prompt nonthermal spectrum, where the photon spectral shape is given by <cit.>
\begin{equation}
\label{band_function}
\begin{array}{lr}
F(E) = F_0 (E/E_c)^{\beta_1} e^{-E/E_0}, & E \le E_c \\
F(E) = F_0 (E/E_c)^{\beta_2} e^{\beta_2-\beta_1}, & E > E_c
\end{array}
\end{equation}
The quantities $E_0$ and $E_c$ are related to the peak energy by $E_0 = E_p/(1+\beta_1)$ and $E_c = (\beta_1-\beta_2)E_0$. Let the spectral indices be constant, and suppose the parameters $E_p$ and $F_0$ vary as power laws in time, i.e.
\begin{equation}
\label{Ep_powerlaw}
E_p = E_n t^{\alpha_p}
\end{equation}
\begin{equation}
\label{F0_powerlaw}
F_0 = F_n t^{\alpha_0}
\end{equation}
where $E_n$ and $F_n$ are normalization constants. In the low and high energy limits, we have respectively $F(E \ll E_p) \propto t^{\alpha_0-\alpha_p \beta_1}$ and $F(E \gg E_p) \propto t^{\alpha_0-\alpha_p \beta_2}$. Joint XRT and BAT observations fit with a Band model give $\alpha_p = -1.6$, $\beta_1 \approx -0.13$, and $F_{BAT} \sim F(E \gg E_p) \propto t^{-2.0}$ <cit.>. There is considerable uncertainty in $\beta_2$, so we do not fix its value, but rather try several values in the observed range. Choosing $\beta_2 = -1.9$ and $\alpha_0 =1$, we have $F(E \ll E_p) \propto t^{0.8}$, giving a model that roughly reproduces the time behaviour and relative flux in the XRT and BAT bands (see Figure <ref>). These choices are consistent with observations of the early XRT light curve, which show $F_{XRT} \propto t^{0.7}$.
Light curves obtained using a Band spectrum with time-varying parameters $E_p \propto t^{-1.6}$ and $F_0 \propto t^{1.0}$, and constant spectral indices $\beta_1 = -0.13$ and $\beta_2=-1.9$. Model light curves in the 0.3$-$10 keV band (green) and the 15$-$150 keV band (red) are compared with XRT data from <cit.> (black diamonds) and BAT data from <cit.> (red squares). Both light curves are fit well after 400 s, although the 15–150 keV flux is too high at early times.
§.§.§ Towards an inverse Compton model
The presence of a thermal component in the XRT band with a time-scale and luminosity comparable to the nonthermal component motivates consideration of an inverse Compton (IC) mechanism for the prompt emission. Before investigating physical situations that might lead to strong IC upscattering, we first discuss some features generic to IC models for GRB 060218. For now, we assume only that some hot electrons are present in the burst environment, but place no condition on the location of these electrons or the mechanism of their acceleration.
First, if the primary photon source is approximately monochromatic (as is the case for the observed constant-temperature thermal component), then the quickly decaying peak energy implies that the scattering electrons are rapidly decelerating in the comoving frame. If the source frequency is $\nu_0 \sim k_B T_0$ in the observer frame, and the electrons dominating the radiation have comoving Lorentz factor $\gamma_p$, then the IC component will peak at $E_p \sim \gamma_p^2 k T_0$. This implies $\gamma_p \propto t^{-0.8}$, since $E_p \propto t^{-1.6}$ <cit.> and $T_0$ is approximately constant <cit.>. Suppose the electrons are accelerated to characteristic Lorentz factor $\gamma_m$, and can cool to Lorentz factor $\gamma_c$ in time $t$. If cooling is slow (i.e., $\gamma_m > \gamma_c$), then $\gamma_p \sim \gamma_m$. On the other hand, if cooling is rapid ($\gamma_c > \gamma_m$), then $\gamma_p \sim \gamma_c$ when $p < 3$, or $\gamma_p \sim \gamma_m$ when $p > 3$, because the $\nu F_\nu$ spectrum goes as $\nu^{(3-p)/2}$ between $\nu_m \equiv \gamma_m^2 k T_0$ and $\nu_c \equiv \gamma_c^2 k T_0$ <cit.>. $E_p$ evolves from $\sim 40$ keV at first detection to $\sim 1$ keV at $2000$ s <cit.>, and $k_B T_0 \simeq\,0.17$ keV for most times, implying that $\gamma_p$ varies from $\sim 15$ to $2$ throughout the prompt phase.
Second, the scattering medium is at most moderately optically thick, i.e. the electron scattering optical depth is $\tau_e \la 1$. This follows from the observation of distinct thermal and nonthermal components, since when $\tau_e \gg 1$, essentially all of the photons undergo multiple scatterings, resulting in a single nonthermal peak. A rough estimate of $\tau_e$ can be made by comparing the number of thermal and nonthermal photons. The thermal component, with peak luminosity $L_{th} \sim 1 \times 10^{46}$ erg s$^{-1}$ and time-scale $t_{th} \sim 3000 $ s, carries $N_{th} \sim L_{th} t_{th}/kT_0 \sim 1 \times 10^{59}$ photons. The nonthermal component has $L_{nt} \sim 3 \times 10^{46}$ erg s$^{-1}$ at peak, duration $t_{nt} \sim 1000$ s, and peak energy $E_p \sim 1$ keV near maximum light, and therefore contains $N_{nt} \sim 2 \times 10^{58}$ photons. This implies $\sim 1/6$ of thermal photons are scattered by electrons with $\gamma_e \simeq \gamma_p$, i.e. $\tau_e \sim 0.2$. Note that $N_{th} \propto L_{th} t \propto t^{1.8}$ according to our model for thermal emission in Section <ref>, while $N_{nt} \propto F_0 t \propto t^2$ according to the Band function model described earlier in this section. Hence $\tau_e \propto t^{0.2}$ is approximately constant in time in the simplest description.
Third, the nonthermal luminosity exceeds the thermal luminosity throughout most of the evolution. This implies that the Compton parameter $y \sim \tau_e \langle \gamma_e^2 \rangle \ga 1$. $\langle \gamma_e^2 \rangle$ represents the average gain per scattering. Because $\tau_e \la 1$, we require $\langle \gamma_e^2\rangle \ga \tau_e^{-1} \ga 1$ to get $y \ga 1$, suggesting at least mildly relativistic electrons. Since the total nonthermal luminosity is $L_{nt} \sim \tau_e \langle \gamma_e^2 \rangle L_{th}$, we can write $y \sim L_{nt}/L_{th}$. Near maximum light, when the nonthermal component peaks in the XRT band, we can estimate $y$ directly: at 1000 s we have $L_{nt}/L_{th} \sim y \sim 6$, and at 3000 s we have $L_{nt} \approx L_{th}$ and $y \sim 1$. In our simple Band function description, $y \propto E_p F_0/L_{th} \propto t^{-1.4}$, consistent with the values above.
Finally, the prompt spectrum holds information about the distribution of scattering electrons. For electrons distributed as a power law in energy with index $p$, the spectral slope above $\nu_c$ and $\nu_m$ is given by $F_\nu \propto \nu^{-p/2}$. The high energy spectrum in GRB 060218 has typical index $\beta_2 \approx -1.5$, but $\beta_2$ varies from $-3$ to $-1$, implying $p$ is in the range $2$–$6$ with typical value $p \approx 3$. While most GRBs have $p$ values closer to 2, $p=3$ is not outside of the observed spread in $p$ values <cit.>.
In summary, any IC model for GRB 060218 should be in the limit of moderately small scattering optical depth but appreciable energy gain, so that the thermal component carries most of the photons (i.e., $N_{nt}/N_{th} \sim \tau_e \la 1$), but the nonthermal component carries most of the energy (i.e., $E_{nt}/E_{th} \sim y \ga 1$). $\tau_e$ most likely varies slowly in time, while $y$ and $\gamma_p$ decrease rapidly. We now look at how well three different IC models satisfy these criteria.
§.§.§ The photospheric model
In the past several years, prompt thermal X-rays have been inferred from spectral fits in a number of GRBs, prompting the investigation of Comptonized photospheric models for the primary radiation <cit.>. In this picture, some of the bulk kinetic energy is dissipated into leptons within the jet, perhaps by internal shocks or magnetic effects. Depending on the optical depth at which the dissipation occurs, different emergent spectra are possible. Numerical simulations by <cit.> have shown that, for leptons initially distributed as a power law $N(E) \propto E^{-p}$ (as typically assumed for shock heating), the observed spectrum takes the form of a thermal component with a high-energy power-law tail for relatively low dissipation optical depth $\tau_{diss} \sim 0.01-0.1$. This type of spectrum is qualitatively similar to the observed spectrum of GRB 060218. If the dissipation optical depth is decreasing with time, then one expects the peak energy to continuously decrease and the nonthermal component to become relatively weaker until only the thermal component remains. This, too, is qualitatively consistent with GRB 060218, where the nonthermal component gradually fades, leaving a blackbody-dominated spectrum by $\sim 7000$ s.
However, the photospheric view is not without its problems. For one, it is unclear how quickly $E_p$ decays. In cosmological GRBs, the peak of the nonthermal component and the temperature of the thermal component are observed to be correlated, i.e. $E_p \propto T_0^{\alpha_T}$ with $\alpha_T$ typically $1$–$2$, and $T_0$ typically evolves with time <cit.>. GRB 060218 does not obey this correlation, nor does it show evidence for an evolving temperature. Nevertheless, the photospheric model provides a reasonable framework to interpret the early spectrum in GRB 060218, and is deserving of further attention. More work is needed to understand subphotospheric emission, particularly in the low-luminosity and small Lorentz factor limit, and in the case where a dense envelope surrounds the progenitor star, but detailed photospheric modeling is beyond the scope of the the current paper.
§.§.§ IC emission from external shocks in a long-lived jet
A long-lived central engine not only serves as a possible source of strong photospheric emission, but also influences ejecta dynamics by driving shocks into the surrounding medium. Here, we investigate IC scattering of prompt thermal photons from relativistic electrons in an engine-sustained reverse shock as a source for the observed prompt nonthermal emission.
First, we argue against the forward shock (FS) as the predominant IC emission site, because in this case obtaining a rapid decline of both the peak energy and the high energy light curve is difficult. Let the density of circumstellar material (CSM) as a function of radius be $\rho \propto r^{-\alpha}$. A steep $E_p$ decline implies a rapid deceleration, which in turn suggests a flat CSM density profile so that the FS sweeps up mass more quickly. Yet, the optical depth through the shocked region $\tau \sim \kappa \rho r \propto r^{1-\alpha}$ actually increases with time when $\alpha < 1$. Compounding this with the rising thermal luminosity $L_{th} \propto t^k$, the peak spectral luminosity ($L_{\nu,max} \propto L_{th} \tau \propto t^k r^{1-\alpha}$ in the optically thin case) rises sharply with time in a flat density distribution. Since the BAT band lies above the peak energy, the luminosity there scales as $L_{BAT} \propto L_{\nu,max} \nu_c^{1/2} \nu_m^{(p-1)/2}$ <cit.>. Whether the peak is due to $\nu_c$ or $\nu_m$, the rising $L_{\nu,max}$ makes it difficult to ever obtain a BAT flux that declines faster than the peak energy in the case where the FS dominates emission.
This problem is alleviated by considering the reverse shock (RS) as the emission site instead. An immediate question is how to attain a declining peak energy at the RS, since generally this shock would go to higher Lorentz factor (in the fluid frame) as the flow is decelerated. As it turns out, a long-lived central engine can help in this regard. In the limit where the engine deposits mass into the RS more quickly than the FS sweeps mass from the CSM, the dynamics differ markedly from the typical GRB case. To illustrate this, consider ejecta with mass $M_{ej}$ and Lorentz factor $\gamma_{ej}$ interacting with a cold CSM. Previous authors <cit.> have shown that the outflow undergoes a dynamical transition when the mass swept by the forward shock, $M_{sw}$, becomes equal to $\sim M_{ej}/\gamma_{ej}$. When $M_{sw} \ll M_{ej}/\gamma_{ej}$ the shocked shell at the ejecta-CSM interface coasts with a constant Lorentz factor $\gamma_{sh} \approx \gamma_{ej}$, and when $M_{sw}\gg M_{ej}/\gamma_{ej}$, $\gamma_{sh}$ evolves with time. In the typical GRB case, where there is no continued mass or energy input from the central engine, $M_{ej}$ is constant while $M_{sw}$ grows over time; the system begins in the coasting state, and the shell starts to decelerate once $M_{sw}$ becomes sufficiently large. However, for continuous mass ejection, $M_{ej} \propto \int \dot{M}_j dt$ also increases with time. The dynamics will be altered if $M_{ej}$ grows faster than $M_{sw}$, which is possible for a steep CSM density gradient. In that limit $M_{sw} > M_{ej}/\gamma_{ej}$ initially, and the shell accelerates, eventually reaching a terminal Lorentz factor $\gamma_{sh} \approx \gamma_{ej}$ once $M_{sw} \ll M_{ej}/\gamma_{ej}$. In this scenario, the RS decelerates steadily in the contact discontinuity frame. Therefore, $E_p \propto \nu_m$ falls off quickly if the emission comes from a rapid-cooling reverse shock. In a steep density gradient, $\tau \propto r^{1-\alpha}$ also decreases with time, making it possible for $L_{BAT}$ to decline faster than $E_p$ if the emission comes from the RS.
While this rough example serves to illustrate the benefit of prolonged engine activity, calculating the XRT and BAT light curves requires a more thorough model. For an engine-driven outflow the resulting forward-reverse shock system can be divided into four regions – the cold CSM, the shocked CSM, the shocked outflow, and the unchecked outflow – which we label with subscripts 1 to 4, respectively. Regions 2 and 3 are separated by a contact discontinuity. Taking a constant adiabatic index of $4/3$, and assuming that the bulk Lorentz factor and internal energy in regions 2 and 3 do not change much between the shock and contact discontinuity, the dynamics of this system are governed by the equations <cit.>:
\begin{equation}
\label{fdefine}
f \equiv \bar{\rho}_4(R_{rs})/\rho_1(R_{fs}),
\end{equation}
\begin{equation}
\label{fgeneral}
f = \dfrac{(4\gamma_2+3)(\gamma_2-1)}{(4\bar{\gamma}_3+3)(\bar{\gamma}_3 -1)},
\end{equation}
\begin{equation}
\label{gamma3general}
\bar{\gamma}_3 = \gamma \gamma_2 (1-\beta \beta_2),
\end{equation}
\begin{equation}
\label{radius}
R_{fs} \approx R_{rs} \approx \dfrac{\beta_2 ct}{1-\beta_2},
\end{equation}
where $R_{fs}$ is the radius of the FS and $R_{rs}$ is the radius of the RS. These equations are valid in both the ultrarelativistic and mildly relativistic limit, but break down as the forward shock becomes nonrelativistic since in that case $R_{rs} \ll R_{fs}$.
To close the system of equations (<ref>)–(<ref>), expressions for the densities $\rho_1$ and $\bar{\rho}_4$ as functions of radius $r$ are needed. The outer wind density we parametrize by
\begin{equation}
\label{rho1}
\rho_1 = 5 \times 10^{11} A_* r^{-2} (r/R_{ext})^{2-\alpha} \text{\,g\,cm}^{-3}.
\end{equation}
Here, $A_*$ is a parameter scaled to a pre-explosion mass-loss rate of $\dot{M}_{wind} = 10^{-5}\,M_\odot$ yr$^{-1}$ and velocity $v_{wind} = 1000$ km s$^{-1}$. While $A_* =1$ is typical for Wolf-Rayet progenitors, a star with an extended envelope could have a different mass loss history. $\alpha$ determines the slope of the CSM density profile, with $\alpha=2$ corresponding to the usual wind profile. For a given $A_*$, winds with different $\alpha$ are scaled to the same density at $R_{ext}$. The density of the inner wind is given by
\begin{equation}
\label{rho4}
\bar{\rho}_4 = \dfrac{1}{4 \pi r^2 \beta \gamma} \dot{M}_{iso}(t_{emit}),
\end{equation}
with $\dot{M}_{iso}$ given by equation (<ref>). The time $t_{emit} =(\beta-\beta_2)t/\beta(1-\beta_2)$ takes into account the delay between the arrival of photons and matter from the central engine at the reverse shock. Note that our model is only valid when $\alpha < 3$ (i.e., when the swept CSM mass is not negligible) and $k > -2$ (i.e., when the energy input from the central engine is not negligible).
We solved numerically the system of equations (<ref>)–(<ref>) to determine the dynamical variables. For reference, we also present analytical solutions of these equations in the limit $\gamma \gg 1$ in Appendix <ref>. Once $\gamma_2$, $\bar{\gamma}_3$ (or $\bar{\beta}_3$), and R are known, the spectral parameters $\nu_m$, $\nu_c$, and $L_{\nu,max}$ for the forward- and reverse-shocked regions can be determined using the standard theory, as outlined in Appendix <ref>, where we give analytical expressions for the maximum spectral power $L_{\nu,max}$ and break frequencies $\nu_c$ and $\nu_m$. In calculating the spectrum, we approximate the thermal photons as monochromatic with frequency $\sim 3 kT_0$. We take the blackbody temperature to be $0.14$ keV as in <cit.>, as we find this gives a better fit than the higher value inferred by <cit.>. The nonthermal spectrum is taken to have a power-law form with breaks at $\nu_m$ and $\nu_c$, and the BAT and XRT light curves are determined by integrating the spectrum over $15$–$150$ keV and $0.3$–$10$ keV, respectively. Our best fitting models have $p>3$, so in all cases discussed here $E_p \simeq h\nu_m$. The free parameters of our models are $\alpha$, $p$, $A_*$, $\gamma$, and the fractions of postshock energy $\epsilon_{e2}$ and $\epsilon_{e3}$ going into relativistic electrons in the FS and RS respectively. We fix $k$, $L_0$, and $\xi$ to the values inferred from the observed thermal component, as in Section <ref>.
We calculated $L_{BAT}$, $L_{XRT}$, and $E_p$ over a range of parameter space. In order to match the observed slope of $E_p$ and $L_{BAT}$, we fixed $\alpha$ to 2.7; for other values of $\alpha$, the fit for these observables is typically worse, although the fit to $L_{XRT}$ is sometimes improved. We varied $p$ from 3.0–4.0 in steps of 0.1, $\log A_*$ from 2.0–5.0 in steps of 0.0125, $\gamma_e$ from 6.0–12.0 in steps of 0.025, and $\log \epsilon_{e3}$ from -3.5 to -1.5 in steps of 0.025. In no case were we able to fit the spectrum reasonably with FS emission, so rather than vary $\epsilon_{e2}$, we fit the light curves with the sum of thermal emission and nonthermal RS emission, and then calculate an upper limit on $\epsilon_{e2}$ by assuming the FS contributes less than 30% of the flux in XRT and BAT. In addition, we place an upper limit on $\epsilon_{B}$ by demanding that the comoving energy density ($u_{rad}$) of thermal radiation at the RS be higher than the comoving energy density in magnetic fields ($u_B$). For each model, we calculated the reduced chi-squared $\chi^2$ via a joint fit to the observed XRT luminosity from <cit.> and the BAT luminosity and peak energy measured by <cit.>. We do not include the XRT data after $4000$ s in the fit.
In some cases – particularly when $A_*$ is rather large – the optical depth of the shocked regions exceeds unity. Our model, which assumes single scattering, is not valid in that case. Thus, we discard models with high optical depth, keeping only models that become optically thin prior to $300$ s. After that cut is applied, our best-fitting model has $p=3.8$, $A_* = 4900$, $\gamma = 10.8$, and $ \epsilon_{e3} = 6.0 \times 10^{-3}$. This model is shown in Figure <ref>. We find that the FS does not contribute substantially to the emission when $\epsilon_{e2} \la 0.05 \epsilon_{e3}$. $u_{rad}/u_B > 1$ at all times in this model if $\epsilon_B < 2 \times 10^{-5}$. If we relax the condition to $u_{rad}/u_B > 1$ only after 300 s, then $\epsilon_B \la 10^{-4}$. While the low upper limit on $\epsilon_B$ is somewhat troubling, we note that other authors <cit.> have also found a low value of $\epsilon_B$ compared to $\epsilon_e$ in GRB 060218.
Top: XRT (0.3$-$10 keV) and BAT (15$-$150 keV) light curves for our reverse shock IC model. When fixing $k=0.66$, $\xi=1$, $L_{0,46}=1$, and $\alpha=2.7$, the best-fitting parameters are $\epsilon_{e3} = 6.0 \times 10^{-3}$, $\gamma=10.8$, $A_* = 4900$, and $p=3.8$ The thermal, nonthermal, and total XRT luminosities are shown in blue, green, and orange respectively, while the BAT luminosity is drawn in red. Dashed curves show the contribution of the forward shock to the BAT (red) and XRT (green) light curves, assuming $\epsilon_{e2}/\epsilon_{e3} = 0.05$. The black diamonds are XRT data from <cit.>, and the red squares are BAT data from <cit.>. Bottom: The peak energy in our model, as compared to data from <cit.>.
While this model can plausibly fit the light curves, it cannot explain the low-frequency spectral shape: for $\nu_c < \nu < \nu_m$, we have $F_\nu \propto \nu^{-1/2}$, steeper than the observed spectrum $F_\nu \propto \nu^{-0.1}$. However, we note that <cit.> found a different spectral shape at low energies, $F_\nu \propto \nu^{-0.4}$, when using a cut-off power-law to fit the data instead of a Band function, so the observed $\beta_1$ seems to depend in part on the assumed spectral model. We also find a high value of $p\simeq3.8$ that, while roughly consistent with the observed value of <cit.>, is large compared to the value in typical GRBs <cit.>. An additional issue with our model is that it slightly underpredicts the XRT flux near peak by up to a factor of 2, and slightly overestimates the peak energy. Furthermore, our model is only one-dimensional, and it does not take into any effect of collimation or sideways expansion of the jet.
Despite these issues and the crudeness of the model, the reverse shock IC interpretation does a reasonable job of capturing the basic behaviour of the light curves, and it has some attractive features. Notably, of all the models we consider, this model is the only one that provides a natural explanation for the steep decline of $E_p$ and $L_{BAT}$. Additionally, this type of emission is expected when a long-lived, dissipative jet is present, and should therefore contribute to the emission on some level (although, IC emission from the RS only dominates the contribution from external shocks under certain circumstances, as described above). However, if other lepton populations (e.g., those excited by internal shocks or other dissipation in the jet interior) also strongly contribute to the emission, the external shock emission may not be observed.
Having an independent estimate for the outflow Lorentz factor allows us to break the degeneracy of our radio model discussed in Sections <ref> and <ref>, by calculating $E_{iso}$ directly. Assuming $t_L$ corresponds to the turn-off time of the engine, we find $M_{iso} = 4.9 \times 10^{-3}\,M_\odot$ and $E_{iso} = 8.7 \times 10^{52}$ ergs via equations (<ref>) and (<ref>). Thus, the high energy, high Lorentz factor radio model is preferred. The low upper limits on $\epsilon_B$ and $\epsilon_e$ in this section are consistent with the ranges inferred from the radio (see discussion at the end of Section <ref>). The jet is cold and radiates inefficiently, with $\eta_\gamma = 6.9 \times 10^{-4}$. Applying $\theta_0 \ga \gamma^{-1}$, we find that the true jet energy is $E_j \ga 4 \times 10^{50}$ ergs. These results have interesting implications when compared to standard GRBs. $E_{iso}$ and $E_j$ fall within the range typical for cosmological bursts <cit.>, suggesting that the total kinetic energy released in GRB 060218 is not unusual, although it is released over a longer time. The main factor that distinguishes GRB 060218 from the bulk of observed GRBs is therefore its radiative efficiency: whereas most bursts have $E_{\gamma,iso} \approx E_{iso}$ <cit.>, our model for GRB 060218 predicts $E_{\gamma,iso} \ll E_{iso}$. This fact is closely linked to the low values we deduced for the $\epsilon$ parameters, which may perhaps be related to the lower bulk Lorentz factor or the long engine lifetime. For now, this is only speculation, but the possibility that standard GRBs are a corner case where the radiative efficiency is high (due, perhaps, to a higher Lorentz factor or a more abrupt deposition of kinetic energy), while most collapsar events go unobserved because of a much lower radiative efficiency, is intriguing.
In addition, we can deduce some properties of the CSM near the progenitor from the inferred value of $A_*$. If it extends in to $R_{ext}$, the wind is optically thick to electron scattering, with total optical depth $\tau_w = \int_{R_{ext}}^{\infty} \rho_1 \kappa dr \approx 30$. However, because of the steep density gradient, the high optical depth is due mostly to material very close to $R_{ext}$. In fact, the shock radius in our model is $R \approx 5 \times 10^{13}$ cm at the time of first observation; our model does not constrain the wind density at radii less than this. The optical depth of the wind is small compared to the envelope optical depth, so the addition of such a wind does not affect the optical model discussed in Section <ref>. This wind cannot be the origin of the high value of $N_H$, however, as the absorbing column through the wind changes as the shock propagates outward, while the observed $N_H$ is constant.
The extent of the wind is not known, but the equipartition radius $R_N = 1.3 \times 10^{16}$ cm at day 5 gives an upper limit, since as discussed in Section <ref> the radio observations imply a constant density CSM. The total mass of the wind is therefore $M_w < \int_{R_{ext}}^{R_N} 4 \pi r^2 \rho_1 dr = 3.6 \times 10^{-3} M_\odot$. This is comparable to the isotropic mass of the jet, so it is possible that the jet undergoes some deceleration while sweeping the outer layers of the envelope. In addition, even though the terminal Lorentz factor is $\gamma_2 \approx \gamma = 10.8$ in our model, the transition to the coasting state is quite gradual: we find $\gamma_2$ only reaches $\approx 5$ by the end of the prompt phase at $t_L$. It therefore seems plausible that the jet could decelerate to $\beta\gamma \sim 1$ by day 5, as implied by radio observations. We stress, however, that there is still some tension in producing the flat radio light curve with a mildly relativistic outflow.
After the source of thermal photons fades away, the emission from external shocks will be dominated by synchrotron radiation. Since the overall SED appears incompatible with a single synchrotron spectrum, this component should not overwhelm the optical emission and dust echo afterglow emission observed at the same time. Since the jet begins to decelerate shortly after $t_L$ in our model, the synchrotron emission peaks near $t_L$. At that time we find that the critical synchrotron frequencies are $\nu_m \simeq 6 \times 10^{-9}$ Hz and $\nu_c \simeq 4 \times 10^{12}$ Hz; both are far below the optical band. We calculate a peak synchrotron $\nu F_\nu$ flux of $5 \times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ $2 \times 10^{-17}$ erg cm$^{-2}$ s$^{-1}$, respectively at $5 \times 10^{14}$ Hz and $1$ keV. This is far below the observed $\nu F_\nu$, which is $\sim 10^{-11}$ erg cm$^{-2}$ s$^{-1}$ in both the X-ray and optical bands <cit.>. The main reason synchrotron emission from the jet is so weak compared to cooling envelope emission and dust scattering of the prompt light is the low values of $\epsilon_e$ and $\epsilon_B$. This answers the question posed at the end of Section <ref>, suggesting that the primary reason a dust echo is observed in GRB 060218 is because of low values of the microphysical parameters.
§.§.§ Other models for jet emission
Here we briefly consider some other possibilities for the prompt emission, but each is problematic. Thus, we prefer an inverse Compton interpretation for the prompt emission in GRB 060218.
Synchrotron emission from external and/or internal shocks is also expected for relativistic jets. The standard FS synchrotron model, with constant external density and no continuous energy injection (i.e., $\alpha=0$ and $k=-2$), gives $E_p \propto \nu_m \propto t^{-3/2}$ and $F(E \gg E_p) \propto t^{(2-3p)/4}$ in the rapid cooling limit <cit.>. When $p\simeq 3$, this gives a time behaviour similar to the observed one. However, the peak energy in this case is much too low to explain observations, even for unphysically high explosion energies.
Internal shock models, in which ejecta shocked by the collision of successive engine-launched shells radiate via synchrotron, remain a prominent model for the prompt radiation in cosmological GRBs <cit.>. This picture provides a natural interpretation of the high degree of variability in GRB light curves, as the many shell collisions give rise to multiple peaks. GRB 060218, with its smooth, single-peaked light curve, may therefore be hard to explain in an internal shock context, unless an additional mechanism acts to smooth out the light curve. It is unclear, as well, how the presence of an extended envelope could affect the internal shock signature.
In some cases, steep decays in the prompt GRB light curve have been attributed to a kinematical effect, wherein the observer continues to see emission from high-latitude parts of the curved emission region after the prompt emission process ends. This phenomenon, known as the curvature effect, leads to fainter and softer emission over time because of relativistic beaming. <cit.> already investigated curvature effects for GRB 060218, and showed that the simultaneous steep decay of the peak energy and high energy light curve is inconsistent with this interpretation.
§ DISCUSSION
We have presented a model for the peculiar GRB 060218 in which the prompt X-ray emission arises from a low-power jet and the early optical emission is powered by fast SN ejecta interacting with a low-mass circumstellar envelope. Our picture has some features in common with the recent model of <cit.>, where the prompt X-ray and optical emission is produced by a choked jet interacting with a circumstellar envelope. In both cases, a jet is needed to decouple the mildly relativistic outflow from the SN, and an extended envelope of similar mass ($\sim 0.01$ $M_\odot$) and radius ($\sim 100$ $R_\odot$) is inferred. Both models provide a reasonable explanation for the radio afterglow flux, although Nakar's model has an advantage in explaining the shallow slope of the light curve. Neither model can account for the X-ray afterglow through external shock synchrotron radiation alone. There are several key differences between the models, however. We differ on the jet properties (we suggest a low-power, long-duration jet, whereas Nakar uses a more typical GRB jet), the origin of the prompt X-rays (we prefer a dissipative jet and some Compton scattering process, whereas Nakar posits shock breakout), and the power source for the cooling envelope emission seen in the optical band (we suggest that it is driven by the underlying SN, whereas Nakar proposes a smothered jet explosion). A detailed discussion of the strengths and weaknesses of each model is therefore warranted.
An advantage of Nakar's model is that the luminosity and time-scale of the jet take on typical GRB values. In addition, shock interaction naturally produces a smooth, single peaked light curve in X-rays, as observed <cit.>. This model also helps to explain the lack of a jet break in the radio, since the jet outflow becomes quasi-spherical before leaving the envelope. A wind-like CSM profile is also inferred for afterglows powered by shock breakout <cit.>, which is expected for a WR star progenitor. On the other hand, the high value ($\sim 50$ keV) and slow decay ($\propto t^{-(0.5-1)}$) of the prompt peak energy that one expects in the shock breakout scenario <cit.> seem hard to reconcile with direct measurements of the peak energy that show it declines steeply as $t^{-1.6}$ and with a value of $\sim$ a few keV throughout most of the prompt phase <cit.>. The fact that the prompt optical and X-ray emission are observed simultaneously, and that they each evolve smoothly from the earliest observation, also seems hard to interpret in a shock interaction model where the observed radiation evolves from a nonequilibrium state toward thermal equilibrium. A better understanding of the expected X-ray signal from mildly relativistic shocks in low-mass envelopes is needed to determine whether these problems can be resolved. The origin of the prompt thermal X-ray component is also unclear in the choked jet model, since the inferred photospheric radius is considerably smaller than the envelope radius. Finally, as discussed above, there is some question of whether an ultrarelativistic jet can truly be sphericized in a 0.01 $M_\odot$ envelope. Detailed hydrodynamical simulations will be crucial to fully understand how such an envelope affects jet propagation and sideways expansion.
Our low-luminosity jet model comes with its own merits and drawbacks. A jet origin for the prompt X-rays and $\gamma$-rays places GRB 060218 at the low-luminosity, long-duration end of a continuum of GRB processes. In this unified picture, similarities to cosmological bursts (such as satisfying the Amati and lag-luminosity correlations) are perhaps not surprising, although as in typical GRBs the physical origin of these correlations is not well understood. None the less, these coincidences are not easily accounted for in the shock interaction view. Furthermore, the presence of a thermal component is expected for a dissipative jet, and decoupling the prompt X-ray and optical emission removes problems with the X-ray to optical evolution. However, the low-power jet interpretation inherits one usual problem with jet models, namely that the prompt emission mechanism in relativistic jets is still not well understood. Also, since a low-luminosity jet stays collimated while it drills through the circumstellar envelope, our model requires the jet to become nonrelativistic in the CSM, which may be difficult unless some additional mass close to the star helps to decelerate the jet. While we infer a wind-like CSM at small radii where the prompt X-rays are emitted, we find that a uniform circumburst density is needed beyond $ 10^{16}$ cm where the radio is emitted. This is contrary to usual expectations for a WR progenitor. Finally, we require an unusually long-lived, low-power central engine, the origin of which is unclear.
This last point deserves more discussion. A shortcoming of our model, as with many engine driven models, is the need to prescribe unknown properties of the central engine. A simple parametrization glosses over many of the finer details of compact object formation and jet launching, the physics of which are not yet fully understood. In particular, producing a long-duration, low-luminosity engine from a nascent black hole presents problems: <cit.> have shown in their black hole simulations that a minimum jet luminosity of $\sim 10^{49} \text{ erg s}^{-1}$ is needed to overcome the ram pressure of accreting material, and black hole-driven engines tend to operate on time-scales much shorter than $\sim 10^3$ seconds. However, the SN might clear away infalling material, thus allowing a lower luminosity jet to propagate. It is unclear whether the formation of a neutron star (or magnetar) could drive the type of mildly relativistic outflow we require, but the longer time-scales involved are more consistent with the long-lasting prompt emission observed <cit.>. Magnetar-powered scenarios are particularly intriguing in light of the recent result of <cit.>, who claim the detection of magnetar-driven superluminous SN associated with the ultra-long GRB 111209A. Here, we only aim to show that a low-luminosity outflow, if present, can explain many features of the prompt thermal and nonthermal emission. Note that several other bursts, such as GRB 130925A <cit.> and the ultra-long bursts discussed by <cit.>, may also require long-lived central engines, so this problem is not unique to GRB 060218.
The lack of variability in the light curves of GRB 060218 and GRB 100316D merits further investigation, as well. If the typical GRB variability originates from relativistic turbulence, then the smooth light curves observed in some LLGRBs could be ascribed to the lack of highly relativistic material <cit.>. Even in the absence of relativistic effects, a light jet lifting heavier external material would give rise to Rayleigh-Taylor instabilities that may induce light curve fluctuations. This could be circumvented by, e.g., a Poynting-flux dominated jet, but a matter-dominated jet is required to produce the prompt nonthermal X-rays through IC processes. The smooth light curve also constrains the degree of clumpiness which can be present in the CSM. Detailed numerical simulations will be needed to characterize the amount of variability expected from a mildly relativistic jet as it penetrates the star and envelope, breaks out from the envelope's edge, and sweeps the surrounding CSM.
We note that, because our model involves an on-axis jet, we cannot appeal to geometric effects to increase the rate of 060218-like events. However, our model does imply a unique, non-standard progenitor different from the usual high-mass WR stars thought to give rise to most cosmological LGRBs. Thus, our explanation for the high rate of LLGRBs is simply that LLGRB progenitors are intrinsically $10$–$100$ times more common. Assuming that the presence of a long-lived, low-luminosity jet is also somehow tied to the progenitor structure, such jets might also be more common than ultrarelativistic, short-lived ones, but we are biased against observing them due to their low power. In the model of <cit.>, the progenitor is again different from the standard one, but the prompt emission is roughly isotropic as well, so that the higher event rate is due to some combination of geometric and intrinsic effects.
It may be possible to construct a "hybrid" model that retains some of the best features of both our model and Nakar's. This speculative model is depicted in Figure <ref>. Suppose that the central engine switches off while the jet is traversing the envelope (as in Nakar's model), but let the envelope mass be smaller (as in our model) so that the outflow decelerates significantly and expands sideways somewhat, but does not have time to sphericize before breaking out. The explosion then breaks out aspherically, with shock breakout emission expected from near the poles. If dissipation continues to occur after the cessation of engine activity, a thermal component might also be observable once the ejecta clear out. After breakout, the ejecta expand into the low-density CSM, eventually producing the radio synchrotron emission as electrons are accelerated by the external shocks. Since the outflow expands preferentially into the CSM post-breakout, a quasi-toroidal envelope remnant is left behind, which is shocked by the fast SN ejecta and then emits the prompt optical emission as it cools. The X-ray afterglow is produced by dust scattering, as in Section <ref>. As with our model, this scenario gives a possible explanation for the thermal emission, and since the optical and X-ray are decoupled there are no concerns with the X-ray to optical spectral evolution. Yet, as with Nakar's model, this case generates a smooth prompt X-ray light curve via shock breakout, and more easily explains the radio because the initial outflow is wider than a jet. However, the expected signal from an aspherical shock breakout in the relativistic limit has not been calculated in detail, which is an important caveat.
A hybrid model for the prompt emission. Upper left: A jet is launched with short time-scale compared to the envelope breakout time, as in <cit.>. However, the jet does not have time to become quasispherical before breaking out; it undergoes significant deceleration, and possibly a small degree of lateral spreading, but the explosion breaks out primarily in the forward direction, leaving the envelope mostly intact. Thermal emission could be observed, e.g. from the walls of the jet cavity, once material clears out along the line of sight. Upper right: As in our model, the fast SN ejecta heat the remaining envelope, which cools through optical radiation. Lower left: The X-ray afterglow is produced from dust scattering, as described in Section <ref>. Lower right: The radio afterglow comes from a nonrelativistic, quasispherical blast wave. Because the ejecta are already decelerated to $\beta \gamma \sim 1$ by the envelope, a spherical flow is more readily achieved than in our jet breakout model.
Moving past the prompt emission, we have also shown that a dust echo model gives a reasonable fit to the X-ray afterglow light curve and spectral index evolution. The dust echo model used only an empirical fit to the prompt light curve and spectrum, and therefore is insensitive to the mechanism of prompt emission. Moreover, the scattering angle from the dust grains, $\theta_d \approx (2ct/R_d)^{1/2} \sim 1\degree (R_d/30 \text{ pc})^{-1/2} (t/10 \text{ days})^{1/2}$, is small, so the echo emission depends only on the prompt radiation roughly along the observer's line of sight. Thus, the dust echo interpretation applies equally well whether the prompt emission originates from a low-luminosity jet or from shocked gas. If the reason for the small radiated energy in bursts like GRB 060218 is small values of $\epsilon_e$ and $\epsilon_B$, as our model suggests, then dust echo type afterglows should commonly accompany this class of bursts, because the synchrotron emission from external shocks will be weak. So far, this is borne out by observations, as the afterglow GRB 100316D is also consistent with a dust echo. Why the synchrotron efficiency is poor, and whether this is related somehow to the long burst duration, remains to be worked out.
If our picture for GRB 060218 is correct, one would expect to observe broad-lined Type Ic SNe with accompanying mildly relativistic radio afterglows, but without a prompt X-ray component, when viewing GRB 060218-like events off-axis. The global rate of such events would be some 10–100 times greater than the on-axis rate, assuming wide opening angles in the range of $\sim$10–30 degrees. Such events might be uncovered by radio follow-up of Type Ic SNe. Future survey projects such as the Large Synoptic Survey Telescope should detect more Type Ic SNe with a double-peaked signature of cooling envelope emission, expanding the number of potential interesting targets for radio follow-up.
Whether or not the long prompt emission is tied to the presence of a circumstellar envelope is an interesting open question. Clearly, this is so for the model of <cit.>. For our model, though, the prompt X-ray and optical emission have different origins, so it may be possible to observe an X-ray signal akin to that of GRB 060218 with no prompt optical counterpart. (On the other hand, if the envelope plays a crucial role in jet dissipation, it may still be needed.) The high-$T_{90}$, high-variability light curves of ultra-long GRBs do seem to suggest the possibility of intrinsically long-lasting jet emission. Interestingly, several ultra-long bursts (e.g., GRB 101225A, GRB 111209A, and GRB 121027A) also show an early optical peak that may be consistent with shock cooling <cit.>. In other ultra-long GRBs (e.g., GRB 130925A and GRB 090417B), no optical light was detected, but the presence of early optical emission cannot be ruled out due to the high extinction to those events <cit.>. Overall, prompt optical emission is observed more often than not in very long bursts, hinting at one of two possibilities: either the engine duration is long because a circumstellar envelope is present, or a circumstellar envelope is present because the progenitors of long-duration engines also tend to have circumstellar envelopes. This topic is of considerable theoretical interest going forward.
Regardless of whether objects like GRB 060218 are powered by a jet or a shocked envelope, circumstellar interaction clearly has a role to play in explaining these unusual LLGRBs. Both our model and the <cit.> model can be taken as further indirect evidence for the existence of a dense environment immediately surrounding the progenitor star, indicative of strong pre-explosion mass loss or binary evolution. The mechanism driving this mass loss is unclear, but possibilities include late unstable nuclear burning <cit.>, gravity wave-driven mass loss <cit.>, or common envelope evolution <cit.>. Alternately, the circumstellar envelope could arise from a stripped binary scenario as in Type IIb SNe. We emphasize that the progenitor's pre-explosion history is a crucial factor in determining the observed radiation's characteristics, and that this theme applies broadly to many transients including Type Ia-CSM, Type IIn, and Type IIb SNe. Understanding the late phases of intermediate- to high-mass stellar evolution will play a critical role as our ability to detect transient phenomena continues to evolve.
§ CONCLUSIONS
We have presented a comprehensive model for the unique LLGRB GRB 060218 that provides reasonable explanations for each of its features. The model includes a peculiar engine-driven jet with a low luminosity ($L_{iso} \sim 3 \times 10^{49} \text{ erg s}^{-1}$) and a long duration ($t_j \sim 3000$ s), properties that we suggest are related to a non-standard progenitor. We have shown that, if the jet dissipates some modest fraction of its kinetic energy into thermal radiation, Comptonization of seed thermal photons by hot electrons can explain features of the prompt spectrum, light curve, and peak energy evolution. We investigated different emission sites for the IC process, and found that scattering from electrons in the reverse-shocked gas can roughly account for the prompt X-ray light curve and peak energy decay, if the fraction of energy put into magnetic fields and into electrons in the forward-shocked gas is small. Scattering from a nonthermal electron population within a dissipative jet outflow also remains a possibility for the prompt emission. Scattering from forward shock electrons can be ruled out, as the light curves and peak energy cannot be reproduced in this case. We also argued against a synchrotron origin for the prompt emission.
We analysed constraints on the jet properties from the prompt thermal emission, the radio afterglow, and dynamical considerations. There exists a region of parameter space that can fit both the radio afterglow and the prompt thermal emission without violating other constraints, although there is considerable degeneracy that prevents precise determination of the parameters. The early thermal emission and the late-time radio afterglow can be explained either by a cold jet with relatively high energy and Lorentz factor and relatively little postshock energy in electrons and magnetic fields, or by a hot jet with lower energy and Lorentz factor and standard choices for $\epsilon_e$ and $\epsilon_B$. Our IC model for the prompt emission breaks this degeneracy, strongly preferring the former scenario. We derived the jet parameters $E_k \simeq 4 \times 10^{50}$ ergs, $\gamma \simeq 11$, and $\theta_0 \simeq 0.1$, and find that the immediate circumstellar environment has a density profile of $\rho_1 \propto r^{-2.7}$ and wind parameter $A_* \simeq 4900$. The inferred microphysical parameters of the reverse shock are $\epsilon_{e3} \simeq 6 \times 10^{-3}$ and $p=3.8$. Combining the radio and prompt X-ray models, we constrained the magnetic parameter to $10^{-5.5} \la \epsilon_B \la 10^{-4}$ and the electron energy fraction in the forward shock to $10^{-5.5} \la \epsilon_{e2} \la 10^{-3.5}$. Radio observations constrain the density at $r > 10^{16}$ cm to be constant, with $n \sim 10^{3.5}$–$10^{5.5}$ cm$^{-3}$ depending on the values of $\epsilon_{e2}$ and $\epsilon_{B}$. However, there is some concern that the outflow will not have time to sphericize prior to the radio observations, which makes the shallow radio light curve difficult to interpret. Our results suggest that GRB 060218 may be an engine-driven event that has the same kinetic energy coupled to relativistic ejecta as in typical GRBs, but radiates very inefficiently in comparison. This result has interesting implications considering the high volumetric rate of LLGRBs.
We have shown as well that the early peak in optical/UV can be powered by interaction of the fast outer SN layers with a low-mass extended envelope surrounding the progenitor star. With the SN parameters inferred for SN 2006aj, and the luminosity and blackbody radius implied by the measured host extinction, we derive the envelope parameters $M_{ext} \approx 4 \times 10^{-3} M_\odot$ and $R_{ext} \approx 9 \times 10^{12}$ cm. SN 2006aj is perhaps the best case so far of a double-peaked light curve characterized by cooling envelope emission, as described by <cit.>.
We also tested the idea that the unusual X-ray afterglow in GRB 060218 is a dust echo of the prompt emission, as suggested by <cit.> to explain the extremely soft afterglow spectrum. Using the available prompt emission data as an input, we modeled the expected dust echo emission from a shell of dust at $R_d$, with scattering optical depth $\tau_d$ at 1 keV. Assuming dust grains distributed uniformly in size from a minimum radius $a_- = 0.005 \text{ }\mu\text{m}$ to a maximum radius $a_+ = 0.25 \text{ }\mu\text{m}$, we found that $\tau_d = 0.006$ and $R_d \simeq 35$ pc gave a good fit to the afterglow light curve and the spectral index evolution. Because the echo emission does not depend on the prompt emission mechanism, this result is robust, making GRB 060218 quite a convincing case for a dust echo. The echo model implies only a modest amount of dust consistent with the dust content of the ISM. That the dust echo dominates over the usual synchrotron afterglow can be explained in this case by a low value of $\epsilon_e$ and $\epsilon_B$, consistent with our radio estimates and prompt X-ray modeling. We compared our results for GRB 060218 to the other bursts with soft afterglow spectra identified by <cit.>, and found that two distinct classes of echo-dominated afterglows are indicated: one requiring a typical amount of dust (like GRB 060218), and one requiring an unusually high amount of dust (like GRB 130925A).
We conclude by noting that our understanding of the class of low-luminosity, ultra-long GRBs with smooth light curves is severely hindered by the the small sample size – presently, GRB 060218 and GRB 100316D are the only constituent members of this class. In addition, because GRB 100316D lacks a detection of prompt optical emission or clear-cut evidence for prompt thermal emission, we are unable to draw any firm conclusions about it in our model. More observations of this unique class of objects is needed to settle questions about the prompt emission mechanism and the transition from beamed to spherical outflow, to better constrain the properties of the progenitor, envelope, jet, and CSM. With the rates estimated by <cit.>, Swift should turn up a new burst in this class every several years or so. In the meantime, advancing our theoretical understanding of shock-envelope interaction, the emission mechanism in relativistic jets, and the propagation of jets in complex circumstellar environments can furnish testable predictions for the next observed event.
§ ACKNOWLEDGEMENTS
We thank R. Barniol-Duran and B. Morsony for helpful discussions. This research was supported in part by NASA Grant NNX12AF90G.
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§ SHOCK DYNAMICS OF A RELATIVISTIC OUTFLOW INTERACTING WITH A POWER-LAW CSM
Analytical solutions of equations (<ref>)$-$(<ref>) are available when $\gamma \gg 1$. We consider the general case where the outer density profile is a power law in radius as in equation (<ref>), and the luminosity of thermal photons and the kinetic luminosity of the jet vary as power laws in time, i.e. $L_{th} =L_0 (t/t_L)^k$ and $L_{iso} =\mathcal{L}_0 (t/t_M)^s$. We therefore have $M_{iso}(t_{emit}) \approx L_{iso} \gamma^{-1} c^{-2} (t_{emit}/t)^s$. Combining equations (<ref>), (<ref>), and (<ref>) with the above expressions leads to
\begin{equation}
\label{f}
f = C_0 A_*^{-1} L_{iso,48} \gamma^{-2} \left(\dfrac{t_{emit}}{t} \right)^s \left( \dfrac{R}{R_{ext}} \right)^{\alpha-2} \left( \dfrac{R_{fs}}{R_{rs} }\right)^2.
\end{equation}
$C_0 = 5.9 \times 10^3$ is a dimensionless constant determined by scaling the density to $A_*$ and $L_{iso}$ to $10^{48}$ ergs s$^{-1}$. Note that, in our model, $L_{th}$ and $L_{iso}$ are related by equation (<ref>); when $\gamma \gg 1$ we have
\begin{equation}
\label{Liso_48}
L_{iso,48} \approx 2.3 \times 10^{-3} \xi^{-2} L_{th,46}^{1/2} \gamma^4.
\end{equation}
For the sake of convenience and generality we do not make this substitution yet.
Three dynamical limits are possible, depending on the relative value of $f$ and $\gamma$ <cit.>:
* The coasting regime ($f \gg \gamma^2$): The FS coasts with an approximately constant Lorentz factor, and the RS is Newtonian with $\bar{\beta}_3 \ll 1$:
\begin{equation}
\label{coasting}
\begin{array}{lr}
\gamma_2 \approx \gamma \\
\bar{\beta}_3 \approx \left( \dfrac{8\gamma^2}{7f}\right)^{1/2} \\
R = \dfrac{\beta_2 ct}{1-\beta_2} \approx 2 \gamma^2 ct \\
t_{emit} = \dfrac{\beta-\beta_2}{\beta(1-\beta_2)} t \approx 2 \bar{\beta}_3 t .
\end{array}
\end{equation}
The shocked regions are thin; the forward shock's size is $\sim R/\gamma^2$, and the reverse shock is even thinner by a factor $\bar{\beta_3}$, so that $R_{rs} \approx R \approx R_{fs}$ is a good approximation.
* The decelerating (or accelerating) regime ($\gamma^{-2} \ll f \ll \gamma^{2}$): The FS and the RS are both relativistic:
\begin{equation}
\label{accel}
\begin{array}{lr}
\gamma_2 \approx \left(\dfrac{f \gamma^2}{4}\right)^{1/4} \\
\bar{\gamma}_3 \approx \left(\dfrac{\gamma^2}{4f}\right)^{1/4} \\
R \approx 2 \gamma_2^2 ct \\
t_{emit} \approx t.
\end{array}
\end{equation}
Accelerating or decelerating cases are possible, depending on the evolution of $f$. As in the coasting case, the shocked regions are thin compared to their radius, so that $R_{rs} \approx R \approx R_{fs}$ applies.
* The nonrelativistic regime ($f \ll \gamma^{-2}$): A third solution is also possible in which the FS becomes nonrelativistic and $R_{rs} \ll R_{fs}$. As the requisite high CSM density and low engine Lorentz factor are unlikely to be encountered in GRB 060218, we do not discuss this scenario further.
The CSM density $\rho_1 \propto R^{-\alpha} \propto (\gamma_2^2 t)^{-\alpha}$ depends on time implicitly through $\gamma_2$. It is useful to separate out the explicit time dependence by defining $B_* = A_* (t/t_{ext})^{2-\alpha}$, with $t_{ext} \equiv R_{ext}/c$. Additionally, we make the convenient definition $\ell = 2-\alpha$. Then, by substituting equation (<ref>) into equation (<ref>) or (<ref>), one obtains solutions for the dynamical variables after some algebra:
\begin{equation}
\label{fdyn}
f = \left\{
\begin{array}{lr}
\left[C_1 L_{iso,48} B_*^{-1} \gamma^{-(\ell+2)+(s-\ell)}
\right]^{2/(s+2)},
& f > \gamma^2 \\
\left[C_0 L_{iso,48} B_*^{-1} \gamma^{-(\ell+2)}
\right]^{2/(\ell+2)},
& \gamma^{-2} < f < \gamma^2 \\
\end{array}
\right. ,
\end{equation}
\begin{equation}
\label{gamma2dyn}
\gamma_2 = \left\{
\begin{array}{lr}
\gamma,
& f > \gamma^2 \\
2^{-1/2} \left[ C_0 L_{iso,48} B_*^{-1} \right]^{1/2(\ell+2)},
& \gamma^{-2} < f < \gamma^2 \\
\end{array}
\right. ,
\end{equation}
\begin{equation}
\label{gamma3dyn}
\bar{\beta}_3 \bar{\gamma}_3 = \left\{
\begin{array}{lr}
\left(\frac{8}{7}\right)^{1/2} \left[C_1 L_{iso,48} B_*^{-1} \gamma^{-2(\ell+2)} \right]^{-1/(s+2)},
& f > \gamma^2 \\
2^{-1/2} \left[C_0 L_{iso,48} B_*^{-1} \gamma^{-2(\ell+2)}\right]^{-1/2(\ell+2)},
& \gamma^{-2} < f < \gamma^2 \\
\end{array}
\right. .
\end{equation}
We have defined $C_1 = (32/7)^{s/2} 2^{-\ell} C_0$, as this quantity appears repeatedly. We see that the essential dynamical variables all depend on $B_*^{-1} L_{iso} \propto t^{s-\ell}$, allowing for two possibilities. If $s < \ell$ we have the typical case where the forward shock begins in a coasting state and starts to decelerate once $f \sim \gamma^2$. If $s > \ell$, however, the forward shock starts with $\gamma_2 < \gamma$ and accelerates until it reaches a terminal Lorentz factor $\gamma$ when $f \sim \gamma^2$. When $s=\ell$, the shock velocity is constant in time; this special case generalizes the result of <cit.>, who studied a constant luminosity outflow ($s=0$) in a wind density profile ($\ell=0$). The usual afterglow dynamics <cit.> can be recreated with $s=-1$ (negligible energy input) and $\ell = 2$ (constant density CSM). The transition between dynamical regimes occurs when $f \simeq \gamma^2$, at an approximate time
\begin{equation}
\label{t_f}
t_f \simeq
\left[C_0^{-1} \mathcal{L}_{0,48}^{-1} A_* \gamma^{2(\ell+2)} t_{ext}^{-\ell} t_M^s
\right]^{1/(s-\ell)},
\end{equation}
although the exact time of transition differs for $f$, $\gamma_2$, and $\bar{\gamma}_3$ due to different leading numerical factors.
To obtain the spectral parameters in Appendix <ref>, it is useful to have expressions for $N_2$ and $N_3$, the number of electrons contained in regions 2 and 3. The number of electrons swept into the forward shock can be found by integrating over the CSM density profile: $N_2 = \chi_e \int_0^R 4 \pi r^2 \rho_1 dr/m_p $, where $m_p$ is the proton mass and $\chi_e$ is the average number of electrons per nucleon. We take $\chi_e = 0.5$, appropriate for hydrogen-free gas, which leads to
\begin{equation}
\label{N2}
N_2 = \dfrac{5.6 \times 10^{49}}{ (\ell+1)} \times \left\{
\begin{array}{lr}
2^{\ell+1} B_* \gamma^{2(\ell+1)} t_3,
& f > \gamma^2 \\
\left[C_0^{\ell+1} L_{iso,48}^{\ell+1} B_* \right]^{1/(\ell+2)} t_3,
& \gamma^{-2} < f < \gamma^2 \\
\end{array}
\right. .
\end{equation}
$t_3$ is the time in units of $10^3$ s. The number of electrons in the reverse-shocked region is $N_3 = \chi_e \int_0^{t_{emit}} L_{iso} \gamma^{-1} dt/m_p c^2$, which gives
\begin{equation}
\label{N3}
N_3 = \dfrac{5.6 \times 10^{49}}{(s+1)} \times \left\{
\begin{array}{lr}
2^\ell \left(\frac{32}{7}\right)^{1/2} \left[C_1 L_{iso,48} B_*^{s+1} \gamma^{2(\ell+2)(s+1)-(s+2)}
\right]^{1/(s+2)} t_3,
& f > \gamma^2 \\
C_0 L_{iso,48} \gamma^{-1} t_3,
& \gamma^{-2} < f < \gamma^2 \\
\end{array}
\right. .
\end{equation}
Note that $N_3 \sim f^{1/2} N_2$. The comoving optical depth of region 2 can be estimated as $\tau_2 \sim (\sigma_T N_2)/(4 \pi R^2)$, and likewise for region 3.
§ INVERSE COMPTON RADIATION FROM A RELATIVISTIC OUTFLOW INTERACTING WITH A POWER-LAW CSM
The FS and RS, if at least mildly relativistic, will produce relativistic electrons and strong magnetic fields that give rise to nonthermal emission. We adopt the standard theory, wherein fractions $\epsilon_e$ and $\epsilon_B$ of the total postshock energy density go into relativistic electrons and magnetic fields, respectively. The postshock electron energies are assumed to be distributed as a power law, $N_{\gamma_e} \propto (\gamma_e-1)^{-p}$, above some minimum Lorentz factor $\gamma_m$. We have $\gamma_{m2} = 610 \epsilon_{e2} g_p (\gamma_2-1)$ and $\gamma_{m3} = 610 \epsilon_{e3} g_p (\bar{\gamma}_3-1)$ for regions 2 and 3, respectively <cit.>. $g_p = 3(p-2)/(p-1)$ scales the results to $p=2.5$.
Let $P(\gamma_e)$ be the power radiated by a relativistic electron, and $\nu(\gamma_e)$ be the frequency of that radiation. We have $P(\gamma_e) = (4/3) \sigma_T c \gamma_e^2 \gamma_2^2 u_{rad} $,where $\sigma_T$ is the Thomson cross section and $u_{rad} = L_{th}/(4 \pi R^2 \gamma_2^2 c)$ is the photon energy density in the comoving frame. An electron with $\gamma_e$ emits at frequency $\nu(\gamma_e) = \gamma_2 \gamma_e^2 \nu_{rad} $, where $\nu_{rad} \sim k_B T_0/h\gamma_2$ is the frequency of a typical thermal photon in the shock frame <cit.>. Electrons above the critical Lorentz factor $\gamma_c = (3 m_e c)/(4 \sigma_T \gamma_2 u_{rad} t)$ can cool in time $t$ <cit.>. $\gamma_c $ is the same for regions 2 and 3 because the energy density and bulk Lorentz factor are equal across the contact discontinuity. In the single scattering limit, the maximum spectral power emitted by an ensemble of $N_e$ electrons will be $L_{\nu,max} \approx N_e P(\gamma_e)/\nu(\gamma_e)$ <cit.>. Ignoring the self-absorption frequency $\nu_a$, which falls well below the X-ray band, the spectrum will have two breaks, at $\nu_c = \nu(\gamma_c)$ and $\nu_m = \nu(\gamma_m)$, and two possible shapes depending on whether $\nu_m < \nu_c$ (slow cooling) or $\nu_c < \nu_m$ (fast cooling). The form of $L_\nu$ in either case is given by equations (7) and (8) in <cit.>.
If the above expressions give $\gamma_{m2} < 1$, $\gamma_{m2} \approx 1$ should be used. However, in that case, only a fraction $N_{rel}/N_2 \approx [610\epsilon_{e2} g_p (\gamma_2-1)]^{p-1}$ of the electrons are relativistic with $\gamma_e -1 \ge 1$. We make the approximation that nonrelativistic electrons do not contribute significantly to the emission at $\nu > kT_0$. Then the spectrum above $\nu_{m2}$, where $L_\nu \propto L_{\nu,max2} \nu_{m2}^{(p-1)/2} \propto N_{rel} \gamma_{m2}^{p-1}$, is unchanged whether $\gamma_{m2} > 1$ or $\gamma_{m2} \approx 1$. The same applies for $\gamma_{m3}$.
With the above assumptions and the dynamical equations of Appendix <ref>, one can compute the spectral parameters. For IC, the characteristic frequencies in region 2 and 3 are
\begin{equation}
\label{num2_ic}
h \nu_{m2} = 630 \text{ keV} \times \left\{
\begin{array}{lr}
g_p^2 \epsilon_{e2,-1}^2 \xi \gamma^2,
& f > \gamma^2 \\
\frac{1}{2} g_p^2 \epsilon_{e2,-1}^2 \xi \left[C_0 L_{iso,48} B_*^{-1} \right]^{1/(\ell+2)},
& \gamma^{-2} < f < \gamma^2 \\
\end{array}
\right. .
\end{equation}
\begin{equation}
\label{num3_ic}
h \nu_{m3} = 630 \text{ keV} \times \left\{
\begin{array}{lr}
\left(\frac{4}{7}\right)^2 g_p^2 \epsilon_{e3,-1}^2 \xi \left[C_1L_{iso,48} B_*^{-1} \gamma^{-2(\ell+2)}
\right]^{-4/(s+2)},
& f > \gamma^2 \\
\frac{1}{2} g_p^2 \epsilon_{e3,-1}^2 \xi \left[ C_0 L_{iso,48} B_*^{-1} \gamma^{-2(\ell+2)}\right]^{-1/(\ell+2)},
& \gamma^{-2} < f < \gamma^2 \\
\end{array}
\right. .
\end{equation}
Assuming IC is the dominant cooling process, the cooling frequency is
\begin{equation}
\label{nuc_ic}
h \nu_c = 3.0 \times 10^{-6} \text{ keV} \times \left\{
\begin{array}{lr}
L_{th,46}^{-2} \xi \gamma^{10} t_3^2,
& f > \gamma^2 \\
\frac{1}{32} L_{th,46}^{-2} \xi \left[C_0 L_{iso,48} B_*^{-1} \right]^{5/(\ell+2)} t_3^2,
& \gamma^{-2} < f < \gamma^2 \\
\end{array}
\right. .
\end{equation}
The peak IC spectral power in region 2 (in cgs units) is
\begin{equation}
\label{Lnu2_ic}
L_{\nu,max2} = \dfrac{1.1 \times 10^{27}}{ (\ell+1)} \times \left\{
\begin{array}{lr}
2^{\ell-1} L_{th,46} \xi^{-1} B_* \gamma^{2(\ell-1)} t_3^{-1},
& f > \gamma^2 \\
L_{th,46} \xi^{-1} \left[C_0^{\ell-1} L_{iso,48}^{\ell-1} B_*^3
\right]^{1/(\ell+2)} t_3^{-1},
& \gamma^{-2} < f < \gamma^2 \\
\end{array}
\right. .
\end{equation}
and in region 3 it is
\begin{equation}
\label{Lnu3_ic}
L_{\nu,max3} = \dfrac{1.1 \times 10^{27}} {(s+1) } \times \left\{
\begin{array}{lr}
2^{\ell-2} \left(\frac{32}{7}\right)^{1/2} L_{th,46} \xi^{-1}
\left[C_1 L_{iso,48} B_*^{s+1} \gamma^{2(\ell+2)(s+1)-5(s+2)}
\right]^{1/(s+2)} t_3^{-1},
& f > \gamma^2 \\
L_{th,46} \xi^{-1} \left[C_0^{\ell} L_{iso,48}^{\ell} B_*^2
\gamma^{-(\ell+2)} \right]^{1/(\ell+2)} t_3^{-1},
& \gamma^{-2} < f < \gamma^2 \\
\end{array}
\right. .
\end{equation}
Table <ref> summarizes the time behaviour of the spectral parameters. We point out that, for $s > \ell$, $\nu_m$ tends towards a steep decay $\propto t^{-4(s-\ell)/(s+2)}$ as the coasting regime is approached, making this model particularly well-suited to describing GRB 060218 or other objects where a rapid decline in $E_p$ is observed.
Quantity Coasting ($f > \gamma^2$) Accelerating ($f < \gamma^2$)
$\nu_{m2}$ 0 $\frac{s-\ell}{\ell+2}$
$\nu_{m3}$ $\frac{-4(s-\ell)}{s+2}$ $\frac{-(s-\ell)}{\ell+2}$
$\nu_c$ $-2(k-1)$ $-2(k-1) + \frac{5(s-\ell)}{\ell+2}$
$L_{\nu,max2}$ $(k-1)+\ell$ $(k-1)+s -\frac{3(s-\ell)}{\ell+2}$
$L_{\nu,max3}$ $(k-1) + \ell + \frac{s-\ell}{s+2}$ $(k-1)+s-\frac{2(s-\ell)}{\ell+2}$
Temporal evolution of the spectral parameters. Each parameter in the leftmost column evolves as a power-law in time, with the power-law index in the coasting and accelerating regimes given in the centre and right columns, respectively.
Given equations (<ref>)–(<ref>), the spectrum of the forward (reverse) shock emission can be constructed for any ordering of $\nu_c$, $\nu_{m2}$ ($\nu_{m3}$), and the observed frequency $\nu$ according to <cit.>. If the flux density at $\nu$ and the peak energy are measured at $t \ll t_f$ or $t \gg t_f$, and the relationship between $\nu$, $\nu_c$, and $\nu_m$ is known, then we can invert the model expressions for flux and peak energy to solve for two of the parameters $\gamma$, $A_*$, and $\epsilon_e$ in terms of observables and the third parameter. As a practical example, consider the case where the reverse shock dominates the emission at $t \ll t_f$ and $\nu_c < \nu < \nu_{m3}$. In this limit we have $L_\nu = L_{\nu,max3} \nu_c^{1/2} \nu^{-1/2}$ and $E_p = \nu_{m3}$; $\nu_{m3}$, $\nu_c$ and $L_{\nu,max3}$ are given by equations (<ref>), (<ref>), and (<ref>) respectively, taking the solution for $f < \gamma^2$. The luminosity integrated over a frequency range $\nu_1$–$\nu_2$ (with $\nu_c < \nu_1$ and $\nu_2 < \nu_{m3}$) is
\begin{equation}
\label{lightcurve}
L_{int} = 8.1 \times 10^{40} \text{ erg s}^{-1} (s+1)^{-1} C_{int} \xi^{-1/2} \left[C_1^{2\ell+5} L_{iso,48}^{2\ell+5} B_*^{-1}
\right]^{1/2(\ell+2)} \gamma^{-1},
\end{equation}
where $C_{int} = 2\left[(h\nu_2/\text{keV})-(h\nu_1/\text{keV})\right]$. Defining two dimensionless quantities that depend on the observables, $\tilde{L}_{int} = L_{int}/(8.1 \times 10^{40} \text{\,erg\,s}^{-1})$ and $\tilde{E}_p = E_p/(7.6 \times 10^{19} \text{\,Hz})$, considerably simplifies the algebra. Equations (<ref>) and (<ref>) can be rewritten to eliminate either $B_*$ or $L_{iso}$:
\begin{equation}
\label{simplify1}
\tilde{L}_{int}^2 \tilde{E}_p = (s+1)^{-2} C_{int}^2 \left(g_p \epsilon_{e3,-1}\right)^2 \left(C_0 L_{iso,48}\right)^2
\end{equation}
\begin{equation}
\label{simplify2}
\tilde{L}_{int} \tilde{E}_p^{(2\ell+5)/2} = (s+1)^{-1} C_{int} \left(g_p \epsilon_{e3,-1}\right)^{2\ell+5} \xi^{\ell+2}
B_* \gamma^{2(\ell+2)}.
\end{equation}
Finally, if the thermal photons come from a dissipative outflow as described in Section <ref>, we can substitute equation (<ref>) for $L_{iso,48}$ and solve for $\gamma$ and $B_*$, with the results
\begin{equation}
\label{gamma_solve}
\gamma = (13.6)^{-1/4} (s+1)^{1/4} C_{int}^{-1/4} \left(g_p \epsilon_{e3,-1}\right)^{-1/4} L_{th,46}^{-1/8} \xi^{1/2}
\tilde{L}_{int}^{1/4} \tilde{E}_p^{1/8}
\end{equation}
\begin{equation}
\label{B_solve}
B_* = (13.6)^{(\ell+2)/2} (s+1)^{-\ell/2} C_{int}^{\ell/2} \left(g_p \epsilon_{e3,-1}\right)^{-(3\ell+8)/2} L_{th,46}^{(\ell+2)/4}
\xi^{-2(\ell+2)} \tilde{L}_{int}^{-\ell/2} \tilde{E}_p^{(3\ell+8)/4}.
\end{equation}
In GRB 060218, we have $s=k/2$ by equation (<ref>), and since $k=0.66$ <cit.>, $s=0.33$. We can estimate $\alpha$ and $p$ by assuming the coasting regime has been reached by late times. We have $\nu_{m3} \propto t^{-4(s-\ell)/(s+2)}$ for $f > \gamma^2$, so in order to get $E_p \propto \nu_{m3} \propto t^{-1.6}$ we require $\ell=-0.6$ and $\alpha=2.6$. This in turn gives $\nu_c \propto t^{0.68}$ and $L_{\nu,max3} \propto t^{-0.54}$ for $t \gg t_f$. To obtain a light curve $L_{\nu,max3} \nu_c \nu_{m3}^{(p-1)/2} \propto t^{-2}$ at high energies, $p=3.25$ is needed, implying $g_p = 1.67$. The corresponding high energy spectral index is $\beta_2 \approx -1.6$, consistent with the data of <cit.>. Taking the same values for $\alpha$ and $s$, we have $L_{\nu,max3} \propto t^{-1.33}$ and $\nu_c \propto t^4$ at early times when $f < \gamma^2$. In this limit the light curve for $\nu_c < \nu < \nu_m$ goes as $L_{\nu,max3} \nu_c^{1/2} \propto t^{0.67}$, consistent with the early rise in the XRT and BAT light curves. We conclude that $t < t_f$ at early times, and since the 0.3–10 keV band is well below $E_p$ at early times (and presumably above $h\nu_c$), we can apply the model described above. Taking $L_{int} = L_{XRT}$, we calculate $C_{BAT} = 5.23$. At $t=300$ s, the 0.3–10 keV luminosity was $L_{XRT} \approx 1 \times 10^{46}$ erg s$^{-1}$ <cit.> and the peak energy was $E_p \approx 23$ keV <cit.>. Thus, we have $\tilde{L}_{int}= 1.2 \times 10^{5}$ and $ \tilde{E}_p = 7.3 \times 10^{-2}$. Finally, we have $L_{th,46} \approx 0.2$ and $\xi \approx 1$ at 300 s <cit.>, $t_M = t_L \approx 2800$ s <cit.>, and $t_{ext} = 300$ s (Section <ref>) so $B_* \approx A_*$ at 300 s. Plugging all of this into equations (<ref>) and (<ref>), we find
\begin{equation}
\label{gamma_060218}
\gamma = 5.3 \epsilon_{e3,-1}^{-1/4}
\end{equation}
\begin{equation}
\label{A_060218}
A_* = 0.28 \epsilon_{e3,-1}^{-(3\ell+8)/2}.
\end{equation}
When $\epsilon_{e3} = 10^{-2.5}$, we obtain $\gamma \approx 13$ and $A_* \approx 1.3 \times 10^4$, reasonably close to the results of the best-fitting numerical model.
We stress that this model is not self-consistent unless IC is more important than synchrotron, and the emission is dominated by the RS. Here we check whether each of these conditions is satisfied. With $\gamma$ and $A_*$ as above, we calculate $L_{iso,48} \approx 26$, $f \approx 0.36$, $\gamma_2 \approx 1.9$, $\bar{\gamma}_3 \approx 1.7$, $R \approx 6.5 \times 10^{13}$ cm, and $\rho_1(R) \approx 4.7 \times 10^{-13}$ g cm$^{-3}$ at 300 s. The comoving energy density of thermal photons at the shock radius is $u_{rad} = L_{th}/4\pi R^2 \gamma_2^2 c = 3.5 \times 10^{5}$ erg cm$^{-3}$, while the energy density in magnetic fields is $u_B = 4 \gamma_2^2 \rho_1 c^2 \epsilon_B = 6.1 \times 10^9 \epsilon_B$ erg cm$^{-3}$ <cit.>. Synchrotron is not too important if $u_B/u_{rad} \la 1$, implying $\epsilon_B \la 6 \times 10^{-5}$, similar to the numerically inferred value.
We have assumed $\nu_c < \nu < \nu_{m3}$, so that the RS luminosity is $L_{RS} = L_{\nu,max3} \nu_c^{1/2} \nu^{-1/2}$. Whether the FS or RS dominates the emission depends on the value of $\nu_{m2}$. If $\nu_{m2} > \nu$, then $L_{RS}/L_{FS} = (L_{\nu,max3} \nu_c^{1/2} \nu^{-1/2})/(L_{\nu,max2} \nu_c^{1/2} \nu^{-1/2}) = L_{\nu,max3}/L_{\nu,max2} \sim N_3/N_2 \sim f^{1/2}$. This cannot be the case, since $f < 1$ at 300 s in our model. Instead, we require $\nu_{m2} < \nu$, so that $L_{FS} = L_{\nu,max2} \nu_c^{1/2} \nu_{m2}^{(p-1)/2} \nu^{-p/2}$, and $L_{RS}/L_{FS} \sim f^{1/2} (\nu/\nu_{m2})^{(p-1)/2}$. Substituting $h \nu_{m2} \simeq [610 g_p \epsilon_{e2} (\gamma_2-1)]^2 kT_0$, we obtain $L_{RS}/L_{FS} \ga 1$ when $\epsilon_{e2} \la 9 \times 10^{-4} (h \nu/kT_0)^{1/2}$. This is qualitatively similar to the numerical result in that it also suggests $\epsilon_{e2} < \epsilon_{e3}$, although the numerical model produced a tighter upper limit on $\epsilon_{e2}$.
|
1511.00382
|
Department of Mathematics, UCLA, Los Angeles, CA 90095-1555.
Email. heilman@math.ucla.edu
Acknowledgement. Thanks to Oded Regev for helpful discussions. Thanks to Elchanan Mossel for encouraging me to publish these results.
We study the Gaussian noise stability of subsets $A$ of Euclidean space satisfying $A=-A$. It is shown that an interval centered at the origin, or its complement, maximizes noise stability for small correlation, among symmetric subsets of the real line of fixed Gaussian measure. On the other hand, in dimension two and higher, the ball or its complement does not always maximize noise stability among symmetric sets of fixed Gaussian measure. In summary, we provide the first known positive and negative results for the Symmetric Gaussian Problem.
§ INTRODUCTION
Gaussian noise stability is a well-studied topic with connections to geometry of minimal surfaces <cit.>, hypercontractivity and invariance principles <cit.>, isoperimetric inequalities <cit.>, sharp Unique Games hardness results in theoretical computer science <cit.>, social choice theory, learning theory <cit.>, and communication complexity <cit.>.
In applications, it is often desirable to maximize noise stability. A sample result is the following well-known Theorem of Borell, which has recently been re-proven and strengthened in various ways:
Among all subsets of Euclidean space $\R^{n}$ of fixed Gaussian measure, a half space maximizes noise stability (for positive correlation).
Here a half space is any set of points lying on one side of a hyperplane.
A well-known Corollary of Theorem <ref> says: among all subsets of Euclidean space $\R^{n}$ of fixed Gaussian measure, a half space has minimal Gaussian surface area. This statement may be surprising if one has only seen the isoperimetric inequality for Lebesgue measure. The latter inequality says: among all subsets of Euclidean space $\R^{n}$ of fixed Lebesgue measure, a ball has minimal surface area.
The present paper concerns a variant of Theorem <ref> where we only consider symmetric sets. We say a subset $A$ of $\R^{n}$ is symmetric if $A=-A$. Such a variant of Theorem <ref> is a conjecture.
Among all symmetric subsets of $\R^{n}$ of fixed Gaussian measure, the ball centered at the origin or its complement maximizes noise stability (for positive correlation).
If Conjecture <ref> were true, then a Corollary would be: among all symmetric subsets of Euclidean space $\R^{n}$ of fixed Gaussian measure, a ball centered at the origin or its complement has minimal Gaussian surface area. So, by restricting our attention to symmetric sets, the isoperimetric sets for the Gaussian measure and Lebesgue measure become essentially the same.
We show that Conjecture <ref> holds in the case $n=1$, and it does not hold in general in the case $n\geq2$. That is, we provide the first known positive and negative results for Conjecture <ref>.
§.§ Previous Work
It is natural to expect that that the approaches for proving Theorem <ref> taken e.g. in <cit.> would apply to Conjecture <ref>. However, this does not seem to be the case. The approaches of <cit.> in proving Theorem <ref> all seem to use the following property of a half-space: when a half space is translated, it still maximizes noise stability (with a different measure constraint). This translation-invariance property goes away when we consider Conjecture <ref>.
The restriction that the subset $A$ is symmetric in Conjecture <ref>, i.e. that $A=-A$, immediately removes any translation invariance property of the maximizers for this problem. That is, if a set $A$ satisfies $A=-A$ and the set $A$ maximizes noise stability among all symmetric subsets of $\R^{n}$ with Gaussian measure $1/2$, then an arbitrary translation of $A$ will no longer be a symmetric set. So, this translated set cannot maximize noise stability among symmetric sets. In short, we need to use some approach different from <cit.> in our investigation of Conjecture <ref>. The approach of <cit.> was designed to avoid this translation-invariance issue, and we can similarly apply the approach of <cit.> to Conjecture <ref>. When applying this approach to Conjecture <ref>, we first maximize noise stability for the correlation $\rho=0$. Then, when the correlation $\rho$ is small, one shows that the first variation condition for noise stability essentially defines a contractive mapping in a neighborhood of a global maximum. On the other hand, the approach of <cit.> currently seems special to the low correlation regime, whereas the approaches of <cit.> work for Theorem <ref> for any correlation value $\rho\in(-1,1)$.
Also, as used in various other works on isoperimetry with respect to the Gaussian measure (see e.g. <cit.>), one may try to solve Conjecture <ref> by solving the analogous problem on the unit $n$-dimensional unit sphere $S^{n}$ equipped with its normalized Haar measure. Solving this analogous problem on $S^{n}$ and letting $n\to\infty$ could potentially solve Conjecture <ref> itself. In fact, <cit.> mentions this strategy for considering Conjecture <ref>. However, as communicated to us by K. Oleszkiewicz (and noted in <cit.>), this strategy seems infeasible for proving Conjecture <ref>. There is a symmetric torus in $S^{3}$ of Haar measure $1/2$ which has less surface area than two spherical caps of total measure $1/2$. Therefore, two spherical caps of total measure $1/2$ cannot maximize noise stability on the sphere $S^{3}$ for all correlation values $\rho\in(-1,1)$. (If we normalize correctly, the derivative of noise stability at $\rho=1$ is equal to surface area. That is, maximizing noise stability for $\rho\to1$ corresponds to minimizing surface area.) It is still possible that spherical caps maximize noise stability on the sphere $S^{n}$ as $n\to\infty$, but this example for $S^{3}$ suggests the situation could be complicated for any fixed $n$.
In contrast to Conjecture <ref>, <cit.> shows that, if we choose a modified definition of Gaussian surface area, then symmetric strips have minimal modified Gaussian surface area among all sets of fixed Gaussian measure. Here a symmetric strip is a symmetric set bounded by two parallel hyperplanes. Since <cit.> uses a modified definition of Gaussian surface area, <cit.> does not contradict Conjecture <ref>.
Lastly, it is tempting to try to prove Conjecture <ref> by a symmetrization argument, as in <cit.>, but it is unclear how to construct such an argument in this setting. A symmetrization argument would begin with an arbitrary set $A$, and then construct a new set which is “more symmetric,” and whose noise stability would be larger than that of $A$. In other words, a symmetrization argument begins with a set $A$ and then moves $A$ in a direction of “increasing gradient” of noise stability. Symmetrization arguments are best suited for statements such as Theorem <ref>, where the set of maximum noise stability is unique, up to rotations. However, since we expect Conjecture <ref> to have at least two local maxima (namely the ball centered at the origin and its complement), a symmetrization argument seems more difficult to implement.
Noise stability can be interpreted as a nonlocal interaction energy <cit.>. Note that a theory of nonlocal minimal surfaces has been developed <cit.>, but it does not seem to apply in the present setting.
For the reasons including those mentioned in <cit.>, Conjecture <ref> appears to be a difficult problem to solve in general. Furthermore, Conjecture <ref> essentially contains the problem of minimizing “entropy” among self-shrinking solutions to the mean curvature flow. This problem has recently found significant progress <cit.>, building on a sequence of works including <cit.>, but this minimization problem is still not fully resolved. The main result of <cit.> only considers minimizing “entropy” among compact sets, so e.g. cones are ignored in their result.
§.§ Basic Definitions
Let $n$ be a positive integer. Let $A\subset\R^{n}$ be a measurable set. Define the Gaussian measure of $A$ to be
Let $\N\colonequals\{0,1,2,\ldots\}$. For any $x=(x_{1},\ldots,x_{n}),y=(y_{1},\ldots,y_{n})\in\R^{n}$, define $\langle x,y\rangle\colonequals\sum_{i=1}^{n}x_{i}y_{i}$, $\vnorm{x}_{2}\colonequals\sqrt{\langle x,x\rangle}$. For any $f\colon\R^{n}\to\R$, let $\vnorm{f}_{L_{2}(\gamma_{n})}\colonequals(\int_{\R^{n}}\abs{f(x)}^{2}d\gamma_{n}(x))^{1/2}$. Let $L_{2}(\gamma_{n})\colonequals\{f\colon\R^{n}\to\R\colon\vnorm{f}_{L_{2}(\gamma_{n})}<\infty\}$. For any $x\in\R^{n}$ and any $r>0$, let $B(x,r)\colonequals\{y\in\R^{n}\colon\vnorm{x-y}_{2}<r\}$. Let $f\colon\R^{n}\to[0,1]$ and let $\rho\in[-1,1]$, define the Ornstein-Uhlenbeck operator with correlation $\rho$ applied to $f$ by
\begin{equation}\label{oudef}
\end{equation}
$T_{\rho}$ is a parametrization of the Ornstein-Uhlenbeck operator. $T_{\rho}$ is not a semigroup, but it satisfies $T_{\rho_{1}}T_{\rho_{2}}=T_{\rho_{1}\rho_{2}}$, as we will see below. We have chosen this definition since the usual Ornstein-Uhlenbeck operator is only defined for $\rho\in[0,\pi/2]$.
Let $n$ be a positive integer. Let $\rho\in(-1,1)$. Let $A\subset\R^{n}$ be a measurable set. Define the Noise Stability of $A$ with correlation $\rho$ to be
§.§ The Symmetric Gaussian Problem
Conjecture <ref> appeared in <cit.> in relation to the Gap-Hamming-Distance problem in communication complexity. There, the following inequality was proven using concentration of measure techniques. In particular, the following property was used: when $n$ is large, most of the measure of $\gamma_{n}$ is concentrated near the sphere of radius $\sqrt{n}$ centered at the origin.
For all $c,\epsilon>0$, there exists $\delta,N>0$ such that, for all $n>N$, for all $0\leq\rho\leq c/\sqrt{n}$, and for all $A,B\subset\R^{n}$ with $\gamma_{n}(A)\geq e^{-\delta n}$ with $A=-A$, we have
A sharper estimate of the right side would give sharper lower bounds for the Gap-Hamming-Distance problem. Some related versions of Theorem <ref> were investigated in <cit.> and <cit.>.
The following conjecture is suggested in <cit.>. Conjecture <ref> below is a formal re-statement of Conjecture <ref>.
(Symmetric Gaussian Problem) Let $0<a,b<1$, $-1\leq\rho\leq1$ and let $A,B\subset\R^{n}$ with $\gamma_{n}(A)=a$, $\gamma_{n}(B)=b$. Let $r_{a},r_{b},r_{a}',r_{b}'>0$ so that $\gamma_{n}(B(0,r_{a}))=a$, $\gamma_{n}(B(0,r_{b})^{c})=b$, $\gamma_{n}(B(0,r_{a}')^{c})=a$, and $\gamma_{n}(B(0,r_{b}'))=b$. If $\rho>0$, then $(B(0,r_{a}),B(0,r_{b})^{c})$ or $(B(0,r_{a}')^{c},B(0,r_{b}'))$ achieves the following infimum
\begin{equation}\label{six1.9}
\inf_{\substack{A,B\subset\R^{n}\colon\\ \gamma_{n}(A)=a,\gamma_{n}(B)=b,\,A=-A}}\int_{\R^{n}}1_{A}(x)T_{\rho}(1_{B})(x)d\gamma_{n}(x).
\end{equation}
If $\rho<0$, the same result holds, with the additional restriction that $B=-B$ in (<ref>).
To see that Conjecture <ref> is equivalent to that of <cit.>, let $A,B\subset\R^{n}$ and observe
∫_^n 1_A(x)T_ρ(x)1_Bdγ_n(x)
=∫_^n 1_A(x)∫1_B(xρ+y√(1-ρ^2))dγ_n(y)dγ_n(x)
=∬_^n 1_A(x)1_B(xρ+y√(1-ρ^2))dγ_n(y)dγ_n(x)
Here $X=(X^{(1)},\ldots,X^{(n)}),Y=(Y^{(1)},\ldots,Y^{(n)})$ are jointly normal standard $n$-dimensional Gaussian random variables such that the covariances satisfy $\mathbb{E}(X^{(i)}Y^{(j)})=\rho\cdot1_{\{i=j\}}$.
Restricting Conjecture <ref> to the case $a+b=1$ and $A=B^{c}$ gives the following special case of Conjecture <ref>.
(Symmetric Gaussian Problem, Quadratic Version) Let $0< a<1$, $-1\leq\rho\leq1$ and let $A\subset\R^{n}$ with $\gamma_{n}(A)=a$. Let $r_{a},r_{a}'>0$ so that $\gamma_{n}(B(0,r_{a}))=a$, $\gamma_{n}(B(0,r_{a}')^{c})=a$. Then $B(0,r_{a})$ or $B(0,r_{a}')^{c}$ achieves the following supremum
\begin{equation}\label{six1.9b}
\sup_{\substack{A\subset\R^{n}\colon\\ \gamma_{n}(A)=a,\,A=-A}}\int_{\R^{n}}1_{A}(x)T_{\rho}(1_{A})(x)d\gamma_{n}(x).
\end{equation}
§.§ Our Contribution
In this paper, we provide two distinct approaches to Conjectures <ref> and <ref>. In our first approach, we examine the first variation of the noise stability directly, as in <cit.>. This approach is used in our first main result, Theorem <ref>.
In our second approach, we compute the second variation of noise stability for balls and their complements, which reduces to proving certain Gaussian Poincaré-type inequalities in Lemma <ref>. The second-variation approach appears to be the first application of second-variation arguments to noise stability problems. We show that, for certain measure restrictions $0<a<1$, the ball or its complement locally maximizes noise stability. But for other measure restrictions $a$, the ball or its complement does not locally maximize noise stability. As a result, Conjectures <ref> and <ref> are false, for certain measure restrictions $a$.
Here is our first main result.
Let $n=1$, $0<a,b<1$, and let $\abs{\rho}<\min(e^{-40},\min(a^{20},(1-a)^{20})\min(b^{20},(1-b)^{20}))/1000$. Then Conjecture <ref> holds with these parameters. Consequently, Conjecture <ref> holds with these parameters.
The proof of Theorem <ref> adapts the strategy of <cit.>, though the case $n=1$ of Conjecture <ref> provides several simplifications compared to the fairly intricate geometric arguments of <cit.>. Also, the proof in <cit.> was only able to handle correlations $\rho>0$, whereas the present paper can handle both positive and negative correlations $\rho$.
Unfortunately, already when $n=2$, Conjecture <ref> is incorrect (and consequently Conjecture <ref> is incorrect). To see why, let $A\subset\R^{2}$ and define
\begin{equation}\label{zero1}
\end{equation}
If Conjecture <ref> is correct, then by differentiating twice with respect to $\rho$ at $\rho=0$ (see (<ref>) below for details), Conjecture <ref> for $n=2$ implies that the ball or its complement maximizes $F(A)$ among all symmetric sets $A\subset\R^{2}$ with $\gamma_{2}(A)=a$. (Without loss of generality, if $A$ maximizes noise stability, then we may apply a rotation to $A$ if necessary to ensure that $\int_{A}x_{1}x_{2}d\gamma_{2}(x)=0$.) However, $F(A)$ is not always maximized by the ball or its complement. To see this, define
$$A=\{(x_{1},x_{2})\in\R^{2}\colon x_{1}^{2}/(2.5)^{2}+x_{2}^{2}/(2.31394)^{2}\leq1\}.$$
Let $r=2.4$. Then $\gamma_{2}(B(0,r))=1-e^{-r^{2}/2}\approx .943865$. And from Lemma <ref> below, $F(B(0,r))=F(B(0,r)^{c})=\frac{1}{2}r^{4}e^{-r^{2}}\approx.0522732$.
And if $r'=\sqrt{-2\log(1-e^{-2.88})}$, then $\gamma_{2}(B(0,r'))=\gamma_{2}(B(0,r)^{c})$, and again from Lemma <ref> below, $F(B(0,r'))=\frac{1}{2}(r')^{4}e^{-(r')^{2}}\approx.0059468$.
Finally, a numerical computation shows that $\gamma_{2}(A)\approx.943865$, and $F(A)=F(A^{c})\approx .0524720>.0522732$.
That is, $\gamma_{2}(B(0,r)^{c})=\gamma_{2}(B(0,r'))\approx\gamma_{2}(A^{c})$, but
That is, Conjectures <ref> and <ref> are false. A few more details are provided for this numerical calculation in Remark <ref> below. Furthermore, note that if $A'=\{(x_{1},x_{2})\in\R^{2}\colon \abs{x_{1}}\leq 1.90999\}$, then a numerical calculation shows that $\gamma_{2}(A')\approx.943865$, and $F(A')=F((A')^{c})\approx.0604796$. So, for this measure restriction, the strip actually has larger value than the ball, or the complement of a ball, or the ellipse $A$. So, at least for this measure restriction, the symmetric set maximizing $F$ seems to be the strip $A'$. If so, this result would agree with the S-inequality (formerly the S-conjecture) proven in <cit.>, which implies in particular that: for any symmetric convex set $A$ with $\gamma_{2}(A)=\gamma_{2}(A')$, and for any $t\geq1$, we have $\gamma_{2}(tA)\geq\gamma_{2}(tA')$.
The fact that Conjectures <ref> and <ref> are false for $n\geq2$ is especially surprising since they are more or less known to be true in the limit when $\rho\to1$, by e.g. <cit.>. At very least, the boundary of a set $A$ which maximizes noise stability in the limit $\rho\to1$ should be a minimal surface, i.e. a surface of constant mean curvature. On the other hand, the example above and Theorem <ref> suggest that strips could maximize noise stability for small $\rho$, as in the main result of <cit.>. In fact, it is entirely unclear which symmetric set maximizes $F$, and it is unclear which symmetric set maximizes noise stability. It could be the case that noise stability among symmetric sets of fixed Gaussian measure is maximized when $A\subset\R^{n}$ has boundary which is a dilation of the set $S^{k}\times\R^{n-k-1}$ for some $0\leq k\leq n-1$. However, this statement could also be false.
We did not arrive at the above counterexamples by accident. In fact it is generally true that when a ball or its complement has a large radius, then that set does not maximize the quantity (<ref>). This fact is made rigorous by computing the second variation of the quantity (<ref>). Before presenting this second variation result, we establish some notation.
Let $A\subset\R^{n}$ be a set with smooth boundary, and let $N\colon\partial A\to S^{n-1}$ denote the unit exterior normal to $\partial A$. Let $X\colon\R^{n}\to\R^{n}$ be a vector field. Let $\Psi\colon\R^{n}\times(-1,1)\to\R^{n}$ such that $\Psi(x,0)=x$ and such that $\frac{d}{dt}\Psi(x,t)=X(\Psi(x,t))$ for all $x\in\R^{n},t\in(-1,1)$. For any $t\in(-1,1)$, let $A^{(t)}=\Psi(A,t)$. Note that $A^{(0)}=A$. Let $G\colon\R^{n}\times\R^{n}\to\R$ be a Schwartz function, e.g. to investigate noise stability we let $G(x,y)=(2\pi)^{-n}e^{\frac{-\vnorm{x}_{2}^{2}-\vnorm{y}_{2}^{2}+2\rho\langle x,y\rangle}{2(1-\rho^{2})}}$ $\forall$ $x,y\in\R^{n}$. Or to investigate the functional in (<ref>), we let $G(x,y)=\sum_{i=1}^{n}(1-x_{i}^{2})(1-y_{i}^{2})\gamma_{n}(x)\gamma_{n}(y)$. Define
$$V(x,t)\colonequals\int_{A^{(t)}}G(x,y)dy,\qquad V\colon\R^{n}\times(-1,1)\to\R.$$
$$F(A)\colonequals \int_{\R^{n}}\int_{\R^{n}} 1_{A}(x)G(x,y)1_{A}(y)dxdy.$$
When integrating on a surface $\partial A$, we let $dx$ denote the restriction of Lebesgue measure to the surface $\partial A$.
The second variation formula of Theorem <ref> essentially appears in <cit.>, though their statement differs a bit from ours. Nevertheless, their proof immediately gives Theorem <ref>. We reproduce the details of the proof of <cit.> in the Appendix, Section <ref>. Theorem <ref> does not seem to have been applied to noise stability before. In particular, optimizing noise stability has typically focused on either first variation arguments, or on heat flow methods. So, we consider our application of the second variation formula to the noise stability functional to be one of the main contributions of this work.
Combining Theorem <ref> with a Poincaré-type inequality on the sphere (Lemma <ref> below), we deduce the following second variation calculation for the functional $F$ when $G(x,y)=\sum_{i=1}^{n}(1-x_{i}^{2})(1-y_{i}^{2})\gamma_{n}(x)\gamma_{n}(y)$ $\forall$ $x,y\in\R^{n}$. This second variation computation constitutes our second main result.
Let $n\geq2$. Let $r>0$ such that $r^{2}\leq n+2$. Assume that $\frac{d}{dt}|_{t=0}\gamma_{n}(A^{(t)})=0$ and $\frac{d^{2}}{dt^{2}}|_{t=0}\gamma_{n}(A^{(t)})=0$. Then if $A^{(0)}=B(0,r)$ or if $A^{(0)}=B(0,r)^{c}$, we have
That is, the sets $B(0,r)$ and $B(0,r)^{c}$ locally maximize the sum of the squares of their second-degree Fourier coefficients among symmetric sets of fixed Gaussian measure.
However, if $r^{2}> n+2$, then $\exists$ symmetric sets $\{A^{(t)}\}_{-1<t<1}$ so that $\frac{d}{dt}|_{t=0}\gamma_{n}(A^{(t)})=0$, $\frac{d^{2}}{dt^{2}}|_{t=0}\gamma_{n}(A^{(t)})=0$, with $A^{(0)}=B(0,r)$ or $A^{(0)}=B(0,r)^{c}$, and such that
That is, the sets $B(0,r)$ and $B(0,r)^{c}$ do not locally maximize the sum of the squares of their second-degree Fourier coefficients among symmetric sets of fixed Gaussian measure.
The equality $\frac{d}{dt}F(A^{(t)})|_{t=0}=0$ follows readily from (<ref>) below, and the second variation calculation is contained in Corollary <ref> below.
The condition $r^{2}\leq n+2$ unfortunately does not hold for all measure restrictions as $n\to\infty$. That is, there are many measure restrictions $a=\gamma_{n}(B(0,r))$ where $r^{2}>n+2$. To see this, let $r(s,n)=\sqrt{n+s\sqrt{2n}}$ for any $s>0$, $n\in\N$. Then $\lim_{n\to\infty}\gamma_{n}(B(0,r(s,n)))=\int_{-\infty}^{s}\gamma_{1}(t)dt$, which follows from the Central Limit Theorem. And $\lim_{n\to\infty}[(r(s,n))^{2}-n-2]<0$ if and only if $s<0$. That is, the ball $B(0,r(s,n))$ only locally maximizes $F$ for sufficiently large $n$ when $\lim_{n\to\infty}\gamma_{n}(B(0,r(s,n)))\leq1/2$. And the complement $B(0,r(s,n))^{c}$ only locally maximizes $F$ for sufficiently large $n$ when $\lim_{n\to\infty}\gamma_{n}(B(0,r(s,n))^{c})\geq1/2$.
As we observed in the case $n=2$, with $r=2.4$ and with $r'=\sqrt{-2\log(1-e^{-2.88})}\approx.3399$, we had $\gamma_{2}(B(0,r)^{c})=\gamma_{2}(B(0,r'))$ with $F(B(0,r)^{c})>F(B(0,r'))$. Theorem <ref> then says that Conjectures <ref> and <ref> are false in a fairly strong sense, since if the radius of the ball or complement is sufficiently large, then that set does not locally maximize $F$. It therefore seems natural to try to formulate a weaker version of Conjecture <ref> which only identifies sets of large noise stability as $n\to\infty$. Such a statement may still be suitable for applications to the Gap-Hamming-Distance problem as well.
For any $a\in(0,1)$, $n\geq1$, let $B_{n,a}\subset\R^{n}$ be the ball centered at the origin such that $\gamma_{n}(B_{n,a})=a$.
If $A=-A$, and if $\gamma_{n}(A)=a$, is it true that
We would like to change the above question, replacing $\sup_{n\geq1}$ with $\lim_{n\to\infty}$ or $\limsup_{n\to\infty}$. Unfortunately, the following calculation shows that, for any $0<a<.15$, quantity $F(B_{n,a})$ decreases monotonically when $n$ is very large.
From the Central Limit Theorem with error bound (also known as the Edgeworth Expansion) <cit.>, for any $s\in\R$, the following asymptotic expansion holds as $n\to\infty$:
And an asymptotic expansion in Lemma <ref> shows that, as $n\to\infty$, if $B(0,\sqrt{n+s\sqrt{2n}})\subset\R^{n}$, we then have
For $n$ sufficiently large, if $B(0,\sqrt{n+s\sqrt{2n}})\subset\R^{n}$ and $s<-1$, then $\gamma_{n}(B(0,\sqrt{n+s\sqrt{2n}}))$ decreases as $n$ increases, and $F(B(0,\sqrt{n+s\sqrt{2n}}))$ increases as $n$ increases. And if $s>1$, then $\gamma_{n}(B(0,\sqrt{n+s\sqrt{2n}})^{c})$ increases as $n$ increases, and $F(B(0,\sqrt{n+s\sqrt{2n}}))$ decreases as $n$ increases. Therefore, if $a<.15$, $F(B_{n,a})<F(B_{n,1-a}^{c})$, and $B_{n,1-a}^{c}$ does not locally maximize $F$, by Theorem <ref>. That is, when $0<a<.15$, when $n$ is large, and when $\abs{\rho}$ is near zero, Conjectures <ref> and <ref>.
If $r>0$, then since $B(0,r)\subset\R^{n}$ is rotationally symmetric, we conclude that $F(B(0,r))=\frac{1}{n}(\int_{B(0,r)}(n-\sum_{i=1}^{n}x_{i}^{2})d\gamma_{n}(x))^{2}$. So, if $n$ is fixed, $\max_{a>0}F(B_{n,a})=F(B(0,\sqrt{n}))$. Also, if $B(0,\sqrt{n})\subset\R^{n}$, then $F(B(0,\sqrt{n}))< F(B(0,\sqrt{n+1}))$, by Lemma <ref> below. And since $\lim_{n\to\infty}F(B(0,\sqrt{n}))=\frac{1}{\pi}$, a variant of Question <ref> can be:
If $A\subset\R^{n}$ with $A=-A$ (with no restriction on the measure of $A$), is it true that
If Question <ref> is incorrect, then Conjecture <ref> is false in a much stronger sense than mentioned above. That is, if Question <ref> is incorrect, then there exists some $A\subset\R^{n}$ such that $\partial A$ is a the level set of a degree two polynomial such that $A$ has larger noise stability than any ball or ball complement in any Euclidean space of any dimension (for noise stability with small correlation $\rho$). If Question <ref> is correct, then Question <ref> can be interpreted as an “infinite-dimensional” special case of the following weakened “infinite-dimensional” version of Conjecture <ref>:
Let $0<\rho<1$. For any $a\in(0,1)$, $n\geq1$, let $B_{n,a}\subset\R^{n}$ be the ball centered at the origin such that $\gamma_{n}(B_{n,a})=a$.
If $A=-A$, and if $\gamma_{n}(A)=a$, is it true that
If $A=-A$, and if $\gamma_{n}(A)=a$, is it true that
Despite the negative results mentioned above, including Theorem <ref>, we present some evidence toward positive answers to Question <ref> and <ref>. First, Theorem <ref> can be extended to handle the noise stability functional, as long as $\rho$ is small. Before stating this Theorem, for any $A\subset\R^{n}$, and for any $-1<\rho<1$, let
Below, we continue to use the notational conventions of Theorem <ref>.
Assume $\int_{\partial A}\abs{\langle X(x),N(x)\rangle}^{2}dx=1$.
Let $r>0$ such that $r^{2}\leq n+2$. Let $0<a<1$. Assume that $\gamma_{n}(A^{(0)})=a$. Then there exists $\rho_{0}=\rho_{0}(a,r,n)>0$ such that, for all $\abs{\rho}<\rho_{0}$, the following holds. Assume that $\frac{d}{dt}|_{t=0}\gamma_{n}(A^{(t)})=0$ and $\frac{d^{2}}{dt^{2}}|_{t=0}\gamma_{n}(A^{(t)})=0$. Then if $A^{(0)}=B(0,r)$ or if $A^{(0)}=B(0,r)^{c}$, we have
That is, the sets $B(0,r)$ and $B(0,r)^{c}$ locally maximize noise stability among symmetric sets of fixed Gaussian measure.
However, if $r^{2}> n+2$, if $0<a<1$, then there exists $\rho_{0}=\rho_{0}(a,r,n)>0$ such that, for all $\abs{\rho}<\rho_{0}$, the following holds. There exist symmetric sets $\{A^{(t)}\}_{-1<t<1}$ so that $\gamma_{n}(A^{(0)})=a$, $\frac{d}{dt}|_{t=0}\gamma_{n}(A^{(t)})=0$, $\frac{d^{2}}{dt^{2}}|_{t=0}\gamma_{n}(A^{(t)})=0$, with $A^{(0)}=B(0,r)$ or $A^{(0)}=B(0,r)^{c}$, and such that
That is, the sets $B(0,r)$ and $B(0,r)^{c}$ do not locally maximize noise stability among symmetric sets of fixed Gaussian measure.
Let $A\subset\R^{n}$ with $A=-A$. Since $A=-A$, $\frac{d}{d\rho}|_{\rho=0}\int_{\R^{n}}1_{A}(x)T_{\rho}1_{A}(x)d\gamma_{n}(x)=0$ (see (<ref>) below). Moreover, we have the Taylor series estimate
\begin{equation}\label{newone}
\abs{\int_{\R^{n}}1_{A}(x)T_{\rho}1_{A}(x)d\gamma_{n}(x)
\leq\abs{\rho}^{3}.
\end{equation}
So, the second derivative of noise stability at $\rho=0$ is most significant when $\rho$ is near zero. In fact, as shown in (<ref>) below,
\frac{d^{2}}{d\rho^{2}}|_{\rho=0}\int_{\R^{n}}1_{A}(x)T_{\rho}1_{A}(x)d\gamma_{n}(x)
+2\sum_{i,j\in\{1,\ldots,n\}\colon i\neq j}(\int_{A}x_{i}x_{j}d\gamma_{n}(x))^{2}.
However, the last sum can be ignored for the following reason. If we interpret $x\in\R^{n}$ as a column vector, consider the $n\times n$ matrix $M\colonequals\int_{A}xx^{T}d\gamma_{n}(x)$. Then if $Q$ is an orthogonal $n\times n$, matrix, a change of variables shows that $QMQ^{T}=\int_{A}(Qx)(Qx)^{T}d\gamma_{n}(x)=\int_{QA}xx^{T}d\gamma_{n}(x)$. So, to diagonalize $M$ with an orthogonal matrix $Q$, it suffices to replace $A$ with the set $QA$. So, when we maximize $\int_{\R^{n}}1_{A}(x)T_{\rho}1_{A}(x)d\gamma_{n}(x)$ or when we consider variations of the noise stability, we can and will assume that $\int_{A^{(t)}}x_{i}x_{j}d\gamma_{n}(x)=0$ for any $i,j\in\{1,\ldots,n\}$ with $i\neq j$, for any $t\in(-1,1)$. In particular (using $t=0$), we can and will assume that $\int_{A}x_{i}x_{j}d\gamma_{n}(x)=0$ for any $i,j\in\{1,\ldots,n\}$ with $i\neq j$
§.§ General Framework
Though Conjecture <ref> and other noise stability optimization problems concern the optimization of a very specific functional, i.e. noise stability, our treatment of Conjecture <ref> uses a fairly general strategy. That is, we can consider our approach to Conjecture <ref> within the following general context:
* We are given some Banach space $V$, and for each $\rho\in(-1,1)$, we have a function $F_{\rho}\colon V\to\R$ to be maximized.
* The maximum of $F_{0}$ over $V$ is equivalent to maximizing $F_{0}$ over a finite-dimensional manifold.
* We would like to show: if $v_{0}\in V$ maximizes $F_{0}$, then $v_{0}$ also maximizes $F_{\rho}$ for all $\rho$ close to $0$.
It is generally impossible that the final statement holds. For example, suppose we are asked to maximize $F_{\rho}(v)=-(v-\rho)^{2}$ where $v\in \R$. Then $v=0$ maximizes $F_{0}$, but $v=0$ does not maximize $F_{\rho}$ when $\rho\neq0$.
Our main strategy in proving Theorem <ref> is to try to relate the first variation (i.e. first derivative) of $F_{\rho}$ to that of $F_{0}$ when $\rho$ is near $0$.
(i) Prove some stability estimate for $F_{0}$. (If $v$ nearly maximizes $F_{0}$, then $v$ is close to $v_{0}$.)
(ii) Show that if $\rho$ is close to $0$, then the first variation of $F_{\rho}$ is close to that of $F_{0}$.
(iii) Assume that $F_{\rho}$ depends continuously on $\rho$, if $v$ maximizes $F_{\rho}$, then $v$ nearly maximizes $F_{0}$. So, $v$ is close to $v_{0}$ by (ii).
(iv) Since $v$ is close to $v_{0}$ by (iii), an appropriate version of $(ii)$ implies that $v$ is very close to $v_{0}$. Then, by iterating $(ii)$ an infinite number of times, we conclude that $v=v_{0}$, as desired.
This strategy was used in <cit.> to show that if $\rho$ is close to zero, then the maximum noise stability of three sets partitioning $\R^{n}$ each with Gaussian measure $1/3$ occurs when the three sets are cones each with cone angle $2\pi/3$. This appeared to be the first use of this strategy applied to noise stability problems. However, a similar strategy has been used for perturbations of perimeter functionals <cit.>
In the present paper, we will consider the Banach $V$ consisting of symmetric bounded functions: $f\colon\R^{n}\to[0,1]$ with $f(-x)=f(x)$. So, if $A$ is symmetric, i.e. $A=-A$ and $A\subset\R^{n}$, then $1_{A}\in V$. We will also let $F_{\rho}$ be the noise stability.
The strategy depicted above, as used in <cit.>, however has some shortcomings for Conjecture <ref> when $n\geq2$. In particular, part (iv) of the above strategy seems most natural only when we impose the additional restriction that the set $A\subset\R^{n}$ satisfies $\int_{A}(1-x_{i}^{2})d\gamma_{n}(x)=\int_{A}(1-x_{j}^{2})d\gamma_{n}(x)$ for all $i,j\in\{1,\ldots,n\}$. This assumption imposes additional constraints beyond the assumption that $A=-A$. It is possible to prove a version of Theorem <ref> under this additional constraint, but we choose not to do so, since this constraint seems too restrictive to be of interest.
In any case, in order to prove Theorems <ref> and <ref>, we abandon the strategy of <cit.>, and we instead use the second variation formula Theorem <ref>. That is, we change the itemized strategy above to the following.
(i) Compute the second variation of $F_{0}$ is negative at a ball centered at the origin.
(ii) Show that if $\rho$ is close to $0$, then the second variation of $F_{\rho}$ is close to that of $F_{0}$.
§.§ Organization
Sections <ref> through <ref> provide supporting lemmas for the proof of Theorem <ref>, which appears in Section <ref>.
Theorems <ref> and <ref> are proven in Section <ref>.
§.§ Some Hermite-Fourier Analysis
Let $\lambda>0$. Recall that the Hermite polynomials $h_{0},h_{1},h_{2},\ldots$ of one variable are defined by
$$e^{\lambda x-\lambda^{2}/2}\equalscolon\sum_{\ell\in\N}\lambda^{\ell}h_{\ell}(x),\qquad\forall\,x\in\R.$$
Note that $\int_{\R}h_{\ell}(x)^{2}d\gamma_{1}(x)=1/\ell!$, and $\{\sqrt{\ell!}\,h_{\ell}\}_{\ell\in\N}$ is an orthonormal basis of $L_{2}(\gamma_{1})$ with respect to the inner product $\langle f,g\rangle=\int_{\R}f(x)\overline{g(x)}d\gamma_{1}(x)$, $f,g\colon\R\to\R$. Set $f(x)\colonequals e^{\lambda x-\lambda^{2}/2}$. A routine computation <cit.> shows that $T_{\rho}(f)(x)=e^{(\lambda\rho)x-(\lambda\rho)^{2}/2}$, $\forall$ $x\in\R$, $\forall$ $\rho\in(-1,1)$.
We therefore have the relation
\begin{equation}\label{six1}
\end{equation}
So, by linearity, $T_{\rho}h_{\ell}(x)=\rho^{\ell}h_{\ell}(x)$, $\forall$ $x\in\R$, $\forall$ $\ell\in\N$, $\forall$ $\rho\in(-1,1)$.
We now extend the above observations to higher dimensions. Let $f\in L_{2}(\gamma_{n})$, so that $f=\sum_{\ell\in\N^{n}}\langle f,h_{\ell}\sqrt{\ell!}\rangle h_{\ell}\sqrt{\ell!}$ in the $L_{2}(\gamma_{n})$ sense, where $\ell=(\ell_{1},\ldots,\ell_{n})\in\N^{n}$ and $h_{\ell}(x)=\prod_{i=1}^{n}h_{\ell_{i}}(x_{i})$ $\forall$ $x\in\R^{n}$. Write $\vnorm{\ell}_{1}\colonequals \ell_{1}+\cdots+\ell_{n}$ and $\ell!\colonequals (\ell_{1}!)\cdots(\ell_{n}!)$. Then $T_{\rho}$ satisfies $T_{\rho}h_{\ell}=\rho^{\vnorm{\ell}_{1}}h_{\ell}$ for any $\ell\in\N^{n}$, and for any $\rho\in(-1,1)$,
\begin{equation}\label{six1.8}
\qquad\forall\,x\in\R^{n}.
\end{equation}
Let $f,g\in L_{2}(\gamma_{n})$. By Plancherel's Theorem and (<ref>) we have
\begin{equation}\label{one1}
\int_{\R^{n}} f(x) T_{\rho}g(x)d\gamma_{n}(x)=\sum_{\ell\in\N^{n}}\rho^{\vnorm{\ell}_{1}}
\int_{\R^{n}} f(x)\sqrt{\ell!}h_{\ell}(x)d\gamma_{n}(x)
\int_{\R^{n}} g(y)\sqrt{\ell!}h_{\ell}(y)d\gamma_{n}(y).
\end{equation}
By formally taking the second derivative $d^{2}/d\rho^{2}$ of (<ref>), we get
\begin{equation}\label{one2.5}
\frac{d^{2}}{d\rho^{2}}\int f T_{\rho}fd\gamma_{n}
=\sum_{\ell\in\N^{n}}\vnorm{\ell}_{1}(\vnorm{\ell}_{1}-1)\rho^{\vnorm{\ell}_{1}-2}\abs{\int f\sqrt{\ell!}h_{\ell}d\gamma_{n}}^{2}.
\end{equation}
Evaluating (<ref>) at $\rho=0$,
\begin{equation}\label{one3}
\left.\frac{d^{2}}{d\rho^{2}}\right|_{\rho=0}\int f T_{\rho}fd\gamma_{n}
=2\sum_{\ell\in\N^{n}\colon\vnorm{\ell}_{1}=2}\abs{\int f\sqrt{\ell!}h_{\ell}d\gamma_{n}}^{2}.
\end{equation}
Suppose $f(x)=f(-x)$, $\forall$ $x\in\R^{n}$. Applying this property and then changing variables,
\begin{equation}\label{one3.1}
\int_{\R^{n}}h_{\ell}(x)f(x)d\gamma_{n}(x)=(-1)^{\vnorm{\ell}_{1}}\int_{\R^{n}}h_{\ell}(-x)f(-x)d\gamma_{n}(x)
\end{equation}
§ MAXIMIZING SECOND DEGREE FOURIER COEFFICIENTS
We begin with the following adaptation of <cit.>. The following Lemma provides existence and first variation conditions (<ref>) for maximizing the second degree term in (<ref>), or equivalently the second derivative term in (<ref>).
Let $0<a<1$. Then there exists $A\subset\R^{n}$ such that
\sum_{\substack{\ell\in\N^{n}\colon\\ \vnorm{\ell}_{1}=2}}\left(\int_{\R^{n}}1_{A}(x)h_{\ell}(x)\sqrt{\ell!}d\gamma_{n}(x)\right)^{2}
=\sup_{\substack{\{f\colon\R^{n}\to[0,1],\\ \int_{\R^{n}} f(x)d\gamma_{n}(x)=a\}}}
\sum_{\substack{\ell\in\N^{n}\colon\\ \vnorm{\ell}_{1}=2}}\left(\int_{\R^{n}}f(x)h_{\ell}(x)\sqrt{\ell!}d\gamma_{n}(x)\right)^{2}.
Moreover, $A=-A$, and there exists $c\in\R$ such that
\begin{equation}\label{one4}
\int_{\R^{n}}1_{A}(y)\left[\sum_{i=1}^{n}(x_{i}^{2}-1)(y_{i}^{2}-1)+2\sum_{i\neq j}(x_{i}x_{j}y_{i}y_{j})\right]d\gamma_{n}(y)\geq c\right\}.
\end{equation}
The set $C\colonequals\{f\colon\R^{n}\to[0,1],\int_{\R^{n}} f(x)d\gamma_{n}(x)=a\}$ is a norm closed, convex and norm bounded subset of the Hilbert space $L_{2}(\gamma_{n})$. Therefore, $C\subset L_{2}(\gamma_{n})$ is weakly closed. Also, $C$ is weakly compact by the Banach-Alaoglu Theorem. Define $T\colon C\to\R$ by
\begin{equation}\label{one8}
T(f)\colonequals\sum_{\ell\in\N^{n}\colon\vnorm{\ell}_{1}=2}\abs{\int_{\R^{n}} f(x)\sqrt{\ell!}h_{\ell}(x)d\gamma_{n}(x)}^{2}.
\end{equation}
Since $\vnormf{h_{\ell}\sqrt{\ell!}}_{L_{2}(\gamma_{n})}=1$, $T$ is a finite sum of weakly continuous functions. Therefore, $T$ is weakly continuous on the weakly compact set $C\subset L_{2}(d\gamma_{n})$. So, there exists $f\in C$ such that $T(f)=\max_{g\in C}T(g)$.
Now, the function $f_{s}(x)\colonequals(f(x)+f(-x))/2$ satisfies
\int_{\R^{n}} f_{s}(x)\sqrt{\ell!}h_{\ell}(x)d\gamma_{n}(x)
\begin{cases}
\int_{\R^{n}} f(x)\sqrt{\ell!}h_{\ell}(x)d\gamma_{n}(x) & ,\vnorm{\ell}_{1}\,\mbox{even, }\ell\in\N^{n}\\
0 & ,\vnorm{\ell}_{1}\,\mbox{odd, }\ell\in\N^{n}.
\end{cases}
So, $T(f_{s})\geq T(f)$. Let $C^{s}\colonequals\{f\colon\R^{n}\to[0,1],f(x)=f(-x),\int_{\R^{n}} f(x)d\gamma_{n}(x)=a\}$. We have just shown that
\begin{equation}\label{two3}
\max_{g\in C}T(g)=\max_{g\in C^{s}}T(g).
\end{equation}
We therefore try to maximize $T(g)$ on $C^{s}$. We now show that $T$ is convex on $B^{s}$. Let $f_{1},f_{2}\in C^{s}$, and let $\lambda\in[0,1]$. Then
\begin{equation}\label{two4}
\lambda T(f_{1})+(1-\lambda)T(f_{2})-T(\lambda f_{1}+(1-\lambda)f_{2})
\stackrel{\eqref{one8}}{=}\lambda(1-\lambda)T(f_{1}-f_{2})
\geq0.
\end{equation}
So, $T$ is a weakly continuous convex function on the weakly compact set $C^{s}\subset L_{2}(\gamma_{n})$. Therefore, there exists $A\subset\R^{n}$ such that $1_{A}\in C^{s}$ satisfies $T(1_{A})=\max_{g\in C^{s}}T(g)$ <cit.>. Combining this observation with (<ref>), $T(1_{A})=\max_{g\in C^{s}}T(g)=\max_{g\in C}T(g)$, and $A=-A$ since $1_{A}\in C^{s}$. The existence of $A$ is therefore proven.
We now prove (<ref>). We argue by contradiction. Define
\stackrel{\eqref{six1.8}}{=}\sum_{\ell\in\N^{n}\colon\vnorm{\ell}_{1}=2}\left(\int_{\R^{n}}f(y)\sqrt{\ell!}h_{\ell}d\gamma_{n}(y)\right)\sqrt{\ell!}h_{\ell}(x).$$
Note that $\int_{\R^{n}} f(x)\overline{T}f(x)d\gamma_{n}(x)=T(f)$. Suppose there exists $x_{1},x_{2}\in\R^{n}$, $x_{1}\notin A,x_{2}\in A$ such that $\overline{T}1_{A}(x_{1})>\overline{T}1_{A}(x_{2})$. Let $U_{1}\subset\R^{n}$ be a small ball around $x_{1}$ and let $U_{2}$ be a small ball around $x_{2}$ such that $\overline{T}1_{A}(u_{1})>\overline{T}1_{A}(u_{2})$, $\forall$ $u_{1}\in U_{1},u_{2}\in U_{2}$. Also, assume that $U_{1}\cap U_{2}=\emptyset$ and $\gamma_{n}(U_{1})=\gamma_{n}(U_{2})$. Define $A'\colonequals (A\setminus U_{2})\cup U_{1}$. Then $1_{A'}=1_{A}-1_{U_{2}}+1_{U_{1}}$, and for $U_{1},U_{2}$ sufficiently small,
=∫_^n 1_A(x)T1_A(x)dγ_n(x)
>∫_^n 1_A(x)T1_A(x)dγ_n(x).
This inequality contradicts the maximality of $A$. We conclude that no such $x_{1},x_{2}$ exist, so (<ref>) holds.
One difficulty in proving Question <ref> is that there are many potential critical points for the noise stability. For example, if the boundary of $A$ is of the form $S^{m}\times\R^{n-m}$, with $0\leq m\leq n$, then $A$ satisfies (<ref>). Also, if the boundary of the set $A$ is any Simons-Lawson cone, then $A$ satisfies (<ref>). That is, in the limit $\rho\to0$, $A$ is a candidate critical point of noise stability if the boundary of $A$ is equal to
$$\{(x_{1},\ldots,x_{2n})\in\R^{2n}\colon \sum_{i=1}^{n}x_{i}^{2}=\sum_{i=n+1}^{2n}x_{i}^{2}\}.$$
§ ITERATIVE ESTIMATES
The following inequality for Hermite polynomials will be useful in the sequel.
For $\ell=(\ell_{1},\ldots,\ell_{n})\in\N^{n}$ and $x=(x_{1},\ldots,x_{n})\in\R^{n}$,
\leq\vnorm{\ell}_{1}^{n}3^{\vnorm{\ell}_{1}}\prod_{i=1}^{n}\max(1,\abs{x_{i}}^{\ell_{i}}).
Below we will also require the following bounds on $T_{\rho}$ applied to the indicator function of an interval.
Let $B=B(0,r)\subset\R$ with $\gamma_{1}(B)=a$. Let $x\in\R$ with $\abs{x}\leq\sqrt{-4\log\abs{\rho}}$, and let $\abs{\rho}<e^{-40}$. Then
\bigg|\frac{d}{dx}T_{\rho}1_{B}(x)+\rho^{2}\sqrt{\frac{2}{\pi}}xre^{-r^{2}/2}\bigg|\leq\min(a^{1/2},(1-a)^{1/2})10\abs{\rho}^{15/4}.
Also, for any $f\colon\R\to[-1,1]$,
\frac{d}{dx}T_{\rho}f(x)=\frac{\rho}{\sqrt{1-\rho^{2}}}\int_{\R} yf(x\rho+y\sqrt{1-\rho^{2}})d\gamma_{1}(y).
Recall that $h_{\ell}(x)=\sum_{m=0}^{\lfloor \ell/2\rfloor}\frac{x^{\ell-2m}(-1)^{m}2^{-m}}{m!(\ell-2m)!}$. So, $h_{1}(x)=x$, $h_{2}(x)=(1/2)(x^{2}-1)$, $(d/dx)h_{\ell}=h_{\ell-1}$ for $\ell\geq1$, and $\int_{\R} 1_{B}(x)h_{2}(x)\sqrt{2}d\gamma_{1}(x)=-re^{-r^{2}/2}/\sqrt{\pi}$.
Since $\gamma_{1}(B)=a$, $\vnorm{1_{B}}_{L_{2}(\gamma_{n})}=a^{1/2}$ and $\vnorm{1_{B^{c}}}_{L_{2}(\gamma_{n})}=(1-a)^{1/2}$. Then, for $h_{\ell}$ with $\ell\geq1$, the Cauchy-Schwarz inequality implies that
\begin{equation}\label{two6}
\begin{aligned}
\abs{\int_{\R} 1_{B}(x)h_{\ell}(x)\sqrt{\ell!}d\gamma_{1}(x)}
&=\min\left(\,\abs{\int_{\R} 1_{B}(x)h_{\ell}(x)\sqrt{\ell!}d\gamma_{1}(x)},\abs{\int_{\R} 1_{B^{c}}(x)h_{\ell}(x)\sqrt{\ell!}d\gamma_{1}(x)}\,\right)\\
\end{aligned}
\end{equation}
By (<ref>), and using $\int_{\R}1_{B}(x)h_{1}(x)d\gamma_{1}(x)=\int_{\R}1_{B}(x)h_{3}(x)d\gamma_{1}(x)=0$, which follows from (<ref>), we have for any $x\in\R$,
Then, by (<ref>) and Lemma <ref>, if $\abs{x}\leq\sqrt{-4\log\rho}$,
≤10 min(a^1/2,(1-a)^1/2)ρ^15/4.
§ PERTURBATION OF FOURIER COEFFICIENTS
When $n=1$, we would like to show: if $B'\subset\R$ exists such that $f=1_{B'}$ nearly maximizes (<ref>), then $B'$ is close to a ball $B\subset\R$ centered at the origin. When $n=1$, this statement amounts to a simple rearrangement argument.
For any $x\in\R$, let $g(x)=1-x^{2}$, and let $0<c<d$. Then
g(c)\geq\frac{1}{\gamma_{1}([c,d])}\int_{c}^{d}g(x)d\gamma_{1}(x)\geq g(d-(d-c)/3).
Let $n=1$. Let $B=B(0,r)$ such that $\gamma_{1}(B)=a$ and
\begin{equation}\label{three0.0}
\int_{\R}1_{B}(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x)
=\inf_{\substack{\{f\colon\R\to[0,1],\\ \int_{\R} f(x)d\gamma_{1}(x)=a\}}}
\int_{\R} f(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x).
\end{equation}
Let $B'\subset\R$. Assume that there is an $\epsilon>0$ such that $B'$ satisfies
\begin{equation}\label{three1.0}
\int_{\R}1_{B'}(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x)
<\inf_{\substack{\{f\colon\R\to[0,1],\\ \int_{\R} f(x)d\gamma_{1}(x)=a\}}}
\int_{\R} f(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x)
+\epsilon a.
\end{equation}
\begin{equation}\label{three2.0}
\int_{\R}\abs{1_{B'}(x)-1_{B}(x)}d\gamma_{1}(x)<10\epsilon^{1/2}.
\end{equation}
We use a rearrangement argument. Note that
$$\gamma_{1}(B'\setminus B)=\gamma_{1}(B')-\gamma_{1}(B'\cap B)=\gamma_{1}(B)-\gamma_{1}(B'\cap B)=\gamma_{1}(B\setminus B').$$
Since $B=(B'\cap B)\cup(B\setminus B')$ and $B'=(B'\cap B)\cup(B'\setminus B)$,
\begin{equation}\label{three3}
\int_{B}(1-x^{2})d\gamma_{1}(x)-\int_{B'}(1-x^{2})d\gamma_{1}(x)
=\int_{B\setminus B'}(1-x^{2})d\gamma_{1}(x)-\int_{B'\setminus B}(1-x^{2})d\gamma_{1}(x).
\end{equation}
Let $r_{0}\in[0,r)$ such that $\gamma_{1}([r_{0},r])=(1/2)\gamma_{1}(B\setminus B')$, and let $r_{1}\in(r,\infty]$ such that $\gamma_{1}([r,r_{1}])=(1/2)\gamma_{1}(B'\setminus B)$. Then, since $(B\setminus B')\subset B=B(0,r)$,
$$\int_{B\setminus B'}(1-x^{2})d\gamma_{1}(x)
\geq2\int_{r_{0}}^{r}(1-x^{2})d\gamma_{1}(x).
Also, since $(B'\setminus B)\subset B^{c}=B(0,r)^{c}$,
\int_{B'\setminus B}(1-x^{2})d\gamma_{1}(x)
\leq2\int_{r}^{r_{1}}(1-x^{2})d\gamma_{1}(x).
Let $f(x)\colonequals1-x^{2}$. From (<ref>) and Lemma <ref>,
\begin{equation}\label{three4}
\begin{aligned}
\geq2\int_{r_{0}}^{r}(1-x^{2})d\gamma_{1}(x)-2\int_{r}^{r_{1}}(1-x^{2})d\gamma_{1}(x)\\
&\quad=\gamma_{1}(B\setminus B')\left(\frac{1}{\gamma_{1}([r_{0},r])}\int_{r_{0}}^{r}(1-x^{2})d\gamma_{1}(x)
&\quad\geq\gamma_{1}(B\setminus B')(f(r-(r-r_{0})/3)-f(r)\\
&\quad\geq(4r/3)(r-r_{0})(1/3)\gamma_{1}(B\setminus B')
\geq(4/9)\sqrt{2\pi}r\gamma_{1}(B\setminus B')\gamma_{1}([r_{0},r])\\
&\quad\geq(2/9)\sqrt{2\pi}r\gamma_{1}(B\setminus B')^{2}
\geq(2\pi/9)\gamma_{1}(B\setminus B')^{2}a.
\end{aligned}
\end{equation}
Finally, by (<ref>) we have
\begin{equation}\label{three6}
\int_{B}(x^{2}-1)d\gamma_{1}(x)-\int_{B'}(x^{2}-1)d\gamma_{1}(x)
\leq-a(1/6)(\int\abs{1_{B'}(x)-1_{B}(x)}d\gamma_{1}(x))^{2}.
\end{equation}
So, combining (<ref>), (<ref>) and (<ref>), $\int_{\R}\abs{1_{B'}(x)-1_{B}(x)}d\gamma_{1}(x)<10\epsilon^{1/2}$.
Let $n=1$. Let $B=B(0,r')^{c}$ such that $\gamma_{1}(B)=a$ and
\begin{equation}\label{three0}
\int_{\R}1_{B}(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x)
=\sup_{\substack{\{f\colon\R\to[0,1],\\ \int_{\R} f(x)d\gamma_{1}(x)=a\}}}
\int_{\R} f(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x).
\end{equation}
Let $B'\subset\R^{1}$. Assume that there is an $\epsilon>0$ such that $B'$ satisfies
\begin{equation}\label{three1}
\int_{\R}1_{B'}(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x)
>\sup_{\substack{\{f\colon\R\to[0,1],\\ \int_{\R} f(x)d\gamma_{1}(x)=a\}}}
\int_{\R} f(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x)
\end{equation}
\begin{equation}\label{three2}
\int_{\R}\abs{1_{B'}(x)-1_{B}(x)}d\gamma_{1}(x)<10\epsilon^{1/2}.
\end{equation}
Apply Lemma <ref> to $B^{c}$.
§ EXISTENCE LEMMA
We prove the existence of two sets which minimize Gaussian correlation. The argument below is almost identical to Lemma <ref>.
Let $\rho\in(0,1)$, $0<a,b<1$. Then there exist $A,B\subset\R^{n}$ with $A=-A$, $B=-B$ such that
\begin{equation}\label{zero0}
\int_{\R^{n}}1_{A}(x)T_{\rho}1_{B}(x)d\gamma_{n}(x)
=\inf_{\substack{\{f,g\colon\R^{n}\to[0,1],\int_{\R^{n}} f(x)d\gamma_{n}(x)=a\\\int_{\R^{n}} g(x)d\gamma_{n}(x)=b,f(x)=f(-x)\,\forall\,x\in\R^{n}\}}}
\int_{\R^{n}} f(x)T_{\rho}g(x)d\gamma_{n}(x).
\end{equation}
If $\rho\in(-1,0)$, the same result holds, with the additional restriction $g(x)=g(-x)$ $\forall$ $x\in\R^{n}$ in (<ref>).
Define the set $C\colonequals\{f,g\colon\R^{n}\to[0,1],\int_{\R^{n}} f(x)d\gamma_{n}(x)=a,
\int_{\R^{n}} g(x)d\gamma_{n}(x)=b,f(x)=f(-x)\,\forall x\in\R^{n}\}$. Then $C$ is a norm closed, convex and norm bounded subset of the Hilbert space $L_{2}(\gamma_{n})\oplus L_{2}(\gamma_{n})$. Therefore, $C\subset L_{2}(\gamma_{n})\oplus L_{2}(\gamma_{n})$ is weakly closed. Also, $C$ is weakly compact by the Banach-Alaoglu Theorem. Define $T\colon C\to\R$ by
\begin{equation}\label{zero8}
T(f,g)\colonequals\int_{\R^{n}} f(x)T_{\rho}g(x)d\gamma_{n}(x).
\end{equation}
From the Cauchy-Schwarz inequality and (<ref>), $\abs{T(f,g)}\leq\vnorm{f}_{L_{2}(\gamma_{n})}\vnorm{g}_{L_{2}(\gamma_{n})}$. That is, $T$ is a strongly bounded bilinear function, so $T$ is weakly continuous. So, $T$ is weakly continuous on the weakly compact set $C\subset L_{2}(d\gamma_{n})\oplus L_{2}(d\gamma_{n})$. And there exist $f,g\in C$ such that $T(f,g)=\min_{(f',g')\in C}T(f',g')$.
From (<ref>), we have the following absolutely convergent sum
\begin{equation}\label{zero2}
\begin{aligned}
\int_{\R^{n}} f(x)\sqrt{\ell!}h_{\ell}(x)d\gamma_{n}(x)\int_{\R^{n}} g(y)\sqrt{\ell!}h_{\ell}(y)d\gamma_{n}(y)\\
\int_{\R^{n}} f(x)\sqrt{\ell!}h_{\ell}(x)d\gamma_{n}(x)\int_{\R^{n}} g(y)\sqrt{\ell!}h_{\ell}(y)d\gamma_{n}(y).
\end{aligned}
\end{equation}
Since $f(x)=f(-x)$ for all $x\in\R^{n}$, the sum over odd terms in (<ref>) is zero by (<ref>).
Now, the function $g_{s}(x)\colonequals (g(x)+g(-x))/2$ satisfies
\int_{\R^{n}} g_{s}(x)\sqrt{\ell!}h_{\ell}(x)d\gamma_{n}(x)
\begin{cases}
\int_{\R^{n}} g(x)\sqrt{\ell!}h_{\ell}(x)d\gamma_{n}(x) & ,\vnorm{\ell}_{1}\,\mbox{even}\\
0 & ,\vnorm{\ell}_{1}\,\mbox{odd}.
\end{cases}
So, $T(f,g_{s})\leq T(f,g)$. (If $\rho<0$, then we have already assumed that $g(x)=g(-x)$ for all $x\in\R^{n}$, so that $g_{s}=g$.) Let $C^{s}\colonequals\{f,g\colon\R^{n}\to[0,1],f(x)=f(-x),g(x)=g(-x),\forall x\in\R^{n},\int_{\R^{n}} f(x)d\gamma_{n}(x)=a,\int_{\R^{n}} g(x)d\gamma_{n}(x)=b\}$. We have just shown that
\begin{equation}\label{zero3}
\min_{(f',g')\in C}T(f',g')=\min_{(f',g')\in C^{s}}T(f',g').
\end{equation}
We therefore try to minimize $T$ on $C^{s}$. But $T$ is linear in each of its arguments, and $T$ is a weakly continuous function on the weakly compact set $C^{s}\subset L_{2}(\gamma_{n})\oplus L_{2}(\gamma_{n})$. Therefore, there exist $A,B\subset\R^{n}$ such that $1_{A},1_{B}\in C^{s}$ satisfy $T(1_{A},1_{B})=\min_{(f',g')\in C^{s}}T(f',g')$. Combining this fact with (<ref>), $T(1_{A},1_{B})=\min_{(f',g')\in C^{s}}T(f',g')=\min_{(f',g')\in C}T(f',g')$, and $A=-A,B=-B$ since $(1_{A},1_{B})\in C^{s}$.
§ PERTURBATION LEMMA
Similar to Section <ref>, we show: if two sets $A,B\subset\R$ nearly minimize the product of their second-order Hermite-Fourier coefficients, then these sets are close to a ball and the complement of a ball, respectively.
Let $n=1$, $0<a,b<1$. Let $(A,B)=(B(0,r_{a}),B(0,r_{b}')^{c})$ or let $(A,B)=(B(0,r_{a}')^{c},B(0,r_{b}))$ such that $\gamma_{1}(A)=a,\gamma_{1}(B)=b$ and such that
\begin{equation}\label{four0}
\begin{aligned}
&\qquad=\inf_{\substack{\{f,g\colon\R\to[0,1],\\ \int_{\R} f(x)d\gamma_{1}(x)=a,\int_{\R} g(y)d\gamma_{1}(y)=b\}}}
\left(\int_{\R} f(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x)\right)\left(\int_{\R} g(y)\sqrt{2!}h_{2}(y)d\gamma_{1}(y)\right).
\end{aligned}
\end{equation}
Let $A',B'\subset\R^{1}$ with $\gamma_{1}(A')=a$ and $\gamma_{1}(B')=b$. Assume that there is an $\epsilon>0$ such that
\begin{equation}\label{four1}
\begin{aligned}
&\qquad<\inf_{\substack{\{f,g\colon\R\to[0,1],\\ \int_{\R} f(x)d\gamma_{1}(x)=a,\int_{\R} g(y)d\gamma_{1}(y)=b\}}}
\left(\int_{\R} f(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x)\right)\left(\int_{\R} g(y)\sqrt{2!}h_{2}(y)d\gamma_{1}(y)\right)\\
\end{aligned}
\end{equation}
\begin{equation}\label{four2.3}
\min\left(\int_{\R}\abs{1_{A}(x)-1_{A'}(x)}d\gamma_{1}(x),\int_{\R}\abs{1_{A^{c}(x)}-1_{A'}(x)}d\gamma_{1}(x)\right)<10\epsilon^{1/2}\min(a,1-a)^{-1/4},
\end{equation}
\begin{equation}\label{four2.4}
\min\left(\int_{\R}\abs{1_{B}(y)-1_{B'}(y)}d\gamma_{1}(y),\int\abs{1_{B^{c}}(y)-1_{B'}(y)}d\gamma_{1}(y)\right)<10\epsilon^{1/2}\min(b,1-b)^{-1/4}.
\end{equation}
Suppose without loss of generality that $(A,B)=(B(0,r_{a}')^{c},B(0,r_{b}))$. First, note that there exists $\widetilde{B}\subset\R$ with $\gamma_{1}(\widetilde{B})=b$ such that
$$\int_{\R} 1_{\widetilde{B}}(y)\sqrt{2!}h_{2}(y)d\gamma_{1}(y)=\inf_{\substack{\{g\colon\R\to\R,0\leq g\leq 1\\ \int_{\R} g(y)d\gamma_{1}(y)=b\}}}\int_{\R} g(y)\sqrt{2!}h_{2}(y)d\gamma_{1}(y).$$
So, using $\vnormf{1_{\widetilde{B}}}_{L_{2}(\gamma_{1})}=\sqrt{b}$ and the Cauchy Schwarz inequality,
\begin{equation}\label{four2.9}
\begin{aligned}
&\bigg|\inf_{\substack{\{g\colon\R\to\R,0\leq g\leq 1\\ \int_{\R} g(y)d\gamma_{1}(y)=b\}}}\int_{\R} g(y)\sqrt{2!}h_{2}(y)d\gamma_{1}(y)\bigg|\\
&\qquad=\abs{\min\left(\int_{\R} 1_{\widetilde{B}}(y)\sqrt{2!}h_{2}(y)d\gamma_{1}(y),\int_{\R} 1_{\widetilde{B}^{c}}(y)\sqrt{2!}h_{2}(y)d\gamma_{1}(y)\right)}
\leq\min(\sqrt{b},\sqrt{1-b}).
\end{aligned}
\end{equation}
By (<ref>),
\begin{equation}\label{four2.1}
\int_{\R}1_{A'}(x)h_{2}(x)\sqrt{2!}d\gamma_{1}(x)>\sup_{\substack{\{f\colon\R\to[0,1],\\ \int_{\R} f(x)d\gamma_{1}(x)=a\}}}
\left(\int_{\R} f(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x)\right)-\epsilon\min(\sqrt{a},\sqrt{1-a}),
\end{equation}
\begin{equation}\label{four2.2}
\int_{\R}1_{B'}(y)h_{2}(y)\sqrt{2!}d\gamma_{1}(y)<\inf_{\substack{\{g\colon\R\to\R,0\leq g\leq 1,\\ \int_{\R} g(y)d\gamma_{1}(y)=b\}}}
\left(\int_{\R} g(y)\sqrt{2!}h_{2}(y)d\gamma_{1}(y)\right)+\epsilon\min(\sqrt{b},\sqrt{1-b}).
\end{equation}
For example,
if (<ref>) is false, then (<ref>) implies
∫_ g(y)dγ_1(y)=b}
(∫_ g(y)√(2!)h_2(y)dγ_1(y))
∫_ f(x)dγ_1(x)=a}
(∫_ f(x)√(2!)h_2(x)dγ_1(x))-ϵmin(√(a),√(1-a))]·
∫_ g(y)dγ_1(y)=b}
(∫_ g(y)√(2!)h_2(y)dγ_1(y))
∫_ f(x)dγ_1(x)=a,∫_ g(y)dγ_1(y)=b}
(∫_ f(x)√(2!)h_2(x)dγ_1(x))(∫_ g(y)√(2!)h_2(y)dγ_1(y))
This inequality contradicts (<ref>), so that (<ref>) holds. Similarly, (<ref>) holds.
So, (<ref>), (<ref>) and Lemmas <ref> and <ref> imply (<ref>) and (<ref>).
§ FIRST VARIATION
The following first variation argument is well-known.
Let $\rho\in(0,1)$ and let $0<a,b<1$. From (<ref>) and Lemma <ref>, let $(A,B)\subset\R^{n}\times\R^{n}$ with $A=-A,B=-B$ such that
\begin{equation}\label{two5.0}
\int_{\R^{n}}1_{A}(x)T_{\rho}1_{B}(x)d\gamma_{n}(x)=
\inf_{\substack{\{f,g\colon\R^{n}\to[0,1],\int_{\R^{n}} f(x)d\gamma_{n}(x)=a\\ \int_{\R^{n}} g(x)d\gamma_{n}(x)=b,f(x)=f(-x)\,\forall\,x\in\R^{n}\}}}
\int_{\R^{n}}f(x)T_{\rho}g(x)d\gamma_{n}(x).
\end{equation}
Then there exist $c,c'\in\R$ such that
\begin{equation}\label{two5pp}
A=\{x\in\R^{n}\colon T_{\rho}1_{B}(x)\leq c\}\,\wedge\,B=\{x\in\R^{n}\colon T_{\rho}1_{A}(x)\leq c'\}.
\end{equation}
If $\rho\in(-1,0)$, the same result holds, with the additional restriction $g(x)=g(-x)$ $\forall$ $x\in\R^{n}$ in (<ref>).
We argue by contradiction. Suppose there exists $x_{1},x_{2}\in\R^{n}$, $x_{1}\notin A,x_{2}\in A$ such that $T_{\rho}1_{B}(x_{1})>T_{\rho}1_{B}(x_{2})$. Let $U_{1}\subset\R^{n}$ be a small ball around $x_{1}$ and let $U_{2}$ be a small ball around $x_{2}$ such that $T_{\rho}1_{B}(u_{1})>T_{\rho}1_{B}(u_{2})$, $\forall$ $u_{1}\in U_{1},u_{2}\in U_{2}$. Also, assume that $U_{1}\cap U_{2}=\emptyset$ and $\gamma_{n}(U_{1})=\gamma_{n}(U_{2})$.
Define $A'\colonequals (A\setminus U_{2})\cup U_{1}$. Then $1_{A'}=1_{A}-1_{U_{2}}+1_{U_{1}}$, and for $U_{1},U_{2}$ sufficiently small,
=∫_^n 1_A(x)T_ρ1_B(x)dγ_n(x)
>∫_^n 1_A(x)T_ρ1_B(x)dγ_n(x).
This inequality contradicts the maximality of $A$. We conclude that no such $x_{1},x_{2}$ exist, so (<ref>) holds.
§ FIRST MAIN THEOREM
Let $n=1$, $0<a,b<1$, and let $\abs{\rho}<\min(e^{-40},a^{20},(1-a)^{20},b^{20},(1-b)^{20})/1000$. By Lemma <ref>, let $(A,B)=(B(0,r_{a}),B(0,r_{b}')^{c})$ or let $(A,B)=(B(0,r_{a}')^{c},B(0,r_{b}))$ such that $\gamma_{1}(A)=a,\gamma_{n}(B)=b$ and such that
\begin{equation}\label{four10}
\begin{aligned}
\left(\int_{\R}1_{B}(y)\sqrt{2!}h_{2}(y)d\gamma_{1}(y)\right)\\
&\qquad=\inf_{\substack{\{f,g\colon\R\to[0,1],\\ \int_{\R} f(x)d\gamma_{1}(x)=a,\int_{\R} g(y)d\gamma_{1}(y)=b\}}}
\left(\int_{\R} f(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x)\right)
\left(\int_{\R} g(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x)\right).
\end{aligned}
\end{equation}
From Lemma <ref>, let $A',B'\subset\R$ such that $\gamma_{1}(A')=a,\gamma_{1}(B')=b$ and such that
\begin{equation}\label{four11}
\int_{\R} 1_{A'}(x)T_{\rho}1_{B'}(x)d\gamma_{1}(x)
=\inf_{\substack{\{f,g\colon\R^{n}\to[0,1],\int_{\R} f(x)d\gamma_{1}(x)=a,\\
\int_{\R} g(x)d\gamma_{1}(x)=b,\,f(x)=f(-x),\forall\,x\in\R\}}}
\int_{\R}f(x)T_{\rho}g(x)d\gamma_{n}(x).
\end{equation}
If $\rho>0$, then $(A,B)=(A',B')$. If $\rho<0$, the same result holds, with the additional restriction $g(x)=g(-x)$ $\forall$ $x\in\R^{n}$ in (<ref>).
Without loss of generality $(A,B)=(B(0,r_{a}),B(0,r_{b}')^{c})$.
Step 1. Approximating Noise Stability using second order Hermite-Fourier coefficients.
From (<ref>), and using that $A'=-A'$ with (<ref>)
\begin{equation}\label{four12}
\begin{aligned}
\end{aligned}
\end{equation}
From (<ref>) and (<ref>),
∫_ f(x)dγ_1(x)=a,
∫_ g(x)dγ_1(x)=b,f(x)=f(-x) ∀x∈}
∫_ f(x)T_ρg(x)dγ_1(x)-ab/ρ^2
Similarly, from (<ref>)
∫_ f(x)dγ_1(x)=a,
∫_ g(x)dγ_1(x)=b,f(x)=f(-x) ∀x∈}
∫_ f(x)T_ρg(x)dγ_1(x)-ab/ρ^2
Combining (<ref>), (<ref>) and (<ref>),
\begin{equation}\label{four12.9}
\begin{aligned}
&\qquad\leq\inf_{\substack{\{f,g\colon\R\to[0,1],\\ \int_{\R} f(x)d\gamma_{1}(x)=a,\int_{\R} g(y)d\gamma_{1}(y)=b\}}}
\left(\int_{\R} f(x)\sqrt{2!}h_{2}(x)d\gamma_{1}(x)\right)\left(\int_{\R} g(y)\sqrt{2!}h_{2}(y)d\gamma_{1}(y)\right)\\
\end{aligned}
\end{equation}
Step 2. Optimal sets are close to balls or their complement.
From (<ref>) and Lemma <ref>,
\begin{equation}\label{four13}
\int_{\R}\abs{1_{A}(x)-1_{A'}(x)}^{2}d\gamma_{1}(x)<10\abs{\rho}^{7/8}\,\wedge\,\int_{\R}\abs{1_{B}(y)-1_{B'}(y)}^{2}d\gamma_{1}(y)<10\abs{\rho}^{7/8}.
\end{equation}
Then, by the Cauchy-Schwarz inequality, for every $\ell\in\N$,
\begin{equation}\label{four14}
\begin{aligned}
\abs{\int_{\R}(1_{A}(x)-1_{A'}(x))\sqrt{\ell!}h_{\ell}(x)d\gamma_{1}(x)}&<\sqrt{10}\abs{\rho}^{7/16}\\
\abs{\int_{\R}(1_{B}(y)-1_{B'}(y))\sqrt{\ell!}h_{\ell}(y)d\gamma_{1}(y)}&<\sqrt{10}\abs{\rho}^{7/16}.
\end{aligned}
\end{equation}
Step 3. Estimating $T_{\rho}1_{B'}$.
Let $g=1_{B}-1_{B'}$. Recall that $b=2\int_{r_{b}}^{\infty}e^{-x^{2}/2}dx/\sqrt{2\pi}$. Then $\min(b,1-b)/10\leq r_{b}\leq\sqrt{-3\log\min(b,1-b)}$. Since $0<\abs{\rho}<\min(b,1-b)<1$, we have $-\log\abs{\rho}>-\log\min(b,1-b)$. Let $\abs{x}\leq\sqrt{-4\log\abs{\rho}}$. Since $\abs{\rho}<e^{-10}$,
\begin{equation}\label{fourzero}
\abs{\ell}3^{\abs{\ell}}(-4\log\abs{\rho})^{\abs{\ell}/2}<1.
\end{equation}
By (<ref>), $\int_{\R}g(x)h_{3}(x)d\gamma_{1}(x)=0$. So, using Lemma <ref>,
√(ℓ!)h_ℓ(x)∫_ h_ℓ(x)√(ℓ!)g(x)dγ_1(x)
Lemma <ref>≤ρ^3∑_ℓ∈ℓ≥4ρ^ℓ-3
ℓ3^ℓmax(1,x^ℓ)∫_ h_ℓ(x)√(ℓ!)g(x)dγ_1(x)
That is, for any $\abs{x}\leq\sqrt{-4\log\abs{\rho}}$,
\begin{equation}\label{four15}
\absf{T_{\rho}1_{B}(x)-T_{\rho}1_{B'}(x)-\rho^{2}\frac{\sqrt{2}}{2}(x^{2}-1)\int_{\R}(1_{B}(x)-1_{B'}(x))\sqrt{2}h_{2}(x)d\gamma_{1}(x)}
\end{equation}
Similarly, for any $\abs{x}\leq\sqrt{-3\log\abs{\rho}}$,
\begin{equation}\label{four15.5}
\absf{\frac{d}{dx}T_{\rho}(1_{B}-1_{B'})(x)-\rho^{2}\sqrt{2}x\int_{\R}(1_{B}(y)-1_{B'}(y))\sqrt{2}h_{2}(y)d\gamma_{1}(y)}\leq\abs{\rho}^{11/4}
\end{equation}
Step 4. Finding the level sets of $T_{\rho}1_{B'}$
We now apply Lemma <ref>. Let $\min(b,1-b)/10\leq\abs{x}\leq\sqrt{-3\log\abs{\rho}}$. Then, using that $\min(b,1-b)/10\leq r_{b}'\leq\sqrt{-2\log\min(b,1-b)}$,
\begin{equation}\label{four16}
\mathrm{sign}(x)\cdot\frac{d}{dx}T_{\rho}1_{B}(x)
\geq\abs{x}\rho^{2}\min(b,(1-b))/10
\geq\rho^{2}\min(b,1-b,a,1-a)^{2}/10.
\end{equation}
Let $\min(b,1-b)/10\leq r_{0}\leq\sqrt{-3\log\abs{\rho}}$. By (<ref>), using that $\mathrm{sign}(x)\frac{d}{dx}T_{\rho}1_{B}(x)<0$ for all $x\neq0$, there is a $\lambda=\lambda(r_{0})\in\R$ such that
Also, we may take $\lambda$ to be a continuous, strictly increasing function of $r_{0}$. By (<ref>), (<ref>) and (<ref>), and using $\abs{\rho}<\min(e^{-40},a^{20},(1-a)^{20},b^{20},(1-b)^{20})/1000$.
\begin{equation}\label{four16.9}
\begin{aligned}
&x\in B(0,r_{0}-(1/10)\min(e^{-40},b,1-b,a,1-a))\Longrightarrow T_{\rho}1_{B'}(x)\leq\lambda,\\
&x\in B(0,\sqrt{-3\log\abs{\rho}})\setminus B(0,r_{0}+(1/10)\min(e^{-40},b,1-b,a,1-a))\\
&\qquad\qquad\qquad\Longrightarrow T_{\rho}1_{B'}(x)>\lambda.
\end{aligned}
\end{equation}
By Lemma <ref>, there exists $c_{1},c_{2}\in\R$ such that
\begin{equation}\label{four000}
A'=\{x\in\R\colon T_{\rho}1_{B'}(x)\leq c_{1}\}\,\wedge\,
B'=\{x\in\R\colon T_{\rho}1_{A'}(x)\leq c_{2}\}.
\end{equation}
Since $\gamma_{1}(B(0,\sqrt{-3\log\abs{\rho}})^{c})<\min(a,1-a)$, $B(0,\sqrt{-3\log\abs{\rho}})\cap A'\neq\emptyset$. So, by (<ref>) there exists an $x\in B(0,\sqrt{-3\log\abs{\rho}})$ such that $T_{\rho}1_{B'}(x)\leq c_{1}$. So, there exists $r_{0}$ such that $\lambda(r_{0})=c_{1}$. Rewriting (<ref>),
\begin{equation}\label{four17}
\begin{aligned}
&B(0,r_{0}-(1/10)\min(e^{-40},b,1-b,a,1-a))\subset\{x\in\R\colon T_{\rho}1_{B'}(x)\leq c_{1}\}\\
&\,\wedge\,(B(0,\sqrt{-3\log\abs{\rho}})\setminus B(0,r_{0}+(1/10)\min(e^{-40},b,1-b,a,1-a)))\\
&\qquad\qquad\qquad\qquad\cap\{x\in\R\colon T_{\rho}1_{B'}(x)\leq c_{1}\}=\emptyset.
\end{aligned}
\end{equation}
Combining (<ref>) and (<ref>), we have $\frac{d}{dx}T_{\rho}1_{B'}(x)\mathrm{sign}(x)>0$ for all $x$ such that $\sqrt{-3\log\abs{\rho}}\geq\abs{x}\geq\min(b,1-b,a,1-a)/10$. Using this fact and (<ref>), there exists $\min(b,1-b)/10\leq r_{1}\leq r_{b}'$ such that
\begin{equation}\label{four18.5}
\begin{aligned}
&B(0,r_{1})\subset\{x\in\R\colon T_{\rho}1_{B'}(x)\leq c_{1}\}\\
&\qquad\,\wedge\,[B(0,\sqrt{-3\log\abs{\rho}})\setminus B(0,r_{1})]\cap\{x\in\R\colon T_{\rho}1_{B'}(x)\leq c_{1}\}=\emptyset.
\end{aligned}
\end{equation}
Repeating the above implications with the roles of $A'$ and $B'$ reversed, there exists $\min(a,1-a,b,1-b)/10\leq r_{2}\leq r_{a}$ such that
\begin{equation}\label{four18.8}
\begin{aligned}
&B(0,r_{2})\cap\{x\in\R\colon T_{\rho}1_{A'}(x)\leq c_{2}\}=\emptyset\\
&\qquad\,\wedge\,[B(0,\sqrt{-3\log\abs{\rho}})\setminus B(0,r_{2})]\subset\{x\in\R\colon T_{\rho}1_{A'}(x)\leq c_{2}\}.
\end{aligned}
\end{equation}
Step 5. A final iterative argument to eliminate points far from the origin.
We now construct an iteration. Let $k\in\N$. It is given that
\begin{equation}\label{four20}
\begin{aligned}
&B(0,r_{1})\subset\{x\in\R\colon T_{\rho}1_{B'}(x)\leq c_{1}\}\\
&\qquad\,\wedge\,[B(0,\sqrt{-(k+2)\log\abs{\rho}})\setminus B(0,r_{1})]\cap\{x\in\R\colon T_{\rho}1_{B'}(x)\leq c_{1}\}=\emptyset,\\
&B(0,r_{2})\cap\{x\in\R\colon T_{\rho}1_{A'}(x)\leq c_{2}\}=\emptyset\\
&\qquad\,\wedge\,[B(0,\sqrt{-(k+2)\log\abs{\rho}})\setminus B(0,r_{2})]\subset\{x\in\R\colon T_{\rho}1_{A'}(x)\leq c_{2}\}.
\end{aligned}
\end{equation}
We then conclude that
\begin{equation}\label{four21}
\begin{aligned}
&B(0,r_{1})\subset\{x\in\R\colon T_{\rho}1_{B'}(x)\leq c_{1}\}\\
&\qquad\,\wedge\,[B(0,\sqrt{-(k+3)\log\abs{\rho}})\setminus B(0,r_{1})]\cap\{x\in\R\colon T_{\rho}1_{B'}(x)\leq c_{1}\}=\emptyset,\\
&B(0,r_{2})\cap\{x\in\R\colon T_{\rho}1_{A'}(x)\leq c_{2}\}=\emptyset\\
&\qquad\,\wedge\,[B(0,\sqrt{-(k+3)\log\abs{\rho}})\setminus B(0,r_{2})]\subset\{x\in\R\colon T_{\rho}1_{A'}(x)\leq c_{2}\}.
\end{aligned}
\end{equation}
(Note that $k=2$ for (<ref>) is exactly (<ref>) and (<ref>).)
Let $x$ with $\sqrt{-(k+2)\log\abs{\rho}}\leq\abs{x}\leq\sqrt{-(k+3)\log\abs{\rho}}$. From Lemma <ref> and (<ref>),
Similarly, $(\sqrt{1-\rho^{2}}/\abs{\rho})\frac{d}{dx}T_{\rho}1_{B'}(x)>0$. Therefore, (<ref>) implies that (<ref>) holds. So, let $k\to\infty$ in (<ref>). Combining (<ref>) and (<ref>) then completes the theorem.
B(0,r_{2})=\{x\in\R\colon T_{\rho}1_{A'}\leq c_{1}\}=B'\,\wedge\,
B(0,r_{1})=\{x\in\R\colon T_{\rho}1_{B'}\leq c_{1}\}=A'.
§ A SECOND VARIATION FORMULA
In preparation for later sections, we now investigate a second variation formula for quadratic functionals. Lemma <ref> below essentially appears in <cit.>. However, their statement and proof are slightly different than we require. We prove Lemmas <ref> and Lemma <ref> in the Appendix, Section <ref> (see Lemmas <ref> and <ref>.)
Let $A\subset\R^{n}$ be a set with smooth boundary, and let $N\colon\partial A\to S^{n-1}$ denote the unit exterior normal to $\partial A$. Let $X\colon\R^{n}\to\R^{n}$ be a vector field. Let $\Psi\colon\R^{n}\times(-1,1)$ such that $\Psi(x,0)=x$ and such that $\frac{d}{dt}|_{t=0}\Psi(x,t)=X(\Psi(x,t))$ for all $x\in\R^{n},t\in(-1,1)$. For any $t\in(-1,1)$, let $A^{(t)}=\Psi(A,t)$. Note that $A^{(0)}=A$. Define
$$V(x,t)\colonequals\int_{A^{(t)}}G(x,y)dy,\qquad \forall\,x\in\R^{n},\,\forall\,t\in(-1,1).$$
Let $G\colon\R^{n}\times\R^{n}\to\R$ be a Schwartz function. For any $A\subset\R^{n}$, let $F(A)\colonequals \int_{\R^{n}}\int_{\R^{n}} 1_{A}(x)G(x,y)1_{A}(y)dxdy$. Then
$$\frac{d}{dt}|_{t=0}\gamma_{n}(A^{(t)})=\int_{\partial A}\langle X(x),N(x)\rangle \gamma_{n}(x)dx.$$
$$\frac{d^{2}}{dt^{2}}|_{t=0}\gamma_{n}(A^{(t)})=\int_{\partial A}(\mathrm{div}(X(x))-\langle X(x),x\rangle)\langle X(x),N(x)\rangle \gamma_{n}(x)dx.$$
§.§ Noise Stability
As our first application of Lemma <ref>, we study the second variation of the noise stability.
For any $x,y\in\R^{n}$, let $G(x,y)=\frac{e^{-\vnorm{\rho x-y}^{2}/[2(1-\rho^{2})]}\gamma_{n}(x)}{(1-\rho^{2})^{n/2}(2\pi)^{n/2}}=\frac{e^{\frac{-\vnorm{x}_{2}^{2}-\vnorm{y}_{2}^{2}+2\rho\langle x,y\rangle}{2(1-\rho^{2})}}}{(1-\rho^{2})^{n/2}(2\pi)^{n}}$. Suppose Assumption <ref> holds. For any $A\subset\R^{n}$, for any $\rho\in(-1,1)$, define
$$F_{\rho}(A)\colonequals\int_{\R^{n}}\int_{\R^{n}} 1_{A}(x)G(x,y)1_{A}(y)dxdy
=\int_{\R^{n}}\int_{\R^{n}} 1_{A}(x)T_{\rho}1_{A}(x)d\gamma_{n}(x).
Then, for any $t\in(-1,1)$ and for any $x\in\R^{n}$, using Assumption <ref> we have
\begin{equation}\label{Afour1}
\end{equation}
Let $\rho\in(-1,1)$. Assume Assumption <ref> holds with $\frac{d^{2}}{dt^{2}}|_{t=0}\gamma_{n}(A^{(t)})=0$. Assume also that $T_{\rho}1_{A}(x)$ is constant for all $x\in\partial A$. Then
\begin{equation}\label{newfive}
\begin{aligned}
\frac{1}{2}\frac{d^{2}}{dt^{2}}F(A^{(t)})|_{t=0}
&=\int_{\partial A}\int_{\partial A}G(x,y)\langle X(x),N(x)\rangle\langle X(y),N(y)\rangle dxdy\\
&\quad+\int_{\partial A}\langle\nabla T_{\rho}1_{A}(x),X(x)\rangle\langle X(x),N(x)\rangle dx.
\end{aligned}
\end{equation}
Applying Lemma <ref>,
\begin{equation}\label{Afour2}
\begin{aligned}
\frac{1}{2}\frac{d^{2}}{dt^{2}}F(A^{(t)})|_{t=0}
&=\int_{\partial A}\int_{\partial A}G(x,y)\langle X(x),N(x)\rangle\langle X(y),N(y)\rangle dxdy\\
&\qquad+\int_{\partial A}\mathrm{div}(V(x,0)X(x))\langle X(x),N(x)\rangle dx\\
&\stackrel{\eqref{Afour1}}{=}\int_{\partial A}\int_{\partial A}G(x,y)\langle X(x),N(x)\rangle\langle X(y),N(y)\rangle dxdy\\
&\qquad+\int_{\partial A}\mathrm{div}(T_{\rho}1_{A}(x)\gamma_{n}(x)X(x))\langle X(x),N(x)\rangle dx\\
&=\int_{\partial A}\int_{\partial A}G(x,y)\langle X(x),N(x)\rangle\langle X(y),N(y)\rangle dxdy\\
&\qquad+\int_{\partial A}(\sum_{i=1}^{n}T_{\rho}1_{A}(x)\frac{\partial}{\partial x_{i}}X^{(i)}(x)-x_{i}T_{\rho}1_{A}(x)X^{(i)}(x)\\
&\qquad\qquad+\frac{\partial}{\partial x_{i}}T_{\rho}1_{A}(x)X^{(i)}(x))\langle X(x),N(x)\rangle \gamma_{n}(x)dx.
\end{aligned}
\end{equation}
Using Lemma <ref>, and that $T_{\rho}1_{A}(x)$ is constant when $x\in\partial A$, we then get (<ref>).
§.§ A Poincaré-Type Inequality
In our investigation of the second variation of the ball or its complement in Section <ref> below, we require the following Poincaré-type inequality.
Let $r>0$ and let $B(0,r)\subset\R^{n}$. Let $f\colon \partial B(0,r)\to\R$ with $\int_{\partial B(0,r)}f(x)dx=0$.
\begin{equation}\label{ten4}
\sum_{i=1}^{n}\left(\int_{\partial B(0,r)}x_{i}^{2}f(x)dx\right)^{2}\leq \frac{2r^{n+3}\mathrm{Vol}(S^{n-1})}{n(n+2)}\int_{\partial B(0,r)}(f(x))^{2}dx.
\end{equation}
Moreover, equality occurs when $f(x)=(n-1)x_{1}^{2}-\sum_{j=2}^{n}x_{j}^{2}$ for any $x\in \partial B(0,r)$.
For any $n\geq1$, write $\mathrm{Vol}(S^{n-1}) = 2\pi\prod_{\ell=1}^{n-2}\int_{0}^{\pi} \sin^{\ell} (x)dx$. We first claim
\begin{equation}\label{newthree}
\int_{\partial B(0,r)}x_{1}^{4}dx=\frac{3r^{n+3}\mathrm{Vol}(S^{n-1})}{n(n+2)},
\qquad \int_{\partial B(0,r)}x_{1}^{2}x_{2}^{2}dx=\frac{r^{n+3}\mathrm{Vol}(S^{n-1})}{n(n+2)}.
\end{equation}
Indeed, using (hyper)-spherical coordinates,
=3·2π/8 r^n+3∏_ℓ=5^n+2∫_0^πsin^ℓ(x)dx
A similar calculation proves the other part of (<ref>).
Now, for any $i\in\{1,\ldots,n\}$, let $g_{i}\colon\R^{n}\to\R$ so that $g_{i}(x)=(n-1)x_{i}^{2}-\sum_{j\neq i}x_{j}^{2}$. It then follows from (<ref>) that
\begin{equation}\label{newten}
\int_{\partial B(0,r)}(g_{i}(x))^{2}dx
=n(n-1)\left(\int_{\partial B(0,r)}x_{1}^{4}dx-\int_{\partial B(0,r)}x_{1}^{2}x_{2}^{2}dx\right)
\end{equation}
And if $j\in\{1,\ldots,n\}$ with $j\neq i$, then
\begin{equation}\label{neweleven}
\int_{\partial B(0,r)}g_{i}(x)g_{j}(x)dx
=n\left(-\int_{\partial B(0,r)}x_{1}^{4}dx+\int_{\partial B(0,r)}x_{1}^{2}x_{2}^{2}dx\right)
\end{equation}
So, the functions $g_{1},\ldots,g_{n}$ span an $(n-1)$-dimensional vector space, $\sum_{i=1}^{n}g_{i}=0$, and $\langle g_{i},g_{j}\rangle\colonequals\int_{\partial B(0,r)}g_{i}(x)g_{j}(x)dx=-2(n-1)\frac{r^{n+3}\mathrm{Vol}(S^{n-1})}{n(n+2)}$ if $i,j\in\{1,\ldots,n\}$ with $i\neq j$.
We now proceed to prove (<ref>). Using $\int_{\partial B(0,r)}f(x)dx=0$, we have
So, to prove (<ref>), we can equivalently prove that
\begin{equation}\label{newseven}
\sum_{i=1}^{n}\left(\int_{\partial B(0,r)}g_{i}(x)f(x)dx\right)^{2}
\leq\frac{2n}{n+2}r^{n+3}\mathrm{Vol}(S^{n-1})\int_{\partial B(0,r)}(f(x))^{2}dx.
\end{equation}
Since the polynomials $g_{1},\ldots,g_{n}$ are polynomials which are homogeneous of degree $2$, by expanding $f$ in spherical harmonics, it suffices to assume that $f$ is also a polynomial which is homogeneous of degree $2$. Then, the left side of (<ref>) is the sum of the squared lengths of the projections of $f$ onto $g_{1},\ldots,g_{n}$. Since $\sum_{i=1}^{n}g_{i}=0$, $g_{1},\ldots,g_{n}$ span an $(n-1)$-dimensional space, and $\langle g_{i},g_{j}\rangle$ is a constant for any $i,j\in\{1,\ldots,n\}$ with $i\neq j$, we conclude that the left side of (<ref>) is bounded by a multiple of the right side. More specifically, if $\int_{\partial B(0,r)}(f(x))^{2}dx$ is fixed, then the left side of (<ref>) is maximized whenever $f$ is in the span of $g_{1},\ldots,g_{n}$. In particular, the left side of (<ref>) is maximized when $f$ is a multiple of $f_{1}$. And indeed, equality holds in (<ref>) in this case, since if $f=g_{1}$, we have by (<ref>) and (<ref>),
And by (<ref>),
\sum_{i=1}^{n}\int_{\partial B(0,r)}(f(x))^{2}dx
\frac{\sum_{i=1}^{n}\left(\int_{\partial B(0,r)}g_{i}(x)f(x)dx\right)^{2}}{\int_{\partial B(0,r)}(f(x))^{2}dx}
That is, (<ref>) holds, and the proof is complete.
Let $k$ be a positive integer. Then
\begin{equation}\label{five20}
\int_{0}^{\pi}\sin^{k}(\theta)d\theta
=\begin{cases}\pi\frac{(k-1)!!}{(k)!!}&\mbox{, if $k$ is even}\\
2\frac{(k-1)!!}{(k)!!}&\mbox{, if $k$ is odd.}
\end{cases}
\end{equation}
Let $k\geq0$. Let $c_{k}\colonequals\int_{0}^{\pi}\sin^{k}(\theta)d\theta$. Then
c_k =-∫_0^πsin^k-1(θ)d/dθcos(θ) dθ=∫_0^π(k-1)sin^k-2(θ)cos^2(θ) dθ
=∫_0^π(2k-1)sin^k-2(θ)(1-sin^2(θ)) dθ=(k-1)(c_k-2-c_k).
So, if $d_{k}=c_{0}=\pi$ when $k$ is even, and $d_{k}=c_{1}=2$ when $k$ is odd,
Let $r>0$. Then
\begin{equation}\label{neweight}
\int_{B(0,r)}(1-x_{1}^{2})d\gamma_{n}(x)=\frac{\mathrm{Vol}(S^{n-1})r^{n} e^{-r^2/2}}{n(2\pi)^{n/2}}.
\end{equation}
For $\alpha>0$, define
Then, using the product rule,
=\mathrm{Vol}(S^{n-1})\left(\alpha^{\frac{n}{2}}\int_{0}^{r}(-s^{n+1}/2)e^{-\alpha s^{2}/2}\frac{ds}{(2\pi)^{n/2}}
+\frac{n}{2}\alpha^{\frac{n}{2}-1}\int_{0}^{r}s^{n-1}e^{-\alpha s^{2}/2}\frac{ds}{(2\pi)^{n/2}}\right).
Plugging in $\alpha=1$, we then get
Also, applying the Fundamental Theorem of Calculus to the definition of $g$,
Combining the two formulas for $g'(1)$, we get
That is, (<ref>) holds.
§.§ Sum of Squared Fourier Coefficients
For our second application of Lemma <ref>, we consider the sum of squared second-order Hermite-Fourier coefficients of a set.
Suppose Assumption <ref> holds with $\frac{d^{2}}{dt^{2}}|_{t=0}\gamma_{n}(A^{(t)})=0$. For any $A\subset\R^{n}$, define
$$F(A)\colonequals\sum_{i=1}^{n}\int_{\R^{n}}\int_{\R^{n}} 1_{A}(x)(1-x_{i}^{2})(1-y_{i}^{2})\gamma_{n}(x)\gamma_{n}(y)1_{A}(y)dxdy.$$
Assume there exists $r>0$ such that $A=B(0,r)$ or $A=B(0,r)^{c}$. Then
\begin{equation}\label{newsix}
\begin{aligned}
\frac{1}{2}\frac{d^{2}}{dt^{2}}F(A^{(t)})|_{t=0}
&=\sum_{i=1}^{n}\left(\int_{\partial A}(1-x_{i}^{2})\langle X(x),N(x)\rangle\gamma_{n}(x)dx\right)^{2}\\
%&\qquad+\sum_{i=1}^{n}(\int_{A} (1-y_{i}^{2})d\gamma_{n}(y))\int_{\partial A}(1-x_{i}^{2})\langle-x,X(x)\rangle\langle X(x),N(x)\rangle \gamma_{n}(x)dx\\
&\qquad+2\sum_{i=1}^{n}\int_{\partial A}(-x_{i})X^{(i)}\langle X(x),N(x)\rangle d\gamma_{n}(x)(\int_{A} (1-y_{i}^{2})d\gamma_{n}(y)).
\end{aligned}
\end{equation}
Let $i\in\{1,\ldots,n\}$ and let $G_{i}(x,y)=(1-x_{i}^{2})(1-y_{i}^{2})\gamma_{n}(x)\gamma_{n}(y)$ for any $x,y\in\R^{n}$. Define
$$F_{i}(A^{(t)})\colonequals\int_{\R^{n}}\int_{\R^{n}} 1_{A^{(t)}}(x)G_{i}(x,y)1_{A^{(t)}}(y)=(\int_{A^{(t)}} (1-x_{i}^{2})d\gamma_{n}(x))^{2}.$$
Then $F(A^{(t)})=\sum_{i=1}^{n}F_{i}(A^{(t)})$. Define
\begin{equation}\label{Athree1}
\end{equation}
Applying Lemma <ref>,
\begin{equation}\label{Athree2}
\begin{aligned}
\frac{1}{2}\frac{d^{2}}{dt^{2}}F_{i}(A^{(t)})|_{t=0}
&=\int_{\partial A}\int_{\partial A}(1-x_{i}^{2})(1-y_{i}^{2})\langle X(x),N(x)\rangle\langle X(y),N(y)\rangle \gamma_{n}(x)\gamma_{n}(y)dxdy\\
&\qquad+\int_{\partial A}\mathrm{div}(V_{i}(x,0)X(x))\langle X(x),N(x)\rangle dx\\
&\stackrel{\eqref{Athree1}}{=}(\int_{\partial A}(1-x_{i})^{2}\langle X(x),N(x)\rangle \gamma_{n}(x)dx)^{2}\\
&\qquad+\int_{\partial A}\mathrm{div}((1-x_{i}^{2})\gamma_{n}(x)X(x))\langle X(x),N(x)\rangle dx(\int_{A}(1-y_{i}^{2})d\gamma_{n}(y)).
\end{aligned}
\end{equation}
We compute the last term as follows.
\begin{equation}\label{Athree3}
\begin{aligned}
=\sum_{j=1}^{n}\frac{\partial}{\partial x_{j}}((1-x_{i}^{2})\gamma_{n}(x)X^{(j)}(x))\\
&\qquad=\sum_{j=1}^{n}(1-x_{i}^{2})((-x_{j})X^{(j)}(x)+\frac{\partial}{\partial x_{j}}X^{(j)}(x))\gamma_{n}(x)-2\cdot 1_{\{i=j\}}x_{i}\gamma_{n}(x)X^{(j)}(x)\\
&\qquad=(1-x_{i}^{2})\Big(\langle -x,X(x)\rangle+\mathrm{div}(X(x))\Big)\gamma_{n}(x)-2x_{i}\gamma_{n}(x)X^{(i)}(x).
\end{aligned}
\end{equation}
We now combine (<ref>) and (<ref>) and sum over $i\in\{1,\ldots,n\}$. Since $A=B(0,r)$ or $A=B(0,r)^{c}$, the first term from (<ref>) vanishes by Lemma <ref> after it is integrated, resulting in (<ref>).
We continue to use the assumptions and notation from Assumtion <ref>. Also, for any $A\subset\R^{n}$, define
$$F(A)\colonequals\sum_{i=1}^{n}\int_{\R^{n}}\int_{\R^{n}} 1_{A}(x)(1-x_{i}^{2})(1-y_{i}^{2})\gamma_{n}(x)\gamma_{n}(y)1_{A}(y)dxdy.$$
Let $r>0$. Assume $A=A^{(0)}= B(0,r)$ or $A=A^{(0)}=B(0,r)^{c}$. Let $f(x)=\langle X(x),N(x)\rangle$ for any $x\in\partial B(0,r)$. Assume that $\int_{\partial B(0,r)}f(x)dx=0$ and $\int_{\partial B(0,r)}\abs{f(x)}^{2}dx=1$. Then
Moreover, equality holds when $f$ is a multiple of the function $x\mapsto (n-1)x_{1}^{2}-\sum_{j=2}^{n}x_{j}^{2}$.
Suppose $A=A^{(0)}=B(0,r)$, and let $f(x)=\langle X(x),N(x)\rangle$ for any $x\in \partial B(0,r)$. Then, using Lemma <ref> we get
-2re^-r^2/2/(2π)^n/2∫_B(0,r) (1-y_1^2)dγ_n(y))∫_∂B(0,r)(f(x))^2 dx.
Since $\int_{\partial B(0,r)}f(x)dx=0$, Lemma <ref> applies, yielding
-(2re^-r^2/2/(2π)^n/2∫_B(0,r)(1-y_1^2)dγ_n(y))∫_∂B(0,r)(f(x))^2 dx
≤2r^n+3e^-r^2Vol(S^n-1)/n(n+2)(2π)^n∫_∂B(0,r)(f(x))^2 dx
-(2re^-r^2/2/(2π)^n/2∫_B(0,r)(1-y_1^2)dγ_n(y))∫_∂B(0,r)(f(x))^2 dx.
So, if $\int_{\partial B(0,r)}(f(x))^{2} dx=1$, we have
\begin{equation}\label{eight7}
\frac{1}{2}\frac{d^{2}}{dt^{2}}F(A^{(t)})|_{t=0}\\
\leq\frac{2r^{n+3}e^{-r^{2}}\mathrm{Vol}(S^{n-1})}{n(n+2)(2\pi)^{n}}
\end{equation}
Then, substituting Lemma <ref>,
\begin{equation}
\begin{aligned}
\frac{1}{2}\frac{d^{2}}{dt^{2}}F(A^{(t)})|_{t=0}
&=\frac{2r^{n+1}e^{-r^{2}}\mathrm{Vol}(S^{n-1})}{n(n+2)(2\pi)^{n}}\left(r^{2}-n-2\right). %
\end{aligned}
\end{equation}
The equality case then comes directly from Lemma <ref>, if we can find a vector field $X$ such that $\langle X(x),N(x)\rangle=(n-1)x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2}$ for any $x\in\partial A$ and such that $\frac{d^{2}}{dt^{2}}|_{t=0}\gamma_{n}(A^{t})=0$. (Since $\int_{\partial A}[(n-1)x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2}]dx=0$, we know $\frac{d}{dt}|_{t=0}\gamma_{n}(A^{t})=0$ by Lemma <ref>.)
We can construct the vector field $X$ explicitly. For any $1\leq i\leq n$, define $g_{i}(x)\colonequals(x_{1}^{2}+\cdots+x_{n}^{2}-r^{2})x_{i}/2$, $x\in\R^{n}$. Then $\frac{\partial}{\partial x_{i}}g_{i}(x)=x_{i}^{2}$ and $g_{i}(x)=0$ when $x\in\partial A$. We therefore define
$$X(x)\colonequals r\big((n-1)(x_{1}+g_{1}(x)),-x_{2}-g_{2}(x),\ldots,-x_{n}-g_{n}(x)\big),\qquad\forall\,x\in\R^{n},$$
Then if $x\in\partial A$, we have
So, $\frac{d^{2}}{dt^{2}}|_{t=0}\gamma_{n}(A^{(t)})=0$ by Lemma <ref>, as desired. Lastly, note that $N(x)=\pm x/\vnorm{x}_{2}$ for all $x\in\partial A$, so $\pm\langle X(x),N(x)\rangle=(n-1)x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2}$ for any $x\in\partial A$ since $g_{i}(x)=0$ for any $x\in\partial A$, and for any $1\leq i\leq n$.
§ LOCAL OPTIMALITY
We continue to use the assumptions and notation from Assumtion <ref>. Also, for any $A\subset\R^{n}$, define
$$F(A)\colonequals\sum_{i=1}^{n}\int_{\R^{n}}\int_{\R^{n}} 1_{A}(x)(1-x_{i}^{2})(1-y_{i}^{2})\gamma_{n}(x)\gamma_{n}(y)1_{A}(y)dxdy.$$
Apply Corollary <ref>.
Note that there is a “phase transition” that occurs in Theorem <ref> (and in Corollary <ref>), where the ball or its complement changes from locally maximizing $F$ to not locally maximizing $F$. We do not currently have an intuitive explanation for this phenomenon.
Here we provide some more details for the calculation demonstrated in the Introduction which demonstrated the incorrectness of Conjecture <ref>.
Let $r=2.4$. Then $\gamma_{2}(B(0,r))=1-e^{r^{2}/2}\approx .943865$. And from Lemma <ref>, $(\int_{B(0,r)}(1-x_{1}^{2})d\gamma_{2}(x))^{2}+(\int_{B(0,r)}(1-x_{2}^{2})d\gamma_{2}(x))^{2}=\frac{1}{2}r^{4}e^{-r^{2}}\approx.0522732$.
Let $A=\{(x_{1},x_{2})\in\R^{2}\colon x_{1}^{2}/(2.5)^{2}+x_{2}^{2}/(2.31394)^{2}\leq1\}$. A numerical computation shows that $\gamma_{2}(A)\approx.943865$, $\int_{A}(1-x_{1}^{2})d\gamma_{2}(x)\approx .143076$ and $\int_{A}(1-x_{2}^{2})d\gamma_{2}(x)\approx .178889$, so $(\int_{A}(1-x_{1}^{2})d\gamma_{2}(x))^{2}+(\int_{A}(1-x_{2}^{2})d\gamma_{2}(x))^{2}\approx .0524720>.0522732$.
That is, if $F(A)\colonequals(\int_{A}(1-x_{1}^{2})d\gamma_{2}(x))^{2}+(\int_{A}(1-x_{2}^{2})d\gamma_{2}(x))^{2}$, then $F(A^{c})\approx .0524720>.0522732\approx F(B(0,r)^{c})$. And if $r'=\sqrt{-2\log(1-e^{-2.88})}$, then $\gamma_{2}(B(0,r'))=\gamma_{2}(B(0,r)^{c})$, and again from Lemma <ref>, $F(B(0,r'))=\frac{1}{2}(r')^{4}e^{-(r')^{2}}=2(\log(1-e^{-2.88}))^{2}(1-e^{-2.88})^{2}\approx.0059468$. In summary,
$$F(A)>\max(F(B(0,r)^{c}),F(B(0,r'))),\qquad \gamma_{2}(B(0,r)^{c})=\gamma_{2}(B(0,r'))\approx\gamma_{2}(A).$$
That is, Conjectures <ref> and <ref> are false.
In fact, this behavior is generic for other measure restrictions when $n=2$. If $1-e^{-r^{2}/2}=e^{-(r')^{2}/2}$, then $r'=\sqrt{-2\log(1-e^{-r^{2}/2})}$, $F(B(0,r))=\frac{1}{2}r^{4}e^{-r^{2}}$, and $F(B(0,r')^{c})=2(\log(1-e^{-r^{2}/2}))^{2}(1-e^{-r^{2}/2})^{2}\geq F(B(0,r))$ for all $0<r<\sqrt{2}$. So, $\max(F(B(0,r)),F(B(0,r')))=F(B(0,r'))$. And from Corollary <ref>, $B(0,r')$ locally maximizes $F$ only when $r'<2$. That is, $B(0,r')$ locally maximizes $F$ only when $r>\sqrt{4-2\log(e^{2}-1)}\approx .53928$.
In summary, if $0<r< .53928$, and if $a=\gamma_{2}(B(0,r))$, then Conjectures <ref> and <ref> are false, since $F(B(0,r))<F(B(0,r')^{c})$, and $B(0,r')^{c}$ does not maximize $F$ by Corollary <ref>, since $r'>2$. Moreover, as mentioned in the Introduction, if $A'=\{(x_{1},x_{2})\in\R^{2}\colon x_{1}^{2}\leq1.90999\}$, then a numerical computation shows $\gamma_{2}(A')\approx .943865$, and $F(A')=F((A')^{c})\approx.0604796$. That is,
Let $\rho\in(-1,1)$. Let $G(x,y)=e^{-\frac{\vnorm{\rho x-y}^{2}}{2(1-\rho^{2})}}\gamma_{n}(x)=e^{\frac{-\vnorm{x}_{2}^{2}-\vnorm{y}_{2}^{2}+2\rho\langle x,y\rangle}{2(1-\rho^{2})}}$ for any $x,y\in\R^{n}$. For any $A\subset\R^{n}$, define
$$F_{\rho}(A)\colonequals\int_{\R^{n}}\int_{\R^{n}} 1_{A}(x)G(x,y)1_{A}(y)dxdy
=\int_{\R^{n}}\int_{\R^{n}} 1_{A}(x)T_{\rho}1_{A}(x)d\gamma_{n}(x).
F(A)\colonequals\int_{\R^{n}}\int_{\R^{n}} 1_{A}(x)\left(\sum_{i=1}^{n}(1-x_{i}^{2})(1-y_{i}^{2})\right)\gamma_{n}(x)\gamma_{n}(y)1_{A}(y)dxdy.
We require the following well-known Gaussian/Mehler heat kernel expansion for $G$, which appears e.g. in <cit.>: for any $x,y\in\R^{n}$, and for any $\rho\in(-1,1)$,
\begin{equation}\label{Csix}
\sum_{k=0}^{\infty}\rho^{k}\sum_{\substack{\ell\in\N^{n}\colon\\ \ell_{1}+\cdots+\ell_{n}=k}}h_{\ell}(x)h_{\ell}(y)\ell!
=(1-\rho^{2})^{-n/2}e^{-\frac{(\|x\|_{2}^{2}+\|y\|_{2}^{2}-2\rho\langle x,y\rangle)}{2(1-\rho^{2})}}.
\end{equation}
Combining (<ref>) with the bound for Hermite polynomials in Lemma <ref>, for any $x,y\in\R^{n}$,
\begin{equation}\label{Cseven}
\begin{aligned}
+\rho^{2}\sum_{\substack{\ell\in\N^{n}\colon\\ \ell_{1}+\cdots+\ell_{n}=2}}h_{\ell}(x)h_{\ell}(y)\ell!\Big)\Big|\\
\leq\gamma_{n}(x)\gamma_{n}(y)\sum_{k=3}^{\infty}\rho^{k}(n+k)^{n}3^{k}k^{n}(1+\vnorm{x}_{2}^{k})(1+\vnorm{y}_{2}^{k})\Big).
\end{aligned}
\end{equation}
Below, $C$ denotes a large constant that depends on $n,r,\rho$, whose value can change each time it appears. Also, $A$ denotes $B(0,r)$ or $B(0,r)^{c}$. We now show that the formulas from Lemmas <ref> and <ref> are close in a precise sense.
Write $X(x)=f(x)N(x)$, where $x\in\partial B(0,r)$. Since $\frac{d}{dt}\gamma_{n}(A^{(t)})=0$, Lemma <ref> implies that $\int_{\partial B(0,r)}f(x)dx=0$. Also, recall that if $A\subset\R^{n}$ is symmetric, then (<ref>) implies that $\int_{A}x_{i}d\gamma_{n}(x)=0$ for any $i\in\{1,\ldots,n\}$. Combining these facts with (<ref>) and choosing $\rho$ sufficiently small (depending on $n$ and $r$), we have
\begin{equation}\label{end1}
\begin{aligned}
&\Big|\int_{\partial A}\int_{\partial A}G(x,y)f(x)f(y) dxdy
-\rho^{2}\int_{\partial A}\int_{\partial A}\sum_{\substack{\ell\in\N^{n}\colon\\ \ell_{1}+\cdots+\ell_{n}=2}}h_{\ell}(x)h_{\ell}(y)\ell!f(x)f(y)\Big|\\
&\leq C\abs{\rho}^{5/2}(\int_{\partial A}\abs{f(x)}\gamma_{n}(x)dx)^{2}
\leq C\abs{\rho}^{5/2}(\int_{\partial A}\abs{f(x)}^{2}\gamma_{n}(x)dx).
\end{aligned}
\end{equation}
Similarly, it follows from (<ref>) that, $\forall$ $x\in\R^{n}$,
\Big|T_{\rho}1_{A}(x)-\gamma_{n}(a)-\rho^{2}\sum_{\substack{\ell\in\N^{n}\colon\\ \ell_{1}+\cdots+\ell_{n}=2}}\ell! h_{\ell}(x)(\int_{A}h_{\ell}(y)d\gamma_{n}(y))\Big|
\leq C\abs{\rho}^{5/2}(1+\vnorm{x}_{2}^{k}).
\Big|\frac{\partial}{\partial x_{i}}T_{\rho}1_{A}(x)-\rho^{2}\sum_{\substack{\ell\in\N^{n}\colon\\ \ell_{1}+\cdots+\ell_{n}=2}}\ell! h_{\ell}'(x)(\int_{A}h_{\ell}(y)d\gamma_{n}(y))\Big|
\leq C\abs{\rho}^{5/2}(1+\vnorm{x}_{2}^{k}).
From Remark <ref>, we may assume that $\int_{A^{(t)}}x_{i}x_{j}d\gamma_{n}(x)=0$ whenever $i,j\in\{1,\ldots,n\}$ with $i\neq j$, and for all $t\in(-1,1)$. Also, from (<ref>) below (where the $G$ we use there is $G(x,y)\colonequals y_{i}y_{j}$ for all $x,y\in\R^{n}$), we may assume that $\int_{\partial A}x_{i}x_{j}f(x)d\gamma_{n}(x)=0$ whenever $i,j\in\{1,\ldots,n\}$ with $i\neq j$. Consequently,
\begin{equation}\label{end3}
\begin{aligned}
&\Big|\int_{\partial A}\langle\nabla T_{\rho}1_{A}(x),X(x)\rangle\langle X(x),N(x)\rangle dx\\
&\quad-\sum_{i=1}^{n}\int_{\partial A}(-x_{i})X^{(i)}\langle X(x),N(x)\rangle d\gamma_{n}(x)(\int_{A} (1-y_{i}^{2})d\gamma_{n}(y))\Big|\\
&\qquad\qquad\qquad\leq C\abs{\rho}^{5/2}(\int_{\partial A}\abs{f(x)}^{2}dx).
\end{aligned}
\end{equation}
So, combining (<ref>) and (<ref>) with Lemmas <ref> and <ref>, (and $\int_{\partial A}x_{i}x_{j}f(x)\gamma_{n}(x)=0$ if $i,j\in\{1,\ldots,n\}$ with $i\neq j$,)
\leq C\abs{\rho}^{1/2}\int_{\partial B(0,r)}(f(x))^{2}dx.$$
That is, for $\rho$ sufficiently small (depending on $n$ and $r$), Theorem <ref> follows from Theorem <ref>. That is, the ball or its complement is a local maximum of noise stability among symmetric sets.
§ ASYMPTOTICS FOR SECOND DEGREE FOURIER COEFFICIENTS
Let $m\geq3$ be an integer. Then there exists $\lambda_{m}$ such that
and such that for any integer $n\geq6$,
For $n\in\N$ and $s\in\R$ define $r(s,n)\colonequals\sqrt{n+s\sqrt{2n}}$. Then, as $n\to\infty$, the following asymptotic holds:
Moreover, in the case $s=0$, the quantity $\sum_{i=1}^{n}(\int_{B(0,\sqrt{n})}(1-x_{i}^{2})d\gamma_{n}(x))^{2}$ strictly increases as $n$ increases.
We begin with the first statement. Using Lemma <ref>,
Taking the logarithm of the fraction, we get
Combining these estimates proves the asymptotic formula.
We now consider the case $s=0$. For any $t>2$, define
Let $x=1/(t-2)$ so that $t-2=1/x$, $t=2+1/x$, and $1/t=x/(2x+1)$.
Let $h(x)=(1/3)x^{2}+\log(1+2x)-x-x/(2x+1)$. Then $h'(x)=(2/3)x+2/(2x+1)-1-[(2x+1)-2x]/(2x+1)^{2}=(2/3)x+2/(2x+1)-1-1/(2x+1)^2$. Then $(2x+1)^{2}h'(x)=(2/3)x(2x+1)^{2}+2(2x+1)-(2x+1)^{2}-1=(8/3)x^{3}+(8/3)x^{2}+(2/3)x+4x+2-4x^{2}-4x-1-1=(8/3)x^{3}-(4/3)x^{2}+(2/3)x=(1/3)x(8x^{2}-4x+2)$. And the function $x\mapsto 8x^{2}-4x+2$ has a positive minimum at $x=1/4$. So, $h'(x)>0$ for all $x\in(0,1)$. That is, $g'(t)\geq0$ for any $t\geq3$. Therefore, the quantity $\sum_{i=1}^{n}(\int_{B(0,\sqrt{n})}(1-x_{i}^{2})d\gamma_{n}(x))^{2}$ strictly increases as $n$ increases, if $n\geq6$. (The case $1\leq n< 6$ follows by direct computation.)
§ APPENDIX: PROOF OF THE SECOND VARIATION FORMULA
Let $A\subset\R^{n}$ be a set with smooth boundary, and let $N\colon\partial A\to S^{n-1}$ denote the unit exterior normal to $\partial A$. Let $X\colon\R^{n}\to\R^{n}$ be a vector field. Let $\Psi\colon\R^{n}\times(-1,1)$ such that $\Psi(x,0)=x$ and such that $\frac{d}{dt}|_{t=0}\Psi(x,t)=X(\Psi(x,t))$ for all $x\in\R^{n},t\in(-1,1)$. For any $t\in(-1,1)$, let $A^{(t)}=\Psi(A,t)$. Define
\begin{equation}\label{Btwo5.1}
\end{equation}
Let $G\colon\R^{n}\times\R^{n}\to\R$ be a Schwartz function.
\begin{equation}\label{Bone6}
\frac{d}{dt}|_{t=0}\int_{\R^{n}} 1_{A^{(t)}}(y)G(x,y)dy
=\int_{\partial A}G(x,y)\langle X(y),N(y)\rangle dy.
\end{equation}
In particular, setting $G(x,y)=\gamma_{n}(y)$, we get
$$\frac{d}{dt}|_{t=0}\gamma_{n}(A^{(t)})=\int_{\partial A}\langle X(y),N(y)\rangle \gamma_{n}(y)dy.$$
Let $G\colon\R^{n}\times\R^{n}\to\R$ be a Schwartz function. Then
1/2d^2/dt^2|_t=0∫_^n∫_^n 1_A^(t)(x)G(x,y)1_A^(t)(y)dy
Write $\Psi$ and $X$ in their components as $\Psi=(\Psi^{(1)},\ldots,\Psi^{(n)})$, $X=(X^{(1)},\ldots,X^{(n)})$. We use subscript notation to denote partial derivatives, and we let $\mathrm{div}(X)=\sum_{i=1}^{n}X_{i}^{(i)}$ denote the divergence of $X$. Let $J\Psi(y,t)$ denote $\abs{\det D\Psi(y,t)}=\abs{\det(\partial\Psi^{(i)}(y,t)/\partial y_{j})_{1\leq i,j\leq n}}\in\R$.
By assumption,
\begin{equation}\label{Btwo1}
\frac{d\Psi}{dt}|_{t=0}=X(\Psi(x,0))=X(x).
\end{equation}
Since $\Psi$ is smooth, we can write
$$Z\colonequals\frac{d^{2}\Psi}{dt^{2}}|_{t=0},\quad Z^{(i)}=\sum_{j=1}^{n}X_{j}^{(i)}X^{(j)}.$$
We then have the determinant expansion
Since $J\Psi(x,t)=\abs{\det(D\Psi(x,0))}$, we therefore have
\begin{equation}\label{Btwo2}
\end{equation}
\begin{equation}\label{Btwo3}
\end{equation}
\begin{equation}\label{Btwo4}
\frac{d^{2}\Psi^{(i)}}{dt^{2}}|_{t=0}=\sum_{j=1}^{n}X_{j}^{(i)}X^{(j)}.
\end{equation}
\begin{equation}\label{Btwo5}
\frac{d^{2}}{dt^{2}}J\Psi(x,t)|_{t=0}=\mathrm{div}((\mathrm{div}(X))X).
\end{equation}
\begin{equation}\label{Btwo5.2}
F(A^{(t)})=\int_{\R^{n}} 1_{A^{(t)}}(x)G(x,y)1_{A^{(t)}}(y)dxdy=\int_{A^{(t)}}V(x,t)=\int_{A}V(\Psi(x,t),t)J\Psi(x,t)dx.
\end{equation}
In the sequel, we will use the chain rule and divergence theorem repeatedly.
\begin{equation}\label{Btwo6}
\begin{aligned}
\frac{d}{dt}F(A^{(t)})
\end{aligned}
\end{equation}
Step 1. Computing the Second Derivative of $F(A^{(t)})$ with respect to $t$.
\begin{equation}\label{Btwo7}
\begin{aligned}
+V_{tt}(\Psi(x,t),t)J\Psi(x,t) dx.
\end{aligned}
\end{equation}
\begin{equation}\label{Btwo8}
\begin{aligned}
\stackrel{\substack{\eqref{Btwo1}\wedge\eqref{Btwo2}\wedge\\ \eqref{Btwo3}\wedge\eqref{Btwo4}\wedge\eqref{Btwo5}}}{=}
\int_{A}\sum_{i,j=1}^{n}V_{x_{i}x_{j}}(x,0)X^{(i)}(x)X^{(j)}(x)\\
+V_{tt}(x,t) dx.
\end{aligned}
\end{equation}
From (<ref>), $V_{t}(x,0)=\int_{\partial A}G(x,y)\langle X(y),N(y)\rangle dy$. So, combining the second and fifth terms of (<ref>), then applying the divergence theorem,
\begin{equation}\label{Btwo9}
\begin{aligned}
&\int_{A}2\langle \nabla_{x}V_{t}(x,0),X(x)\rangle+2V_{t}(x,0)\mathrm{div}(X(x))dx\\
=2\int_{\partial A}V_{t}(x,0)\langle X(x),N(x)\rangle dx\\
&\qquad=2\int_{\partial A}\int_{\partial A}G(x,t)\langle X(x),N(x)\rangle\langle X(y),N(y)\rangle dxdy.
\end{aligned}
\end{equation}
Combining the first, third and fourth terms of (<ref>), and using the divergence theorem,
\begin{equation}\label{Btwo10}
\begin{aligned}
+\sum_{i=1}^{n}V_{x_{i}}(x,t)X^{(i)}(x)\mathrm{div}(X(x)) dx\\
&\qquad=\int_{A}\mathrm{div}(\langle\nabla_{x}V(x,0),X(x)\rangle X(x))dx
=\int_{\partial A}\langle \nabla_{x}V(x,0),X(x)\rangle\langle X(x),N(x)\rangle dx
\end{aligned}
\end{equation}
Combining the sixth term and one of the fourth terms of (<ref>), then applying the divergence theorem,
\begin{equation}\label{Btwo11}
\begin{aligned}
&\int_{A} V(x,0)\mathrm{div}(\mathrm{div}(X(x))X(x))dx
+\langle\nabla_{x} V(x,0),X(x)\rangle\mathrm{div}(X(x))dx\\
&\qquad=\int_{\partial A}V(x,0)(\mathrm{div}(X(x)))\langle X(x),N(x)\rangle dx
\end{aligned}
\end{equation}
Step 2. Combining the Terms.
Now, substituting (<ref>), (<ref>) and (<ref>) into (<ref>),
\begin{equation}\label{Btwo12}
\begin{aligned}
=2\int_{\partial A}\int_{\partial A}G(x,y)\langle X(x),N(x)\rangle\langle X(y),N(y)\rangle dxdy\\
&\quad+\int_{\partial A}\langle \nabla_{x}V(x,0),X(x)\rangle\langle X(x),N(x)\rangle dx
+\int_{\partial A}V(x,0)(\mathrm{div}(X(x)))\langle X(x),N(x)\rangle dx\\
&\quad+\int_{A}V_{tt}(x,t) dx.\\
&\,=2\int_{\partial A}\int_{\partial A}G(x,y)\langle X(x),N(x)\rangle\langle X(y),N(y)\rangle dxdy\\
&\quad+\int_{\partial A}\mathrm{div}(V(x,0)X(x))\langle X(x),N(x)\rangle dx+\int_{A}V_{tt}(x,0) dx.
\end{aligned}
\end{equation}
Step 3. Computing the final term, $V_{tt}$.
It therefore remains to compute $\int_{A}V_{tt}(x,t)dx$. From (<ref>),
V_t(x,t) =d/dt∫_AG(x,Ψ(y,t))JΨ(y,t)dy
=∫_A⟨∇_zG(x,Ψ(y,t))(d/dt)Ψ(y,t)⟩JΨ(y,t)+G(x,Ψ(y,t))(d/dt)JΨ(y,t) dy
So, applying the Chain rule, and then the divergence theorem,
\begin{equation}\label{Btwo13}
\begin{aligned}
&\,=\frac{d}{dt}|_{t=0}\int_{A}\int_{A}\langle\nabla_{z}G(x,\Psi(y,t))(d/dt)\Psi(y,t)\rangle J\Psi(y,t)+G(x,\Psi(y,t))(d/dt)J\Psi(y,t) dydx\\
&\,=\frac{d}{dt}|_{t=0}\int_{A}\int_{A}\langle\nabla_{y}G(x,\Psi(y,t))[D\Psi(y,t)]^{-1}(d/dt)\Psi(y,t)\rangle J\Psi(y,t)\\
&\qquad\qquad+G(x,\Psi(y,t))(d/dt)J\Psi(y,t) dydx\\
&\,=\frac{d}{dt}|_{t=0}-\int_{A}\int_{A}G(x,\Psi(y,t))\mathrm{div}([D\Psi(y,t)]^{-1}(d/dt)\Psi(y,t)\rangle J\Psi(y,t)) dydx\\
&\quad+\int_{A}\int_{\partial A}\langle G(x,\Psi(y,t))[D\Psi(y,t)]^{-1}(d/dt)\Psi(y,t)\rangle J\Psi(y,t), N(y)\rangle dydx\\
&\quad+\int_{A}\int_{A}G(x,\Psi(y,t))(d/dt)J\Psi(y,t) dydx
\end{aligned}
\end{equation}
We now differentiate the three terms in (<ref>).
\begin{equation}\label{Btwo14.0}
D\Psi=I+tD X+O(t^{2}),\quad [D\Psi]^{-1}=I-tD X+O(t^{2}).
\end{equation}
\begin{equation}\label{Btwo14}
\end{equation}
\begin{equation}\label{Btwo15}
\begin{aligned}
&(d/dt)|_{t=0}G(x,\Psi(y,t))\mathrm{div}([D\Psi(y,t)]^{-1}(d/dt)\Psi(y,t)\rangle J\Psi(y,t))\\
&\stackrel{\substack{\eqref{Btwo1}\wedge\eqref{Btwo4}\wedge\\ \eqref{Btwo3}\wedge\eqref{Btwo14}\wedge\eqref{Btwo14.0}}}{=}
\langle \nabla_{y}G(x,y),X(y)\rangle\mathrm{div}(X(y))\\
&\qquad\qquad\qquad+G(x,y)\mathrm{div}(-(DX)X+(\sum_{j=1}^{n}X_{x_{j}}^{(i)}X^{(j)})_{i}+X\mathrm{div}(X)) \\
&\qquad=\langle \nabla_{y}G(x,y),X(y)\rangle\mathrm{div}(X(y))+G(x,y)\mathrm{div}(X\mathrm{div}(X))\\
\end{aligned}
\end{equation}
As in (<ref>),
\begin{equation}\label{Btwo16}
\begin{aligned}
&(d/dt)|_{t=0}\langle G(x,\Psi(y,t))[D\Psi(y,t)]^{-1}(d/dt)\Psi(y,t)\rangle J\Psi(y,t), N(y)\rangle\\
&\qquad= \langle \nabla_{y}G(x,y),X(y)\rangle X(y)+G(x,y)X(y)\mathrm{div}(X(y))\\
&\qquad= X(y)\mathrm{div}_{y}(G(x,y)X(y)).
\end{aligned}
\end{equation}
\begin{equation}\label{Btwo17}
\begin{aligned}
&(d/dt)|_{t=0}G(x,\Psi(y,t))(d/dt)J\Psi(y,t) \\
&\qquad\stackrel{\eqref{Btwo3}\wedge\eqref{Btwo5}}{=}\langle \nabla_{y}G(x,y),X(y)\rangle\mathrm{div}(X(y))+G(x,y)\mathrm{div}(X(y)\mathrm{div}(X(y)))\\
\end{aligned}
\end{equation}
Substituting (<ref>), (<ref>) and (<ref>) into (<ref>) and noting that (<ref>) and (<ref>) cancel,
\begin{equation}\label{Btwo18}
\begin{aligned}
\int_{A}V_{tt}(x,0)dx
&= \int_{A}\int_{\partial A}\mathrm{div}_{y}(G(x,y)X(y))\langle X(y),N(y)\rangle dydx\\
&=\int_{\partial A}\mathrm{div}_{y}\left[\left(\int_{A}G(x,y)dx\right) X(y)\right]\langle X(y), N(y)\rangle dy\\
&\stackrel{\eqref{Btwo5.1}}{=}\int_{\partial A}\mathrm{div}(V(x,0) X(x))\langle X(x),N(x)\rangle dx
\end{aligned}
\end{equation}
Step 4. Combining all terms together
Substituting (<ref>) into (<ref>), we finally get
$$\frac{d^{2}}{dt^{2}}|_{t=0}\gamma_{n}(A^{(t)})=\int_{\partial A}(\mathrm{div}(X(x))-\langle X(x),x\rangle)\langle X(x),N(x)\rangle \gamma_{n}(x)dx.$$
Let $G(x,y)\colonequals\gamma_{n}(x)\gamma_{n}(y)$ for any $x,y\in\R^{n}$. Then for any $A\subset\R^{n}$, $\int_{A}\int_{A}G(x,y)dxdy=(\gamma_{n}(A))^{2}$. By Lemma <ref>,
$$\frac{d}{dt}|_{t=0}\gamma_{n}(A^{(t)})=\int_{\partial A}\langle X(y),N(y)\rangle \gamma_{n}(y)dy.$$
And (<ref>) says $V(x,t)\colonequals\int_{A^{(t)}}G(x,y)dy=\gamma_{n}(x)\gamma_{n}(A^{(t)})$ for any $x\in\R^{n},t\in\R$. So,
$$\mathrm{div}(V(x,0)X(x))=(\mathrm{div}(X(x))-\langle X(x),x\rangle)\gamma_{n}(x)\gamma_{n}(A),\qquad\forall\,x\in\R^{n}.$$
Then, by the Chain Rule and Lemma <ref>,
The Lemma follows.
|
1511.00223
|
Let $G$ be a polycyclic, metabelian or soluble of type (FP)$_{\infty}$ group such that the class $Rat(G)$ of all rational subsets of $G$ is a boolean algebra. Then $G$ is virtually abelian. Every soluble biautomatic group is virtually abelian.
§ INTRODUCTION
The topic of this paper is two important concepts: rational sets and biautomatic structure. We study finitely generated soluble groups $G$ such that the class $Rat(G)$ of all rational subsets of $G$ is a boolean algebra. We conjecture that every such group is virtually abelian. Note that every finitely generated virtually abelian group satisfies to this property. We confirm this conjecture in the case where $G$ is a polycyclic, metabelian or soluble group of type FP$_{\infty}$. This conjecture remains open in general case. It appeared that the notion FP$_{\infty}$ helps to prove by the way that every soluble biautomatic group is virtually abelian. Thus, we give answer to known question posed in <cit.>.
We provide full proofs of four theorems attributed to Bazhenova, a former student of the author, stating that some natural assumptions imply that a soluble group is virtually abelian. The original proofs was given by her in collaboration with the author more than 14 years ago and had never been published. Now we fill this gap by presenting improved versions of these proofs.
An excellent introduction to rational sets is <cit.>, where the reader can find out basic definitions and fundamental results in this area.
Much of the basic theory of automatic and biautomatic groups is presented by Epstein and al. in <cit.>. One of the major open questions in group theory is whether or not an automatic group is necessarily biautomatic. The answer is not known even in the class of soluble groups. Note that some known results about biautomatic groups remain open questions for automatic groups.Gersten and Short initiated in <cit.> the study of the subgroup structure of biautomatic groups. Among other results they established that a polycyclic subgroup of a biautomatic group is virtually abelian. Also they proved that if a linear group is biautomatic, then every soluble subgroup is (finitely generated) virtually abelian.
§ PRELIMINARIES
Given an alphabet $\Sigma $, recall that a regular language $R$ is a certain subset of the free monoid $\Sigma^{\ast}$ generated by $\Sigma ,$ which is empty or can be obtained by taking singleton subsets of $\Sigma $, and perform, in a finite number of steps, any of the three basic (rational) operations: taking union, string concatenation (product), and the Kleene star (generating of submonoid).
The construction of a set like $R$ is still possible when $\Sigma^{\ast}$ is changed by any monoid $M.$ Let $S$ be the set of all singleton subsets of $M.$ Consider the closure $Rat(M)$ of $S$ under the rational operations of union, product, and the formation of a submonoid of $M.$ In other words, $Rat(M)$ is the smallest subset of $M$ such that
* $\emptyset \in Rat(M)$,
* $A, B \in Rat(M) $ imply $A\cup B \in Rat(M),$
* $A,B \in Rat(M)$ imply $AB \in Rat(M),$ where $AB =\{ab| a\in A, b \in B\}$,
* $A \in Rat(M)$ implies $A^{\ast} \in Rat(M),$ where $A^{\ast}$ is the submonoid of $M$ generated by $A.$
A rational set of $M$ is an element of $Rat(M).$
If $\Sigma^{\ast}$ is a finite generated free monoid, then a rational set of $\Sigma^{\ast}$
is also called regular language.
The rational sets of a monoid $M$ are precisely the subsets accepted by finite automata over $M.$ A finite automaton $\Gamma $ over $M$ is a finite directed graph with a distinguished initial vertex, some distinguished terminal vertices, and with edges labelled by elements from $M.$ The set accepted by $\Gamma $ is the collection of labels of paths from the initial vertex to a terminal vertex, where label $\mu (p)$ of a path $p$ is the product of labels of sequential edges in $p$
Recall two auxillary assertions.
(The Pumping Lemma (see <cit.>).
Let $M$ be a monoid, $R\in Rat(M).$ Then either $R$ is finite, or it contains a set of the form $aq^{\ast}b = \{aq^nb| n \geq 0\}, a,q,b \in M, q \not= 1.$ Moreover, if $M$ is a group, then the subset $aq^{\ast}b$ of $M$ can be written in the form $abb^{-1}q^{\ast}b = ab(b^{-1}qb)^{\ast}.$ So we might assume $b=1.$
Let $G$ be a group, $H\leq G$ be a subgroup. If $R \subseteq H$ is rational subset of $G$, then $R$ is a rational subset of $H.$
As we have <ref>, we don't have to care if a set is rational as a subset of a larger group or of a smaller one. Note that with monoids which are not groups the situation may be different.
(see <cit.>).
Let $G$ be a group and $H \leq G.$ Then $H \in Rat(G)$ if and only if $H$ is finitely generated. If $R \in Rat(G)$ then gp$(R)\in Rat(G)$, and so finitely generated.
(see <cit.>).
Let $G$ be a group, $T \lhd G,$ and $\varphi : G \rightarrow G/T$ is the standard homomorphism. Then for every $R \in Rat(G)$ we have $\varphi (R) \in Rat(G/T).$ If $T$ is finitely generated then for every $S \in Rat(G/T)$ we have $\varphi^{-1}(S) \in Rat(G).$
§ POLYCYCLIC GROUPS
The goal of this section is to prove this:
Let $G$ be a polycyclic group. If $Rat(G)$ is a boolean algebra, then $G$ is virtually abelian.
Let us first consider a special case, that is the 'core' of the problem, in the sense concentrates all the difficult points of it. So, let a group $G$ is a semidirect product $A
\lambda H$, where $A \simeq \mathbb{Z}^r$ is a normal free abelian of rank $r$ subgroup of $G$, $H=$ gp$(h)$ is a cyclic subgroup of $G.$ Suppose that $Rat(G)$ is a boolean algebra. We are to prove that $G$ is virtually abelian.
We may assume $h$ to have infinite order. First prove that some nontrivial element $g \in A$ and some exponent $h^m, m> 0,$ commute. Take an arbitrary nontrivial element $x \in A.$ Consider the set
$$R = \{h^{-n}xh^n | n \in \mathbb{Z}\}.$$
If $R$ is finite, then we have $h^{-n}xh^n = h^{-t}xh^t$ for some $n > t$, which implies $[x, h^{n-t}] = 1$, hence we get what we need. Now assume that $R$ is infinite. Note that $R$ is rational, because it is equal to $(h^{-1})^{\ast}xh^{\ast} \cap A.$ Then by lemma <ref>, $R$ contains a subset $P = aq^{\ast}, a, q \in A, q \not= 1.$ Let $I$ be the set of indices such that $P = \{h^{-i}xh^i| i\in I\}$. Then $S= \{h^i| i \in I\} = h^{\ast} \cap (x^{-1}h^{\ast}P)$ is infinite rational set. So it contains a subset $T = h^k(h^l)^{\ast}, k, l \in \mathbb{Z}, l > 0.$ The set $Q= \{f^{-1}xf| f \in T\} = (T^{-1}xT) \cap A$ is rational and subset of $P.$ We can assume that $k = 0.$ In other case we change
$R, P, a$ and $q$ to $h^{-k}Rh^{k}, h^{-k}Ph^{k}, h^{-k}ah^{k}$ and $h^{-k}qh^{k},$ respectively. Now $T = (h^l)^{\ast}.$
Since $A$ is isomorphic to the free abelian group of rank $r$, we may regard it as a lattice of $\mathbb{R}^r.$ Let $| \cdot |$ be any standard norm on $\mathbb{R}^r.$ Since now we will use additive notation for the operation on $A$ as well as multiplicative one.
Take an arbitrary real $\varepsilon > 0.$ Pick a positive integer $m$ such that $aq^m$ (or, in additive notation, $a + mq$ ) belongs to $Q$ and the inequality
$$1/m|(h^{-l}ah^l) - a| < \varepsilon $$
holds. This is possible, because $Q$ is infinite. Since $aq^m \in Q$, the element $h^{-l}(aq^m)h^l$ is in $Q$, so it can be written in the form $aq^p, p \in \mathbb{Z}, p \geq 0.$ Then
$$h^{-l}ah^l + m(h^{-l}qh^l) = a + pq,$$
$$m(h^{-l}qh^l)- pq = a- h^{-l}ah^l,$$
|h^{-l}qh^l- (p/m)q|= (1/m)|a- h^{-l}ah^l| < \varepsilon .$$
It follows that $h^{-l}qh^l$ is a limit of elements of the form $sq, s \in \mathbb{R}, s \geq 0,$ so it has this form too. Clearly, $s$ is nonzero and rational, because $sq \in \mathbb{Z}^r \setminus \{0\}.$ Let $u$ be the greatest positive integer with the property: $q$ has the form $sq', q' \in A.$ Then $v = su$ is the greatest positive integer with the property: $sq$ has the form $vq', q' \in A.$ Since there exists
an automorphism of $A$ which takes $q$ to $sq,$ we get $u = sm$, so $s = 1.$ Then we have $h^{-l}qh^{l} = q,$ so we get what we need.
Let us finish the proof. Let $f \in A, f \not= 1.$ Let $g, h^u$ ($g \not= 1, g \in A, u > 0$) commute. If $r,s$ are nonzero integers such that $g^r = f^s,$ then $(h^{-u}fh^u)^{rs}=
(h^{-u}f^sh^u)^{r} = g^{r^2}= f ^{rs}.$ As $A$ is abelian and torsion-free, $h^{-u}fh^u = f,$ so $f$ and $h^u$ commute.
Now suppose that $g^r=f^s$ cannot hold unless $r = s =0,$ or, equivalently, $g^rf^s = g^nf^l$ cannot hold unless $r=n, s = l.$ Let $w = h^u.$ Consider the set
$$R = (wg)^{\ast}f^{\ast} \cap w^{\ast}(gf)^{\ast} = \{w^ng^nf^n | n \geq 0\}.$$
$$S = R(g^{\ast}\cup (g^{-1})^{\ast}) \cap w^{\ast}f^{\ast} =$$
$$\{w^nf^n| n \geq 0\}.$$
Take a subset $aq^{\ast} \subseteq S, q \not= 1.$ Let $a = w^nf^n, aq = w^mf^m, n \not= m.$ Then
$$aq^2 =(aq)a^{-1}(aq) = w^{2m-n}(w^{n-m}f^{m-n}w^{m-n})f^m$$
also has the form $w^tf^t, t \geq 0.$ Then $t = 2m-n.$ Hence
$$f^{2m-n} = f^m(w^{n-m}f^{m-n}w^{m-n}),$$
$$f^{m-n}= w^{n-m}f^{m-n}w^{m-n},$$
$$f = w^{n-m}fw^{m-n}.$$
Then $f$ and $h^{m-n}$ commute.
We proved that for each $f \in A$ there exists $t \in \mathbb{Z}, t > 0,$ such that $f$ and $h^t$ commute. Let $f_1, ..., f_r$ be free generators for $A.$ Pick some nonzero $t_1, ..., t_r$ such that for each $i$ the elements $f_i$ and $h^{t_i}$ commute. Let $N = t_1 ... t_r.$ Then $h^N$ and every $f_i$ commute. Hence the group $M$ generated by $A$ and $h^N$ is abelian. Clearly, $M$ has finite index in $G.$ Thus $G$ is virtually abelian.
Now we are ready to prove the theorem by using induction on solubility length. If $G$ is abelian the statement is trivial. Let $G$ be nonabelian.The derived subgroup $G'$ has smaller solubility length, and, as it is finitely generated, its rational subsets are a boolean algebra. Then $G'$ is virtually abelian. Let $G = $ gp$(G', g_1, ..., g_j)$. Consider the series of subgroups $G' = G_0 \leq G_1 \leq G_j = G$, where $G_{i+1} = $ gp$(G_i, g_{i+1})$ for $i = 0, ..., j-1.$ Clearly, all $G_i$ are normal in $G.$ Prove by induction on $i$ that all $G_i$ are virtually abelian. Suppose that $H_i$ is a finite index normal abelian subgroup of $G_i.$ We have proved above that gp$(H_i, g_{i+1}$ is virtually abelian. Hence $H_{i+1}$ is virtually abelian.
§ METABELIAN GROUPS
Recall that a group $G$ is said to have the Howson property (or to be a Howson group) if the intersection $H \cap K$ of any two finitely generated subgroups $H, K$ of $G$ is finitely generated subgroup. Let $G$ be a group in which $Rat(G)$ is a boolean algebra. Then $G$ has the Howson property. Indeed, a subgroup of arbitrary group is a rational set if and only if it is finitely generated. By our assumption the intersection $H\cap K$ is rational set. Hence, $H \cap K$ is finitely generated subgroup.
All finitely generated metabelian nonpolycyclic Howson groups are characterized as follows.
( Kirkinskij <cit.>).
Let $G$ be a finitely generated metabelian nonpolycyclic group. Then the following properies are equivalent:
* $G$ has the Howson property,
* the finitely generated nonpolycyclic subgroups of $G$ have finite indexes,
* $G$ has a subgroup $H$ of finite index containing a normal finite subgroup $T$ such that $H/T \simeq $ gp$(x, a| [a,a^{x^i}] =1, i \in \mathbb{Z}, a^{f(x)} =1)$ with $f(x)$ being irreducible over $\mathbb{Z}$ polynomial with integral coefficients such that deg$f(x)\geq 1 $ and for every $n \in \mathbb{N}$ this polynomial does not divide any polynomial in $x^n$ of degree deg$f(x)-1.$ If $f(x) = q_0x^m + q_1x^{m-1} + ... + q_m,$ then $a^{f(x)}$ means $(a^{x^m})^{q_0}(a^{x^{m-1}})^{q_1} ... a^{q_m}.$
Let $G$ be a finitely generated metabelian group such that $Rat(G)$ is a boolean algebra. Then $G$ is virtually abelian group.
If $G$ is polycyclic the statement follows by theorem <ref>.
Suppose $G$ is not polycyclic. Then by theorem <ref> $G$ has a series $1 \leq T \lhd H \leq G$. Since $H$ is finitely generated, then $Rat(H)$ is a boolean algebra. Since $T$ is finite then by lemma <ref> $Rat(H/T)$ is a boolean algebra.
Let by theorem <ref> $H/T \simeq $ gp$(x, a| [a,a^{x^i}] =1, i \in \mathbb{Z}, a^{f(x)} =1),$ $f(x) = q_0x^m + q_1x^{m-1} + ... + q_m.$
Note that every element $g \in H/T$ can be expressed as $g = x^ka^{\frac{r(x)}{x^l}},$ where $k, l \in \mathbb{Z}, l \geq 0,$ and $r(x)$ is a polynomial with integer coefficients. One has $g = 1$ if and only if $k = 0$ and $r(x)$ divides to $f(x)$ in the polynomial ring $\mathbb{Z}[x].$
Fix some numbers $p, d \in \mathbb{Z}, p, d > 0.$ Define the following rational sets in $H/T$:
R_1 = ((a^dx^p)^{-1})^{\ast}(\{a^d, a^{d+1}\}x^p)^{\ast},$$
$$R_2 = (x^{-p})^{\ast}(\{1, a\}x^p)^{\ast},$$
$$R_3 = ((a^dx^{-p})^{-1})^{\ast}(\{a^d, a^{d+1}\}x^{-p})^{\ast},$$
\label{eq:48}
R_4 = (x^p)^{\ast}(\{1, a\}x^{-p})^{\ast}.
By our assumption all intersections $R_i\cap R_j$ for $i,j = 1, ..., 4,$ are rational.
Any element of $R_1$ can be written in the form:
where $l, k \in \mathbb{Z}, l, k \geq 0,$ and $\epsilon_i = 0$ or $\epsilon_i = 1.$
Any element of $R_2$ can be written in the form:
$l, k \in \mathbb{Z}, l, k \geq 0,$ $\epsilon_i = 0$ or $\epsilon_i = 1.$
Any element of $R_3$ can be written in the form:
(a^dx^{-p})^{-l+k}(a^{\epsilon_k})^{x^{-p}}(a^{\epsilon_{k-1}})^{x^{-2p}} ... (a^{\epsilon_1})^{x^{-kp}},
$l, k \in \mathbb{Z}, l, k \geq 0,$ where $\epsilon_i = 0$ or $\epsilon_i = 1.$
Any element of $R_4$ can be written in the form:
x^{(l-k)p}(a^{\epsilon_k})^{x^{-p}}(a^{\epsilon_{k-1}})^{x^{-2p}} ... (a^{\epsilon_1})^{x^{-kp}},
$l, k \in \mathbb{Z}, l, k \geq 0,$ $\epsilon_i = 0$ or $\epsilon_i = 1.$ Also note that for $n> 0$ we have:
(a^dx^p)^n = x^{np}(a^{d})^{x^{np}} ... (a^d)^{x^p},
(a^dx^p)^{-n} = x^{-np}(a^{-d})^{x^{-(n-1)p}} ... (a^{-d})^{x^{-p}}a^{-d},
(a^dx^{-p})^n = x^{-np}(a^{d})^{x^{-np}} ... (a^d)^{x^{-p}},
(a^dx^{-p})^{-n} = x^{np}(a^{-d})^{x^{(n-1)p}} ... (a^{-d})^{x^p}a^{-d}.
The sets $R_1$ and $R_2$ contain the elements
$$a^{x^{kp}} = ((a^dx^p)^{-1})^k\cdot (a^{d+1}x^p)(a^dx^p)^{k-1}=$$
$$(x^{-p})^k\cdot (ax^p)(x^p)^{k-1}, k = 1, 2, ... .$$
Similarly, the sets $R_3$ and $R_4$ contain the elements $a^{x^{-kp}}, k = 1, 2, ... .$
Let $N = $ ncl$(a)$ be the normal closure of the element $a$ in $H/T$ (that is the minimal normal subgroup of $N/T,$ containing $a$). If the sets $S_1 = R_1 \cap R_2$ and $S_2 = R_3 \cap R_4$ lie in $N,$ then the subgroup $M$ generated by $S_1 \cup S_2 \cup \{a\},$ is the normal closure of $a$ in the subgroup generated by $a$ and $x^p.$ By lemma <ref> every subgroup generated by a rational set is rational and finitely generated. Since $N$ is generated by a finite set of subgroups that are conjugate to $M,$ it is finitely generated too. In the case $H/T$ and $G$ are polycyclic. We get contradiction to our assumption. Hence at least one of the subsets $S_i, i = 1, 2,$ does not lie in $N.$ Then one of the following equalities is true:
(a^{\epsilon_n})^{x^{np}} ...
\begin{equation}
\label{eq:57}
(a^{d+\epsilon_{k-1}})^{x^{(k-1)p}} ... (a^{d+\epsilon_1})^{x^{p}} = 1,
\end{equation}
\begin{equation}
\label{eq:58}(a^{\epsilon_n})^{x^{np}}...(a^{\epsilon_{1}})^{x^{p}}(a^{-d})(a^{-d})^{x^{-p}}...(a^{-d})^{x^{-1p}} = 1,
\end{equation}
\begin{equation}
\label{eq:59}
(a^{d+\epsilon_{1}})^{x^{-p}} ... (a^{d+\epsilon_k})^{x^{-kp}}
(a^{\epsilon_{k+1}})^{x^{-(k+1)p}} ... (a^{\epsilon_n})^{x^{-np}} = 1,
\end{equation}
\begin{equation}
\label{eq:60}
(a^{-d})^{x^{lp}} ...
(a^{-d})^{x^{p}}(a^{-d})(a^{\epsilon_1})^{x^{-p}}(a^{\epsilon_k})^{x^{-kp}} = 1,
\end{equation}
$l, k, n \in \mathbb{Z}, l \geq 0, n \geq k > 0, \epsilon_i \in \{-1, 0, 1\}.$ If the absolute value $\mu $ of one of the coefficients $q_0, q_m$ in $f(x)$ is greater than $3$, we may assume that chosen number $d$ is such that $d-1, d$ and $d+1$ do not divide to $\mu .$ It follows that all equalities <ref> – <ref> failed. Further, both of the coefficients $q_0, q_m$ cannot be $\pm 1,$ because in the case $H/T$ is polycyclic. Thus $q_0$ or $q_m$ is equal to $2$ or $3.$ We set $d = 2^2\cdot 3^2 + 2 = 38.$ Then
$d-1, d, $ and $d+1$ do not divide to $\mu^2.$ We can assume that $p > m+1.$ Then everyone of the equalities <ref> – <ref> implies that all the coefficients of $f(x)$ divide to $\mu ,$ and $|q_0| = |q_m| = \mu .$ Then the normal closure $K=$ ncl$(a^{\mu }) \lhd H/T$ is finitely generated. By lemma <ref> $Rat((H/T)/K)$ is a boolean algebra. The quotient $(H/T)/K$ is a homomorphic image of the wreath product $Z_{\mu } wr Z,$ where $\mu $ is prime. Hence, either $(H/T)/K$ is polycyclic, or is $Z_{\mu } wr Z.$ In the first case $(H/T)/K$ satisfies to the ascending chain condition, and since $K$ is finitely generated abelian, then $H/T$ satisfies to the ascending chain condition too. Every soluble group with the ascending chain condition is polycyclic (see for instance <cit.>). Hence $H/T$ is polycyclic. The second case is impossible, because $Z_{\mu } wr Z$ is not Howson group (see <cit.>).
§ SOLVABLE GROUPS OF TYPE $FP_{\INFTY}$ WITH BOOLEAN ALGEBRAS OF RATIONAL SUBSETS.
A group $G$ is said to be of type FP$_{\infty}$ if and only if there is a projective resolution
\begin{equation}
\label{eq:61} ... \rightarrow P_j \rightarrow ... \rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow \mathbb{Z} \rightarrow 0
\end{equation}
of finite type: that is, in which every $P_j$ is finitely generated.
A group $G$ has finite cohomological dimension if and only if there is a projective resolution
\begin{equation}
\label{eq:62} ... \rightarrow P_n \rightarrow ... \rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow \mathbb{Z} \rightarrow 0
\end{equation}
of finite length: that is, in which the $P_i$ are zero from some point on.
The following remarkable theorem is a base in our proof of the main result of this section.
(Kropholler <cit.>, see also <cit.>).
If $G$ is a soluble group of type FP$_{\infty}$ then vcd$(G) < \infty $.
Also we need in a standard statement as follows.
If cd$(G) < \infty $, then:
1) $G$ is torsion-free,
2) there is $n > 0,$ such that: if $A \leq G, A \simeq \mathbb{Z}^k,$ then $k\leq n.$
At this section we will specialize to soluble groups of type FP$_{\infty}$ and establish:
If $G$ is a finitely generated soluble group of type (FP)$_{\infty},$ such that $Rat(G)$ is a boolean algebra. Then $G$ is virtually abelian group.
By theorem <ref> there is a subgroup of finite index $H \leq G$, that has finite cohomological dimension. The subgroup $H$ is finitely generated and thus rational in $G.$ Then $Rat(H)$ is a boolean algebra. By lemma <ref> $H$ is torsion-free. Hence, every finitely generated abelian subgroup of $H$ is a free abelian of bounded rank. By Kargapolov's theorem (see <cit.> or <cit.>), $H$ has finite rank, i.e., there is a finite number $r$ such that every finitely generated subgroup of $H$ can be generated by $r$ elements and $r$ is least such integer (Prüfer or Mal'cev rank).
Thus, $H$ is a soluble torsion-free group of finite rank. By Robinson-Zaǐcev theorem (see <cit.>), every finitely generated soluble group with finite rank is a minimax group. It means there is a subnormal series
\begin{equation}
\label{eq:63}
1 = K_0 \leq K_1 \leq ... \leq K_n = K,
\end{equation}
in which every quotient $K_{i+1}/K_i$ satisfies to the ascending or descending condition. Also we know (see <cit.>), that every soluble torsion-free minimax group is nilpotent-by-(virtually abelian).
Let $N$ be a nilpotent normal subgroup of $H,$ such that $H/N$ is virtually abelian. We will prove that $N$ should be abelian. Let $1 \leq \zeta_1(N) \leq \zeta_2(N) \leq ... \leq \zeta_j(N) = N$ be the upper central series of $N.$ If $u \in N, y \in \zeta_2(N)\setminus \zeta_1(N)$ do not commute, then $[u, y] \in \zeta_1(H)$ has infinite order. We know that the quotient of torsion-free nilpotent group by the center is torsion-free (see <cit.>). Then the subgroup gp$(g,y)$ is isomorphic to the free nilpotent of rank $2$ and class $2$ group (UT$_3(\mathbb{Z})$ or the Heizenbergh group). But UT$_3(\mathbb{Z})$ is obviously non virtually abelian, hence $Rat(UT_3(\mathbb{Z}))$ is not boolean algebra by <cit.>. It follows that $\zeta_2(N) = \zeta_1(N),$ thus $N$ is abelian. Then $H$ is extension of the abelian normal subgroup $N$ with virtually abelian group $H/N.$ Then by theorem <ref> $G$ is virtually abelian.
§ SOLVABLE BIAUTOMATIC GROUPS.
The class of automatic groups is one of the main classes studying by geometric group theory. Different properties of automatic and biautomatic groups are described in the classical monography <cit.>. See also <cit.>. We give the main definitions.
Let $(A, \lambda , L)$ be a rational structure for $G.$ Recall, that $A$ is a finite alphabet, $\lambda $ is a homomorphism of the free monoid $A^{\ast}$ onto $G,$ $L$ is a regular language in $A^{\ast}$ such that $\lambda (L) = G.$ The set $A$ is a generating set for $G,$ considered as a monoid. We assume that $A$ is symmetric: that is $A$ contains with every its element $a$ its formal inverse $a^{-1}.$ We assume that homomorphisms of $A^{\ast}$ to groups map formally inverse elements to inverse images. The set $L$ is considered as the set of normal forms of expressions of elements of $G.$ We add to $A$ a new symbol $\$.$ Consider alphabet $A^{2\$},$ consisting of the pairs $(b, c),$ where $b, c \in A \cup \$.$ Take the corresponding free monoid $(A^{2\$})^{\ast}.$ The homomorphism $\lambda $ is naturally extended to homomorphism of the free monoid $(A^{2\$})^{\ast}$ onto $G^2.$ One has $\lambda (\$) = 1.$
A rational structure $(A, \lambda , L)$ is called automatic for $G$ under the following conditions are satisfied:
\begin{equation}
\label{eq:}
\{(u, v) \in L^{2\$}: \lambda (u) = \lambda (v)\}
\end{equation}
and for every $a \in A$ the language
\begin{equation}
\label{eq:}
\{(u, v) \in L^{2\$} : \lambda (u) = \lambda (va)\}
\end{equation}
is regular in $A^{2\$}.$
A group $G$ is said to be automatic if $G$ has an automatic structure $(A, \lambda , L).$
An automatic structure is said to be biautomatic, if for every $a \in A$ the language
\begin{equation}
\label{eq:66}
\{(u, v) \in L^{2\$} : \lambda (u) = \lambda (av)\}
\end{equation}
is regular in $A^{2\$}.$
A group $G$ is said to be biautomatic if $G$ has a biautomatic structure $(A, \lambda , L).$
In <cit.> a question is formulated: is every biautomatic group virtually abelian? We give a positive answer to this question by the following theorem. This result has been obtained by Bazhenova, Noskov, Remeslennikov and the author with using of information received them from Kropholler.
(Bazhenova, Noskov, Remeslennikov, Roman'kov).
Let $G$ be a finitely generated soluble biautomatic group. Then $G$ is virtually abelian.
By <cit.>, theorem 10.2.6, every soluble biautomatic group has type FP$_{\infty}.$ Hence by theorem <ref> $G$ has a subgroup of finite index $H$ with cd$(H) < \infty .$ By lemma <ref> $H$ is torsion-free. Moreover, all abelian subgroups of $H$ has bounded rank. Hence by Kargapolov's theorem (see <cit.> or <cit.>) $H$ has finite rank. By Robinson-Zaǐcev theorem (see <cit.>), every finitely generated soluble group with finite rank is a minimax group. Also we know (see <cit.>), that every soluble torsion-free minimax is nilpotent-by-(virtually abelian). Let $N$ be a nilpotent normal subgroup of $H$ such that $H/N$ is virtually abelian.
We will prove that $H$ is abelian. Let $1 \leq \zeta_1(N) \leq \zeta_2(N) \leq ... $ be the upper central series in $N.$ Let $u \in N, y \in \zeta_2(N)\setminus \zeta_1(N)$ don't commute. Then $[u, y] \in \zeta_1(H)$ is a nontrivial element of infinite order.
Then the subgroup gp$(g,y)$ is isomorphic to the free nilpotent of rank $2$ and class $2$ group UT$_3(\mathbb{Z}).$ It cannot happen since UT$_3(\mathbb{Z})$ is polycyclic but not virtually abelian. Indeed, by <cit.> every polycyclic subgroup of biautomatic group is virtually abelian.
Thus, $\zeta_2(H) = \zeta_1(H), $ and $H =\zeta_1(H)$ is abelian.
The group $H$ is finitely generated and virtually metabelian. Every subgroup of a finite index in a biautomatic group is biautomatic <cit.>. Thus, $H$ is biautomatic. Then $H$ satisfies to the minimal condition for centralizers (see <cit.> or <cit.>). It means: if
${\cal\bf S}(H)$ be a class of subgroups of $H$ of type $M = C_H(X) = \{h \in H: \forall x \in X \ [h,x] = 1\},$ where $X \subseteq H,$ then descending sequence $M_1 \geq M_2 \geq ... \geq M_l ... $ of subgroups in ${\cal\bf S}(H)$ stabilizes on a finite step. Then there is a number $l$ such that $M_l = M_{l+1} = ... .$ By <cit.> each biautomatic group with this property satisfies to the maximal condition on abelian subgroups. By Mal'cev's theorem (see <cit.>) a soluble group with this condition is polycyclic. Hence $H$ is polycyclic. Then by <cit.> $H$ is virtually abelian. Hence $G$ is virtually abelian.
Acknowledgment. The author was supported in part by Russian Research Fund, grant №14-11-00085.
ECHLPT D.B.A. Epstein, J.W. Cannon, D.F. Holt, S. Levy, M.S. Patterson, W. Thurston, Word processing in groups, Jones and Bartlett, 1992.
B G. Bazhenova, Rational sets in finitely generated nilpotent groups, Algebra and Logic, 39(4), 2000, 379-394
GS S. Gersten and H. Short, Rational subgroups of biautomatic groups, Annals of Math., 134 (1991), 125-158.
Gil R.H. Gilman, Formal languages and infinite groups, in Geometric and computationalperspectives on infinite groups
(Minneapolis, MN and New Brunswick, NJ, 1994), DIMACS Ser. Discrete Math. Theor. Comp. Sci., 25 (Amer. Math. Soc., Providence, RI, 1996), 27-51.
KM M.I. Kargapolov and Yu.I. Merzlyakov, Foundations of the theory of groups, Springer, 1979.
Kir A.S. Kirkinskij, Intersections of finitely generated subgroups in metabelian groups, Algebra and Logic, 20(1) (1981), 24-36.
K1 P.H. Kropholler, Soluble groups of type (FP)$_{\infty}$ have finite torsion-free rank, Bull. London Math. Soc., 25(6) (1993), 558-566.
K1 P.H. Kropholler, Hierarchical decompositions, generalized Tate cohomology, and groups of type (FP)$_{\infty}$, in Combinatorial and Geometric Group Theory (Duncan, A.J., Gilbert, N.D. and Howie, J., Editors) (Proc. Conf. Edinburgh, 1993), London Mathematical Society Lecture Note Series, 204 (Cambridge University Press, Cambridge, UK, 1995), 190-216.
K2 P.H. Kropholler, Soluble groups of type (FP)$_{\infty}$ have finite torsion-free rank, Bull. London Math. Soc., 25(6) (1993), 558-566.
K3 P.H. Kropholler, On groups of type (FP)$_{\infty}$, J. Pure and Applied Algebra, 90(1), 55-67.
LR J.C. Lennox and D.J.S. Robinson, The Theory of Infinite Soluble Groups, Clarendon Press, Oxford, 2004.
R D.J.S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Springer-Verlag, Berlin, 1972.
(Романьков <cit.>, проблема 18.85).
Конечно порожденная разрешимая группа $G,$ в которой семейство $Rat(G)$ всех рациональных подмножеств является булевой алгеброй, почти абелева.
M L. Mosher, Central quotients of biautomatic groups, Comment. Math. Helvetici, 72(1) (1997), 16-29.
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1511.00160
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§ INTRODUCTION
Exploring properties of QCD at finite temperatures and densities is one of the exciting topics in particle physics.
Lattice QCD has proved to be a powerful nonperturbative method at finite temperatures,
but its direct application at finite densities is extremely hard because of the sign problem;
for nonzero chemical potentials, the action becomes complex and
the integrand becomes a highly oscillatory function, which invalidates the importance sampling.
Usual Monte Carlo algorithms break down and we can not extract information of high density region of QCD.
Various methods have been applied for studying QCD at finite density, which include, e.g.,
Taylor expansion method, multi-reweighting method, use of imaginary chemical potential, study of SU(2) theory, complex Langevin method
and so on<cit.>.
In this work, we try to apply the Lefschetz thimble method<cit.> to the lattice (0+1) dimensional Thirring model at finite density, which is proposed in Ref. <cit.>.
This model has several similarities to lattice QCD at finite chemical potentials:
(1) it is described by compact variables,
(2) the sign problem occurs from its Dirac determinant,
(3) the fermion number density and the chiral condensate are physically important observables.
Moreover, differently from QCD,
(4) the partition function of this model can be evaluated exactly.
We firstly locate the critical points (saddle points) of the gradient flows within the subspace of time-independent (complex) link field, and study the thiemble structure and the Stokes phenomenon to identify the thimbles which contribute to the path-integral. We next perform HMC simulations on the single dominant thimble and compare the results to the exact solution.
This sutdy of the Thirring model with the thimble method will give us
useful lessons for its future application to the (3+1) dimensional QCD at finite density.
§ MODEL DEFINITION AND APPLICATION OF THE LEFSCHETZ THIMBLE METHOD
In this investigation, we use the lattice action<cit.>
S=∑_n=1^L βN_f (1-cosA_n) - ∑_i=1^N_f logD_i,
D_i nm = 1/2 [ e^μ_i +iA_n δ_n+1,m - e^-μ_i +iA_m δ_n,m+1 ] + m_i δ_n,m,
where $L$ is the lattice size, $N_{f}$ is the flavor number, $\beta$ is the inverse coupling squared $\beta=1/2g^{2}$, $m_{i}$ is the fermion mass, and $\mu_{i}$ is the chemical potential.
The auxiliary field $A_{n}$ is defined as a compact link-auxiliary field $e^{i A_{n}}$ on links,
and takes the values in the range, $-\pi \le A_{n}< \pi$.
In our study, we focus only on the single flavor, $N_{f}=1$.
The partition function can be evaluated exactly,
Z= e^-Lβ/2^L-1 [ I_1(β)^L coshL μ+ I_0(β)^L coshLm̂ ], m̂ =sinh^-1 m ,
where $I_{0}(x)$ and $I_{1}(x)$ are the modified Bessel functions.
From this exact form, we can obtain the exact number density and chiral condensate by differentiating the partition function with respect to the chemical potential and the fermion mass, respectively,
⟨n ⟩≡1/L logZ/μ = I_1(β)^L sinhL μ/I_1(β)^L coshL μ+ I_0(β)^L coshL m̂,
⟨χ̅ χ⟩≡1/L logZ/m = I_1(β)^L sinhL μ/ [ I_1(β)^L coshL μ+ I_0(β)^L coshL m̂] coshm̂.
(a) $\mu=0$
(b) $\mu=0.8$
Thimble structure in the zero-mode subspace at (a) $\mu=0$ and (b) $\mu=0.8$.
The green points and red points denote critical points and fermion-zeros, respectively.
The blue arrows indicate the downward flows.
Here, we apply the Lefschetz thimble method to the Thirring model.
We firstly extend the real variables $A_{n}$ to complex numbers, $A_{n} \rightarrow z_{n}$.
Note that the originally compact variables $e^{i A_{n}}$ become non-compact after this extension.
Then, we solve the complexified saddle point equation,
S[z]/z_n =0 n,
to find out the critical points.
We denote a set of critical points as $\{ \sigma \}$.
The Lefschetz thimble ${\cal J}_\sigma$ associated to a critical point $\sigma$ is defined
as a union of the downward flow in the complexified configuration space:
d z_n/dt = S̅[z̅]/z_n z →σ t →- ∞.
Along a flow line the imaginary part of the action ${\rm Im} S$ remains constant.
The downward flows extend to the region where $h=-{\rm Re}S$ becomes $-\infty$.
since the action diverges at the zero points of the fermion determinant,
D[z] =0,
the thimbles generally can end at those fermion-zeros.
Indeed the zero point condition (<ref>) can be solved exactly.
When we restrict that $z_{n}$ is independent of $n$, the solutions are written
z_zero = i (μ∓m̂) + 2 ℓ+1/L π, ℓ∈ℤ_L.
Figure <ref> shows the thimble structure within the zero-mode subspace at $\mu=0$ and $0.8$.
For $\mu=0$ the downward flows emanating from the origin run into the other critical points at $z=\pm \pi$,
while for $\mu=0.8$ they flow into the determinant zeros.
The thimble structure depends on parameters in the theory.
To decompose the original integration contour into the thimbles,
we need to select the contributing thimbles, which have the
nonzero intersection numbers determined by the dual cycles ${\cal K}_\sigma$
sharing the same critical point $\sigma$.
The dual cycle ${\cal K}_\sigma$ is defined by the following equation,
d z_n/dt = S̅[z̅]/z_n z →σ t →+ ∞.
${\cal K}_\sigma$ emanates from either of the points $\pm i \infty$, where $h=+\infty$.
Roughly, the intersection number $n_\sigma$ is unity (zero) if its dual cycle ${\cal K}_\sigma$
does (not) intersect with the original contour.
In general it is hard to determine all intersection numbers because we need whole knowledge on the global thimble structure in the complexified configuration space.
Furthermore, the set of thimbles which contribute to the partition function changes depending on $\mu$.
In the next section, we study the Stokes phenomenon to obtain information of change of intersection numbers.
§ STOKES PHENOMENON AND INTERSECTION NUMBERS
Here we study the Stokes phenomenon which provides information of change of intersection numbers.
In order for an intersection number to change (with increasing $\mu$),
the endpoint of ${\cal K}_\sigma$ must change, e.g.,
from $-i\infty$ to $+i\infty$. Inbetween there is a critical value $\mu^*$
at which ${\cal K}_\sigma$ is connected to another critical point $\sigma'$.
At this $\mu^*$, ${\cal K}_\sigma$ is overlapping with ${\cal J}_{\sigma'}$.
When two critical points are connected by a flow, we say that the Stokes phenomenon occurs.
A necessary condition for the Stokes phenomenon is as follows:
ImS_σ = ImS_σ^' + 2 πk,
with $k \in {\mathbb Z}$.
In ordinary cases such as $\lambda \phi^{4}$ theory the second term on the r.h.s.
is not necessary, but it is needed in our case because of the branch cuts
caused by the $\log \det D$ term in the action.
(a) Critical points and fermion-zeros
(b) values of ${\rm Im}S_{\sigma}$
Critical points and determinant zeros at $L=4,N_{f}=1,\beta=3,m=1$ and values of ${\rm Im}S_{\sigma}$ at those critical points.
In the left panel, the green and red circles are critical points and fermion-zeros, respectively.
The red lines running from the fermion-zeros denote branch cuts caused by $\log \det D$ term in the action.
We investigate the thimble structure of the right-half plane surrounded by a blue square below.
In the right panel, $\mu^{*}$ denotes the value at which the Stokes phenomenon occurs.
Figure <ref> shows the critical points $\sigma_i$ and zero points in the left panel,
and the values of ${\rm Im}S_{\sigma}$ on the critical points in the zero-mode subspace
with $L=4,N_{f}=1,\beta=3,m=1$ in the right panel.
For this parameter set, there are six $\mu^{*}$ at which the Stokes phenomenon occurs.
(a) $\mu<\mu^{*}$
(b) $\mu=\mu^{*}$
(c) $\mu>\mu^{*}$
Lefschetz thimble structure around some $\mu_4^{*}$.
The green and red points denote critical points and fermion-zeros, respectively.
The solid (dotted) lines show the thimbles (dual cycles).
Figure <ref> shows the thimble structures around $\mu_4^{*}$.
The dual cycle ${\cal K}_{\sigma_1}$ intersects
the real axis at $\mu<\mu^{*}$ but it does not at $\mu>\mu^{*}$.
Hence, $\mu_4^{*}$ is the critical value for change of the intersection number $n_{\sigma_1}$.
(a) $\mu^{*}_{1}<\mu<\mu^{*}_{2}$
(b) $\mu^{*}_{2}<\mu<\mu^{*}_{3}$
(c) $\mu^{*}_{3}<\mu<\mu^{*}_{4}$
(d) $\mu^{*}_{4}<\mu<\mu^{*}_{5}$
(e) $\mu^{*}_{5}<\mu<\mu^{*}_{6}$
(f) $\mu^{*}_{6}<\mu$
$\mu$-dependence of the thimble structure in the right-half plain of the zero-mode configuration space at $L=4,N_{f}=1,\beta=3,m=1$.
The contour of the zero-mode partition function can be constructed as follows:
(a)${\cal C} = {\cal J}_{\sigma_{0}}+{\cal J}_{\sigma_{2}}+{\cal J}_{\sigma_{\bar{2}}}$, (b)${\cal C}= {\cal J}_{\sigma_{0}}+{\cal J}_{\sigma_{2}}+{\cal J}_{\sigma_{\bar{2}}}$, (c)${\cal C}= {\cal J}_{\sigma_{0}}+{\cal J}_{\sigma_{1}}+{\cal J}_{\sigma_{\bar{1}}}$, (d)${\cal C}= {\cal J}_{\sigma_{0}}$, (e)${\cal C}= {\cal J}_{\sigma_{0}}+{\cal J}_{\sigma_{2}}+{\cal J}_{\sigma_{\bar{2}}}$, (f)${\cal C}= {\cal J}_{\sigma_{0}}$.
We can construct the integration contour of the zero-mode partition function by thimbles.
In Fig. <ref>, we present the right-half plain of the zero-mode configuration space at several values of $\mu$.
From this figure, we find that the thimble ${\cal J}_{\sigma_{0}}$ always contributes to the partition function.
Actually, by comparing ${\rm Re}S$ at $\sigma_i$, this thimble ${\cal J}_{\sigma_0}$ is the most dominant.
One of the important questions then is how accurately we can reproduce the exact solutions only by
this dominant thimble ${\cal J}_{\sigma_0}$.
§ HMC SIMULATION
We perform lattice simulations on the single thimble ${\cal J}_{0}$
in order to test how accurately the dominant thimble can reproduce the exact solutions.
We have employed the HMC algorithm proposed in Ref.<cit.> with a few
improvements by
introducing a scale parameter $\lambda$ to rescale the variables as $z_{n} \rightarrow \lambda z_{n}$
and by using the adaptive step size in the Runge-Kutta method.
Figure <ref> shows the residual phase averages
at $L=4,8$, $\beta=1,3,6$ and $m=1$.
At those parameters, the averages of the residual phase
stabilize around $\langle \exp(i \theta) \rangle \sim 0.8$-$1$.
Hence, we can rely on the phase reweighting to estimate observables.
Numerical results for the number density and the fermion condensation
are plotted in Figs. <ref> and <ref>.
At $\beta=3,6$, the numerical results are in good agreement with the exact solutions within the $\mu$-range considered.
However, at $\beta=1$, we find discrepancies between the numerical results and the exact ones in the crossover region.
(a) $L=4$
(b) $L=8$
Residual phase at (a) $L=4$ and (b) $L=8$.
The fermion mass is fixed at $m=1$.
The red, green and blue colors denote the results at $\beta=1,3$ and $6$, respectively.
The real (imaginary) part is denoted by $\circ$ ($\triangledown$).
(a) $L=4$
(b) $L=8$
$\mu$-dependence of the number density at (a)$L=4$ and (b)$L=8$ with a unit fermion mass.
The squared coupling constant is chosen as $\beta=1$ (red), $3$ (green), and $6$ (blue).
We represent numerical and exact results using colored points and solid lines, respectively.
(a) $L=4$
(b) $L=8$
$\mu$-dependence of the fermion condensation at (a)$L=4$ and (b)$L=8$ with a unit fermion mass.
The squared coupling constant is chosen as $\beta=1$ (red), $3$ (green), and $6$ (blue).
We represent numerical and exact results using colored points and solid lines, respectively.
§ SUMMARY
We have applied the Lefschetz thimble method to the (0+1) dimensional Thirring at finite density.
We firstly investigate thimble structure, and then study the Stokes phenomenon to identify the contributing thimbles to the evaluation of the partition function within the $n$-independent complex field subspace.
We next performed HMC simulations on the dominant thimble ${\cal J}_{\sigma_0}$, which ends at the determinant-zeros
at finite chemical potential $\mu$.
At large $\beta$ the simulation results reproduce the exact one very well,
but at small $\beta$ we have observed the discrepancies in the crossover region of $\mu$.
This numerical result is consistent with the analytical study of the model's thimble structure
and shows that
in the crossover region
we need to take into account the contributions from the subdominant thimbles
to reproduce the exact results appropriately.
P. de Forcrand,
PoS LAT 2009, 010 (2009)
[arXiv:1005.0539 [hep-lat]].
D. Sexty,
PoS LATTICE 2014, 016 (2014)
[arXiv:1410.8813 [hep-lat]].
E. Witten,
arXiv:1001.2933 [hep-th].
M. Cristoforetti et al. [AuroraScience Collaboration],
Phys. Rev. D 86, 074506 (2012)
[arXiv:1205.3996 [hep-lat]].
H. Fujii, D. Honda, M. Kato, Y. Kikukawa, S. Komatsu and T. Sano,
JHEP 1310, 147 (2013)
[arXiv:1309.4371 [hep-lat]].
J. M. Pawlowski, I. O. Stamatescu and C. Zielinski,
arXiv:1402.6042 [hep-lat].
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arXiv:1509.09141 [hep-lat].
|
1511.00296
|
Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo ON N2L 2Y5, Canada
LD - Research, Pappelallee 78/79, 10437 Berlin, Germany
Darwinian evolution can be modeled in general terms as a flow in the space of fitness (i.e. reproductive rate) distributions. In the limit where mutations are infinitely frequent and have infinitely small fitness effects (the “diffusion approximation"), Tsimring et al. have showed that this flow admits “fitness wave" solutions: Gaussian-shape fitness distributions moving towards higher fitness values at constant speed. Here we show more generally that evolving fitness distributions are attracted to a one-parameter family of distributions with a fixed parabolic relationship between skewness and kurtosis. Unlike fitness waves, this statistical pattern encompasses both positive and negative (a.k.a. purifying) selection and is not restricted to rapidly adapting populations. Moreover we find that the mean fitness of a population under the selection of pre-existing variation is a power-law function of time, as observed in microbiological evolution experiments but at variance with fitness wave theory. At the conceptual level, our results can be viewed as the resolution of the “dynamic insufficiency" of Fisher's fundamental theorem of natural selection. Our predictions are in good agreement with numerical simulations.
“In general, inference in biology depends critically on understanding the nature of limiting distributions." — S. A. Frank <cit.>.
§ INTRODUCTION
Evolution is the self-sustaining, open-ended process arising wherever entities (organisms, algorithms, memes, etc.) are subject to differential reproduction, heredity and variation. The specifics of this process are as diverse as life itself, but its basic structure—the principle of the “survival of the fittest"—is abstract and universal. Does this general principle translate into a general pattern—a pattern which could be predicted mathematically and tested empirically, in the same way in which, say, energy conservation translates into the Boltzmann distribution in statistical mechanics? Are there falsifiable laws of evolution?
Experience in the physical and statistical sciences suggests that emergent patterns often arise through suitable “coarse-gainings", i.e. after large numbers of different configurations of the system are grouped according to dynamically relevant macroscopic variables. In practice, pinpointing these relevant variables is no easy task, as the history of thermodynamics and the discovery of energy as a conserved quantity shows; in biology, it was not until Darwin's Origin of Species that such a variable—reproductive rate or Malthusian fitness[Malthusian fitness is also known as log-fitness, because it is the logarithm of the Wrightian fitness—the expected number of viable offsprings per capita. In this paper “fitness" always refers to Malthusian fitness; the conversion to Wrightian fitness is immediate (normal distributions become log-normal etc.)]—was identified. The formal analogy between these two coarse-grained variables, energy and fitness, has been repeatedly highlighted in recent years <cit.>, but an equivalent to the Boltzmann distribution for evolutionary dynamics has yet to be identified.
These considerations encourage us to push Darwin's logic to its end and treat populations as collections of fitness classes—groups of individuals with the same reproductive rate irrespective of their phenotype; we can then ask about the structure of fitness distributions in evolving systems. To be sure, this approach amounts to a dramatic reduction of biological (or algorithmic, or cultural) reality: important ingredients such as genotype, phenotype, but also evolutionarily stable strategies, etc., are entirely left out of the analysis. Such may be, however, the price to pay to lift ourselves off from system-dependent properties and extract a robust prediction from the Darwinian paradigm.
To our knowledge, the fitness space approach was first explored by Eshel in 1971 within a discrete-time framework <cit.>; in continuous time, it was introduced by Tsimring et al. <cit.>. Using a “diffusion approximation" familiar from non-equilibrium statistical mechanics, these authors went on to identify a class of “fitness waves" solutions: Gaussian-shape fitness distributions moving at constant speed towards higher fitness values <cit.>. Subsequent literature has developed methods to compute the speed of these fitness waves in terms of the frequency, effect and fixation probability of new mutations <cit.>. The major finding of these studies is the extreme sensitivity of that speed to stochastic effects of rare mutations: in Daniel Fisher's words, fitness waves describe the evolutionary process as “a dog led by its mutational nose" <cit.>.
Fitness waves hint at a general statistical pattern, but they are incomplete in one key respect: they disregard the importance of negative selection in evolution. Positive (or directional) selection consists in the growth of newly discovered high-fitness traits at the expense of slower reproducers, resulting in significant gains for the population mean fitness; by contrast, negative selection is the removal of low-fitness individuals without any new beneficial traits being introduced into the population. A condition for positive selection to sustain itself through time is that beneficial mutations are sufficiently frequent: the “diffusion approximation" of Tsimring et al. expresses this assumption in extreme form by imposing that new mutations are infinitely frequent with infinitely small fitness effects (both beneficial and deleterious). This can potentially capture aspects of the evolutionary process in phases of rapid adaptation, but not much more—real mutations (in biology and in other fields) are rare, mostly deleterious, and can occasionally have major fitness effects.
In this paper we reconsider the dynamics of fitness distributions under less restrictive assumptions, with the goal of identifying more general patterns of evolution. Our approach consists in proving limit theorems for fitness distributions, analogous to the H-theorem in statistical physics or the central limit theorem in probability theory. Limit theorems are powerful tools which cut through the complexity of statistical phenomena to extract their emergent, or “universal" properties; in this sense they are the mathematician's “extra sense" famously envied by Darwin.[“I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics; for men thus endowed seem to have an extra sense." — C. Darwin <cit.>] A key ingredient underlying all limit theorems is the distinction between a probability distribution and its type. Technically, the type of distribution is its equivalence class under an affine transformation of its argument; in practice, this means the shape of the distribution, i.e. all information in the distribution except its location and scale. As a rule, limit theorems show that distribution types are more strongly constrained by large-number effects than location or scale. In the H-theorem, for instance, the mean and variance of the Boltzmann distribution depend on temperature, hence are system-dependent; its exponential structure is not.[The same is true in the central limit theorem: the mean and variance of the sum of many independent random variables depend of the variables' distributions, but their normal type does not.] That is, types, not distributions, are subject to emergence and universality.
We combine an exact solution of the general replication-mutation equation in fitness space with suitable asymptotic estimates to prove that evolving fitness distributions are attracted to a one-parameter family of universal types. This pattern breaks down into two sub-patterns: under positive selection (of pre-existing variation or of new mutations), fitness is normally distributed; under negative selection and weak mutations, fitness has a reverted gamma distribution. A generic evolution trajectory consists of crossovers between these types. We check these findings with numerical simulations of the Wright-Fisher process and with a simple genetic algorithm.
§ EVOLUTION AS TRANSPORT
We consider an infinite population with a continuous distribution of (Malthusian) fitness $p_t(x)$. We assume that fitness-changing new mutations occur with a rate $U$, and that their fitness effects $x\mapsto x+\Delta$, with $\Delta>0$ (resp. $\Delta<0$) for beneficial (resp. deleterious) mutations, are distributed according to some distribution of fitness effects (DFE) $m(\Delta)$, a common assumption in evolutionary biology <cit.>. (For clarity of the presentation we assume that the mutation rate $U$ and DFE $m(\Delta)$ are fixed, but our results can be straightforwardly generalized to time-dependent and/or stochastic mutational effects to allow for changing fitness landscapes, changing environments, etc.)
The dynamical equation for the evolution of fitness distributions under selection and mutation can be formulated both in continuous and in discrete time, without it making any difference for our purposes. For definiteness we focus on the continuous case,[See the Appendix for the corresponding results in discrete time.] where it reads
\begin{equation}\label{floweq}
\frac{\partial p_t(x)}{\partial t}=(x-\mu_t)p_t(x)+U\int d\Delta\, m(\Delta)\,[p_t(x-\Delta)-p_t(x)]
\end{equation}
with $\mu_t\equiv\int dx\, x\,p_t(x)$ the mean fitness at time $t$. The first term in this equation expresses natural selection, i.e. the population effect of differential fitness; the second term is the mutation term, responsible for the introduction of new variations in fitness distributed according to the DFE $m(\Delta)$.
After Ref. <cit.> this “replicator-mutator equation"[Replicator-mutator equations are often written in genotype or trait space <cit.>; here by contrast it is introduced as an equation for fitness distributions.] is often approximated by a reaction-diffusion equation, with the mutation integral replaced by a term proportional to $\partial^2p_t(x)/\partial x^2$, but this step is neither well justified—real mutations are not infinitely frequent—nor necessary. Indeed we can obtain the general solution of (<ref>) in closed form for any DFE. It suffices for that to transform (<ref>) into an equation for the cumulant-generating-function (CGF) of the fitness distribution, defined by $\psi_t(s)\equiv\ln[\int dx\, e^{sx}p_t(x)]$. This gives a simple transport-like equation, with explicit solution (Appendix <ref>)
\begin{equation}\label{solution}
\psi_t(s)=\psi_0(s+t)-\psi_0(t)+U\int_0^tdu\, \big(\chi_{m}(s+u)-\chi_{m}(u)\big)
\end{equation}
where $\chi_m(s)\equiv\int d\Delta\, e^{s\Delta}m(\Delta)$ is the moment-generating-function (MGF) of the DFE. This reformulation is intuitively appealing: in $(t,s)$-space, natural selection corresponds to the transport of the initial CGF $\psi_0(s)$ towards progressively lower values of $s$, and mutations to a source term, see Fig. <ref>.
Evolution as transport in $(t,s)$ space. The CGF $\psi_t(s)$ (curved surface) is transported along the characteristics $s+t=\textrm{constant}$ (arrows), with a mutational source term $U(\chi_m(s)-1)$ (grey shading, with darker colors indicating larger values of $\chi_m(s)$, hence stronger mutational effects). The cumulants of the fitness distribution are given by the $s$-derivatives of $\psi_t(s)$ at $s=0$ (red line). For this plot we took $p_0(x)=S_{0,1,0}(x)$ and $m(\Delta)=S_{0,.1,-1}(\Delta)$ with $S_{\mu,\sigma,\xi}$ the skew-normal distribution with location $\mu$, scale $\sigma$ and shape $\xi$.
The explicit solution (<ref>) allows us to distinguish between two different regimes of evolution. Indeed depending on the relative magnitude of the terms above, the evolutionary trajectory can either be dominated by the selection of pre-existing variation, in which case
\begin{equation}\label{preexisting}
\psi_t(s)\simeq\psi_0(s+t)-\psi_0(t),
\end{equation}
or by the selection of new mutations, and then we have
\begin{equation}\label{new}
\psi_t(s)\simeq U\int_0^tdu\, \big(\chi_m(s+u)-\chi_m(u)\big).
\end{equation}
Note that Eq. (<ref>) is not the solution of (<ref>) without the selection term on the right-hand side.
§ FOUR LIMIT THEOREMS
We now analyze these two regimes separately, using the tools of asymptotic analysis.
Henceforth the symbol $\sim$ stands for “asymptotically equivalent up to a multiplicative constant".
§.§ Selection of pre-existing variation
We begin by focusing on the case where Eq. (<ref>) holds, either because the mutation rate $U$ is small or because the initial fitness distribution $p_0(x)$ is broad. This regime, the selection of pre-existing variation, is covered by the two limit theorems below.[In the final stages of this work we discovered that some of the results in Sec. <ref> are contained in theorems obtained in a very different context by Balkema, Klüppelberg, and Resnick <cit.>.]
The first theorem is concerned with initial fitness distributions with unbounded support, i.e. such that $\sup_x\{p_0(x)>0\}=\infty$. This condition is the mathematical counterpart of the biological notion of positive (or directional) selection, expressing the idea that ever-higher fitness individuals continuously take over the population at the expense of more common but less fit variants.
If Eq. (<ref>) holds and the initial fitness distribution $p_0(x)$ has unbounded support with
$$-\ln\int_x^\infty dy\,p_0(y)\underset{x\to\infty}{\sim} x^\alpha$$
for any $\alpha>1$, then the fitness distribution $p_t(x)$ becomes asymptotically normal as $t$ grows. Furthermore, the mean $\mu_t$ and variance $\sigma_t^2$ of $p_t(x)$ scale as
$$ \mu_t\underset{t\to\infty}{\sim} t^{\overline{\alpha}-1}\quad\textrm{and}\quad\sigma^2_t\underset{t\to\infty}{\sim} t^{\overline{\alpha}-2}$$
where $\overline{\alpha}$ is the exponent conjugate to $\alpha$, i.e. $1/\alpha+1/\overline{\alpha}=1$.
We make several remarks concerning this limit theorem. First, and in spite of an obvious formal similarity, Thm. 1 is not a consequence of the central limit theorem. This is apparent from its proof in Appendix <ref>, which involves a different kind of asymptotic estimate; it is also clear from the scaling behavior of the mean $\mu_t$ and variance $\sigma_t^2$: in contrast with the central limit theorem, where the mean and variance are linear functions of the sample size, here $\mu_t$ and $\sigma_t^2$ do not grow with time at the same rate. Second, the thin-tail condition on $p_0(x)$ in Thm. 1 is a general one, which encompasses many classical distributions (such as the Weibull family); it can moreover be generalized further in terms of the notion of “regular variation at infinity" <cit.>. Third, a lower bound on the rate of convergence of the fitness distribution to the normal type can be estimated as a function of the tail index $\alpha$, see Appendix <ref>.
Our second theorem deals with the case of fitness distributions with a finite right endpoint $x_+\equiv\sup_x\{p_0(x)>0\}<\infty$, corresponding to the negative selection of pre-existing variation.
If Eq. (<ref>) holds and $p_0(x)$ has a finite right endpoint $x_+$ with
$$p_0(x_+-x)\underset{x\to x_+}{\sim} (x_+-x)^\beta$$
for some $\beta\geq 0$, then the fitness distribution $p_t(x)$ becomes asymptotically a reversed Gamma distribution with shape parameter $1+\beta$ as $t$ grows, i.e. converges in type to the distribution with density function
$$g_\beta(x)\equiv\frac{(1+\beta)^{(1+\beta)/2}}{\Gamma(1+\beta)}\,e^{-(1+\beta)^{1/2}[(1+\beta)^{1/2}-x]}\Big[(1+\beta)^{1/2}-x\Big]^\beta \quad\textrm{for}\quad x\leq (1+\beta)^{1/2}.
Furthermore, the variance in fitness $\sigma_t^2$ eventually decreases as $\sigma_t^2\sim t^{-2}.$
The reversed gamma distributions above interpolate between a reversed exponential distribution for $\beta=0$ and a normal distribution for $\beta\to\infty$. In this sense, the situations of positive selection (Thm. 1) and negative selection (Thm. 2) are unified in a single continuous one-parameter family of distribution types, plotted in Fig. <ref>. Given this, we can interpret the parameter $1/\beta$ as selection negativity: low values of $\beta$ correspond to highly skewed distributions strongly dominated by the high-fitness individuals; high values of $\beta$, on the other hand, correspond to the situation where the most frequent individuals in the population have sub-optimal fitness.
Attractors for the selection of pre-existing variation, described by Eq. (<ref>). Left: convergence of various initial distributions to the $g_\beta$ family (red curve) in the skewness-kurtosis plane. The continuous lines correspond to initial distributions with unbounded support (three skew-normal distributions $S_\xi$ with shape parameter $\xi$); the dashed lines correspond to initial distributions with bounded support (three beta distributions $B_\xi$ with shape parameter $\xi$.) Right: shape of the attracting distributions $g_\beta$ for various values of $\beta$, in terms of Malthusian fitness $x$ (top) and of Wrigtian fitness $w=e^x$ (bottom).
§.§ Selection of new mutations
In the second regime of evolution, captured mathematically by Eq. (<ref>), the structure of the fitness distribution is determined by the constant stream of new mutations rather than by initial conditions. The nature of evolutionary dynamics in this regime depends on whether these mutations are sometimes beneficial (positive selection) or always deleterious or neutral (negative selection).
Our next theorem shows that, provided at least some mutations are beneficial, $\Delta_+\equiv \sup_\Delta \{m(\Delta)>0\}>0$, the fitness distribution converges to the normal type for any DFE. This is a strong form of universality in evolutionary dynamics.
If Eq. (<ref>) holds and at least some mutations are beneficial, $\Delta_+> 0$, the fitness distributions becomes asymptotically normal as $t$ grows independently of the distribution of fitness effects.
Finally, the case where all mutations are deleterious or neutral ($\Delta_+\leq 0$) was treated by Eshel in the context of discrete generations <cit.>. We obtain the following result.
If Eq. (<ref>) holds and all mutations are either deleterious or neutral, $\Delta_+\leq 0$, the fitness distributions $p_t(x)$ converges to the unique distribution with mean $\mu_\infty=x_+-U$, variance $\sigma_\infty^2=U\mu_m$ and higher standardized cumulants
$$K_\infty^{(p)}=-U^{1-p/2}\,\frac{\mu_m^{(p-1)}}{\mu_m^{p/2}}\quad\textrm{for}\quad p\geq 3$$
where $\mu_m$ is the mean fitness effect and $\mu_m^{(p)}$ the higher moments of the DFE $m(\Delta)$. In particular:
* the asymptotic mean fitness is independent of the distribution of fitness effects, as noted by Eshel <cit.>, and
* the asymptotic fitness distribution becomes normal in the limit of large mutation rates ($U\to\infty$) or small fitness effects ($\sigma_m\to 0$).
Note that, unlike the situation in Theorems 1-3, which all describe to phases of adaptative evolution (the mean fitness increases), the mutation-selection balance in Theorem 4 does not lead to universal fitness distributions, except when fitness effects are very small or mutation rates very high.
§ DISCUSSION
§.§ Crossovers
The theorems above capture the dynamics of fitness distributions in different limits, none of which holds exactly true in a real system. However these idealizations can serve as a basis to describe realistic evolutionary trajectories. Consider a large population starting with some pre-existing variation in fitness $\sigma_0$, subject to rare mutations (small $U$) with small fitness effects (small $\sigma_m$). Moreover suppose that the initial mean fitness $\mu_0$ is far from the maximal fitness $x_+$ available to the system, $\mu_0\ll x_+$. The qualitative behavior of the fitness distribution $p_t(x)$ at future times is then completely prescribed by Theorems 1-4.
In a first phase of evolution, the dynamics is dominated by the positive selection of pre-existing variation and the fitness distribution converges to the normal type, with the scaling of its mean $\mu_t$ and variance $\sigma_t^2$ determined by the high-fitness tail of $p_0(x)$, as stated in Thm. 1. When the of pre-existing variation is exhausted, i.e. when $x_t\simeq x_+$, selection becomes negative: the type of the distribution undergoes a rapid crossover to one of the $g_\beta(x)$ distributions in Thm. 2; the variance in fitness then starting to decay as $\sigma_t^2\sim t^{-2}$. This situation continues until mutations start becoming dominant, $\sigma_t\simeq \mu_m$. At that point, the future behavior of the fitness distribution depends on the whether mutations are partly beneficial ($\Delta_+>0$) or entirely deleterious ($\Delta_+\leq 0$). In the former case, Thm. 3 applies and the distribution undergoes a second crossover, taking it back to the normal type; in the latter case, a non-universal mutation-selection balance is reached, with $p_t(x)$ converging to a fixed limit $p_\infty(x)$ entirely determined by the DFE $m(x)$.
This sequence of patterns and crossovers between them is illustrated in the skewness-kurtosis plane in Fig. <ref>. The skewness $S_t$ and kurtosis $K_t$ of the fitness distributions are defined in terms of its third and fourth cumulants $\kappa_t^{(3)}$ and $\kappa_t^{(4)}$ as $S_t\equiv \kappa_t^{(3)}/\sigma_t^3$ and $K_t\equiv\kappa_t^{(4)}/\sigma_t^4$; this plane allows a very convenient low-dimensional representation of distributions types.
Crossover between different attractors, as follows from to the general solution (<ref>) . Here we took a standard normal distribution truncated at $x_+=10$ as initial fitness distribution and as skew-normal distribution with location $0$, scale $\sigma_m$ and shape parameter $-1$ as DFE. The blue-shaded region is the subset of the skewness-kurtosis plane available to unimodal distributions according to the Klaassen-Mokveld-van Es inequality <cit.>. The red curve is the $\beta$-family of attractors in Thm. 1-3, with then normal at $(S,K)=(0,3)$ corresponding to positive selection of pre-existing variation and of new mutations. Note that the attractiveness of the red curve is an increasing function of $U/\sigma_m$.
§.§ Dynamic sufficiency and Fisher's fundamental theorem
It is well known (and easy to check) that the dynamical equation (<ref>) is equivalent to the infinite tower of cumulant equations
\begin{equation}\label{tower}
\frac{\partial\kappa_t^{(p)}}{\partial t}=\kappa_t^{(p+1)}+U\mu_m^{(p)} \quad\textrm{for}\ \ p\geq 1
\end{equation}
where $\kappa_t^{(p)}$ is the $p$-th cumulant of the fitness distribution and $\mu_m^{(p)}$ is the $p$-th moment of the DFE. The equation for $p=1$, which relates the growth of the mean fitness to the variance in fitness (and mutational effects), is known as Fisher's fundamental theorem of natural selection <cit.>. The tower of equations (<ref>) being unclosed, it has been claimed that Fisher's theorem (or its generalization) is “dynamically insufficient", i.e. that it has no predictive power; see <cit.> and references therein for a review of this literature.
Our results in this paper show that this view is overly pessimistic. To begin with, the solution (<ref>) translates into exact expressions for the cumulants in terms of the initial CGF and of the DFE, namely (Appendix <ref>)
\begin{equation}
\kappa_t^{(p)}=\psi_0^{(p)}(t)+U\big(\chi_m^{(p-1)}(t)-\chi_m^{(p-1)}(0)\big)\quad\textrm{for}\ \ p\geq 1.
\end{equation}
While not solely expressed in terms of cumulants, this solution does predict the future evolution of all $\kappa_t^{(p)}$ given an initial fitness distribution and a DFE. Secondly, Theorems 1-3 imply very tight asymptotic relationships between the cumulants of the fitness distribution along an adaptive evolutionary trajectory. In particular,
In each one of the three adaptive regimes described by Theorems 1-3, the skewness $S_t\equiv\kappa_t^{(3)}/\sigma_t^3$ and kurtosis $K_t\equiv \kappa_t^{(4)}/\sigma_t^4$ of the fitness distribution are attracted towards the universal relationship
$K_t\sim3+\frac{3}{2}S_t^2$ with $-2\leq S_t\leq 0$.
This universal relationship—consistent with the mathematical inequality $K\geq S^2 +189/125$ holding for any unimodal distribution <cit.>—is a property of gamma and normal distributions. Along an evolutionary trajectory, it holds true at all times when the mean fitness increases (except during rapid crossover phases). Crucially, this relationship closes the tower of cumulant equations (<ref>): the mean fitness is determined by the variance in fitness, which is determined by the skewness, which is determined by the kurtosis—which in turn is determined by the skewness.
§.§ Scaling and drift
We have emphasized that robust patterns should not be expected at the level of the mean fitness $\mu_t$ or the variance in fitness $\sigma_t^2$. This is intuitively clear: the mean fitness of a population depends on the contingent history of mutational effect along its particular evolutionary trajectory—a single strongly beneficial mutation can have major effects on the whole population. The recent literature on fitness waves cited in the introduction has also strongly emphasized this point. For this reason, we have not attempted in this paper to characterize the dynamics of $\mu_t$ and $\sigma_t^2$ in any detail: this would require a stochastic treatment dealing with the statistics of fixation and drift.
There is however one case is which we expect the infinite-population limit to be relevant for the evolution of the mean fitness and its variance: the selection of pre-existing variation. In this case, indeed, we can assume that all mutations have already been fixed in the population. In this regime, Thm. 1 implies that both $\mu_t$ and $\sigma_t^2$ should be power-law functions of time, with an exponent which depends on the fat-tailedness of the initial fitness distribution (captured by the tail index $\alpha$). This prediction is at variance with fitness wave theory, which predicts a linear growth of the mean fitness <cit.>, but it matches with the results of a long-term evolution experiment with E. coli <cit.>. Whether pre-existing variation in fitness was indeed the main determinant of the fitness gains observed by Lenski et al., or new mutations did play a critical role in the evolution of E. coli populations, is unclear to us.
Also note that Thm. 1 implies the existence of a critical value of the tail index, $\alpha=2$, below which the variance in fitness increases under pure selection. This means that, if the initial distribution of fitness is sufficiently fat-tailed, selection does not necessarily imply a loss of variation in fitness. It would be interesting to see if this somewhat counter-intuitive behavior is realized in real evolving systems, at least for sufficiently long transients during which new mutations do not play a significant role.
§ NUMERICAL EXPERIMENTS
Numerical simulations with the WF process (purple) and with a simple GA (green). Left: trajectories of fitness distributions in the skewness-kurtosis plane for the WF and GA simulations described in the text, with the $g_\beta(x)$ attractors represented as the red parabolic curve; the inset plots the mean (continuous) and maximum (dashed) fitness in the population as a function of time. Right: the initial and final fitness distributions, overlaid with a best-fit distribution from the $g_\beta(x)$ family; the insets are P-P plot showing the good match between the empirical distributions and their fit.
§.§ Wright-Fisher simulations
We tested our results with Wright-Fisher (WF) simulations (Appendix <ref>). The WF process is a Markov chain representing the evolutionary dynamics of non-overlapping generations; it is a useful model to assess the importance of finite-population (drift) effects. We considered a (purposely relatively small) population of $10^5$ individuals grouped in $500$ distinct fitness classes, with Wrightian fitness (number of offspring) $w=e^x$ ranging between $w_{\textrm{min}}=1$ and $w_{\textrm{max}}=10$. Mutations were introduced with a rate $U=10^{-3}$, and their distribution of fitness effects $w\mapsto w'$ was assumed to be a fixed function of the selection coefficient $w'/w-1$, namely a Laplace distribution with location $-10^{-2}$ and scale $10^{-2}$. The fitness of the initial population was uniformly distributed between $w_{\textrm{min}}$ and the mid-point $(w_{\textrm{min}}+w_{\textrm{max}})/2$. We let this population evolve for $100$ generations and extracted the skewness and kurtosis of the resulting Malthusian fitness distributions.
The results, plotted in Fig. <ref>, are in good agreement with the patterns predicted in sec. <ref>. The pre-existing variation in fitness was selected first, and the fitness distribution was correspondingly attracted to the reverted exponential, with skewness $S=-2$ and kurtosis $K=6$. After this initial adaptation phase, mutations started feeding new fitness gains, which translated into the fitness distribution being attracted to the normal type with $(S,K)=(0,3)$. A fit of the final standardized distribution, at generation $n=100$, with the parametric family $g_\beta(x)$ gave a best-fit value $\beta=69.6$ with a value of $9.10^{-3}$ for the Kolmogorov-Smirnov statistic.[The corresponding $p$-value is irrelevant here: finite-population errors are not mere sampling errors from a fixed underlying distribution—the distribution itself is stochastic.]
§.§ Genetic algorithm
The relevance of our limit theorems is not restricted to biology. To illustrate this point we ran a genetic algorithm (GA) solving the linear integer optimization problem (Appendix <ref>)
\begin{equation}
\max\{c\cdot y\,;\, y\in\{1,\cdots,Q\}^L\ \textrm{and}\ b\cdot y\leq d\}
\end{equation}
where $L,Q\in\mathbb{N}$ and $d>0$ are fixed numbers, $b$ and $c$ are two randomly chosen vectors in $[0,1]^L$, and $\cdot$ denotes the dot product of vectors. We coded $10^5$ candidate solutions to this problem (with $Q=10$, $L=100$ and $d=1$) as strings of integers (“bases"), and generated an evolutionary trajectory from a random initial population with a mutation rate per base per generation $u=10^{-2}$. Unlike the WF model, this is a setting where the evolutionary dynamics is not a priori guaranteed to follow the replicator-mutator dynamics; in particular the DFE is unknown and possibly ill-defined (Appendix <ref>). In spite of this, we found that the fitness distribution quickly converged to the $g_\beta(x)$ attractors, see Fig. <ref>. In this case, pre-existing variation in fitness was negligible and no crossover was observed. A fit of the empirical fitness distribution after $n=100$ generations with the $g_\beta(x)$ family gave $\beta=4.7$ with a value of $6.10^{-3}$ for the Kolmogorov-Smirnov statistic.
§ CONCLUSION
selection pre-existing variation new mutations
positive normal normal
negative reverted gamma DFE-dependent (but normal if $\sigma_m/U\to 0$)
Four modes of natural selection and the types of the corresponding fitness distributions.
We have studied the dynamics of fitness distributions in four regimes of evolution, as summarized in Table I. In the regimes where evolution is adaptive, the attracting types belong to a single one-parameter family of distributions, parametrized by a “selection negativity" parameter. In terms of Malthusian fitness, these distributions are all negatively skewed and leptokurtic, with a universal parabolic relationship between kurtosis and skewness. We argue that these limiting distributions play a role similar to that of the Boltzmann distribution in equilibrium statistical mechanics or the normal distribution in statistics: they act as a benchmark against which evolving fitness distributions are to be assessed.
By relaxing strong assumptions on the frequency and effect of new mutations, our results go well beyond fitness-wave theory. As we saw with our numerical experiments, a generic evolutionary trajectory involves a phase of negative selection during which the effect of mutations on the fitness distribution is negligible. A consequence of this is that the low-fitness tail of the distribution becomes fatter than Gaussian. This means that there are in general many more low-fitness individuals in a population than predicted by fitness wave theory. Reciprocally, our results suggest that measuring the skewness of the fitness distribution is a test of positive vs. negative selection: if this measured skewness is significantly negative, we can say that the population is undergoing negative selection. The $\beta$ parameter in Theorem 2 is a quantitative measure of such “selection negativity".
Our approach also highlights the difference between universal and system-dependent features of evolving fitness distributions. Like temperature in Boltzmann's H-theorem or the mean and variance in the central limit theorem, we find that the mean, the variance and (to a lesser extent[The skewness of evolving fitness distributions must belong to the interval $[-2,0]$.]) the skewness—the first three cumulants—of evolving fitness distributions depend on the system under study, its underlying fitness landscape, etc.; all higher cumulants, on the other hand, are completely fixed once the latter are. This observation resolves a long-standing debate regarding the content and interpretation of Fisher's famous “fundamental theorem of natural selection". It also constitute a definite statistical prediction of Darwin's theory of evolution through natural selection; empirical tests of this prediction with published microbiological fitness data are in progress and will be reported elsewhere.
We thank M. Harper, A. Riello and S. A. Frank for useful critical comments on an early version of this manuscript. Research at the Perimeter Institute is supported in part by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
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§ EVOLUTION AS TRANSPORT
The replicator-mutator equation in fitness space reads
\begin{equation}\label{flow}
\dot{p}_t(x)=(x-\mu_t)p_t(x)+U\int d\Delta\, m(\Delta)\,[p_t(x-\Delta)-p_t(x)]
\end{equation}
where $p_t(x)$ is the distribution of Malthusian fitness $x$, $\mu_t$ the mean fitness at time $t$, $U$ is the mutation rate, $m(\Delta)$ a fixed distribution of fitness effects $\Delta$ of new mutations and dot means $\partial/\partial t$. In the limit where these mutations are infinitely frequent with infinitely small effects, this equation can be reduced to a reaction-diffusion equation, as in Refs. <cit.>. This (usually unrealistic) approximation is unnecessary, however, as (<ref>) can be solved exactly in terms of generating functions.
\begin{equation}
\chi_t(s)=\int dx\, e^{sx}p_t(x)
\end{equation}
be the moment-generating-function (MGF) of $p_t(x)$, such that $\mu_t=\chi_t'(0)$, and let $\psi_t(x)=\ln \chi_t(s)$ be its cumulant-generating-function (CGF). Denote $\chi_m(s)$ and $\psi_m(s)$ the corresponding quantities for the the DFE $m(\delta)$. From (<ref>) we have
\begin{equation}
\dot{\chi}_t(s)=\chi'_t(s)-\chi'_t(0)\chi_t(s)+U(\chi_m(s)-1)\chi_t(s),
\end{equation}
where prime means $\partial/\partial s$, hence
\begin{equation}\label{transport}
\dot{\psi}_t(s)=\psi'_t(s)-\psi'_t(0)+U(\chi_m(s)-1).
\end{equation}
This transformed version of the replicator-mutator equation has two remarkable features: it is linear and, except for the second term on the right-hand side, it has the structure of a transport equation.[A transport equation in one-dimension is a first-order PDE of the form $\dot{f}+vf'=g$, where $v$ and $g$ are two functions called drift and source respectively.] This observation suggests that its general solution can be written in terms of the characteristics $s+t=\textrm{constant}$, and indeed we can check that
\begin{equation}\label{sol}
\psi_t(s)=\psi_0(s+t)-\psi_0(t)+U\int_0^tdu\, \big(\chi_m(s+u)-\chi_m(u)\big)
\end{equation}
is the general solution of (<ref>). The interpretation which results is compelling: the evolution of fitness distributions through selection and mutations is equivalent to the transport of their CGF towards progressively lower values of $s$, with sources given by the mutational MGF. This is illustrated in Fig. <ref>.
The solution (<ref>) can be used to write an explicit formula for the evolved fitness distribution $p_t(s)$ using an inverse Laplace transform, namely
\begin{equation}
p_t(s)=\frac{1}{2i\pi}\int_{c-i\infty}^{c+i\infty} ds\, \frac{\chi_0(s+t)}{\chi_0(t)}\, \exp\left(U\int_0^tdu\, \big(\chi_m(s+u)-\chi_m(u)\big)-st\right).
\end{equation}
More usefully for our purposes, it also gives its cumulants $\kappa_t^{(p)}\equiv \psi_t^{(p)}(0)$ as
\begin{equation}\label{cum}
\kappa_t^{(p)}=\psi_0^{(p)}(t)+U\big(\chi_m^{(p-1)}(t)-\chi_m^{(p-1)}(0)\big)\quad\textrm{for}\ \ p\geq 1.
\end{equation}
Formula (<ref>), and its decomposition into pre-existing variation (first term on the RHS) and new mutations (second term on the RHS) is the basis for all our limit theorems.
§ DISCRETE TIME
All steps above can be repeated in the discrete-time formulation the replicator-mutator dynamics. In this case we use an integer generation label $n$ instead of time $t$, and assume that an individual with fitness $x$ has $w=e^x$ offsprings per generation. (The quantity $w$ is the Wrightian fitness.) Then we can write
\begin{equation}
p_{n+1}(x)=\frac{\int d\Delta\, M(\Delta)\, e^{x-\Delta}p_n(x)}{\int dx'e^{x'}p_n(x')}.
\end{equation}
with $M(\delta)\equiv (1-U)\delta(\Delta)+Um(\Delta)$. In terms of CGFs this reads
\begin{equation}
\psi_{n+1}(s)=\psi_{n}(s+1)-\psi_n(1)+\ln\left[(1-U)+U\chi_m(s)\right]
\end{equation}
which, as already noted by Eshel <cit.>, can be solved by a straightforward recursion:
\begin{equation}
\psi_n(s)=\psi_0(s+n)-\psi_0(n)+\sum_{j=0}^{n-1}\ln\left[(1-U)+U\chi_m(s+j)\right].
\end{equation}
As before the cumulants of the fitness distribution follow immediately.
§ PROOFS
The proofs of Theorems 1-4 all follow the same pattern:
* Consider the standardized (“renormalized", “comoving") fitness distribution
\begin{equation}
g_t(x)\equiv \sigma_t\, p_t(\sigma_t x+\mu_t),
\end{equation}
in which $\mu_t$ and $\sigma_t$ are the mean and standard deviation of $p_t(x)$. The cumulants $K_t^{(p)}$ of $g_t(x)$ are given by
\begin{equation}\label{standardizedcum}
K_t^{(p)}\equiv \frac{\kappa_t^{(p)}}{\sigma_t^p}=\frac{\psi_t^{(p)}(0)}{\psi_t''(0)^{p/2}}.
\end{equation}
* Use asymptotic estimates of generating functions for $s\to\infty$ to obtain the limit of (<ref>) as $t\to\infty$.
* Identify the limit distribution $\lim_{t\to\infty} g_t(x)$ from its cumulants $\lim_{t\to\infty} K_t^{(p)}$.
§.§ Theorem 1
*Asymptotic normality.
The Kasahara Tauberian theorem <cit.> states that, if $p_0(x)$ is a distribution with
\begin{equation}
-\ln\int_x^\infty dy\,p_0(y)\underset{x\to\infty}{\sim} x^\alpha,
\end{equation}
for some $\alpha>1$, then its associated CGF $\psi_0(s)$ satisfies
\begin{equation}\label{asympCGF}
\psi_0(s)\underset{s\to\infty}{\sim} s^{\overline{\alpha}}
\end{equation}
where $\overline{\alpha}$ is the exponent conjugate to $\alpha$, i.e. $1/\alpha+1/\overline{\alpha}=1$; see also <cit.>. For fitness distributions in the regime of selection of pre-existing variation, we have $\psi_t(s)=\psi_0(s+t)-\psi_0(s)$, hence from (<ref>),
\begin{equation}\label{conv}
\end{equation}
Since $\overline{\alpha}>1$, the exponent on the right-hand side is negative for all $p\geq 3$, hence
\begin{equation}
K_t^{(1)}=0,\quad K_t^{(2)}=1\quad \textrm{and}\quad\lim_{t\to\infty} K_t^{(p)}=0\ \textrm{for}\ p\geq3.
\end{equation}
This implies that $g_t(x)$ converges (weakly hence uniformly[Weak convergence to a continuous distribution implies uniform convergence.]) to the standard normal distribution. Moreover we have
\begin{equation}
\mu_t=\psi_0'(t)\underset{t\to\infty}{\sim} t^{\overline{\alpha}-1}\quad\textrm{and}\quad \sigma_t^2=\psi_0''(t)\underset{t\to\infty}{\sim} t^{\overline{\alpha}-2},
\end{equation}
as announced in Theorem 1.
*Rate of convergence.
We close by showing that the rate of uniform convergence to the normal is $\mathcal{O}(t^{-\overline{\alpha}/2})$, following the steps in the proof of the Berry-Esseen theorem in <cit.>. For this purpose we introduce the cumulative distribution functions
\begin{equation}
G_t(x)\equiv\int_{-\infty}^xdx'\, g_t(x')\quad\textrm{and}\quad\mathcal{N}(x)\equiv\int_{-\infty}^xdx'\, \frac{e^{-x'^2/2}}{\sqrt{2\pi}}.
\end{equation}
We start from Esseen's inequality, according to which for any $T>0$,
\begin{equation}\label{esseen}
\sup_{x\geq0} \vert G_t(x)-\mathcal{N}(x)\vert \leq \frac{2}{\pi}\int_0^T\frac{\vert f_t(u)-e^{-u^2/2}\vert}{u}\,du+\frac{24}{\pi T\sqrt{2\pi}},
\end{equation}
where $f_t(u)\equiv\exp \sum_{p=2}^{\infty}K_t^{(p)}(iu)^p/p!$ is the characteristic function of the standardized fitness distribution. Next, we use $\vert e^{z}-1\vert\leq \vert z\vert\, e^{\vert z\vert}$ to note that
\begin{equation}\label{useful}
\vert f_t(u)-e^{-u^2/2}\vert\leq e^{-u^2/2} \epsilon_t(u)\,e^{\epsilon_t(u)}
\end{equation}
\begin{equation}
\epsilon_t(u)\equiv\Big\vert \sum_{p=3}^{\infty}K_t^{(p)}\frac{(iu)^p}{p!}\Big\vert\leq u^3\,\vert K_t^{(3)}\vert\, \sum_{q=0}^{\infty}\Big\vert\frac{K_t^{(q+3)}}{K_t^{(3)}}\Big\vert \frac{T^q}{(q+3)!}\quad\textrm{for}\ u\in[0,T].
\end{equation}
From (<ref>), we have $K_t^{(3)}=\mathcal{O}(t^{-\overline{\alpha}/2})$ and
\begin{equation}
\Big\vert\frac{K_t^{(q+3)}}{K_t^{(3)}}\Big\vert=\mathcal{O}(t^{-q\overline{\alpha}/2})\prod_{k=0}^{q+2}(\overline{\alpha}-q).
\end{equation}
The series
\begin{equation}
\sum_{q=0}^{\infty}\frac{1}{(q+3)!}\prod_{k=0}^{q+2}(\overline{\alpha}-q)
\end{equation}
being summable, we can take $T=C_1n^{\overline{\alpha}/2}$ to obtain
\begin{equation}
\epsilon_t(u)=u^3\mathcal{O}(t^{-\overline{\alpha}/2}).
\end{equation}
Let $C_2$ be a constant such that $\epsilon_t(u)\leq C_2 u^3 t^{-\overline{\alpha}/2}$. We have
\begin{equation}
-\frac{u^2}{2}+\epsilon_t(u)\leq -\frac{u^2}{4}+u^2\left(C_2 T n^{-\overline{\alpha}/2}-\frac{1}{4}\right)\leq -\frac{u^2}{4}+\left(C_1C_2-\frac{1}{4}\right),
\end{equation}
hence choosing $C_1=1/4C_2$ gives
\begin{equation}
-\frac{u^2}{2}+\epsilon_t(u)\leq -\frac{u^2}{4}.
\end{equation}
Combining (<ref>) with (<ref>), we have obtained
\begin{equation}
\sup_{x\geq0} \vert G_t(x)-\mathcal{N}(x)\vert \leq\mathcal{O}(t^{-\overline{\alpha}/2})\int_0^{C_1n^{\overline{\alpha}/2}}u^2 e^{-u^2/4}\, du+\mathcal{O}(t^{-\overline{\alpha}/2}).
\end{equation}
The integral is bounded from above by $\int_0^{\infty}t^2 e^{-t^2/4}\, dt=2\sqrt{\pi}$ hence is $\mathcal{O}(1)$, and the conclusion follows.
§.§ Theorem 2
In the previous section we excluded the case $\overline{\alpha}=1$ in (<ref>). By the Paley-Wiener theorem, this case corresponds to fitness distributions with compact support (that is, with finite right endpoint $x_+$). If $p(x_+)\neq 0$, it is easy to see using Laplace's method that
\begin{equation}
\chi(s)=\int_0^{x_+} e^{sx}p(x)dx\underset{s\to\infty}{\sim} \frac{e^{sx_+}}{s}.
\end{equation}
In particular,
\begin{equation}
\psi(s)=\ln\chi(s)\underset{s\to\infty}{\sim} x_+s-\ln s.
\end{equation}
It follows that the standardized cumulants in (<ref>) satisfy
\begin{equation}
K_n^{(p)}\underset{t\to\infty}{\sim} \frac{-\ln^{(p)}(t)}{\ln''(t)^{p/2}}=(-1)^p(p-1)!.
\end{equation}
This sequence of moments characterizes the reversed exponential distribution $e^{-1+x}$, to which $g_t$ therefore converges (weakly and uniformly) as $t\to\infty$. (This can be seen by resumming the cumulant series, which gives the moment-generating function $e^{s}/(1+s)$.)
This computation can be generalized to the case where $p(x_+)=0$. Suppose there exists $\beta>0$ such that[We conjecture that it is sufficient that $F(x)$ is regularly varying at $x=x_+$ with index $\beta+1$, but do not know how to prove the theorem in this case.]
\begin{equation}
p_0(x)\underset{x\to x_+}{\sim} (x_+-x)^{\beta}
\end{equation}
Then a refined version of Laplace's method <cit.> gives
\begin{equation}
\chi_0(s)=\int_0^{x_+} e^{sx}p(x)dx\underset{s\to\infty}{\sim} \frac{e^{sx_+}}{s^{1+\beta}}
\end{equation}
It follows that
\begin{equation}\label{asympfinite}
\psi_t(s)\underset{s\to\infty}{\sim} x_+s-(1+\beta)\ln s
\end{equation}
hence for the standardized cumulants
\begin{eqnarray}
K_t^{(p)}\underset{t\to\infty}{\sim} (1+\beta)^{1-p/2}\frac{-\ln^{(p)}(t)}{\ln''(t)^{p/2}}=(1+\beta)^{1-p/2} (-1)^p\,(p-1)!.
\end{eqnarray}
Resumming the cumulant series gives the CGF
\begin{eqnarray}
\sum_{p=0}^\infty K_t^{(p)}\frac{s^p}{p!}&\underset{t\to\infty}{\sim}&(1+\beta)\sum_{p=2}^\infty \frac{(-1)^p}{p}\,(1+\beta)^{-p/2}s^p\\&=&(1+\beta)\Big[s(1+\beta)^{-1/2}-\ln\Big(1+s(1+\beta)^{-1/2}\Big)\Big].
\end{eqnarray}
We can obtain the associated density function $\lim_{t\to\infty}g_{t}(x)$ by means of an inverse Laplace transform, which gives
\begin{equation}\label{newdistribution}
\lim_{t\to\infty}g_{t}(x)=\frac{(1+\beta)^{(1+\beta)/2}}{\Gamma(1+\beta)}\,e^{-(1+\beta)^{1/2}[(1+\beta)^{1/2}-x]}\Big[(1+\beta)^{1/2}-x\Big]^\beta\quad\textrm{for}\ x<(1+\beta)^{1/2}.
\end{equation}
This, together with the observation that
\begin{equation}
\mu_t=\psi_0'(t)\underset{t\to\infty}{\sim} x_+\quad\textrm{and}\quad \sigma_t^2=\psi_0''(t)\underset{t\to\infty}{\sim} t^{-2},
\end{equation}
concludes the proof of Theorem 2.
§.§ Theorem 3
In the regime of positive selection of new mutations, (<ref>) implies for the standardized cumulants
\begin{equation}\label{cummut}
K_t^{(p)}=U^{1-p/2}\, \frac{\chi_m^{(p-1)}(t)-\chi_m^{(p-1)}(0)}{[\chi_m'(t)-\chi_m'(0)]^{p/2}}.
\end{equation}
We assume for simplicity that the DFE has a bounded support with finite right end-point $\Delta_+>0$. Using the same Laplace estimate as above, we have
\begin{equation}
\chi_m(s)\underset{s\to\infty}{\sim}\frac{e^{s\Delta_+}}{s^{1+\beta}}.
\end{equation}
It follows that
\begin{equation}
\end{equation}
which goes to zero for all $p\geq3$. Thus $g_t$ converges again to a normal distribution.
§.§ Theorem 4
When all mutations are deleterious, $\Delta_+<0$, the first in (<ref>) goes to zero as $t\to\infty$, hence
\begin{equation}
\lim_{t\to\infty}K_t^{(p)}=-U^{1-p/2}\,\frac{\chi_m^{(p-1)}(0)}{[-\chi_m'(0)]^{p/2}}
\end{equation}
as claimed in Thm. 4.
§ WRIGHT-FISHER SIMULATIONS
The Wright-Fisher process is a well-known stochastic model of asexual evolution through selection and mutations in finite populations. It is based on non-overlapping generations, whose composition is determined probabilistically by $(i)$ the composition of the earlier generation, and $(ii)$ a fixed mutation rate $U$.
In this work we set up a Wright-Fisher process in the following way. First, we consider a discretized fitness space consisting of $F$ bins $i$ with Wrightian fitness (number of offsprings) $w(i)=w_{\textrm{min}} +(w_{\textrm{max}}-w_{\textrm{min}})i/F$. The number of individuals with fitness $f_i$ at generation $n$ is denoted $N_n(i)$, and the total population $N=\sum_iN_n(i)$ is fixed. Finally we assume that a distribution of fitness effects has been given, in the form of fixed transition matrix $M(j,i)$ giving the probability of a mutation changing the fitness from $w(j)$ to $w(i)$.
The composition of generation $n$ then determines the probability that a randomly chosen individual at generation $n+1$ is the fitness bin $i$ as
\begin{equation}
p_{n+1}(i)=\frac{(1-U)w(i) N_{n}(i)+U\sum_j M(j,i)w(j)N_{n}(j)}{\sum_i[ (1-U)w(i) N_{n}(i)+U\sum_j M(j,i)w(j)N_{n}(j)]}.
\end{equation}
In words, this individual is either (with probability $1-U$) the non-mutated offspring of a parent in the same bin $i$, which comes with a weight $w(i) N_{n}(i)$, or (with probability $U$) the mutated offspring of a parent in one of the other bins $j$, which comes with a weight $M(j,i)w(j)N_n(j)$. The probabilities $p_{n+1} (i)$ then determine the probability of the number distribution $\{N_{n+1}\}$ according to the multinomial expression:
\begin{equation}
\textrm{Prob}(\{N_{n+1}\})=\frac{N!}{\prod_i N_{n+1}(i)}\prod_i p_{n+1}(i)^{N_{n+1}(i)}.
\end{equation}
In the main text we present the results of our simulations with $F=500$, $U=10^{-2}$, total population $N=10^5$ and a transition matrix of the form
\begin{equation}
\end{equation}
where $h$ is a Laplace distribution $h(z)=e^{-\vert x-\mu_m\vert/\sigma_m}/2\sigma_m$.
§ GENETIC ALGORITHM
We ran a genetic algorithm (GA) to test the applicability of our Theorems in a setting where the notion of “distribution of fitness effects of new mutations" is not a priori well-defined. The GA performs an iterative search for the solution of the linear optimization problem
\begin{equation}
\max\{c\cdot y;\, y\in\{1,\cdots,Q\}^L\ \textrm{and}\ b\cdot y\leq d\}
\end{equation}
where $L,Q\in\mathbb{N}$ and $d>0$ are fixed numbers, $b$ and $c$ are two randomly chosen vectors in $[0,1]^L$, and $\cdot$ denotes the dot product of vectors.
Our GA proceeds by maintaining a population of $N$ strings $y$ (“genes") of $L$ integers $y_i\in\{1,\cdots,Q\}$ (“bases"). Each genome is assigned the Wrightian fitness
\begin{equation}
w(y)=\exp\left[\frac{\min(c\cdot y,\,[c\cdot y-(b\cdot y-d)]_+)}{a}\right]
\end{equation}
where $[\,\cdot\,]_+\equiv \max(\,\cdot\,,0)$ and $a$ is fixed positive number ($1/a$ is the “selective pressure"). The exponential function ensures that a random population of genes does not has a fitness distribution resembling the attractors in Theorems 1-3.
Then at each new generation the following two steps are taken:
* Selection. A new population is generated by choosing $N$ genes $y$ from the previous populations weighted by their Wrightian fitness $w(y)$.
* Mutation. With probability $u<1$, each base of each gene is mutated into a randomly chosen new base in $\{1,\cdots,Q\}$.
At each generation the distribution $p_n(x)$ of Malthusian fitness $x(y)=\ln w(y)$ is measured and its cumulants $\kappa_n^{(p)}$ extracted. We also tested whether the GA has a single, well-defined DFE— it does not, see Fig. <ref>.
The GA does not have a single, well-defined DFE. In this figure each plot represents a histogram of the fitness effects $\Delta$ found in the mutational neighborhood of a randomly chosen gene. In some cases all mutations are almost neutral; in other cases they are mostly deleterious; in some rare cases all mutations are beneficial.
§ CLASSICAL DISTRIBUTIONS
For the reader's convenience we collect here the definitions of the classical probability distributions mentioned in the main text, in standard form. The following special functions are used:
* The Gamma function,
$$\Gamma(s)\equiv\int_0^\infty t^{s-1}e^{-s}\,dt$$
* The (lower) incomplete Gamma function,
$$\gamma(s,x)\equiv\int_0^x t^{s-1}e^{-s}\,dt$$
* The Beta function,
* The incomplete Beta function,
* The Owen T function,
$$T(h;a)\equiv\frac{1}{2\pi}\int_0^a\frac{e^{-h^2(1+t^2)/2}}{1+t^2} \,dt$$
Name Cumulative density function Probability density function Support Shape parameters
Normal $\frac{1}{2}\left[1+\textrm{erf}(x/\sqrt{2})\right]$ $\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ $x\in\mathbb{R}$
Skew-normal $\frac{1}{2}\left[1+\textrm{erf}(x/\sqrt{2})\right]-2T(x,\alpha)$ $\frac{1}{\sqrt{2\pi}}e^{-x^2/2} \left[1+\textrm{erf}(\alpha x/\sqrt{2})\right]$ $x>0$
Log-normal $\frac{1}{2}\left[1+\textrm{erf}(\ln x/\sqrt{2})\right]$ $\frac{1}{x\sqrt{2\pi}}e^{-(\ln x)^2/2}$ $x>0$
Gamma $\frac{1}{\Gamma(\alpha)}\,\gamma(\alpha,x)$ $\frac{1}{\Gamma(\alpha)}\,x^{\alpha-1}e^{-x}$ $x>0$ $\alpha>0$
Beta $\frac{B(x;\alpha,\beta)}{B(\alpha,\beta)}$ $\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$ $x\in(0,1)$ $\alpha,\beta>0$
Weibull $1-e^{-x^\alpha}$ $\alpha x^{\alpha-1}e^{-x^\alpha}$ $x>0$ $\alpha>0$
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1511.00467
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Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente, P. O. Box 217, Enschede 7500 AE, The Netherlands
Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente, P. O. Box 217, Enschede 7500 AE, The Netherlands
Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente, P. O. Box 217, Enschede 7500 AE, The Netherlands
Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente, P. O. Box 217, Enschede 7500 AE, The Netherlands
Philips Lighting, High Tech Campus 44, Eindhoven 5656 AE, The Netherlands
Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente, P. O. Box 217, Enschede 7500 AE, The Netherlands
We have studied the transport of light through phosphor diffuser plates that are used in commercial solid-state lighting modules (Fortimo). These polymer plates contain $\Y$ phosphor particles that elastically scatter light and Stokes shifts it in the visible wavelength range (400-700 nm). We excite the phosphor with a narrowband light source, and measure spectra of the outgoing light. The Stokes shifted light is separated from the elastically scattered light in the measured spectra and using this technique we isolate the elastic transmission of the plates. This result allows us to extract the transport mean free path $\lt$ over the full wavelength range by employing diffusion theory. Simultaneously, we determine the absorption mean free path $\la$ in the wavelength range 400 to 530 nm where $\Y$ absorbs. The diffuse absorption $\ma =\frac{1}{\la}$ spectrum is qualitative similar to the absorption coefficient of $\Y$ in powder, with the $\ma$ spectrum being wider than the absorption coefficient. We propose a design rule for the solid-state lighting diffuser plates.
45.15.Eq Optical system design, 42.25.Fx Diffraction and scattering, 42.25.Dd Wave propagation in random media
§ INTRODUCTION
Energy efficient generation of white light is attracting much attention in recent years, since it is important for lighting and for medical and biological applications <cit.>. One of the main directions is the technology of solid-state lighting <cit.>, that was recognized with the 2014 Nobel Prize in physics <cit.>. Solid-state lighting provides superior energy efficiency and flexibility in terms of color temperature. Conventional solid-state lighting employs a blue semiconductor light emitting diode (LED) in combination with a phosphor layer to realize a white-light emitting diode. The phosphor layer plays two important roles in a white LED: first, the phosphor layer absorbs blue light emitted by the LED, and efficiently converts part of the blue light into the additional colors green, yellow and red light. The desired mixture of blue, green, yellow and red light results in white outgoing light. Secondly, the phosphor layer multiply scatters all colors, thereby diffusing the outgoing light, resulting in an even lighting without hot spots, and with a uniform angular color distribution, as required for lighting applications. In addition the scatterers enhance the color conversion by increasing the path blue light travels in phosphor layer. In state-of-the art solid-state lighting technology the phosphor layer is engineered to have a complex internal structure<cit.>. Light inside this layer may be multiply scattered not only by phosphor, but also by other scatterers.
fig:tr1tr_a_1111.pdf(a) Normalized absorption spectra (blue circles) and emission spectrum (green squares) of the $\Y$ phosphor used in our study. The spectral range where emission and absorption spectra overlap is indicated with a green bar between $\lal$ and $\lar$. (b) Transmission spectra obtained by using the narrowband (red squares) and the broadband light sources (green circles) for the polymer plate with 4 wt $\%$ phosphor particles. Arrows point to the relevant ordinate for the data. (c) CIE 1931 (x,y) chromaticity diagram <cit.>. Monochromatic colors are located at the perimeter of the diagram. In the middle of the diagram the white color is located. The dashed gray area represent the region where emission and absorption of $\Y$ overlap. The overlap range was previously inaccessible and it is made accessible in this work.3 in
In spite of the wide use of solid-state lighting in everyday life, and the apparent simplicity of the physical processes occurring in the phosphor layer, there is no analytical theory that predicts the spectra of white LEDs. The main challenge arises from the lack of physical understanding of systems, where multiple scattering and absorption of blue light coexist with emission of light in a broadband wavelength range. Typically, numerical methods such as ray-tracing and Monte Carlo techniques are used <cit.>, that do not have the predictive power of analytical theory. These simulations require a set of heuristic parameters that is derived from measurements on LEDs with a wide range of structural and optical parameters. The resulting heuristic parameters are often adjusted, thereby further hampering the predictive power of these methods. Moreover, simulations are time consuming and computationally demanding. All these aspects hamper the efficient design of new white LEDs.
In this paper we - for the first time - extract optical properties of the phosphor layers typically used for the solid-state lighting in the visible wavelength range (400-700 nm). We use a narrowband light source and record the spectra of the transmitted light through the phosphor layer. The transmitted light contain both elastically scattered light and Stokes shifted light; they are separated spectrally. We extract the diffuse transmission from the elastically scattered light, and calculate the optical properties of the phosphor layer using the diffusion theory. Using a broadband light source a similar approach was previously applied to calculate optical properties of the diffuser plates <cit.>, and phosphor plates <cit.>. This approach fails when Stokes shifted light overlaps spectrally with the elastically scattered light. In Fig. <ref>(c) we show the overlap region for the plates studied here in $x-y$ chromaticity diagram. This spectral range correspond to the green part of the white LED spectrum where the human eye is most sensitive<cit.>. We present a new measurement technique that allow us to separate the elastically scattered and Stokes shifted light in the overlap range for phosphor diffuser plates that are used in commercial white LEDs. As a result we now close the ”the green gap” and extract the relevant transport and absorption parameters for solid-state lighting in the whole visible spectral range.
We use analytical theory originating from nanophotonics, wherein propagation of light is described from first principles <cit.>. Such $ab$ $initio$ theory supplies fundamental physical insights on the light propagation inside solid-state lighting device <cit.>. Extracting the optical parameters from theory are less time consuming than performing many simulations, and more importantly, the resulting parameters are robust and predictions can be made beyond the parameter range that was initially studied. For instance, knowledge of the absorption spectra provides us with the design guidelines for the solid-state light units. The design parameters such as the thickness of the diffuse plates and the phosphor concentration can be directly extracted from the absorption spectra depending on the blue pump wavelength of a white LED.
§ THEORY
§.§ Total transmission with energy conversion
Multiple light scattering is usually studied by measuring the total transmission through a slab of a complex, multiple scattering medium <cit.>. Total transmission, or diffuse transmission, is the transmission of an incident collimated beam with intensity $\Io $ that is multiple scattered and integrated over all outgoing angles at which light exits from a medium. The total transmission carries information on the transport mean free path $\lt$ and on the absorption mean free path $\la$, which are the crucial parameters that describe multiple light scattering <cit.>. The transport mean free path $\lt$ is the distance it takes for the direction of light to become randomized while performing a random walk in a scattering medium. The absorption mean free path $\la$ is the distance it takes for light to be absorbed to a fraction $(1/e)$ while light performs a random walk in a scattering medium.
Phosphor particles do not only scatter light, but also convert blue light into other colors by absorbing blue and re-emitting other colors of light. Therefore, from here on we will refer to the measured total transmission in presence of energy conversion as the $total$ $relative$ $intensity$ $\Tr$
\begin{equation}
\Tr =\frac{\It }{\Io}\mathrm{,}
\label{eq:trel}
\end{equation}
where $\It$ is the integrated intensity that is collected at the back side of the diffusion plate. In the emission range of a phosphor $(\lambda\geq\lal)$ the collected intensity $\It$ can be written as a sum of the diffuse intensity $I$ and the Stokes shifted intensity $\Ie$.
Thus, the total relative intensity can be separated into two parts <cit.>
\begin{equation}
\Tr=\T+\Tee=\frac{I (\lambda)+\Ie}{\Io}\mathrm{,}
\label{eq:trel1}
\end{equation}
where the first term $T$ is the total transmission, and the second term $\Tee $ is the emission that accounts for the energy conversion of light in the diffuse absorptive medium.
In Ref. Leung14 these two terms could not be distinguished in the overlap range $\lal < \lambda < \lar$.
The central question in this paper is how to distinguish the total transmission $\T$ from the total relative intensity $\Tr$ as this allows one to obtain both the transport mean free path $\lt$ and the absorption mean free path $\la$.
To access the total transmission we employ a tunable narrowband light source and spectrally resolve the narrowband transmitted light. Since the light that is converted by the phosphor exhibits a Stokes shift $\Ie$, this part $\Tee$ of the total relative intensity is filtered, hence we obtain the desired total transmission $\T$ that we interpret with diffusion theory.
§.§ Total transmission in absence of energy conversion
According to diffusion theory for light, the total transmission $\T$ through a slab, even in the presence of absorption, is a function of the slab thickness $L$, the wavelength $\lambda$, the transport mean free path $\lt$, and the absorption $\la$ mean free path, and can be expressed as <cit.> [1]:
\begin{equation}
T\left( L, \lai , \lt, \la\right) =Q^{-1}\left[ \mathrm{sinh}\left( \ma z_{p}\right) +\ma\ze \mathrm{cosh}\left(\ma z_{p} \right) \right] \mathrm{,}
\label{eq:tr}
\end{equation}
\begin{equation}
Q\left( L, \lai , \lt, \la\right)\equiv \left( 1+ \ma^{2}\ze^{2}\right) \mathrm{sinh}\left( \ma L\right) +2\ma\ze \mathrm{cosh}\left(\ma L \right) \mathrm{,}
\label{eq:ex}
\end{equation}
where the extrapolation lengths are equal to
\begin{equation}
\zeo \left(\lt\right)=\zet \left(\lt\right)=\ze \left(\lt\right)= \frac{2}{3} \lt \frac{1+\overline{R_{1,2}}}{1-\overline{R_{1,2}}}\mathrm{.}
\label{eq:ex1}
\end{equation}
Here $\zp$ is the diffuse penetration depth of light, $\ma \equiv 1/\la$ the inverse absorption mean free path, $\overline{R_{1,2}}$ is the angular and polarization averaged reflectivity of the respective boundaries <cit.>. For a normal incident collimated beam the penetration depth becomes $\zp=\lt$ <cit.>, and $\overline{R_{1,2}}$=0.57 for polymer plates with an average refractive index $n=1.5$ <cit.>.
For samples with no absorption ($\ma = 0$), Eq. (<ref>) simplifies to the optical Ohm's law <cit.>
\begin{equation}
T\left( L, \lai , \lt\right)=\frac{\lt +\ze}{L+2\ze}\mathrm{.}
\label{eq:tr1}
\end{equation}
In the range of zero phosphor absorption the total transmission is a function of the sample thickness $L$, the incoming wavelength $\lai$, and the transport mean free path $\lt$, $T=T(L,\lai , \lt)$. Therefore, we can extract $\lt$ using Eq. (<ref>) from measurements of the total transmission $\T$ in the range of no absorption $(\lai\geq \lar)$. In the range of strong phosphor absorption $(\lai\leq\lar)$ the total transmission also depends on the absorption mean free path $\la$: $T=T(L,\lai , \lt, \la)$. Therefore the transport mean free path $\lt$ has to be derived separately, which we can do by extrapolating the $\lt$ values extracted in the zero absorption wavelength range $(\lai\geq\lar)$ into the strong absorption wavelength range, since $\lt$ is a monotoneously increasing function of $\lambda$ for size-polydisperse scatterers <cit.>. By measuring the total transmission in the range of strong absorption, we thus obtain $\la$ using extrapolated values of $\lt$, by using Eq. (<ref>).
§ EXPERIMENTAL DETAILS
We have studied the light transport through polymer plates that are used in Fortimo solid-state lighting units <cit.>. The polymer plates consist of a polycarbonate matrix (Lexan 143R) with $\Y$ ceramic phosphor particles that are widely used in white LEDs. The phosphor particles have a broad size distribution with center around 10$\ \mu$m <cit.>, and a $\mathrm{Ce^{3+}}$ concentration in the $\Y$ of 3.3 wt $\%$.
The emission and absorption spectra of the $\Y$ in powder form, which was used in the polymer diffusion plates are shown in Fig. <ref>(a). The absorption and emission bands have peaks at 458 nm and 557 nm, respectively, and overlap in the spectral range between $\lal$=490 nm and $\lar$=520 nm. As a result, we distinguish three spectral ranges in the visible spectrum where different physical processes are taking place: (1) in the spectral range up to $\lal=$490 nm, light is partly elastically scattered and partly absorbed. This range has already been studied in Ref. Leung14. (2) In the spectral range between $\lal=$490 nm and $\lar=$520 nm, externally incident light is elastically scattered and absorbed, while light is also emitted by the internal phosphor, and subsequently elastically scattered. This is the overlap range that is central to the present work. (3) In the spectral range beyond $\lar=$520 nm, output light is the sum of externally incident light that is elastically scattered and of internally emitted light by the phosphor that is also elastically scattered. This range has also already been studied in Ref. Leung14.
Here we present the transmission measurements on five such polymer plates with a phosphor concentration ranging from $\phi=2.0$ to $4.0$ wt$\%$. The corresponding volume fractions range from $\phi=0.5$ to $1$ vol$\%$, which is in the limit of low scatterer concentration. The plates were prepared using injection molding where a powder of $\Y$ particles is mixed with the polymer powder and the mixture is melted and pressed into a press-form. The polymer plates, shown in Fig. <ref>(a), are circular with a diameter of 60 mm and a thickness of 2 mm.
Figure <ref>(b) shows a drawing of the setup for measuring the spectrally resolved total transmission $\Tt$ with a narrowband incident light beam. We illuminate the sample with two different light sources: a tunable narrowband light source and a broadband light source. The narrowband light source consists of a Fianium supercontinuum white-light source (WL-SC-UV-3), which is spectrally filtered to a bandwidth of less than $\Delta\lambda=$2.4 nm by a prism monochromator (Carl-Leiss Berlin-Steglitz). The wavelength of the narrowband source is tunable in the wavelength range between 400 and 700 nm, as shown in Fig <ref>.
The infrared part above 700 nm of the supercontinuum laser source is filtered by a neutral density filter (NENIR30A) and a dichroic mirror (DMSP805). The spectrum of the supercontinuum source after filtering the infrared light is shown in Fig. <ref>. The spectrum vary drastically in intensity at different wavelengths.
The incident beam illuminates the phosphor plate at normal incidence and the plate is placed at the entrance port of an integrating sphere. We verified that the entrance port of the integrating sphere is sufficiently large to collect all intensities that emanated from the strongest scattering sample. The intensity of the outgoing light entering the integrating sphere is analyzed with a fiber-to-chip spectrometer (AvaSpec-USB2-ULS2048L) with a spectral resolution of $\Delta\lambda=$2.4 nm.
An example of the measured spectra for three incident wavelengths $\lai$=527, 550 and 634 nm are shown in Fig. <ref> with the red squares and red dotted lines. The peak intensity varies significantly across the spectral range as a result of the spectral variation of the supercontinuum source. The integration time is changed at every wavelength during measurements to maintain a fixed reference intensity.
The alternative broadband light source consist of a white LED (Luxeon LXHL-MW1D) (see Fig. <ref>) with an emission spectrum covering the range from 400 nm to 700 nm as shown in Fig. <ref>. This source is not filtered.
fig:setSetup11.pdfNarrowband measurement setup. (a) A polymer slab with a 4 wt$\%$ $\Y$ compared to a $\texteuro{1}$ coin. (b) Supercontinuum white light source Fianium, NDF: neutral density filter, DM: dichroic mirror, L1: achromatic doublet (AC080-010-A-ML, f=10 mm), L2: achromatic doublet (f=50 mm), M: mirror, I: integrating sphere, S: spectrometer, P: prism spectrometer ($f_{\sharp} = 4.6$).
fig:spspec11.pdfNormalized reference spectra of the light sources used in the experiment. Blue circles - normalized reference spectrum of the supercontinuum source (SS) after being filtered by a neutral density filter and a dichroic mirror that removes the infrared part of the spectra. Black squares - normalized reference spectrum of a broadband source (BS) that was not filtered. Red squares - spectrally filtered narrowband light source (NS) normalized to the initial BS spectrum represent. We show narrowband spectra for three incident wavelengths $\lai=486,551,634$ nm. Intensity normalized to the SS.
For all phosphor plates, we measured the transmission spectra with both the narrowband and the broadband light source to check the consistency in the spectral range where both methods can be applied ($\lambda\leq\lal$ and $\lambda\geq\lar$).
For the broadband light source the total transmission is obtained by normalizing the measured spectrum $I(\lambda )$ to a reference spectrum $\Io$ measured in the absence of a sample. For the narrowband light source the total transmission is determined as the ratio of the transmitted intensity $I(\lai )$ and a reference intensity $I_0 (\lai )$ without the sample at the designated wavelength $\lai$ that is set with the monochromator. The total transmission is reproducible to within a few percent points on different measurements with different light sources.
§ RESULTS
§.§ Transmission measurements
In order to measure the total transmission $\T$ of the polymer plates we tune the narrowband light source to an incident wavelength $\lai$, and measure the transmitted intensity of the outgoing light. In Fig. <ref> we show the normalized transmitted intensity for three incident wavelengths $\lai$. For an incident wavelength $\lai$=490 nm, we see a pronounced peak with the maximum at $\lambda$=490 nm. This peak contains mostly elastically scattered photons, because inelastically scattered photons are Stokes shifted to longer wavelengths. Indeed between 500 nm and 650 nm the intensity profile reveals a broad peak that represents the Stokes shifted intensity, since the intensity profile has a shape similar to that of the $\Y$ in powder form (black dashed curve). Both emission spectra have a maximum at 557 nm. For $\Y$ in powder form the intensity is slightly higher at longer wavelengths $\lambda > 600$ nm than for the phosphor plate.
The peak value of the Stokes shifted light amounts to 3 $\%$ of the transmitted intensity at the incident wavelength $\lai$=490 nm. We calculated the amount of Stokes shifted light $\Ie$ in the elastic peak at $\lai$=490 nm by extrapolating the emission curve to this wavelength range. We find that $\Ie$ amounts to less than 1 $\%$ of the transmitted intensity I($\lambda$), and can be neglected safely. The total Stokes shifted intensity decreases drastically for longer incident wavelengths $\lai$ inside the overlap range, and can thus be neglected too. For incident light in the middle of the overlap range at $\lai$=505 nm, we see that the emitted intensity is even weaker with a normalized intensity $\Ie$ less than 1$\%$ of I($\lambda$). At the edge of the absorption band at the incoming wavelength $\lai$=520 nm we observe an even smaller Stokes shifted intensity $\Ie$. The reason for this decrease is that the absorption cross section decreases drastically in the overlap range, and only a very little amount of light is being absorbed, and as a result is Stokes shifted. Throughout the overlap range the contribution from the Stokes shifted light $\Ie$ is less then 1 $\%$ of I($\lambda$) at the incident wavelength $\lai$. We thus conclude that the Stokes shifted intensity contribution can be neglected throughout the overlap range. Therefore, we have distinguished the elastically scattered (or absorbed) fraction of light from the Stokes shifted light. As a result we can now measure total transmission at any desired wavelength.
We have scanned the incident wavelength $\lai$ through the wavelength range of interest. In Fig. <ref>(c) the total transmission for the slab with 4 wt$\%$ of $\Y$ has been obtained from these scans (red). The total relative intensity $\Tr$ measured with the broadband source is also shown in Fig. <ref>(c) for comparison (blue). For short wavelengths, both transmission spectra coincide within a few percent. The blue spectrum reveals a deep trough with a minimum at 458 nm. The trough matches well with the peak of the absorption band of $\Y$ in Fig. <ref>(a), and reveals that a significant fraction of the light in this wavelength range is absorbed by the phosphor. At wavelengths longer than $\lar$=520 nm both transmission spectra are flat, but the spectrum measured with the broad band light source is 10 $\%$ larger than the spectrum measured with the narrowband light source. The blue triangle spectrum contains a significant contribution of the Stokes shifted light $\Ie$ in this spectral range, as most of the emission occurs in this spectral range (see Fig. <ref>(a)). The narrowband spectrum on the contrary does not have this contribution. The difference in transmission between these two spectra at long wavelengths is equal to $\Tee$, in Eq. (<ref>). In the overlap region ($\lal=490$ nm$<\lambda<\lar=520$ nm) both spectra reveal a sharp rise. The total relative intensity spectrum deviates from the total transmission spectrum due to the contribution of the Stokes shifted light $\Ie$ in this wavelength range. In Ref. Leung14 it was not possible to separate elastically scattered and Stokes shifted light.
Finally, we filtered the broadband light source with a longpass filter at $\lambda$=520 nm, which ensures that the phosphor is not excited (we thus have zero emitted intensity $\Tee$=0). Hence, the measured total relative intensity equals the total transmission $\Tr=\T$ in this spectral range. In Fig. <ref>(b) we compare the total transmission measured with the filtered broadband light source (green), and the total transmission measured with the narrowband light source (red). The total transmission measured with the narrowband light source agrees within a few percent points with the total transmission measured with the filtered broadband light source (see Fig. <ref>(c)) for $\lai\geq\lar$. In summary, we have for the first time extracted the total transmission $\T$ for a diffuser plate with phosphor in the whole visible range, including the previously inaccessible <cit.> overlap range.
fig:emem_111.pdfIntensity profile of the signal that we measure in the range where emission and absorption overlap for three different pump wavelengths. The orange dashed line is the emission spectra from Fig. <ref>(a). The blue curve was normalized to 7791 counts and the green and the red curves to 15480 and 20297 counts respectively.
§.§ Transport mean free path
fig:tr2trm_211.pdfTransport mean free path. (a) Transport mean free path as a function of wavelength for different phosphor concentrations. Dashed lines represent linear fits to the measured data with parameters listed in Tab. <ref>. (b) Inverse transport mean free path as a function of concentration for two different incoming wavelengths. Dashed lines are linear fits to the data.
In the range of low absorption we have extracted the transport mean free path from the transmission data using Eq. <ref> and plotted the result in Fig. <ref>(a). We see that the transport mean free path increases linearly with wavelength at constant phosphor concentration. In highly polydisperse non-absorbing media, a similar relationship between the transport mean free path and wavelength was found and interpreted <cit.>. Therefore, we have fitted the transport mean free path with a line for every phosphor concentration. Parameters of the linear fits are listed in appendix. We linearly extrapolate $\lt$ to the absorption range $\lambda\leq\lar=520$ nm, and use the extrapolated values of the transport mean free path $\lt$ to obtain the absorption mean free path $\la$ in the range of strong absorption.
C (wt$\%$) a b
2 -0.57 $\pm$ 0.07 0.00358 $\pm$ 0.00001
2.5 -0.55$\pm$ 0.05 0.00305 $\pm$ 0.00008
3 -0.43$\pm$ 0.03 0.00225 $\pm$ 0.00055
3.5 -0.47$\pm$ 0.02 0.00214 $\pm$ 0.00038
4 -0.54$\pm$ 0.02 0.00218$\pm$ 0.00038
Parameters of the linear models of the transport mean free path versus wavelength: $\lt = a + b\lambda$. The parameters $a$ and $b$ depend on the phosphor concentration $C$, and are shown with their standard errors.
In the limit of low concentration each scatterer can be treated independently. In this case $1/\lt$ is proportional to the concentration of the scatterers <cit.>. Indeed in Fig. <ref>(b) we see that the transport mean free path increases inversely proportional with the increasing phosphor concentration at a fixed wavelength. We observe that with increasing wavelength the scattering cross section increases similarly to what was obtained earlier <cit.>.
fig:trmT_labs111.pdfDetermination of the absorption mean free path. (a) Look-up tables presented in a form of plots for the Eq. <ref> for three different wavelengths, and 4 wt$\%$ concentration of the phosphor. ($\lt$=0.40;0.44;0.57 $\mathrm{mm}$ respectively) (b) Absorption mean free path extracted from the look-up tables for the plate with 4 wt$\%$ phosphor concentration. Red circles and red dashed lines indicate the process of mapping $\ma$ from look-up table to the Fig. <ref>(b). Blue triangles indicate the inverse absorption length measured in Ref. <cit.>
§.§ Absorption mean free path
By using the transport mean free path $\lt$ extrapolated to the wavelength range between 400 and 530 nm where the phosphor absorbs light, we now derive the absorption mean free path $\la$ from the measured total transmission (see Fig. <ref>(c)). Since we do not have analytic inverse function of Eq. (<ref>) , we have solved the inversion numerically and made look-up tables for each phosphor concentration and at each wavelength. Fig. <ref>(a) shows three inverted curves of $\ma$ versus total transmission for three different wavelengths ($\lambda=$430, 450, 510 nm) at a phosphor concentration $C$=4 wt$\%$. We plot $\ma$ - rather then $\la$ - since this quantity tends to zero for vanishing absorption.
Fig. <ref>(a) shows that $\ma$ (and thus the absorption) increase. In the limit of strong absorption, all total transmission curves tend to zero. In the limit of vanishing $\ma$, the total transmission equals the (extrapolated) values that decrease with decreasing wavelength (Fig. <ref>(a)). The vertical dashed lines indicate the measured total transmission, and the intersections with the curves yield the corresponding $\ma$ for each wavelength at this phosphor concentration.
Fig. <ref>(b) shows the extracted absorption profile $\ma$ for the polymer plate with the highest phosphor concentration $C$=4 wt$\%$ studied here. The FWHM of this curve is 64.5 nm. The dashed purple line in Fig. <ref>(b) indicates the inverse thickness of the sample. The absorption mean free path $\la$ is shorter than the thickness of the sample $L$ between 418 and 501 nm. This means that incident light is effectively absorbed in the volume of the sample, and the density of the phosphor is optimized for use in a white-light LED. At the edges of the absorption range at 400 and 530 nm $\ma$ tends to zero, as expected from the known absorption (Fig. <ref>(a)). We note that our present $\ma$ values differ from previous results obtained on the same samples <cit.> (see Fig. <ref>(b)).
The absorption mean free path varies significantly with wavelength in our case. We notably attribute the difference to the use of an incorrect diffusion equation in Ref. Leung14. The spectral shape obtained at present is in much better agreement with the phosphor absorption spectra than previously, which is gratifying.
In Fig. <ref>(a) we have plotted $\ma(\lambda)$ for three wavelength $\lambda$ = 430, 460 and 500 nm as a function of phosphor concentration. We see that $\ma(\lambda)$ increases linearly with increasing phosphor concentration, which agrees with the assumption that each absorber is independent. Figure <ref>(a) shows that the steepest slope appears at the wavelength $\lambda$=460 nm, which corresponds to the peak of the absorption curve. The maximum absorption cross section is $\sigma_{\mathrm{abs}}=$30 $\mu m^{2}$, which agrees reasonably well with the typical measured absorption cross section for $\Y$ of the order of 10 $\mu m^{2}$ (Ref. Liu10,Kaczmarek99,Zhao03,Mares07,Mihokova07,Kucera08).
Figure <ref>(b) shows the extracted normalized absorption spectrum of $\ma$ for the polymer plates with different phosphor concentrations. All absorption curves are normalized to their maximum, and they coincide with each other within a few percent points. Absorption mean free path $\la$ scales linearly with the phosphor concentration. All absorption curves tend to zero outside $400<\lambda<530$ nm.
fig:trm1abs_con1.pdfAbsorption mean free path. (a) $\ma$ as a function of concentration is shown for the three different wavelength. (b) Line shapes of the absorption coefficient is shown for the phosphor in powder (dot-dashed black line) and in polymer plates with different phosphor concentrations.
In Fig. <ref>(b) we also compare the shape of the absorption curve of the phosphor in powder form that was used to manufacture the samples to $\ma$ curves. The black dot-dashed line shows absorption spectra of the $\Y$ in powder. These two sets of data were normalized to their maxima, so the positions of maximums of these two graphs coincide. The $\ma$ spectrum appear to have broader tails compared to the $\Y$ in powder form. The absorption spectra of $\Y$ in powder has a FWHM=54 nm, that is 10 nm smaller than the FWHM of the measured absorption spectra. One possible reason is that light is multiply internally reflected in the $\Y$ particles <cit.>, so particles can not be treated as an independent point scatterers.
Finally, let us place our approach in the context with previous work: Vos et al. reported the transport properties measurements in $\mathrm{TiO}_{2}$ scattering plates using a broadband light source <cit.>. They showed that the transport mean free path $\lt$ linearly depends on the wavelength in the visible wavelength range. This method can not be applied to the $\Y$ plates, as these plates have ranges with strong absorption, emission and an overlapping range. Leung et al. <cit.> reported the transport properties measurements in $\Y$ plates using a filtered broadband light source, where the linear dependence of $\lt$ was exploited to calculate $\la$. Initially, the approximation used to analyze total transmission $\T$ from Ref. <cit.> was used outside its range of validity. Secondly, the described method is limited to the region of strong absorption or emission, but not in the overlap region.
In this paper we have been able to separate light of the same wavelength yet originated from different physical processes occurring in the polymer plates of the solid-state light units. The separate measurement of elastically scattered and Stokes shifted light allowed us to extract the transport and absorption mean free path of the given polymer plates, which are the important parameters required for modeling and predicting the color spectra of solid-state lighting devices. The optimal parameters of the solid-state lighting units can be directly extracted from the measured absorption curves.
We vary the thickness of the plate $L$ or the phosphor concentration depending on the desired level of pump absorption using absorption curve in Fig. <ref>(b).
§ SUMMARY AND OUTLOOK
We have developed a new technique to measure the light transport of white-light LED plates in the visible range based on narrowband illumination and spectrally sensitive detection. We compare the data obtained with the new technique to the data measure with the broadband light source. The two sets of data coincide in the range without absorption $\lambda\geq\lar$. We extracted the total transmission in the overlap range, that was previously inaccessible.
We used diffusion theory to extract transport $\lt$ and absorption $\la$ mean free paths from these data in the previously inaccessible range. The shape of the absorption coefficient measured for the $\Y$ powder, and $\Y$ powder in polymer matrix have similar trends. Although for the polymer plates the curve is broader then for the $\Y$ powder. Both $\ma$ and $1/\lt$ are proportional to the concentration of phosphor, which reveals that elastic and inelastic processes do not influence each other.
By exploiting narrow band light source and interpreting the resulting total transmission by diffusion theory, we are able to extract for the first time light transport parameters for white LEDs in the whole visible wavelength range. However, theory only gives an analytical solution for simple sample geometries, such as a slab, a sphere, or a semi-infinite medium. Therefore, to efficiently model a real white LED with a complex geometry we must in the end supplement an $ab$ $initio$ theory with a numerical method, such as ray-tracing.
We would like to thank Cornelis Harteveld for technical support, Vanessa Leung for contribution early on in the project and Teus Tukker, Oluwafemi Ojambati, Ravitej Uppu, Diana Grishina for discussions. This work was supported by the Dutch Technology
Foundation STW (contract no. 11985), and by FOM and NWO, and the ERC (279248).
[1]In Ref. Garcia92 an approximation was used, that is only valid for very small absorption. This approximation lead to the slightly different result in Ref. Leung14. Here we use exact expression and our results are valid for very high $\ma$.
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1511.00129
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71.15.Dx, 71.15.Qe, 73.22.-f
Calculating the quasiparticle (QP) band structure of two-dimensional (2D) materials within the GW self-energy approximation has proven to be a rather demanding computational task. The main reason is the strong $\mathbf{q}$-dependence of the 2D dielectric function around $\mathbf{q} = \mathbf{0}$ that calls for a much denser sampling of the Brillouin zone than is necessary for similar 3D solids. Here we use an analytical expression for the small $\mathbf{q}$-limit of the 2D response function to perform the BZ integral over the critical region around $\mathbf{q} = \mathbf{0}$. This drastically reduces the requirements on the $\mathbf{q}$-point mesh and implies a significant computational speed-up. For example, in the case of monolayer MoS$_2$, convergence of the $G_0W_0$ band gap to within $\sim\SI{0.1}{eV}$ is achieved with $12\times 12$ $\mathbf{q}$-points rather than the $36\times 36$ mesh required with discrete BZ sampling techniques. We perform a critical assessment of the band gap of the three prototypical 2D semiconductors MoS$_2$, hBN, and phosphorene including the effect of self-consistency at the GW$_0$ and GW level. The method is implemented in the open source GPAW code.
Center for Atomic-scale Materials Design (CAMD), Department of Physics and Center for Nanostructured Graphene (CNG), Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark
§ INTRODUCTION
The past few years have witnessed an explosion in research on atomically thin two-dimensional (2D) materials. Of particular interest are the 2D semiconductors including the family of transition metal dichalcogenides, which have been found to exhibit a number of unique opto-electronic properties<cit.>. For understanding and predicting these properties the electronic band structure of the material is of fundamental importance. The GW method<cit.>, introduced by Hedin<cit.> in 1965 and first applied to real solids in an ab-initio framework by Hybertsen and Louie<cit.> and Godby, Sham, and Schlüter<cit.>, has become the “gold standard” for calculating quasi-particle (QP) band structures. Over the years its performance has been thoroughly established for bulk materials<cit.> and more recently also for molecules<cit.>. In comparison, critical assessments of the accuracy and numerical convergence of GW calculations for 2D materials are rather scarce<cit.>. Nevertheless these studies have shown that (i) it is essential to use a truncated Coulomb interaction to avoid long range screening between periodically repeated layers which reduces the QP band gap, and (ii) when a truncated Coulomb interaction is used, the convergence of the GW calculation with respect to the number of $\mathbf{k}$-points becomes much slower than is the case for similar bulk systems.
The slow $\mathbf{k}$-point convergence of the GW band structure is directly related to the nature of electronic screening in 2D which is qualitatively different from the well known 3D case.<cit.> Specifically, while the dielectric function, $\varepsilon(\mathbf{q})$, of bulk semiconductors is approximately constant for small wave vectors, the dielectric function of a 2D semiconductor varies sharply as $\mathbf{q}\to\mathbf{0}$.<cit.> As a consequence, the number of $\mathbf{q}$-points required to obtain a proper sampling of the screened interaction $W(\mathbf{q})$ over the Brillouin zone (BZ) is much higher for the 2D material than what would be anticipated from the 3D case. For example, the band gap of bulk MoS$_2$ is converged to within $\sim\SI{0.1}{eV}$ with an in-plane $\mathbf{k}$-point grid of $12\times 12$ while the same accuracy for monolayer MoS$_2$ requires a grid of $36 \times 36$ when standard BZ sampling schemes are applied.
Importantly, supercell calculations only describe the characteristic features of screening in 2D materials correctly when the unphysical screening between the periodically repeated layers is removed, e.g. using a truncated Coulomb interaction. Without this, the dielectric function does not approach unity for $\mathbf{q}\to 0$ and the band gap can be significantly underestimated (by around 0.5eV for MoS$_2$ with 10Å vacuum<cit.>) as a result of over screening. Since in this case, the screening is really intermediate between 3D and 2D, the GW gap shows faster convergence with $k$-points than is observed using a truncated Coulomb interaction. This is presumably the reason why most earlier GW calculations for 2D semi-conductors have been performed with $k$-point grids ranging from $6\times 6$ to $15\times 15$ which are much too coarse for describing the truly isolated 2D material.
Here we show that the slow $\mathbf{k}$-point convergence of the GW self-energy in 2D materials can be alleviated by performing the BZ integral of $W(\mathbf{q})$ analytically in the critical region around $\mathbf{q} = \mathbf{0}$ where $\varepsilon(\mathbf{q})$ varies most strongly. The analytical expression for $W(\mathbf{q})$ is obtained from a lowest order expansion in $\mathbf{q}$ of the head, $\chi^0_\mathbf{00}(\mathbf{q})$, and wings, $\chi^0_\mathbf{0G}(\mathbf{q})$, of the non-interacting density response function. This simple scheme reduces the required number of $\mathbf{q}$-points by an order of magnitude without loss of accuracy.
§ THE GW SELF-ENERGY
We split the GW self-energy into the exchange and correlation part, respectively. The former does not present particular problems in 2D materials and is calculated using a Wigner-Seitz truncated Coulomb interaction as described elsewhere<cit.>. In a plane wave expansion the correlation part of the self-energy, evaluated for an electronic state $\ket{n\mathbf{k}}$, takes the general form<cit.>
\begin{equation}
\begin{split}
\bra{n\mathbf{k}} \Sigma^c(\omega) \ket{n\mathbf{k}} = \frac{1}{(2\pi)^3}\int_\text{BZ} d\mathbf{q} \sum_{\mathbf{GG}'} \frac{i}{2\pi} \int_{-\infty}^\infty d\omega' \overline{W}_{\mathbf{GG}'}(\mathbf{q}, \omega') \\
\times \sum_m \frac{[\rho_{n,\mathbf{k}}^{m,\mathbf{k} + \mathbf{q}}(\mathbf{G})] [\rho_{n\mathbf{k}}^{m,\mathbf{k}+\mathbf{q}}(\mathbf{G}')]^*}{\omega + \omega' - \epsilon_{m\mathbf{k} + \mathbf{q}} - i\eta\,\text{sgn}(\epsilon_{m\mathbf{k} + \mathbf{q}}-\mu)}, \label{eq:self-energy}
\end{split}
\end{equation}
where $m$ runs over all electronic bands, $\epsilon_{m\mathbf{k}+\mathbf{q}}$ are single particle energies, $\mu$ is the chemical potential and $\eta$ is a broadening parameter. The pair densities are defined as $\rho_{n\mathbf{k}}^{m\mathbf{k} + \mathbf{q}}(\mathbf{G}) = \bra{n\mathbf{k}} e^{i(\mathbf{G} + \mathbf{q}) \cdot \mathbf{r}} \ket{m\mathbf{k} + \mathbf{q}}$, and $\overline{W}_{\mathbf{GG}'}(\mathbf{q}, \omega)$ is the correlation part of the dynamical screened potential given by
\begin{equation}
\overline{W}_{\mathbf{GG}'}(\mathbf{q}, \omega) = v_\mathbf{G}(\mathbf{q}) \left[\varepsilon^{-1}_{\mathbf{GG}'}(\mathbf{q}, \omega) - \delta_{\mathbf{GG}'}\right], \label{eq:screened-potential}
\end{equation}
where $v_\mathbf{G}(\mathbf{q})=4\pi/|\mathbf{G}+\mathbf{q}|^2$ is the Coulomb interaction. In most implementations the BZ integral is evaluated numerically with a standard quadrature method using a regular $\mathbf{q}$-point grid matching the $\mathbf{k}$-point grid of the ground state DFT calculation. Since the screened potential, Eq. (<ref>), diverges for $\mathbf{G} = 0,\;\mathbf{q} = \mathbf{0}$ (for bulk materials) this point is generally handled separately, so the integral may be written
\begin{equation}
\int_\text{BZ} d\mathbf{q} \mathcal{S}(\mathbf{q}, \omega) \to \frac{\Omega}{N_\mathbf{q}}\sum_{\mathbf{q}_n \neq \mathbf{0}}\mathcal{S}(\mathbf{q}_n, \omega) + \int_{\Omega_0} d\mathbf{q} \mathcal{S}(\mathbf{q}, \omega), \label{eq:sigma_num_int}
\end{equation}
where $\mathcal{S}(\mathbf{q}, \omega)$ denotes the entire integrand, $\Omega$ is the volume of the BZ, $N_\mathbf{q}$ is the total number of $\mathbf{q}$-points in the grid and $\Omega_0$ denotes a small region around $\mathbf{q} = \mathbf{0}$. For bulk systems $\Omega_0$ is normally defined as a sphere centered at $\mathbf{q} = \mathbf{0}$. For a 2D material, the BZ integral in Eq. (<ref>) reduces to a 2D integral with in-plane sampling of $\mathbf{q}$, and $\Omega_0$ represents an area. We now focus on how to calculate the contribution to the integral around the special point $\mathbf{q} = \mathbf{0}$ in the 3D versus the 2D case.
Within the random phase approximation (RPA) the dielectric matrix is given by
\begin{equation}\label{eq:epsilon}
\varepsilon_{\mathbf{GG}'}(\mathbf{q}, \omega) = \delta_{\mathbf{GG}'} - v_\mathbf{G}(\mathbf{q})\chi^0_{\mathbf{GG}'}(\mathbf{q}, \omega).
\end{equation}
For a solid with a finite band gap it can be shown that the head of the non-interacting response function $\chi^0_\mathbf{00}(\mathbf{q},\omega) \propto q^2$ for small $q$<cit.>. Since $v_\mathbf{0}(\mathbf{q}) = 4\pi/q^2$ it follows that in 3D the head of the dielectric function $\varepsilon_\mathbf{00}(\mathbf{q}, \omega)$ converges to a finite value $>1$ when $q\to 0$. Moreover, this value is typically a reasonable approximation to $\varepsilon_\mathbf{00}(\mathbf{q}, \omega)$ in a relatively large region around $\mathbf{q} = \mathbf{0}$. This means that in the BZ integration in Eq. (<ref>) around the singular point $\mathbf{G}=\mathbf{G}’=\mathbf{q}=\mathbf{0}$ all factors, except $4\pi/q^2$, can be assumed to be constant and the integral can be performed analytically over a sphere centred at $\mathbf{q} = \mathbf{0}$<cit.> or numerically on a fine real space grid<cit.>.
For GW calculations of 2D materials performed with periodic boundary conditions in the out-of-plane direction, the direct use of Eq. (<ref>) leads to significant overscreening due to the interaction between the repeated images<cit.>. One way of dealing with this is to subtract the artificial interlayer contribution calculated from a classical electrostatic model<cit.>. A more direct way of avoiding the spurious interactions is to truncate the Coulomb potential in the direction perpendicular to the layers. Thus in Eqs. (<ref>) and (<ref>), $v_\mathbf{G}(\mathbf{q})$ should be replaced by<cit.>
\begin{equation}\label{eq:2Dcoulomb}
v^\text{2D}_\mathbf{G}(\mathbf{q}_\parallel) = \frac{4\pi}{|\mathbf{q}_\parallel + \mathbf{G}|^2} \left[ 1 - e^{-|\mathbf{q}_\parallel + \mathbf{G}_\parallel| L / 2} \cos(|G_z| L / 2) \right],
\end{equation}
where only in-plane $\mathbf{q}$ are considered. $L$ is the length of the unit cell in the non-periodic direction and the truncation distance is set to $L/2$, which simplifies the expression. In the long wavelength limit, $\mathbf{G} = \mathbf{0}$, $\mathbf{q}_\parallel \to \mathbf{0}$, the truncated interaction becomes $v^\text{2D}_\mathbf{0}(q_\parallel)\approx \frac{2\pi L}{q_\parallel}$. We see that the $\mathbf{q}=\mathbf{0}$ divergence in the truncated Coulomb potential is reduced by a power of $\mathbf{q}$ compared to that of the full Coulomb interaction. As will be shown in the following this has important consequences for the form of the screened interaction. However, before presenting the form of the screened interaction of a 2D semiconductor evaluated using the full expression for the response function and truncated Coulomb interaction, it is instructive to consider a simplified description of the 2D material.
Let us consider a strict 2D model of a homogeneous and isotropic semiconductor. In the small $\mathbf{q}$ limit, the density response function takes the form $\chi_0(\mathbf{q})=\alpha_\text{2D} q^2$ where $\alpha_\text{2D}$ is the 2D polarizability<cit.>. Using that the 2D Fourier transform of $1/r$ equals $2\pi/q$, the leading order of the dielectric function becomes
\begin{equation}\label{eq:2Deps}
\varepsilon^{2D}(\mathbf{q}) \approx 1 + 2\pi \alpha_\text{2D} q.
\end{equation}
(Color online) Static macroscopic dielectric functions of a representative set of 2D semiconductors as a function of $\mathbf{q}$ along the $\Gamma \to M$ direction for the hexagonal structures and along the path from $\Gamma$ to $X$ or $Y$ in the case of phosphorene.
Some examples of macroscopic dielectric functions for a representative set of 2D semiconductors are shown in Figure <ref> (see Ref. huser_how_2013 for a precise definition of this quantity). The linear form (<ref>) is clearly observed in the small $\mathbf{q}$ regime. Importantly, if we use the same strategy for evaluating the BZ integral in Eq. (<ref>) as in 3D, i.e. assuming $\epsilon^{-1}(\mathbf{q})$ to be a slowly varying function and evaluating it on the discrete $\mathbf{q}$-point grid, we obtain zero contribution for the $\mathbf{q} = 0$ term because $1/\epsilon^{2D}-1=0$ for $\mathbf{q}=0$, see Eq. (<ref>). On the other hand, it is clear that the screened interaction takes the form $\overline{W}^{2D}(\mathbf{q}) = -4\pi^2 \alpha_\text{2D}/(1 + 2\pi \alpha_\text{2D} q)$ for small $\mathbf{q}$. In particular, $\overline{W}^{2D}(\mathbf{q})$ takes a finite value for $\mathbf{q}=0$ which is qualitatively different from the 3D case where $\overline{W}(\mathbf{q})$ diverges for $\mathbf{q} \to \mathbf{0}$.
In Appendix <ref> we show, following an analysis similar to that of Ref. freysoldt_dielectric_2007 adapted to the case of a truncated Coulomb interaction, that for a general non-isotropic 2D material, the small $\mathbf{q}_\parallel$ limit of the head of the screened potential takes the form
\begin{equation}
\begin{split}
\overline{W}_\mathbf{00}(\mathbf{q}_\parallel) ={}& - \left(\frac{4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2})}{|\mathbf{q}_\parallel|} \right)^2 \\
& \times \frac{\hat{\mathbf{q}_\parallel} \cdot \mathsf{A}\hat{\mathbf{q}_\parallel}}{1 + 4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2}) \hat{\mathbf{q}_\parallel} \cdot \mathsf{A} \hat{\mathbf{q}_\parallel}},\label{eq:exact}
\end{split}
\end{equation}
where $\hat{\mathbf{q}_\parallel} = \mathbf{q}_\parallel / |\mathbf{q}_\parallel|$ and $\mathsf{A}$ is a second rank tensor which also depends on the frequency. We see that we have $\overline{W}_\mathbf{00}(\mathbf{q}_\parallel = \mathbf{0}) = -(2\pi L)^2 \hat{\mathbf{q}_\parallel} \cdot \mathsf{A} \hat{\mathbf{q}_\parallel}$. The expression $\hat{\mathbf{q}_\parallel} \cdot \mathsf{A}\hat{\mathbf{q}_\parallel}$ is closely related to the slope of the dielectric function and the 2D polarizability but includes local field effects. In addition to Eq. (<ref>) there are similar expressions for the wings and body of the screened interaction, see Eq.(<ref>) to Eq.(<ref>). These expressions must be integrated over the $\Omega_0$-region, that we now define as the primitive cell in the 2D BZ that surrounds the $\mathbf{q}_\parallel =\mathbf{0}$ point. The expression is simplified to one that can be integrated analytically as shown in Appendix <ref>.
(Color online) The head of the static component of the screened potential (subtracted the bare interaction) of monolayer a) h-BN b) MoS$_2$ and c) phosphorene as a function of $\mathbf{q}$ along the $\Gamma\to M$ direction or $\Gamma\to X$ and $\Gamma\to Y$ in the case of phosphorene. The crosses are the numerical values obtained on a fine $\mathbf{q}$-point grid while the circles or triangles represent the values obtained on a coarse $\mathbf{q}$-point grid. The bars represent a simple numerical approximation to the BZ integral of $\overline{W}_{00}(\mathbf{q})$ performed on the coarse $\mathbf{q}$-point grid. The value of the screened potential for $\mathbf{q} = \mathbf{0}$ is set to the analytical result Eq. (<ref>). The full curve represents the analytical small $\mathbf{q}$ approximation, Eq. (<ref>), and the hatched area shows its contribution to the integral.
The full expression for $\overline{W}$ in Eq. (<ref>) is therefore evaluated numerically on a discrete sub-grid, constructed as a Monkhorst-Pack grid within $\Omega_0$, and the simplified expression in Eq. (<ref>) is only used for $\mathbf{q}_\parallel = \mathbf{0}$ on the sub-grid. The limit of the integral is now given by the radius $r_{\Omega_0}$, defined as $\pi r^2_{\Omega_0} = \Omega_0 / N_{q_0}$, where $N_{q_0}$ is the number of grid points in the sub-grid. This approach ensures a smooth evaluation of $\overline{W}$, that converges fast with $N_{q_0}$. It is found to be necessary to have $N_{q_0}\approx 100$ when $\mathbf{q}_\parallel = \mathbf{0}$ is evaluated using Eq. (<ref>) for both iso- and anisotropic materials where as $N_{\mathbf{q_0}}\approx 10^5$ is needed if the analytical correction at $\mathbf{q}_\parallel = \mathbf{0}$ is omitted.
§.§ Results
To investigate how this method performs we have carried out test calculations for the three monolayers h-BN, MoS$_2$ and phosphorene, which have quite different dielectric functions as seen on Figure <ref>. h-BN is a large gap dielectric with low screening ability leading to a small slope of the dielectric function at $\mathbf{q} = \mathbf{0}$, while MoS$_2$ has a larger dielectric function and quite steep slope at $\mathbf{q} = \mathbf{0}$. Phosphorene has a dielectric function similar to MoS$_2$ in size and steepness but is anisotropic with slopes varying by $\sim 40\%$ between the two high symmetry directions, $\Gamma\to X$ and $\Gamma\to Y$.
All the calculations were performed using the GPAW electronic structure code<cit.>. The structures used in the present calculations are relaxed with DFT using the PBE exchange-correlation (xc) functional<cit.>. The resulting lattice constant for h-BN is 2.504Å, the in-plane lattice constant for MoS$_2$ is 3.184Å with a S-S distance of 3.127Å. For phosphorene the in-plane unit cell is 4.630Å by 3.306Å, the in-plane P-P-P angle is 95.8 and the layer thickness is 2.110Å. A convergence test with respect to the amount of vacuum between repeating periodic images was carried out and 15Å of vacuum was necessary for h-BN and Phosphorene where as only 10Å was needed for MoS$_2$. The PBE eigenvalues and wavefunctions were calculated with a plane wave basis cut-off energy of 600eV and used as input in the GW calculations. For the initial investigation of the $\mathbf{q}$-point convergence, the dielectric function and the correlation self-energy were calculated using a cutoff of 50eV. This cutoff is insufficient to ensure properly converged quasi-particle energies, but it is adequate to describe the trends related to the improved $\mathbf{q}$-point sampling relevant for this study. The following fully converged calculations were carried out using a $1/N_\text{pw}$ extrapolation to the complete basis set limit using cutoff energies of up to 200eV<cit.>.
In Figure <ref> we compare the analytical small $\mathbf{q}$ expression, Eq. (<ref>), for the head of the screened potential $\overline{W}_\mathbf{00}(\mathbf{q})$ with the numerical values obtained using a fine and coarse $\mathbf{q}$-point sampling. In all the cases the $\mathbf{q} = \mathbf{0}$ value has been set to the analytical value. It is evident that the screened potential falls off quickly and thus for a coarse $\mathbf{q}$-point sampling the $\mathbf{q} = \mathbf{0}$ contribution to the integral is by far the largest and should therefore not be neglected. Similarly, using only the exact value in $\mathbf{q}=\mathbf{0}$ could also pose a problem as the contribution will be grossly overestimated due to the convex nature of potential. We note that the analytical expression follows the numerical results quite closely and is even accurate far away from the $\Gamma$-point – for MoS$_2$ we have an almost perfect agreement for the points shown. Thus using the analytical limit within the region around $\mathbf{q}=\mathbf{0}$ is reasonable. We notice that the anisotropy of phosphorene makes $\overline{W}_\mathbf{00}(\mathbf{q})$ ill-defined at $\mathbf{q} = \mathbf{0}$ (different limit values depending on the direction of $\mathbf{q}$). For larger $\mathbf{q}$ the dielectric anisotropy becomes negligible. However, because of the relatively large weight of the $\mathbf{q} = \mathbf{0}$ contribution to the BZ integral, the anisotropy should be taken into account for accurate GW calculations.
We note that a similar approach to the treatment of the $\mathbf{q} = \mathbf{0}$ term of the screened potential was suggested in Ref. ismail-beigi_truncation_2006. That particular method was based on fitting to an empirical expression for $\varepsilon(q)$ calculated from the value at a small but finite $\mathbf{q}$. The method outlined here is different in that the analytical expression for $\overline{W}(\mathbf{q})$ is obtained from a lowest order expansion of the head, $\chi^0_\mathbf{00}(q)$, and wings, $\chi^0_\mathbf{0G}(q)$, of the non-interacting density response function<cit.> and thus can be obtained without fitting or using empirical parameters. This also ensures that the effect of in-plane dielectric anisotropy is explicitly included.
(Color online) The G$_0$W$_0$ quasi-particle band gap of monolayer (a) 2H-MoS$_2$ (b) h-BN and (c) phosphorene, calculated using two different treatments of the $\mathbf{q} = \mathbf{0}$ term in Eq. (<ref>). The dashed (green) line shows the contribution obtained when the head and wing elements of the $\mathbf{q} = \mathbf{0}$ term are neglected corresponding to the standard treatment used for 3D systems. The solid (blue) line shows the contribution obtained when using the analytical results, Eq. (<ref>), to perform the integral over the $\mathbf{q} = \mathbf{0}$ element. The insets shows the results for the largest $\mathbf{k}$-point grids on a reversed linear scale in $1/N_\mathbf{k}$. Notice the zero point is at the right side of the $x$-axis.
In Fig. <ref> we show the minimum QP band gap of monolayer h-BN, MoS$_2$ and phosphorene as a function of $1/N_\mathbf{k}$ where $N_\mathbf{k}$ is the total number of $\mathbf{k}$-points in the BZ sampling (the $\mathbf{q}$ point grid for the GW integration is the same as the $\mathbf{k}$-point grid used in DFT). We compare the results obtained using two methods: i) neglecting the $\mathbf{q} = \mathbf{0}$ contribution to head and wings of the screened potential and ii) evaluating Eq. (<ref>) as described. It is clear that method i) in all cases underestimates the correlation self-energy due to the underestimation of the screening; In order to get the band gap converged to within $\sim\SI{0.1}{eV}$ one would have to use a $\mathbf{k}$-point sampling of minimum $36\times 36$ for h-BN, $36\times 36$ for MoS$_2$ and $22\times 30$ for phosphorene. We also note that for large $\mathbf{k}$-point grids the band gaps using this method converge approximately as $1/N_\mathbf{k}$ as the missing contribution is almost proportional to the area of the $\mathbf{q} = \mathbf{0}$ region. Clearly, the latter approach varies significantly less with the $\mathbf{k}$-point grid and in fact the gap is converged to within 0.2eV already for a $\mathbf{k}$-point grid in the order of $6\times 6$ and to within $\sim\SI{0.1}{eV}$ with a $12\times 12$ grid (in the worst case). We have performed test calculations for other 2D semiconductors and obtained similar conclusions although the number of $\mathbf{k}$-points required to reach convergence within 0.1eV following the conventional approach ($\mathbf{q} = \mathbf{0}$ term neglected) is somewhat system dependent; materials with efficient screening, e.g. MoS$_2$ and NiS$_2$, require larger $\mathbf{k}$-point grids than materials with poor screening, e.g. h-BN and HfO$_2$ (see Fig. <ref>).
(Color online) The band gap of monolayer 2H-MoS$_2$ calculated with different amounts of vacuum between repeated layers. The solid and dashed lines are with and without the $\mathbf{q}=0$ correction, respectively. As the vacuum is increased, the weight of the correction is decreased and it is necessary to use denser in-plane $\mathbf{k}$-point sampling to achieve convergence.
To obtain converged band gaps it is necessary to use a unit cell with enough vacuum between repeated layers to avoid an artificial interaction. This is true even when a truncated Coulomb interaction is used as the finite vacuum affects wave functions and energies, in particular for higher lying unbound states. As the amount of vacuum is increased, the Brillouin zone shrinks and the analytical correction around $\mathbf{q} = 0$, applied only for $\mathbf{G}=0$, has smaller weight. This means a slower convergence with respect to in-plane $\mathbf{k}$-points. This is shown in Fig. <ref> for MoS$_2$, where it is clear that the correction is less effective for larger vacuum. The calculations converge toward the same value indicating that for MoS$_2$ 10Å of vacuum is sufficient. The most efficient procedure to obtain converged band gaps is therefore to first converge the amount of vacuum at a low $\mathbf{k}$-point sampling without applying the correction and then afterwards converge the $\mathbf{k}$-point sampling with the correction at the given vacuum.
In Table <ref> we report the converged values for the quasiparticle band gaps. For h-BN the band gap is indirect between the $K$- and $\Gamma$-point, for MoS$_2$ and phosphorene it is direct at the $K$- and $\Gamma$-point respectively. For these calculations we used $18\times 18$ $\mathbf{k}$-points for h-BN, $18\times 18$ $\mathbf{k}$-points for MoS$_2$ and $10\times 14$ for phosphorene with the analytical integration of $W(\mathbf{q})$ around $\mathbf{q} = \mathbf{0}$. According to Fig. <ref> this is sufficient to ensure convergence to within 0.05eV. We note that spin-orbit interactions are not included in the reported values. Inclusion of spin-orbit interactions split the valence band of MoS$_2$ at the $K$ point by 0.15eV thereby lowering the QP gap by around 0.07eV<cit.>. Spin-orbit interactions have no effect for h-BN and phosphorene.
For MoS$_2$ the converged G$_0$W$_0$@PBE band gap of 2.54eV agrees well with our previously reported value of 2.48eV (with spin-orbit coupling) obtained using a Wigner-Seitz truncated Coulomb interaction and $30\times 30$ $\mathbf{k}$-points<cit.>. Other reported gaps range from 2.402.82eV<cit.>.
However, these calculations were performed i) without the use of a truncated Coulomb interaction and including 15-25 Å vacuum, ii) employing relatively small $\mathbf{k}$-point grids of 6x6 to 18x18, and iii) using different in-plane lattice constants varying between 3.15 and 3.19 Å. These different settings can affect the band gap by as much as 0.5 eV <cit.>, and therefore we refrain from providing detailed comparison of our result to these earlier calculations. An overview of previous GW results for MoS$_2$ can be found in Ref. huser_quasiparticle_2013.
In Ref. qiu_erratum:_2015 a G$_0$W$_0$@LDA band gap for MoS$_2$ of 2.70eV is reported using a truncated Coulomb interaction and a calculation of the screened potential at $\mathbf{q} = \mathbf{0}$ based on the method in Ref. ismail-beigi_truncation_2006. In that study, the lattice constant of MoS$_2$ was 3.15Å. With this lattice constant we obtain a gap of 2.64eV, which is in fair agreement with Ref. qiu_erratum:_2015. Our result is very close to the experimental value of 2.5eV inferred from photo current spectroscopy<cit.>. Performing partially self-consistent GW$_0$ the band gap increases to 2.65eV (2.58eV including spin-orbit coupling).
For h-BN, we obtained a G$_0$W$_0$ band gap of 7.06eV which increases to 7.49eV with GW$_0$. In Ref. nieminen-hbn the G$_0$W$_0$ band gap was calculated to be 7.40eV. Instead of a truncated Coulomb interaction the band gap was extrapolated to infinite vacuum. The treatment of the $\mathbf{q}=0$ term is not mentioned nor is the size of the $\mathbf{k}$-point grid. Despite the difference at the G$_0$W$_0$ level, they report a similar increase of the band gap of 0.4eV when doing a GW$_0$ calculation.
For phosphorene we calculate a G$_0$W$_0$ band gap of 2.03eV which agrees well with the previously reported value of 2.0eV<cit.> using the method of Ref. ismail-beigi_truncation_2006. The band gap increases to 2.29eV with GW$_0$.
Transition DFT-PBE G$_0$W$_0$@PBE GW$_0$@PBE
h-BN $K\rightarrow \Gamma$ 4.64 7.06 7.49
$K\rightarrow K$ 4.72 7.80 8.25
2H-MoS$_2$ $K\rightarrow K$ 1.65 2.54 2.65
Phosphorene $\Gamma\rightarrow \Gamma$ 0.90 2.03 2.29
Band gaps in eV calculated with DFT-PBE, G$_0$W$_0$@PBE and GW$_0$@PBE using the PBE-relaxed structures. The GW calculations were performed using analytic integration of $\overline{W}(\mathbf{q})$ around $\mathbf{q} = \mathbf{0}$ without including spin-orbit interactions. 10Å of vacuum was used for MoS$_2$ and 15Å for h-BN and phosphorene. The following $\mathbf{k}$-point grids were used; h-BN: $18\times 18$, 2H-MoS2: $18\times 18$ and phosphorene: $10\times 14$.
§ CONCLUSION
In conclusion, we have discussed the connection between the form of the $\mathbf{q}$-dependent dielectric function of a 2D semiconductor and the slow $\mathbf{k}$-point convergence of the GW band structure. We have derived an analytical expression for the $\mathbf{q} \to \mathbf{0}$ limit of the screened potential of a semiconductor when a 2D truncation of the Coulomb potential is used. The method accounts for dielectric anisotropy and does not rely on any additional parameters or fitting. Using this expression we have shown that convergence of the GW self-energy with respect to the size of the $\mathbf{k}$-point grid is drastically improved. For the specific case of monolayer MoS$_2$, we found that the use of the analytical form alone reduces the $\mathbf{k}$-point grid required to achieve convergence of the GW self-energy contribution to the band gap to within $\sim\SI{0.1}{eV}$ from around $36\times 36$ to $12\times 12$ – a reduction in the number of $\mathbf{k}$-points by an order of magnitude. This method may therefore enable future large-scale GW calculations for 2D materials without compromising accuracy.
We acknowledge support from the Danish Council for Independent Research’s Sapere Aude Program, Grant No. 11-1051390. The Center for Nanostructured Graphene is sponsored by the Danish National Research Foundation, Project DNRF58.
§ CALCULATION OF THE $\MATHBF{Q} \TO \MATHBF{0}$ LIMIT OF THE SCREENED POTENTIAL
In the following we derive the analytical form of the screened potential, Eq. (<ref>), for 2D materials in the limit $\mathbf{q}_\parallel \to \mathbf{0}$. We largely follow the approach of Ref. freysoldt_dielectric_2007 where the same limit for bulk systems was considered. As explained in the main text we use a truncated Coulomb interaction of the form
\begin{equation}
v(\mathbf{r}_\parallel, z) = \frac{\theta(R - |z|)}{\sqrt{|\mathbf{r}_\parallel|^2 + z^2}}.
\end{equation}
Using this potential we effectively turn off interaction between electrons on different 2D layers of the supercell calculation. We choose $R$ to be half the height of the unitcell, $R = L/2$, so that an electron in the center of the layer will not interact with electrons located in the neighboring unitcell. This means that the 2D truncated coulomb interaction of Eq. (16) in <cit.> reduces to
\begin{equation}
v^\text{2D}_\mathbf{G}(\mathbf{q}_\parallel) = \frac{4\pi}{|\mathbf{q}_\parallel + \mathbf{G}|^2} \left[ 1 - e^{-|\mathbf{q}_\parallel + \mathbf{G}_\parallel| L / 2} \cos(|G_z| L / 2) \right], \label{eq:coulomb-2d}
\end{equation}
where only in-plane $\mathbf{q}$ is considered. We note that in the limit $L \to \infty$ it takes the usual 3D form, $v_\mathbf{G}(\mathbf{q}) = \frac{4\pi}{|\mathbf{q} + \mathbf{G}|^2}$. In the long wavelength limit it has the asymptotic behavior
\begin{align}
v^\text{2D}_\mathbf{0}(\mathbf{q}_z = 0, \mathbf{q}_\parallel \to \mathbf{0}) = \frac{2\pi L }{|\mathbf{q}_\parallel|}, \label{eq:coulomb-2d-limit}
\end{align}
diverging slower than the full Coulomb potential with profound consequences for the properties of 2D materials.
In the long wavelength limit $\mathbf{q} \to \mathbf{0}$ the non-interacting density response function or irreducible polarizability has the following behavior<cit.>
\begin{align}
\chi^0_{\mathbf{00}'}(\mathbf{q} \to \mathbf{0}) ={}& \mathbf{q} \cdot \mathsf{P} \mathbf{q} = |\mathbf{q}|^2 \hat{\mathbf{q}} \cdot \mathsf{P} \hat{\mathbf{q}} \label{eq:chi-head}\\
\chi^0_{\mathbf{G0}}(\mathbf{q}\to \mathbf{0}) ={}& \mathbf{q} \cdot \mathbf{p}_\mathbf{G} = |\mathbf{q}| \hat{\mathbf{q}} \cdot\mathbf{p}_\mathbf{G}, \label{eq:chi-wings-rows},\\
\chi^0_{\mathbf{0G}}(\mathbf{q}\to \mathbf{0}) ={}& \mathbf{q} \cdot \mathbf{s}_\mathbf{G} = |\mathbf{q}| \hat{\mathbf{q}} \cdot\mathbf{s}_\mathbf{G}, \label{eq:chi-wings-cols}
\end{align}
where $\mathsf{P}$ is a second rank tensor, $\mathbf{p}_\mathbf{G}$ and $\mathbf{s}_\mathbf{G}$ are proper vectors and $\hat{\mathbf{q}} = \mathbf{q} / |\mathbf{q}|$. The density response function, and therefore also $\mathsf{P}$ and $\mathbf{p}_\mathbf{G}$, has a frequency dependence which here and through the rest of this section has been left out to simplify the notation.
Within the random phase approximation the dielectric function is given by (schematically)
\begin{equation}
\varepsilon = 1 - v \chi^0.
\end{equation}
Due to technical reasons<cit.> it is easier to work with a similar symmetrized version given in Fourier space by
\begin{equation}
\tilde{\varepsilon}_{\mathbf{GG}'}(\mathbf{q}) = \delta_{\mathbf{GG}'} - \sqrt{v_\mathbf{G}(\mathbf{q})} \chi^0_{\mathbf{GG}'}(\mathbf{q}) \sqrt{v_{\mathbf{G}'}(\mathbf{q})}.
\end{equation}
Inserting the Coulomb potential, Eq. (<ref>), and the expressions for the non-interacting response function Eqs. (<ref>)-(<ref>), the head and wings of the symmetrized dielectric function are
\begin{align}
\tilde{\varepsilon}_\mathbf{00}(\mathbf{q}_\parallel \to \mathbf{0}) ={}& 1 - v^\text{2D}_\mathbf{0}(\mathbf{q}_\parallel) |\mathbf{q}_\parallel|^2 \hat{\mathbf{q}_\parallel} \cdot \mathsf{P}\hat{\mathbf{q}_\parallel} \nonumber\\
={}& 1 - 4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2}) \hat{\mathbf{q}_\parallel} \cdot \mathsf{P} \hat{\mathbf{q}_\parallel} \\
\tilde{\varepsilon}_{\mathbf{G0}}(\mathbf{q}_\parallel \to \mathbf{0}) ={}& -\sqrt{v^\text{2D}_\mathbf{G}(\mathbf{0})} \sqrt{v_\mathbf{0}^\text{2D}(\mathbf{q}_\parallel)} \hat{\mathbf{q}_\parallel} \cdot \mathbf{p}_\mathbf{G} \nonumber\\
={}& -\sqrt{v^\text{2D}_\mathbf{G}(\mathbf{0})} \sqrt{4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2})} \hat{\mathbf{q}_\parallel} \cdot \mathbf{p}_\mathbf{G}\\
\tilde{\varepsilon}_{\mathbf{0G}}(\mathbf{q}_\parallel \to \mathbf{0}) ={}& -\sqrt{v^\text{2D}_\mathbf{G}(\mathbf{0})} \sqrt{v_\mathbf{0}^\text{2D}(\mathbf{q}_\parallel)} \hat{\mathbf{q}_\parallel} \cdot \mathbf{s}_\mathbf{G} \nonumber\\
={}& -\sqrt{v^\text{2D}_\mathbf{G}(\mathbf{0})} \sqrt{4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2})} \hat{\mathbf{q}_\parallel} \cdot \mathbf{s}_\mathbf{G}.
\end{align}
To determine the inverse dielectric function we write the dielectric function as a block matrix in the $\mathbf{G}, \mathbf{G}'$ components with head, wings and body of the form
\begin{equation}
\tilde{\boldsymbol{\varepsilon}} = \begin{pmatrix} H & \mathbf{w}^\intercal \\ \mathbf{v} & \mathbf{B} \end{pmatrix}
\end{equation}
The inverse is then given by
\begin{equation}
\tilde{\boldsymbol{\varepsilon}}^{-1} = \begin{pmatrix} (H - \mathbf{w}^\intercal \mathbf{B}^{-1} \mathbf{v})^{-1} & -(H - \mathbf{w}^\intercal \mathbf{B}^{-1} \mathbf{v})^{-1} \mathbf{w}^\intercal \mathbf{B}^{-1} \\ -\mathbf{B}^{-1} \mathbf{v} (H - \mathbf{w}^\intercal \mathbf{B}^{-1} \mathbf{v})^{-1} & \mathbf{B}^{-1} + \mathbf{B}^{-1} \mathbf{v} (H - \mathbf{w}^\intercal \mathbf{B}^{-1} \mathbf{v})^{-1} \mathbf{w}^\intercal \mathbf{B}^{-1} \end{pmatrix}
\end{equation}
From this we see that
\begin{align}
\tilde{\varepsilon}^{-1}_\mathbf{00} ={}& \left[ \tilde{\varepsilon}_\mathbf{00} - \sum_{\mathbf{G}, \mathbf{G}' \neq \mathbf{0}} \tilde{\varepsilon}_\mathbf{0G} B^{-1}_{\mathbf{GG}'} \tilde{\varepsilon}_{\mathbf{G}'\mathbf{0}} \right]^{-1} \\
\tilde{\varepsilon}^{-1}_\mathbf{G0} ={}& -\tilde{\varepsilon}^{-1}_\mathbf{00} \sum_{\mathbf{G}' \neq \mathbf{0}} B^{-1}_{\mathbf{GG}'} \tilde{\varepsilon}_{\mathbf{G}'\mathbf{0}} \\
\tilde{\varepsilon}^{-1}_\mathbf{0G} ={}& -\tilde{\varepsilon}^{-1}_\mathbf{00} \sum_{\mathbf{G}' \neq \mathbf{0}} \tilde{\varepsilon}_{\mathbf{0}\mathbf{G}'} B^{-1}_{\mathbf{G}'\mathbf{G}} \\
\tilde{\varepsilon}^{-1}_{\mathbf{GG}'} ={}& B^{-1}_{\mathbf{GG}'} + \tilde{\varepsilon}^{-1}_{\mathbf{00}} \left( \sum_{\mathbf{G}'' \neq \mathbf{0}} B^{-1}_{\mathbf{GG}''} \tilde{\varepsilon}_{\mathbf{G}''\mathbf{0}} \right) \left( \sum_{\mathbf{G}'' \neq \mathbf{0}} \tilde{\varepsilon}_{\mathbf{0G}''} B^{-1}_{\mathbf{G}''\mathbf{G}'} \right)
\end{align}
Introducing the vectors $\mathbf{a}_\mathbf{G}$, $\mathbf{b}_\mathbf{G}$ and the tensor $\mathsf{A}$ given by
\begin{align}
\mathbf{a}_\mathbf{G} ={}& -\sum_{\mathbf{G}' \neq \mathbf{0}} B^{-1}_{\mathbf{GG}'} \sqrt{v^\text{2D}_{\mathbf{G}'}(\mathbf{0})} \mathbf{p}_{\mathbf{G}'} \\
\mathbf{b}_\mathbf{G} ={}& -\sum_{\mathbf{G}' \neq \mathbf{0}} \sqrt{v^\text{2D}_{\mathbf{G}'}(\mathbf{0})} \mathbf{s}_{\mathbf{G}'} B^{-1}_{\mathbf{G}'\mathbf{G}} \\
\mathsf{A} ={}& -\mathsf{P} - \sum_{\mathbf{G} \neq \mathbf{0}} \sqrt{v^\text{2D}_\mathbf{G}(\mathbf{q}_\parallel)} \mathbf{s}_\mathbf{G} \otimes \mathbf{a}_\mathbf{G},
\end{align}
where $\otimes$ denotes the tensor product, the long wavelength limit of the inverse dielectric function is seen to be given by
\begin{align}
\tilde{\varepsilon}^{-1}_\mathbf{00}(\mathbf{q}_\parallel \to \mathbf{0}) ={}& \frac{1}{1 + 4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2}) \hat{\mathbf{q}_\parallel} \cdot \mathsf{A}\hat{\mathbf{q}_\parallel}} \\
\tilde{\varepsilon}^{-1}_{\mathbf{G0}}(\mathbf{q}_\parallel \to \mathbf{0}) ={}& -\frac{\sqrt{4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2})} \hat{\mathbf{q}_\parallel} \cdot \mathbf{a}_\mathbf{G}}{1 + 4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2}) \hat{\mathbf{q}_\parallel} \cdot \mathsf{A} \hat{\mathbf{q}_\parallel}} \\
\tilde{\varepsilon}^{-1}_{\mathbf{0G}}(\mathbf{q}_\parallel \to \mathbf{0}) ={}& -\frac{\sqrt{4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2})} \hat{\mathbf{q}_\parallel} \cdot \mathbf{b}_\mathbf{G}}{1 + 4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2}) \hat{\mathbf{q}_\parallel} \cdot \mathsf{A} \hat{\mathbf{q}_\parallel}} \\
\tilde{\varepsilon}^{-1}_{\mathbf{GG}'}(\mathbf{q}_\parallel \to \mathbf{0}) ={}& B^{-1}_{\mathbf{GG}'} \nonumber\\
& + \frac{4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2}) (\hat{\mathbf{q}_\parallel} \cdot \mathbf{a}_\mathbf{G}) (\hat{\mathbf{q}_\parallel} \cdot \mathbf{b}_{\mathbf{G}'})}{1 + 4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2}) \hat{\mathbf{q}_\parallel} \cdot \mathsf{A} \hat{\mathbf{q}_\parallel}}
\end{align}
Inserting these expression in the equation for the screened potential, Eq. (<ref>), we see that the head and wings are given by
\begin{align}
\overline{W}_\mathbf{00}(\mathbf{q}_\parallel \to \mathbf{0}) ={}& v^\text{2D}_\mathbf{0}(\mathbf{q}_\parallel) \left[ \tilde{\varepsilon}^{-1}_\mathbf{00}(\mathbf{q}_\parallel) - 1\right] \nonumber\\
={}& -\left( \frac{4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2})}{|\mathbf{q}_\parallel|} \right)^2 \frac{\hat{\mathbf{q}_\parallel} \cdot \mathsf{A}\hat{\mathbf{q}_\parallel}}{1 + 4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2}) \hat{\mathbf{q}_\parallel} \cdot \mathsf{A} \hat{\mathbf{q}_\parallel}} \label{eq:Wheadfull}\\
\overline{W}_\mathbf{G0}(\mathbf{q}_\parallel \to \mathbf{0}) ={}& \sqrt{v^\text{2D}_\mathbf{G}(\mathbf{0})} \tilde{\varepsilon}^{-1}_\mathbf{G0}(\mathbf{q}_\parallel) \sqrt{v^\text{2D}_\mathbf{0}(\mathbf{q}_\parallel)} \nonumber\\
={}& -\frac{4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2})}{|\mathbf{q}_\parallel|} \frac{\sqrt{v^\text{2D}_\mathbf{G}(\mathbf{0})} \hat{\mathbf{q}_\parallel} \cdot \mathbf{a}_\mathbf{G}}{1 + 4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2}) \hat{\mathbf{q}_\parallel} \cdot \mathsf{A} \hat{\mathbf{q}_\parallel}} \\
\overline{W}_\mathbf{0G}(\mathbf{q}_\parallel \to \mathbf{0}) ={}& \sqrt{v^\text{2D}_\mathbf{G}(\mathbf{0})} \tilde{\varepsilon}^{-1}_\mathbf{G0}(\mathbf{q}_\parallel) \sqrt{v^\text{2D}_\mathbf{0}(\mathbf{q}_\parallel)} \nonumber\\
={}& -\frac{4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2})}{|\mathbf{q}_\parallel|} \frac{\sqrt{v^\text{2D}_\mathbf{G}(\mathbf{0})} \hat{\mathbf{q}_\parallel} \cdot \mathbf{b}_\mathbf{G}}{1 + 4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2}) \hat{\mathbf{q}_\parallel} \cdot \mathsf{A} \hat{\mathbf{q}_\parallel}}.
\end{align}
and the body also gets a correction and becomes
\begin{align}
\overline{W}_{\mathbf{GG}'}(\mathbf{q}_\parallel\to\mathbf{0}) ={}& \sqrt{v^\text{2D}_\mathbf{G}(\mathbf{0}) v^\text{2D}_{\mathbf{G}'}(\mathbf{0})} \left[\varepsilon^{-1}_{\mathbf{GG}'}(\mathbf{q}_\parallel) - \delta_{\mathbf{GG}'}\right] \nonumber\\
\begin{split} ={}& \sqrt{v^\text{2D}_\mathbf{G}(\mathbf{0}) v^\text{2D}_{\mathbf{G}'}(\mathbf{0})} \left[ B^{-1}_{\mathbf{GG}'} - \delta_{\mathbf{GG}'} + \frac{4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2}) (\hat{\mathbf{q}_\parallel} \cdot \mathbf{a}_\mathbf{G}) (\hat{\mathbf{q}_\parallel} \cdot \mathbf{b}_{\mathbf{G}'})}{1 + 4\pi (1 - e^{-|\mathbf{q}_\parallel| L/2}) \hat{\mathbf{q}_\parallel} \cdot \mathsf{A} \hat{\mathbf{q}_\parallel}} \right]. \label{eq:Wbodyfull}
\end{split}
\end{align}
Taking the limit $\mathbf{q}_\parallel \to \mathbf{0}$ we see that the head of the screened potential is $\overline{W}_\mathbf{00}(\mathbf{q}_\parallel) \to \mathbf{0}) = (2\pi L)^2 \hat{\mathbf{q}_\parallel} \cdot \mathsf{A}\hat{\mathbf{q}_\parallel}$ which is a finite value.
Defining the dimensionless quantity $\mathbf{x} = \mathbf{q}_\parallel L/2$ and the rotational average $A = \frac{1}{2\pi}\int_0^{2\pi}\hat{\mathbf{x}}(\phi) \cdot \mathsf{A} \hat{\mathbf{x}}(\phi) d\phi$ it is possible to rewrite the head of the screened potential as
\begin{align} \label{eq:w_dimless}
\tilde{w}(\mathbf{x}) &= \frac{\overline{W}(2\mathbf{x}/L)}{(2\pi L)^2 A} \nonumber \\
&= - \left( \frac{1-e^{-|\mathbf{x}|}}{|\mathbf{x}|} \right)^2 \frac{1}{1+4\pi A (1-e^{-|\mathbf{x}|})}.
\end{align}
It is evident that the polar integral of Eq. (<ref>), over a small circle with radius $r_{\Omega_0}$, cannot be evaluated analytically:
\begin{align}
&\int_{\Omega_0} \tilde{w}(\mathbf{x})\,\mathrm{d}\mathbf{x} = \nonumber \\
& -2\pi \int_0^{r_{\Omega_0}} \left( \frac{1-e^{-\mathbf{x}}}{\mathbf{x}} \right)^2 \frac{1}{1+4\pi A (1-e^{-\mathbf{x}})} \mathbf{x}\,\mathrm{d}\mathbf{x}.
\end{align}
It is however noticed that the function $\tilde{y}(\mathbf{x}) = \frac{1}{1+(1+4\pi A)\mathbf{x}}$ agrees very well with the integrand for small $\mathbf{x}$. It has the same first order Taylor expansion and it is integrable. This yields
\begin{align} \label{eq:w_taylor}
&\int_{\Omega_0} \tilde{w}(\mathbf{x})\,\mathrm{d}\mathbf{x} \approx
-2\pi \int_0^{r_{\Omega_0}} \frac{\mathbf{x}}{1+(1+4\pi A)\mathbf{x}} \,\mathrm{d}\mathbf{x} = \nonumber \\
& \frac{-2\pi (4\pi A r_{\Omega_0} + r_{\Omega_0} - \ln(4\pi A r_{\Omega_0} + r_{\Omega_0} + 1)}{(4\pi A + 1)^2}.
\end{align}
Since the expression in Eq. (<ref>) only holds for small $\mathbf{x}$, it is generally not valid in the entire $\Omega_0$ region. Also, the expression is not valid for non-isotropic materials, where it is not justified to take the rotational average of the $\mathsf{A}$ tensor.
|
1511.00272
|
6 Georgian Circle, Newark, DE 19711
Department of Mathematics
University of Florida
P. O. Box 118105
Gainesville, FL 32611
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716
$^*$This work was partially supported by a grant from the Simons Foundation (#204181 to Peter Sin)
The $n$-cube graph is the graph on the vertex set
of $n$-tuples of $0$s and $1$s, with two vertices joined by
an edge if and only if the $n$-tuples differ in exactly one component.
We compute the Smith group of this graph, or, equivalently, the
elementary divisors of an adjacency matrix of the graph.
§ INTRODUCTION
Let $Q_n$ be the $n$-cube graph, with vertex set $\{0,1\}^n$ and
two vertices joined if they differ in one component. In the language of association
schemes, $Q_n$ is the distance 1 graph of the binary Hamming scheme.
It is of interest to compute linear algebraic invariants of a graph, such as
its eigenvalues and the invariant factors of an adjacency matrix or
Laplacian matrix. In the case of $Q_n$, previous work includes
<cit.> and <cit.>, where many of these invariants have been computed
and some conjectures made about others. Here we shall consider the
Smith group. If $X$ is an $m\times n$ integral matrix, then the Smith
group of $X$ is the abelian group defined as the quotient of $\Z^m$ by the subgroup
spanned by the columns $X$; that is, the abelian group whose invariant factor decomposition
is given by the Smith normal form of $X$.
If $A$ is the adjacency matrix (with respect to any ordering
of the vertices) of a graph, then the Smith group of the graph
is defined as the Smith group of $A$, and does not depend on the ordering
on the vertices.
We recall that two integral matrices $X$ and $Y$ are integrally equivalent
if there exist unimodular integral matrices $U$ and $V$ such that
\begin{equation}\label{UXV}
\end{equation}
As is well known, $X$ and $Y$ are integrally equivalent if
and only if $Y$ can be obtained from $X$ by a finite sequence of integral
unimodular row and column operations.
A diagonal form for $X$
is a matrix integrally equivalent to $X$ that has nonzero entries only
on the leading diagonal. The Smith normal form of $X$ is one such diagonal form.
Another way to describe the Smith group is in terms of the $p$-elementary divisors of $X$ with respect to primes $p$.
Any diagonal form for $X$ gives a cyclic decomposition of the Smith group,
so in a certain sense, the various diagonal forms carry the same information
as the list of $p$-elementary divisors as $p$ varies over all primes.
The notion of integral equivalence can be generalized to $\Z_{(p)}$-equivalence, where $\Z_{(p)}$
is the ring of $p$-local integers, by requiring that the matrices $U$ and $V$
appearing in (<ref>) be invertible over $\Z_{(p)}$. We can also consider the
$p$-elementary divisors of any matrix $X$ with entries in $\Z_{(p)}$.
If $X$ happens to have integer entries then its $p$-elementary
divisors are the same whether it is considered as a matrix over $\Z$
or $\Z_{(p)}$
Let $A$ be an adjacency matrix for $Q_n$.
It was proved in <cit.> that for every odd prime $p$, $A$ is $\Z_{(p)}$-equivalent to
the diagonal matrix of the eigenvalues (all of which are integers).
When $n$ is odd, all the eigenvalues are odd integers, so $\det(A)$
is an odd integer. Thus, the eigenvalue matrix is a diagonal form when $n$ is odd.
When $n$ is even,
there remains the problem of finding the $2$-elementary divisors
of $A$. A conjecture for the multiplicity of each power $2^e$ as
a $2$-elementary divisor was stated in <cit.>. (See Conjecture <ref> below.)
The purpose of this paper is to give a proof of this conjecture.
As a consequence of the conjecture, we obtain the following
description of the Smith group of $Q_n$ when $n$ is even.
Suppose that $n=2m$ is even. Then the adjacency matrix
$A$ of the $n$-cube $Q_n$ is integrally equivalent to a diagonal matrix
with $\tbinom{n}{m}$ diagonal entries equal to zero and
whose nonzero diagonal entries are the integers $k=1$,$2$,…$m$, in which the multiplicity of $k$ is $2\tbinom{n}{m-k}$.
§ INCLUSION OF SUBSETS OF A FINITE SET
Let $n$ be a positive integer and
$X=\{1,2,\ldots, n\}$. For brevity, we shall use the term
$k$-subsets for the subsets of $X$ of size $k$.
For $k\leq n$, let $M_k$ denote the free $\Z$-module on the set of $k$-subsets
and for $t$, $k\leq n$ let
\eta_{t,k}: M_t\to M_k
be the incidence map, induced by inclusion. Thus, if $t\leq k$
a $t$-subset is mapped to the sum of all $k$-subsets containing it,
while if $t\geq k$ the image of a $t$-subset is the sum of
all $k$-subsets which it contains.
For each $k\leq n$, if we fix ordering on the $k$-subsets,
we can think of elements of $M_k$ as row vectors. Let $W_{t,k}$ denote the $\tbinom{n}{t}\times\tbinom{n}{k}$ matrix of $\eta_{t,k}$ with respect to these ordered bases of $M_t$ and $M_k$.
§ CANONICAL BASES FOR SUBSET MODULES
The notion of the rank of a subset was introduced by Frankl <cit.>.
We shall only need the concept of a $t$-subset of rank $t$, for $t\leq \frac{n}{2}$.
let $T=\{i_1,i_2,\ldots, i_t\}\subseteq X$, with the elements in increasing order.
Then $T$ has rank $t$ if and only if $i_j\geq 2j$ for all $j=1$,…,$t$.
A $t$-subset is of rank $t$ if an only if it is
the set of entries in the second row of a standard
Young tableau of shape $[n-t,t]$. This is one way to see that
the number of $t$-subsets of rank $t$ is $\tbinom{n}{t}-\tbinom{n}{t-1}$.
Assume $0\leq j\leq k\leq \frac{n}{2}$.
Let $E_{j,k}$ denote the $[\tbinom{n}{j}-\tbinom{n}{j-1}]\times \tbinom{n}{k}$
submatrix of $W_{j,k}$ formed from the
the rows labeled by $j$-subsets of rank $j$, and let
$E_k$ be the $\tbinom{n}{k}\times \tbinom{n}{k}$ matrix formed by
stacking the $E_{j,k}$, $0\leq j\leq k$, with $j$ increasing as we move down the
matrix $E_k$.
Wilson <cit.> found a diagonal form for $W_{t,k}$. We shall
state his result for $t\leq k\leq n/2$.
There exist unimodular matrices $U_{t,k}$
and $V_{t,k}$ such that
\begin{equation}\label{UV}
U_{t,k}W_{t,k}V_{t,k}= D_{t,k},
\end{equation}
where the diagonal form $D_{t,k}$ has diagonal entries
$\tbinom{k-j}{t-j}$, with multiplicity $\tbinom{n}{j}-\tbinom{n}{j-1}$, for
$j=0$, …, $t$.
Bier <cit.> refined Wilson's results, showing that
we can take $U_{t,k}=E_t$ for all $k$ and $V_{t,k}={E_k}^{-1}$ for all $t$.
The additional uniformity will be important for us.
Assume $k\leq \frac{n}{2}$. Then the matrix $E_k$ is unimodular.
Furthermore, for $t\leq k$, we have
\begin{equation}
E_t W_{t,k} {E_k}^{-1} = D_{t,k},
\end{equation}
where $D_{t,k}$ is Wilson's diagonal form.
We shall refer to the basis of $M_k$ corresponding to
the rows of $E_k$ as the canonical basis of $M_k$.
It consists of all vectors of the form $\eta_{j,k}(J)$, where
$0\leq j\leq k$ and $J$ is a $j$-subset of rank $j$.
§ THE $N$-CUBE
Let $Q_n$ denote the $n$-cube graph. The vertex set of $Q_n$ is $\{0,1\}^n$
and $(a_1,\ldots,a_n)$ is adjacent to $(b_1,\ldots,b_n)$ if and only if
there is exactly one index $j$ with $a_j\neq b_j$. There is clearly a bijection
of the vertex set with the set of subsets of $X=\{1,2,\dots,n\}$,
under which a vertex corresponds to the subset of indices where the vertex has entry $1$.
We use this bijection and our fixed ordering of $k$-subsets
for $k\leq n$ to order the vertices of $Q_n$, taking the subsets in order of increasing
size. Let $A$ denote the adjacency matrix, with respect to this ordering.
Next we review the results of <cit.>.
By viewing the vertex set of $Q_n$ as $\F_2^n$, and transforming $A$ by the
character table of the additive group $\F_2^n$
the eigenvalues of $Q_n$ are found to be $n-2\ell$,
with multiplicity $\tbinom{n}{\ell}$ for $0\leq\ell\leq n$.
When $n$ is odd, we see that $\det A$ is odd; hence all elementary divisors are odd.
Then since $\F_2^n$ is an abelian $2$-group, the same
discrete Fourier transform method yields the elementary divisors.
When $n$ is even, one still obtains the $p$-elementary divisors for all
odd $p$. The $2$-elementary divisors were not computed in <cit.>, but
the following conjecture was stated:
<cit.> Suppose $n$ is even. Then the multiplicity
of $2^i$ as a $2$-elementary divisor of $A$ is equal to the number of eigenvalues
of $A$ whose exact $2$-power divisor is $2^{i+1}$.
§ BASES FOR THE FREE MODULE ON $Q_N$ AND MATRIX REPRESENTATIONS
OF ADJACENCY
Let $\Z^{Q_n}$ denote the free $\Z$-module on the set of vertices of $Q_n$.
The matrix $A$ can be viewed as an endomorphism of $\Z^{Q_n}$,
sending a vertex to the sum of all adjacent vertices.
It is important for us to adopt a slightly different point of view. We can
think of $\Z^{Q_n}$ as the ring of $\Z$-valued functions on the set of vertices of $Q_n$.
Then the matrix $A$ defines the map $\alpha$ such that for any function
$f\in\Z^{Q_n}$ we have
\begin{equation}
\alpha(f)(a_1,\ldots,a_n)=\sum_{i=1}^n f(a_1,\ldots a_{i-1},1-a_i,a_{i+1},\ldots,a_n),
\end{equation}
for $(a_1,\ldots,a_n)\in Q_n$.
If we further regard the set $\{0,1\}^n$ of vertices of $Q_n$ as a subset of
$\Z^n$, then functions are restrictions of polynomials and
we have a ring isomorphism of $\Z^{Q_n}$ with
\Z[X_1,\ldots,X_n]/({X_i}^2-X_i,\, 1\leq i\leq n).
Now a different
natural basis becomes evident, namely the set of
monomials $X_I=\prod_{i\in I}X_i$, for $I\subseteq X=\{1,\ldots,n\}$.
With respect to the monomial basis we have
\begin{equation}
\alpha(X_I)= \sum_{i\in I} (X_{I\setminus\{i\}}-X_I) + \sum_{i\notin I} X_I
=(n-2\abs{I})X_I + \sum_{\begin{smallmatrix}J\subset I\\ \abs{J}=\abs{I}-1\end{smallmatrix}
\end{equation}
Therefore, if we order monomials in the same way as we ordered
subsets, the matrix of $\alpha$ with respect to this basis has the form
\begin{equation*}
\tilde A=
\begin{pmat}[{|||||||}]
\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots\cr\-
\end{pmat}.
\end{equation*}
Assume from now on that $n=2m$ is even.
\begin{equation}\label{MNmatrix}
\tilde A=\begin{bmatrix}M&0\\0&N
\end{bmatrix}
\end{equation}
\begin{equation*}
\begin{pmat}[{|||||||}]
\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots\cr\-
\end{pmat}.
\end{equation*}
\begin{equation*}
\vdots&\ddots&\ddots&W_{n-4,n-3}&0&0&0\cr\-
\vdots&\vdots&\ddots&-(n-6)I&W_{n-3,n-2}&0&0\cr\-
\vdots&\vdots&\hdots&0&(-(n-4)I&W_{n-2,n-1}&0\cr\-
\vdots&\vdots&\hdots&0&0&-(n-2)I&W_{n-1,n}\cr\-
\end{pmat}.
\
\end{equation*}
Due to the block form (<ref>) of $\tilde A$, the multiplicity
of a prime power as an elementary divisor of $\tilde A$
is the sum of its multiplicites in $M$ and $N$.
Up to now the choice of orderings on the $j$-susbsets, $0\leq j\leq n$
used in the definition of the inclusion matrices $W_{t,k}$
has been an arbitrary (but fixed) one. Any choice would result in matrices of the form
$M$ and $N$ as above, but the rows and columns of the submatrices
$W_{t,k}$ would be permuted.
Now we shall specify these orderings more carefully.
The matrix $M$ involves only the matrices $W_{t,k}$ with $0\leq t<k\leq m$,
while the matrix $N$ involves only the matrices $W_{t,k}$ with
$m\leq t<k\leq n$. We start from the arbitrary but
fixed ordering on the $j$-subsets with $0\leq j\leq m$ that led to
the matrix $M$. Then for $0\leq j<m$ we choose the ordering of $(n-j)$-subsets
to be the order induced by the complementation map. In this way
we have specified an ordering on the $j$-subsets, for all $j$.
Finally we wish to consider a second ordering on $m$-subsets,
namely, the ordering defined from the given ordering by complementation.
We use the first ordering on $m$-sets to define the submatrix $W_{m-1,m}$ of $M$
and the second ordering to define the submatrix $W_{m,m+1}$ of $N$.
From the block form (<ref>), we see that the matrix $\tilde A$
thus constructed differs from the one in which the same ordering on
$m$-sets is used for both $W_{m-1,m}$ and $W_{m,m+1}$
only by a permutation of the rows in the first row-block of $N$, so
the two matrices would be integrally equivalent.
The reason we have been careful to choose the ordering as above is that
we now have, for $0\leq t<k\leq m$,
\begin{equation}\label{transpose}
\end{equation}
If we reverse the order of the block-rows and block-columns of $N$
and then take the transpose, we obtain
\begin{equation*}
\begin{aligned}
\begin{pmat}[{|||||||}]
\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots\cr\-
\end{pmat}\\
\begin{pmat}[{|||||||}]
\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots\cr\-
\end{pmat}.
\end{aligned}
\end{equation*}
Thus, $N'$ differs from $M$ only by the sign of the diagonal
entries. In fact, to see that $N'$ is integrally
equivalent to $M$, we perform the following simple sequence of unimodular operations.
First mutliply the first block-column by -1, then multiply the second block-row
by -1, then the third block-column, etc., until we reach the bottom-right
of the matrix, at which point $N'$ has been converted to $M$.
We have established the following reduction.
Let $M$ and $N$ be the matrices in (<ref>).
Then $M$ and $N^t$ are integrally equivalent.
In particular the multiplicity of an elementary divisor of $\tilde A$
(and hence of $A$) is twice its multplicity as an elementary divisor of $M$.
From now, we focus on the matrix $M$.
Let $E_j$ for $0\leq j\leq m$ be defined as in <ref>
and for $0\leq k\leq m$, set
\begin{equation*}
\end{equation*}
Then from Theorem <ref> we immediately obtain:
\begin{equation}\label{Jmatrix}
E(m-1)\cdot M\cdot E(m)^{-1}=
\begin{pmat}[{|||||||}]
\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots\cr\-
\end{pmat}.
\end{equation}
For example when $n=4$, we have
\begin{equation*}
\begin{pmat}[{|...|.....}]
4& 1 &1& 1& 1& 0& 0& 0 &0 &0& 0\cr\-
0& 2& 0& 0& 0& 1& 1& 0& 1& 0& 0\cr
0& 0& 2& 0& 0& 1& 0& 1& 0& 1& 0\cr
0& 0& 0& 2& 0& 0& 1& 1 &0 &0 &1\cr
0& 0& 0& 0& 2& 0& 0& 0& 1& 1& 1\cr
\end{pmat}
\end{equation*}
\begin{equation*}
\begin{pmat}[{|...}]
1&0& 0& 0& 0\cr\-
0&1 &1 &1 &1\cr
0&0 &1 &0 &0\cr
0&0 &0 &1 &0\cr
0&0 &0 &0 &1\cr
\end{pmat}
\end{equation*}
\begin{equation*}
\begin{pmat}[{|...|.....}]
\end{pmat}
\end{equation*}
\begin{equation*}
E(1)\cdot M\cdot E(2)^{-1}=
\begin{pmat}[{|...|.....}]
\end{pmat}
\end{equation*}
We denote the matrix in (<ref>) by $B$ and let $B'$ be the
matrix obtained by zeroing out the diagonal. Thus,
\begin{equation}\label{Jprimematrix}
\begin{pmat}[{||||||}]
\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots\cr
\end{pmat}.
\end{equation}
We examine the matrix $D_{i-1,i}$ more closely.
It has $\tbinom{n}{i-1}$ rows and $\tbinom{n}{i}$ columns
and has the form
\begin{equation}\label{Dij}
\begin{pmat}[{||||||}]
\mathbf{i}&\mathbf{0}&\mathbf{0}&\hdots&\mathbf{0}&\mathbf{0}&\mathbf{0}\cr\-
\mathbf{0}&\mathbf{i-1}&\mathbf{0}&\hdots&\mathbf{0}&\mathbf{0}&\mathbf{0}\cr\-
\mathbf{0}&\mathbf{0}&\mathbf{i-2}&\hdots&\mathbf{0}&\mathbf{0}&\mathbf{0}\cr\-
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\cr\-
\mathbf{0}&\mathbf{0}&\mathbf{0}&\hdots&\mathbf{2}&\mathbf{0}&\mathbf{0}\cr\-
\mathbf{0}&\mathbf{0}&\mathbf{0}&\hdots&\mathbf{0}&\mathbf{1}&\mathbf{0}\cr
\end{pmat},
\end{equation}
where a bold number $\mathbf{s}$, $s\neq 0$ represents a scalar matrix $sI$
of the appropriate size and $\mathbf{0}$ denotes a zero block of the appropriate size.
The block sizes are readily found; if $n_j=\tbinom{n}{j}-\tbinom{n}{j-1}$, then $D_{i-1,i}$ has $i+1$ block-columns and $i$ block-rows, and the $k$-th block-column of $D_{i-1,i}$ contains $n_{k-1}$ columns, for $1\leq k\leq i+1$, while for $1\leq \ell\leq i$,
the $\ell$-th block-row contains $n_{\ell-1}$ rows.
Also, let $\mathbf{(c)}$ denote a scalar matrix (more precisely the class of scalar matrices) whose
scalar is a multiple of $c$. (We introduce this notation because all that we shall
use about the diagonal entries is that they are even, and working with
the entire class of such matrices will facilitate the use of mathematical induction.)
Then we can write $B$ in the following form (more precisely, $B$ lies in the given class of matrices).
\begin{equation}\label{BigJmatrix}
\begin{pmat}[{|.|..|...|.|....|}]
\mb{(2)}&\mb{1}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\hdots&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}\cr\-
\mb{0}&\mb{(2)}&\mb{0}&\mb{2}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\hdots&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}\cr
\mb{0}&\mb{0}&\mb{(2)}&\mb{0}&\mb{1}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\hdots&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}\cr\-
\mb{0}&\mb{0}&\mb{0}&\mb{(2)}&\mb{0}&\mb{0}&\mb{3}&\mb{0}&\mb{0}&\mb{0}&\hdots&\hdots&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}\cr
\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{(2)}&\mb{0}&\mb{0}&\mb{2}&\mb{0}&\mb{0}&\hdots&\hdots&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}\cr
\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{(2)}&\mb{0}&\mb{0}&\mb{1}&\mb{0}&\hdots&\hdots&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}\cr\-
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\cr
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\cr
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\cr\-
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\cr
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\cr\-
\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\hdots&\mb{(2)}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{m}&\mb{0}&\hdots&\mb{0}&\mb{0}&\mb{0}\cr
\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\hdots&\mb{0}&\mb{(2)}&\hdots&\mb{0}&\mb{0}&\mb{0}&\mb{m-1}&\hdots&\mb{0}&\mb{0}&\mb{0}\cr
\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\hdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\cr
\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\hdots&\mb{0}&\mb{0}&\hdots&\mb{(2)}&\mb{0}&\mb{0}&\mb{0}&\hdots&\mb{2}&\mb{0}&\mb{0}\cr
\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\mb{0}&\hdots&\hdots&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{(2)}&\mb{0}&\mb{0}&\hdots&\mb{0}&\mb{1}&\mb{0}\cr
\end{pmat}.
\end{equation}
$B$ and $B'$ are equivalent over the $2$-local integers $\Z_{(2)}$.
Let $\overline B(m)$ denote a matrix (actually the class of matrices) in which all the blocks (bold numbers) in $B$ are replaced by the plain numbers, and each $\mb{(c)}$ is replaced by $(c)$, representing
a multiple of the $2$-local integer $c$. It suffices to show that any matrix of the form $B(m)$ is
$2$-locally equivalent to the matrix obtained by zeroing out its diagonal.
Indeed, if there is a sequence of $\Z_{(2)}$-unimodular row and column operations
which zero out the diagonal of $\overline B(m)$, then the same sequence
of the blockwise versions of these operations will kill the diagonal of $B$.
So we may discard $B$ and work only with $\overline B(m)$ from now on.
\begin{equation}\label{Jbar}
\overline B(m)=
\begin{pmat}[{|||||||}]
(2)I&\overline D_1&0&0&0&\hdots&0&0\cr\-
0&(2)I&\overline D_2&0&0&\hdots&0&0\cr\-
0&0&(2)I&\overline D_3&0&\hdots&0&0\cr\-
0&0&0&(2)I&\overline D_4&\hdots&0&0\cr\-
\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots\cr\-
0&\hdots&\hdots&\hdots&\ddots&(2)I&\overline D_{m-1}&0\cr\-
0&\hdots&\hdots&\hdots&\hdots&0&(2)I&\overline D_{m}\cr
\end{pmat},
\end{equation}
\begin{equation}
\overline D_{i}=
\begin{pmat}[{.....|}]
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\cr
\end{pmat}
\end{equation}
is the “condensed” version of $D_{i-1,i}$, consisting of a diagonal $i\times i$ matrix augmented by a column of zeros.
From now on it will be convenient to refer to the entries of $\overline B(m)$ (and
matrices derived from it) by their positions relative to the submatrices
$\overline D_i$. Note this is a change in the way we are indexing blocks, compared
to how we did it in the original matrix $M$.
The row of the main matrix containing
the $k$-th row of $\overline D_i$ is assigned the label $[i,k]$
and the column containing the $\ell$-th column of $\overline D_{j}$
is assigned the label $[j,\ell]$, while the first column is
labeled $[0,1]$. Thus, in a row index $[i,k]$ we have $1\leq i\leq m$
and $1\leq k\leq i$, while a column index $[j,\ell]$ has
$0\leq j\leq m$ and $1\leq \ell\leq j+1$.
We shall perform some row and column operations
using the odd entries on the main diagonals of the $\overline D_i$s to kill
diagonal entries of the main matrix. Each odd entry of $\overline D_i$
for $1\leq i\leq m-1$ will be used to kill the two diagonal entries of the main matrix, one in the same row as the odd entry and one in the same column. The odd entries of $\overline D_m$ will be used to kill the diagonal entry in the same row. This procedure will create new entries $(4)$ at locations where there previously were zeroes.
To be precise, if the odd entry is at $([i,k],[i,k])$ then the diagonal entry of the main matrix in the same row is at $([i,k],[i-1,k])$. The diagonal entry of the main matrix in the same column as $([i,k],[i,k])$ is at $([i+1,k],[i,k])$ (with no such
entry if $i=m$). By multiplying column $[i,k]$
by a suitable element of $(2)$ and subtracting it from column $[i-1,k]$, we
kill off the entry at $([i,k],[i-1,k])$ and create a new entry $(4)$ at $([i+1,k],[i-1,k)$,
if $i\leq m-1$. The new entry is $(4)$ because the
entry being subtracted from zero is a $(2)$-multiple of the $(2)$ at $([i+1,k],[i,k])$
on the main diagonal. No new entry is created if $i=m$.
Then, we can subtract a $(2)$-multiple of row $[i,k]$ from row $[i+1,k]$ to kill off the
diagonal entry at $([i+1,k],[i,k])$, without creating any new nonzero entries, if
$i\leq m-1$, while there is nothing to be done when $i=m$.
The following figure (for $m=5$) shows the resulting matrix after performing these operations with the entry 1 at $([3,3],[3,3])$. The block indices are specified by numbered braces.
\begin{equation*}
\begin{array}{rl}
\overbrace{\phantom{\begin{pmat}.{}.(0)\cr \end{pmat}}}^{\mbox{0}}&
\overbrace{\phantom{\begin{pmat}.{}.(0)&0\cr\end{pmat}}}^{\mbox{1}}&\overbrace{\phantom{\begin{pmat}.{}.(0)&(0&0\cr\end{pmat}}}^{\mbox{2}}&\overbrace{\phantom{\begin{pmat}.{}.(0)&(0)&0&0\cr\end{pmat}}}^{\mbox{3}}&\overbrace{\phantom{\begin{pmat}.{}.(0)&(0)&(0)&(0&0\cr\end{pmat}}}^{\mbox{4}}&
\overbrace{\phantom{\begin{pmat}.{}.0&0&0&0&(0)\cr\end{pmat}}}^{\mbox{5}}
\end{matrix}\\
\begin{pmat}.{}.
\coolleftbrace{1}{Y}\cr
\coolleftbrace{2}{Y\\Y}\cr
\coolleftbrace{3}{Y\\ Y\\Y}\cr
\coolleftbrace{4}{Y\\Y\\Y\\Y}\cr
\coolleftbrace{5}{Y\\ Y\\ Y\\ Y\\ Y\\Y}\cr
\end{pmat}%
\begin{pmat}[{|.|..|...|....|.....}]
\end{pmat}
\end{array}
\end{equation*}
We do this for all odd entries of all the $\overline D_i$. In our $m=5$ example,
after the operations the matrix looks as follows.
\begin{equation*}
\begin{pmat}[{|.|..|...|....|.....}]
\end{pmat}
\end{equation*}
At this point, the rows and columns of the main matrix that correspond to an odd
entry of some $\overline D_i$ have no other nonzero entries. The submatrix $D(m)$
formed from these rows and columns is a square diagonal matrix with odd entries,
and the main matrix is the block sum of this diagonal matrix with the submatrix $A(m)$
formed from the remaining rows and columns. (Since permuting rows and columns results
in an integrally equivalent matrix, the reader may find it helpful, for easier
visualization of this block sum decomposition, to renumber the rows and columns in order to put $D(m)$ at the bottom right of the main matrix.)
We observe that $A(m)$ will have entries $(4)$ all along its main diagonal,
and each row of $A(m)$ has one nonzero entry off the main diagonal (coming from the
even entries on the main diagonals of the $\overline D_i$.)
In the picture below, we show $A(5)$.
\begin{equation*}
\begin{pmat}[{||.|.|..|..}]
\end{pmat}
\end{equation*}
We next note that if $i$ and $j$ have different parity then
all entries at locations $([i,k],[j,\ell])$ are zero,
so $A(m)$ is the block sum of the submatrix $A''(m)$ corresponding to
rows $[i,k]$ and columns $[j,\ell]$ for odd indices $i$ and $j$,
and the submatrix $A'(m)$ formed from the rows and columns corresponding to the even
indices. For better visualization, we can reorder the rows and columns.
The block decomposition of $A(5)$ after reording rows and columns is
shown below.
\begin{equation*}
\begin{pmat}[{|.|..||.|..}]
\end{pmat}
\end{equation*}
We can see from this picture that $A'(5)=2\overline B(2)=A''(5)$.
In general, it is easy to see that if we delete the rows and columns
of $\overline D_i$ which contain odd entries, the resulting matrix is
$2\overline D_{i'}$ where $i'=\lfloor\frac{i}{2}\rfloor$ (with
the understanding that $\overline D_{0}$ is the empty matrix).
It follows that $A'(m)=2\overline B(m')$, where $m'=\lfloor\frac{m}{2}\rfloor$,
and $A''(m)=2\overline B(m'')$, and $m''=\lfloor\frac{m-1}{2}\rfloor$. (More formally, $A'(m)$ belongs to the class $2B(m')$ and $A''(m)$ belongs to the class $2B(m'')$.)
Thus, we have shown that
$\overline B(m)$ is $2$-locally equivalent to a matrix $C$ which is the diagonal block sum
of $2\overline B(m')$ and $2\overline B(m'')$ and the diagonal matrix $D(m)$.
Arguing by induction on $m$, there are $\Z_{(2)}$-unimodular row and column operations
on $C$ that kill the diagonals of $A'(m)$ and of $A''(m)$ while leaving all other entries
of $C$ unchanged. The resulting matrix is equal, up to reordering rows and columns, to the matrix obtained from $\overline B(m)$ by zeroing out its diagonal.
Suppose that $n=2m$ is even. Then the adjacency matrix
$A$ of the $n$-cube $Q_n$ is $\Z_{(2)}$-equivalent to a diagonal form
with $\tbinom{n}{m}$ diagonal entries equal to zero and whose nonzero diagonal entries are the integers $k=1$,$2$,…$m$, in which the multiplicity of $k$ is $2\tbinom{n}{m-k}$.
By Lemma <ref>, it suffices to find a diagonal form for $M$,
hence for $B'$, with which we have shown $M$ to be integrally equivalent.
The nonzero entries of $B'$ are the integers $1$ to $m$. The integer $k$
occurs in $D_{i-1,i}$ only when $k\leq i$ and then its
multiplicity in $D_{i-1,i}$ is $\tbinom{n}{i-k}-\tbinom{n}{i-1-k}$.
Therefore, the multiplicity of $k$ in $B'$ is
\begin{equation}
\sum_{i=k}^{m}\tbinom{n}{i-k}-\tbinom{n}{i-1-k} =
\sum_{s=0}^{m-k}\tbinom{n}{s}-\tbinom{n}{s-1}=\tbinom{n}{m-k}.
\end{equation}
Conjecture <ref> is true.
By Lemma <ref>, the multiplicity of the prime power
$p^e$ as an $p$-elementary divisor of $\tilde A$, and hence also of
$A$, is twice the sum of the binomial coefficients $\tbinom{n}{m-k}$, taken over
those $k$ with $1\leq k\leq m$ that are exactly divisible by $p^e$.
On the other hand, we know ( <ref>)
that the nonzero eigenvalues of $A$ are the integers $\pm 2k$ for $1\leq k\leq m$,
and the multiplicity of $2k$ is $\tbinom{n}{m-k}$.
Comparing these numbers for $p=2$, we see that Conjecture <ref> is true.
§.§ Proof of Theorem <ref>
By Theorem <ref>, it suffices to show, for every odd prime $p$, that $A$ is $\Z_{(p)}$-equivalent to a diagonal matrix whose nonzero entries are $k=1$,…,$m$, where
$k$ has multiplicity $2\tbinom{n}{m-k}$. Let $p$ be given. We know from <cit.> that $A$ is
$\Z_{(p)}$-equivalent to a diagonal matrix whose nonzero are $n-2\ell$
with multiplicity $\tbinom{n}{\ell}$, for $0\leq \ell\leq n$. The latter is easily seen to be
integrally equivalent to a diagonal matrix whose nonzero entries are $2k=1$,…,$m$, where
$k$ has multiplicity $2\tbinom{n}{m-k}$, and hence $\Z_{(p)}$-equivalent to
the diagonal form given in Theorem <ref>.
§.§ Final remarks
It would be of interest to find a diagonal form for the Laplacian matrix $nI-A$ of $Q_n$.
The Smith group of this matrix is called the critical group of $Q_n$.
By the results of <cit.>, only the $2$-Sylow subgroup of the critical group
remains to be determined, for both odd and even $n$.
We do not have any conjecture about its exact structure.
However, we note that if two integral matrices are equal modulo $p^s$ then for $i<s$
the multiplicity of $p^i$ as a $p$-elementary divisor is the same for both matrices.
Thus, for example, when $n=2^s$, Theorem <ref> gives part of the cyclic
decomposition of the $2$-Sylow subgroup of the critical group.
Bai Hua Bai, On the critical group of the n-cube. Linear Algebra Appl. 369 (2003) 251-–261.
DJ J. Ducey, D. Jalil, Integer invariants of abelian Cayley graphs, Linear Algebra Appl. 445 (2014) 316–325
Thomas Bier, Remarks on recent formulas of Wilson and Frankl,
European J. Combin. 14 (1993), no. 1, 1–8.
P. Frankl, Intersection theorems and mod p rank of inclusion matrices,
Journal of Combinatorial Theory, Series A, 54 (1990), 85–-94
Richard M. Wilson, A diagonal form for the incidence matrices of
$t$-subsets vs. $k$-subsets, European J. Combin. 11 (1990),
no. 6, 609–615.
0 0 0 0 0 0 (2) 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 (2) 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 (2) 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 (2) 0 0 0 0 0 0 0 0 0
|
1511.00118
|
This paper introduces a new watermarking algorithm based on discrete chaotic iterations. After defining some
coefficients deduced from the description of the carrier medium, chaotic discrete iterations are used to mix the
watermark and to embed it in the carrier medium. It can be proved that this procedure generates topological
chaos, which ensures that desired properties of a watermarking algorithm are satisfied.
§ INTRODUCTION
Information hiding has recently become a major security technology,
especially with the increasing importance and widespread distribution of
digital media through the Internet. It includes several techniques, among which is
digital watermarking. The aim of digital watermarking is to embed a piece of
information into digital documents, like pictures or movies for example. This is for a
large panel of reasons, such as: copyright protection, control utilization,
data description, integrity checking, or content authentication. Digital
watermarking must have essential characteristics including imperceptibility
and robustness against attacks. Many
watermarking schemes have been proposed in recent years, which can be
classified into two categories: spatial domain <cit.> and frequency
domain watermarking <cit.>, <cit.>. In spatial domain
watermarking, a great number of bits can be embedded without inducing too
clearly visible artifacts, while frequency domain watermarking has been
shown to be quite robust against JPEG compression, filtering, noise
pollution, and so on. More recently, chaotic methods have been proposed to
encrypt the watermark, or embed it into the carrier image for security
In this paper, a new watermarking algorithm is given. It is based on the commonly
named chaotic iterations and on the choice of relevant coefficients
deduced from the description of the carrier medium. This new algorithm
consists of two basic stages: a mixture stage and an embedding stage. At
each of these two stages, the proposed algorithm offers additional steps
that allow the authentication of relevant information carried by the
medium or the extraction of the watermark without knowledge about the original image.
This paper is organized as follows: firstly, some basic definitions concerning
chaotic iterations is recalled. Then, the
new chaos-based watermarking algorithm is introduced in Section <ref>.
Section <ref> is constituted by the evaluation of our algorithm: a case study is
presented, some classical attacks are executed and the results are presented and commented on.
The paper ends by a conclusion section where our contribution is summarized,
and planned future work is discussed.
§ BASIC RECALLS: CHAOTIC ITERATIONS
In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$, $V_{i}$
denotes the $i^{th}$ component of a vector $V$ and $f^{k}=f\circ ...\circ f$
denotes the $k^{th}$ composition of a function $f$. Finally, the following
notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.
Let us consider a system of a finite number $\mathsf{N}$ of
cells, so that each cell has a boolean state. Then a sequence of
length $\mathsf{N}$ of boolean states of the cells corresponds to a
particular state of the system. A sequence which elements belong in $%
\llbracket 1;\mathsf{N} \rrbracket $ is called a strategy. The set of
all strategies is denoted by $\mathbb{S}.$
Let $S\in \mathbb{S}$. The shift function is defined by $\sigma
:(S^{n})_{n\in \mathds{N}}\in \mathbb{S}\longrightarrow (S^{n+1})_{n\in %
\mathds{N}}\in \mathbb{S}$ and the initial function $i$ is the map
which associates to a sequence, its first term: $i:(S^{n})_{n\in \mathds{N}%
}\in \mathbb{S}\longrightarrow S^{0}\in \llbracket1;\mathsf{N}\rrbracket$.
The set $\mathds{B}$ denoting $\{0,1\}$, let $f:\mathds{B}^{\mathsf{N}%
}\longrightarrow \mathds{B}^{\mathsf{N}}$ be a function and $S\in \mathbb{S}
$ be a strategy. Then, the so-called chaotic iterations are defined by $x^0\in \mathds{B}^{\mathsf{N}}$ and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket,$
\begin{equation}
\begin{array}{ll}
x_i^{n-1} & \text{ if }S^n\neq i \\
\left(f(x^{n-1})\right)_{S^n} & \text{ if }S^n=i.%
\end{array}%
\right.%
\label{chaotic iterations}
\end{equation}
§ A NEW CHAOS-BASED WATERMARKING ALGORITHM
§.§ Most and Least Significant Coefficients
Let us first introduce the definitions of most and least significant
coefficients of an image.
For a given image, the most significant coefficients (in short MSCs), are
coefficients that allow the description of the relevant part of the image,
i.e. its most rich part (in terms of embedding information), through
a sequence of bits.
For example, in a spatial description of a grayscale image, a definition of
MSCs can be the sequence constituted by the first three bits of each pixel.
By least significant coefficients (LSCs), we mean a translation of some
insignificant parts of a medium in a sequence of bits (insignificant can be
understand as: “which can be altered without sensitive damages”).
The LSCs are used during the embedding stage: some of the least significant
coefficients of the carrier image will be chaotically chosen and replaced by
the bits of the (possibly mixed) watermark.
The MSCs are only useful in case of authentication, mixture and embedding
stages will then depend on them. Hence, a coefficient should not be defined
at the same time both as a MSC and a LSC: the LSC can be altered, while the
MSC is needed to extract the watermark (in case of authentication).
§.§ Stages of the Algorithm
Our watermarking scheme consists of two classical stages: the mixture of
the watermark and its embedding into a cover image.
§.§.§ Watermark mixture
For security reasons, the watermark can be mixed before its embedding. A
common way to achieve this stage is to use the bitwise exclusive or (XOR),
for example, between the watermark and a logistic map. In this paper, we will
introduce a mixture scheme based on chaotic iterations. Its chaotic
strategy will be highly sensitive to the MSCs, in case of an authenticated
watermark <cit.>. For the details of this stage see the Paragraph <ref> in Section <ref>.
§.§.§ Watermark Embedding
This stage can be done either by applying the logical negation of some LSCs, or by replacing them by the bits of the possibly mixed watermark.
To choose the sequence of LSCs to be changed, a number of integers, less
than or equals to the number $N$ of LSCs, corresponding to a chaotic
sequence $\left( U^{k}\right)_{k}$, is generated from the chaotic strategy
used in the mixture stage and possibly the watermark. Thus, the $U^{k}-th
$ least significant coefficient of the carrier image is either switched, or
substituted by the $k^{th}$ bit of the possibly mixed watermark. In case
of authentication, such a procedure leads to a choice of the LSCs which
are highly dependent on the MSCs.
On the one hand, when the switch is chosen, the watermarked image is
obtained from the original image, whose LSCs $L = \mathds{B}^{\mathsf{N}}$
are replaced by the result of some chaotic iterations. Here, the iterate
function is the vectorial boolean negation, defined by $f_{0}: \mathds{B}^{\mathsf{N}} \longrightarrow \mathds{B}^{\mathsf{N}},
(x_{1}, \hdots,x_{\mathsf{N}}) \longmapsto (\overline{x_{1}},\hdots,%
\overline{x_{\mathsf{N}}})$, the initial state is $L$ and strategy is equal to $\left( U^{k}\right)_{k}$. In this case, it is possible to prove that the whole embedding stage satisfies topological chaos
properties <cit.>, but the original medium is needed to extract the watermark.
On the other hand, when the selected LSCs are substituted by the watermark,
its extraction can be done without the original cover. In this case, the
selection of LSCs still remains chaotic, because of the use of a chaotic
map, but the whole process does not satisfy topological chaos <cit.>: the use of
chaotic iterations is reduced to the mixture of the watermark.
See the Paragraph <ref> in Section <ref> for
more details.
§.§.§ Extraction
The chaotic sequence $U^k$ can be regenerated, even in the case of an
authenticated watermarking: the MSCs have not been changed during the stage
of embedding watermark. Thus, the altered LSCs can be found. So, in case of
substitution, the mixed watermark can be rebuilt and “decrypted”. In case
of negation, the result of the previous chaotic iterations on the
watermarked image, is the original image.
If the watermarked image is attacked, then the MSCs will change.
Consequently, in case of authentication and due to the high
sensitivity of the embedding sequence, the LSCs designed to receive the
watermark will be completely different. Hence, the result of the decrypting
stage of the extracted bits will have no similarity with the original
§ A CASE STUDY
§.§ Stages and Details
§.§.§ Images Description
Carrier image is the famous Lena, which is a 256 grayscale image and the
watermark is the $64\times 64$ pixels binary image depicted in Fig. <ref>a.
The embedding domain will be the spatial domain. The selected MSCs are the
four most significant bits of each pixel and the LSCs are the three
following bits (a given pixel will at most be modified by four levels of
gray by an iteration). The last bit is then not used.
Lastly, LSCs of Lena are substituted by the bits of the mixed watermark.
(a) Watermark.
(b) Watermarked Lena.
Watermark and watermarked Lena.
§.§.§ Mixture of the Watermark
The initial state $x^{0}$ of the system is constituted by the
watermark, considered as a boolean vector. The iteration function is the
vectorial logical negation $f_{0}$ and the chaotic strategy $(S^{k})_{k\in %
\mathds{N}}$ will depend on whether an authenticated watermarking method is
desired or not, as follows.
A chaotic boolean vector is generated by a number $T$ of iterations of a
logistic map ($(\mu ,U_{0})$ parameters will constitute the private key).
Then, in case of unauthenticated watermarking, the bits of the chaotic
boolean vector are grouped six by six, to obtain a sequence of integers
lower than 64, which will constitute the chaotic strategy. In case of
authentication, the bitwise exclusive or (XOR) is made between the chaotic
boolean vector and the MSCs and the result is converted into a chaotic
strategy by joining its bits as above. Thus, the mixed watermark is the
last boolean vector generated by the chaotic iterations.
§.§.§ Embedding of the Watermark
To embed the watermark, the sequence $(U^{k})_{k\in \mathds{N}}$ of altered
bits taken from the LSCs must be defined. To do so, the strategy $%
(S^{k})_{k\in \mathds{N}}$ of the mixture stage is used as follows
\begin{equation}
\left\{
\begin{array}{lll}
U^{0} & = & S^{0} \\
U^{n+1} & = & S^{n+1}+2\times U^{n}+n ~ (\textrm{mod } \textsf{M}).
\end{array}%
\right.
\end{equation}
To obtain the result depicted in Fig. <ref>b.
Remark that the map $\theta \mapsto 2\theta $ of the torus, which is a
famous example of topological Devaney's chaos <cit.>, has been chosen
to make $(U^{k})_{k\in \mathds{N}}$ highly sensitive to the chaotic strategy.
As a consequence, $(U^{k})_{k\in \mathds{N}}$ is highly sensitive to the
alteration of the MSCs: in case of authentication, any significant
modification of the watermarked image will lead to a completely different
extracted watermark.
§.§ Simulation Results
To prove the efficiency and the robustness of the proposed algorithm, some
attacks are applied to our chaotic watermarked image. For each attack, a
similarity percentage with the watermark is computed, this percentage is the
number of equal bits between the original and the extracted watermark.
§.§.§ Zeroing Attack
In this kind of attack, some pixels of the image are put to 0. In this case, the results in Table <ref> have been obtained. We can conclude that in case of unauthentication, the
watermark still remains after a cropping attack: the desired robustness is
reached. In case of authentication, even a small change of the carrier image leads to a very different extracted watermark.
In this case, any attempt to alter the carrier image will be signaled.
2|c||UNAUTHENTICATION 2c|AUTHENTICATION
Size (pixels) Similarity Size (pixels) Similarity
10 99.08% 10 89.81%
50 97.31% 50 54.54%
100 92.43% 100 52.24%
Zeroing attacks.
§.§.§ Rotation Attack
Let $r_{\theta }$ be the rotation of angle $\theta $ around the center $%
(128, 128)$ of the carrier image. So, the transformation $r_{-\theta }\circ
r_{\theta }$ is applied to the watermarked image.
The good results in Table <ref> are obtained.
2|c||UNAUTHENTICATION 2c|AUTHENTICATION
Angle Similarity Angle Similarity
5° 94.67% 5° 59.47%
10° 91.30% 10° 54.51%
25° 80.85% 25° 50.21%
Rotation attacks.
§.§.§ JPEG Compression
A JPEG compression is applied to the watermarked image, depending on a
compression level. Let us notice that this attack leads to a change of
the representation domain (from spatial to DCT domain). In this case, the
results in Table <ref> have been found.
A good authentication through JPEG attack is obtained. As for the
unauthentication case, the watermark still remains after a compression level
equal to 10. This is a good result if we take into account the fact that we
use spatial embedding.
2|c||UNAUTHENTICATION 2c|AUTHENTICATION
Ratio Similarity Ratio Similarity
2 82.95% 2 54.39%
5 65.23% 5 53.46%
10 60.22% 10 50.14%
JPEG compression attacks.
§.§.§ Gaussian Noise
Watermarked image can be also attacked by the addition of a Gaussian noise,
depending on a standard deviation. In this case, the results in Table <ref> have
been found.
2|c||UNAUTHENTICATION 2c|AUTHENTICATION
Standard dev. Similarity Standard dev. Similarity
1 74.26% 1 52.05%
2 63.33% 2 50.95%
3 57.44% 3 49.65%
Gaussian noise attacks.
§ DISCUSSION AND FUTURE WORK
In this paper, a new way to generate watermarking methods is proposed. The
new procedure depends on a general description of the carrier medium to
watermark, in terms of the significance of some coefficients we called MSC
and LSC.
Its mixture and also the selection of coefficients to alter are based on
chaotic iterations, which generate topological chaos in the sense of Devaney. Thus,
the proposed algorithm possesses expected desirable properties for
a watermarking algorithm. For example, strong authentication of the
carrier image, security, resistance to attacks, and discretion.
The algorithm has been evaluated through attacks and the results expected
by our study have been experimentally obtained. The aim was not to find the
best watermarking method generated by our general algorithm, but to give a
simple illustrated example to prove its feasibility.
In future work, other choices of iteration functions and chaotic strategies will be explored. They will be compared in order to increase authentication and resistance to attacks. Lastly, frequency domain representations will be used to select the MSCs and LSCs.
|
1511.00418
|
We propose an uncoordinated medium access control (MAC) protocol, called all-to-all broadcast coded slotted ALOHA (B-CSA) for reliable all-to-all broadcast with strict latency constraints. In B-CSA, each user acts as both transmitter and receiver in a half-duplex mode. The half-duplex mode gives rise to a double unequal error protection (DUEP) phenomenon: the more a user repeats its packet, the higher the probability that this packet is decoded by other users, but the lower the probability for this user to decode packets from others. We analyze the performance of B-CSA over the packet erasure channel for a finite frame length. In particular, we provide a general analysis of stopping sets for B-CSA and derive an analytical approximation of the performance in the EF region, which captures the DUEP feature of B-CSA. Simulation results reveal that the proposed approximation predicts very well the performance of B-CSA in the EF region. Finally, we consider the application of B-CSA to vehicular communications and compare its performance with that of carrier sense multiple access (CSMA), the current MAC protocol in vehicular networks. The results show that B-CSA is able to support a much larger number of users than CSMA with the same reliability.
§ INTRODUCTION
Random access protocols based on slotted ALOHA <cit.> are widely used in wireless communication systems in order to support uncoordinated transmissions from a large number of users. These protocols offer low latency in scenarios in which each user is only intermittently transmitting. In slotted ALOHA, time is divided into slots and users select a single slot at random for transmission. If two packets are transmitted in the same slot, the respective receiver observes a collision and the colliding packets are considered lost, which significantly limits the efficiency of slotted ALOHA.
In <cit.>, it was suggested to repeat packets twice in randomly selected slots, thus slightly increasing the probability of a successful transmission. In <cit.>, it was further suggested to utilize SIC, as explained in the following. The system operates in frames, where each frame is a periodically occurring structure that consists of a predefined number of slots. All users are assumed to be frame-synchronized. Each user transmits multiple copies (two or three) of its packet in a single frame, each copy in a different slot. Each copy of a packet contains pointers to all other copies of a packet. Once one copy is successfully received, the positions of the other copies are obtained and their interference in the respective slots is subtracted. Exploiting SIC in <cit.> provides significant performance improvement with respect to slotted ALOHA.
SIC is also used in many other applications, e.g., <cit.> to combat the hidden terminal problem in wireless networks, or in <cit.>, where it is combined with network coding.
In <cit.>, it was proposed to use different repetition factors for different users. To that end, users choose their repetition factor by drawing a random number according to a predefined distribution. It was recognized in <cit.> that SIC for the described protocol is similar to decoding of graph-based codes over the binary erasure channel. Hence, the theory of codes on graphs can be used to design good distributions. In <cit.>, it was shown that using the so-called soliton distribution allows transmitting one packet in each slot when the frame length goes to infinity, which can be seen as the “capacity” of the protocol in <cit.>. Coding over packets was used in <cit.> in the protocol termed CSA in order to achieve high efficiency under transmit energy constraints. A protocol without a fixed frame structure was proposed in <cit.>.
The protocols in <cit.> are designed for unicast transmission, i.e., when several users transmit to a common receiver (several receivers are possible in <cit.>). In this paper, we consider a scenario where users exchange messages between each other, which is referred to as an all-to-all broadcast scenario. This is a standard scenario used as a context for distributed consensus algorithms <cit.>. However, we chiefly draw our motivation from the emerging wireless scenario of VC, in which cars exchange safety messages. We propose a novel MAC protocol based on CSA, which we call ABCSA. In particular, each user is equipped with a half-duplex transceiver, so that a user cannot receive packets in the slots it uses for transmission.[If full-duplex communication is possible, the analysis of the all-to-all broadcast scenario is identical to that of the unicast scenario, since each receiver operates undisturbed by its own transmissions.] The half-duplex mode gives rise to a DUEP property: the more the user repeats its packet, the higher the chance for this packet to be decoded by other users, but the lower the number of available slots to receive in and, hence, the lower the chance to decode packets of others.
The proposed protocol provides a reliable access for a large number of devices under strict latency constraints, thus satisfying the needs of VC and other applications of future communications systems <cit.>. Since low latency is crucial in such applications, we analyze the performance of B-CSA in the finite frame length regime, which causes the appearance of an EF in the performance of B-CSA. The EF is due to stopping sets, which are harmful graph structures <cit.> that prevent iterative decoding. Stopping sets are extensively analyzed for graph-based codes. In particular, stopping sets for a specific graph are well studied in <cit.> and references therein. CSA, on the other hand, is represented by a random graph and can be seen as a code ensemble. Stopping sets for code ensembles were analyzed in <cit.>, however, the obtained results are intractable for code lengths of interest and irregular codes. A similar approach to <cit.> was applied to CSA in <cit.>, where quite loose bounds were obtained. In <cit.> we proposed a low complexity EF approximation for CSA at low-to-moderate channel load based on the heuristically determined “dominant” stopping sets, which is very accurate in the EF region. Here, we extend the analysis in <cit.> and propose a systematic way of determining dominant stopping sets together with their probabilities. The proposed analysis is able to capture the DUEP feature of B-CSA. The analytical approximation shows good agreement with the simulation results for low-to-moderate channel loads and can be used to optimize the parameters of B-CSA.
Finally, we compare the performance of B-CSA with that of CSMA, currently adopted as the MAC protocol for VC over the PEC. The use of the PEC is justified in that it provides a simplified model of the fading channel<cit.>, which allows for a tractable analysis. Moreover, as we show in the paper, the all-to-all broadcast communication with half-duplex operation can be modeled as a PEC. More accurate channel models were considered in <cit.>, however, they do not allow for a system optimization due to computational complexity. Our analysis shows that ABCSA significantly outperforms CSMA for channel loads of interest and that it is more robust to channel erasures due to the inherent time diversity.
The contributions are summarized in the following. (a) All-to-all broadcast CSA is proposed;
(b) The analysis of CSA over the PEC in <cit.> is extended to the all-to-all broadcast scenario;
(c) A more accurate analysis of the performance of CSA compared to <cit.> is presented, which includes a rigorous analysis of the probability of stopping sets and a systematic search of dominant stopping sets;
(d) An analytical treatment of the DUEP property for large frame lengths is presented;
(e) A comparison of B-CSA with CSMA for VC over the PEC is carried out.
§ SYSTEM MODEL
We consider a network of $m+1$ users, which exchange messages between each other using half-duplex transceivers. We assume that time is divided into frames, users are frame synchronized,[The synchronization can be achieved by means of, e.g., GPS, which provides an absolute time reference for all users.] and each user transmits one packet per frame. Frames are divided into $n$ slots, each slot matching the packet length.
The transmission phase of the ABCSA protocol is identical to that of unicast CSA <cit.>, and is briefly described in the following. Every user draws a random number $l$ based on a predefined probability distribution, maps its message to a PHY packet, and then repeats it $l$ times in randomly and uniformly selected slots within one frame, as shown in fig:system_model(a). Such a user is called a degree-$l$ user. Every packet contains pointers to its copies, so that, once a packet is successfully decoded, full information about the location of the copies is available.[The pointers can be efficiently represented by a seed for a random generator to reduce the overhead, as suggested in <cit.>. We therefore ignore the overhead in this paper.]
[Users' transmissions in a B-CSA system. Rectangles represent transmitted packets. The time slots in which user $\uA$ cannot receive are shown with gray. Erased packets due to the PEC are shown with hatched rectangles.]
[Original graph $\setGt$. The dashed lines correspond to the packets erased due to the PEC, i.e., the solid lines show the PEC induced graph $\setG$. ]
[PEC induced graph $\setG$. The dashed lines show the nodes and the packets erased due to broadcast, i.e., solid lines show the broadcast induced graph $\setG^{(\rxdeg)}$ for user $\uA$.]
System model.
The main difference of the proposed ABCSA protocol compared to unicast CSA is that every user is also a receiver. Whenever a user does not transmit, it buffers the received signal. Without loss of generality, we focus on the performance of a single user, denoted by $\uA$, also referred to as the receiver. $\mathcal{U}$ denotes the set of the other $m$ users, termed neighbors of user $\uA$.
Since the frames are independent, it is sufficient to analyze the system within one frame. The received signal buffered by user $\uA$ in slot $i$ is
\begin{equation*}
y_i = \sum_{j \in \setU_i} h_{i,j} a_{j},
\end{equation*}
where $a_j$ is a packet of the $j$-th user in $\mathcal{U}$, $h_{i,j}$ is the channel coefficient between user $j$ and the receiver, and $\setU_i \subset \mathcal{U}$ is the set of user $\uA$'s neighbors that transmit in the $i$-th slot.
The $i$-th slot is called a singleton slot if it contains only one packet. If it contains packets from more that one user, we say that a collision occurs in the $i$-th slot. Decoding proceeds as follows. First, user $\uA$ decodes the packets in singleton slots and obtains the location of their copies. We assume decoding is possible if the corresponding channel coefficient satisfies $|h_{i,j}| > C$, where $C$ is a threshold that depends on the physical layer implementation. If $|h_{i,j}| \le C$, we assume that the packet is erased, i.e., it cannot be decoded and does not cause any interference. The channel coefficients are assumed to be independent across users and slots and identically distributed such that $\Pr{|h_{i,j}| \le C} = \epsilon$ and $\Pr{|h_{i,j}| > C} = 1- \epsilon$, i.e., $\epsilon$ is the probability of a packet erasure. We refer to such a channel as a PEC.
Using data-aided methods <cit.>, the channel coefficients corresponding to the copies are then estimated. After subtracting the interference caused by the identified copies, decoding proceeds until no further singleton slots are found. We assume perfect interference cancellation, which is justified by physical layer simulation results in <cit.>. We remark that decoding is always performed in singleton slots, such that the code rates of different users do not have to satisfy the rate constraint for joint decoding <cit.>.
The system can be represented by a bipartite graph and can be analyzed using the theory of codes on graphs <cit.>. In the graph, each user corresponds to a VN and represents a repetition code, whereas slots correspond to CN and can be seen as single parity-check codes. In the following, the terms “users” and “VN” are used interchangeably. A bipartite graph is defined by $\setG =({\setV}, {\setC}, {\setE})$, where $\setV$, $\setC$, and $\setE$ represent the sets of VN, CN, and edges connecting them, respectively. The number of edges connected to a node is called the node degree. An important parameter in the graph is the VN degree distribution <cit.>
\begin{equation}\label{eq:distr_orig}
\lambda(x) = \sum_{l = 0}^{\maxd}\lambda_{l}x^{l},
\end{equation}
where $\lambda_l$ is the probability of a VN to have a degree $l$ (i.e., the probability that the user transmits $l$ copies of its packet) and $\maxd$ is the maximum degree. We define a vector representation of (<ref>) as $\bm{{\lambda}} = [{\lambda}_0, \dots, {\lambda}_q]$.[In this paper, we consider only distributions with finite $q$ to avoid some technical problems in the following. This condition is always satisfied for all distributions of practical interest.] Furthermore, we define the graph profile as the vector $\bm{v}(\setG) = [ v_0(\setG), v_1(\setG), \dots, v_q(\setG)]$, where $v_l(\setG)$ is the number of degree-$l$ VN in $\setG$. The total number of VNs in $\setG$ is denoted by $\nu(\setG)$ and the total number of CNs is denoted by $\mu(\setG)$, i.e., $\nu(\setG) = m$ and $\mu(\setG) = n$.
In this paper, we focus on the broadcast PLR, defined as the probability that a user in $\setU$ is not resolved by the receiver. Since all users are independent, the PLR can be calculated as
\begin{equation}\label{eq:plr_first}
\bar{p} = \frac{\bar{w}}{m},
\end{equation}
where $\bar{w}$ is the average number of users that are not successfully decoded by user $\uA$, termed unresolved users. Note that the PLR gives the probability that a packet of a user is not successfully received within a frame and does not refer to the copies sent by the user.
We define the channel load as the ratio of contending users and the number of slots, i.e., $g = (m+1)/n$. It should be noted that, in the unicast scenario, the channel load is calculated as $g = m/n$, since the receiver is not contending.
§ INDUCED DISTRIBUTION AND PACKET LOSS RATE
For transmission over a PEC, we showed in <cit.> that the performance of unicast CSA can be accurately approximated based on an ID observed by the receiver. The fact that users in ABCSA cannot receive in the slots they use for transmission can also be modeled as packet erasures. Therefore, its performance can also be analyzed by means of the ID. In this section, we derive the ID in the general case of ABCSA over the PEC. To this end, we first find the degree distribution after the PEC and then the degree distribution perceived by user $\uA$. Throughout the paper, $l$ and $d$ denote the original and the induced degrees of a user in $\setU$, respectively, and $r$ denotes the degree of the receiver.
§.§ Induced Distribution
For ABCSA over the PEC, three different graphs can be defined. The first one is the original graph, denoted by $\setGt$, that contains the edges $\tilde{\setE} = \{e_{i,j}: 1\le i \le n, \,\forall j \in \setU_i\}$. We call its degree distribution the original distribution (the one used by the users for transmission) and denote it by $\lambdat(x)$. The original graph corresponds to that of unicast CSA <cit.> and its distribution is in the hands of the system designer.
The PEC induced graph, denoted by $\setG$, includes only the edges $e_{i,j} \in \tilde{\setE}$, for which $|h_{i,j}| > C$. In other words, $\setG$ is obtained from $\setGt$ by removing the edges corresponding to the erased packets. Since all elements of $\setG$ are contained in $\setGt$, we call $\setG$ a subgraph of $\setGt$ and write $\setG \subset \setGt$ <cit.>. The VN degree distribution of the PEC induced graph is called the PEC ID and is denoted by $\lambda(x)$. The graph $\setG$ is what a base station in unicast CSA would observe after the PEC. However, only part of this graph is available to user $\uA$ due to the half-duplex operation. Assuming that user $\uA$ selects degree $\rxdeg$, we denote its available subgraph by $\setG^{(\rxdeg)}$ and call it a broadcast induced graph.
$\setG^{(\rxdeg)}$ can be obtained from $\setG$ by removing the $\rxdeg$ CNs corresponding to the slots where user $\uA$ transmits and their adjacent edges. We call the degree distribution of this graph the broadcast ID and denote it by $\lambda^{(\rxdeg)}(x)$. The number of check nodes in the broadcast induced graph, $n^{(\rxdeg)} = \mu(\setG^{(\rxdeg)}) = n - \rxdeg$, is called the induced frame length. For the example in fig:system_model(a), $\setGt$, $\setG$, and $\setG^{(\rxdeg)}$ are shown in Figs. <ref>(b) and <ref>(c).
We now derive the PEC ID. Let a user from the set $\setU$ repeat its packet $l$ times. Each copy of this packet is erased with probability $\epsilon$. Hence, its degree in graph $\setG$ is $k \le l$ with probability $\binom{l}{k} \epsilon^{l-k} (1-\epsilon)^{k}$. Averaging over the original distribution $\tilde{\lambda}(x)$ leads to the PEC ID
\begin{equation*}\label{eq:induced_distribution_pre}
\lambda(x) = \sum_{l = 0}^{\maxd}\tilde{\lambda}_{l} \sum_{k = 0}^{l} \binom{l}{k} \epsilon^{l-k} (1-\epsilon)^{k}x^{k},
\end{equation*}
which can be written in the standard form (<ref>), where
\begin{equation}\label{eq:lambda}
\lambda_{l} = \sum_{k = l}^{\maxd}\binom{k}{l} \epsilon^{k-l} (1-\epsilon)^{l}\tilde{\lambda}_{k}.
\end{equation}
Note that $\lambda_l$ is the fraction of users of degree $l$ after the PEC.
Assuming that user $\uA$ selects degree $\rxdeg$, another user that has degree $l$ after the PEC is perceived by user $\uA$ as a degree-$d$ user if its non-erased transmissions take place in $l-d$ slots that are also selected by user $\uA$, which occurs with probability $\binom{n-\rxdeg}{d}\binom{\rxdeg}{l-d}/\binom{n}{l}$. Given the constraint $0\le l-d \le \rxdeg$, the broadcast ID $\lambda^{(\rxdeg)}(x)$ observed by user $\uA$ can be written as
\begin{equation*}
\lambda^{(\rxdeg)}(x) = \sum_{d = 0}^{\maxd}\lambda^{(\rxdeg)}_{d}x^{d},
\end{equation*}
\begin{equation}
\lambda^{(\rxdeg)}_d = \sum_{l = d}^{\min\{q, \rxdeg+d\}} \frac{\binom{n-\rxdeg}{d}\binom{\rxdeg}{l-d}}{\binom{n}{l}}\lambda_{l}\label{eq:t_induced}
% & =\sum_{l = d}^{\min\{q, \rxdeg+d\}} \frac{\binom{n-\rxdeg}{d}\binom{\rxdeg}{l-d}}{\binom{n}{l}}\sum_{k = l}^{\maxd}\binom{k}{l} \epsilon^{k-l} (1-\epsilon)^{l}\tilde{\lambda}_{k}\label{eq:induced_final}
\end{equation}
is the fraction of users of degree $d$ as observed by user $\uA$ if it selects degree $\rxdeg$. Clearly, the broadcast ID depends on $n$, as opposed to the original and the PEC ID. The IDs in <cit.> and <cit.> are a special case of (<ref>) when $\rxdeg=0$ and $\epsilon = 0$, respectively.
For the original distribution
\begin{equation} \label{eq:distr_example}
\lambdat(x) = 0.5 x^2 + 0.5 x^4,
\end{equation}
the PEC ID for $\epsilon = 0.01$ is
\begin{equation}\label{eq:distr_example_pec}
\lambda(x) = 0.00005+ 0.0099x + 0.49x^2 + 0.019 x^3 + 0.48 x^4.
\end{equation}
The broadcast IDs depend on $n$. For $n = 100$, the broadcast ID are
\begin{align}
\lambda^{(2)}(x) &= 0.0004 + 0.03x + 0.47x^2 + 0.06 x^3 + 0.44 x^4,\label{eq:ex_dist1}\\
\lambda^{(4)}(x) &= 0.001 + 0.05x + 0.46x^2 + 0.09 x^3 + 0.41 x^4.\label{eq:ex_dist2}
\end{align}
for a receiver degree 2 and 4, respectively.
The coefficient in front of $\lambda_l$ in (<ref>) can be written as
\begin{equation*}
\frac{(n-\rxdeg)! (n-l)!}{n! (n - \rxdeg - d)!}\frac{\rxdeg! l! }{d! (\rxdeg - l + d)! (l- d)!}\propto n^{d- l},
\end{equation*}
i.e., it tends to zero as $n \to \infty$ if $l > d$. Since $l \ge d$, it can be shown that for any finite $\rxdeg$, $\lambda_d^{(\rxdeg)} = \lambda_d$ for all $\rxdeg$ and $d$ when $n \rightarrow \infty$, i.e., $\lambda^{(\rxdeg)}(x) \approx \lambda(x)$. This means that the effect of the broadcast nature of communications is negligible if the number of slots is large enough.
However, when the number of slots is small, which is the case in delay critical applications, the difference between the PEC ID and the broadcast ID is significant, especially if $\rxdeg$ is large.
§.§ Packet Loss Rate
Let $p_d^{(\rxdeg)}$ denote the probability that a user with degree $d$ in the broadcast induced graph $\setG^{(\rxdeg)}$ is not resolved by a receiver of degree $\rxdeg$. We refer to $p_d^{(\rxdeg)}$ as the degree-$d$ PLR. It can be calculated as
\begin{equation}\label{eq:plr_tot}
p_d^{(\rxdeg)} = \frac{\bar{w}^{(\rxdeg)}_d}{\bar{m}^{(\rxdeg)}_{d}} = \frac{\bar{w}^{(\rxdeg)}_d}{m \lambda^{(\rxdeg)}_{d}},
\end{equation}
where $\bar{m}^{(\rxdeg)}_{d}$ and $\bar{w}^{(\rxdeg)}_d$ are the average number of total and unresolved users of degree $d$ in $\setG^{(\rxdeg)}$, respectively. For $d = 0$, $p_0^{(\rxdeg)} = 1$ and $\bar{w}^{(\rxdeg)}_0 = m \lambda^{(\rxdeg)}_0$. For other degrees, we show how $\bar{w}^{(\rxdeg)}_d$ can be approximated based on the broadcast ID in the next section. The probability that a degree-$\rxdeg$ receiver cannot resolve a user is called the average PLR and can be obtained by averaging (<ref>) over the broadcast ID as
\begin{equation}\label{eq:plr_rx}
p^{(\rxdeg)} = \sum_{d = 0}^{q}\lambda_{d}^{(\rxdeg)} p_{d}^{(\rxdeg)}. % = \sum_{l = 0}^{q}\lambdat_{l} \tilde{p}_{l}^{(\rxdeg)}.
\end{equation}
The probability that the original degree-$l$ user is not resolved by a degree-$\rxdeg$ receiver can be obtained as
\begin{align}
\tilde{p}_{l}^{(\rxdeg)}
%&= \sum_{k = 0}^{l} \binom{l}{k}\epsilon^{l-k} (1 - %\epsilon)^{k}\check{p}_k^{(\rxdeg)}\\
&= \sum_{k = 0}^{l} \binom{l}{k}\epsilon^{l-k} (1 - \epsilon)^{k} \hspace{-0.7cm} \sum_{d = \mathrm{max}\{k -\rxdeg, 0\}}^{k} \hspace{-0.3 cm} \frac{\binom{n-\rxdeg}{d}\binom{\rxdeg}{k-d}}{\binom{n}{k}} p_{d}^{(\rxdeg)},\label{eq:plr_tx_rx}
\end{align}
by reversing the operations in eq:lambdaeq:t_induced. Finally, the broadcast PLR in (<ref>) is obtained as
\begin{equation}\label{eq:plt_rx_aver}
\bar{p} = \sum_{\rxdeg = 0}^{q}\lambdat_{\rxdeg} p^{(\rxdeg)}.
\end{equation}
From the equations above it is clear that the performance of a ABCSA system depends on both the receiver and the transmitter degrees. We call this property DUEP and formalize it in the following lemmas.
For a given original distribution $\lambdat(x)$ and any $n$, $\tilde{p}_l^{(\hat{\rxdeg})} \ge \tilde{p}_l^{(\rxdeg)}$ if $\hat{\rxdeg} > \rxdeg$.
We prove the lemma by contradiction. Assume that the opposite holds, i.e., $\tilde{p}_l^{(\hat{\rxdeg})} < \tilde{p}_l^{(\rxdeg)}$ if $\hat{\rxdeg} > \rxdeg$. This implies that a degree-$\rxdeg$ receiver can improve its performance by ignoring $(\hat{\rxdeg}-\rxdeg)$ randomly selected slots, which cannot be true, as ignoring the slot information is the worst possible way to use the slot. This leads to a contradiction.
Lemma <ref> describes the DUEP from the receiver perspective. The DUEP from the transmitter perspective is discussed in Lemma 2 in the following section.
§ FINITE FRAME LENGTH ANALYSIS
From eq:plr_toteq:plt_rx_aver it follows that the degree-$d$ PLR $p_d^{(\rxdeg)}$ is sufficient to describe all performance metrics for ABCSA. $p_d^{(\rxdeg)}$ only depends on the distribution $\lambda^{(\rxdeg)}(x)$ and the induced frame length $n^{(\rxdeg)}$ seen by the receiver. The nature of these parameters is immaterial for the performance analysis. The receiver can be a user in a broadcast scenario that sees the broadcast ID $\lambda^{(\rxdeg)}(x)$. Alternatively, it can be thought of as a receiver in a unicast scenario in which the contending users use $\lambda^{(\rxdeg)}(x)$ as the original distribution with the frame length $n^{(\rxdeg)}$. Therefore, for the sake of simplicity, in this section we consider a unicast scenario, we omit superscript $(\rxdeg)$ and analyze $p_d$ in (<ref>) for frame length $n$ and an arbitrary distribution $\lambda(x)$ used to generate graph $\setG$.
For $n \to \infty$, the typical performance of CSA exhibits a threshold behavior, i.e., all users are successfully resolved if the channel load is below a certain threshold value, which can be obtained via DE <cit.>. The threshold, denoted by $g^*(\bm{\lambda})$, depends only on the degree distribution $\lambda(x)$.
We use the analysis in <cit.> to describe the DUEP from the transmitter perspective in the following lemma.
For a given distribution $\lambda(x)$, any load $0 < g \le 1$, and sufficiently large $n$, $p_{\hat{d}}< p_{d}$ if $\hat{d} > d$.
We denote the degree-$d$ PLR at the $\rho$-th decoding iteration, $\rho \ge 1$, as a function of $n$ by $p_d(n, \rho)$.
According to the analysis in <cit.>,
\begin{equation} \label{eq:DE_UEP}
\lim_{n \to \infty} p_d(n, \rho) = (\xi_\rho)^d
\end{equation}
for any finite $d$, where $\xi_\rho$ is the probability of not removing an edge at the $\rho$-th decoding iteration. This probability is obtained recursively as <cit.>
\begin{equation}\label{eq:DE_eq}
\xi_\rho = 1 - \exp{\left(-g\lambda'(\xi_{\rho-1})\right)},
\end{equation}
where the prime denotes the derivative and $\xi_0 = 1$. (<ref>) asserts that for any $\theta >0$ there exists an $n$ such that
\begin{equation}\label{eq:de_ineq}
(\xi_\rho)^d - \theta \le p_d(n, \rho) \le (\xi_\rho)^d + \theta.
\end{equation}
It is easy to show from (<ref>) that $\xi_\rho < 1$ for $0<g\le 1$ and any $\rho \ge 1$. Hence, $(\xi_\rho)^{\hat{d}}< (\xi_{\rho})^{d}$ and we can always find $\theta > 0$ such that $(\xi_\rho)^{\hat{d}} +\theta< (\xi_{\rho})^{d} - \theta$. Thus, according to (<ref>) there exists an $n$ such that $p_{\hat{d}} < p_{d}$ at any iteration.
In practice, it is sufficient that $n\gg q$ to guarantee $p_{\hat{d}} < p_{d}$ if $\hat{d}> d$.
Lemma <ref> states that users that transmits more have a higher probability of being decoded by the receiver for sufficiently large frame lengths.
The finite frame length regime gives rise to an EF in the PLR performance of CSA. This EF is due to stopping sets in the graph $\setG$. In this section, we first define stopping sets and analyze their contribution to the PLR. We then identify the stopping sets that contribute the most to the EF and propose an analytical approximation to the performance in the EF region.
§.§ Stopping Sets and Their Contribution to Packet Loss Rate
Since erased packets are accounted for in the ID, the only source of errors in the considered model is represented by the harmful structures in the graph $\mathcal{G}$. For example, when two degree-$2$ users transmit in the same slots (see fig:error_events(a)), the receiver is not able to resolve them. Such harmful structures are commonly referred to as stopping sets <cit.>.
A connected bipartite graph $\setS$ is a stopping set if all CN in $\setS$ have a degree larger than one.
We say that a stopping set $\setS$ has profile $\bm{v}(\setS)$ and contains $\nu(\setS)$ VNs and $\mu(\setS)$ CNs. For example, for the stopping set in fig:error_events(a), the graph profile is $\bm{v}(\setS) = [0, 0, 2,0,\dots, 0]$, where the number of zeros in the end depends on $q$, $\nu(\setS) = 2$, and $\mu(\setS) = 2$. Stopping sets are referred to as “loops” in <cit.>. However, stopping sets do not necessarily form a loop if degree-$1$ users are present. To analyze stopping sets, we define a VN-induced graph as follows.
[Stopping set $\setS$.]
[Graph $\setG$.]
Number of stopping sets $\setS$ in graph $G$.
A graph consisting of a subset of VNs of a graph $\setG$ and all their neighbors is called a VN-induced graph.
Since the graph $\setG$ and its profile $\bm{v}(\setG)$ are random, the average number of unresolved degree-$d$ users in (<ref>) can be expressed using stopping sets as
\begin{equation}\label{eq:wl_super_general}
\bar{w}_d = \expect{\setG}{\sum_{\setS \in \setA}v_d(\setS) \hat{w}(\setS, \setG)},
\end{equation}
where $\setA$ is the set of all possible stopping sets for given $n$ and $m$, $v_d(\setS)$ is the number of degree-$d$ VNs in $\setS$, and $\expect{x}{\cdot}$ denotes the expectation over the random variable $x$. $\hat{w}(\setS, \setG)$ in (<ref>) is the number of VN-induced graphs in a graph realization $\setG$ which are both: i) isomorphic with $\setS$ <cit.>; ii) not a subgraph of a VN-induced graph isomorphic with another stopping set in $\setA$. With a slight abuse of notation, we refer to $\hat{w}(\setS, \setG)$ as the number of stopping sets $\setS$ in a graph $\setG$. The following example explains the definition of $\hat{w}(\setS, \setG)$.
Consider the stopping set $\setS$ in fig:error_events(a) and the graph $\setG$ in fig:error_events(b). There are three VN-induced graphs in $\setG$ isomorphic with $\setS$, namely, the graphs induced by VNs $\{1,2\}$, $\{4,5\}$, and $\{5,6\}$. However, the two latter ones are subgraphs of another VN-induced graph isomorphic with a larger stopping set induced by the VNs $\{4,5,6\}$. Hence $\hat{w}(\setS, \setG) = 1$.
The averaging in (<ref>) can be done in two steps as
\begin{equation}\label{eq:wl_general}
\bar{w}_d = \expect{\bm{v}(\setG)}{\sum_{\setS \in \setA} v_d(\setS) w(\setS, \setG)},
\end{equation}
\begin{equation*}
w(\setS, \setG) = \expect{\hat{\setG}: \bm{v}(\hat{\setG}) = \bm{v}(\setG)}{\hat{w}(\setS, \hat{\setG})}
\end{equation*}
is the average number of stopping sets $\setS$ in graphs with a particular profile $\bm{v}(\setG)$.
From the definition of a stopping set it follows that, for a given profile $\bm{v}(\setG)$, the number of stopping sets can be expressed as
\begin{equation}\label{eq:w4graph}
w(\setS, \setG) = \alpha(\setS, \setG) \beta(\setS) \gamma(\setS) \delta(\setS, \setG),
\end{equation}
\begin{equation}\label{eq:multi_general}
\alpha(\setS, \setG) = \prod_{d = 1}^{\maxd} \binom{v_d(\setG)}{v_d(\setS)}
\end{equation}
is the number of ways to select VNs needed to create $\setS$ in a graph with profile $\bm{v}(\setG)$ and
\begin{equation}\label{eq:beta}
\beta(\setS) = \binom{n}{\mu(\setS)}
\end{equation}
is the number of ways to select $\mu(\setS)$ CNs out of $n$ CNs. $\gamma(\setS)$ in (<ref>) is the probability of the selected VNs to be connected to the selected CNs so that $\setS$ is created. $\gamma(\setS)$ can be written as the ratio of the number of stopping sets $\setS$ that the selected VNs can create over the total number of graphs they can create, i.e.,
\begin{equation}\label{eq:gamma_general}
\gamma(\setS) = \frac{c(\setS)} {\prod_{d = 1}^{q}\binom{n}{d}^{v_d(\setS)}},
\end{equation}
where $c(\setS)$ is the number of graphs isomorphic with $\setS$ that the selected VNs can create. Unfortunately, deriving a closed-form expression for this constant does not seem to be straightforward. However, it can be found numerically according to its definition, as demonstrated in the following example.
To find $c(\setS)$, we need to enumerate all combinations of connecting the VNs in $\setS$ to the CNs in $\setS$ so that the resulting graph is isomorphic with $\setS$. Consider stopping set $\setS$ in fig:gamma_ex(a). The only two graphs that are isomorphic with $\setS$ are shown in Figs. <ref>(b) and <ref>(c). Hence, $c(\setS) = 3$. To find $c(\setS)$, all graphs with a given profile $v(\setS)$ need to be generated and their isomorphism with $\setS$ tested <cit.>.
Isomorphic graphs for a stopping set in (a).
$\delta(\setS, \setG)$ in (<ref>) is the probability that the other $m-\nu(\setS)$ VNs are connected to CNs in such a way that another stopping set $\hat{\setS} \supset \setS$ is not created (see fig:error_events). It does not have a closed-form expression in general and we resort to an upper bound. By setting $\delta(\setS, \setG) = 1$ in (<ref>), substituting the result into (<ref>) and bringing the expectation inside the summation, we obtain the upper bound given by
\begin{equation}
\bar{w}_d \le {\sum_{\setS \in \setA} v_d(\setS) \alpha(\setS) \beta(\setS) \gamma(\setS)}, \label{eq:ub}
\end{equation}
where $\alpha(\setS)= \expect{\bm{v}(\setG)}{\alpha(\setS, \setG)}$, which can be expressed as <cit.>
\begin{equation}\label{eq:alpha_general}
\alpha(\setS) = \frac{m!}{(m - \nu(\setS))!} \prod_{d = 1}^{\maxd}\frac{\lambda_{d}^{v_{d}(\setS)}}{v_d(\setS)!}.
%\alpha(\setS) = \binom{m}{\nu(\setS)}p_{\mathsf{mn}}(\bm{v}(\setS), \bm{\lambda}, \nu(\setS)).
\end{equation}
The upperbound in (<ref>) can be seen as a union bound, in which the occurrences of stopping sets are no longer exclusive events.
We now show that the bound in (<ref>) is tight for large $n$. $\delta(\setS, \setG)$ in (<ref>) can be lower-bounded by the probability that none of these VNs is connected to the selected CNs. For a user of degree $d$, the probability of not being connected to the selected CNs is
$\frac{\binom{n-\mu(\setS)}{ d}}{ \binom{n}{d}}$
for $n \ge d + \mu(\setS)$, which together with the expression for the binomial coefficient gives
\begin{equation}\label{eq:delta_tight_lb}
\delta(\setS, \setG) \ge \prod_{d = 1}^{q} \left( \prod_{k = 0}^{d - 1} \frac{n - \mu(\setS) - k}{n - k} \right)^{(v_d(\setG) - v_d(\setS))^{+}},
\end{equation}
where $x^{+} = \max{}(0, x)$ and $n \ge q + \mu(\setS)$ (since the inequality $n \ge d + \mu(\setS)$ has to hold for any $d$). A looser lower bound can be obtained additionally assuming that all these VNs have degree $q$,
\begin{align}
\delta(\setS, \setG) &\ge \left(\prod_{k = 0}^{q-1} \frac{n - \mu(\setS) - k}{n - k} \right)^{(m - \nu(\setS))^{+}}\nonumber\\
&\ge \prod_{k = 0}^{q-1} \left( \frac{n - \mu(\setS) - q + 1}{n - q +1} \right)^{(m - \nu(\setS))^{+}}\nonumber\\
&= \left( 1 - \frac{\mu(\setS)}{n - q +1} \right)^{q(m - \nu(\setS))^{+}} \nonumber\\
&\ge \left( 1 - \frac{\mu(\setS)}{n - q +1} \right)^{qm} \nonumber\\
&\ge \exp{\left( -\frac{q \mu(\setS)m}{n - q + 1 - \mu(\setS)}\right)}, \label{eq:delta_lb}
\end{align}
where the last step follows from the inequality $\log(x) \ge 1- 1/x$ for $x > 0$. From (<ref>) it follows that, for large $n$ ($n \gg q + \mu(\setS)$) and low channel loads ($n \gg q \mu(\setS) m$), $\delta(\setS, \setG) \approx 1$, which shows that (<ref>) is tight for large $n$.
For the distribution $\lambda(x) = x$ (slotted ALOHA), i.e., all VNs have degree 1, all factors in (<ref>) can be easily calculated and the exact expression for (<ref>) can be obtained. Since all VN have degree 1, the profile of $\setG$ is deterministic with $\bm{v}(\setG) = [0,\,m]$, and the expectation in (<ref>) is trivial. Furthermore, all stopping sets have one CN with $s$, $2 \le s \le m$, VNs connected to it. Such a stopping set, denoted by $\setS_s$, has parameters
\begin{align*}
&\bm{v}(\setS_s) = [0,\,s],\quad \mu(\setS_s) = 1, \quad \alpha(\setS_s, \setG) = \binom{m}{s}, \nonumber\\
&\beta(\setS_s) = \binom{n}{1}, \, \gamma(\setS_s) = \phi^{s},
\, \delta(\setS_s, \setG) = (1-\phi)^{m-s},
\end{align*}
where $\phi = 1/n$. We remark that the expression for $\delta(\setS_s, \setG)$ coincides with the bound in (<ref>) since the VNs ouside $\setS_s$ cannot be connected to $\setS_s$ without creating a larger stopping set and $c(\setS_s) = 1$ for any $s$.
Since only degree-$1$ VNs are present in the graph, $\bar{w}_1$ (see (<ref>)) can be calculated exactly as
\begin{equation}\label{eq:w_aloha}
\bar{w}_1 = \sum_{s = 2}^{m} s \binom{m}{s} \binom{n}{1}
\phi^{s} (1 - \phi)^{m-s}.
\end{equation}
Finally, by using the properties of the binomial distribution, we can obtain the exact expression for the PLR in (<ref>) as
\begin{multline}\label{eq:plr_sa}
p = p_1 = \frac{\bar{w}_1}{m} = \frac{n}{m} \sum_{s = 2}^{m} s \binom{m}{s}
\phi^{s} (1 - \phi)^{m-s}\\
= g^{-1} (m\phi - m\phi(1-\phi)^{m-1}) = 1 - (1-\phi)^{ng-1},
\end{multline}
which corresponds to the well known expression for the PLR of framed slotted ALOHA <cit.>.[Eq. (3) in <cit.> gives the number successfully transmitting users.]
§.§ Dominant Stopping Sets and Error Floor Approximation
Identifying all stopping sets and calculating the corresponding $\gamma(\setS)$ in a systematic way for an arbitrary $n$ is not possible in general. We therefore determine stopping sets that contribute the most to the PLR for low channel loads, i.e., in the EF region. It is clear that, for low channel loads ($n \gg m$), stopping sets with a small number of nodes are more likely to occur. We therefore focus on stopping sets with few CNs. Furthermore, to reduce the number of considered stopping sets, we define minimal stopping sets as follows.
A minimal stopping set is a stopping set that does not contain a nonempty stopping set of smaller size.
It can be seen from (<ref>) and (<ref>) that for low channel loads (more precisely, when $\sum_{d} d v_d(\setS) \ll n$), each edge in $\setS$ gives a factor $n^{-1}$ in $\gamma$ and each CN gives a factor $n$ in $\beta(\setS)$. Assume now two stopping sets $\setS$ and $\hat{\setS}$ such that $\setS \subset \hat{\setS}$. To obtain $\setS$ from $\hat{\setS}$, more edges than CNs need to be removed from $\hat{\setS}$ since each CN in $\hat{\setS}$ is connected to at least two edges. Hence, the contribution of a non-minimal stopping set $\hat{\setS}$ to the PLR is smaller than that of $\setS$ for low channel loads. We therefore only consider minimal stopping sets in our analysis. Simulation results in sec:num_example justify restricting to minimal stopping sets.
We run an exhaustive search of minimal stopping sets with $\mu(\setS)$ up to five. For $\mu(\setS) \le 4$, there are 31 minimal stopping sets with the corresponding parameters given in tab:stopping_sets. For $\mu(\setS) = 5$, there are 111 minimal stopping sets (not included in this paper). We remark that the complexity of finding $c(\setS)$ depends on the size of a stopping set <cit.>. Therefore, we restrict the analysis to $\mu(\setS) \le 4$. The most dominant stopping sets presented in <cit.> are a subset of the stopping sets in tab:stopping_sets. Considering more stopping sets can improve the PLR approximation for moderate channel loads.
Parameters of minimal stopping sets with $\mu(\setS)\le 4$.
$\bm{v}(\setS)$ $\mu(\setS)$ $c(\setS)$
$[0,\,2,\,0,\,0,\,0]$ $1$ $1$
$[0,\,0,\,2,\,0,\,0]$ $2$ $1$
$[0,\,2,\,1,\,0,\,0]$ $2$ $2$
$[0,\,0,\,0,\,2,\,0]$ $3$ $1$
$[0,\,1,\,1,\,1,\,0]$ $3$ $3$
$[0,\,0,\,3,\,0,\,0]$ $3$ $6$
$[0,\,0,\,2,\,1,\,0]$ $3$ $6$
$[0,\,3,\,0,\,1,\,0]$ $3$ $6$
$[0,\,2,\,2,\,0,\,0]$ $3$ $12$
$[0,\,0,\,0,\,0,\,2]$ $4$ $1$
$[0,\,1,\,0,\,1,\,1]$ $4$ $4$
$[0,\,0,\,2,\,0,\,1]$ $4$ $6$
$[0,\,0,\,1,\,2,\,0]$ $4$ $12$
$[0,\,0,\,1,\,1,\,1]$ $4$ $12$
$[0,\,0,\,0,\,3,\,0]$ $4$ $24$
$[0,\,0,\,0,\,2,\,1]$ $4$ $12$
$[0,\,2,\,1,\,0,\,1]$ $4$ $12$
$[0,\,2,\,0,\,2,\,0]$ $4$ $24$
$[0,\,1,\,2,\,1,\,0]$ $4$ $24$
$[0,\,1,\,2,\,1,\,0]$ $4$ $24$
$[0,\,1,\,2,\,0,\,1]$ $4$ $24$
$[0,\,1,\,1,\,2,\,0]$ $4$ $48$
$[0,\,0,\,3,\,1,\,0]$ $4$ $24$
$[0,\,0,\,3,\,0,\,1]$ $4$ $24$
$[0,\,0,\,4,\,0,\,0]$ $4$ $72$
$[0,\,0,\,3,\,1,\,0]$ $4$ $144$
$[0,\,0,\,2,\,2,\,0]$ $4$ $48$
$[0,\,0,\,2,\,2,\,0]$ $4$ $48$
$[0,\,4,\,0,\,0,\,1]$ $4$ $24$
$[0,\,3,\,1,\,1,\,0]$ $4$ $72$
$[0,\,2,\,3,\,0,\,0]$ $4$ $144$
Constraining the set of the considered stopping sets in (<ref>) to minimal stopping sets with $\mu(\setS) \le 4$ and combining it with (<ref>) gives the following approximation of the degree-$d$ PLR in the EF region
\begin{equation}\label{eq:final_aprx}
p_d \approx \frac{1}{m\lambda_d}\sum_{\setS \in \mathcal{A}_{31}}{v_{d}(\setS) \alpha(\setS) \beta(\setS)}\gamma(\setS),
\end{equation}
where $\mathcal{A}_{31}$ is the set of 31 minimal stopping sets in tab:stopping_sets with $\alpha(\setS)$, $\beta(\setS)$, and $\gamma(\setS)$ given in (<ref>), (<ref>), and (<ref>), respectively. Since the considered stopping sets include VNs of degrees up to four, the approximation in (<ref>) can only be used for $d \in \{1,\dots, 4\}$. If the PLR for larger degrees needs to be estimated, the set of considered stopping sets should be extended.
We remark that, in practice, distributions with large fractions of low-degree VN are most commonly considered since they achieve high thresholds. For instance, the soliton distribution <cit.>, for which $g^*(\bm{\lambda}) = 1$, has $\lambda_2 = 0.5$ and $\lambda_3 = 0.17$ (and $\lambda_0 = \lambda_1 = 0$). Moreover, IDs for ABCSA and/or the PEC have relatively high fractions of users of low degrees due to packet erasures. We therefore conclude that the approximation in (<ref>) is accurate for estimating the average performance of most CSA systems of practical interest. This is supported by extensive numerical simulations, some of which are presented in the next section.
§.§ Asymptotic Analysis of Stopping Sets
Even though deriving exact expressions for $\gamma(\setS)$ and $\delta(\setS)$ is not possible in general, it is possible to find useful bounds on their product that can provide some insight into the asymptotic behavior of stopping sets. To that end, we express the PLR in (<ref>) as
\begin{equation}\label{eq:plr_contribution}
p_d = \sum_{\setS \in \setA} \frac{v_d(\setS)}{\lambda_d}\contrib(\setS),
\end{equation}
\begin{equation}\label{eq:contribution}
\contrib(\setS) = \expect{\bm{v}(\setG)}{\frac{w(\setS, \setG)}{m}}
\end{equation}
is the contribution of $\setS$.
The PLR in (<ref>) can be expressed in terms of $\contrib(\setS)$ as
\begin{equation}\label{eq:plr_aver_contribution}
p = \sum_{\setS \in \setA} \nu(\setS) \contrib(\setS).
\end{equation}
The asymptotic behavior of the contribution (<ref>) is characterized in the following theorem.
For any stopping set $\setS$, we have
\begin{equation}\label{eq:contrib_sim}
\contrib(\setS) \propto n^{\mu(\setS) + \nu(\setS) - \sum_{d = 1}^{q} d {v_d(\setS)}-1}
\end{equation}
for large $n$.
The proof is given in Appendix <ref>.
The contribution of a stopping set $\setS$ does not vanish as $n \rightarrow \infty$ if, and only if, $\setS$ is a tree.
Since a stopping set is a connected graph, the relation between the number of edges and the number of nodes can be written as
\begin{equation}\label{eq:nodes_graph}
\mu(\setS) + \nu(\setS) - \sum_{d = 1}^{q} d {v_d(\setS)}-1 \le 0,
\end{equation}
which holds with equality if a stopping set is a tree <cit.>.
The “only if” part of Corollary 1 is similar to <cit.>, where the probability of observing non-tree-like stopping sets for ensembles of LDPC codes are analyzed.
The typical performance of CSA exhibits a threshold behavior, i.e., all users are successfully resolved if the channel load is below a certain threshold value obtained via DE <cit.> when $n \rightarrow \infty$. The threshold depends only on the degree distribution $\lambda(x)$ and is denoted by $g^*(\bm{\lambda})$. The peculiarity of the induced degree distribution in CSA gives rise to the following corollary.
For distributions $\lambda(x)$ with $\lambda_0 \neq 0$ and/or $\lambda_1 \neq 0$, $g^*(\bm{\lambda}) = 0$.
If $\lambda_0 \neq 0$, the threshold is obviously zero, since $p \ge \lambda_0$ for any $g \ge 0$. Assume that $\lambda_0 = 0$ and $\lambda_1 \neq 0$. A stopping set can be a tree only if it contains degree-$1$ users. According to Lemma <ref>, if $\lambda_1 \neq 0$, there exist stopping sets, whose contribution is bounded away from zero for any $g>0$.
The DE and the stopping set analysis for LPDC codes predicts the ensemble average performance. Even though the actual performance of a randomly selected code is close to the predicted one according to the concentration theorem <cit.>, this analysis does not give the performance of an actual code. For CSA however this analysis gives the performance of the actual system, since the system itself can be seen as an ensemble.
§ NUMERICAL RESULTS
In this section, we first present different aspects of the ABCSA performance for the distributions in Example <ref>. We then show how the proposed EF approximation can be used for the optimization of the original distribution.
§.§ Induced Distribution and Packet Loss Rate
In fig:duep, we show the simulated PLRs $p_d^{(\rxdeg)}$ and $\tilde{p}_l^{(\rxdeg)}$ (solid lines) for the system described in Example <ref>. fig:duep(a) shows the PLR for users of different induced degrees and illustrates the DUEP property. Lines with circles show $p_d^{(2)}$ for the distribution in (<ref>) and lines with diamonds show $p_d^{(4)}$ for the distribution in (<ref>) and $d = 1,\dots,4$. It can be seen that for a given distribution, the larger the transmitter degree $d$, the better the performance, as Lemma <ref> states.
fig:duep(b) shows the simulated PLR $\tilde{p}_{l}^{(\rxdeg)}$ for the original transmitter degree $l$. As it can be seen from the figure, for a given $l$, the receiver with a smaller degree $\rxdeg$ has better performance in accordance with Lemma <ref>. On the other hand, for a given $\rxdeg$, the transmitter with a smaller degree $l$ has worse performance. The rationale behind the DUEP is that the chance of a user to be resolved by other users increases if the user transmits more, but at the same time the chance to resolve other users decreases.
We remark that the curves for $\tilde{p}_{l}^{(\rxdeg)}$ in fig:duep(b) can be alternatively obtained from the solid curves in fig:duep(a) via (<ref>). The resulting curves would then appear exactly on top of the simulation results in fig:duep(b), which confirms the correctness of (<ref>) and the derived IDs.
PLR performance of B-CSA for the scenario in Example <ref> ($n =100$). Solid curves show simulation results and dashed curves show analytical EF approximations. Circles and diamonds show the PLR for $\rxdeg = 2$ and $\rxdeg = 4$, respectively.
§.§ Error Floor Approximation
fig:duep also shows the proposed analytical EF approximation (<ref>) (dashed lines). fig:duep(a) shows (<ref>) for the distribution in (<ref>), $\rxdeg = 2$ and $d = 1,2,3,4$, whereas fig:duep(b) shows (<ref>) used together with the approximation (<ref>) for $l = 2,4$ and $\rxdeg = 2,4$. The analytical EF approximations demonstrate good agreement with the simulation results for low to moderate channel loads. This justifies the approximation in (<ref>) and the use of minimal stopping sets. It can also be seen that the approximations for $d = 1,2$ are more accurate than those for $d = 3, 4$ since the stopping sets in tab:stopping_sets contain mostly users of low degrees.
fig:plr_over_frame_length shows the dependency of the EF on the frame length for the distributions in (<ref>) and (<ref>), which correspond to $\epsilon = 0$ and $\epsilon = 0.01$, respectively, in the unicast scenario.[For large values of $n$, we use the built-in approximation in the Matlab function for the calculation of binomial coefficients.] It can be observed that without channel erasures (green curve), the EF decays exponentially with $n$. When erasures are present (red curve),
the performance first decays exponentially for small $n$ and then approaches the value predicted by DE (black dot) as $n$ grows. For $n \lessapprox 10^3$, the finite frame length is the main cause of the EF, whereas for $n \gtrapprox 10^3$ the PEC is the dominant factor causing the EF. In this case, increasing $n$ does not improve the performance. Markers show simulation results, which agree well with the analytical approximation.
PLR performance of unicast CSA versus $n$ for the distributions in eq:distr_exampleeq:distr_example_pec and $g = 0.5$. Solid lines show the analytical approximation (<ref>) with (<ref>) and markers show simulation results. The PLR value $2\cdot 10^{-4}$ predicted by DE is shown with a black dot.
Finally, in fig:DE_UEP we show how the proposed EF approximation compares with the DE results for large $n$. The solid lines show the PLR $\lim_{n \to \infty} p_d(n, \infty)$ obtained via DE (<ref>) for the distribution in (<ref>) and $d = 1,\dots, 4$ in the unicast scenario.
The proposed EF approximation (<ref>) for $n = 10^{7}$ is shown with dashed lines and agrees well with the DE curves. The lower the degree, the larger the range of channel loads for which the agreement is good. We remark that in all considered examples, the proposed analytical EF approximation always underestimates the PLR. This can be improved by including more stopping sets in (<ref>).
PLR performance of unicast CSA with the distribution in (<ref>) and $n = 10^7$. Solid lines show the PLR predicted by DE and dashed lines show the proposed EF approximation.
§.§ Distribution Optimization for All-to-All Broadcast Coded Slotted ALOHA
In this subsection, we concentrate on the broadcast scenario and discuss the optimization of the degree distribution for finite frame lengths using the proposed EF approximation. To this end, we constrain the original distribution to have the form $\tilde{\lambda}(x) = \tilde{\lambda}_2 x^2 + \tilde{\lambda}_3 x^3 + \tilde{\lambda}_4 x^4 + \tilde{\lambda}_8 x^8$. Such distributions have a good performance in the unicast scenario <cit.> and are typical for low-density parity-check codes <cit.>.
Ideally, we would like to minimize the PLR around the values of $g$ at which the PLR curve switches from the waterfall region to the EF region. However, analytical tools to predict the PLR at such channel loads are missing. The proposed EF approximation (<ref>) is accurate only for low to moderate channel loads. If the EF approximation is the only optimization objective, the optimal distribution is always $\tilde{\lambda}(x) = x^8$. However, such a distribution has $g^*(\bm{\lambda}) = 0.54$, hence, bad performance at channel loads of interest. Here, we use a linear combination of the broadcast PLR in (<ref>) and the threshold as the optimization objective, which corresponds to scalarization of a multidimensional objective function <cit.>.
For notational convenience, we write the broadcast PLR in (<ref>) as $\bar{p} (\bm{\tilde{\lambda}})$ to highlight that it depends on the distribution $\bm{\tilde{\lambda}}$ and formulate the following optimization problem
\begin{equation*}
\begin{aligned}
& \underset{\bm{\tilde{\lambda}}}{\text{minimize}}
& & -g^*(\bm{\tilde{\lambda}}) + \eta \bar{p} (\bm{\tilde{\lambda}})\\
& \text{subject to}& & \tilde{\lambda}_i \ge 0 \; i = 2, 3, 4, 8,\\
& & & \tilde{\lambda}_i = 0 \; \text{otherwise},\\
& & & \lambdat_2 + \lambdat_3 + \lambdat_4 + \lambdat_8 = 1,\\
\end{aligned}
\end{equation*}
where $\eta$ is a weighting coefficient. We numerically solve this optimization problem[We solve the optimization problem by means of the Nelder-Mead simplex algorithm <cit.>. Global optimality is therefore not guaranteed.] by using the EF approximation (<ref>) to calculate $\bar{p} (\bm{\tilde{\lambda}})$ for different values of $\eta$ and obtain the EF vs. threshold tradeoff shown in fig:tradeoff(a). The corresponding optimal distributions are shown in fig:tradeoff(b). As it can be seen from fig:tradeoff(a), the optimal distributions around $g^* = 0.85$ provide relatively high threshold for relatively low EF values ($\approx 10^{-5}$). We pick the distribution
\begin{align}\label{eq:optim_distr}
\tilde{\lambda}(x) &= 0.86 x^3 + 0.14 x^8
\end{align}
and use it in the next section when we compare ABCSA with CSMA. We remark that the choice of $g^*$ for selecting the distribution depends on the required reliability. Since global optimality of the results in fig:tradeoff is not guaranteed, better distributions can potentially be obtained. However, the presented distributions are the best known distributions that provide the EF vs. threshold tradeoff.
[EF vs. threshold tradeoff.]
[Optimal distributions.]
Distribution optimization for $n = 500$, $g = 0.5$, and $\epsilon = 0$.
The results in fig:tradeoff are obtained for $n = 500$, $g = 0.5$, and $\epsilon = 0$. We remark that the distribution in (<ref>) is close to the one presented in <cit.> for the unicast scenario. Furthermore, we observed that the optimization problem is not very sensitive to the choice of parameters $g$, $n$, and $\epsilon$.
§ BROADCAST CODED SLOTTED ALOHA IN VEHICULAR NETWORKS
In this section, we evaluate the performance of B-CSA in a vehicular network and compare it with the currently used CSMA protocol. The physical layer parameters used here are taken from <cit.> and are given in tab:params. We consider transmission of CAM <cit.> packets that are sent periodically every $\tframe$ seconds. We set the frame duration equal to $\tframe$ and we assume that the network does not change during this period. We assume that all packets have length $\tpack$, which depends on the packet size $\packsize$, transmission rate $\rdata$, and the length of the preamble added to every packet $\tpream$, i.e., $\tpack = \tpream + \packsize/\rdata$. In addition, the slot duration $\tslot = \tpack + \tguard$, where $\tguard$ accounts for timing inaccuracy. The number of slots is determined as $n = \lfloor \tframe/ \tslot \rfloor$. The inclusion of a guard interval, $\tguard$, reduces the number of slots in a frame and thus worsens the performance of B-CSA.
The PHY parameters. The values in the upper part are taken from <cit.>; the values in the lower part are derived.
Parameter Variable 2c|Value Units
Data rate $\rdata$ 2c|6 Mbps
PHY preamble $\tpream$ 2c|40 $\mu$s
CSMA slot duration $\tcsma$ 2c|13 $\mu$s
AIFS time $\taifs$ 2c|58 $\mu$s
Frame duration $\tframe$ 2c|100 ms
Guard interval $\tguard$ 2c|5 $\mu$s
Packet size $\packsize$ 200 400 byte
Packet length $\tpack$ 312 576 $\mu$s
Slot duration $\tslot$ 317 581 $\mu$s
Number of slots $n$ 315 172
§.§ Carrier Sense Multiple Access
CSMA is used as MAC protocol for vehicular networks <cit.>. We analyze the following system model that can be compared with B-CSA. We consider a network with $m+1$ users indexed by $j = 1,\dots, m+1$, where every user is within all other users transmission range, i.e., no collision occurs due to the hidden terminal problem. $m$ can be thought of as the instantaneous number of neighbors for a given user. We assume that a user can always sense other users transmissions. Collisions are assumed to be destructive. If no collision occurs, each user may not able to decode a packet with probability $\epsilon$ due to noise-induced errors. We say that such a packet is erased. Note that the backoff protocol is not affected by channel erasures and partial collisions are not possible.
The set of users is denoted by $\mathcal{V}$ and time is denoted by $t$. At the beginning of the contention ($t = 0$), every user selects a real random number $\tau_j \in [0,\,\,\tframe)$, which represents the time when the $j$-th user attempts to transmit its first packet invoking the CSMA procedure from <cit.>. The contention window size is selected to be 511 <cit.>. $\taifs$ is the sensing period during which the users sense the channel to determine whether it is busy of not, where AIFS stands for arbitration interframe space. $\tcsma$ is the duration of a backoff slot. The values of these parameters are specified in <cit.> (see tab:params for the values used in simulations).
To overcome the effect of packet erasures, each user attempts $\kappa$ transmissions of each packet. At time instant $\tau_j + (k-1) \tframe/\kappa + (i-1) \tframe$, the $j$-th user makes the $k$-th attempt, $k = 1,\dots, \kappa$, to transmit its $i$-th packet, $i = 1,\,2,\,\dots$. If by the time $\tau_j + i \tframe$ none of the copies of the $i$-th packet is transmitted, the packet is dropped. For $\kappa = 1$, the described protocol is considered in <cit.>.
The channel load is defined as the ratio of the number of users, $m+1$, and $\tframe$ (expressed in slots) to match the definition of the channel load for ABCSA, i.e., $g = (m + 1)/(\tframe/\tslot) = (m+1)/n$. The PLR for user $j$ is defined as
\begin{equation*}
p_j = 1 - \frac{\expect{\tau_1,\dots, \tau_m}{\sum_{\substack{i \in \mathcal{V}\\ i\neq j}} \eta_{i,j}}}{ m},\label{eq:plr_csma}
\end{equation*}
where $\eta_{i,j} \in \{0, 1, 2 \}$ is the number of packets of user $i$ successfully received by user $j$ over the time interval $[t_0,\,\, t_0 + \tframe)$. To estimate the performance, we introduce a time offset $t_0 = 2\tframe$ in order to remove the transient in the beginning of the contention. As the performance does not depend on the particular user, without loss of generality, we select user $j = 1$.
The performance of CSMA for different values of $\epsilon$, $\kappa$, and the frame lengths in tab:params is shown in fig:csma_pec. We can observe that the performance of CSMA degrades as $n$ increases due to the increase of sensing overhead. Furthermore, for a given $n$, increasing $\kappa$ reduces the achievable throughput but improves the performance at low channel loads. The PLR curves approach the value $\epsilon^{\kappa}$ for $g = 0$, hence, the number of repetitions can be predicted based on $\epsilon$ and the required reliability.
PLR performance for CSMA over the PEC. Solid and dashed lines correspond to $\epsilon = 0$ and $\epsilon = 0.01$, respectively. Circles and diamonds show the PLR for $n = 172$ and $n = 315$, respectively.
§.§ Carrier Sense Multiple Access vs. All-to-all Broadcast Coded Slotted ALOHA
Even though the reliability requirements are not specified in <cit.> and depend on the particular application, for the sake of comparison we assume that the broadcast PLR of interest is in the range $10^{-2}\, -\, 10^{-3}$. From fig:csma_pec it follows that for CSMA, $\kappa = 2$ provides good performance for the PEC with $\epsilon \le 10^{-2}$. For ABCSA we select the distribution in (<ref>).
The broadcast PLR of the two protocols is shown in fig:comparison for $n = 172$ and $n = 315$ (see tab:params). Simulation results for ABCSA and CSMA are shown with solid and dashed lines, respectively. The dash-dotted lines show the broadcast PLR obtained using the approximation (<ref>). The figure shows that the protocols react differently to the increase of the frame length: The CSMA performance degrades when $n$ increases whereas the performance of ABCSA improves as $n$ grows large. This gives an extra degree of freedom when designing a ABCSA system, as increasing the bandwidth will decrease the packet length and, hence, increase the number of slots. Moreover, ABCSA is robust to packet erasures for any channel load as opposed to CSMA, which suffers significantly at low channel loads. We also point out that standard CSMA with $\kappa = 1$ fails in providing the required level of reliability over the PEC with $\epsilon = 0.01$ (see fig:csma_pec).
PLR comparison of optimized CSMA and B-CSA systems for different values of $n$ and $\epsilon$. Solid and the dashed lines show the performance of B-CSA and CSMA, respectively. The dash-dotted lines show the analytical PLR approximation obtained using (<ref>).
It can also be seen from fig:comparison that ABCSA significantly outperforms CSMA for medium to high channel loads. For example, for $\epsilon = 0$, ABCSA achieves a PLR of $10^{-3}$ at channel loads $g = 0.68$ and $g = 0.73$ for $n = 172$ and $n = 315$, respectively. CSMA achieves the same reliability at $g = 0.36$ and $g = 0.32$ for $n = 172$ and $n = 315$, respectively, i.e., ABCSA can support approximately twice as many users as CSMA for this reliability. When erasures are present, the gains are even larger. For $\epsilon = 0$, CSMA yields better PLR than ABCSA for low channel loads only ($g \lessapprox 0.3$). Moreover, CSMA shows better performance for heavily loaded networks ($g>0.84$). However, in this case both protocols provide a poor reliability (PLR of around $0.4$), which is unacceptable in VC.
We remark that the high user mobility prohibits the use of acknowledgements in VC, and thereby methods for mitigating the hidden terminal problem. Thus, the performance of CSMA will be severely affected by the hidden terminal problem in a real vehicular network. On the other hand, the problem of hidden terminals does not exist in B-CSA since collisions are used for decoding and no sensing is required.
§ CONCLUSION AND DISCUSSION
In this paper, we proposed a novel uncoordinated MAC protocol for a message exchange in the all-to-all broadcast scenario. Furthermore, we analyzed its performance over the PEC for finite frame length and proposed an accurate analytical approximation of the PLR performance in the EF region. The proposed analytical approximation can be used to optimize the degree distribution for CSA in the finite frame length regime. The analysis shows that ABCSA is robust to packet erasures and is able to support a much higher number of users that can communicate reliably than the state-of-the-art MAC protocol currently used for VC.
In order to guarantee high reliability over the PEC, the PHY layer and the MAC protocol of a communication system should offer a certain level of time diversity. For protocols that do not exploit collisions, such as CSMA, increasing time diversity leads to channel congestion at low channel loads. Hence, exploiting collisions is inevitable for systems that require high reliability for high channel loads over the PEC. ABCSA is an elegant way of utilizing collisions. The obtained gains come at the expense of increased computation complexity that a system designer should be ready to pay if they wants to satisfy the stringent requirements of VC.
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1511.00544
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In this paper, we study a broker-based TV white space market, where unlicensed white space devices (WSDs) purchase white space spectrum from TV licensees via a third-party geo-location database (DB), which serves as a spectrum broker, reserving spectrum from TV licensees and then reselling the reserved spectrum to WSDs.
We propose a contract-theoretic framework for the database's spectrum reservation under demand stochasticity and information asymmetry.
In such a framework, the database offers a set of contract items in the form of reservation amount and the corresponding payment, and each WSD chooses the best contract item based on its private information.
We systematically study the optimal reservation contract design (that maximizes the database's expected profit) under two different risk-bearing schemes: DB-bearing-risk and WSD-bearing-risk, depending on who (the database or the WSDs) will bear the risk of over reservation.
Counter-intuitively, we show that the optimal contract under DB-bearing-risk leads to a higher profit for the database and a higher total network profit.
TV White Space Networks, Reservation, Contract Theory, Game Theory
§ INTRODUCTION
§.§ Background and Motivations
Nowadays, radio spectrum is becoming more congested and scarce with the explosive development of wireless services and networks.
Dynamic spectrum sharing can effectively improve the spectrum efficiency and alleviate the spectrum scarcity,
by allowing unlicensed secondary devices access to the licensed spectrum in an opportunistic manner.
TV white space network is one of the promising paradigms of dynamic spectrum sharing <cit.>, where unlicensed devices (called white space devices, WSDs) exploit the un-used or under-utilized broadcast television spectrum (called TV white spaces, TVWS[For convenience, we simply use spectrum to represent TVWS in this paper.]) opportunistically.
In order to fully utilize TVWS while not harming licensed devices, regulatory bodies (e.g., FCC in the US and Ofcom
in the UK)
have advocated a database-assisted spectrum access solution,
which relies on a third-party white space database called geo-location
<cit.>.[Based on the database-assisted solution proposed by the regulators, The IEEE 802.22 <cit.>, CEPT ECC <cit.>, and ETSI <cit.> proposed corresponding standards for WSDs operating in a database-assisted TV white space network.]
In this solution, WSDs obtain the available spectrum information through querying the geo-location database, instead of performing spectrum sensing.
More specifically, WSDs periodically report their location information and other optional information (e.g., spectrum demand) to a geo-location database, and then the database returns the available spectrum in the respective locations and time periods to WSDs.
In general, there are two types of different TV white space spectrum resources.
The first type is the TV spectrum not registered to any TV licensee or Programme Making and Special Events (PMSE) at a particular location.
This type of spectrum is usually for the open and shared usage among unlicensed WSDs, according to the regulators' policies <cit.>.
The second type is the TV spectrum already registered to some TV licensees and PMSE, but not fully utilized by those licensees.
Hence, the licensees can temporarily lease these idle spectrum to unlicensed WSDs for the exclusive usage.
In such a secondary spectrum market, the geo-location database can act as an intermediary (e.g., a broker) between the licensees (sellers) and the WSDs (buyers), due to its proximity to both sides of the market.[This model is currently employed by real-world geo-location database operators such as SpectrumBridge (<https://spectrumbridge.com/>) in the US and COGEU (<http://www.ict-cogeu.eu/>) in Europe.]
In this work, we focus on the secondary sharing and trading of the second type spectrum resource, i.e., those registered but under-utilized spectrum.
Such spectrum can be exclusively used by a WSD (with the permission of the licensees), hence are particularly suitable for supporting applications that require a high QoS.
§.§ Market Model and Problem
Broker-based spectrum reservation market. In Step 0,
the geo-location database (broker) reserves spectrum from TV licensees and PMSE for every reservation period (e.g., one day).
In Step 1, each WSD (master) reports its location and demand in every access period (e.g., one hour).
In Step 2, the database sells the corresponding spectrum to the WSD in every access period.
In Step 3, the WSD serves end-users (slaves) in every access period using the obtained spectrum.
Notice that Steps 1-3 will occur repeatedly within every reservation period, as one reservation period consists of many access periods.
Specifically, we study a broker-based secondary spectrum market, where TV licensees lease their idle spectrum to unlicensed WSDs via a spectrum broker acted by a geo-location database.
As a broker, the database purchases spectrum from TV licensees in advance, and then resells the leased spectrum to WSDs.
Figure <ref> illustrates such a broker-based spectrum reservation market model.
As the TV towers have fixed locations and TV programs have well planned schedules, the reservation period of TV spectrum can be relative long <cit.>.
Thus, we model and analyze a spectrum reservation market, where the database reserves spectrum from TV licensees in advance for a relatively large time period (e.g., more than one day), called the reservation period. Then, within each reservation period, the database sells the reserved spectrum to WSDs periodically with a relatively small time period (e.g., one hour), called the access period. Namely, the spectrum reservation decision is made at the beginning of the reservation period, which consists of many access periods. [Please refer to Section <ref> for the detailed model.]
In such a spectrum reservation market, the database needs to reserve spectrum in advance, without knowing the actual future demands from WSDs.
Therefore, an important problem for the database in this market is:
* How much bandwidth should the database reserve for each WSD, aiming at maximizing the database's profit?
The problem is challenging due to the demand stochasticity and the information asymmetry.
(i) Demand Stochasticity.
Due to the stochastic nature of end-users' activities and requirements, each WSD's spectrum demand (for serving its end-users) is a random variable, and cannot be precisely predicted by the WSD or the database in advance.
Therefore, there is inevitably a risk of reservation mismatch, e.g., spectrum over-reservation or under-reservation. Accordingly, the database's reservation decision depends on the risk-bearing scheme, namely, who will bear the risk of over-reservation: the database (called DB-bearing-risk) or the WSD (called WSD-bearing-risk)?
In the former case, the WSD only pays for the spectrum it actually purchases in every access period; while in the latter case, the WSD has to pay for the reserved spectrum(even if it is more than actually needed) in every access period.[
The DB-bearing-risk scheme is widely used in manufacturing outsourcing systems such as <cit.>, while the WSD-bearing-risk scheme is widely used in many Newsvendor models and practical retailing markets such as <cit.>.
(ii) Information Asymmetry.
The above mentioned demand information is asymmetric between the database and WSDs.
Due to the proximity to end-users, the WSD usually has more information (i.e., with less uncertainty) about the spectrum demand than the database.
This implies that the database can potentially make a better reservation decision, if it is able to know the WSD's private information regarding the demand.
However, without proper incentives, the WSD may not be willing to share its private information with the database. As will be shown in Section 5, the WSD may even report a false information to the database intentionally, as long as such a misreport can increase the WSD profit.
§.§ Results and Contributions
We propose a contract-theoretic reservation framework, in which the database offers a list of contract items in the form of reservation amount and the corresponding payment, and each WSD chooses the best contract item based on its private demand information (from its served end-users).
We first study the incentive compatible contract design, under which each WSD will disclose its private demand information credibly, by choosing the contract item intended for its private information.
With the incentive compatibility, we further derive the optimal reservation contracts that maximize the database expected profit under both DB-bearing-risk and WSD-bearing-risk schemes.
For clarity, we summarize the key results regarding the optimal contract design in Table <ref>.
As far as we know, this is the first paper that systematically studies the contract-based reservations under different risk-bearing schemes for TV white space markets.
The proposed market model, together with the derived reservation solutions, can offer the proper economic incentives for the database operator, and support the practical and commercial deployment of TV white space networks.
The main contributions of this paper are summarized as follows.
* Novel modeling and solution techniques:
We study a generic spectrum reservation market under demand stochasticity and information asymmetry, and propose a contract-theoretic reservation framework, which ensures that WSDs disclose their private information truthfully, and meanwhile maximizes the database profit.
* Optimal contract design:
We analytically derive the optimal reservation contract design under DB-bearing-risk and WSD-bearing-risk schemes, and numerically compare their performances.
Through these numerical comparisons, we characterize the impacts of risk-bearing scheme, demand stochasticity, and information asymmetry on the reservation solutions.
* Numerical results and insights:
Our numerical results show that the optimal contract under the DB-bearing-risk scheme can achieve a higher database profit and a higher total network profit, compared to the optimal contract under the WSD-bearing-risk scheme.
The intuition is that the WSD is more risk-averse than the database.
Key Results in This Paper
2|c|Information & Sharing
Spectrum Reservation Decision
Symmetry (Benchmark)
The database makes the reservation decision based on the WSD's knowledge about demand.
%= \Da + {G^{-1} \Big( \frac{w-c}{w} \Big)}
$ in Eq. (<ref>)
(Scheme I)
Asymmetry (Benchmark)
No Sharing
The database makes the reservation decision based on its own knowledge about demand.
\Kdbasy
%= (F \times G )^{-1} \Big( \frac{w-c}{w} \Big),
$ in Eq. (<ref>)
(Our Focus)
Credibly Sharing via Contract
The database offers a spectrum reservation contract, and each WSD chooses a proper contract item (reservation-payment pair).
%\CTRdb^* = \{ \langle
\Kdbctr
%(\Da),{\Pdbctr}(\Da) \rangle \}_{\forall \Da}
$ in Theorem <ref>
Each WSD makes the reservation decision based on its knowledge about demand.
$ in Eq. (<ref>)
(Scheme II)
No Sharing
Each WSD makes the reservation decision based on its knowledge about demand.
$\Kmsasy $ in Eq. (<ref>)
(Our Focus)
Credibly Sharing via Contract
The database offers a spectrum reservation contract, and each WSD chooses a proper contract item (reservation-payment pair).
$\Kmsctr$ in Theorem <ref>
The rest of this paper is organized as follows. In Section
<ref>, we review the related literature.
In Section <ref>, we present the system model.
In Sections <ref>, we provide the integrated optimal reservation solution as a benchmark.
In Sections <ref> and <ref>, we study the decentralized reservations without information sharing and with information sharing (via contract), respectively.
We provide numerical results in Section <ref>, and finally conclude in Section <ref>.
§ RELATED WORK
In the recent regulator's policy <cit.>, the databases are allowed to determine their own pricing schemes for operating the TVWS.
This motivates researchers to study the economic issues in TVWS<cit.>.
In <cit.>, Feng et al. studied the hybrid pricing scheme for the database manager.
In <cit.>, Luo et al. studied the pricing strategy of oligopoly competitive WSDs.
However, none of the existing work considered the bandwidth reservation problem under information asymmetry.
Some recent studies <cit.> proposed the pure and hybrid information models for TV white spaces, which focus on unlicensed TV white space.
Our work is related to the supply chain contract design in the operations management and marketing science literature.
Supply chain contract is widely used as a mechanism to coordinate production quantity and pricing, so that the performance of decentralized supply chain is close or the same as that of an integrated one.
In <cit.>, Cachon et al. considered the stochastic nature of demand and prescribed analytical remedies for credible information sharing between a supplier and a manufacturer.
$\ddot{O}$zer et al. in <cit.> extended Cachon's work and further examined how a supplier can screen buyers's private information by offering a menu of contracts.
However, the above work considered the case where the contract designer bears all of the risk of over-reservation. We consider both cases where the contract designer (the database) and the buyer (WSD) bears the risk of over-reservation, respectively.
Recently, the concept of contract was also introduced into the spectrum trading model (e.g. <cit.>).
In <cit.>, Gao et al. proposed a quality-price contract for the spectrum trading in a monopoly spectrum market.
In <cit.>, Duan et al. proposed a contract-based cooperative spectrum sharing mechanism to promote the cooperation of a primary user and a secondary user.
In <cit.>, Sheng et al. proposed a contract for a primary license holder to sell its excess spectrum capacity to potential secondary users.
In this paper, we propose a contract-based mechanism for the spectrum reservation problem.
In our model, the demand of a WSD consists of two parts: one is unknown by both the database and the WSD, and the other is only known by the WSD (hence is the WSD's private information).
Thus, the optimal contract design needs to consider not only the truthful information disclosure of the WSD, but also the uncertainty of demand for both the database and the WSD.
This makes our contract design much more challenging than existing contract designs.
§ SYSTEM MODEL
§.§ System Overview
We consider a TV white space network where unlicensed
WSDs exploit the under-utilized broadcast television spectrum (called TV white space, or spectrum for simplicity) via a
geo-location database.
Each WSD is an infrastructure-based device (e.g., a base station), and serves a set of unlicensed end-users/devices called “slave” devices.
We assume that the number of unlicensed WSDs is large enough, so that the spectrum demand of a particular WSD does not affect other WSDs' demand.
This allows us to concentrate on the interaction between the database and each WSD.
We focus on the secondary sharing and trading of the under-utilized licensed spectrum of TV licensees.
In particular, we model a broker-based secondary spectrum market, where the geo-location database acts as a spectrum broker, reserving spectrum from TV licensees in advance and then reselling the reserved spectrum to unlicensed WSDs.
§.§ Broker-based Spectrum Reservation Market
Spectrum reservation and access processes.
Step 0: the database reserves for every reservation period $T$;
Step 1: the WSD reports the realized demand in every access period $t$;
Step 2: the database returns to the WSD in every access period $t$;
Step 3: the WSD serves end-users in every access period $t$.
Now we discuss the proposed spectrum reservation market more detailedly.
Let $c$ denote the unit price (cost) at which the database reserves spectrum from TV licensees.
Let $w$ denote the unit price (wholesale price) at which the database sells spectrum to the WSD.
Let $r$ and $s$ denote the unit price (market price) at which the WSD serves the subscribed and un-subscribed end-users, respectively.[In Section <ref>, We will discuss the two types of users in details.]
In order to concentrate on the reservation problem, we consider a fixed spectrum trading model, that is, the trading prices $c,\ w,\ r,$ and $s$ are fixed system parameters.[Our model does allow the possibility of changing the prices over a longer time horizon. Specifically, we can divide the whole time period into multiple frames, each lasting for certain time (say several hours).
At the beginning of each frame, the WSD can adjust the trading price of $r$ and $s$ according to the congestion level of spectrum.
Then trading prices remain fix during a frame, and our results and analysis characterize the system within this frame.]
This implies that our proposed reservation framework does not need to alter the trading process, and thus is compatible with many existing spectrum market mechanism designs.
Moreover, to make the trading model meaningful, we assume that $\min\{r,s\} > w > c$, i.e., both the database and the mater will benefit from the trading process.
We illustrate the detailed spectrum reservation and trading/access processes in Figure <ref> and Algorithm <ref>.
It is notable that the spectrum reservation process (Step 0) is performed at a relatively large time period (e.g, oncen every day or every week), called the reservation period (denoted by $T$); while the spectrum trading/access processes (Steps 1-3) are performed at a relatively small time period (e.g., once per hour), called the access period (denoted by $t$).
each reservation period $T = 1, 2, \ldots$
Step 0: The database reserves $\K$ unit of from TV licensees at a unit price $c$, for each reservation period;
each access period $t=1,...,T$
Step 1: The WSD collects the realized end-user demand $\D$, and requests $\D$ units of spectrum from the database in each access period;
Step 2: The database sells $\min\{\K, \D\}$;
Step 3: The WSD serves end-users using the received at a market price $r$ or $s$ in each access period.
Algorithmic statement for the three-stage hierarchical model.
We focus on the following database's reservation problem: how to determine the proper reservation amount $\K$ to maximize the database profit?
The problem is challenging due to the demand stochasticity (see Section <ref>) as well as the information asymmetry (see Section <ref>).
Moreover, the reservation decision also depends on the risk-bearing scheme (see Section <ref>), namely, who (i.e., the database or the WSD) will bear the risk of over-reservation. This further complicates the problem.
§.§ Demand Stochasticity
In each access period, a WSD $\n \in \Nset$ uses the purchased spectrum to serve its end-users.
We consider two types of end-users for each WSD: registered end-users (called subscribers) and unregistered end-users (called random access users or random users).
Let $ \J_{\n}$ and $ \I_{\n}$ denote the sets of WSD $\n$'s subscribers and random users, respectively.
Specifically, subscribers characterize the residents in the WSD's serving area, and these users can sign a service contract with the WSD in advanced. Because of this, the WSD has a good knowledge regarding the demand of these users based on the long-term interactions. The random end-users characterize the travelers to the WSD's serving area,
and these users do not have any prior contractual relationship with the WSD. It is difficult for the WSD to predict the demand from these users.
Naturally, we assume that subscribers have a higher priority in obtaining service than random users.
That is, when the received by the WSD (from the database) is not enough to meet all end-users' demand,
the WSD will satisfy the subscribers's demand first, and then serve the random users using the remaining spectrum.
Recall that $r$ and $s$ are the unit prices (of ) for serving subscribers and random users, respectively.
Due to the high priority of subscribers, it is reasonable to assume that $r > s$.
Let $\Da_{\n,j} $ and $\Db_{\n,i} $ denote the demands
of a subscriber $j\in\J_{\n}$ and a random user $i\in\I_{\n}$ (to WSD $n$) in one access period, respectively.
We assume that (i) $\Da_{\n,j} $ keeps unchanged within each reservation period $T$ (but may vary across $T$), which implies that each contract's validity is larger than one access period;
and (ii) $\Db_{\n,i}$ keeps unchanged within each access period $t$ (but may vary across $t$), which implies that each random user's average QoS and wireless characteristic remain constant in each access period.[Although the small scale fading coherence time can be much smaller than one access period, we can use proper modulation and coding schemes to combat the impact of fast fading. The assumption on demand $\Db_{\n,i}$ implies that the large scale fading does not change faster than one access period (e.g., users do not move often).]
The total demand (of all subscribers and random users) of WSD $\n$ in one access period is:
\begin{equation}\label{total_demand}
\textstyle
\D_{\n} = \sum_{j \in \J_{\n}} \Da_{\n,j} + \sum_{i \in \I_{\n}} \Db_{\n,i} \triangleq \Da_{\n} + \Db_{\n},
\end{equation}
where $\Da_{\n} \triangleq \sum_{j \in \J_{\n}} \Da_{\n,j}$ is total subscriber demand, and
${\Db_{\n}} \triangleq \sum_{i \in \I_{\n}} \Db_{\n,i} \triangleq \Da_{\n}$ is total random user demand.
For convenience, we refer to $\Da_{\n} $ as the scheduled demand of WSD $n$ (as it is known at the beginning of each reservation period, and keeps unchanged during the whole reservation period), and refer to ${\Db_{\n}}$ as the bursty demand of WSD $n$ (as it is known only at the beginning of each access period, and changes randomly in different access periods).[Note that such a two-fold demand formulation in Eq. (<ref>) is widely used
in economic literature to characterize the asymmetry of demand information (see, e.g., <cit.>-<cit.>).
It can represent a lot of practical demand scenarios, such as (i) the two-stage demand used in the electricity market, where $\Da_{\n}$ is the pre-ordered demand and $\Db_{\n}$ is the real-time replenishment, and (ii) the forecast demand with error, where $\Da_{\n}$ is the estimated demand and $\Db_{\n}$
is the forecast error.]
Based on the assumptions mentioned above, the scheduled demand $\Da_{\n}$ is a random variable changing each reservation period $T$, and the bursty demand $\Db_{\n}$ is a random variable changing each access period $t$.
For simplicity, we assume that $\Da_{\n}$ and $\Db_{\n}$ are independent and identically distributed (i.i.d) in different reservation periods and access periods, respectively.
Let $f(\Da )$ and $F(\Da)$ denote the probability
density function (pdf) and cumulative distribution function
(cdf) of $\Da$, and $g(\Db )$ and $G(\Db)$ denote the pdf and cdf of $\Db$, respectively.
As in many mechanism design literature (see, e.g., <cit.>), we assume that such distribution information are public information to both the database and the WSD.
In practice, they can be obtained through machine learning in a sufficiently long time period.
As mentioned previously, the number of WSDs is large enough so that one WSD's strategy is independent of others.
Hence, we can concentrate on the interaction between the database and one WSD.
Since the total demand $\D$ changes randomly in each access period $t$, while the reservation is performed at the beginning of each reservation period $T$, the database or the WSD faces a reservation problem under demand stochasticity.
Obviously, a higher reservation can serve more demand potentially, but may also lead to a higher risk of over-reservation. A lower reservation, however, may lead to a higher loss due to the spectrum under-reservation.
Next we draw some useful properties of the scheduled demand $\Da$ and the bursty demand $\Db$.
First, we notice that the random users' bursty demand usually depends on the real-time market price $s$ and end-users' wireless characteristics.
As an example commonly used in the networking literature (e.g., <cit.>-<cit.>), a random user $i$'s utility $\pi_i$ can be defined as the difference between the achiavable data rate (e.g., the Shannon capacity assuming high SNR <cit.>) and the payment, e.g.,[This is just an illustrative example. Our analysis applies to more generic utility functions. ]
\begin{equation*}\label{random_user_profit}
\textstyle
\pi_i = \beta \cdot \Db_i \cdot \ln\left(\frac{P_i|h_i |^2 }{ \Db_i n_0 } \right)- s\cdot \Db_i ,
\vspace{-1mm}
\end{equation*}
where $\left|h_i\right|$ is the channel gain, $P_i$ is the
transmission power, $n_0$ is the noise power per unit bandwidth, and $\beta$ denotes the monetary income per unit of data rate.
Based on the above utility definition, the optimal bursty demand for a random user $i$ that maximizes its payoff $\pi_i$ is
\begin{equation*}
\textstyle
\Db_i =\frac{P_i\cdot e^{-{(1+s/\beta)}} \cdot |h_i |^2 }{ n_0 } .
\end{equation*}
Notice that the channel coefficient $h_i$ satisfies: (i) $h_i \sim \mathcal{C}(0,1 )$, the complex normal distribution (when the channel experiences the Rayleigh fading), and (ii) $h_i$ is i.i.d for different users $i \in \I_{\n}$.
Therefore, both $\Db_i $ and $\Db $ follow the chi-square distribution <cit.> (with different degrees of freedom).
Note, however, that our analysis also holds for other demand distributions such as the normal distribution.
Second, the subscribers' scheduled demand $\Da$ is a long-term average demand (changing every reservation period, e.g., one day), and usually independent of the short-term wireless characteristics.
Our analysis holds for arbitrary $\Da$ distribution with the increasing failure rate (IFR), i.e., $\frac{f(\Da)}{1- F(\Da)}$ is increasing in $\Da$.[Such an IFR constraint is widely used in the mechanism design literature (e.g., <cit.>).
Many commonly used distributions, such as
the uniform distribution, exponential distribution, and normal distribution, satisfy the IFR constraint.]
§.§ Information Asymmetry
Due to the different proximities to end-users, the database and the WSD usually have different knowledge about the scheduled demand $\Da$ and the bursty demand $\Db$.
Table <ref> illustrates the difference between the database's knowledge and the WSD's knowledge regarding the end-user demand at the beginning of each reservation period $T$ (when making the reservation decision). Specifically,
* Bursty demand $\Db$ of random users: Notice that $\Db$ changes randomly every access period.
neither the WSD nor the database knows the exact value of $\Db$ at the beginning of the reservation period.
That is, both the WSD and the database only know the distribution of ${\Db}$.
* Scheduled demand $\Da$ of subscribers:
Notice that $\Da$ keeps unchanged within each reservation period.
Thus, the WSD is able to know the exact value of $\Da$ (e.g., through bilateral agreements signed with subscribers) at the beginning of the reservation period.
The database, however, does not know the exact value of $\Da$ unless the WSD shares such information.
That is, the database only knows the distribution of ${\Da}$.
We refer to the difference between the database's knowledge and the WSD's knowledge regarding demand information as information asymmetry.
The co-existence of these two types of end-users and the information asymmetry provide incentives for the WSD to misreport its private information.
Without the existence of random users, the WSD would request the database to reserve spectrum equal to the demand of subscribed users.
With the existence of random end-users, the WSD would reserve the amount of spectrum larger than the demand of subscribed end-users, in order to gain more revenue by serving both the subscribed end-users and the random end-users.
However, the exact value of the random end-users demand is unknown by the WSD at the beginning of the reservation period.
to maximize its own profit, the WSD would optimize the value to be reported to the database, instead of truthfully revealing his information of the certain demand of subscribed users. Such strategic misreporting will make it difficult for the database to make the optimal reservation decision to maximize the database's payoff.
To hedge information asymmetry, it is important to design an incentive compatible mechanism to elicit the WSD's private demand information (i.e., $\Da$).
In this work, we will propose contract-theoretic reservation mechanisms to achieve this goal.
§.§ Risk-Bearing Scheme
Due to the demand stochasticity, there is a risk of over-reservation.[Note that spectrum under-reservation will hurt the profits of both the database and the WSD directly, and thus there is no need to discuss the risk sharing under spectrum under-reservation.
Under spectrum over-reservation, however, the database and the WSD must decide who will pay for the over-reserved spectrum.]
Thus, the reservation decision depends greatly on the risk-bearing scheme.
Namely, who will bear the risk of over-reservation, i.e., the database or WSDs?
We refer to the former scheme as DB-bearing-risk (Scheme I) and the latter scheme as WSD-bearing-risk (Scheme II).
* DB-bearing-risk (Scheme I): In this case, the WSD only pays for the spectrum it actually purchases in each access period, and thus the database bears all the risk of over-reservation.
That is, in each access period, the WSD will only pay for $\min\{\K,\D\}$ units of spectrum that it consumes.
* WSD-bearing-risk (Scheme II): In this case, the WSD pays for all the spectrum reserved, and thus the WSD bears all the risk of over-reservation. That is, in each access period, the WSD will pay for all $\K$ units of reserved spectrum, even if the total demand $\D$ is smaller than $\K$.
In this paper, we will study the reservation problem under both risk-bearing schemes systematically.
In the following sections, we first study the centralized/integrated reservation solution as a (centralized) benchmark (Section <ref>).
Then we study the decentralized reservation solution without information sharing as another (decentralized) benchmark (Section <ref>), and show that it may lead to a poor performance (in terms of database profit and network profit) due to the asymmetry of information.
To this end, we study the decentralized reservation solution with contract-based credible information sharing (Section <ref>).
To facilitate the understanding, we have listed the key results of this work in Table <ref>.
§ INTEGRATED RESERVATION SOLUTION
In this section, we consider an integrated system, where the database and the WSD act as an integrated decision maker to maximize their aggregate profit (called network profit, denoted by $\Utot$).
We will study this integrated/centralized optimal reservation as the centralized benchmark.
Obviously, in this case the integrated player (database and WSD) knows the precise value of $\Da$ and the distribution of ${\Db} $.
Moreover, there is no difference between the DB-bearing-risk scheme and the WSD-bearing-risk scheme.
Specifically, given any reservation $\K$,
the expected network profit is
\begin{equation}\label{centralized_profit}
\begin{aligned}
\textstyle
\Utot
(\K,\Da) = & r \cdot {\min{ \{\K,\Da \}}} + s \cdot \EX_{\Db} \big[ \min\big\{{\Db} , (\K-{\Da})^{+} \big\} \big]
- c\cdot {\K} ,
\end{aligned}
\end{equation}
where $(x)^+= \max\{x,0\}$.
This formula implies that the WSD will satisfy the subscribers' scheduled demand first (1st term), and then satisfy the random users' bursty demand using the remaining spectrum (2nd term).
Next we study the centralized optimal reservation $\Kso$ that maximizes the network profit defined in (<ref>).
Notice that when $\K \leq \Da$, we have $\frac{\partial{\Utot(\K,\Da)}}{\partial{\K}} = r-c >0 $, which implies that the optimal $\K$ cannot be smaller than $\Da$; when $\K \geq \Da$, we have (i) $\frac{\partial{\Utot(\K,\Da)}}{\partial{\K}} = s\big[1-G(\K-\Da)\big] - c$,
and (ii) $\frac{\partial^2{\Utot(\K,\Da)}}{\partial{\K}^2} =-s \cdot g\big( \K -\Da \big) \leq 0$.
Thus, the centralized optimal reservation $\Kso$ is given by the first-order condition $\frac{\partial{\Utot(\K,\Da)}}{\partial{\K}} = 0$, and more formally,
\begin{equation}
\label{centralized_bandwidth}
\textstyle
\Kso = \Da + {G^{-1}\Big( \frac{s-c}{s}\Big)}.
\vspace{-1mm}
\end{equation}
Intuitively, $\Kso$ consists of two parts: (i) the scheduled demand $\Da$, and (ii) the best response to the bursty demand $\Db$.
Note that the centralized optimal reservation $\Kso$ is a function of $\Da$, but not a function of $\Db$.
This is because the integrated player knows the precise value of $\Da$, but not the value of $\Db$.
§ DECENTRALIZED RESERVATION – NO INFORMATION SHARING
Now we consider a general decentralized system, where the database and the WSD make decisions independently, aiming at maximizing their individual profits.
In this section, we will study the decentralized reservation solution under information symmetry and under information asymmetry without information sharing as the decentralized benchmarks.
§.§ Scheme I: DB-Bearing-Risk
Under the DB-bearing-risk scheme, the WSD only pays for the spectrum it actually uses, and thus the database bears all the risk of over-reservation.
That is, in each access period, the WSD will only purchase $\min\{\K,\D\}$ units of .
(a) Spectrum Reservation vs Scheduled Demand $\Da$, and (b) Network Profit vs Wholesale Price $w$. Here, $\sigma_{\Da}$ denotes the variance of the scheduled demand $\Da$..
§.§.§ Information Symmetry
We first study the database's optimal reservation solution under information symmetry, where the database is assumed to know the precise value of $\Da$.
Specifically, for any reservation $\K$, the WSD's and the database's (ex-ante) expected profits are, respectively,
\begin{equation}\label{asym_decentral_WSD_profit}
\begin{aligned}
\textstyle
\Ums (\K,\Da ) = & (r-w) \cdot \min \{\K, \Da \} \\
& \quad{} + (s-w)\cdot \EX_{\Db} \big[ \min\big\{{\Db} , (\K-\Da )^{+}\big\} \big],
\end{aligned}
\end{equation}
\begin{equation}\label{sym_dencentral_DB_profit}
\textstyle
\Udb (\K, \Da ) =
w \cdot \EX_{\Db} \big[ \min \{ \Db + \Da, \K \} \big] - c\cdot {\K}. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{equation}
The optimal reservation for the database
(i.e., that maximizes its profit defined in (<ref>)) is
\begin{equation}\label{sym_dencentral_bandwidth}
\textstyle
\Kdbsym = \Da +
{G^{-1} \Big( \frac{w-c}{w} \Big)}.
\end{equation}
Similar to the centralized optimal reservation $\Kso $, the above decentralized optimal reservation $\Kdbsym $ under information symmetry is also a function of $\Da$.
§.§.§ Information Asymmetry
In practice, the demand information is asymmetric between the database and the WSD as discussed in Section <ref>.
Now we study the database's optimal reservation solution under information asymmetry, where the database does not know the precise value of $\Da$.
We first show that the reservation solution $\Kdbsym $ in (<ref>) under information symmetry may not be the database's optimal solution in this case, as it
cannot ensure that the WSD shares its private information $\Da$ with the database credibly.
Notice that (i) the WSD profit $\Ums (\K,\Da )$ in (<ref>) increases with the reservation $\K$, and (ii) the database's optimal reservation $\Kdbsym$ in (<ref>) is linear to $\Da$.
This implies that the WSD has an incentive to inflate its private information $\Da$.
The key reason behind this phenomenon is that the database bears all the risk of over-reservation.
As a consequence, the database will not trust the information (i.e., the value of $\Da$) informed by the WSD, and therefore will act based on its own prior distribution information of $\Da$ and $\Db$. That is, it will maximize the following expected profit:
\begin{equation} \label{asym_decentral_DB_profit}
\Udba(\K) \triangleq \EX_{\Da} \big[ \Udb (\K, \Da )\big] =
w \cdot \EX_{\Da,\Db} \big[ \min \{ \Db + \Da, \K \} \big] - c\cdot {\K},
\end{equation}
where the expectation $\EX_{\Da,\Db}$ is taken over the distribution of $\Da$ and $\Db$.
The optimal reservation for the database that maximizes its expected profit defined in (<ref>) is
\begin{equation}
\label{asym_decentral_bandwidth}
\textstyle
\Kdbasy= (F \times G
)^{-1} \Big( \frac{w-c}{w} \Big),
\end{equation}
$F \times G$ is the joint c.d.f. of $\Da +
{\Db} $.
Note that $\Kdbasy$ is not a function of $\Da$, which is different from (<ref>) and (<ref>). This implies that the database cannot adjust its reservation decision to account for the WSD's private information. Therefore, both parties's profits may reduce due to the ignorance of information $\Da$ (that WSD has) in the reservation.
To solve this problem, we will propose a spectrum reservation contract to achieve the credible information sharing between the database and the WSD in Section <ref>.
§.§ Scheme II: WSD-Bearing-Risk
Under the WSD-bearing-risk scheme, the WSD pays for all the spectrum reserved, and thus the WSD bears all the risk of over-reservation. That is, in each access period, the WSD will pay for all $\K$ units of reserved spectrum, even if the total demand $\D $ is smaller than $\K$.
§.§.§ Information Symmetry
Similarly, we first study the WSD's optimal reservation decision under information symmetry.
Specifically, for any reservation $\K$, the WSD's and the database's (ex-ante) expected profits are, respectively,
\begin{equation}\label{asym_decentral_WSD_profit2}
\begin{aligned}
\textstyle
%\Ums (\K,\Da ) = & r \cdot \min \{\K, \Da \} - w \cdot \K \\
%& + s\cdot \EX_{\Db} \big[ \min\big\{{\Db} , (\K-\Da )^{+}\big\} \big],
\Ums (\K,\Da ) = & r \cdot \min \{\K, \Da \} + s\cdot \EX_{\Db} \big[ \min\big\{{\Db} , (\K-\Da )^{+}\big\} \big] - w \K,
\end{aligned}
\end{equation}
\begin{equation}\label{sym_dencentral_DB_profit2}
\begin{aligned}
\textstyle
\Udb (\K, \Da ) =
(w - c ) \cdot {\K} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{aligned}
\end{equation}
Note that if the WSD bears the risk, then the WSD will determine the reservation amount. Otherwise, the database will always choose a very large reservation as it does not bear the risk of over-reservation.
Accordingly, the optimal reservation for the WSD (i.e., that maximizes its profit defined in (<ref>)) is
\begin{equation}\label{sym_dencentral_bandwidth2}
%= \Da +
% {G^{-1} \Big( \frac{s-w}{s} \Big)},
\Kmssym= \Da +
{G^{-1} \Big( \frac{s-w}{s} \Big)},
\end{equation}
which is also a function of $\Da$.
§.§.§ Information Asymmetry
Since the WSD itself holds the private information under information asymmetry, the WSD's expected profit under information asymmetry is exactly same as (<ref>).
Thus, the optimal reservation for the WSD under information asymmetry is same as that under information symmetry, i.e.,
\begin{equation}\label{asym_decentral_bandwidth2}
\Kmsasy =
\Kmssym= \Da +
{G^{-1} \Big( \frac{s-w}{s} \Big)}.
\end{equation}
Notice that the database profit $\Udb(\K, \Da)$ defined in (<ref>) is increasing in the reservation $\K$.
This implies that it is possible for the database to improve its profit by incentivizing the WSD to increase the reservation $\K$.
In Section <ref> , we will propose a spectrum reservation contract to maximize the database profit under the WSD-bearing-risk scheme.
§.§ Comparison
Now we compare the above decentralized optimal reservations (without information sharing).
It is easy to see that these decentralized solutions deviate from the integrated optimal solution (<ref>), due to the “double marginalization” effect as well as the lack of information on the database side under information asymmetry.
§.§.§ Performance under Information Symmetry
We first compare two reservation solutions under information symmetry, i.e., $ \Kdbsym$ and $\Kmssym $.
There exists a critical wholesale price $w^{*} = \sqrt{sc}$ such that
* when $w < w^{*}$, then $\Kso > \Kmssym > \Kdbsym$;
* when $w > w^{*}$, then $\Kso > \Kdbsym> \Kmssym$.
We illustrate the reservation solutions vs scheduled demand $\Da$ in Figure <ref>.a, where $s=0.8$, $w=0.5$, $c=0.2$, and obviously, $w >\sqrt{sc}=0.4$.
It is easy to see that $\Kdbsym$ under DB-bearing-risk (the blue triangle curve) is always larger than $\Kmssym$ under WSD-bearing-risk (the red square curve). This is because with a large wholesale price (e.g., $w>\sqrt{sc}$), the risk of over-reservation that the WSD bears under WSD-bearing-risk is higher than that the database bears under DB-bearing-risk, and thus the WSD will reserve less spectrum than the database.
We can further see that $\Kdbsym$ and $\Kmssym$ are smaller than $\Kso$ in the integrated system (the green circle curve) .
The gap between $\Kdbsym$ (or $\Kmssym$) and $\Kso$ is caused by the double marginalization effect.
Under information symmetry, there exists a critical wholesale price $w^{*} = \sqrt{sc}$ such that
* when $w < w^{*}$,
the optimal network profit under WSD-bearing-risk (i.e., under $\Kmssym$) is larger than that under DB-bearing-risk (i.e., under $\Kdbsym$);
* when $w > w^{*}$, the optimal network profit under WSD-bearing-risk (i.e., under $\Kmssym$) is smaller than that under DB-bearing-risk (i.e., under $\Kdbsym$)
<ref> can be obtained by Lemma
<ref>, together with the fact that the network profit increases with $k$ when $k \leq \Kso $.
For clarity, we illustrate the network profit under different reservation solutions vs wholesale price $w$ in Figure <ref>.b.
We can see that (i) the centralized optimal network profit (the green circle curve) does not depend on the wholesale price $w$, and (ii) the decentralized optimal network profit under DB-bearing-risk (the blue triangle curve) increases with the wholesale price $w$, while the decentralized optimal network profit under WSD-bearing-risk (the red square curve) decreases with the wholesale price.
This is because with a larger wholesale price, the database will reserve more spectrum under DB-bearing-risk (hence, the network profit increases), while the WSD will reserve less spectrum under WSD-bearing-risk (hence, the network profit decreases).
§.§.§ Performance under Information Asymmetry
We now compare two reservation solutions under information asymmetry, i.e., $ \Kdbasy$ and $\Kmsasy $.
From Figure <ref>.a, we can see that
$\Kdbasy$ (blue dashed curve with mark “x”) under DB-bearing-risk is independent of $\Da$, while $\Kmsasy$ (red dashed curve with mark “$+$”) under WSD-bearing-risk increases linearly with $\Da$.
Obviously, $\Kdbasy > \Kmsasy $ when $\Da$ is small (e.g., $\Da < 14$), while $\Kdbasy < \Kmsasy $ when $\Da$ is large (e.g., $\Da > 14$).
This is because the database makes the reservation decision $\Kdbasy$ without knowing the exact value of $\Da$, and thus it is likely to over-reserve spectrum when $\Da$ is small, while under-reserve spectrum when $\Da$ is large.
Similarly, from Figure <ref>.b, we can see that (i) the decentralized optimal network profits under DB-bearing-risk (the blue dash curves with mark “x”) increases with the wholesale price $w$, while the decentralized optimal network profit under WSD-bearing-risk (the red dash curve with mark “$+$”, overlapping with the
red square curve) decreases with $w$.
The reason is similar to that under information symmetry, i.e., a larger wholesale price will increase the database's reservation $\Kdbasy $ under DB-bearing-risk, but reduce the WSD's reservation $\Kmsasy $ under WSD-bearing-risk.
Moreover, we can see that the decentralized optimal network profit under DB-bearing-risk (the blue dash curves with mark “x”) decreases with the variance of scheduled demand $\Da$ (denoted by $\VDa$).
This is because the database's spectrum reservation $\Kdbasy $ under DB-bearing-risk does not consider the exact value of $\Da$; hence, a larger variance of $\Da$ will lead to a larger network profit loss.
§.§ Observation
By the above comparison, we can see that performances of the decentralized optimal solution under information asymmetry (i.e., $\Kdbasy$ in (<ref>) and $\Kmsasy$ in (<ref>)) depend on the wholesale price $w$ and the variance of scheduled demand $\Da$. Moreover, both of these solutions
may lead to low profits for both the database and the WSD (comparing with the centralized benchmark),
due to the lack of information and/or the double marginalization effect.
§ DECENTRALIZED RESERVATION – CONTRACT-THEORETIC APPROACH
In the previous section, we have shown that lacking of information and/or the double marginalization effect may result in profit losses for both the database and the WSD. In this section, we will propose a contract-theoretic approach to achieve credible information sharing and hedge double marginalization in reservation.
§.§ Contract under DB-Bearing-Risk
As shown in (<ref>), under the DB-bearing-risk scheme, the profit loss under information asymmetry is mainly due to the lack of information $\Da$ (when the database makes the reservation decision).
Therefore, we propose a Spectrum Reservation Contract to achieve the credible information sharing between the database and the WSD.
We derive the optimal contract that maximizes the database profit under information asymmetry analytically.
Simulations demonstrate that with the optimal contract, the total network profit can also be improved, comparing with that (under information asymmetry) without credible information sharing.
§.§.§ Contract Design
The key idea of a spectrum reservation contract is as follows.
To motivate the WSD credibly reveal its private information $\Da$,
the database put an additional charge on the WSD for spectrum reservation (on top of the wholesale charge of $w \cdot \min{[ \K, \Da ]}$).
This forces the WSD to share the cost of over-reservation, such that the WSD has no incentive to inflate the value of $\Da$.
Based on this idea, we design the following contract: $\CTRdb \triangleq \{ \langle \K (\Da ), \P (\Da ) \rangle \}_{\forall \Da}$, which consists of a menu of contract items, $\langle \K(\Da), \P(\Da)\rangle$, each intending for a possible scheduled demand $\Da$. Here, $\K(\Da)$ and $\P(\Da)$ denote the reservation and the WSD's payment to the database, respectively, when the scheduled demand is $\Da$.[Note that $\P(\Da)$ is the WSD's payment for reserving spectrum via the database, and is not the total cost of using spectrum.]
The detailed reservation process is as follows.
* Before reserving spectrum, the database announces the contract $\CTRdb = \{ \langle \K (\Da ), \P (\Da ) \rangle \}_{\forall \Da}$;
* The WSD selects the contract item $\langle \K(\hDa), \P(\hDa)\rangle$ that maximizes its expected profit, based on its private information $\Da$;
* The database reserves $\K(\hDa)$ for one reservation period, and charges the WSD a reservation fee $\P(\hDa)$ (Step 0 in Figure <ref>);
* The database sells spectrum to the WSD in each access period (Steps 1-3 in Figure <ref>).
When the WSD with information $\Da$ chooses a contract item $\langle \K(\hDa), \P(\hDa)\rangle$ (i.e., that intended for information $\hDa$), the WSD profit, the database profit, and the aggregate profits (network profit) are, respectively,
\begin{equation}
\label{asy_contract_WSD_profit_new}
\begin{aligned}
\textstyle
& \Ums (\K(\hDa), \P(\hDa),\Da ) = (r-w) \cdot \min \{ \K(\hDa), \Da \} \\
& \qquad{} + (s-w) \cdot \EX_{\Db} \big[ \min\big\{{\Db} , (\K(\hDa)-\Da )^{+}\big\} \big] - \P(\hDa) ,
\end{aligned}
\end{equation}
\begin{equation}
\label{asy_contract_database_profit}
\begin{aligned}
\Udb (\K(\hDa), \P(\hDa), \Da ) = & ~
w \cdot \EX_{\Db} \big[ \min \{ \Db + \Da, \K(\hDa)\} \big] \\
& ~ - c \cdot \K(\hDa) + \P(\hDa),~~~~~~~~~~
\end{aligned}
\end{equation}
\begin{equation}
\label{asy_contract_total_profit_new}
\begin{aligned}
& \Utot (\K(\hDa), \P(\hDa), \Da) = r\cdot {\min{ \{\K(\hDa), \Da \}}} \\
& \qquad{} + s \cdot \EX_{\Db} \big[ \min\big\{{\Db} , (\K(\hDa) -{\Da})^{+} \big\} \big] - c\cdot \K(\hDa).~~~~~~~~~
\end{aligned}
\end{equation}
We define a feasible contract as follows.
A contract is feasible, if and only if
* Incentive Compatibility (IC):
The WSD with any information $\Da$ prefers the contract item $\langle \K( {\Da}), \P( {\Da})\rangle$ (that is intended for $\Da$) than all other contract items $\langle \K( {\hDa}), \P( {\hDa})\rangle, \forall \hDa \neq \Da$. Formally, we have
\begin{equation}\label{IC_requirement}
\begin{aligned}
\Ums (\K( {\Da}), \P( {\Da}),\Da ) \geq \Ums (\K(\hDa), \P(\hDa),\Da ),\ \forall \hat{\Da}, \Da.
\end{aligned}
\end{equation}
* Individual Rationality (IR):
The WSD can achieve a minimum acceptance profit $\Umsmin$ when choosing $\langle \K( {\Da}), \P( {\Da})\rangle$. Formally, we have
\begin{equation}
\label{AC_requirement}
\Ums (\K( {\Da}), \P( {\Da}),\Da ) \geq \Umsmin,\ \forall \Da.~~~~~~~~~~~~~~~~~~
\end{equation}
Moreover, we define an optimal contract, denoted by $\CTRdb^* = \{ \langle \Kdbctr (\Da ), \Pdbctr (\Da ) \rangle \}_{\forall \Da}$, as follows.
The contract $\CTRdb^* = \{ \langle \Kdbctr (\Da ), \Pdbctr (\Da ) \rangle \}_{\forall \Da}$ is optimal if this contract is feasible and maximizes the database expected profit.
Formally, the optimal contract is given by
\begin{equation}
\label{optimal_contract_asy}
\begin{aligned}
\max_{\langle \K (\Da ), \P (\Da ) \rangle, \forall \Da } & \EX_{\Da} \big[ \Udb (\K( {\Da}), \P( {\Da}), \Da ) \big],\\
\mathrm{subject~to:~~} & \mathrm{IC~and~IR~in~(\ref{IC_requirement})~and~(\ref{AC_requirement}).}
\end{aligned}
\end{equation}
In the following, we first provide the necessary and sufficient conditions for a feasible contract.
Then, we derive the optimal contract systematically.
For clarity, we present all of the detailed proofs in <cit.>.
§.§.§ Feasibility
Suppose that a contract $\CTRdb = \{ \langle \K (\Da ), \P (\Da ) \rangle \}_{\forall \Da}$ is feasible.
Then, the following necessary conditions hold.
$$ \K (\Da_1) > \K (\Da_2), \mathrm{~~if~and~only~if~~} \P (\Da_1) > \P (\Da_2).$$
$$\K (\Da_1) \geq \K (\Da_2),\quad \forall \Da_1 > \Da_2.$$
Proposition <ref> implies that in a feasible contract,
a larger reservation $\K(\cdot)$ must correspond to a larger reservation fee $\P(\cdot)$.
This is quite intuitive, as the WSD's profit is increasing in $\K(\cdot)$ but decreasing in $\P(\cdot)$.
Proposition <ref> implies that the reservation $\K(\cdot)$ increases with the value of scheduled demand $\Da$.
For convenience, we denote $\Ums ( \Da ) \triangleq \Ums (\K( {\Da}), \P( {\Da}),\Da )$ as the WSD profit when choosing the contract item intended for its true private information $\Da$.
Given any feasible $\K(\Da)$ (i.e., those non-decreasing with $\Da$), we have the following necessary conditions for the feasible $\P(\Da)$, or equivalently, for the WSD profit $\Ums ( \Da )$.
$$\Ums (\Da_1) \geq \Ums (\Da_2),\quad \forall \Da_1 > \Da_2.$$
\begin{equation*}\label{eq:nece4}
\begin{aligned}
\Ums (\Da) = & \Ums (\underline{\Da}) + (r-s)\cdot (\Da-\underline{\Da}) \\
&~ + \int_{\underline{\Da}}^{\Da}
(s-w ) \cdot G\big( \K(x) - x \big)\,\mathrm{d}{x} .
\end{aligned}
\end{equation*}
Proposition <ref> implies that in a feasible contract, the WSD profit increases with the value of $\Da$.
Proposition <ref> further gives the detailed form of the WSD profit in a feasible contract, given any feasible $\K(\Da)$.
Note that the third term on the r
Here, $\underline{\Da}$ is the minimum achievable value of scheduled demand $\Da$, i.e., $g(\Da) = 0$ if $\Da < \underline{\Da}$.
By Proposition <ref>, we can get the following feasible reservation fee $\P(\Da)$ directly:
\begin{equation}
\label{eq:opt-p}
\begin{aligned}
\P (\Da) = & - { \Ums ( \Da )} + (r-w) \cdot \min \{\K( {\Da}), \Da \} \\
& + (s-w) \cdot \EX_{\Db} \big[ \min\big\{{\Db} , (
\K( {\Da})-\Da )^{+}\big\} \big],
\end{aligned}
\end{equation}
where $ { \Ums ( \Da )}$ is given in Proposition <ref>.
We have shown the necessary conditions for a feasible contract through Propositions <ref>-<ref>.
Next we show that these conditions are also sufficient for a contract to be feasible.
A contract $\CTRdb = \{ \langle \K (\Da ), \P (\Da ) \rangle \}_{\forall \Da}$ is feasible, if the following conditions hold:
* $\K (\Da )$ is non-decreasing in $\Da$ (i.e., Necessary Condition II in Proposition <ref>),
* $\P (\Da )$ is given by (<ref>) (i.e., Necessary Condition IV in Proposition <ref>),
* $\Ums (\underline{\Da}) \geq \Umsmin $ (i.e., IR Condition).
Intuitively, the first two conditions guarantee the IC condition for the contract, and the last condition guarantees the IR condition for the contract. Therefore, the conditions in Proposition <ref> are sufficient.
§.§.§ Optimality
Now we study the database's optimal contract characterized by (<ref>).
By (<ref>) and (<ref>), we notice that the total profit can be freely transferred between the database and the WSD through the reservation fee $\P(\Da)$.
Therefore, to maximize the database profit, we need to shrink the WSD's profit as much as possible.
This leads to the following optimality condition immediately.
\begin{equation*}\label{eq:opt1}
\Ums (\underline{\Da} ) = \Umsmin.
\end{equation*}
Proposition <ref> implies that in the optimal contract, the database will assign the minimal acceptable profit to the WSD.
Intuitively, if the WSD profit $\Ums (\underline{\Da} ) = X > \Umsmin$, then the database can immediately improve its profit by increasing the reservation fee $\P(\Da)$ by a constant $(X-\Umsmin)$ for all $\Da$.
Denote $\Udb ( \Da ) \triangleq \Udb ( \K( {\Da}), \P( {\Da}),\Da )$ and $\Utot ( \Da ) \triangleq \Utot ( \K( {\Da}), \P( {\Da}),\Da )$.
By (<ref>)-(<ref>), we can write the database's profit as $\Udb ( {\Da} ) = \Utot ( {\Da} ) - \Ums ({\Da} )$.
Together with Proposition <ref> and Proposition <ref>, we can rewrite the database profit maximization problem (<ref>) as follows.
\begin{equation*}
\begin{aligned}
\max_{ \K (\Da ) , \forall \Da } ~~
\EX_{\Da} \big[ \Udb ( \Da ) \big] & \triangleq \textstyle \int_{\underline{\Da}}^{\bar{\Da}} \phidb \big(\K(\Da),\Da \big) \cdot f (\Da ) \mathrm{d}{\Da} - \Umsmin, \end{aligned}
\end{equation*}
\begin{equation}
\label{asy_optimal_contract_formulation}
\begin{aligned}
\mathrm{subject~to:~~} & \K(\Da) \mathrm{~is~non\mbox{-}decreasing~in~} \Da,
\end{aligned}
\end{equation}
\textstyle
\phidb \big(\K(\Da),\Da \big) \triangleq \Utot ( {\Da} ) - \frac{1-F (\Da )}{f (\Da )}\big[ r-s + (s-w) \cdot G\big(\K(\Da) - \Da\big)\big].
We first notice that $\phidb (\K(\Da),\Da ) $ is related to a particular $\Da$ only, and is independent of other $\hDa \neq \Da$. Thus, the optimal solution of (<ref>)
can be obtained by maximizing $\phidb\big(\K(\Da),\Da\big)$ for each $\Da$ independently (as long as the non-decreasing condition is not violated).
due to the non-convexity of $G(\cdot)$, $\phidb (\K(\Da),\Da ) $ is non-convex in $\K(\Da)$, and thus the classic Karush-Kuhn-Tucker (KKT) analysis cannot be directly applied here.[As an example mentioned in Section <ref>, the bursty demand $\Db$'s distribution $G(\cdot)$ is the chi-square distribution, which is non-convex.]
Next we can show that $\phidb (\K(\Da),\Da ) $ has the nice property of piecewise convexity.
Based on this, the maximizer of $\phidb (\K(\Da),\Da ) $ is unique, and it satisfies the first-order condition: $\frac{\partial {\phidb (\K, \Da )}}{\partial{\K } }=0$.
Formally, the optimal $\K(\Da), \forall \Da$, is given by
\begin{equation}
\label{asy_optimal_K_in_contract}
\begin{aligned}
\textstyle
\frac{\partial {\phidb (\K, \Da )}}{\partial{\K } } = & s \cdot [1-G ( \K(\Da) -\Da ) ] - c
\\
& - \frac{1-F(\Da)}{f(\Da)}\cdot ( s-w )\cdot g( \K(\Da) - \Da ) = 0 .
\end{aligned}
\end{equation}
We can further check that optimal $\K(\Da)$ given by (<ref>) is indeed non-decreasing in $\Da$, due to the IFR assumption for $F(\cdot)$, i.e., $ \frac{1-F(\Da)}{f(\Da)} $ decreases with $\Da$.
Therefore, we have the following optimal contract under DB-bearing-risk.
Under DB-bearing-risk, the database's optimal contract $ \CTRdb^* = \{ \langle \Kdbctr(\Da),{\Pdbctr}(\Da) \rangle \}_{\forall \Da}$ is given by: $\forall \Da \in [\underline{\Da}, \bar{\Da}]$,
* $\Kdbctr(\Da)$ is given by (<ref>), and
* ${\Pdbctr}(\Da)$ is given by (<ref>) with $\Ums (\underline{\Da}) = \Umsmin $.
Now we provide some useful properties for the optimal contract $ \CTRdb^* = \{ \langle \Kdbctr(\Da),{\Pdbctr}(\Da) \rangle \}_{\forall \Da}$. Specifically,
\begin{equation}\label{eq:marginal-wholesale-price-BR}
\begin{aligned}
\textstyle
\frac{\mathrm{d} \Pdbctr }
{\mathrm{d} \Kdbctr }
\frac{ \mathrm{d} \Pdbctr / \mathrm{d}{\Da} }
{ \mathrm{d} \Kdbctr / \mathrm{d}{\Da}}
( s-w ) \cdot \big[ 1 - G( \Kdbctr-\Da ) \big] \geq 0,~~~
\end{aligned}
\end{equation}
\begin{equation}\label{eq:marginal-wholesale-price-SOC-BR}
\begin{aligned}
\textstyle
\frac{\mathrm{d}^2 \Pdbctr~}
{ \mathrm{d}~{\Kdbctr}^2 }
& \textstyle
\frac{ \mathrm{d}
\big(\frac{\mathrm{d} \Pdbctr }
{\mathrm{d} \Kdbctr }\big) / \mathrm{d}{\Da}}
{\mathrm{d} \Kdbctr / \mathrm{d}{\Da}}
\frac{ -( s-w ) \cdot g( \Kdbctr-\Da ) \cdot \left( {\mathrm{d} \Kdbctr / \mathrm{d}{\Da}} - 1 \right)}{\mathrm{d} \Kdbctr / \mathrm{d}{\Da}} \leq 0.
\end{aligned}
\end{equation}
The above properties show that $\Pdbctr$ is concavely increasing in $\Kdbctr$ (which can be seen from Figure <ref>.a).
This implies that the database's reservation fee charge for each additional unit of reservation will decrease with the total amount of reservation.
§.§ Contract under WSD-Bearing-Risk
Comparing (<ref>) and (<ref>), we can see that under WSD-bearing-risk, the gap between the centralized optimal reservation $\Kso$ and the decentralized optimal reservation $\Kmsasy $ (under information asymmetry without information sharing) is mainly due to the double marginalization effect, which further leads to some loss in both the database profit and the total network profit.
The perfect coordination of the WSD's optimal solution (<ref>) and the centralized optimal solution (<ref>) requires the wholesale price to be as low as the cost (i.e., $w = c$). This is obviously undesirable for a profit-maximizing database.
To this end, we propose a Spectrum Reservation Contract to mitigate the double marginalization effect in this case.
Similarly, we analytically derive the optimal contract that maximizes the database profit under information asymmetry. Simulations demonstrate that with the optimal contract, the total network profit can also be improved, comparing with that (under information asymmetry) without credible information sharing.
§.§.§ Contract Design
The detailed contract formulation under WSD-bearing-risk is similar to that under DB-bearing-risk (in Section <ref>).
Specifically, to motivate the WSD to order according to the database's profit-maximizing objective, the database charges
the WSD for the spectrum reservation (in addition of the wholesale charge of $w \cdot \K$).[Note that this wholesale charge is different from that under DB-bearing-risk. The latter is $w \cdot \min{[ \K, \Da ]}$, as the WSD only needs to pay for the spectrum it actually purchases.]
This forces the database to share the cost of over-reservation, such that the WSD operates as the database desired.
Similarly, we design the following contract: $\CTRms \triangleq \{ \langle \K (\Da ), \P (\Da ) \rangle \}_{\forall \Da}$, where each contract item $\langle \K(\Da), \P(\Da)\rangle$ specifies a reservation level $\K(\Da)$ and the corresponding WSD's payment $\P(\Da)$.
The detailed reservation process is the same as that in Section <ref>.
However, the definitions for the database's and the WSD profits are different, due to the different risk-bearing schemes.
Specifically, when the WSD with information $\Da$ chooses a contract item $\langle \K(\hDa), \P(\hDa)\rangle$ (i.e., that intended for $\hDa$), the WSD's profit, the database profit, and the aggregate profits (network profit) are, respectively,
\begin{equation}
\label{asy_contract_WSD_profit_ad}
\begin{aligned}
& \Ums (\K(\hDa), \P(\hDa),\Da ) = r \cdot \min \{ \K(\hDa), \Da \} - w \cdot \K(\hDa) \\
& \qquad{} \qquad{} + s \cdot \EX_{\Db} \big[ \min\big\{{\Db} , (\K(\hDa)-\Da )^{+}\big\} \big] - \P(\hDa),
\end{aligned}
\end{equation}
\begin{equation}
\label{asy_contract_database_profit_ad}
\begin{aligned}
\Udb (\K(\hDa), \P(\hDa), \Da ) = &
(w - c) \cdot \K(\hDa) + \P(\hDa),~~~~~~~~
\end{aligned}
\end{equation}
\begin{equation}
\label{asy_contract_total_profit_new_ad}
\begin{aligned}
& \Utot (\K(\hDa), \P(\hDa), \Da) = r\cdot {\min{ \{\K(\hDa), \Da \}}} \\
& \qquad{} \qquad{} + s \cdot \EX_{\Db} \big[ \min\big\{{\Db} , (\K(\hDa) -{\Da})^{+} \big\} \big] - c\cdot \K(\hDa).
\end{aligned}
\end{equation}
Obviously, the aggregate profit in (<ref>) is same as that in (<ref>), that is,
the network profit does not depend on the choice of the risk-bearing scheme.
Similar as in Definition <ref> and <ref>, we first define the contract feasibility and optimality.
The contract $\CTRms = \{ \langle \K (\Da ), \P (\Da ) \rangle \}_{\forall \Da}$ is feasible, if and only if it satisfies the following conditions.
\begin{equation}
\label{IC_requirement_ad}
\mathrm{IC:}~~\Ums (\K( {\Da}), \P( {\Da}),\Da ) \geq \Ums (\K(\hDa), \P(\hDa),\Da ),\ \forall \hat{\Da}, \Da;
\end{equation}
\begin{equation}
\label{AC_requirement_ad}
\mathrm{IR:}~~\Ums (\K( {\Da}), \P( {\Da}),\Da ) \geq \Umsmin,\ \forall \Da.~~~~~~~~~~~~~~~~~~
\end{equation}
We denote the optimal contract by $\CTRms^* = \{ \langle \Kmsctr (\Da ), \Pmsctr (\Da ) \rangle \}_{\forall \Da}$, which is defined below.
The contract $\CTRms^* = \{ \langle \Kmsctr (\Da ), \Pmsctr (\Da ) \rangle \}_{\forall \Da}$ is optimal if this contract is feasible and maximizes the database expected profit. Formally, the optimal contract is given by
\begin{equation}
\label{optimal_contract_asy_ad}
\begin{aligned}
\max_{\langle \K (\Da ), \P (\Da ) \rangle, \forall \Da } & \EX_{\Da} \big[ \Udb (\K( {\Da}), \P( {\Da}), \Da ) \big],\\
\mathrm{subject~to:~} & \mathrm{IC~and~IR~in~(\ref{IC_requirement_ad})~and~(\ref{AC_requirement_ad}).}
\end{aligned}
\end{equation}
§.§.§ Feasibility
It is easy to check that the necessary conditions II and III in Propositions <ref>-<ref> also hold for the feasible contract under WSD-bearing-risk.
However, the necessary condition IV in Proposition <ref> is a bit different.
Given a feasible $\K( {\Da})$, the WSD's expected profit is
\begin{equation*}\label{eq:nece4_ad}
\begin{aligned}
\Ums (\Da) = & \Ums (\underline{\Da}) + (r-s)\cdot (\Da-\underline{\Da}) + \int_{\underline{\Da}}^{\Da}
s \cdot G\big( \K(x) - x \big)\,\mathrm{d}{x} .
\end{aligned}
\end{equation*}
Accordingly, the feasible reservation fee $\P (\Da)$ is
\begin{equation}
\label{eq:opt-p-ad}
\begin{aligned}
\P (\Da) = & - {\Ums ( \Da )} + r \cdot \min \{\K( {\Da}), \Da \} \\
& + s \cdot \EX_{\Db} \big[ \min\big\{{\Db} , (\K( {\Da})-\Da )^{+}\big\} \big] - w \cdot \K ( {\Da}),
\end{aligned}
\end{equation}
where $\Ums ( \Da )$ is given in Proposition <ref>.
§.§.§ Optimality
Notice that the optimality condition in Proposition <ref> also holds for the WSD-bearing-risk scheme.
Thus, we can similarly rewrite the database profit
maximization problem (<ref>) as
\begin{equation*}
\begin{aligned}
\max_{ \K (\Da ) , \forall \Da } & \textstyle ~~ \EX_{\Da} \big[ \Udb ( \Da ) \big] \triangleq \int_{\underline{\Da}}^{\bar{\Da}} \phims \big(\K(\Da),\Da \big) \cdot f (\Da ) \mathrm{d}{\Da} - \Umsmin,
\end{aligned}
\end{equation*}
\begin{equation}
\label{asy_optimal_contract_formulation_AD}
\begin{aligned}
\mathrm{subject~to:~~} & \K(\Da) \mathrm{~is~non\mbox{-}decreasing~in~} \Da,
\end{aligned}
\end{equation}
\textstyle
\phims \big(\K(\Da),\Da \big) \triangleq \Utot ( {\Da} ) - \frac{1-F (\Da )}{f (\Da )} \cdot \big[ r-s + s \cdot G\big(\K(\Da) - \Da\big)\big].
Using a similar analysis as in Section <ref>, we can show that the optimal solution
of (<ref>) can be obtained by maximizing $\phims \big(\K(\Da),\Da \big)$ for each $\Da$ independently.
Moreover, the optimal $\K(\Da)$ satisfies the first-order condition: $ \frac{\partial {\phims (\K, \Da )}}{\partial{\K } } =0$. Formally,
\begin{equation}
\label{asy_optimal_K_in_contract_AD}
\begin{aligned}
\textstyle
\frac{\partial {\phims (\K(\Da), \Da )}}{\partial{\K } } = & s \cdot [1-G ( \K(\Da) -\Da ) ] - c \\
& - \frac{1-F(\Da)}{f(\Da)}\cdot s \cdot g( \K(\Da) - \Da ) = 0 .
\end{aligned}
\end{equation}
Therefore, the optimal contract under the WSD-bearing-risk scheme is given in the following theorem.
Under WSD-bearing-risk, the optimal contract $ \CTRms^* = \{ \langle \Kmsctr(\Da),{\Pmsctr}(\Da) \rangle \}_{\forall \Da}$ is given by: $\forall \Da \in [\underline{\Da}, \bar{\Da}]$,
* $\Kmsctr(\Da)$ is given by (<ref>), and
* ${\Pmsctr}(\Da)$ is given by (<ref>) with $\Ums (\underline{\Da}) = \Umsmin $.
We provide some useful properties for the optimal
contract $\CTRms^* = \{ \langle \Kmsctr (\Da ), \Pmsctr (\Da ) \rangle \}_{\forall \Da}$. Specifically,
\begin{equation}\label{eq:marginal-wholesale-price-AP}
\begin{aligned}
\textstyle
\frac{\mathrm{d} \Pmsctr }
{\mathrm{d} \Kmsctr }
\frac{ \mathrm{d} \Pmsctr / \mathrm{d}{\Da} }
{ \mathrm{d} \Kmsctr / \mathrm{d}{\Da}}
s\cdot \big[ 1 - G( \Kmsctr-\Da ) \big] - w,~~~~~~~~~
\end{aligned}
\end{equation}
\begin{equation}\label{eq:marginal-wholesale-price-SOC-AP}
\begin{aligned}
\textstyle
\frac{\mathrm{d}^2 \Pmsctr~}
{ \mathrm{d}~{\Kmsctr}^2 }
& \textstyle
\frac{ \mathrm{d}
\big(\frac{\mathrm{d} \Pmsctr }
{\mathrm{d} \Kmsctr }\big) / \mathrm{d}{\Da}}
{\mathrm{d} \Kmsctr / \mathrm{d}{\Da}}
\frac{ -s \cdot g( \Kmsctr-\Da ) \cdot \left( {\mathrm{d} \Kmsctr / \mathrm{d}{\Da}} - 1 \right)}{\mathrm{d} \Kmsctr / \mathrm{d}{\Da}} \leq 0.
\end{aligned}
\end{equation}
The second property shows that $\Pmsctr$ is concave in $\Kmsctr$, and the first property shows that $\Pmsctr$ is non-monotonous in $\Kmsctr$. More precisely, $\Pmsctr$ first increases with $\Kmsctr$ and then decreases with $\Kmsctr$, as illustrated in Figure <ref>.a.
§.§ Comparison
Now we compare the optimal contract $\CTRdb^* = \{ \langle \Kdbctr (\Da ), \Pdbctr (\Da ) \rangle \}_{\forall \Da}$ under the DB-bearing-risk scheme (in Theorem <ref>) and the optimal contract $\CTRms^* = \{ \langle \Kmsctr (\Da ), \Pmsctr (\Da ) \rangle \}_{\forall \Da}$ under the WSD-bearing-risk scheme (in Theorem <ref>).
Figure <ref>.a compares
the structures of both contracts, by showing the relationships of reservation and reservation fee under both optimal contracts.
* For the optimal contract $\CTRdb^*$ under DB-bearing-risk, we can see that the reservation fee $p^*$ monotonically increases with the reservation $k^*$. This is because the WSD always benefits from a larger spectrum reservation level (as it does not need to bear the risk); hence, the database can charge a higher reservation fee for a higher reservation level.
* For the optimal contract $\CTRms^*$ under WSD-bearing-risk, we can see that the reservation fee $p^*$ first increases and then decreases with the reservation $k^*$. This is because the WSD's profit first increases with the reservation level, and then decreases with the reservation level (due to the high risk of over-reservation); hence, the reservation fee first increases with the reservation level, and then decreases with the reservation level.
We can further see that under the same reservation level $k^*$, the reservation fee under DB-Bear-Risk is larger than that under WSD-Bear-Risk, hence charges
a higher reservation fee to compensate its expected cost due to over-reservation.
(a) Illustration of Optimal Contracts, (b) Contract-based Spectrum Reservations vs Scheduled Demand $\Da$. Here, $\sigma_{\Da}$ denotes the variance of scheduled demand $\Da$.
Then we compare the reservations under both contracts.
By Proposition <ref>, both $\Kmsctr (\Da)$ and $\Kdbctr (\Da)$ are increasing in $\Da$.
By (<ref>) and (<ref>), we further have the following observation.
\Kmsctr (\Da) \leq \Kdbctr (\Da) \leq \Kso(\Da),\ \ \forall \Da \in [\underline{\Da}, \bar{\Da}],
and $\Kmsctr({\Da}) = \Kdbctr (\Da)
= \Kso( {\Da})$ only when $\Da = \bar{\Da}$.
That is, only when the realized scheduled demand $\Da$ reaches its maximum value (i.e., $\Da = \bar{\Da}$), the reservations under both optimal contracts are identical, and are equal the integrated optimal reservation.
Under other values of $\Da$, the reservation in the contract $\CTRms^*$ (under WSD-bearing-risk) is smaller than that in the contract $\CTRdb^*$ (under DB-bearing-risk), which is further smaller than the integrated optimal reservation.
We illustrate the result of Lemma <ref> in Figure <ref>.b. Intuitively,
When the database bears the risk, it has an incentive to charge a high reservation fee in order to force the WSD to shoulder some of the potential cost.
When the WSD bears the risk, however, the database has less incentive to charge a high reservation fee.
Hence, for the same $\Da$, we find that $\Pdbctr(\Da) > \Pmsctr(\Da)$.
Combined with Proposition <ref>, we have $\Kmsctr(\Da) < \Kdbctr(\Da)$.
§ NUMERICAL RESULTS
In this section, we provide numerical results to compare the performances of the proposed contract-based reservation mechanisms.
Practically speaking, the database's contract choice depends on many factors, among which the reservation decision and the resulting (expected) profit are the most important ones.
Hence, we will present the expected profits (of the database, WSD, and the aggregated one) under different contracts associated with different risk-bearing schemes.
Unless specified otherwise, we assume the following spectrum trading parameters: $r=1$,
$s=0.8$, $w=0.5$, and $c=0.2$.
We further assume that the scheduled demand $\Da$ follows the normal distribution, and the bursty demand $\Db$ follows the chi-square distribution.[The parameter setting is for an illustrative purpose; similar insights can be obtained using other parameter settings.]
(a) Database Profit vs Wholesale Price, and (b) Network Profit vs Wholesale Price.
§.§ Profit vs Wholesale Price
Figure <ref> illustrates (a) the database profit and (b) the network profit (aggregate profit) achieved in different reservation solutions (associated with information asymmetric under different wholesale prices $w$.)
In this simulation, we assume that $\Da$ follows the normal distribution with mean $\EDa = 30$ and variance $\VDa^2 = 64$, and $\Db$ follows the chi-square distribution with mean $\EDb = 30$ and variance $\VDb^2 = 60$.
From Figure <ref>.a, we have the following observations regarding the database profit.
* Under both risking bear-schemes, the contract-based reservation leads to a much higher profit for the database, compared to the reservation solution without information sharing.
* The database can achieve a higher profit with the optimal reservation contract under DB-bearing-risk (the blue triangle curve) than that under WSD-bearing-risk (the red square curve).
This is quite counter-intuitive. The reason is that the WSD is more risk-averse than the database.
From Figure <ref>.b, we have the following observations.
* Centralized Optimal Network Profit:
The green circle curve denotes the optimal network profit achieved in the centralized reservation solution $\Kso$ given in (<ref>), which is independent of the wholesale price $w$, and serves as an upper-bound of the network profit under any other reservation solution.
* Network Profit under DB-Bear-Risk:
The blue “x” (dash) curve and blue triangle (solid) curve denote the network profit achieved under DB-Bearing-Risk, without and with contract, respectively. Specifically, the former one is achieved from the reservation solution without information sharing, i.e., $\Kdbasy $ given in (<ref>). The latter one is achieved from the optimal reservation contract $\CTRdb^*$ given in Theorem <ref>.
Obviously, information sharing based on the optimal reservation contract proposed in this paper improves the total network profit up to $5\%$.
* Network Profit under WSD-Bear-Risk:
The red “$+$” (dash) curve and red square (solid) curve denote the network profit achieved under WSD-Bearing-Risk, with and without contract, respectively.
Specifically, the former one is achieved from the reservation solution without information sharing, i.e., $\Kmsasy $ given in (<ref>). The latter one is achieved from the optimal reservation contract $\CTRms^*$ given in Theorem <ref>.
Different with the DB-Bearing-Risk scheme, we can see that only when the wholesale price $w$ is large (e.g., $w>0.62$ in this example), the performance under the optimal reservation contract is better than that without information sharing.
This is because the purpose of contract under the WSD-bearing-risk is to reduce the double marginalization effect. Hence, the network profit under WSD-Bearing-Risk contract is independent of the wholesale price.
However, as the objective of contract is maximizing the database profit, the database would charge an equivalent high “wholesale price" from the WSD. As shown by the Figure <ref>.b, such equivalent “wholesale price" lies between $0.6$ and $0.7$. This high equivalent high wholesale price decreases the performance of social welfare.
Our results provide the following important insight for a general reservation problem: it is not only individually better, but also socially better to leave the over-reservation risk to the less risk-averse decision maker.
§.§ Profit vs Scheduled Demand Variance
Figure <ref> illustrates (a) the database profit and (b) the network profit achieved in different reservation solutions (associated with information asymmetry), under different scheduled demand variance $\VDa^2$.
Notice that $\VDa^2$ reflects the degree of information asymmetry. That is, a higher $\VDa^2$ implies a larger variance of $\Da$, and thus a higher uncertainty of the database regarding $\Da$.
In this simulation, we assume that $\Da$ follows the normal distribution with mean $\EDa = 30$ (and with different variances), and $\Db$ follows the chi-square distribution with mean $\EDb = 30$ and variance $\VDb^2 = 60$.
(a) Database Profit vs Scheduled Demand Variance, and (b) Network Profit vs Scheduled Demand Variance.
From Figure <ref>.a, we can further see that under both risk-bearing schemes, the optimal contracts ($\CTRdb^*$ and $\CTRms^*$) can greatly improve the database profit. Moreover, the database can achieve a slightly higher profit with the optimal contract $\CTRdb^*$ under DB-bearing-risk, than the optimal contract $\CTRms^*$ under WSD-bearing-risk.
Figure <ref>.b leads to a similar observation as Figure <ref>.b. Specifically, under DB-bearing-risk, the optimal contract $\CTRdb^*$ can always increase with the total network profit; while under WSD-bearing-risk, the optimal contract $\CTRms^*$ can only increase the total network profit when $\VDa^2$ is small (i.e., when the degree of information asymmetry is low).
We can further see that the profits under both optimal contracts decrease with $\VDa^2$.
This is because with a larger $\VDa^2$, the scheduled demand $\Da$ varies more dramatically. As the scheduled demand $\Da$ is the private information of the WSD, the larger variance of $\Da$ means that the database needs to pay a higher information rent to the WSDs.
§ CONCLUSION
We propose a broker-based spectrum reservation market model for TV white space network, under stochastic demand and information asymmetry.
To solve the problem, we propose a contract-based reservation framework, which ensures WSDs share their private information credibly.
We analyze the incentive compatibility of contracts, and further derive the optimal contracts under different risk-bearing schemes.
Our analysis and extensive simulations indicate that the optimal contract under DB-bearing-risk
leads to a higher database profit and higher network profit than that under WSD-bearing-risk.
As there is no large-scale commercializatin of TV white space network with detailed spectrum reservation scheme, our work can serve as a first step to give theoretical insights into the problem of risk-bearing between the database and the WSD, and promote the economic study of such a network.
In this work, we have focused on the TV white space network, where the primary users are the TV broadcasters. As the TV towers have fixed locations and TV programs have well planned schedules, the database has full information regarding the primary usage of TV spectrum ahead of time. This allows us to focus on the demand uncertainty from unlicensed users in this paper.
On the other hand, the issue of primary usage uncertainly becomes much more important,
if we consider the Licensed Shared Access (LSA) and Authorised Shared Access (ASA) models, where unlicensed users may access specific non-TV band (e.g., $3.5$ GHz band in the United States and $2.3$ GHz band in Europe).
This is because these bands are used for ship- and air-borne radar systems which are critical to the operation of the national defense.
Our model can be directly extended to analyze the LSA/ASA systems, if
there is no penalty to the database and the WSD for not being able to serve all demands.
However, when the expected payoffs of the database and the WSD depend on both the demand randomness and the available spectrum randomness, it would be much more challenging to obtain theoretical results by solving the contract design problem.
We will consider the issue of two-sided uncertainty and the interaction among the licensee, the database, and the WSDs in our future work.
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§ APPENDIX
§.§ Total Spectrum Reservation
In this section, we will show how the database determines the optimal aggregate reservation after knowing WSDs' contract item choices.
§.§.§ Aggregate Reservation under DB-Bearing-Risk
Under the DB-bearing-risk scheme, a WSD only purchases the spectrum that it actually needs in each access period, and hence the database can sell the over-reserved spectrum to other co-located WSDs in each access period.
Consider a set $\Nset $ of co-located WSDs.
Suppose that in a particular reservation period, the scheduled demand of each WSD $n \in \Nset$ is $\Da_{\n}$.
By the incentive compatibility of the optimal contract $\CTRdb^* = \{ \langle \Kdbctr (\Da ), \Pdbctr (\Da ) \rangle \}_{\forall \Da}$ (defined in Theorem <ref>), each WSD $n $ will choose the reservation amount $\Kdbctr (\Da_n)$ intended for its demand type.
Without the further optimization on the aggregate reservation, the database will simply reserve an amount $\Kdbctr (\Da_n)$ for each WSD $n$, hence the total reservation is $\TKdb = \sum_{n\in\Nset}\Kdbctr (\Da_n)$.
Next we study how to further optimize the aggregate reservation for the database.
Let $\TDa = \sum_{\n \in \Nset} \Da_{\n}$ denote the aggregated scheduled demand of all WSDs in $\Nset$,
and let $\TDb = \sum_{\n \in \Nset} \Db_{\n}$ denote the aggregated bursty demand of all WSDs in $\Nset$.
Note that $\TDa$ does not change during the whole reservation period, while $\TDb$ changes every access period.
Moreover, the database can deduce the previous value of the scheduled demand $\Da_{\n}$ of each WSD $n$ (hence the total scheduled demand $\TDa$) from the WSDs' selections.
However, neither the database nor the WSDs can obtain the precise value of $\TDb$, as it changes every access period.
Let $h(\TDb)$ and $H(\TDb)$ denote the p.d.f and c.d.f. of $\TDb$, respectively.
It is easy to see that when the realized total demand $\TD = \TDa + \TDb$ is smaller than the total reservation $\TKdb$, the database can reduce the reservation cost by reducing the reservation amount.
With a reduced aggregate reservation, on the other hand, the database may need to replenish the spectrum reservation (with a higher replenishment cost) when the realized total demand $\TD $ is larger than the aggregate reservation.
Let $\OTKdb$ denote that the reduced aggregate reservation of the database (for WSDs $\Nset$).
Let $\TKdb = \sum_{\n \in \Nset} \Kdbctr(\Da_{\n})$ denote the total reservation requests of all WSDs, and we have $\OTKdb \leq \TKdb $ as each WSD will not purchase an amount larger than its request.
The database profit depends on the actual realization of total demand $\TD$.
Let $c$ be the database's reservation cost, and let $ c^{\textsc{ex}}>c$ be the replenishment cost.
Then, we have,
(a) when $\TD \leq \OTKdb $, the database can reduce the reservation cost by $c \cdot ( \TKdb - \OTKdb )$;
(b) when $\OTKdb \leq \TD \leq \TKdb$, the database needs to replenish a reservation amount $\TD - \OTKdb$ to meet the requirements of WSDs, which will introduce a total replenishment cost of $\ce \cdot (\TD - \OTKdb) $; (c) when $ \TKdb \leq \TD$, the database needs to replenish a reservation amount $\TKdb - \OTKdb$ to meet the requirements of WSDs, which will introduce a total replenishment cost of $\ce \cdot (\TKdb - \OTKdb) $.
Based on the above discussion, the expected increasing profit of the database is
\begin{equation}\label{eq:db_revenue_add}
\begin{aligned}
%\AUdb = & \EX_{\TDb}\bigg[ c \cdot ( \TKdb - \OTKdb ) \cdot \Pr( \TD \leq \OTKdb \leq \TKdb ) \\
% & \qquad{}- \ce \cdot (\TD - \OTKdb) \cdot \Pr( \OTKdb \leq \TD \leq \TKdb ) \\
% & \qquad{} - \ce\cdot ( \TKdb - \OTKdb ) \cdot \Pr( \OTKdb \leq \TKdb\leq \TD ) \\
% & \qquad{} - \c \cdot( \OTKdb - \TKdb )\cdot \Pr( \TD \leq \TKdb \leq \OTKdb ) \\
% & \qquad{} + \w \cdot (\TD - \TKdb )\cdot \Pr( \TKdb \leq \TD \leq \OTKdb ) \\
% &\qquad{} - \c \cdot( \OTKdb - \TD ) \cdot \Pr( \TKdb \leq \TD \leq \OTKdb ) \\
% & \qquad{} + \w \cdot ( \OTKdb - \TKdb ) \cdot \Pr( \TKdb \leq \OTKdb \leq \TD )
% \bigg]
% \AUdb = & \EX_{\TDb}\bigg[ c \cdot ( \TKdb - \OTKdb ) \cdot \Pr( \TD \leq \OTKdb ) - \ce \cdot (\TD - \OTKdb) \cdot \Pr( \OTKdb \leq \TD \leq \TKdb ) \\
% & \qquad{} - \ce\cdot ( \TKdb - \OTKdb ) \cdot \Pr( \TKdb\leq \TD )
% - \c \cdot( \OTKdb - \TKdb )\cdot \Pr( \TD \leq \TKdb \leq \OTKdb ) \\
% & \qquad{} + \w \cdot (\TD - \TKdb )\cdot \Pr( \TKdb \leq \TD \leq \OTKdb ) - \c \cdot( \OTKdb - \TD ) \cdot \Pr( \TKdb \leq \TD \leq \OTKdb ) \\
% & \qquad{} + \w \cdot ( \OTKdb - \TKdb ) \cdot \Pr( \TKdb \leq \OTKdb \leq \TD )
\AUdb = & \EX_{\TDb}\bigg[ c \cdot ( \TKdb - \OTKdb ) \cdot \Pr( \TD \leq \OTKdb \leq \TKdb ) \\
& \qquad{} - \ce \cdot (\TD - \OTKdb) \cdot \Pr( \OTKdb \leq \TD \leq \TKdb ) \\
& \qquad{} - \ce\cdot ( \TKdb - \OTKdb ) \cdot \Pr( \OTKdb \leq \TKdb\leq \TD )
\bigg]
\end{aligned}
\end{equation}
The database's optimal aggregate reservation $\OTKdb^*$ that maximizes (<ref>) satisfies
\begin{equation}\label{eq:db_total_optimal}
\begin{aligned}
% &\ce + \w + \c \cdot ( \TKdb - \OTKdb ) \cdot h( \OTKdb - \TDa ) \\
% & \qquad{} \qquad{} - ( \c + \ce + \w ) \cdot H( \OTKdb - \TDa ) = 0
% &\ce + \w + \c \cdot ( \TKdb - \OTKdb ) \cdot h( \OTKdb - \TDa ) - ( \c + \ce + \w ) \cdot H( \OTKdb - \TDa ) = 0.
&\ce + \c \cdot ( \TKdb - \OTKdb^{*} ) \cdot h( \OTKdb^{*} - \TDa ) \\
& \qquad{} \quad{} - ( \c + \ce ) \cdot H( \OTKdb^{*} - \TDa ) = 0.
\end{aligned}
\end{equation}
Obviously, $\OTKdb^{*} $ is a function of $\TDa$.
Figure <ref> illustrates the database profit with and without aggregate reservation optimization under different numbers of WSDs.
The blue-square and the red-circle curves denote the database profit with and without further optimization on the aggregate reservation, respectively.
In this simulation, we assume that different WSDs' scheduled demands $\Da_{\n}$, $\n \in \Nset$ are i.i.d., and different WSDs' bursty demands $\Db_{\n}$, $\n \in \Nset$ are also i.i.d.
The scheduled demand $\Da_{\n}$ of each WSD $n$ follows a normal distribution with mean $\EDa = 9$ and variance $\VDa^2 = 3$.
The bursty demand $\Db_{\n}$ of each WSD $n$ follows a chi-square distribution with different values of the degrees of freedom.[When the degrees of freedom of a chi-square distribution changes, the mean and the variance of this chi-square distribution change accordingly. Specifically, the value of mean is equal to the value of the degrees of freedom, while the value of variance is two times of the value of the degrees of freedom. ]
Database profit vs. numbers of WSDs, where $\EDb$ and $\VDb^2$ denote the mean and variance of the bursty demand of each WSD $n$, respectively.
Figure <ref> shows that with the further optimization, the database can increase its profit up to $12\%$.
This benefit increases with the number of WSDs, as more WSDs submitting their spectrum reservation requirements, more freedom for the database to assign over-reserved spectrum of one WSD to other WSDs in need.
As the mean $\EDb$ and variance $\VDb^2$ of the bursty demand increase,
the difference between the database profit with and without aggregated reservation schedule also increases.
§.§.§ Aggregate Reservation under WSD-Bearing-Risk
Under the DB-bearing-risk scheme, a WSD purchases all the spectrum it requests in each access period, even if the realized demand is smaller than the requested reservation. Hence, the database cannot sell the over-reserved spectrum to other co-located WSDs in each access period. In this case, the database does not need to make the further optimization on the aggregate reservation.
Namely, the database's optimal aggregate reservation $\OTKms^{*}$ is[According to existing regulations <cit.>, WSDs cannot directly communicate with each other to sell their extra spectrum. Hence, we assume that the over-reservation spectrum under the WSD-bearing-risk scheme is wasted. How to let different WSDs trade their over-reservation spectrum through the database will be our future work.],
\begin{equation}\label{eq:ms_total_optimal}
\OTKms^{*} = \sum_{\n \in \Nset} \Kmsctr(\Da_{\n}),
\end{equation}
which is exactly the total requested reservation of all WSDs.
§.§ Proof for Lemma <ref>
Since $s > w > c$, we have $\Kso$ is always larger than $\Kdbsym$ and $\Kmssym$. Note that $\Kdbsym = \Da +
{G^{-1} \Big( {(w-c)}/{w} \Big)}$ is monotonic increasing with wholesale price $w$ while $\Kmssym
= \Da + {G^{-1} \Big( {(s-w)}/{s} \Big)}$ decreases monotonically with $w$. We can easily get the conclusion.
§.§ Proof for Lemma <ref>
Note that (<ref>) and (<ref>) show that the optimal bandwidth $\K \geq \Da$ for any realized $\Da$. Hence, we can write the expected network profit in (<ref>) with respect to $\Da$ as:
\begin{equation}\label{centralized_profit_appendix}
\begin{aligned}
%\EX_{\Da}[\Utot] &= ( r - c) E_{\Da}[\K] - (r - s) ( \EX_{\Da}[\K] - \mu_{\Da} ) \\
%& \quad{} - s \cdot \int_0^{\EX_{\Da}[\K] - \mu_{\Da}} G(u) \mathrm{d}u
\EX_{\Da}[\Utot] = & ( r - c) E_{\Da}[\K] \\
& - (r - s) ( \EX_{\Da}[\K] - \mu_{\Da} ) - s \cdot \int_0^{\EX_{\Da}[\K] - \mu_{\Da}} G(u) \mathrm{d}u
\end{aligned}
\end{equation}
where $\mu_{\Da} = \EX[\Da]$ is the mean of the scheduled demand. Moreover, the first derivative with respect to expected bandwidth $\EX_{\Da}[\K]$ is:
\begin{equation}
\begin{aligned}
\frac{\partial{\EX_{\Da}[\Utot]}}{\partial{\EX_{\Da}[\K]}} &= ( s - c) - s \cdot G( \EX_{\Da}[\K] - \mu_{\Da} )
\end{aligned}
\end{equation}
and the second derivative is $-s \cdot g( \EX_{\Da}[K] - \mu_{\Da} ) \leq 0$.
Hence, the maximum solution of (<ref>) is $\EX_{\Da}[\Kso]$ and the network profit increases with $\EX_{\Da}[\K]$ when $\EX_{\Da}[\K] < \EX_{\Da}[\K]$. By applying Lemma <ref>, we can get the conclusion.
§.§ Proof for Proposition <ref>
We prove it by contradiction.
$\leftarrow$: When $\xi = \xi_1$, the following IC constraint must be satisfied:
\begin{equation}
\begin{aligned}
%&(s-w) \EX_{\Db} \big[ \min\big\{{\Db} , (\K(\Da_2)-\Da_1 )^{+}\big\} \big] \\
%& -(s-w)\EX_{\Db} \big[ \min\big\{{\Db} , (\K({\Da_1})-\Da_1 )\big\} \big] \geq \P(\Da_2)- \P(\Da_1)
&(s-w) \EX_{\Db} \big[ \min\big\{{\Db} , (\K(\Da_2)-\Da_1 )^{+}\big\} \big] \\
&~~ -(s-w)\EX_{\Db} \big[ \min\big\{{\Db} , (\K({\Da_1})-\Da_1 )\big\} \big] \geq \P(\Da_2)- \P(\Da_1)
\end{aligned}
\end{equation}
from which we can find that if $\P(\Da_1) > \P(\Da_2)$, then $\K (\Da_1) > \K (\Da_2)$. For otherwise, the IC cannot be satisfied.
$\rightarrow$: We first differentiate (<ref>) with respect to $\Da$ and get:
\begin{equation}
\begin{aligned}
\frac{\dt{\P (\Da)}}{\dt{\Da}} = (s-w) \cdot [ 1 - G(\K(\Da) - \Da)] \cdot \frac{ \dt{\K(\Da)} }{ \dt{\Da} }
\end{aligned}
\end{equation}
Then we can conclude that if $K (\xi_1) > K (\xi_2)$, then $P (\xi_1) > P (\xi_2)$
§.§ Proof for Proposition <ref>
We prove it by contradiction. Note that:
\begin{equation}
\begin{aligned}
\frac{\partial^2{ \Ums (\K,\P,\Da )}}{\partial{\K}\partial{\Da}} = (s-w) g \big( \K -\Da \big) &> 0 , \mathrm{~~and~~}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\frac{\partial^2{\Ums (\K,\P,\Da )}}{\partial{\K}^2} = -(s-w ) g \big( \K -\Da \big) < 0.
\end{aligned}
\end{equation}
Hence, for any $\Da_1 > \Da_2$, if $\K(\Da_1) < \K(\Da_2)$. That we have:
\begin{equation}
\begin{aligned}
0 &=\left.\frac{\partial{\Ums\left(\K,\P,\Da_1 \right)}}{\partial{\K}}\right|_{\K=\K(\Da_1)} \nonumber\\
&> \left.\frac{\partial{\Ums\left(\K,\P,\Da_1 \right)}}{\partial{\K}}\right|_{\K=\K(\Da_2)} \nonumber\\
&> \left.\frac{\partial{\Ums\left(\K,\P,\Da_2 \right)}}{\partial{\K}}\right|_{\K=\K(\Da_2)},
\end{aligned}
\end{equation}
where the equality is because of IC and the inequalities are because of the sign of the second-order derivatives. But this contradicts the optimality of $\K\left(\Da_2\right)$.
§.§ Proof for Proposition <ref>
The IC constraint implies that $\Ums ({\Da}) = \max_{\hDa} \Ums \big( \K(\hDa), \P(\hDa),\Da \big)$. The envelope theorem further shows that:
\begin{equation}
\begin{aligned}
\frac{\dt{\Ums} (\Da )}{ \dt{\Da}} & = \left.\frac{\partial{\Ums (\K({\hDa}),\P(\hDa),\Da_i )}}{\partial{\Da}}\right|_{\hDa = \Da} \\
& = (r-s ) + (s-w ) G \big( \K( {\Da})-\Da \big) > 0.
\end{aligned}
\end{equation}
Then we have $\Ums(\Da)$ is increasing in $\Da $.
§.§ Proof for Proposition <ref>
By using IC constraint and the envelope theorem, we have:
\begin{equation}\label{asy_contract_master_profit_proof}
\begin{aligned}
\frac{\dt{\Ums} (\Da )}{\dt{\Da}} & = \left.\frac{\partial{\Ums (\K({\hDa}),\P(\hDa),\Da_i )}}{\partial{\Da}}\right|_{\hDa = \Da} \\
& = (r-s ) + (s-w ) G \big( \K( {\Da})-\Da \big).
\end{aligned}
\end{equation}
By integrating both sides, we get the Proposition <ref>.
§.§ Proof for Proposition <ref>
We only have to show that these three conditions imply IC and IR. Noted that (<ref>), $\Ums(\Da) $ is obtained by Lemma (<ref>). We therefore have:
\begin{equation}
\begin{aligned}
%&\Ums \big( \K(\hDa),\P(\hDa),\Da \big) \nonumber\\
%& \quad{} = {\int_{\underline{\Da}}^{\Da}}\frac{\partial{\Ums \big( \K(\hDa),\P(\hDa),x \big)}}{\partial{x}} \dt{x}+ \Ums \big( \K(\hDa),\P(\hDa),\underline{\Da} \big) \nonumber\\
%& \quad{} = \Ums \big( \K(\hDa),\P(\hDa),\hDa \big)+ {\int_{\hDa}^{\Da}}\big[(r-s) +(s-w)\\
%& \qquad{} \qquad{} \qquad{} \qquad{} \qquad{} \qquad{} \quad{} \cdot G\big( \K(\Da) - x \big) \big]\mathrm{d}{x} \nonumber\\
%& \quad{}= \Ums \big( \K(\Da),\P(\Da),\Da \big) + {\int_{\hDa}^{\Da}}(s-w)\big[G\big( \K(\hDa)- x \big) \\
%& \qquad{} \qquad{} \qquad{} \qquad{} \qquad{} \qquad{} \qquad{} -G\big( \K(x)-x \big)\big] \mathrm{d}{x}
&\Ums \big( \K(\hDa),\P(\hDa),\Da \big) \nonumber\\
& \quad{} = {\int_{\underline{\Da}}^{\Da}}\frac{\partial{\Ums \big( \K(\hDa),\P(\hDa),x \big)}}{\partial{x}} \dt{x}+ \Ums \big( \K(\hDa),\P(\hDa),\underline{\Da} \big) \nonumber\\
& \quad{} = \Ums \big( \K(\hDa),\P(\hDa),\hDa \big) \nonumber\\
& \qquad{} + {\int_{\hDa}^{\Da}}\big[(r-s) +(s-w) \cdot G\big( \K(\Da) - x \big) \big]\mathrm{d}{x} \nonumber\\
& \quad{}= \Ums \big( \K(\Da),\P(\Da),\Da \big) \nonumber\\
& \qquad{} + {\int_{\hDa}^{\Da}}(s-w)\big[G\big( \K(\hDa)- x \big) -G\big( \K(x)-x \big)\big] \mathrm{d}{x}
\end{aligned}
\end{equation}
If $\Da > \hDa$, then the above equation is non-positive (because both $\K$ and $G$ are increasing) and hence $\Ums \big( \K(\Da),\P(\Da),\Da \big) \geq
\Ums \big( \K({\hDa}),\P(\hDa),\Da \big)$. This inequality also holds for $\Da < \hDa$ by a similar argument. Therefore, the two condition imply IC.
Evaluate $\Ums (\Da)$ at $\underline{\Da}$ and using (<ref>), we can get IR immediately.
§.§ Proof for Proposition <ref>
Let $i=1,2$ and define ${\hDa}_i^{*} = \arg\max_{\hDa}\Ums \big( \K(\hDa),\P(\hDa),\Da_i \big)$ We therefore have $\Ums \big( \K({\hDa}_1^{*}),\P({\hDa}_1^{*}),\Da_2 \big) \leq \Ums \big( \K({\hDa}_2^{*}),\P({\hDa}_2^{*}),\Da_2 \big)$. Noted that:
\begin{equation}
\begin{aligned}
\frac{\partial{\Ums \big( \K({\hDa}),\P(\hDa),\Da \big)}}{\partial{\Da}} & = (r-s) + (s-w)G\big(\K(\hDa)-\Da \big) \nonumber\\
& \geq 0.
\end{aligned}
\end{equation}
Hence, for any $\Da_1<\Da_2$, We have $\Ums \big( \K({\hDa}_1^{*}),\P({\hDa}_1^{*}),\Da_1 \big) \leq \Ums \big( \K({\hDa}_1^{*}),\P({\hDa}_1^{*}),\Da_2 \big)$. By using the IC constraint, we have ${\hDa}_i^{*} = \Da_i$, and therefore $\Ums \big( \K(\Da_1),\P(\Da_1),\Da_1 \big) \leq \Ums \big( \K(\Da_2),\P(\Da_2),\Da_2 \big)$. Hence IR only needs to be satisfied at $\Da = \underline{\Da}$ and the other participation constraints for $\Da > \underline{\Da}$ are redundant. Hence, we get (<ref>).
§.§ Proof for Theorem <ref>
This Theorem can be simply proved by using Proposition <ref> and Proposition <ref>. Here we jus show that the optimal point of $\phidb (\K(\Da),\Da )$ is unique by proving $\phidb (\K(\Da),\Da ) $ is unimodal.
We let $z(\Da) = \K(\Da)-\Da$, then we rewrite $\phidb (\K(\Da),\Da ) $ as $\phidb (z(\Da))$ and get
\begin{equation}
\begin{aligned}
%\frac{\partial {\phidb (z )}}{\partial{z} } = & s \cdot [1-G ( z ) ] - c \\& \textstyle - \frac{1-F(\Da)}{f(\Da)} ( s-w ) g( z ).
\frac{\partial {\phidb (z )}}{\partial{z} } = & s \cdot [1-G ( z ) ] - c - \frac{1-F(\Da)}{f(\Da)} ( s-w ) g( z ).
\end{aligned}
\end{equation}
To prove that $\phidb (z) $ is unimodal, it suffices to show that ${\partial {\phidb (z )}}/{\partial{z} }$ changes the sign once. Note that $\Db$ follows chi-square distribution with support $[0, +\infty)$. Then we have $\lim_{z \rightarrow 0}{\partial {\phidb (z )}}/{\partial{z} } = s - c >0 $,and $\lim_{z \rightarrow +\infty}{\partial {\phidb (z )}}/{\partial{z} } = - c <0 $. Then we consider the second order derivative of $\phidb (z ) $ with respect to $z$ and we have
\begin{equation}
\begin{aligned}
\frac{\partial^2 {\phidb (z )}}{\partial{z^2} } &= - s \cdot g ( z ) - \frac{1-F(\Da)}{f(\Da)} ( s-w ) g^{'}( z ) \\
&= \frac{z^{\frac{n}{2} - 2} e^{-\frac{z}{2}} }{\Gamma(\frac{n}{2}) 2^{\frac{n}{2}}} \left[ \frac{A(\xi) - 2}{2} z - A(\xi)\left(\frac{n}{2} - 1\right) \right]
\end{aligned}
\end{equation}
where $A(\xi) = { \left( (1-F(\Da))(s-w)\right) }/{(s \cdot f(\Da))}$ and $n$ is the freedom of chi-square distribution. Note that $\lim_{z \rightarrow +\infty}{\partial^2 {\phidb (z )}}/{\partial{z^2} } <0 $, and
case 1: $\frac{A - 2}{2} < 0$, then ${\partial^2 {\phidb (z )}}/{\partial{z^2} } < 0$, and ${\partial {\phidb (z )}}/{\partial{z} }$ changes the sign once;
case 2: $\frac{A - 2}{2} > 0$, then the value of ${\partial^2 {\phidb (z )}}/{\partial{z^2} }$ is first negative, then becomes positive. However, as ${\partial^2 {\phidb (z )}}/{\partial{z^2} }$ is the linear function of $z$, then ${\partial {\phidb (z )}}/{\partial{z} }$ only changes the sign once.
If $\Db$ follows from normal distribution with mean $\mu$ and standard variance $\sigma$, then $\phidb (z) $ is also unimodal as long as $G(x) \rightarrow 0$ when $x<0$. In such case, $\lim_{z \rightarrow 0}{\partial {\phidb (z )}}/{\partial{z} } = s - c >0 $, and $\lim_{z \rightarrow +\infty}{\partial {\phidb (z )}}/{\partial{z} } = - c <0 $. And
\begin{equation*}
\begin{aligned}
\frac{\partial^2 {\phidb (z )}}{\partial{z^2} } &= - s \cdot g ( z ) - \frac{1-F(\Da)}{f(\Da)} ( s-w ) g^{'}( z ) \\
&= \frac{1 - F(\Da) }{f(\Da)} \frac{s-w}{s} \frac{1}{ \sigma \sqrt{2 \pi} } e^{- \frac{ (z - \mu )^2 }{ 2 \sigma^2 }} \left[ \frac{z}{\sigma^2} - \frac{\mu}{\sigma^2} - 1 \right]
\end{aligned}
\end{equation*}
The above equation shows that the value of ${\partial^2 {\phidb (z )}}/{\partial{z^2} }$ first is negative, then become positive. However, as ${\partial^2 {\phidb (z )}}/{\partial{z^2} }$ is the linear function of $z$, then ${\partial {\phidb (z )}}/{\partial{z} }$ only changes the sign once.
§.§ Proof for Proposition <ref>
By using IC constraint and the envelope theorem, we have:
\begin{equation}\label{asy_contract_master_profit_ad_proof}
\begin{aligned}
\frac{\dt{\Ums} (\Da )}{\dt{\Da}} & = \left.\frac{\partial{\Ums (\K({\hDa}),\P(\hDa),\Da_i )}}{\partial{\Da}}\right|_{\hDa = \Da} \\
& = (r-s ) + s \cdot G \big( \K( {\Da})-\Da \big).
\end{aligned}
\end{equation}
By integrating both sides, we get the Proposition <ref>.
§.§ Proof for Theorem <ref>
This Theorem can be simply proved by using Proposition <ref> and Proposition <ref>. Here we jus show that the optimal point of $\phims (\K(\Da),\Da )$ is unique by proving $\phims (\K(\Da),\Da ) $ is unimodal.
We let $z(\Da) = \K(\Da)-\Da$, then we rewrite $\phims (\K(\Da),\Da ) $ as $\phims (z(\Da))$ and get
\begin{equation}
\begin{aligned}
%\frac{\partial {\phims (z )}}{\partial{z} } = & s \cdot [1-G ( z ) ] - c \\& \textstyle - \frac{1-F(\Da)}{f(\Da)} \cdot s \cdot g( z ).
\frac{\partial {\phims (z )}}{\partial{z} } = & s \cdot [1-G ( z ) ] - c - \frac{1-F(\Da)}{f(\Da)} \cdot s \cdot g( z ).
\end{aligned}
\end{equation}
To prove that $\phims (z) $ is unimodal, it suffices to show that ${\partial {\phims (z )}}/{\partial{z} }$ changes the sign once. Note that $\Db$ follows chi-square distribution with support $[0, +\infty)$. Then we have $\lim_{z \rightarrow 0}{\partial {\phims (z )}}/{\partial{z} } = s - c >0 $,and $\lim_{z \rightarrow +\infty}{\partial {\phims (z )}}/{\partial{z} } = - c <0 $. Then we consider the second order derivative of $\phims (z ) $ with respect to $z$ and we have
\begin{equation}
\begin{aligned}
\frac{\partial^2 {\phims (z )}}{\partial{z^2} } &= - s \cdot g ( z ) - \frac{1-F(\Da)}{f(\Da)} \cdot s \cdot g^{'}( z ) \\
&= \frac{z^{\frac{n}{2} - 2} e^{-\frac{z}{2}} }{\Gamma(\frac{n}{2}) 2^{\frac{n}{2}}} \left[ \frac{A(\xi) - 2}{2} z - A(\xi)\left(\frac{n}{2} - 1\right) \right]
\end{aligned}
\end{equation}
where $A(\xi) = { \left( (1-F(\Da)) \cdot s\right) }/{(s \cdot f(\Da))}$ and $n$ is the freedom of chi-square distribution. Note that $\lim_{z \rightarrow +\infty}{\partial^2 {\phims (z )}}/{\partial{z^2} } <0 $, and
case 1: $\frac{A - 2}{2} < 0$, then ${\partial^2 {\phims (z )}}/{\partial{z^2} } < 0$, and ${\partial {\phims (z )}}/{\partial{z} }$ changes the sign once;
case 2: $\frac{A - 2}{2} > 0$, then the value of ${\partial^2 {\phims (z )}}/{\partial{z^2} }$ is first negative, then becomes positive. However, as ${\partial^2 {\phims (z )}}/{\partial{z^2} }$ is the linear function of $z$, then ${\partial {\phims (z )}}/{\partial{z} }$ only changes the sign once.
If $\Db$ follows from normal distribution with mean $\mu$ and standard variance $\sigma$, then $\phims (z) $ is also unimodal as long as $G(x) \rightarrow 0$ when $x<0$. In such case, $\lim_{z \rightarrow 0}{\partial {\phims (z )}}/{\partial{z} } = s - c >0 $, and $\lim_{z \rightarrow +\infty}{\partial {\phidb (z )}}/{\partial{z} } = - c <0 $. And
\begin{equation*}
\begin{aligned}
\frac{\partial^2 {\phims (z )}}{\partial{z^2} } &= - s \cdot g ( z ) - \frac{1-F(\Da)}{f(\Da)} \cdot s \cdot g^{'}( z ) \\
&= \frac{1 - F(\Da) }{f(\Da)} \frac{s-w}{s} \frac{1}{ \sigma \sqrt{2 \pi} } e^{- \frac{ (z - \mu )^2 }{ 2 \sigma^2 }} \left[ \frac{z}{\sigma^2} - \frac{\mu}{\sigma^2} - 1 \right]
\end{aligned}
\end{equation*}
The above equation shows that the value of ${\partial^2 {\phims (z )}}/{\partial{z^2} }$ first is negative, then become positive. However, as ${\partial^2 {\phims (z )}}/{\partial{z^2} }$ is the linear function of $z$, then ${\partial {\phims (z )}}/{\partial{z} }$ only changes the sign once.
§.§ Proof for Lemma <ref>
By comparing (<ref>) and (<ref>), we can easily get the conclusion.
|
1511.00355
|
§ INTRODUCTION
The great success of the LHC during its first run has provided a plethora of data that have tested the Standard Model (SM) to great accuracy. The high precision achieved in many observables, together with their agreement with the SM predictions, has resulted in strong implications for new physics (NP) frameworks, increasing the NP scale or requiring non-trivial flavor structures. In spite of the undisputed success of the SM predictions, this run of LHC has left several hints of NP in semileptonic transitions $b\to s\ell^+\ell^-$. In particular the recent measurement by the LHCb collaboration of the ratio $R_K = \mathrm{Br}(B \rightarrow K \mu^+ \mu^-)/\mathrm{Br}(B\rightarrow K e^+ e^-)$ shows a deviation from the SM prediction at the $2.6\sigma$ level, hinting to a large violation of lepton flavor universality <cit.>. Several global analyses of $b\to s\ell^+\ell^-$ transitions have been performed <cit.>, showing a significant preference towards a NP explanation of the experimental anomalies found in these transitions. Among the many observables entering in the fit, the angular analysis of $B\to K^*\mu^+\mu^-$ decays, also by the LHCb collaboration, presents a clear example of deviation from the SM prediction in the observable $P'_5$ with $2.9\sigma$ significance in two of the bins <cit.>.
The NP necessary to accommodate the $b\to s\ell^+\ell^-$ anomalies should be non-universal in the lepton sector and present flavor changing neutral currents (FCNCs) in the down-quark sector. Various attempts to analize these anomalies in a model-independent way or to accommodate them with specific NP models can be found the literature <cit.>. In this talk I will present a $\mathrm{U(1)}^\prime$ gauge symmetry implementation with a minimal particle content and characterized by having all the flavor violations controlled by the gauge symmetry, which makes them proportional to off-diagonal elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. This model can be obtained by gauging the global symmetry introduced by Branco, Grimus and Lavoura (BGL) in the context of two-Higgs-doublet models (2HDMs) to address the flavor problem of these models while allowing for (controlled) flavor violations <cit.>, providing a solution completely different from the hypothesis of natural flavor conservation <cit.>.
The outline of the talk is as follows: In Section <ref> I introduce the BGL models and their main properties. Section <ref> is devoted to the construction of the gauged $\mathrm{U(1)}_{\mbox{\scriptsize BGL}}$ model. A discussion on the main constraints and phenomenological implications of the new gauge sector is given in Section <ref>. I summarize in Section <ref>.
§ THE BRANCO-GRIMUS-LAVOURA MODEL
BGL models <cit.> provide a class of 2HDMs characterized by the presence of FCNCs at tree-level entirely fixed by CKM matrix elements and the ratio of vevs of the Higgs doublets. This is achieved by the imposition of a global horizontal symmetry that gives rise to a specific set of Yukawa textures. The Yukawa sector of the model is given by
\begin{align}\label{eq:Lagquarks}
-\mathcal{L}^{\mbox{\scriptsize{quark}}}_{\mbox{\scriptsize{Yuk}}}&=\overline{q_L^0}\, \Gamma_i\,
\Phi_i d^0_R+\overline{q_L^0}\, \Delta_i\, \widetilde{\Phi}_i u^0_R+\text{h.c.} \,,
\end{align}
where $\tilde \Phi_i\equiv i\sigma_2\Phi_i^*$, with $\sigma_2$ the Pauli matrix. Both Higgs doublets, $\Phi_i$ ($i=1,2$), acquire vacuum expectation values (vev) $|\langle\Phi^0_i\rangle|=v_i/\sqrt{2}$ with $v\equiv \left(v_1^2+v_2^2\right) = (\sqrt{2} G_F)^{-1/2}\simeq 246$ GeV fixed by measurements of the muon lifetime; as usual I define $\tan \beta = v_2/v_1$. In this talk I will focus on the so-called top-BGL implementation where up and down Yukawa matrices, $\Delta_i$ and $\Gamma_i$, are constrained by the BGL symmetry to have the following structure:
\begin{align}\label{eq:BGLText}
\begin{aligned}
\Gamma_1&=
\begin{pmatrix}
\times & \times & \times\\
\times & \times & \times\\
0 & 0 & 0
\end{pmatrix}\,,\quad
\Gamma_2=
\begin{pmatrix}
0 & 0 & 0\\
0 & 0 & 0\\
\times & \times & \times
\end{pmatrix}\,, \\[0.2cm]
\Delta_1&=
\begin{pmatrix}
\times & \times & 0\\
\times & \times & 0\\
0 & 0 & 0
\end{pmatrix} \,\,,\quad
\Delta_2=
\begin{pmatrix}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & \times
\end{pmatrix} \,.
\end{aligned}
\end{align}
These quark textures introduce FCNCs only in the down-quark sector that are suppressed by quark masses and off-diagonal elements of the third row of the CKM matrix <cit.>, which results in a strong suppression of flavor-changing processes involving light quarks. This symmetry suppression of FCNCs allows top-BGL models to avoid experimental constraints even when the new scalars remain light, with masses of $\mathcal{O}\left(100\right)$ GeV <cit.>.
Given an Abelian symmetry characterized by the field transformation
\begin{align}
\psi\rightarrow e^{iQ^{\psi}}\psi\,,
\end{align}
the most general implementation of the top-BGL Yukawa textures is defined by the following set of charges
\begin{align}
\begin{aligned} \label{eq:system}
Q^\Phi&= \mathrm{diag}\,(X_{\Phi_1}, X_{\Phi_2})= \frac{1}{2}\mathrm{diag}\,\left(X_{uR}-X_{dR},X_{tR}-X_{dR}\right)\,,
\end{aligned}
\end{align}
with $X_{uR}\neq X_{tR}$.
Although the Abelian BGL symmetry can be discrete, it always leads to an enhanced accidental $\mathrm{U(1)}$ global symmetry in the scalar sector, which results in the presence of an undesired Goldstone in the theory <cit.>. Several solutions to this problem of BGL models can be found in Refs. <cit.>. Following Ref. <cit.> I present here a different solution based on the promotion of the BGL symmetry to a local one. This way the gauging of the BGL symmetry serves a two-fold purpose: it provides a natural solution to the Goldstone boson problem in BGL models and introduces at the same time a new gauge boson with a very rich phenomenology, allowing for an explanation of the $b\to s\ell^+\ell^-$ anomalies in terms of symmetry principles.
§ GAUGED BGL SYMMETRY
In this section I extend the SM gauge symmetry with an extra $\mathrm{U(1)}^\prime$ factor that is identified with the BGL symmetry introduced in the previous section. The scalar sector of this model consists of two Higgs doublets and a complex SM singlet, necessary to give a heavy mass to the new gauge boson, while the fermion content remains the same as in the SM. As we are dealing with a chiral symmetry, one should pay special attention to the cancellation of anomalies when gauging the BGL symmetry. In Ref. <cit.> it was shown that in BGL 2HDMs the cancellation is automatic for the QCD currents, i.e. $\mathrm{U(1)}^\prime[\mathrm{SU(3)}_C]^2$. However, this is not the case for the rest of the anomalies,
\begin{align}\label{eq:anomalies}
\begin{aligned}
&\mathrm{U(1)}^\prime [\mathrm{SU(2)}_L]^2\,,\quad \mathrm{U(1)}^\prime [\mathrm{U(1)}_Y]^2\,,
\qquad [\mathrm{U(1)}^\prime]^2 \mathrm{U(1)}_Y \,,\\
&[\mathrm{U(1)}^\prime]^3\,,\quad \qquad \quad \, \mathrm{U(1)}^\prime [\mbox{Gravity}]^2 \,.
\end{aligned}
\end{align}
In particular we find that, with the charge assignments in Eq. (<ref>), there is no solution to all the anomaly cancellation conditions unless we extend the symmetry to the lepton sector. Just like in the SM we find that anomaly cancellation can only be granted through the interplay of quarks and leptons. Taking the most general symmetry implementation, with all the lepton charges being free parameters,
\begin{align}
Q_L^\ell&=\mathrm{diag}\left(X_{eL},X_{\mu L},X_{\tau L}\right)\,,\qquad Q_R^e=\mathrm{diag}\left(X_{eR},X_{\mu R},X_{\tau R}\right)\,,
\end{align}
we find only one solution to the anomaly cancellation conditions. It is characterized by only two free charges, $X_{dR}$ and $X_{\mu R}$, up to lepton-flavor permutations:
\begin{align}
\begin{split}
X_{uR}&=-X_{dR}-\frac{1}{3}X_{\mu R}\,,\quad X_{tR}=-4X_{dR}+\frac{2}{3}X_{\mu R}\,,\\
X_{e L}&=X_{dR}+\frac{1}{6}X_{\mu R}\,,\quad\hspace{6.5pt} X_{e R}=2X_{dR}+\frac{1}{3}X_{\mu R}\,,\\
X_{\tau L}&=\frac{9}{2}X_{dR}-X_{\mu R}\,,\quad\hspace{6pt} X_{\tau R}= 7 X_{dR}-\frac{4}{3}X_{\mu R}\,,\\
X_{\mu_L}&=-X_{dR}+\frac{5}{6}X_{\mu R}\,.
\end{split}
\end{align}
However, one should note that the global scale of the charges is unphysical and only accounts for a rescaling of the $\mathrm{U(1)}_{\mbox{\scriptsize BGL}}$ gauge coupling, $g^\prime$, allowing us to freely remove one of the charges. As we can see, anomaly cancellation conditions determine the extension of the symmetry to the lepton sector in an unique way, with the charged-lepton Yukawa sector of the model taking the form
\begin{align}
-\mathcal{L}^{\mbox{\scriptsize c-leptons}}_{\mbox{\scriptsize{Yuk}}}&=\overline{\ell_L^0}\, \Pi_i\, \Phi_i e^0_R+\text{h.c.} \,,
\end{align}
\begin{equation}
\Pi_1=
\begin{pmatrix}
\times&0&0\\
\end{pmatrix}\,,\quad \Pi_2=
\begin{pmatrix}
\end{pmatrix}\,.
\end{equation}
Since the only source of flavor violation of the model is found in the Yukawa matrices, charged-lepton flavor conservation appears in this model as a natural consequence of the gauge symmetry. I call to attention that the neutrino sector of the model has not been specified. Extensions to account for neutrino masses and mixings can potentially modify the anomaly conditions, opening the possibility for new solutions. A systematic study of the neutrino sector will be presented in a future publication.
For phenomenological purposes it is convenient to eliminate the remaining freedom in the model by fixing $X_{\Phi_2}=0$ (or equivalently $X_{dR}=2/15\, X_{\mu R}$), so that the mixing between neutral gauge bosons is suppressed for large $\tan \beta$. For simplicity, I work in this limit and neglect mixing effects for the rest of the talk, leaving a more general analysis of the model for future work. Finally, without loss of generality, I choose a charge normalization by setting $X_{dR}=1$, with no physical implications. With these choices the $\mathrm{U(1)}_{\mbox{\scriptsize BGL}}$ charges read
\begin{align} \label{eq:charges_model}
\begin{aligned}
\end{aligned}
\end{align}
Permutations of lepton flavors yield six different implementations of the symmetry which we denote as $(e,\mu,\tau)=(i,j,k)$, with the model implementation in Eq. (<ref>) denoted as $(1,2,3)$.
To avoid experimental constraints the new gauge boson associated to the BGL symmetry, $Z^{\,\prime}$, should have a heavy mass of a few TeV. This is achieved through the inclusion of a complex scalar SM singlet, $S$, charged under the new symmetry, that acquires a large vev $|\langle S \rangle | = v_S/\sqrt{2}\gg v$ and spontaneously breaks the extra gauge symmetry. Also note that the charge of the singlet, $X_S$, should be fixed in terms of the other scalar charges in order to avoid undesired Goldstone bosons (for more details see Ref. <cit.>), I choose $X_S=1/2\left(X_{\Phi_1}-X_{\Phi_2}\right)=-9/8$. The Lagrangian for the new gauge sector then reads
\begin{align}
\mathcal{L}_{Z^\prime}\simeq -\frac{1}{4}Z_{\mu\nu}^{\,\prime}Z^{\,\prime\,\mu\nu}+\left|D_\mu\,S\right|^2-V\left(S\right)-J^\mu_{Z^\prime}\,Z^{\,\prime}_\mu \,.
\end{align}
Here $Z_{\mu\nu}^{\,\prime}$ is the $Z^{\,\prime}$ field-strength tensor and the $Z^{\,\prime}$ current is denoted as $J^\mu_{Z^\prime}$. Its fermionic piece takes the form
\begin{align}
J_{Z^\prime}^{\,\mu}\supset g^\prime\,\overline{\psi_{i}}\,\gamma^\mu \left[\widetilde Q^\psi_{L,ij}\,P_L+\widetilde Q^\psi_{R,ij}\,P_R\right] \psi_{j}\,,
\end{align}
where $g^\prime$ is the $Z^{\,\prime}$ gauge coupling and $\widetilde Q^\psi$ stands for the $Z^{\,\prime}$ charges (see Eq. (<ref>)) rotated to the fermion physical eigenbasis
\begin{align}\label{eq:flavorstr}
\widetilde Q^\psi_{R} &= Q_{R}^{\psi}\,,\quad
\widetilde Q^u_{L} = Q_{L}^{q} \,, \quad \widetilde Q^\ell_{L} =
\,,\quad
\widetilde Q^d_{L} = - \frac{5}{4} \mathbb{1} +\frac{9}{4} \, \begin{pmatrix}
|V_{td}|^2 & V_{ts} V_{td}^* & V_{tb} V_{td}^* \\
V_{td} V_{ts}^* & |V_{ts}|^2 & V_{tb} V_{ts}^* \\
V_{td} V_{tb}^* & V_{ts} V_{tb}^*& |V_{tb}|^2
\end{pmatrix} \,.
\end{align}
Note that $Z^{\,\prime}$-mediated flavor violations are only present in the left-handed down-quark sector.
§ PHENOMENOLOGICAL CONSTRAINTS AND MODEL PREDICTIONS
In this section I will only highlight the main constraints and predictions concerning $Z^{\,\prime}$ observables and refer the reader to Ref. <cit.> for an extended discussion on the phenomenology of the model. Since all BGL charges are fixed, $Z^{\,\prime}$ observables are completely determined in terms of just two free parameters, the $Z^{\,\prime}$ mass and gauge coupling.
Constraints on the $Z^{\,\prime}$ from low energy observables are only sensitive to the combination of parameters $M_{Z^\prime}/g^\prime$. Bounds from $B_s$-mixing give the limit $M_{Z^\prime}/g^\prime\gtrsim16$ TeV at $95\%$ CL and we find that other low-energy constraints such as those from neutrino trident production, atomic parity violation, electric dipole moments or anomalous magnetic moments are always weaker than the one from $B_s$-mixing. Also interesting are the LHC bounds on direct searches for a $Z^{\,\prime}$ decaying into a pair of leptons, since they allow to disentangle the two free parameters. Using the model independent analysis provided by the CMS collaboration <cit.> we find the exclusion limit in the mass of the $Z^{\,\prime}$, $M_{Z^\prime}\gtrsim3-4$ TeV, depending on the model implementation. Additionally, requiring the gauge couplings to remain perturbative we obtain an upper limit on the the value of $g^\prime$. The model develops a Landau pole at the see-saw scale, $\Lambda_{\mbox{\scriptsize{LP}}}\gtrsim 10^{14}$ GeV, for $g^\prime\lesssim0.14$ while if we push the Landau pole to the Planck scale, $\Lambda_{\mbox{\scriptsize{LP}}}\gtrsim 10^{19}$ GeV, we find the limit $g^\prime\lesssim0.12$.
Model $C_{10}^{\mbox{\scriptsize{NP}} \mu}/C_{9}^{\mbox{\scriptsize{NP}}\mu}$ $C_{9}^{\mbox{\scriptsize{NP}}e}/C_{9}^{\mbox{\scriptsize{NP}}\mu}$ $C_{10}^{\mbox{\scriptsize{NP}}e}/C_9^{\mbox{\scriptsize{NP}}\mu}$ $C_{9}^{\mbox{\scriptsize{NP}} \tau}/C_{9}^{\mbox{\scriptsize{NP}}\mu}$ $C_{10}^{\mbox{\scriptsize{NP}}\tau}/C_9^{\mbox{\scriptsize{NP}}\mu}$ $\kappa_{9}^{\mu}$
(1,2,3) $3/17$ $9/17$ $3/17$ $-8/17$ $0$ $-1.235$
(1,3,2) $0$ $-9/8$ $-3/8$ $-17/8$ $-3/8$ $0.581$
(2,1,3) $1/3$ $17/9$ $1/3$ $-8/9$ $0$ $-0.654$
(2,3,1) $0$ $-17/8$ $-3/8$ $-9/8$ $-3/8$ $0.581$
(3,1,2) $1/3$ $-8/9$ $0$ $17/9$ $1/3$ $-0.654$
(3,2,1) $3/17$ $-8/17$ $0$ $9/17$ $3/17$ $-1.235$
Correlations among the NP contributions to the effective operators $\mathcal{O}_{9,10}^{\ell}$.
I now turn to the $b\to s\ell^+\ell^-$ anomalies, the effective Hamiltonian for these transitions reads
\begin{align}
\mathcal{H}_{\mbox{\scriptsize eff}}= - \frac{G_F}{\sqrt{2}} \frac{\alpha}{\pi} V_{tb}V_{ts}^*
\sum_{i} \left( C_i^{\ell} \mathcal{O}^{\ell}_i + C_i^{\prime \ell} \mathcal{O}^{\prime \ell}_i
\right),
\end{align}
\begin{align}
\begin{aligned}
\mathcal{O}^{\ell}_9&=\left(\overline{s}\gamma_\mu P_Lb\right)\left(\overline{\ell}\gamma^\mu
\ell\right)\,, &\mathcal{O}_9^{\prime \ell} &= \left(\overline{s}\gamma_\mu P_Rb\right)\left(\overline{\ell}\gamma^\mu \ell\right)\,,\\[5pt] \mathcal{O}^{\ell}_{10}&= \left(\overline{s}\gamma_\mu P_Lb\right)\left(\overline{\ell}\gamma^\mu\gamma_5 \ell\right)\,, &\mathcal{O}_{10}^{\prime \ell}&= \left(\overline{s}\gamma_\mu P_Rb\right)\left(\overline{\ell}\gamma^\mu\gamma_5 \ell\right)\,.
\end{aligned}
\end{align}
The SM contribution to these operators is $C_{9}^{\mbox{\scriptsize{SM}}} \simeq - C_{10}^{\mbox{\scriptsize{SM}}} \simeq 4.2\,\forall\,\ell$, with negligible contributions to the primed operators. Since right-handed quark currents are flavor conserving in our model, $\mathcal{O}_{9,10}^{\prime \ell}$ also receive negligible contributions from the $Z^{\,\prime}$. Its contribution to $\mathcal{O}_{9,10}^\ell$ is given by
\begin{align}
\begin{aligned}
C_{9}^{\mbox{\scriptsize{NP}} \ell}&\simeq - \frac{\pi }{\alpha V_{ts}^* V_{tb} }\; {\widetilde Q}^{\,d}_{L,sb} \left({\widetilde Q}^{\,e}_{L,\ell \ell}+{\widetilde Q}^{\,e}_{R,\ell \ell}\right) \left(\frac{g^\prime v }{M_{Z^\prime} }\right)^2 \,, \\
C_{10}^{\mbox{\scriptsize{NP}} \ell}&\simeq \frac{\pi }{\alpha V_{ts}^* V_{tb} }\; {\widetilde Q}^{\,d}_{L,sb} \left({\widetilde Q}^{\,e}_{L,\ell \ell}-{\widetilde Q}^{\,e}_{R,\ell \ell}\right) \left(\frac{g^\prime v }{M_{Z^\prime} }\right)^2 \,,
\end{aligned}
\end{align}
where $C_{i}^{\ell} \equiv C_{i}^{\mbox{\scriptsize{SM}}} + C_{i}^{\mbox{\scriptsize{NP}} \ell}$. The correlations among the different contributions is shown in Table <ref> where I also provide the value of $C_{9}^{\mbox{\scriptsize{NP}}\mu}$ as a function of $g^\prime/M_{Z^\prime}$, which is given in terms of the following normalization
\begin{align}
C_9^{\mbox{\scriptsize{NP}} \mu} \equiv \kappa_9^{\mu} \times 10^{4} \left( \frac{ g^{\prime}
v }{ M_{Z^{\prime}} } \right)^2=\kappa_9^{\mu} \times605~{\rm TeV}^2 \left( \frac{ g^{\prime}
}{ M_{Z^{\prime}} } \right)^2 \,.
\end{align}
Model $C_{9}^{\mbox{\scriptsize{NP}}\mu} (1\sigma)$ $C_{9}^{\mbox{\scriptsize{NP}}\mu} (2 \sigma)$
(1,2,3) – $[-2.92, -0.61]$
(3,1,2) $[-0.93, -0.43]$ $[-1.16, -0.17]$
(3,2,1) $[-1.20, -0.53]$ $[-1.54, -0.20]$
In figure, model prediction for $R_K$ as a function of $g^\prime/M_{Z^\prime}$. This is shown together with the SM prediction, the experimental measurement by LHCb at $1\,\sigma$ and $2\,\sigma$ and the bound from $B_s$-mixing. In table, bounds on $C_{9}^{\mbox{\scriptsize{NP}}\mu}$ from $R_K$ for the implementations that are able to accommodate the anomaly.
These NP contributions to the effective Hamiltonian can be tested with global fits to the angular distributions of the semileptonic $b\to s\ell^+\ell^-$ transitions. Furthermore, the hadronic ratios
\begin{align} \label{eq:had_ratios}
R_M \equiv \frac{\mathrm{Br}( \bar B \rightarrow \bar M \mu^+ \mu^- )}{\mathrm{Br}( \bar B
\rightarrow \bar M e^+ e^- )}\stackrel{\rm SM}{=}1+\mathcal O(m_\mu^2/m_b^2)\,,
\end{align}
with $M\in\{K, K^*, X_s, K_0(1430),\ldots\}$ <cit.>, provide a precise test on the universality of these transitions. In Figure <ref> I show the model prediction for $R_K$ from the different implementations of the model together with the recent experimental measurement of the ratio by the LHCb collaboration <cit.> and the bound from $B_s$-mixing. As we can see, only two of the implementations are able to explain the anomaly at $1\sigma$ and a third one is able to accommodate it at $2\sigma$. For these models, I show in Table <ref> the bounds on $C_9^{\mbox{\scriptsize{NP}} \mu}$ that are extracted from $R_K$. The values obtained for this operator are in good agreement with those favored by the global fits, as was also noticed in other $Z^{\,\prime}$ models <cit.>. Furthermore, as noted in Ref. <cit.> the absence of flavor violating NP couplings to right-handed quarks, as it happens in this model, implies a strong condition on the ratios defined in Eq. (<ref>), $R_K=R_{K^*}=R_{X_s}=\dots$ This provides an important test on the validity of the model and shows the importance of further measurements of these ratios.
Finally, if a $Z^{\,\prime}$ is discovered in the next runs of LHC a useful test on its universality can be found in the ratios
\begin{align}
\mu_{f/f^{\prime}}&\equiv \frac{\sigma(pp\rightarrow Z^{\,\prime}\rightarrow
f\bar{f})}{\sigma(pp\rightarrow Z^{\,\prime}\rightarrow f^{\prime} \bar f^{\prime})}\,,
\end{align}
that in our model take the following form
\begin{align}
\mu_{b/t} \simeq \frac{X_{bL}^2+X_{bR}^2}{X_{tL}^2+X_{tR}^2} \,, \qquad \mu_{\ell/\ell^{\prime}} \simeq \frac{X_{\ell L}^2+X_{\ell R}^2}{X_{\ell^{\prime} L}^2+X_{\ell^{\prime} R}^2} \,.
\end{align}
We find $\mu_{b/t}\simeq1$ while the ratios $\mu_{\ell/\ell^{\prime}}$ are highly dependent on the model implementation, opening the possibility to test this model and discriminate among the different implementations.
§ SUMMARY
Regions allowed at $1\sigma$ and $2\sigma$ by the $R_K$ measurement in the
$\{M_{Z^{\prime}},g^{\prime}\}$ plane for the models $(1,2,3)$, $(3,1,2)$ and $(3,2,1)$. Exclusion
limits from $Z^{\prime}$ searches at the LHC are shown in gray. The black lines indicate bounds from
perturbativity of $g^\prime$.
In this talk I presented a class of family non-universal $Z^{\,\prime}$ models based on an horizontal gauge symmetry that completely determines the flavor structure of the model, characterized by the presence of tree-level FCNC in the down-quark sector controlled by the CKM matrix, with no flavor violations in the up-quark sector. Anomaly cancellation conditions extend the symmetry to leptons in a precise way, giving rise to flavor-conserving non-universal couplings in the charged-lepton sector and six possible implementations. Moreover, cancellation of anomalies only allows for two free charges which are fixed for phenomenological purposes, leaving only two relevant parameters in the heavy gauge boson sector, the $Z^{\,\prime}$ mass, $M_{Z^\prime}$, and its gauge coupling, $g^\prime$. This renders a highly predictive NP scenario which is able to accommodate the $b\to s\ell^+\ell^-$ anomalies in some of its implementations.
Present data strongly constraints the parameter space of the model: bounds from $B_s$-mixing imply $M_{Z^\prime}/g^\prime\geq 16$ TeV ($95\%$ CL), direct searches at LHC exclude our $Z^{\,\prime}$ for a mass below $3-4$ TeV and perturbativity of the gauge couplings give the upper limit, $g^\prime\lesssim 0.14$. These constraints are shown together with the regions allowed by $R_K$ in Figure <ref>. The model also presents smoking-gun signatures that will be tested in the recent future, such as the equality of all the hadronic ratios defined in Eq. (<ref>), i.e. $R_K=R_{K^*}=R_{X_s}=\dots$ Moreover, if a $Z^{\,\prime}$ is discovered at LHC, measurements of the ratios $\sigma(pp \rightarrow Z^{\,\prime} \rightarrow \ell_i \bar \ell_i )/\sigma(pp\rightarrow Z^{\,\prime} \rightarrow \ell_j\bar \ell_j )$ would be insightful in order to discriminate among the different model variations and from other NP implementations.
I thank the organizers of EPS-HEP 2015 for the opportunity to present these results. I am also grateful to Alejandro Celis, Martin Jung and Hugo Serôdio for the collaboration in the topic presented here and for useful comments during the preparation of this manuscript. This work has been supported in part by the Spanish Government, ERDF from the EU Commission and the Spanish Centro de Excelencia Severo Ochoa Programme [Grants No. FPA2011-23778, FPA2014-53631-C2-1-P, No. CSD2007-00042 (Consolider Project CPAN), SEV-2014-0398]. I also acknowledge VLC-CAMPUS for an “Atracció del Talent" scholarship.
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|
1511.00136
|
We construct the complete invariant for fused links. It is proved that the set of equivalence classes of $n$-component fused links is in one-to-one correspondence with the set of elements of the abelization $UVP_n/UVP_n^{\prime}$ up to conjugation by the elements from the symmetric group $S_n<UVB_n$.
Keywords: Fused links, unrestricted virtual braid group, knot invariant.
§ INTRODUCTION
Knot invariants are functions of knots that do not change under isotopies.
The study of knot invariants is at the core of knot theory. Indeed, the isotopy
class of a knot is, tautologically, a knot invariant. During last years different authors constructed vast number of knot invariants: the (self) linking number, the unknotting number, the knot group, the knot quandle, the Jones polynomial, the Conway polynomial and so on (see, for example, <cit.>). The disadvantage of a large number of easy countable invariants is that they are not complete, i. e. they do not distinguish some knots or links.
At the same time there are few examples of complete knot invariants, but usually it is difficult to understand if the value of the invariant on the knot coincides with the value of this invariant on the another knot. The complement of a knot itself (as a topological space) is known to be a complete invariant of the knot by the theorem of C. Gordon and J. Luecke <cit.> in the sense that it distinguishes the given knot from all other knots up to ambient isotopy and mirror image. Some invariants associated with the knot complement include the knot group which is just the fundamental group of the complement. The knot quandle is also a complete invariant in this sense <cit.> but it is difficult to determine if two quandles are isomorphic.
Thus an important problem in knot theory is to construct a complete knot invariant which can be easily found and used.
Recently some generalizations of classical knots and links were defined and studied: singular links <cit.>, virtual links <cit.>, welded
links <cit.> and fused links <cit.>. The problem of constructing invariants is also important for all of this knot theories.
One of the ways of studying classical links is to study the braid group.
Singular braids <cit.>, virtual braids <cit.>, welded braids <cit.> and unrestricted virtual braids <cit.> were defined
similar to the classical braid group adding the extra generators and relations. Theorem of A. A. Markov <cit.> reduces the problem of classification of links to some algebraic problems of the theory of braid groups. There are generalizations
of Markov theorem for virtual links, welded links and fused links <cit.>.
In the paper we study fused links, which were defined by L. Kauffman and S. Lambropoulou in <cit.>, and their invariants. Fused links are represented as generic immersions of circles
in the plane (fused link diagrams) where double points can be classical (with the usual
information on overpasses and underpasses) or virtual (see Fig. <ref>).
Crossings in the double welded knot diagram
Fused link diagrams are equivalent
under ambient isotopy and some types of local moves (generalized Reidemeister moves):
classical Reidemeister moves (see Fig. <ref>), virtual Reidemeister moves (see Fig. <ref>), mixed Reidemeister
moves (see Fig. <ref>) and Forbidden moves (see. Fig. <ref>).
Classical Reidemmeister moves
Virtual Reidemeister moves
Mixed Reidemeister moves
Forbidden moves
In the theory of fused links every knot is equivalent to the unknot <cit.>. However not every link is equivalent to the trivial link. For example, trivial 2-component link, Hopf link and Hopf link with one virtual crossing and with one classical crossing all are different (see Fig. <ref>).
Different 2-component fused links
The full classification of fused links is not
(completely) trivial. In particular, A. Fish and E. Keyman proved that the fused link with classical crossings only is completely determined by the linking
numbers of each pair of components <cit.>. It means that the set of linking
numbers for each pair of components of fused links is a full invariant for fused links which have only classical crossings.
In the present paper we find all non-equivalent classes of fused links and construct an easy computable full invariant for fused links. We use the following proposition, which is implicitly formulated in <cit.>.
Proposition. There exists a map $\varrho^*: UVB_{\infty}\to UVP_{\infty}$, such that the closures of the braids $\beta$ and $\varrho^*(\beta)$ are equivalent as fused links.
Denote by $T$ the set of coset representatives of $UVP_n/UVP_n^{\prime}$ and for the element $\alpha\in UVP_n$ denote by $\overline{\alpha}\in T$ the unique coset representative $\alpha UVP_n^{\prime}=\overline{\alpha}UVP_n^{\prime}.$
Then the main result of the paper can be formulated in the following form.
Theorem. Let $\alpha$ and $\beta$ be unrestricted virtual braids. Then their closures $\widehat{\alpha}$ and $\widehat{\beta}$ are equivalent as fused links if and only if $\overline{\varrho^*(\alpha)}$ and $\overline{\varrho^*(\beta)}$ are conjugated by the element from $S_n<UVB_n$.
Thus the map $\widehat{\alpha}\to \overline{\varrho^*(\alpha)}^{S_n}$ is a complete invariant for fused links and in order to understand that two fused links $\widehat{\alpha}$ an $\widehat{\beta}$ are equivalent or not we just need to compare two finite sets $\overline{\varrho^*(\alpha)}^{S_n}$ and $\overline{\varrho^*(\beta)}^{S_n}$, or equivalently, to understand that the element $\overline{\varrho^*(\alpha)}$ belongs to the finite set $\overline{\varrho^*(\beta)}^{S_n}$ or not.
At the almost same time B. Audoux, P. Bellingeri, J.-B. Meilhan and E. Wagner found the full classification of fused links independently from the author <cit.>. Using the different from the author's methods they proved that every fused link is completely determined by the set of virtual linking numbers for each pair of components. The result is the same (however in different formulation) and these two approaches seem to us complementary and both interesting.
The author is grateful to Valeriy Bardakov for multiple helpful advices during the work on the paper. Also the author would like to thank the University of Bologna (Italy) where a part of this work was completed and especially Prof. Michele Mulazzani for his help and support.
§ DEFINITIONS AND RESULTS
In this section we fix notation and recall basic definitions and known results about different generalizations of braid groups.
The classical braid group $B_n$ on $n$ strands ($n>1$) is the group generated by the elements $\sigma_1,\dots,\sigma_{n-1}$ (see Fig. <ref>)
Geometric braids representing $\sigma_i$ (on the left) and $\sigma_i^{-1}$ (on the right)
with the following defining relations.
\begin{align}
\sigma_i\sigma_{i+1}\sigma_i&=\sigma_{i+1}\sigma_i\sigma_{i+1}&i=1,\dots,n-2;\tag{$B_1$}\label{B1}\\
\sigma_i\sigma_j&=\sigma_{j}\sigma_i&|i-j|\geq2.\tag{$B_2$}\label{B2}
\end{align}
There exists a homomorphism $\iota: B_n\to S_n$ from the braid group $B_n$ onto the symmetric group $S_n$ on $n$ letters. This homomorphism maps the generator $\sigma_i$ to the transposition $\tau_i=(i,~i+1)$ for $i=1, 2, \dots, n - 1$. The kernel of this homomorphism is called pure
braid group on $n$ strands and denoted by $P_n$.
The virtual braid group $VB_n$ is a group obtained from $B_n$ adding new generators $\rho_1,\dots,\rho_{n-1}$ (see Fig. <ref>)
Geometric braid representing $\rho_i$
and additional relations
\begin{align}
\rho_i\rho_{i+1}\rho_i&=\rho_{i+1}\rho_i\rho_{i+1}&i=1,\dots,n-2;\tag{$P_1$}\label{P1}\\
\rho_i\rho_j&=\rho_{j}\rho_i&|i-j|\geq2;\tag{$P_2$}\label{P2}\\
\rho_i^2&=1&i=1,\dots,n-1;\tag{$P_3$}\label{P3}
\end{align}
\begin{align}
\sigma_i\rho_j&=\rho_{j}\sigma_i&|i-j|\geq2;\tag{$M_1$}\label{M1}\\
\rho_i\rho_{i+1}\sigma_i&=\sigma_{i+1}\rho_i\rho_{i+1}&i=1,\dots,n-2.\tag{$M_2$}\label{M2}
\end{align}
It is easy to verify that the elements $\rho_1,\dots,\rho_{n-1}$ generate the symmetric group $S_n$. Also it is known that the elements $\sigma_1,\dots,\sigma_{n-1}$ generate
the braid group $B_n$.
In the paper <cit.> it is proved that the relations
\begin{align}
\rho_i\sigma_{i+1}\sigma_i&=\sigma_{i+1}\sigma_i\rho_{i+1}&i=1,\dots,n-2;\tag{$F_1$}\label{F1}\\
\rho_{i+1}\sigma_i\sigma_{i+1}&=\sigma_i\sigma_{i+1}\rho_i&i=1,\dots,n-2;\tag{$F_2$}\label{F2}
\end{align}
do not hold in the group $VB_n$. According to <cit.> the welded braid group $WB_n$ on $n$ strands is a quotient of the group $VB_n$ by the forbidden relation (<ref>), i. e. it is a group with the generators $\sigma_1,\dots,\sigma_{n-1},\rho_{1},\dots,\rho_{n-1}$ and relations (<ref>)–(<ref>). If we add to the group $WB_n$ the second forbidden relation (<ref>) the we get the unrestricted virtual braid group $UVB_n$.
The elements $\sigma_1,\dots,\sigma_{n-2},\rho_1,\dots,\rho_{n-2}$ generate the subgroup $UVB_{n-1}$ in the group $UVB_n$. Then we have the following chain of inclusions.
The homomorphism $\iota$ can be extended to the homomorphism $UVB_n\to S_n$ by the rule $\iota:\sigma_i, \rho_i\mapsto\tau_i=(i~i+1)$. The kernel of this homomorphism is called pure unrestricted virtual braid group and is denoted by $UVP_n$. The group $UVP_{\infty}$ is defined analogically to the group $UVB_{\infty}$.
The symmetric group $S_n$ acts on the set $\{1,\dots,n\}$. By the symbol $\pi(k)$ we denote an image of the integer $k\in\{1,\dots,n\}$ under the permutation $\pi$. If for the braid $\alpha$ we have $\iota(\alpha)(k)=k$, then we say that $\alpha$ fixes $k$ or $k$ is fixed by $\alpha$.
We say that the braid $\alpha$ does not involve the strand $j$ if $\alpha$ belongs to the group $\langle\sigma_1,\dots,\sigma_{j-2},\sigma_{j+1},\dots, \sigma_{n-1},\rho_1,\dots,\rho_{j-2},\rho_{j+1},\dots, \rho_{n-1}\rangle$. It is obvious that the braid $\alpha\in\langle\rho_1,\dots,\rho_{n-1}\rangle$ which fixes the strands $n-1,n$ does not involve an $n$-strand.
We say that the braid $\beta\in UVB_{n+1}$ is obtained from the braid $\alpha\in UVB_n$ by right stabilization of positive (negative, virtual) type if $\beta=\alpha\sigma_{n}$ ($\beta=\alpha\sigma_{n}^{-1}$, $\beta=\alpha\rho_{n}$ respectively). In this case we say that the braid $\alpha$ is obtained from the braid $\beta$ by the opposite to the right stabilization of positive (negative, virtual) type transformation. S. Kamada proved an analogue of Markov theorem for welded links in <cit.>.
Closures of two welded braids $\alpha$ and $\beta$ are equivalent as welded links if and only if they are related by the finite sequence of the following transformations.
* A conjugation in the welded braid group,
* A right stabilization of positive, negative or virtual type,
* An opposite to a right stabilization of positive, negative or virtual type transformation.
This theorem also holds for fused links since every relation of welded braid group is fulfilled in the unrestricted virtual braid group.
Let us define some element of $UVB_n$. For $i=1,\dots,n-1:$
\begin{align}\notag\lambda_{i,i+1}&=\rho_i\sigma_i^{-1},\\
\notag\lambda_{i+1,i}&=\rho_i\lambda_{i,i+1}\rho_i=\sigma_i^{-1}\rho_i.
\end{align}
For $1\leq i<j-1\leq n-1$:
\lambda_{j,i}&=&\rho_{j-1}\rho_{j-2}\dots\rho_{i+1}\lambda_{i+1,i}\rho_{i+1}\dots\rho_{j-2}\rho_{j-1}.
\end{matrix}$$
The elements $\lambda_{i,j}$ and $\lambda_{j,i}$ for $i<j$ belong to the pure unrestricted virtual braid group $UVP_n$ and have the following geometric interpretation (see Fig. <ref>).
Geometric braids representing $\lambda_{i,j}$ (on the left) and $\lambda_{j,i}$ (on the right)
The following lemma is proved in <cit.> for the corresponding elements in $VB_n$ and therefore is also true in the quotient $UVB_n$.
The following conjugating rules are fulfilled in $UVB_n$:
* for $1\leq i<j\leq n$ and $k< i-1$ or $i<k<j-1$ or $k>j$:
* for $1\leq i<j\leq n$:
* for $1\leq i<j-1\leq n$:
* for $1\leq i+1<j\leq n$:
* for $1\leq i<j\leq n$:
The following result on the structure of the pure unrestricted virtual braid group $UVP_n$ is presented in <cit.>.
The group $UVP_n$ has a presentation with generators $\lambda_{k,l}$ for $1\leq k\neq l \leq n$, and defining relations: $\lambda_{i,j}$ commutes with $\lambda_{k,l}$ if and only if $\{i,j\}\neq\{k,l\}$.
Let $U_j$ be the subgroup of $UVP_n$ generated by the elements $\{\lambda_{i,j}, \lambda_{j,i}~|~1\leq i<j\}$. Then by Theorem <ref> we have
\begin{equation}\label{eq2}
=F_2\times\dots\times F_2,\end{equation}
\begin{equation}\label{eq4}
UVP_n=U_2\times U_3\times\dots\times U_n.
\end{equation}
The structure of the unrestricted virtual braid group $UVB_n$ follows from Lemma <ref> and the theorem <ref> and is also given in <cit.>.
The group $UVB_n$ is isomorphic to the semidirect product $UVP_n\leftthreetimes S_n$, where the symmetric group $S_n$ acts by permutations on the indices of generators $\lambda_{i,j}$ of $UVP_n$.
Denote by $B_{i,j}=\rho_{j-1}\rho_{j-2}\dots\rho_{i+1}\rho_i$ if $i<j$ and $B_{i,j}=1$ in other cases. Using simple calculations in the group $S_n$ it is easy to see that for $i<j$ the image $\iota(B_{i,j})$ is a cycle $(i~i+1\dots j)$.
§ CONSTRUCTION OF $\VARRHO^*$
In this section we construct the map $\varrho^*:UVB_{\infty}\to UVP_{\infty}$, which is implicitly constructed in <cit.>. At first we prove the following lemma.
Let $\alpha\in UVB_n$ and $s\in\{1,\dots,n\}$ be a maximal number, such that $\iota(\alpha)(s)\neq s$. Then $\alpha$ can be uniquely expressed in the form
$$\alpha=\gamma x_sB_{k_s,s}x_{s+1}x_{s+2}\dots x_n, $$
where $x_i\in U_i$ and the braid $\gamma\in UVB_n$ does not involve the strands $s,s+1,\dots,n$.
Proof. By the theorem <ref> for certain elements $\beta\in UVP_n$ and $\pi\in S_n$ we have
\begin{equation}\label{eq1}
\alpha=\beta\pi.
\end{equation}
Let $k_s=\pi(s)=\iota(\beta\pi)(s)=\iota(\alpha)(s)\neq s$. Since $\iota(\alpha)$ is a bijection of $\{1,\dots,n\}$ and $\iota(\alpha)(s)=k_s$, then we have $\iota(\alpha)(k_s)\neq k_s$. Since $s$ is a maximal number with the condition $\iota(\alpha)(s)\neq s$ then $k_s<s$ and therefore $B_{k_s,s}\neq1$.
Since the permutations $\pi$ and $B_{k_s,s}^{-1}$ act identically on the integers $s+1,\dots,n$, then the permutation $\delta=\pi B_{k_s,s}^{-1}$ also acts identically on the integers $s+1,\dots,n$. Moreover the image of the integer $s$ under the permutation $\delta$ is equal to $s$
$$ \begin{CD}
s @>~\pi~>> k_s @>~B_{k_s,s}^{-1}~>> s,
\end{CD}$$
therefore $\delta=\pi B_{k_s,s}^{-1}$ acts identically on the integers $s,\dots,n$.
By the equality (<ref>) for certain elements $y_j\in U_j$ we have $\beta=y_2y_3\dots y_n$, and therefore we can rewrite the equality (<ref>).
\begin{equation}\label{eq3}
\alpha=\beta\pi= y_2y_3\dots y_n\delta B_{k_s,s}=\delta (y_2y_3\dots y_n)^{\delta} B_{k_s,s}
\end{equation}
Since $(y_2\dots y_n)$ is a pure braid, then $(y_2y_3\dots y_n)^{\delta}$ is a pure braid and therefore by the equality (<ref>) we have $(y_2y_3\dots y_n)^{\delta}=z_2\dots z_n$ for certain elements $z_j\in U_j$. The braids $z_2,\dots,z_{s-1}$ do not involve the strands $s,s+1,\dots,n$, therefore the braid $\gamma=\delta z_2\dots z_{s-1}$ does not involve the strands $s,s+1,\dots,n$. Then the equality (<ref>) can be rewritten in the following form.
\begin{align}
\notag\alpha&=\delta (y_2y_3\dots y_n)^{\delta} B_{k_s,s}=\delta z_2\dots z_{s-1}z_sz_{s+1}\dots z_n B_{k_s,s}\\
\notag&=\gamma z_sz_{s+1}\dots z_n B_{k_s,s}=\gamma z_sB_{k_s,s}(z_{s+1}\dots z_n)^{B_{k_s,s}}=\gamma z_sB_{k_s,s}z_{s+1}^{B_{k_s,s}}\dots z_n^{B_{k_s,s}}
\end{align}
Since $z_j\in U_j=\langle\lambda_{1,j},\lambda_{j,1}\rangle\times\langle\lambda_{2,j},\lambda_{j,2}\rangle\times\dots\times\langle\lambda_{j-1,j},\lambda_{j,j-1}\rangle$, then by the theorem <ref> for every $j=s+1,\dots,n$ we have $z_{j}^{B_{k_s,s}}\in U_j$. Therefore, if we denote $x_s=z_s$, $x_{s+1}=z_{s+1}^{B_{k_s,s}}$, $\dots$, $x_n=z_n^{B_{k_s,s}}$ then we have
$$\alpha=\gamma x_sB_{k_s,s}x_{s+1}\dots x_n.$$
and we proved that the braid $\alpha$ can be written in the form from the formulation of the lemma.
To prove that such a representation is unique we consider another representation of $\alpha$ in the form from the formulation of the lemma
\begin{equation}\label{eq5}
\alpha=\gamma x_sB_{k_s,s}x_{s+1}\dots x_n=\eta y_sB_{t_s,s}y_{s+1}\dots y_n
\end{equation}
and prove that $\gamma=\eta$, $k_s=t_s$ and $x_j=y_j$ for every $j=s,\dots,n$.
From the equality (<ref>) we have
\begin{equation}\label{eq6}
\eta^{-1}\gamma = y_sB_{t_s,s}y_{s+1}\dots y_n(x_sB_{k_s,s}x_{s+1}\dots x_n)^{-1}
\end{equation}
Without loss of generality consider that $t_s\leq k_s$ and look at the images of the braids from the right and left sides of this equality under the homomorphism $\iota$.
The permutation $\iota(y_sB_{t_s,s}y_{s+1}\dots y_n(x_sB_{k_s,s}x_{s+1}\dots x_n)^{-1})$ maps $s$ to $s$ if $t_s=k_s$ and maps $s$ to $t_s$ if $t_s<k_s$
$$ \begin{CD}
s @>~\iota(y_sB_{t_s,s}y_{s+1}\dots y_n)~>> t_s @>~\iota((x_sB_{k_s,s}x_{s+1}\dots x_n)^{-1})~>> t_s.
\end{CD}$$
At the same time since $\eta$ and $\gamma$ does not involve the strands $s,\dots,n$, then $s$ is fixed by the permutation $\iota(\eta^{-1}\gamma)$ and therefore $t_s=k_s$. Then from the equality (<ref>) we have $\iota(y_sB_{t_s,s}y_{s+1}\dots y_n(x_sB_{k_s,s}x_{s+1}\dots x_n)^{-1})=1$ and $\iota(\eta^{-1}\gamma)=1$. Therefore $\eta^{-1}\gamma\in UVP_n$ and since it does not involve the strands $s,s+1,\dots n$, we have
$$\eta^{-1}\gamma\in\langle U_2,\dots,U_{s-1}\rangle.$$
On the other side the braid $y_sB_{t_s,s}y_{s+1}\dots y_n(x_sB_{k_s,s}x_{s+1}\dots x_n)^{-1}$ can be rewritten in the following form.
\begin{align}
\notag y_sB_{t_s,s}y_{s+1}\dots y_n(x_sB_{k_s,s}x_{s+1}\dots x_n)^{-1}&=y_sB_{t_s,s}y_{s+1}\dots y_nx_n^{-1}\dots x_{s+1}^{-1}B_{t_s,s}^{-1}x_s^{-1}\\
\notag&=y_sB_{t_s,s}B_{t_s,s}^{-1}(y_{s+1}\dots y_nx_n^{-1}\dots x_{s+1}^{-1}x_s^{-1})^{B_{t_s,s}^{-1}}x_s^{-1}\\
\notag&=y_s(y_{s+1}\dots y_nx_n^{-1}\dots x_{s+1}^{-1}x_s^{-1})^{B_{t_s,s}^{-1}}x_s^{-1}
\end{align}
By the theorem <ref> for every $j=s+1,\dots,n$ we have $y_j^{B_{t_s,s}^{-1}}\in U_j$, $(x_j^{-1})^{B_{t_s,s}^{-1}}\in U_j$, therefore
$$y_sB_{t_s,s}y_{s+1}\dots y_n(x_sB_{k_s,s}x_{s+1}\dots x_n)^{-1}\in \langle U_s,\dots,U_n\rangle.$$
From the equality (<ref>) we have $\langle U_2,\dots,U_{s-1}\rangle\cap\langle U_s,\dots,U_n\rangle=1$, therefore $\eta=\gamma$ and $x_sB_{k_s,s}x_{s+1}\dots x_n=y_sB_{t_s,s}y_{s+1}\dots y_n$.
The lemma is proved.
The following lemma has a bit different formulation in the paper <cit.> and it completely proved there. However here we formulate the lemma in the other words and repeat the proof from <cit.> since this lemma is extremely important in our paper.
There exists a map $\varrho:UVB_{\infty}\to UVB_{\infty}$ with the following conditions.
* For any $\alpha$ the closures of $\alpha$ and $\varrho(\alpha)$ are equivalent as double welded links.
* If $\alpha\in UVP_{\infty}$, then $\varrho(\alpha)=\alpha$.
* If $\alpha\in UVB_n\setminus UVP_n$, then $\varrho(\alpha)\in UVB_{n-1}$.
Proof. We explicitly construct the image of the braid $\alpha\in UVB_n$ under the map $\varrho$. If $\alpha\in UVP_n$, then $\varrho(\alpha)=\alpha$ and there is nothing to construct.
So let $\alpha\in UVB_n\setminus UVP_n$, then the algorithm of finding $\varrho(\alpha)$ has the following steps.
Step 1. Find a maximal number $s\in\{1,\dots,n\}$ with the condition $\iota(\alpha)(s)=k_s\neq s$.
Step 2. Conjugate the braid $\alpha$ by the element $B_{1,n}^{n-s}$.
$$\alpha_1=B_{1,n}^{s-n}\delta B_{1,n}^{n-s}$$
The fused link $\widehat{\alpha}_1$ is equivalent to $\widehat{\alpha}$, and the permutation induced by the braid $\alpha_1$ maps $n$ to $k_n=n-s+k_s< n$
$$ \begin{CD}
n @>~B_{1,n}^{s-n}~>> s @>~\alpha~>> k_s@>~B_{1,n}^{n-s}~>> n-s+k_s=k_n.
\end{CD}$$
Step 3. By Lemma <ref> express the braid $\alpha_1$ in the form
$$\alpha_1=\gamma w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1},$$
where the braid $\gamma$ does not involve the strand $n$ and $w_i\in\langle\lambda_{i,n},\lambda_{n,i}\rangle$ for $i=1,\dots,n$ (see Fig. <ref>).
The braid $\alpha_1$
By Lemma <ref> we have $\alpha_1=\gamma x_n B_{k_n,n}$, where the braid $\gamma$ does not involve an $n$-strand and $x_n\in U_n$. By the equality (<ref>) we have $x_n=w_1\dots w_{n-1}$ for $w_i\in\langle\lambda_{i,n},\lambda_{n,i}\rangle$. Sine $B_{k_n,n}\neq1$, then $B_{k_n,n}=\rho_{n-1}B_{k_n,n-1}$.
Step 4. Change the braid $\alpha_1$ by the braid
$$\alpha_2=\gamma w_1\dots w_{n-2}\rho_{n-1}B_{k_n,n-1}.$$
From the figure <ref> it is obvious that all the crossings of $w_{n-1}$ belong to the same component of $\widehat{\alpha}_1$, therefore, we can virtualize all the classical crossings of $w_{n-1}\rho_{n-1}$ using Kanenobu's technique <cit.>, i. e. to change $\sigma_{n-1}$ to $\rho_{n-1}$ in the representation of $w_{n-1}$ and to obtain a braid $\alpha_2$ such that $\widehat{\alpha}_2$ and $\widehat{\alpha}_1$ are equivalent as fused links.
Since $w_{n-1}\in \langle\lambda_{n-1,n},\lambda_{n,n-1}\rangle$ and $\lambda_{n-1,n}=\rho_{n-1}\sigma_{n-1}^{-1}$, $\lambda_{n,n-1}=\sigma_{n-1}^{-1}\rho_{n-1}$, then after virtualization of classical crossings the braid $w_{n-1}\rho_{n-1}$ involved an odd number of $\rho_{n-1}$ and we obtain $\rho_{n-1}$ instead of $w_{n-1}\rho_{n-1}$. Therefore the closure of the braid $\alpha_2=\gamma w_1\dots w_{n-2}\rho_{n-1}B_{k_n,n-1}$ is equivalent to the closure of the braid $\alpha_1$ as fused link.
Step 5. Construct $\varrho(\alpha)$.
Note that by Lemma <ref> for all $\lambda_{j,n}$ and $\lambda_{n,j}$ with $j<n-1$ we have:
Therefore $\alpha_2=\gamma \rho_{n-1}w^{\prime}_1\dots w^{\prime}_{n-2}B_{k_n,n-1}$, where $w_i^{\prime}=w_i^{\rho_{n-1}}$ is a braid from $\langle\lambda_{i,n-1},\lambda_{n-1,i}\rangle$ and hence does not involve an $n$-strand (see Fig. <ref>).
The braid $\alpha_2$
In the braid $\alpha_2$ there is only one (virtual) crossing on the $n$-strand, so, using Markov moves we obtain a new braid $\alpha_3=\gamma w^{\prime}_1\dots w^{\prime}_{n-2}B_{k_n,n-1}$ whose closure is again equivalent to $\widehat{\alpha}$ and has $n-1$ strands. We have constructed the braid
$$\varrho(\alpha)=\alpha_3=\gamma w^{\prime}_1\dots w^{\prime}_{n-2}B_{k_n,n-1}.$$
The lemma is proved.
It is obvious that the equality
$\varrho(\alpha)=\varrho(\alpha_1)=\varrho(\alpha_2)$ holds in Lemma <ref>.
Let $\alpha\in UVB_n$, then the sequence $\varrho(\alpha),\varrho^2(\alpha),\dots$ is stabilized on some step. Let $\varrho^*$ be such a map which maps the braid $\alpha$ to the braid $\varrho^k(\alpha)$, where $k$ is a minimal integer such that $\varrho^k(\alpha)=\varrho^{k+1}(\alpha)$. Every step of finding $\varrho(\alpha)$ is clearly defined, therefore $\varrho^*$ is a correct function.
Using simple induction on the number $n$ it is easy to prove the following fact.
Let $\alpha\in UVB_n$ be the braid, such that $\varrho^*(\alpha)\in UVP_m$. If $\alpha=\beta\gamma$ for $\beta\in UVB_n$, $\gamma\in UVP_n^{\prime}$, then $\varrho^*(\alpha)=\varrho^*(\beta)\delta$, where $\delta\in UVP_m^{\prime}$.
From Lemma <ref> We have the following corollary which is formulated and proved in the paper <cit.>.
Let $\alpha\in UVB_n$ be an unrestricted virtual braid, such that its closure $\widehat{\alpha}$ has $m$ components. Then there exists a pure unrestricted virtual braid $\beta\in UVP_m$, such that $\widehat{\alpha}=\widehat{\beta}$.
Proof. $\beta=\varrho^*(\alpha)$.
§ PROOF OF THE MAIN RESULTS
Let $\alpha\in UVP_n$ and $u,v\in \langle\lambda_{n-1,n},\lambda_{n,n-1}\rangle$. Then the closure of the braids $\alpha$ and $\alpha[u,v]$ are equivalent as fused links.
Proof. Let $\gamma$ be the following braid from $UVB_{n+2}$
$$\gamma=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}B_{n-1,n+2}.$$
Here the braid $\alpha u^{-1}\rho_{n-1}$ does not involve the strands $n+1,n+2$ and the braid $u^{\rho_{n}\rho_{n-1}}$ belongs to $\langle\lambda_{n,n+1},\lambda_{n+1,n}\rangle$.
Let us find the braid $\varrho(\gamma)$.
Step 1. Since $\iota(\gamma)(n+2)=n-1$, then the maximal number which is not fixed by $\gamma$ is equal to $n+2$.
Step 2. $\gamma_1=\gamma$.
Step 3. $\gamma=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}B_{n-1,n+2}$, where the braid $\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}$ does not involve the strand $n+2$.
Step 4. $\gamma_2=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}\rho_{n+1}B_{n-1,n+1}$.
Step 5. $\varrho(\gamma)=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}B_{n-1,n+1}$.
Let us find the braid $\varrho^2(\gamma)$.
Step 1. Since $\iota(\varrho(\gamma))(n+1)=n$, then the maximal number which is not fixed by $\varrho(\gamma)$ is equal to $n+1$.
Step 2. $\varrho(\gamma)_1=\varrho(\gamma)$.
Step 3. The braid $\varrho(\gamma)$ can be rewritten
\begin{align}
\notag\varrho(\gamma)&=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}B_{n-1,n+1}\\
\notag&=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}\rho_{n-1}\rho_n\\
\notag&=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}\rho_{n-1}\rho_n=\alpha u^{-1}u^{\rho_{n}}\rho_n,
\end{align}
where the braid $\alpha u^{-1}$ does not involve the strand $n+1$ and $u^{\rho_{n}}$ belongs to $\langle\lambda_{n-1,n+1},\lambda_{n+1,n-1}\rangle$.
Step 4. $\varrho(\gamma)_2=\alpha u^{-1}u^{\rho_{n}}\rho_n$.
Step 5. $\varrho^2(\gamma)=\alpha$.
Since $\varrho^2(\gamma)=\alpha$ is a pure braid, then $\varrho^*(\gamma)=\alpha$. Let $w=v^{\rho_{n-1}B_{n-1,n+1}B_{n-1,n+2}}$, then by Lemma <ref> the braid $w$ belongs to $\langle\lambda_{n+2,n+1},\lambda_{n+1,n+2}\rangle$. Denote by $\delta=w\gamma w^{-1}$ and find the braid $\varrho^*(\delta)$. At first find the braid $\varrho(\delta)$.
Step 1. Since $w$ is a pure braid, then the maximal number, which is not fixed by $\delta$ is equal to the maximal number which is not fixed by $\gamma$ and is equal to $n+2$.
Step 2. $\delta_1=\delta$.
Step 3. The braid $\delta$ can be rewritten in details
\begin{align}
\notag \delta&=w\gamma w^{-1}=w\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}B_{n-1,n+2}w^{-1}\\
\notag&=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}wB_{n-1,n+1}(w^{-1})^{B_{n-1,n+2}^{-1}}B_{n-1,n+2}\\
\notag&=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}w^{B_{n-1,n+1}}(w^{-1})^{B_{n-1,n+2}^{-1}}B_{n-1,n+2}\\
\notag&=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}(w^{-1})^{B_{n-1,n+2}^{-1}}w^{B_{n-1,n+1}}B_{n-1,n+2},
\end{align}
where the braid $\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}(w^{-1})^{B_{n-1,n+2}^{-1}}$ does not involve the strand $n+2$ and the braid $w^{B_{n-1,n+1}}$ belongs to $\langle\lambda_{n+2,n},\lambda_{n,n+2}\rangle$.
Step 4. $\delta_2=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}(w^{-1})^{B_{n-1,n+2}^{-1}}w^{B_{n-1,n+1}}\rho_{n+1}B_{n-1,n+1}$.
Step 5. $\varrho(\gamma)=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}(w^{-1})^{B_{n-1,n+2}^{-1}}w^{B_{n-1,n+1}\rho_{n+1}}B_{n-1,n+1}$.
Let us fin the braid $\varrho^2(\delta)$
Step 1. Since $\iota(\varrho(\delta))(n+1)=n$, then $n+1$ is maximal number which is not fixed by $\varrho(\delta)$.
Step 2. $\varrho(\delta)_1=\varrho(\delta)$.
Step 3. The braid $\varrho(\delta)$ can be rewritten in details
\begin{align}
\notag \varrho(\delta)&=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}B_{n-1,n+1}(w^{-1})^{B_{n-1,n+2}^{-1}}w^{B_{n-1,n+1}\rho_{n+1}}B_{n-1,n+1}\\
\notag&=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}(w^{-1})^{B_{n-1,n+2}^{-1}B_{n-1,n+1}^{-1}}w^{B_{n-1,n+1}\rho_{n+1}B_{n-1,n+1}^{-1}}B_{n-1,n+1}B_{n-1,n+1}\\
\notag&=\alpha u^{-1}\rho_{n-1}u^{\rho_{n}\rho_{n-1}}(w^{-1})^{B_{n-1,n+2}^{-1}B_{n-1,n+1}^{-1}}w^{B_{n-1,n+1}\rho_{n+1}B_{n-1,n+1}^{-1}}\rho_{n-1}\rho_n\\
\notag&=\alpha u^{-1}u^{\rho_{n}}(w^{-1})^{B_{n-1,n+2}^{-1}B_{n-1,n+1}^{-1}\rho_{n-1}}w^{B_{n-1,n+1}\rho_{n+1}B_{n-1,n+1}^{-1}\rho_{n-1}}\rho_n\\
\notag&=\alpha u^{-1}(w^{-1})^{B_{n-1,n+2}^{-1}B_{n-1,n+1}^{-1}\rho_{n-1}}u^{\rho_{n}}w^{B_{n-1,n+1}\rho_{n+1}B_{n-1,n+1}^{-1}\rho_{n-1}}\rho_n,
\end{align}
where the braid $\alpha u^{-1}(w^{-1})^{B_{n-1,n+2}^{-1}B_{n-1,n+1}^{-1}\rho_{n-1}}$ does not involve the strand $n+1$ and the braid $u^{\rho_n}ww^{B_{n-1,n+1}\rho_{n+1}B_{n-1,n+1}^{-1}\rho_{n-1}}$ belongs to $\langle\lambda_{n-1,n+1},\lambda_{n+1,n-1}\rangle$.
Step 4. $\varrho(\delta)_2=\alpha u^{-1}(w^{-1})^{B_{n-1,n+2}^{-1}B_{n-1,n+1}^{-1}\rho_{n-1}}u^{\rho_{n}}w^{B_{n-1,n+1}\rho_{n+1}B_{n-1,n+1}^{-1}\rho_{n-1}}\rho_n$.
Step 5. The braid $\varrho^2(\delta)$ follows
$$\varrho^2(\delta)=\alpha u^{-1}(w^{-1})^{B_{n-1,n+2}^{-1}B_{n-1,n+1}^{-1}\rho_{n-1}}uw^{B_{n-1,n+1}\rho_{n+1}B_{n-1,n+1}^{-1}\rho_{n-1}\rho_n}.$$
Since $w=v^{\rho_{n-1}B_{n-1,n+1}B_{n-1,n+2}}$, then $(w^{-1})^{B_{n-1,n+2}^{-1}B_{n-1,n+1}^{-1}\rho_{n-1}}=v^{-1}$ and
\begin{align}
\notag w^{B_{n-1,n+1}\rho_{n+1}B_{n-1,n+1}^{-1}\rho_{n-1}\rho_n}&=v^{\rho_{n-1}B_{n-1,n+1}B_{n-1,n+2}B_{n-1,n+1}\rho_{n+1}\underline{B_{n-1,n+1}^{-1}}\rho_{n-1}\rho_n}\\
\notag&=v^{\rho_{n-1}B_{n-1,n+1}B_{n-1,n+2}B_{n-1,n+1}\rho_{n+1}\rho_{n-1}\underline{\rho_n\rho_{n-1}\rho_n}}\\
\notag&=v^{\rho_{n-1}\underline{B_{n-1,n+1}B_{n-1,n+2}}B_{n-1,n+1}\rho_{n+1}\rho_{n}\rho_{n-1}}\\
\notag&=v^{\rho_{n-1}B_{n-1,n+2}\underline{B_{n,n+2}B_{n-1,n+1}}\rho_{n+1}\rho_{n}\rho_{n-1}}\\
\notag&=v^{\rho_{n-1}B_{n-1,n+2}\rho_{n+1}\rho_{n-1}\rho_{n+1}\rho_{n}\rho_{n-1}}\\
\notag&=v^{\rho_{n-1}\underline{B_{n-1,n+2}\rho_{n-1}\rho_{n}}\rho_{n-1}}=v^{\rho_{n-1}\rho_{n+1}\rho_{n-1}}=v^{\rho_{n+1}}\\
\notag&=v~~~~~~~~~~~~~~~~(\text{since~}v\in\langle\lambda_{n-1,n},\lambda_{n,n-1}\rangle)
\end{align}
Therefore $\varrho^2(\delta)=\alpha u^{-1}v^{-1}uv=\alpha[u,v]$. Since $\varrho^2(\delta)$ is a pure braid, then $\varrho^*(\delta)=\varrho^2(\delta)=\alpha[u,v]$.
Since the closures of the braids $\gamma$ and $\delta=\gamma^{w^{-1}}$ define equivalent fused links, then the closures of the braids $\varrho^*(\gamma)=\alpha$ and $\varrho^*(\delta)=\alpha[u,v]$ define equivalent fused links. The lemms is proved.
Denote by $T$ the set of coset representatives of $UVP_n/UVP_n^{\prime}$ and for the element $\alpha\in UVP_n$ denote by $\overline{\alpha}\in T$ the unique coset representative $\alpha UVP_n^{\prime}=\overline{\alpha}UVP_n^{\prime}$. The following statement almost immediately follows from Lemma <ref>.
Let $\alpha,\beta\in UVP_{n}$ be unrestricted virtual braids, such that $\overline{\alpha}$ and $\overline{\beta}$ are conjugated by the element from $S_n$. Then the fused links $\widehat{\alpha}$ and $\widehat{\beta}$ are equivalent.
Proof. Since $\overline{\alpha}$ and $\overline{\beta}$ are conjugated by the element from $S_n$ then the closures of the braids $\overline{\alpha}$ and $\overline{\beta}$ are equivalent fused links. Therefore it is enough to prove that the closure of the braid $\alpha$ is equivalent to the closure of the braid $\overline{\alpha}$ and the closure of the braid $\beta$ is equivalent to the closure of the braid $\overline{\beta}$.
Since $\overline{\alpha}UVP_n^{\prime}=\alpha UVP_n^{\prime}$, then $\overline{\alpha}=\alpha[u_1,v_1][u_2,v_2]\dots[u_k,v_k]$ for certain elements $u_i,v_i\in UVP_n$, $i=1,\dots,k$. By Theorem <ref> we can consider that $u_i,v_i\in\langle\lambda_{r_i,s_i},\lambda_{s_i,r_I}\rangle$.
Denote by $\alpha_0=\alpha$, $\alpha_1=\alpha_0[u_1,v_1]$, $\alpha_2=\alpha_1[u_2,v_2]$, $\dots$, $\alpha_k=\alpha_{k-1}[u_k,v_k]=\overline{\alpha}$. If we prove that the closure of $\alpha_i$ and the closure of $\alpha_{i-1}$ define the same fused link for every $i=1,\dots,k$, then we prove that the closures of the braids $\alpha$ and $\overline{\alpha}$ are equivalent as fused links.
Therefore we can consider that $\overline{\alpha}=\alpha[u,v]$ for some $u,v\in\langle\lambda_{r,s},\lambda_{s,r}\rangle$.
The closure of the braid $\overline{\alpha}$ is equivalent to the closure of the braid $\overline{\alpha}^{\mu}$ for any $\mu\in S_n<UVB_n$. If $\mu$ is a permutation which maps $s$ to $n-1$ and maps $r$ to $n$, then the closure of the braid $\overline{\alpha}$ is equivalent to the closure of the braid $\overline{\alpha}^{\mu}=\alpha^{\mu}[u^{\mu},v^{\mu}]$. By Theorem <ref> the braids $u^{\mu},v^{\mu}$ belong to $\langle\lambda_{n-1,n},\lambda_{n,n-1}\rangle$, thus by Lemma <ref> the closure of the braid $\overline{\alpha}$ is equivalent to the closure of the braid $\alpha^{\mu}$ and is also equivalent to the closure of $\alpha$. The closure of $\beta$ is equivalent to the closure of $\overline{\beta}$ by the same reasons.
The corollary is proved.
Let $\alpha, \beta\in UVB_{\infty}$ be unrestricted virtual braids, such that $\widehat{\alpha}$ and $\widehat{\beta}$ are equivalent fused links with $m$ components. Then $\overline{\varrho^*(\alpha)}$ and $\overline{\varrho^*(\beta)}$ are conjugated by the element from $S_n$.
Proof. Since the links $\widehat{\alpha}$ and $\widehat{\beta}$ are equivalent, then by Theorem <ref> the braids $\alpha$ and $\beta$ are related by the finite sequence of Markov's transformations. It means that there is a finite sequence of braids
such that $\alpha_{j+1}$ is obtained from $\alpha_j$ by conjugation in $UVB_{\infty}$, by right stabilization or by inverse to right stabilization transformation.
If we prove that $\overline{\varrho^*(\alpha_j)}$ and $\overline{\varrho^*(\alpha_{j+1})}$ are conjugated by the element from $S_n$ for every $j=0,\dots,k-1$, than we prove that $\overline{\varrho^*(\alpha)}$ and $\overline{\varrho^*(\beta)}$ are conjugated by the element from $S_n$, therefore we can consider that $\beta$ is obtained from $\alpha$ using only one Markov's transformation.
Case 1. The braid $\beta$ is obtained from the braid $\alpha$ by right stabilization of positive, negative or virtual type.
We consider only the case of right stabilization of positive type ($\beta=\alpha\sigma_n$), the cases of right stabilization of negative and virtual type are similar.
Let us count the image of $\beta$ under the map $\varrho$:
Step 1. Since $\beta=\alpha\sigma_n$, then $\iota(\beta)(n+1)=\iota(\alpha\sigma_n)(n+1)=n\neq n+1$
$$ \begin{CD}
n+1 @>~\alpha~>> n+1 @>~\sigma_n~>> n,
\end{CD}$$
therefore $n+1$ is a maximal number which is not fixed by $\beta$.
Step 2. $\beta_1=B_{1,n+1}^{n+1-(n+1)}\beta B_{1,n+1}^{n+1-(n+1)}=\beta$.
Step 3. We can rewrite the braid $\beta_1$ in the following form
where the braid $\alpha$ does not involve the strand $n+1$ and $\lambda_{n,n+1}^{-1}\in\langle\lambda_{n,n+1},\lambda_{n+1,n}\rangle$.
Step 4. $\beta_2=\alpha\rho_n$.
Step 5. $\varrho(\beta)=\alpha$.
Since $\varrho(\beta)=\alpha$, then $\varrho^*(\beta)=\varrho^*(\alpha)$ (and they the braids $\overline{\varrho^*(\beta)}$ and $\overline{\varrho^*(\alpha)}$ are certainly conjugated).
Case 2. The braid $\beta$ is obtained from the braid $\alpha$ by an opposite to a right stabilization of positive, negative or virtual type transformation.
In this case the braid $\alpha$ is obtained from the braid $\beta$ by right stabilization of positive (negative, virtual) type and by the case 1 we have $\varrho^*(\alpha)=\varrho^*(\beta)$.
Case 3. The braid $\beta$ is obtained from the braid $\alpha$ by conjugation in $UVB_n$.
Since the braid $\alpha$ and $\beta$ are conjugated, then for some braid $\theta$ we have $\beta=\alpha^{\theta}$. Since $\sigma_i=\rho_{i}\lambda_{i,i+1}$, then $\theta=y_1y_2\dots y_r$, where
If we denote $\delta_1=\alpha$, $\delta_2=\delta_1^{y_1}$, $\delta_3=\delta_2^{y_2}$, $\dots$, $\delta_{r+1}=\delta_r^{y_r}=\beta$ and prove that $\overline{\varrho^*(\delta_j)}$ is conjugated with $\overline{\varrho^*(\delta_{j+1})}$ by the element from $S_n$ for every $j=1,
\dots,r+1$, then we prove that $\overline{\varrho^*(\alpha)}$ is conjugated with $\overline{\varrho^*(\beta)}$ by the element from $S_n$. Therefore we can consider that $\beta$ is obtained from $\alpha$ conjugating by $\rho_i$ or $\lambda_{i,i+1}^{\pm1}$ for some $i$.
We use the induction by the parameter $n-m$, i. e. by the difference between the number of strands in the braid $\alpha$ and the number components in the link $\widehat{\alpha}$.
If $n-m=0$, then $\alpha$ and $\beta$ are pure braids and $\varrho^*(\alpha)=\alpha$, $\varrho^*(\beta)=\beta$.
Let $\beta=\alpha^{\mu}$ for some braid $\mu\in UVB_n$, then by Theorem <ref> for certain elements $\beta\in UVP_n$ and $\pi\in S_n$ the braid $\mu$ can be presented as a product $\mu=\beta\pi$. Therefore
\begin{align}
\notag\varrho^*(\beta)&=\beta=\alpha^{\mu}=\mu^{-1}\alpha\mu=\pi^{-1}\beta^{-1}\alpha\beta\pi\\
\notag&=\pi^{-1}\alpha\beta^{-1}[\beta^{-1},\alpha]\beta\pi=\pi^{-1}\alpha[\beta^{-1},\alpha]^{\beta}\pi=\pi^{-1}\varrho^*(\alpha)[\beta^{-1},\alpha^{\beta}]\pi
\end{align}
Since $\alpha,\beta\in UVP_n$, then $[\beta^{-1},\alpha^{\beta}]\in UVP_n^{\prime}$, therefore $\overline{\varrho^*(\alpha)[\beta^{-1},\alpha^{\beta}]}=\overline{\varrho^*(\alpha)}$ and $\overline{\varrho^*{(\beta)}}=\overline{\varrho^*(\alpha)}^{\pi}$.
If $n-m>0$, then $\alpha,\beta\in UVB_n\setminus UVP_n$, $\varrho^*(\alpha)\neq\alpha$, $\varrho^*(\beta)\neq\beta$ and we have the following cases
Case 3.1. The braid $\beta$ is obtained from $\alpha$ conjugating by $\rho_i$. Let us find $\varrho(\alpha)$.
Step 1. Let $s$ be a maximal number which is not fixed by $\alpha$ and $\iota(\alpha)(s)=k_s$.
Step 2. Since $\alpha_1=B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}$, then $\iota(\alpha_1)(n)=k_s+n-s=k_n\neq n$.
$$ \begin{CD}
n @>~B_{1,n}^{s-n}~>> s @>~\alpha~>> k_s@>~B_{1,n}^{n-s}~>>k_s+n-s=k_n.
\end{CD}$$
Step 3. Express the braid $\alpha_1$ in the form
\begin{equation}\label{a1}
\alpha_1=B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}=\gamma w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1},
\end{equation}
where the braid $\gamma$ does not involve the strand $n$ and $w_i\in\langle\lambda_{i,n},\lambda_{n,i}\rangle$.
Step 4. $\alpha_2=\gamma w_1\dots w_{n-2}\rho_{n-1}B_{k_n,n-1}.$
Step 5. The braid $\varrho(\alpha)$ has the following form
\begin{equation}\label{a2}
\varrho(\alpha)=\gamma w_1^{\rho_{n-1}}\dots w_{n-2}^{\rho_{n-1}}B_{k_n,n-1},
\end{equation}
where $w_{j}^{\rho_{n-1}}\in U_{n-1}$ does not involve the strand $n$.
Case 3.1.1. $i\leq k_s-2$. Let us count the braid $\varrho(\beta)=\varrho(\rho_i\alpha\rho_i)$.
Step 1. Since $i\leq k_s-2$, then $\rho_i$ fixes $s$ and $k_s$, therefore $s$ is a maximal number which is not fixed by $\beta$ and $\iota(\beta)(s)=k_s$.
$$ \begin{CD}
s @>~\rho_{i}~>> s @>~\alpha~>> k_s@>~\rho_{i}~>> k_s
\end{CD}$$
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$, then the permutation $\iota(\beta)$ maps $n$ to $k_n\neq n$.
$$ \begin{CD}
n @>~B_{1,n}^{s-n}~>> s @>~\beta~>> k_s@>~B_{1,n}^{n-s}~>> k_s+n-s=k_n
\end{CD}$$
Step 3. For $i\leq k_s-2$ we have
\begin{align}
\notag\rho_iB_{1,n}&=\rho_i\rho_{n-1}\dots\rho_1=\rho_i\rho_{n-1}\dots\rho_{i+2}\rho_{i+1}\rho_i\rho_{i-1}\dots\rho_1\\
\notag&=\rho_{n-1}\dots\rho_{i+2}\underline{\rho_i\rho_{i+1}\rho_i}\rho_{i-1}\dots\rho_1\\
\notag&=\rho_{n-1}\dots\rho_{i+2}\rho_{i+1}\rho_{i}\rho_{i+1}\rho_{i-1}\dots\rho_1\\
\notag&=\rho_{n-1}\dots\rho_{i+2}\rho_{i+1}\rho_{i}\rho_{i-1}\dots\rho_1\rho_{i+1}=B_{1,n}\rho_{i+1},
\end{align}
therefore $\rho_iB_{1,n}^{n-s}=B_{1,n}^{n-s}\rho_{i+n-s}$ and hence we have
\begin{align}
\notag\beta_1&=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}=B_{1,n}^{s-n}\rho_i\alpha\rho_i B_{1,n}^{n-s}=\rho_{i+n-s}B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}\rho_{i+n-s}\\
\notag&=\rho_{i+n-s}\gamma w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1}\rho_{i+n-s}\\
\notag&=\gamma^{\rho_{i+n-s}} (w_1\dots w_{n-2})^{\rho_{i+n-s}}w_{n-1}^{\rho_{i+n-s}}\rho_{n-1}^{\rho_{i+n-s}}B_{k_n,n-1}^{\rho_{i+n-s}}
\end{align}
Since $i\leq k_s-2$, then $i+n-s\leq k_s-2+n-s=k_n-2<n-2$, therefore $B_{k_n,n-1}^{\rho_{i+n-s}}=B_{k_n,n-1}$, $w_{n-1}^{\rho_{i+n-s}}=w_{n-1}$, $\rho_{n-1}^{\rho_{i+n-s}}=\rho_{n-1}$ and the braid
$\gamma^{\rho_{i+n-s}}$ does not involve the strand $n$. Since $\rho_{i+n-s}$ fixes $n$, then
$(w_1\dots w_{n-2})^{\rho_{i+n-s}}$ belongs to $\langle\lambda_{1,n},\lambda_{n,1}\rangle\times\dots\times\langle\lambda_{n-2,n},\lambda_{n,n-2}\rangle$. Then we have
$$\beta_1=\gamma^{\rho_{i+n-s}} (w_1\dots w_{n-2})^{\rho_{i+n-s}}w_{n-1}\rho_{n-1}B_{k_n,n-1}.$$
Step 4. $\beta_2=\gamma^{\rho_{i+n-s}} (w_1\dots w_{n-2})^{\rho_{i+n-s}}\rho_{n-1}B_{k_s,n-1}$
Step 5. Since $i+n-s<n-2$, then $\rho_{n-1}$ and $\rho_{i+n-s}$ commute, therefore the braid $\varrho(\beta)$ has the following form
\begin{align}
\notag\varrho(\beta)&=\gamma^{\rho_{i+n-s}} (w_1\dots w_{n-2})^{\rho_{i+n-s}\rho_{n-1}}B_{k_n,n-1}\\
\notag&=\gamma^{\rho_{i+n-s}} (w_1\dots w_{n-2})^{\rho_{n-1}\rho_{i+n-s}}B_{k_n,n-1}\\
\notag &=\gamma^{\rho_{i+n-s}} ((w_1\dots w_{n-2})^{\rho_{n-1}})^{\rho_{i+n-s}}B_{k_n,n-1}^{\rho_{i+n-s}}\\
\notag &=(\gamma(w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1})^{\rho_{i+n-s}}=\varrho(\alpha)^{\rho_{i+n-s}}
\end{align}
Therefore $\varrho(\beta)=\varrho(\alpha)^{\rho_{i+n-s}}$ and by
the induction hypothesis the braids $\overline{\varrho^*(\alpha)}$ and $\overline{\varrho^*(\beta)}$ are conjugated by the element from $S_n$.
Case 3.1.2. $i= k_s-1$. Let us find the braid $\varrho(\beta)$.
Step 1.
Since $k_s\leq s-1$, then $i=k_s-1\leq s-2$ and $\rho_i$ fixes the strands $s$. Then the image of $s$ under the permutation
$\beta=\rho_{k_s-1}\alpha\rho_{k_s-1}$ is equal to $k_s-1$.
$$ \begin{CD}
s @>~\rho_{k_s-1}~>> s @>~\alpha~>> k_s@>~\rho_{k_s-1}~>> k_s-1.
\end{CD}$$
Therefore $s$ is a maximal number which is not foxed by $\beta$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=k_n-1$.
$$ \begin{CD}
n @>~B_{1,n}^{s-n}~>> s @>~\beta~>> k_s-1@>~ B_{1,n}^{n-s}~>> n-s+k_s-1=k_n-1
\end{CD}$$
Step 3. Using the same arguments as in the case 3.1.1 we have
Therefore the braid $\beta_1$ has the following form
\begin{align}
\notag\beta_1&=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}=B_{1,n}^{s-n}\rho_{k_s-1}\alpha\rho_{k_s-1} B_{1,n}^{n-s}=\rho_{k_n-1}B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}\rho_{k_n-1}\\
\notag&=\rho_{k_n-1}\gamma w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1}\rho_{k_n-1}\\
\notag&=\gamma^{\rho_{k_n-1}} (w_1\dots w_{n-2})^{\rho_{k_n-1}}w_{n-1}^{\rho_{k_n-1}}\rho_{n-1}^{\rho_{k_n-1}}B_{k_n,n-1}^{\rho_{k_n-1}}
\end{align}
Since $\rho_{n-1}^{\rho_{k_n-1}}B_{k_n,n-1}^{\rho_{k_n-1}}=\rho_{k_n-1}B_{k_n-1,n}$, then we have
\begin{align}
\notag\beta_1&=\gamma^{\rho_{k_n-1}} (w_1\dots w_{n-2})^{\rho_{k_n-1}}w_{n-1}^{\rho_{k_n-1}}\rho_{k_n-1}B_{n,k_n-1}\\
\notag&=\gamma^{\rho_{k_n-1}}\rho_{k_n-1} (w_1\dots w_{n-2})w_{n-1}B_{n,k_n-1}\\
\notag&=\gamma^{\rho_{k_n-1}}\rho_{k_n-1} (w_1\dots w_{n-2})w_{n-1}\rho_{n-1}B_{n-1,k_n-1}
\end{align}
Since $k_n\leq n-1$, then $k_n-1\leq n-2$, therefore the braid
$\gamma^{\rho_{k_n-1}}\rho_{k_n-1}$ does not involve the strand $n$.
Step 4. $\beta_2=\gamma^{\rho_{k_n-1}}\rho_{k_n-1} (w_1\dots w_{n-2})\rho_{n-1}B_{n-1,k_n-1}$
Step 5. The braid $\varrho(\beta)$ has the following form.
\begin{align}
\notag\varrho(\beta)&=\gamma^{\rho_{k_n-1}}\rho_{k_n-1} (w_1\dots w_{n-2})^{\rho_{n-1}}B_{n-1,k_n-1}\\
\notag&=\gamma^{\rho_{k_n-1}} (w_1\dots w_{n-2})^{\rho_{n-1}\rho_{k_n-1}}\rho_{k_n-1}B_{n-1,k_n-1}\\
\notag&=\gamma^{\rho_{k_n-1}} (w_1\dots w_{n-2})^{\rho_{n-1}\rho_{k_n-1}}B_{n-1,k_n}^{\rho_{k_n-1}}=\varrho(\alpha)^{\rho_{k_n-1}}
\end{align}
Therefore $\varrho(\beta)=\varrho(\alpha)^{\rho_{k_n-1}}$ and by
the induction hypothesis the braids $\overline{\varrho^*(\alpha)}$ and $\overline{\varrho^*(\beta)}$ are conjugated by the element from $S_n$.
Case 3.1.3. $i=k_s$.
Case 3.1.3.1. $k_s\leq s-2$. Let us find the image of the braid $\beta=\rho_{k_s}\alpha\rho_{k_s}$ under the map $\varrho$.
Step 1. Since $k_s\leq s-2$, then and $\rho_{k_s}$ fixes $s$ and the image of $s$ under the permutation $\iota(\beta)$ is equal to $k_s+1$.
$$ \begin{CD}
s @>~\rho_{k_s}~>> s @>~\alpha~>> k_s@>~\rho_{k_s}~>> k_s+1
\end{CD}$$
Therefore $s$ is a maximal number which is not fixed by $\beta$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=k_n+1$.
$$ \begin{CD}
n @>~B_{1,n}^{s-n}~>> s @>~\beta~>> k_s+1@>~B_{1,n}^{n-s}~>> k_n+1,
\end{CD}$$
Step 3. Analogically to the case 3.1.1, since $k_s\leq s-2$, then
$\rho_{k_s}B_{1,n}^{n-s}=B_{1,n}^{n-s}\rho_{k_n}$. Therefore the braid $\beta_1$ has the following form.
\begin{align}
\notag\beta_1&=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}=B_{1,n}^{s-n}\rho_{k_s}\alpha\rho_{k_s} B_{1,n}^{n-s}=\rho_{k_n}B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}\rho_{k_n}\\
\notag&=\rho_{k_n}\gamma w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1}\rho_{k_n}\\
\notag&=\gamma^{\rho_{k_n}} (w_1\dots w_{n-2})^{\rho_{k_n}}w_{n-1}^{\rho_{k_n}}\rho_{n-1}^{\rho_{k_n}}B_{k_n,n-1}^{\rho_{k_n}}
\end{align}
Since $\rho_{n-1}^{\rho_{k_n}}B_{k_n,n-1}^{\rho_{k_n}}\rho_{k_n}B_{k_n+1,n}$, then we have
\begin{align}
\notag\beta_1&=\gamma^{\rho_{k_n}} (w_1\dots w_{n-2})^{\rho_{k_n}}w_{n-1}^{\rho_{k_n}}\rho_{k_n}B_{k_n+1,n}\\
\notag&=\gamma^{\rho_{k_n}}\rho_{k_n} (w_1\dots w_{n-2})w_{n-1} B_{k_n+1,n}.
\end{align}
Since $k_s\leq s-2$, then $k_n\leq n-2$ and the braid
$\gamma^{\rho_{k_n}}\rho_{k_n}$ does not involve the strand $n$.
Step 4. $\beta_2=\gamma^{\rho_{k_n}}\rho_{k_n} (w_1\dots w_{n-2})\rho_{n-1} B_{k_n+1,n-1}$
Step 5. The braid $\varrho(\beta)$ has the form
\begin{align}
\notag\varrho(\beta)&=\gamma^{\rho_{k_n}}\rho_{k_n} (w_1\dots w_{n-2})^{\rho_{n-1}} B_{k_n+1,n-1}\\
\notag&=\gamma^{\rho_{k_n}} (w_1\dots w_{n-2})^{\rho_{n-1}\rho_{k_n}} \rho_{k_n}B_{k_n+1,n-1}\\
\notag&=\gamma^{\rho_{k_n}} (w_1\dots w_{n-2})^{\rho_{n-1}\rho_{k_n}} B_{k_n,n-1}^{\rho_{k_n}}=\varrho(\alpha)^{\rho_{k_n}}
\end{align}
Therefore $\varrho(\beta)=\varrho(\alpha)^{\rho_{k_n}}$ and by
the induction hypothesis the braids $\overline{\varrho^*(\alpha)}$ and $\overline{\varrho^*(\beta)}$ are conjugated by the element from $S_n$.
Case 3.1.3.2. $k_s=s-1$.
Case 3.1.3.2.1. $\iota(\alpha)(s-1)=s$. From the formulas (<ref>) and (<ref>) we have
$$\alpha_1=\gamma w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1},$$
$$\varrho(\alpha)=\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}.$$
Let us rewrite this equalities in more details. Since $\iota(\alpha)(s)=s-1$, then $\iota(\alpha_1)(n)=n-1$ and $B_{k_n,n}=\rho_{n-1}$.
$$ \begin{CD}
n @>~B_{s,n}^{s-n}~>> s @>~\alpha~>> s-1@>~B_{s,n}^{n-s}~>> n-1
\end{CD}$$
Since $\iota(\alpha)(s-1)=s$ then $\iota(\alpha_1)$ maps $(n-1)$ to $n$.
$$ \begin{CD}
n-1 @>~B_{s,n}^{s-n}~>> s-1 @>~\alpha~>> s@>~B_{s,n}^{n-s}~>> n
\end{CD}$$
Therefore, since $\gamma=\alpha_1\left(B_{k_n,n}w_1\dots w_{n-1}\right)^{-1}$, then $\iota(\gamma)(n-1)=n-1$
$$ \begin{CD}
n-1 @>~\alpha_1~>> n @>~\left(B_{k_n,n}w_1\dots w_{n-1}\right)^{-1}~>> n-1
\end{CD}$$
and by Lemma <ref> we have
$$\gamma=\eta v_1\dots v_{n-2},$$
where $\eta$ does not involve the strand $n-1$ and $v_j\in\langle\lambda_{j,n-1},\lambda_{n-1,j}\rangle$ for $j=1,\dots,n-2$.
$$\alpha_1=\eta v_1\dots v_{n-2} w_1\dots w_{n-1}\rho_{n-1},$$
$$\varrho(\alpha)=\eta v_1\dots v_{n-2} (w_1\dots w_{n-2})^{\rho_{n-1}}.$$
Let us find the braid $\varrho(\beta)$.
Step 1. Since $k_s=s-1$ and $\iota(\alpha)(s-1)=s$, then the image of $s$ under the permutation $\iota(\beta)=\iota(\rho_{k_s}\alpha\rho_{k_s})=\iota(\rho_{s-1}\alpha\rho_{s-1})$ is equal to $s-1$.
$$ \begin{CD}
s @>~\rho_{s-1}~>> s-1 @>~\alpha~>> s@>~\rho_{s-1}a~>> s-1,
\end{CD}$$
then $s$ is a maximal number which is not fixed by $\beta$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=n-1$.
$$ \begin{CD}
n @>~B_{1,n}^{s-n}~>> s @>~\beta~>> s-1@>~B_{1,n}^{n-s}~>> n-1,
\end{CD}$$
Step 3. Since $\rho_{s-1}B_{1,n}^{n-s}=B_{1,n}^{n-s}\rho_{n-1}$, then we have
\begin{align}
\notag\beta_1&=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}=B_{1,n}^{s-n}\rho_{s-1}\alpha\rho_{s-1} B_{1,n}^{n-s}=\rho_{n-1}B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}\rho_{n-1}\\
\notag&=\rho_{n-1}\eta v_1\dots v_{n-2} w_1\dots w_{n-1}\rho_{n-1}\rho_{n-1}\\
\notag&=\eta^{\rho_{n-1}} v_1^{\rho_{n-1}}\dots v_{n-2}^{\rho_{n-1}} w_1^{\rho_{n-1}}\dots w_{n-2}^{\rho_{n-1}}w_{n-1}^{\rho_{n-1}}\rho_{n-1}
\end{align}
Since the braid $\eta$ does not involve the strands $n-1,n$, then $\eta^{\rho_{n-1}}=\eta$. Also by Lemma <ref> for $j=1,\dots, n-2$ we have
and $w_{n-1}^{\rho_{n-1}}\in\langle\lambda_{n,n-1},\lambda_{n-1,n}\rangle$. Therefore
$$\beta_1=\eta w_1^{\rho_{n-1}}\dots w_{n-2}^{\rho_{n-1}} v_1^{\rho_{n-1}}\dots v_{n-2}^{\rho_{n-1}} w_{n-1}^{\rho_{n-1}}\rho_{n-1},$$
where the braid
$\eta w_1^{\rho_{n-1}}\dots w_{n-2}^{\rho_{n-1}}$ does not involve the strand $n$, the braid
$v_j^{\rho_{n-1}}$ belongs to $\langle\lambda_{j,n},\lambda_{n,j}\rangle$ for $j=1,\dots,n-2$ and
Step 4. $\beta_2=\eta w_1^{\rho_{n-1}}\dots w_{n-2}^{\rho_{n-1}} v_1^{\rho_{n-1}}\dots v_{n-2}^{\rho_{n-1}} \rho_{n-1}$
Step 5. The braid $\varrho(\beta)$ has the following form
\begin{align}
\notag\varrho(\beta)&=\eta w_1^{\rho_{n-1}}\dots w_{n-2}^{\rho_{n-1}} v_1\dots v_{n-2}\\
\notag&=\eta v_1\dots v_{n-2}w_1^{\rho_{n-1}}\dots w_{n-2}^{\rho_{n-1}}[w_1^{\rho_{n-1}}\dots w_{n-2}^{\rho_{n-1}},v_1\dots v_{n-2}]
\end{align}
Therefore by the remark <ref> the braids $\varrho^*(\beta)$ and $\varrho^*(\alpha)$ are equal modulo $UVP_m^{\prime}$, i. e. $\overline{\varrho^*(\alpha)}=\overline{\varrho^*(\beta)}$ and these braids are certainly conjugated.
Case 3.1.3.2.2. $\iota(\alpha)(s-1)=k_{s-1}\leq s-2$. By the equalities (<ref>) and (<ref>) we have
$$\alpha_1=\gamma w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1},$$
$$\varrho(\alpha)=\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}.$$
Let us rewrite these equalities in more details.
Since $\iota(\alpha_1)(n)=k_n=n-1$, then $B_{k_n,n-1}=1$. Also we have $\iota(\alpha_1)(n-1)=n-s+k_{s-1}=k_{n-1}$
$$ \begin{CD}
n-1 @>~B_{1,n}^{s-n}~>> s-1 @>~\alpha~>> k_{s-1}@>~B_{1,n}^{n-s}~>> n-s+k_{s-1}=k_{n-1},
\end{CD}$$
Since $\gamma=\alpha_1\left(B_{k_n,n}w_1\dots w_{n-1}\right)^{-1}$, then $\iota(\gamma)$ maps $n-1$ to $k_{n-1}$
$$ \begin{CD}
n-1 @>~\alpha_1~>> k_{n-1} @>~\left(B_{k_n,n}w_1\dots w_{n-1}\right)^{-1}~>> k_{n-1},
\end{CD}$$
therefore by Lemma <ref> we have
$$\gamma=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1},$$
where $\eta$ does not involve the strand $n-1,n$ and $v_j$ belongs to $\langle\lambda_{j,n-1},\lambda_{n-1,j}\rangle$ for $j=1,\dots,n-2$.
Therefore we have
$$\alpha_1=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} w_1\dots w_{n-1}\rho_{n-1},$$
$$\varrho(\alpha)=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} (w_1\dots w_{n-2})^{\rho_{n-1}}.$$
Let us count $\varrho^2 (\alpha)$.
Step 1. Since $\eta$ does not involve the strand $n-1$, then $\iota(\varrho(\alpha))(n-1)=k_{n-1}$,
then $n-1$ is a maximal number which is not fixed by $\varrho(\alpha)$.
Step 2. $\varrho(\alpha)_1=\varrho(\alpha)=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} (w_1\dots w_{n-2})^{\rho_{n-1}}$.
Step 3. Let us rewrite the braid $\varrho(\alpha)_1$ in the required form.
\begin{align}
\notag\varrho(\alpha)_1&=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} (w_1\dots w_{n-2})^{\rho_{n-1}}\\
\notag&=\eta v_1\dots v_{n-2} (w_1\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_{n-1},n-1}\\
\notag&=\eta v_1\dots v_{n-2} (w_1\dots w_{k_{n-1}-1}w_{k_{n-1}}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_{n-1},n-1}\\
\notag&=\eta (w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}\\
\notag &~\cdot~ v_1\dots v_{n-3} v_{n-2}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_{n-1},n-1}
\end{align}
By Theorem <ref> the braid
$\eta (w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}$ does not involve the strand $n-1$, the braid $v_j\in\langle\lambda_{j,n-1},\lambda_{n-1,j}\rangle$ for $j=1,\dots,n-3$ and $v_{n-2}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}\in\langle\lambda_{n-1,n-2},\lambda_{n-2,n-1}\rangle$.
Step 4. $\varrho(\alpha)_2=\eta (w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} v_1\dots v_{n-3} \rho_{n-2}B_{k_{n-1},n-2}$
Step 5. The braid $\varrho^2(\alpha)$ follows
\begin{align}\notag\varrho(\alpha)_2&=\eta (w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} (v_1\dots v_{n-3})^{\rho_{n-2}} B_{k_{n-1},n-2}
\end{align}
Let us count $\varrho(\beta)$
Step 1. Since $k_s=s-1$ and $\iota(\alpha)(s-1)=k_{s-1}\leq s-2$, then the image of $s$ under the permutation $\beta=\rho_{k_s}\alpha\rho_{k_s}=\rho_{s-1}\alpha\rho_{s-1}$ is equal to $k_{s-1}$.
$$ \begin{CD}
s @>~\rho_{s-1}~>> s-1 @>~\alpha~>> k_{s-1}@>~\rho_{s-1}~>> k_{s-1},
\end{CD}$$
then $s$ is a maximal number which is not fixed by $\beta$.
The image of $s-1$ under the permutation $\beta=\rho_{k_s}\alpha\rho_{k_s}=\rho_{s-1}\alpha\rho_{s-1}$ is equal to $s$.
$$ \begin{CD}
s-1 @>~\rho_{s-1}~>> s @>~\alpha~>> s-1@>~\rho_{s-1}~>> s,
\end{CD}$$
Step 2. We have $\beta_1=B_{n,1}^{s-n}\beta B_{n,1}^{n-s}$ and
$\iota(\beta_1)(n)=k_{s-1}+n-s=k_{n-1}$, $\iota(\beta_1)(n-1)=n$
$$ \begin{CD}
n @>~B_{n,1}^{s-n}~>> s @>~\beta~>> k_{s-1}@>~B_{1,n}^{n-s}>> k_{s-1}+n-s~=k_{n-1},
\end{CD}$$
$$ \begin{CD}
n-1 @>~B_{n,1}^{s-n}~>> s-1 @>~\beta~>> s@>~B_{n,1}^{s-n}~>> n,
\end{CD}$$
Step 3.Since $k_s=s-1< s$ then $\rho_{s-1}B_{1,n}^{n-s}=B_{1,n}^{n-s}\rho_{n-1}$, therefore we have
\begin{align}
\notag\beta_1&=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}=B_{1,n}^{s-n}\rho_{s-1}\alpha\rho_{s-1}B_{1,n}^{n-s}= \rho_{n-1}B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}\rho_{n-1}\\
\notag&=\rho_{n-1}\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} w_1\dots w_{n-1}\rho_{n-1}\rho_{n-1}\\
\notag&=\eta^{\rho_{n-1}} (v_1\dots v_{n-2})^{\rho_{n-1}}B_{k_{n-1},n-1}^{\rho_{n-1}} (w_1\dots w_{n-1})^{\rho_{n-1}}\rho_{n-1}\\
\notag&=\eta^{\rho_{n-1}} (v_1\dots v_{n-2})^{\rho_{n-1}} (w_1\dots w_{n-1})^{\rho_{n-1}\left(B_{k_{n-1},n-1}^{\rho_{n-1}}\right)^{-1}}B_{k_{n-1},n-1}^{\rho_{n-1}}\rho_{n-1}\\
\notag&=\eta^{\rho_{n-1}} (v_1\dots v_{n-2})^{\rho_{n-1}} (w_1\dots w_{n-1})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}B_{k_{n-1},n}\\
\notag&=\eta^{\rho_{n-1}} (v_1\dots v_{n-2})^{\rho_{n-1}} (w_1\dots w_{n-1})^{B_{k_{n-1},n}^{-1}}B_{k_{n-1},n}\\
\notag&=\eta^{\rho_{n-1}} (v_1\dots v_{n-2})^{\rho_{n-1}} (w_1\dots w_{k_{n-1}-1} w_{k_{n-1}}w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n}^{-1}}B_{k_{n-1},n}\\
\notag&=\eta^{\rho_{n-1}} (w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n}^{-1}}(v_1\dots v_{n-2})^{\rho_{n-1}}w_{k_{n-1}}^{B_{k_{n-1},n}^{-1}}B_{k_{n-1},n}
\end{align}
By Theorem <ref> the braid
$\eta^{\rho_{n-1}} (w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n}^{-1}}$ does not involve the strand $n$,
$v_j^{\rho_{n-1}}\in\langle\lambda_{j,n},\lambda_{n,j}\rangle$ for $j=1,\dots n-2$ and
$w_{k_{n-1}}^{B_{k_{n-1},n}^{-1}}$ belongs to $\langle\lambda_{n-1,n},\lambda_{n,n-1}\rangle$.
Step 4. $\beta_2=\eta^{\rho_{n-1}} (w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n}^{-1}}(v_1\dots v_{n-2})^{\rho_{n-1}}\rho_{n-1}B_{k_{n-1},n-1}$
Step 5. The braid $\varrho(\beta)$ has the following form
\begin{equation}
\label{eq1112}
\varrho(\beta)=\eta^{\rho_{n-1}} (w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n}^{-1}}(v_1\dots v_{n-2})B_{k_{n-1},n-1}
\end{equation}
Let us count $\varrho^2(\beta)$
Step 1. From the equality (<ref>) the image of $n-1$ under the permutation $\iota(\varrho(\beta))$ is equal to $k_{n-1}$,
therefore $n-1$ is a maximal number which is not fixed by $\varrho(\beta)$.
Step 2. $\varrho(\beta)_1=\varrho(\beta$).
Step 3. Rewrite the braid $\varrho(\beta)_1$ in the required form
\begin{align}
\notag\varrho(\beta)_1&=\eta^{\rho_{n-1}} (w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n}^{-1}}(v_1\dots v_{n-2})B_{k_{n-1},n-1}\\
\notag&=\eta (w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n}^{-1}}v_1\dots v_{n-3}w_{n-1}^{B_{k_{n-1},n}^{-1}}v_{n-2}B_{k_{n-1},n-1}
\end{align}
Since $\eta$ does not involve the strands $n-1,n$, then $\eta^{\rho_{n-1}}=\eta$ does not involve the strand $n-1$. By Lemma <ref> the braid $v_{n-2}w_{n-1}^{B_{k_{n-1},n}^{-1}}$ belongs to $\langle\lambda_{n-1,n-2},\lambda_{n-2,n-1}\rangle$, the braid
$(w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n}^{-1}}v_1\dots v_{n-3}$ belongs to $\langle\lambda_{1,n-1},\lambda_{n-1,1}\rangle\times\dots\times \langle\lambda_{n-1,n-2},\lambda_{n-2,n-1}\rangle$.
Step 4. $\varrho(\beta)_2=\eta (w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n}^{-1}}v_1\dots v_{n-3}\rho_{n-2}B_{k_{n-1},n-2}$
Step 5.
$\varrho^2({\beta})=\eta (w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n}^{-1}\rho_{n-2}}(v_1\dots v_{n-3})^{\rho_{n-2}}B_{k_{n-1},n-2}$
Using simple calculations in symmetric group it is easy to show that $B_{k_{n-1},n}^{-1}\rho_{n-2}=\rho_{n-1}B_{k_{n-1},n-1}^{-1}\rho_{n-1}$, therefore
\begin{multline*}(w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n}^{-1}\rho_{n-2}}=\\
=\left((w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}\right)^{\rho_{n-1}}
\end{multline*}
and since $(w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}$ does not involve the strands $n-1$, $n$, then
\begin{multline*}\left((w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}\right)^{\rho_{n-1}}=\\
=(w_1\dots w_{k_{n-1}-1} w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}
\end{multline*}
Therefore $\varrho^2(\alpha)=\varrho^2(\beta)$ and by the induction hypothesis the braids $\overline{\varrho^*(\alpha)}$ and $\overline{\varrho^*(\beta)}$ are conjugated by the element from $S_n$.
Case 3.1.4. $k_s+1\leq i \leq s-2$. Let us count the braid $\varrho(\beta)=\varrho(\rho_{i}\alpha\rho_{i})$.
Step 1. Since $k_s\leq s-3$ and $k_s+1\leq i \leq s-2$, then the braid $\rho_i$ fixes $s$ and $k_s$, and therefore the image of $s$ under the permutation $\iota(\beta)=\iota(\rho_{i}\alpha\rho_{i})$ is equal to $k_s$.
$$ \begin{CD}
s @>~\rho_{i}~>> s @>~\alpha~>> k_s@>~\rho_{i}~>> k_s
\end{CD}$$
Therefore $s$ is a maximal number which is not fixed by $\beta$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=k_n$.
$$ \begin{CD}
n @>~B_{1,n}^{s-n}~>> s @>~\beta~>> k_s@>~B_{1,n}^{n-s}~>> k_n,
\end{CD}$$
Step 3. Since $i\leq s-2$, then $\rho_{i}B_{1,n}^{n-s}=B_{1,n}^{n-s}\rho_{n-s+i},$ and therefore the braid $\beta_1$ follows.
\begin{align}
\notag\beta_1&=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}=B_{1,n}^{s-n}\rho_{i}\alpha\rho_{i} B_{1,n}^{n-s}=\rho_{i+n-s}B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}\rho_{i+n-s}\\
\notag&=\rho_{i+n-s}\gamma w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1}\rho_{i+n-s}\\
\notag&=\gamma^{\rho_{i+n-s}} (w_1\dots w_{n-2})^{\rho_{i+n-s}}w_{n-1}^{\rho_{i+n-s}}\rho_{n-1}^{\rho_{i+n-s}}B_{k_n,n-1}^{\rho_{i+n-s}}
\end{align}
Using simple calculation in the permutation group it is easy to show that
$\rho_{n-1}^{\rho_{i+n-s}}B_{k_n,n-1}^{\rho_{i+n-s}}=B_{k_n,n}^{\rho_{i+n-s}}=\rho_{i+n-s}\rho_{i+n-s-1}B_{k_n,n}$, hence
\begin{align}
\notag\beta_1&=\gamma^{\rho_{i+n-s}} (w_1\dots w_{n-2})^{\rho_{i+n-s}}w_{n-1}^{\rho_{i+n-s}}\rho_{i+n-s}\rho_{i+n-s-1}B_{k_n,n}\\
\notag&=\gamma^{\rho_{i+n-s}}\rho_{i+n-s}\rho_{i+n-s-1} (w_1\dots w_{n-2})^{\rho_{i+n-s-1}}w_{n-1}^{\rho_{i+n-s-1}}B_{k_n,n}
\end{align}
Since $i\leq s-2$, then $i+n-s-1\leq n-3$ and the braid
$\gamma^{\rho_{i+n-s}}\rho_{i+n-s}\rho_{i+n-s-1}$ does not involve the strand $n$. Also by Lemma <ref>
$(w_1\dots w_{n-2})^{\rho_{i+n-s-1}}$ belongs to $\langle\lambda_{1,n},\lambda_{n,1}\rangle\times\dots\times\langle\lambda_{n-2,n},\lambda_{n,n-2}\rangle$ and
Step 4. $\beta_2=\gamma^{\rho_{i+n-s}}\rho_{i+n-s}\rho_{i+n-s-1} (w_1\dots w_{n-2})^{\rho_{i+n-s-1}}\rho_{n-1}B_{k_n,n-1}$
Step 5. The braid $\beta$ has the following form
\begin{align}
\notag\varrho(\beta)&=\gamma^{\rho_{i+n-s}}\rho_{i+n-s}\rho_{i+n-s-1} (w_1\dots w_{n-2})^{\rho_{i+n-s-1}\rho_{n-1}}B_{k_n,n-1}\\
\notag&=\gamma^{\rho_{i+n-s}} (w_1\dots w_{n-2})^{\rho_{i+n-s-1}\rho_{n-1}\rho_{i+n-s-1}\rho_{i+n-s}}\rho_{i+n-s}\rho_{i+n-s-1}B_{k_n,n-1}\\
\notag&=\gamma^{\rho_{i+n-s}} (w_1\dots w_{n-2})^{\rho_{n-1}\rho_{i+n-s}}B_{k_n,n-1}^{\rho_{i+n-s}}=\varrho(\alpha)^{\rho_{i+n-s}}
\end{align}
Therefore $\varrho(\beta)=\varrho(\alpha)^{\rho_{i+n-s}}$ and by the induction hypothesis the braids $\overline{\varrho^*(\alpha)}$ and $\overline{\varrho^*(\beta)}$ are conjugated by the element from $S_n$.
Case 3.1.5. $i=s-1$. In this case we can consider that $k_s< s-1$ since the case when $k_s=s-1=i$ is already solved in the case 3.1.3.2.
Case 3.1.5.1. $\iota(\alpha)(s-1)=s$. By the equalities (<ref>) and (<ref>) the braids $\alpha_1$ and $\varrho(\alpha)$ have the following forms
$$\alpha_1=\gamma w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1},$$
$$\varrho(\alpha)=\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}.$$
Let us rewrite this equalities in more details. Since $\iota(\alpha)(s-1)=s$, then the permutation $\iota(\alpha_1)$ maps $n-1$ to $n$.
$$ \begin{CD}
n-1 @>~B_{1,n}^{s-n}~>> s-1 @>~\alpha~>> s@>~B_{1,n^{n-s}}~>> n
\end{CD}$$
Therefore the permutation $\iota(\gamma)=\iota(\alpha_1\left(w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1}\right)^{-1})$ fixes $n-1$
$$ \begin{CD}
n-1 @>~\alpha_1~>> n @>~\left(w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1}\right)^{-1}~>> n-1,
\end{CD}$$
and by Lemma <ref> we have
$$\gamma=\eta v_1\dots v_{n-2},$$
where $\eta$ does not involve the strands $n-1,n$ and $v_j\in\langle\lambda_{j,n-1},\lambda_{n-1,j}\rangle$.
Therefore the braids $\alpha_1$, $\varrho(\alpha)$ can be rewritten.
$$\alpha_1=\eta v_1\dots v_{n-2} w_1\dots w_{n-1}\rho_{n-1}B_{k_{n},n-1}$$
$$\varrho(\alpha)=\eta v_1\dots v_{n-2} (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1}$$
Let us count $\varrho^2(\alpha)$
Step 1. Since $k_s<s-1$, then $k_n<n-1$, therefore $B_{k_1,n-1}\neq1$,
then $n-1$ is a maximal number which is not fixed by $\varrho(\alpha)$.
Step 2. $\varrho(\alpha)_1=\varrho(\alpha)$.
Step 3.The braid $\varrho(\alpha)_1$ has the following form.
\begin{align}
\notag\varrho(\alpha)_1&=\varrho(\alpha)=\eta v_1\dots v_{n-2} (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1}\\
\notag&=\eta v_1\dots v_{n-3} (w_1\dots w_{n-3})^{\rho_{n-1}}v_{n-2}w_{n-2}^{\rho_{n-1}}B_{k_{n},n-1}
\end{align}
Here the braid $\eta$ does not involve the strand $n-1$, by Lemma <ref> the braid
$v_1\dots v_{n-3} (w_1\dots w_{n-3})^{\rho_{n-1}}$ belongs to $\langle\lambda_{1,n-1},\lambda_{n-1,1}\rangle\times\dots\times\langle\lambda_{1,n-3},\lambda_{n-3,1}\rangle$ and
$v_{n-2}w_{n-2}^{\rho_{n-1}}$ belongs to $\langle\lambda_{n-1,n-2},\lambda_{n-2,n-1}\rangle$.
Step 4. $\varrho(\alpha)_2=\eta v_1\dots v_{n-3} (w_1\dots w_{n-3})^{\rho_{n-1}}{\rho_{n-2}}B_{k_{n},n-2}$
Step 5. The braid $\varrho^2(\alpha)$ follows.
\begin{equation}
\label{eq12}
\varrho^{2}({\alpha})=\eta (v_1\dots v_{n-3})^{\rho_{n-2}} (w_1\dots w_{n-3})^{\rho_{n-1}\rho_{n-2}}B_{k_{n},n-2}
\end{equation}
Let us find the braid $\varrho(\beta)=\varrho(\rho_{i}\alpha\rho_{i})=\varrho(\rho_{s-1}\alpha\rho_{s-1})$.
Step 1. Since $i=s-1$ and $\iota(\alpha)(s-1)=s$, then the image of $s$ under the permutation $\beta=\rho_{s-1}\alpha\rho_{s-1}$ is equal to $s-1$
$$ \begin{CD}
s @>~\rho_{s-1}~>> s-1 @>~\alpha~>> s@>~\rho_{s-1}~>> s-1,
\end{CD}$$
then $s$ is a maximal number which is not fixed by $\beta$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=n-1$.
$$ \begin{CD}
n @>~B_{1,n}^{s-n}~>> s @>~\beta~>> s-1@>~B_{1,n}^{n-s}~>> n-1
\end{CD}$$
Step 3. Since $\rho_{s-1}B_{1,n}^{n-s}=B_{1,n}^{n-s}\rho_{n-1}$, then then we have
\begin{align}
\notag\beta_1&=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}=B_{1,n}^{s-n}\rho_{s-1}\alpha\rho_{s-1} B_{1,n}^{n-s}=\rho_{n-1}B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}\rho_{n-1}\\
\notag&=\rho_{n-1}\eta v_1\dots v_{n-2} w_1\dots w_{n-1}\rho_{n-1}B_{k_{n},n-1}\rho_{n-1}\\
\notag&=\eta^{\rho_{n-1}} (v_1\dots v_{n-2})^{\rho_{n-1}} (w_1\dots w_{n-1})^{\rho_{n-1}}\rho_{n-1}B_{k_{n},n-1}^{\rho_{n-1}}\\
\notag&=\eta^{\rho_{n-1}} (v_1\dots v_{n-2})^{\rho_{n-1}} (w_1\dots w_{n-1})^{\rho_{n-1}}B_{k_{n},n-1}{\rho_{n-1}}\\
\notag&=\eta^{\rho_{n-1}}B_{k_{n},n-1} (v_1\dots v_{n-2})^{\rho_{n-1}B_{k_{n},n-1}} (w_1\dots w_{n-1})^{\rho_{n-1}B_{k_{n},n-1}}{\rho_{n-1}}\\
\notag&=\eta^{\rho_{n-1}}B_{k_{n},n-1} (v_1\dots v_{n-2})^{B_{k_{n},n}} (w_1\dots w_{n-1})^{B_{k_{n},n}}{\rho_{n-1}}\\
\notag&=\eta^{\rho_{n-1}}B_{k_{n},n-1} (v_1\dots v_{n-3})^{B_{k_{n},n}} (w_1\dots w_{n-2})^{B_{k_{n},n}}w_{n-1}^{B_{k_{n},n}}v_{n-2}^{B_{k_{n},n}}{\rho_{n-1}}\\
\notag&=\eta^{\rho_{n-1}}B_{k_{n},n-1}(w_1\dots w_{n-2})^{B_{k_{n},n}} (v_1\dots v_{n-3})^{B_{k_{n},n}} w_{n-1}^{B_{k_{n},n}}v_{n-2}^{B_{k_{n},n}}\rho_{n-1}
\end{align}
Since $\eta$ does not involve the strands $n-1,n$, then $\eta^{\rho_{n-1}}=\eta$ and the braid
$\eta^{\rho_{n-1}}B_{k_{n},n-1}(w_1\dots w_{n-2})^{B_{k_{n},n}}$ does not involve the strand $n$. By Lemma <ref> the braid
$(v_1\dots v_{n-3})^{B_{k_{n},n}} w_{n-1}^{B_{k_{n},n}}$ belongs to $\langle\lambda_{1,n},\lambda_{n,1}\rangle\times\dots\times\langle\lambda_{n-2,n},\lambda_{n,n-2}\rangle$ and
$v_{n-2}^{B_{k_{n},n}}$ belongs to $\langle\lambda_{n-1,n},\lambda_{n,n-1}\rangle$.
Step 4. $\beta_2=\eta^{\rho_{n-1}}B_{k_{n},n-1}(w_1\dots w_{n-2})^{B_{k_{n},n}} (v_1\dots v_{n-3})^{B_{k_{n},n}} w_{n-1}^{B_{k_{n},n}}\rho_{n-1}$
Step 5.
$\varrho(\beta)=\eta^{\rho_{n-1}}B_{k_{n},n-1}(w_1\dots w_{n-2})^{B_{k_{n},n}} (v_1\dots v_{n-3})^{B_{k_{n},n}{\rho_{n-1}}} w_{n-1}^{B_{k_{n},n}{\rho_{n-1}}}$
Let us count the braid $\varrho^2(\beta)=\varrho(\varrho(\beta))$.
Step 1. Since $\eta$ does not involve the strand $n-1,n$ therefore $\eta^{\rho_{n-1}}=\eta$ and $\iota(\eta)$ fixes $n-1$. Moreover since $(w_1\dots w_{n-2})^{B_{k_{n},n}} (v_1\dots v_{n-3})^{B_{k_{n},n}{\rho_{n-1}}} w_{n-1}^{B_{k_{n},n}{\rho_{n-1}}}$ is a pure braid, that the image of $n-1$ under the permutation $\iota(\varrho(\beta))$ is equal to $\iota(B_{k_n,n-1})(n-1)=k_n$. Therefore
$n-1$ is a maximal number which is not fixed by $\varrho(\beta)$.
Step 2. $\varrho(\beta)_1=\varrho(\beta)$.
Step 3. The braid $\varrho(\beta)_1$ has the following form
\begin{align}
\notag \varrho(\beta)_1&=\eta^{\rho_{n-1}}B_{k_{n},n-1}(w_1\dots w_{n-2})^{B_{k_{n},n}} (v_1\dots v_{n-3})^{B_{k_{n},n}{\rho_{n-1}}} w_{n-1}^{B_{k_{n},n}{\rho_{n-1}}}\\
\notag&=\eta^{\rho_{n-1}}(w_1\dots w_{n-2})^{B_{k_{n},n}B_{k_{n},n-1}^{-1}} (v_1\dots v_{n-3})^{B_{k_{n},n}{\rho_{n-1}}B_{k_{n},n-1}^{-1}}\\
\notag&~\cdot~w_{n-1}^{B_{k_{n},n}{\rho_{n-1}}B_{k_{n},n-1}^{-1}}B_{k_{n},n-1}
\end{align}
Since $B_{k_{n},n}B_{k_{n},n-1}^{-1}=\rho_{n-1}$, $B_{k_{n},n}{\rho_{n-1}}B_{k_{n},n-1}^{-1}=\rho_{n-2}\rho_{n-1}$, then we have
\begin{align}
\notag \varrho(\beta)_1&=\eta^{\rho_{n-1}}(w_1\dots w_{n-2})^{\rho_{n-1}} (v_1\dots v_{n-3})^{\rho_{n-2}\rho_{n-1}} w_{n-1}^{\rho_{n-2}\rho_{n-1}}B_{k_{n},n-1}\\
\notag &=\eta^{\rho_{n-1}}(w_1\dots w_{n-3})^{\rho_{n-1}} (v_1\dots v_{n-3})^{\rho_{n-2}\rho_{n-1}} w_{n-2}^{\rho_{n-1}}w_{n-1}^{\rho_{n-2}\rho_{n-1}}B_{k_{n},n-1}
\end{align}
Here the braid $\eta^{\rho_{n-1}}=\eta$ does not involve the strand $n-1$. Also by Lemma <ref> the braid
$(w_1\dots w_{n-3})^{\rho_{n-1}} (v_1\dots v_{n-3})^{\rho_{n-2}\rho_{n-1}}$ belongs to the group $\langle\lambda_{1,n-1},\lambda_{n-1,1}\rangle\times\dots\times\langle\lambda_{n-3,n-1},\lambda_{n-1,n-3}\rangle$ and
Step 4. $\varrho(\beta)_2=\eta^{\rho_{n-1}}(w_1\dots w_{n-3})^{\rho_{n-1}} (v_1\dots v_{n-3})^{\rho_{n-2}\rho_{n-1}} \rho_{n-2}B_{k_{n},n-2}$
Step 5. The braid $\varrho^2(\beta)$ has the following form
\begin{align}\notag\varrho^2(\beta)&=\eta^{\rho_{n-1}}(w_1\dots w_{n-3})^{\rho_{n-1}\rho_{n-2}} (v_1\dots v_{n-3})^{\rho_{n-2}\rho_{n-1}\rho_{n-2}} B_{k_{n},n-2}\\
\notag\varrho^2(\beta)&=\eta^{\rho_{n-1}}(v_1\dots v_{n-3})^{\rho_{n-2}\rho_{n-1}\rho_{n-2}}(w_1\dots w_{n-3})^{\rho_{n-1}\rho_{n-2}}\\
\notag&~\cdot~[(w_1\dots w_{n-3})^{\rho_{n-1}\rho_{n-2}},(v_1\dots v_{n-3})^{\rho_{n-2}\rho_{n-1}\rho_{n-2}}] B_{k_{n},n-2}
\end{align}
By Lemma <ref> it is obvious that $(v_1\dots v_{n-3})^{\rho_{n-2}\rho_{n-1}\rho_{n-2}}=(v_1\dots v_{n-3})^{\rho_{n-2}}$ and therefore by the remark <ref> the braids $\varrho^*(\beta)$ and $\varrho^*(\alpha)$ are equal modulo $UVP_m^{\prime}$, i. e. $\overline{\varrho^*(\alpha)}=\overline{\varrho^*(\beta)}$.
Case 3.1.5.2. $\iota(\alpha)(s-1)= s-1$.
Since $\beta=\rho_{s-1}\alpha\rho_{s-1}$ then $\iota(\beta)$ fixes $s$ and maps $s-1$ to $k_s$.
$$ \begin{CD}
s @>~\rho_{s-1}~>> s-1 @>~\alpha~>> s-1@>~\rho_{s-1}~>> s
\end{CD}$$
$$ \begin{CD}
s-1 @>~\rho_{s-1}~>> s @>~\alpha~>> k_s (\leq s-2)@>~\rho_{s-1}~>> k_s
\end{CD}$$
Therefore $s-1$ is maximal number which is not fixed by $\beta$, and $\beta_1$ has the following form.
\begin{align}\notag \beta_1&=B_{1,n}^{s-1-n}\beta B_{1,n}^{n-s+1}=B_{1,n}^{-1}B_{1,n}^{s-n}\rho_{s-1}\alpha\rho_{s-1}B_{1,n}^{n-s}B_{1,n}\\
\notag&=B_{1,n}^{-1}\rho_{n-1}B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}\rho_{n-1}B_{1,n}=B_{1,n-1}^{-1}\alpha_1B_{1,n-1}
\end{align}
If we denote by $\delta_{1}=\beta_1$, $\delta_{2}=\delta_{1}^{\rho_{1}}$, $\delta_{3}=\delta_{2}^{\rho_{2}}$, $\dots$, $\delta_{n-1}=\delta_{n-2}^{\rho_{n-2}}=\alpha_1$, then it is obvious that $n$ is not fixed by $\iota(\delta_j)$ for every $j$. Since $\delta_{j+1}$ is obtained from $\delta_j$ conjugating by $\rho_j~(j<n-1)$, then by the cases 3.1.1–3.1.4 the braids $\overline{\varrho^*(\delta_{j+1})}$ and $\overline{\varrho^*(\delta_{j})}$ are conjugated by the element from $S_n$, i. e.
Therefore we have
so the braids $\overline{\varrho^*(\beta)}=\overline{\varrho^*(\beta_1)}$ and $\overline{\varrho^*({\alpha})}=\overline{\varrho^*(\alpha_1)}$ are conjugated by the element from $S_n$.
Case 3.1.5.3. $\iota(\alpha)(s-1)\leq s-2$.
Case 3.1.5.3.1. $\iota(\alpha)(s-1)\geq \iota(\alpha)(s)+1$. By the equalities (<ref>) and (<ref>) the braids $\alpha_1$ and $\varrho(\alpha)$ have the following forms.
$$\alpha_1=\gamma w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1}$$
$$\varrho(\alpha)=\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}$$
Let us rewrite this equalities in more details. Since $\iota(\alpha)(s-1)\leq s-2$, then $\iota(\alpha_1)(n-1)=n-s+\iota(\alpha)(s-1)$.
$$ \begin{CD}
n-1 @>~B_{1,n}^{s-n}~>> s-1 @>~\alpha~>> \iota(\alpha)(s-1)@>~B_{1,n}^{n-s}~>> n-s+\iota(\alpha)(s-1)
\end{CD}$$
Therefore since $\iota(\alpha)(s-1)\geq \iota(\alpha)(s)+1$, then the braid $\gamma=\alpha_1\left(w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1}\right)^{-1}$ maps $n-1$ to $k_{n-1}=n-s+\iota(\alpha)(s-1)-1\geq n-s+\iota(\alpha)(s)=k_n$
$$ \begin{CD}
n-1 @>~\alpha_1~>> n-s+\iota(\alpha)(s-1) @>~\left(w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-1}\right)^{-1}~>> n-s+\iota(\alpha)(s-1)-1,
\end{CD}$$
and by Lemma <ref> we have
$$\gamma=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1},$$
where $\eta$ does not involve the strands $n-1,n$ and $v_j$ belongs to $\langle\lambda_{j,n-1},\lambda_{n-1,j}\rangle$ for $j=1,\dots,n-1$.
$$\alpha_1=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} w_1\dots w_{n-1}\rho_{n-1}B_{k_{n},n-1}$$
$$\varrho(\alpha)=\eta v_1\dots v_{n-2} B_{k_{n-1},n-1}(w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1}$$
and $k_{n-1}\geq k_n$.
Let us count $\varrho^2(\alpha)$
Step 1 The permutation $\iota(\varrho(\alpha))$ maps $n-1$ to $k_{n-1}+1=n-s+\iota(\alpha)(s-1)-1+1=n-s+\iota(\alpha)(s-1)\neq n$, therefore $n-1$ is a maximal number which is not foxed by $\varrho(\alpha)$.
Step 2 $\varrho(\alpha)_1=\varrho(\alpha)$.
Step 3 The braid $\varrho(\alpha)_1=\varrho(\alpha)$ can be rewritten.
\begin{align}\notag\varrho(\alpha_1)&=\eta v_1\dots v_{n-2} B_{k_{n-1},n-1}(w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1}\\
\notag&=\eta v_1\dots v_{n-2} (w_1\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_{n-1},n-1}B_{k_{n},n-1}
\end{align}
Since $k_{n-1}\geq k_n$, then $B_{k_{n-1},n-1}B_{k_{n},n-1}=B_{k_n,n-2}B_{k_{n-1}+1,n-1}$ and
\begin{align}\notag\varrho(\alpha_1)&=\eta v_1\dots v_{n-2} (w_1\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_n,n-2}B_{k_{n-1}+1,n-1}\\
\notag&=\eta(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} v_1\dots v_{n-2}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} \\
\notag &~\cdot~B_{k_n,n-2}B_{k_{n-1}+1,n-1}\\
\notag&=\eta(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_n,n-2}\\
\notag&~\cdot~ (v_1\dots v_{n-2})^{B_{k_n,n-2}}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}} B_{k_{n-1}+1,n-1}\\
\notag&=\eta(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{k_{n-1}})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_n,n-2}\\
\notag&~\cdot~ (v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}} v_{n-3}^{B_{k_n,n-2}} B_{k_{n-1}+1,n-1}
\end{align}
By Lemma <ref> the braid
$\eta(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_n,n-2}$ does not involve the strand $n-1$, the braid
$(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}}$ belongs to $\langle\lambda_{1,n-1},\lambda_{n-1,1}\rangle\times\dots\times\langle\lambda_{n-3,n-1},\lambda_{n-1,n-3}\rangle$ and
Step 4 The braid $\varrho(\alpha)_2$ has the following form
\begin{align}
\notag&\varrho(\alpha)_2=\eta(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_n,n-2}\\
\notag&~\cdot~ (v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}} \rho_{n-2} B_{k_{n-1}+1,n-2}
\end{align}
Step 5Finally, the braid $\varrho^2(\alpha)$ follows
\begin{align}
\notag\varrho^2(\alpha)&=\eta(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_n,n-2}\\
\notag& ~\cdot~(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}\rho_{n-2}} B_{k_{n-1}+1,n-2}\\
\notag&=\eta(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}\\
\notag& ~\cdot~(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}B_{k_n,n-2}^{-1}}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}\rho_{n-2}B_{k_n,n-2}^{-1}}\\
\notag&~\cdot~B_{k_n,n-2}B_{k_{n-1}+1,n-2}
\end{align}
Let us count $\varrho(\beta)=\varrho(\rho_{s-1}\alpha\rho_{s-1})$.
Step 1 An image of $s$ under the braid $\beta=\rho_{s-1}\alpha\rho_{s-1}$ is equal to $\iota(\alpha)(s-1)$.
$$ \begin{CD}
s @>~\rho_{s-1}~>> s-1 @>~\alpha~>> \alpha(s-1)@>~\rho_{s-1}~>> \alpha(s-1)
\end{CD}$$
Therefore $s$ is a maximal number which is not fixed by $\beta$.
Step 2 $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}.$
Step 3
Since $\rho_{s-1}B_{1,n}^{n-s}=B_{1,n}^{n-s}\rho_{n-1}$, then we have
\begin{align}
\notag\beta_1&=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}=B_{1,n}^{s-n}\rho_{s-1}\alpha\rho_{s-1} B_{1,n}^{n-s}=\rho_{n-1}B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}\rho_{n-1}=\rho_{n-1}\alpha_1\rho_{n-1}\\
\notag&=\rho_{n-1}\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} w_1\dots w_{n-1}\rho_{n-1}B_{k_{n},n-1}\rho_{n-1}\\
\notag&=\eta^{\rho_{n-1}} (v_1\dots v_{n-2})^{\rho_{n-1}}B_{k_{n-1},n-1}^{\rho_{n-1}} (w_1\dots w_{n-1})^{\rho_{n-1}}B_{k_{n},n}^{\rho_{n-1}}\\
\notag&=\eta^{\rho_{n-1}} (v_1\dots v_{n-2})^{\rho_{n-1}} (w_1\dots w_{n-1})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}B_{k_{n-1},n-1}^{\rho_{n-1}}B_{k_{n},n}^{\rho_{n-1}}
\end{align}
Using simple calculations in the symmetric group, it is easy to see that
\begin{equation}\label{big1}
\end{equation}
\begin{align}
\notag\beta_1&=\eta^{\rho_{n-1}} (v_1\dots v_{n-2})^{\rho_{n-1}} (w_1\dots w_{n-1})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}B_{k_n,n-1}\rho_{n-2}B_{k_{n-1}+1,n}\\
\notag&=\eta^{\rho_{n-1}}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}\\
\notag&~\cdot~(v_1\dots v_{n-2})^{\rho_{n-1}}w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}B_{k_n,n-1}\rho_{n-2}B_{k_{n-1}+1,n}\\
\notag&=\eta^{\rho_{n-1}}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}B_{k_n,n-1}\rho_{n-2}\\
\notag&~\cdot~(v_1\dots v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}}w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}}B_{k_{n-1}+1,n}\\
\notag&=\eta^{\rho_{n-1}}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}B_{k_n,n-1}\rho_{n-2}\\
\notag&~\cdot~(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}}w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}}v_{n-3}^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}}B_{k_{n-1}+1,n}
\end{align}
By Theorem <ref> the braid
$$\eta^{\rho_{n-1}}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}B_{k_n,n-1}\rho_{n-2}$$
does not involve the strand $n$, the braid
$$(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}}w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}}$$
belongs to $\langle\lambda_{1,n},\lambda_{n,1}\rangle\times\dots\times\langle\lambda_{n-2,n},\lambda_{n,n-2}\rangle$ and the braid
$v_{n-3}^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}}$ belongs to $\langle\lambda_{n-1,n},\lambda_{n,n-1}\rangle$.
Step 4 The braid $\beta_2$ follows
\begin{align}
\notag\beta_2&=\eta^{\rho_{n-1}}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}B_{k_n,n-1}\rho_{n-2}\\
\notag&~\cdot~(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}}w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}}\rho_{n-1}B_{k_{n-1}+1,n-1}
\end{align}
Step 5 The braid $\varrho(\beta)$ is the following.
\begin{align}
\notag\varrho(\beta)&=\eta^{\rho_{n-1}}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}B_{k_n,n-1}\rho_{n-2}\\
\notag&~\cdot~(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}}w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}}B_{k_{n-1}+1,n-1}
\end{align}
Let us find the braid $\varrho^2(\beta)$.
Step 1 Since $\eta^{\rho_{n-1}}$ does not involve the strands $n-1,n$, then the image of $n-1$ under the permutation $\iota(\varrho(\beta))$ is equal to the image of $n-1$ under the permutation $\iota(B_{k_n,n-1}\rho_{n-2}B_{k_{n-1}+1,n-1})$ and is equal to $k_n$. Then $n-1$ is a maximal number which is not fixed by $\varrho(\beta)$.
Step 2$\varrho(\beta)_1=\varrho(\beta)$.
Step 3 The braid $\varrho(\beta)_1$ can be rewritten
\begin{align}
\notag\varrho(\beta)_1&=\eta^{\rho_{n-1}}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}\\
\notag&~\cdot~(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}\\
\notag& ~\cdot~ w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}\\
\notag&~\cdot~B_{k_n,n-1}\rho_{n-2}B_{k_{n-1}+1,n-1}
\end{align}
The following equality is faithful in the symmetric group
\begin{equation}\label{big2}
\end{equation}
\begin{align}
\notag\varrho(\beta)_1&=\eta^{\rho_{n-1}}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-1})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}\\
\notag&~\cdot~(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}\\
\notag&~\cdot~w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}\\
\notag&~\cdot~B_{k_{n-1},n-3}B_{k_n,n-1}\\
\notag&=\eta^{\rho_{n-1}}(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}\\
\notag&~\cdot~(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}\\
\notag&~\cdot~w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}\\
\notag&~\cdot~w_{n-1}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}}B_{k_{n-1},n-3}B_{k_n,n-1}
\end{align}
\begin{align}
\notag~~~~~~~~~~~~~~~&=\eta^{\rho_{n-1}}(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}B_{k_{n-1},n-3}\\
\notag&~\cdot~(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}}\\
\notag&~\cdot~w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}B_{k_{n-1},n-3}}\\
\notag&~\cdot~w_{n-1}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}}B_{k_n,n-1}
\end{align}
Here the braid
$\eta^{\rho_{n-1}}(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}B_{k_{n-1},n-3}$ does not involve the strand $n-1$, the braid
\begin{multline*}
(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}}\cdot\\
\cdot w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}B_{k_{n-1},n-3}}
\end{multline*}
belongs to $\langle\lambda_{n-1,1},\lambda_{1,n-1}\rangle\times\dots\times\langle\lambda_{n-3,n-1},\lambda_{n-1,n-3}\rangle$ and
$w_{n-1}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}}$ belongs to $\langle\lambda_{n-2,n-1},\lambda_{n-1,n-2}\rangle$.
Step 4 The braid $\varrho(\beta)_2$ follows.
\begin{align}
\notag\varrho(\beta)_2&=\eta^{\rho_{n-1}}(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}B_{k_{n-1},n-3}\\
\notag&~\cdot~(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}}\\
\notag&~\cdot~w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}B_{k_{n-1},n-3}}\rho_{n-2}B_{k_n,n-2}
\end{align}
Step 5 The braid $\varrho^2(\beta)$ has the following form.
\begin{align}
\notag\varrho^2(\beta)&=\eta^{\rho_{n-1}}(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}B_{k_{n-1},n-3}\\
\notag&~\cdot~(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}\rho_{n-2}}\\
\notag&~\cdot~w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}B_{k_{n-1},n-3}\rho_{n-2}}B_{k_n,n-2}\\
\notag&=\eta^{\rho_{n-1}}(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}\\
\notag&~\cdot~(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}\rho_{n-2}B_{k_{n-1},n-3}^{-1}}\\
\notag&~\cdot~w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}B_{k_{n-1},n-3}\rho_{n-2}B_{k_{n-1},n-3}^{-1}}\\
\notag&~\cdot~B_{k_{n-1},n-3}B_{k_n,n-2}
\end{align}
\begin{align}
\notag~~~~~~~~~~~&=\eta^{\rho_{n-1}}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}\rho_{n-2}B_{k_{n-1},n-3}^{-1}}\\
\notag&~\cdot~(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}\\
\notag&~\cdot~[(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}},\\
\notag&~~~~(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}\rho_{n-2}B_{k_{n-1},n-3}^{-1}}]\\
\notag&~\cdot~w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}B_{k_{n-1},n-3}\rho_{n-2}B_{k_{n-1},n-3}^{-1}}\\
\notag&~\cdot~B_{k_{n-1},n-3}B_{k_n,n-2}
\end{align}
Now we have
\begin{align}
\notag\varrho^2(\beta)&=\eta^{\rho_{n-1}}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}\rho_{n-2}B_{k_{n-1},n-3}^{-1}}\\
\notag&~\cdot~(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}\\
\notag&~\cdot~[(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}},\\
\notag&~~~~(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}\rho_{n-2}B_{k_{n-1},n-3}^{-1}}]\\
\notag&~\cdot~w_{k_{n-1}}^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}B_{k_{n-1},n-3}\rho_{n-2}B_{k_{n-1},n-3}^{-1}}\\
\notag&~\cdot~B_{k_{n-1},n-3}B_{k_n,n-2}\\
\notag\varrho^2(\alpha)&=\eta(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}\\
\notag& (v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}B_{k_n,n-2}^{-1}}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}\rho_{n-2}B_{k_n,n-2}^{-1}}\\
\notag&B_{k_n,n-2}B_{k_{n-1}+1,n-2}
\end{align}
Note that the following equalities are faithful in the symmetric group
\begin{align}
\notag B_{k_{n-1},n-3}B_{k_n,n-2}&=B_{k_n,n-2}B_{k_{n-1}+1,n-2}\\
\notag\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}&=\rho_{n-1}\underline{\rho_{n-2}\rho_{n-3}\rho_{n-2}}\rho_{n-1}\rho_{n-2}\rho_{n-3}\rho_{n-2}\\
\notag&=\rho_{n-1}\rho_{n-3}\rho_{n-2}\underline{\rho_{n-3}\rho_{n-1}}\rho_{n-2}\rho_{n-3}\rho_{n-2}\\
\notag&=\rho_{n-1}\rho_{n-3}\rho_{n-2}\rho_{n-1}\rho_{n-3}\underline{\rho_{n-2}\rho_{n-3}\rho_{n-2}}\\
\notag&=\rho_{n-1}\rho_{n-3}\rho_{n-2}\rho_{n-1}\underline{\rho_{n-3}\rho_{n-3}}\rho_{n-2}\rho_{n-3}\\
\notag&=\underline{\rho_{n-1}\rho_{n-3}}\rho_{n-2}\rho_{n-1}\rho_{n-2}\rho_{n-3}\\
\notag&=\rho_{n-3}\rho_{n-1}\underline{\rho_{n-2}\rho_{n-1}\rho_{n-2}}\rho_{n-3}\\
\notag&=\rho_{n-3}\underline{\rho_{n-1}\rho_{n-1}}\rho_{n-2}\rho_{n-1}\rho_{n-3}\\
\notag&=\rho_{n-3}\rho_{n-2}\underline{\rho_{n-1}\rho_{n-3}}\\
\notag&=\rho_{n-3}\rho_{n-2}\rho_{n-3}\rho_{n-1}\\
\notag B_{k_n,n-2}\rho_{n-2}B_{k_n,n-2}^{-1}&=\rho_{n-3}\rho_{n-2}\rho_{n-3}
\end{align}
Therefore we have
$$(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}=((v_1\dots v_{n-4}v_{n-2})^{\rho_{n-3}\rho_{n-2}\rho_{n-3}})^{\rho_{n-1}},$$
and since $(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-3}\rho_{n-2}\rho_{n-3}}$ does not involve the strands $n-1,n$, then
\begin{multline*}((v_1\dots v_{n-4}v_{n-2})^{\rho_{n-3}\rho_{n-2}\rho_{n-3}})^{\rho_{n-1}}=\\
=(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-3}\rho_{n-2}\rho_{n-3}}=\\
=(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}B_{k_n,n-2}^{-1}}
\end{multline*}
and we have
$$(v_1\dots v_{n-4}v_{n-2})^{\rho_{n-1}B_{k_n,n-1}\rho_{n-2}\rho_{n-1}\rho_{n-2}B_{k_n,n-1}^{-1}}=(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}B_{k_n,n-2}^{-1}}.$$
Also, since
\begin{align}
\notag B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}\rho_{n-2}B_{k_{n-1},n-3}^{-1}&=B_{k_{n-1},n-1}^{-1}\rho_{n-1}\rho_{n-2}\\
\notag&=B_{k_{n-1},n-1}^{-1}\rho_{n-1}\rho_{n-2}\\
\notag&=\rho_{k_{n-1}}\dots\rho_{n-3}\rho_{n-2}\rho_{n-1}\rho_{n-2}\\
\notag&=\rho_{k_{n-1}}\dots\rho_{n-3}\rho_{n-1}\rho_{n-2}\rho_{n-1}\\
\notag&=\rho_{n-1}\rho_{k_{n-1}}\dots\rho_{n-3}\rho_{n-2}\rho_{n-1}=\rho_{n-1}B_{k_{n-1},n-1}^{-1}\rho_{n-1},
\end{align}
\begin{multline*}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}\rho_{n-2}B_{k_{n-1},n-3}^{-1}}=\\
=\left((w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}\right)^{\rho_{n-1}},
\end{multline*}
and since the braid $(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}$ does not involve the strands $n-1,n$, then
\begin{multline*}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{B_{k_{n-1},n-1}^{-1}\rho_{n-1}B_{k_{n-1},n-3}\rho_{n-2}B_{k_{n-1},n-3}^{-1}}=\\
=(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}
\end{multline*}
\begin{multline*}
\end{multline*}
Finally since $\eta$ does not involve the strands $n-1,n$, then $\eta^{\rho_{n-1}}=\eta$. Therefore the braids $\varrho^2(\alpha)$ and $\varrho^2(\beta)$ are equal modulo $UVP_{n-2}^{\prime}$ and by the remark <ref> the braids $\varrho^*(\beta)$ and $\varrho^*(\alpha)$ are equal modulo $UVP_m^{\prime}$, i. e. $\overline{\varrho^*(\alpha)}=\overline{\varrho^*(\beta)}$.
Case 3.1.5.3.2. $\alpha(s-1)< \alpha(s)+1$. This case literally repeats the case 3.1.5.3.1 using the fact that in this case $k_{n-1}<k_n$ and then using the equalities
instead of equalities (<ref>) and (<ref>).
Case 3.1.6. $i=s$. In this case the braid $\beta=\rho_{s}\alpha\rho_{s}$ maps $s+1$ to $k_s$.
s+1 @>~\rho_{s}~>> s @>~\alpha~>> k_s@>~\rho_{s}~>> k_s
\end{CD}$$
Therefore $s+1$ is a maximal number, which is not fixed by $\beta$ and we have
\begin{align}\notag \beta_1&=B_{1,n}^{s+1-n}\beta B_{1,n}^{n-s-1}=B_{1,n}B_{1,n}^{s-n}\rho_{s-1}\alpha\rho_{s-1}B_{1,n}^{n-s}B_{1,n}^{-1}\\
\notag&=B_{1,n}\rho_{n-1}B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}\rho_{n-1}B_{1,n}^{-1}\\
\notag&=B_{1,n}\rho_{n-1}\alpha_1\rho_{n-1}B_{1,n}^{-1}=\rho_{n-2}B_{1,n}\alpha_1B_{1,n}^{-1}\rho_{n-2}
\end{align}
If we denote by $\delta_{1}=\alpha_1$,
$\delta_{n+1}=\delta_n^{\rho_{n-2}}=\beta_1$, then it is obvious, that $\iota(\delta_i)(n)\neq n$ for $i=1\dots n-1$.
Therefore by the previous cases $\overline{\varrho^*(\delta_i)}$ and $\overline{\varrho^*(\delta_{i+1})}$ are conjugated by the element from $S_n$ for every $i=1\dots n-1$.
Also since $\iota(\delta_n)(n-1)\neq n-1$
n-1 @>~B_{1,n}~>> n @>~\alpha_1~>> k_n\neq n@>~B_{1,n}~>> k_n-1~(\text{mod}~n)\neq n-1
\end{CD}$$
then by the previous cases $\overline{\varrho^*(\delta_n)}=\overline{\varrho^*(\delta_{n+1})}^{\mu_n}$ and we have
Therefore the braids $\overline{\varrho^*(\beta)}=\overline{\varrho^*(\beta_1)}$ and $\overline{\varrho^*({\alpha})}=\overline{\varrho^*(\alpha_1)}$ are conjugated by the element from $S_n$.
Case 3.1.7. $i\geq s+1$. In this case the maximal number which is not fixed by the braid $\beta=\rho_{i}\alpha\rho_{i}$ is $s$ and we have
$$ \begin{CD}
s @>~\rho_{i}~>> s @>~\alpha~>> k_s@>~\rho_{i}~>> k_s
\end{CD}$$
Since $i\geq s+2$, then we have
\begin{align}
\notag\rho_iB_{1,n}^{n-s}&=\rho_{i}B_{1,n}^{n-i-1}B_{1,n}B_{1,n}^{i-s}=B_{1,n}^{n-i-1}\rho_{n-1}B_{1,n}B_{1,n}^{i-s}\\
\notag&=B_{1,n}^{n-i-1}B_{1,n-1}B_{1,n}^{i-s}=B_{1,n}^{n-i-1}B_{1,n-1}B_{1,n}B_{1,n}^{i-s-1}\\
\notag&=B_{1,n}^{n-i-1}B_{1,n}B_{2,n}B_{1,n}^{i-s-1}=B_{1,n}^{n-i-1}B_{1,n}B_{2,n}\rho_1\rho_1B_{1,n}^{i-s-1}\\
\notag&=B_{1,n}^{n-i+1}\rho_1B_{1,n}^{i-s-1}=B_{1,n}^{n-i+1}B_{1,n}^{i-s-1}\rho_{i-s}=B_{1,n}^{n-s}\rho_{i-s}
\end{align}
Therefore the braid $\beta_1$ can be rewritten
\begin{align}\notag\beta_1&=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}=B_{1,n}^{s-n}\rho_i\alpha\rho_i B_{1,n}^{n-s}=\rho_{i-s}B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}\rho_{i-s}=\rho_{i-s}\alpha_1\rho_{i-s}
\end{align}
Since $\iota(\alpha_1)(n)\neq n$ and $i-s<n$, then $\overline{\varrho^*(\alpha_1)}$ and $\overline{\varrho^*(\beta_1)}$ are conjugated by the previous cases.
Case 3.2. The braid $\beta$ is obtained from $\alpha$ conjugating by $\lambda_{i,i+1}^{\pm1}$. If $\beta$ is obtained from $\alpha$ conjugating by $\lambda_{i,i+1}^{-1}$, then $\alpha$ is obtained from $\beta$ conjugating by $\lambda_{i,i+1}$ and we can consider that $\beta$ is obtained from $\alpha$ conjugating by $\lambda_{i,i+1}$.
As we already found in the case 3.1 (equalities (<ref>) and (<ref>)) the braids $\varrho(\alpha)$ and $\alpha_1$ has the following forms
$$\alpha_1=\gamma w_1\dots w_{n-1}\rho_{n-1}B_{k_n,n-2},$$
$$\varrho(\alpha)=\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1},$$
where $k_n=n-s+k_s$ and $k_s$ is a maximal number which is not fixed by $\alpha$.
Case 3.2.1. $i\leq k_s-2$. Let us count the braid $\varrho(\beta)=\varrho(\lambda_{i,i+1}^{-1}\alpha\lambda_{i,i+1})$.
Step 1. Since $\lambda_{i,i+1}$ is a pure braid, then the maximal number which is not fixed by $\beta$ is equal to the maximal number which is not fixed by $\beta$ and is equal to $s$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=k_s+n-s=k_n\neq n$.
Step 3. By Lemma <ref> we have $\lambda_{i,i+1}B_{1,n}^{n-s}=B_{1,n}^{n-s}\lambda_{i+n-s,i+n-s+1}$. If we denote by $j=i+n-s$, then the braid $\beta_1$ can be rewritten in more details
\begin{align}
\notag\beta_1&=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}=B_{1,n}^{s-n}\lambda_{i,i+1}^{-1}\alpha\lambda_{i,i+1} B_{1,n}^{n-s}\\
\notag&=\lambda_{j,j+1}^{-1}B_{1,n}^{s-n}\alpha B_{1,n}^{n-s}\lambda_{j,j+1}=\lambda_{j,j+1}^{-1}\alpha_1\lambda_{j,j+1}\\
\notag&=\lambda_{j,j+1}^{-1}\gamma w_1\dots w_{n-1}B_{k_n,n}\lambda_{j,j+1}\\
\notag&=\lambda_{j,j+1}^{-1}\gamma w_1\dots w_{n-1}\lambda_{j,j+1}^{B_{k_n,n}^{-1}}B_{k_n,n}
\end{align}
Since $i<k_s-1$, then $j=i+n-s<k_n-1$ and therefore $\lambda_{j,j+1}^{B_{k_n,n}^{-1}}=\lambda_{j,j+1}$. Therefore we have
$$\beta_1=\lambda_{j,j+1}^{-1}\gamma\lambda_{j,j+1} w_1\dots w_{n-1}B_{k_n,n},$$
where the braid $\lambda_{j,j+1}^{-1}\gamma\lambda_{j,j+1}$ does not involve the strand $n$ and $w_r$ belongs to $\langle\lambda_{r,n},\lambda_{n,r}\rangle$ for $r=1,\dots,w_{n-1}$.
Step 4. $\beta_2=\lambda_{j,j+1}^{-1}\gamma\lambda_{j,j+1} w_1\dots w_{n-2}\rho_{n-1}B_{k_n,n-1}.$
Step 5. The braid $\varrho(\beta)$ has the following form.
\begin{align}
\notag\varrho(\beta)&=\lambda_{j,j+1}^{-1}\gamma\lambda_{j,j+1} (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}\\
\notag&=\lambda_{j,j+1}^{-1}\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}\lambda_{j,j+1}=\varrho(\alpha)^{\lambda_{j,j+1}}
\end{align}
Therefore $\varrho(\alpha)$ and $\varrho(\beta)$ are conjugated and by the induction hypothesis the braids $\varrho^*(\alpha)$ and $\varrho^*(\beta)$ are conjugated by the element from $S_n$.
Case 3.2.2. $i=k_s-1$. Let us find the braid $\varrho(\beta)$.
Step 1. The maximal number which is not fixed by $\alpha$ is equal to $s$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=k_s+n-s=k_n\neq n$.
Step 3. Similarly to the case 3.2.1 we conclude that $\beta_1=\lambda_{k_n-1,k_n}^{-1}\alpha_1\lambda_{k_n-1,k_n}$ and therefore we have
\begin{align}
\notag\beta_1&=\lambda_{k_n-1,k_n}^{-1}\gamma w_1\dots w_{n-1}B_{k_n,n}\lambda_{k_n-1,k_n}\\
\notag&=\lambda_{k_n-1,k_n}^{-1}\gamma w_1\dots w_{n-1}\lambda_{k_n-1,k_n}^{B_{k_n,n}^{-1}}B_{k_n,n}\\
\notag&=\lambda_{k_n-1,k_n}^{-1}\gamma w_1\dots w_{n-1}\lambda_{k_n-1,n}B_{k_n,n}\\
\notag&=\lambda_{k_n-1,k_n}^{-1}\gamma w_1\dots w_{n-2}\lambda_{k_n-1,n}w_{n-1}B_{k_n,n}
\end{align}
Since $k_n<n$, then the braid $\lambda_{k_n-1,k_n}^{-1}\gamma$ does not involve the strand $n$.
Step 4. $\beta_2=\lambda_{k_n-1,k_n}^{-1}\gamma w_1\dots w_{n-2}\lambda_{k_n-1,n}\rho_{n-1}B_{k_n,n-1}.$
Step 5. The braid $\varrho(\beta)$ has the following form.
\begin{align}
\notag\varrho(\beta)&=\lambda_{k_n-1,k_n}^{-1}\gamma (w_1\dots w_{n-2}\lambda_{k_n-1,n})^{\rho_{n-1}}B_{k_n,n-1}\\
\notag&=\lambda_{k_n-1,k_n}^{-1}\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}\lambda_{k_n-1,n-1}B_{k_n,n-1}\\
\notag&=\lambda_{k_n-1,k_n}^{-1}\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}\lambda_{k_n-1,n-1}^{B_{k_n,n-1}}\\
\notag&=\lambda_{k_n-1,k_n}^{-1}\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}\lambda_{k_n-1,k_n}=\varrho(\alpha)^{\lambda_{k_n-1,k_n}}.
\end{align}
Therefore the braids $\varrho(\alpha)$ and $\varrho(\beta)$ are conjugated and by the induction hypothesis the braids $\varrho^*(\alpha)$ and $\varrho^*(\beta)$ are conjugated by the element from $S_n$.
Case 3.2.3. $i=k_s$.
Case 3.2.3.1. $k_s\leq s-2$. Let us count the braid $\varrho(\beta)$.
Step 1. The maximal number which is not fixed by $\alpha$ is equal to $s$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=k_s+n-s=k_n\neq n$.
Step 3. Similarly to the case 3.2.1 we have $\beta_1=\lambda_{k_n,k_n+1}^{-1}\alpha_1\lambda_{k_n,k_n+1}$ and therefore
\begin{align}
\notag\beta_1&=\lambda_{k_n,k_n+1}^{-1}\gamma w_1\dots w_{n-1}B_{k_n,n}\lambda_{k_n,k_n+1}\\
\notag&=\lambda_{k_n,k_n+1}^{-1}\gamma w_1\dots w_{n-1}\lambda_{k_n,k_n+1}^{B_{k_n,n}^{-1}}B_{k_n,n}\\
\notag&=\lambda_{k_n,k_n+1}^{-1}\gamma w_1\dots w_{n-1}\lambda_{n,k_n}B_{k_n,n}\\
\notag&=\lambda_{k_n,k_n+1}^{-1}\gamma w_1\dots w_{n-2}\lambda_{n,k_n}w_{n-1}B_{k_n,n}
\end{align}
Since $k_s<s-1$, then $k_n,k_n+1<n$ and the braid $\lambda_{k_n,k_n+1}^{-1}\gamma$ does not involve the strand $n$.
Step 4. $\beta_2=\lambda_{k_n,k_n+1}^{-1}\gamma w_1\dots w_{n-2}\lambda_{n,k_n}\rho_{n-1}B_{k_n,n-1}.$
Step 5. The braid $\varrho(\beta)$ has the following form.
\begin{align}
\notag\varrho(\beta)&=\lambda_{k_n,k_n+1}^{-1}\gamma (w_1\dots w_{n-2}\lambda_{n,k_n})^{\rho_{n-1}}B_{k_n,n-1}\\
\notag&=\lambda_{k_n,k_n+1}^{-1}\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}\lambda_{n-1,k_n}B_{k_n,n-1}\\
\notag&=\lambda_{k_n,k_n+1}^{-1}\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}\lambda_{n-1,k_n}^{B_{k_n,n-1}}\\
\notag&=\lambda_{k_n,k_n+1}^{-1}\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}\lambda_{k_n,k_n+1}=\varrho(\alpha)^{\lambda_{k_n,k_n+1}}
\end{align}
Therefore the braids $\varrho(\alpha)$ and $\varrho(\beta)$ are conjugated and by the induction hypothesis the braids $\varrho^*(\alpha)$ and $\varrho^*(\beta)$ are conjugated by the element from $S_n$.
Case 3.2.3.2 $k_s=s-1.$
Case 3.2.3.2.1. $\iota(\alpha)(s-1)=s$. Let us find the braid $\varrho(\beta)=\varrho(\lambda_{k_s,k_s+1}^{-1}\alpha\lambda_{k_s,k_s+1})$.
Step 1. The maximal number which is not fixed by $\alpha$ is equal to $s$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=k_s+n-s=k_n\neq n$.
Step 3. Similarly to the case 3.2.1 we have $\beta_1=\lambda_{k_n,k_n+1}^{-1}\alpha_1\lambda_{k_n,k_n+1}$. Since $k_s=s-1$, then $k_n=n-1$, $B_{k_n,n}=\rho_{n-1}$ and $\beta_1=\lambda_{n-1,n}^{-1}\alpha_1\lambda_{n-1,n}$. Then we have
\beta_1=\lambda_{n-1,n}^{-1}\gamma w_1\dots w_{n-1}\rho_{n-1}\lambda_{n-1,n}.
Since $\iota(\alpha)(s-1)=s$, then $\iota(\alpha_1)(n-1)=n$, therefore the braid $\gamma=\alpha_1\left(w_1\dots w_{n-1}\rho_{n-1}\right)^{-1}$ maps $n-1$ to $n-1$. Hence the braids $\gamma$ and $\lambda_{n-1,n}^{-1}$ commute and we have
\begin{align}
\notag\beta_1&=\gamma\lambda_{n-1,n}^{-1} w_1\dots w_{n-1}\rho_{n-1}\lambda_{n-1,n}\\
\notag&=\gamma w_1\dots w_{n-2}\lambda_{n-1,n}^{-1}w_{n-1}\rho_{n-1}\lambda_{n-1,n}\\
\notag&=\gamma w_1\dots w_{n-2}\lambda_{n-1,n}^{-1}w_{n-1}\lambda_{n,n-1}\rho_{n-1}
\end{align}
where the braid $\gamma$ does not involve the strand $n$, the braid $w_r$ belongs to $\langle\lambda_{n,r},\lambda_{r,n}\rangle$ for $r=1,\dots,n-2$ and $\lambda_{n-1,n}^{-1}w_{n-1}\lambda_{n,n-1}$ belongs to $\langle\lambda_{n,n-1},\lambda_{n-1,n}\rangle$.
Step 4. $\beta_2=\gamma w_1\dots w_{n-2}\rho_{n-1}.$
Step 5. $\varrho(\alpha)=\varrho(\beta)$.
Therefore the braids $\varrho(\alpha)$ and $\varrho(\beta)$ are conjugated and by the induction hypothesis the braids $\varrho^*(\alpha)$ and $\varrho^*(\beta)$ are conjugated by the element from $S_n$.
Case 3.2.3.2.2. $\iota(\alpha)(s-1)=k_{s-1}\leq s-2$. As already founded in the case 3.1.3.2.2. for $k_s=s-1$ and $\iota(\alpha)(s-1)=k_{s-1}\leq s-2$ we have
$$\alpha_1=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} w_1\dots w_{n-1}\rho_{n-1},$$
$$\varrho(\alpha)=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} (w_1\dots w_{n-2})^{\rho_{n-1}},$$
$$\varrho^2(\alpha)=\eta (w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} (v_1\dots v_{n-3})^{\rho_{n-2}} B_{k_{n-1},n-2},
where the braid $\eta$ does not involve the strands $n-1,n$ the braid $v_r$ belongs to $\langle\lambda_{r,n-1},\lambda_{n-1,r}\rangle$ for $r=1,\dots,n-2$ and the braid $w_r$ belongs to $\langle\lambda_{r,n},\lambda_{n,r}\rangle$ for $r=1,\dots,n-1$.
Let us count the braid $\varrho(\beta)=\varrho(\lambda_{s-1,s}\alpha\lambda_{s-1,s})$.
Step 1. The maximal number which is not fixed by $\beta$ is equal to $s$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=k_s+n-s=k_n\neq n$.
Step 3. Similarly to the case 3.2.3.2.1. $\beta_1=\lambda_{n-1,n}^{-1}\alpha_1\lambda_{n-1,n}$ and we have
\begin{align}
\notag\beta_1&=\lambda_{n-1,n}^{-1}\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} w_1\dots w_{n-1}\rho_{n-1}\lambda_{n-1,n}\\
\notag&=\eta^{\lambda_{n-1,n}}\lambda_{n-1,n}^{-1} v_1\dots v_{n-2}B_{k_{n-1},n-1} w_1\dots w_{n-1}\lambda_{n,n-1}\rho_{n-1}\\
\notag&=\eta^{\lambda_{n-1,n}} v_1\dots v_{n-2}B_{k_{n-1},n-1}(\lambda_{n-1,n}^{-1})^{B_{k_{n-1},n-1}} w_1\dots w_{n-1}\lambda_{n,n-1}\rho_{n-1}\\
\notag&=\eta^{\lambda_{n-1,n}} v_1\dots v_{n-2}B_{k_{n-1},n-1}\lambda_{k_{n-1},n}^{-1} w_1\dots w_{n-1}\lambda_{n,n-1}\rho_{n-1}
\end{align}
Since the braid $\eta$ does not involve the strands $n-1,n$, then $\eta^{\lambda_{n-1,n}}=\eta$.
Step 4. $\beta_2=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1}\lambda_{k_{n-1},n}^{-1} w_1\dots w_{n-2}\rho_{n-1}.$
Step 5. $\varrho(\beta)=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1}(\lambda_{k_{n-1},n}^{-1} w_1\dots w_{n-2})^{\rho_{n-1}}$.
Let us count the braid $\varrho^2(\beta)$.
Step 1. Since $k_{s-1}\leq s-2$, then $k_{n-1}\leq n-2$, therefore $B_{k_{n-1},n-1}\neq 1$ and the maximal number which is not fixed by $\varrho(\beta)$ is equal to $n-1$.
Step 2. $\varrho(\beta)_1=\varrho(\beta)$.
Step 3. The braid $\varrho(\beta)$ can be rewritten
\begin{align}
\notag\varrho(\beta)&=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1}(\lambda_{k_{n-1},n}^{-1} w_1\dots w_{n-2})^{\rho_{n-1}}\\
\notag&=\eta v_1\dots v_{n-2}(\lambda_{k_{n-1},n}^{-1} w_1\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_{n-1},n-1}\\
\notag&=\eta (w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} v_1\dots v_{n-3}\\
\notag&~\cdot~v_{n-2}\lambda_{{n-1},n-2}^{-1}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_{n-1},n-1}
\end{align}
Step 4. $\varrho(\beta)_2=\eta (w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} v_1\dots v_{n-3}
\rho_{n-2}B_{k_{n-1},n-2}$.
Step 5. $\varrho^2(\beta)=\eta (w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} (v_1\dots v_{n-3})^{\rho_{n-2}}B_{k_{n-1},n-2}$.
Therefore $\varrho^2(\alpha)=\varrho^2(\beta)$ and by the induction hypothesis the braids $\varrho^*(\alpha)$ and $\varrho^*(\beta)$ are conjugated by the element from $S_n$.
Case 3.2.4. $k_s+1\leq i \leq s-2$. Let us find the braid $\varrho(\beta)=\varrho(\lambda_{i,i+1}^{-1}\alpha\lambda_{i,i+1})$.
Step 1. The maximal number which is not fixed by $\beta$ is equal to $s$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=k_s+n-s=k_n\neq n$.
Step 3. By Lemma <ref> we have $\lambda_{i,i+1}B_{1,n}^{n-s}=B_{1,n}^{n-s}\lambda_{j,j+1}$, where $j=i+n-s$. Since $k_s+1\leq i \leq s-2$ then $k_n+1\leq j \leq n-2$ and the braid $\beta$ can be rewritten in more details.
\begin{align}
\notag\beta_1&=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}=B_{1,n}^{s-n}\lambda_{i,i+1}^{-1}\alpha\lambda_{i,i+1} B_{1,n}^{n-s}=\lambda_{j,j+1}^{-1}\alpha_1\lambda_{j,j+1}\\
\notag&=\lambda_{j,j+1}^{-1}\gamma w_1\dots w_{n-1}B_{k_n,n}\lambda_{j,j+1}\\
\notag&=\lambda_{j,j+1}^{-1}\gamma w_1\dots w_{n-1}\lambda_{j,j+1}^{B_{k_n,n}^{-1}}B_{k_n,n}\\
\notag&=\lambda_{j,j+1}^{-1}\gamma w_1\dots w_{n-1}\lambda_{j-1,j}B_{k_n,n}\\
\notag&=\lambda_{j,j+1}^{-1}\gamma\lambda_{j-1,j} w_1\dots w_{n-1}B_{k_n,n},
\end{align}
Where the braid $\lambda_{j,j+1}^{-1}\gamma\lambda_{j-1,j}$ does not involve the strand $n$.
Step 4. $\beta_2=\lambda_{j,j+1}^{-1}\gamma\lambda_{j-1,j} w_1\dots w_{n-2}\rho_{n-1}B_{k_n,n-1}$.
Step 5. The braid $\varrho(\beta)$ follows
\begin{align}
\notag\varrho(\beta)&=\lambda_{j,j+1}^{-1}\gamma\lambda_{j-1,j} (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}\\
\notag&=\lambda_{j,j+1}^{-1}\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}\lambda_{j-1,j}B_{k_n,n-1}\\
\notag&=\lambda_{j,j+1}^{-1}\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}\lambda_{j-1,j}^{B_{k_n,n-1}}\\
\notag&=\lambda_{j,j+1}^{-1}\gamma (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_n,n-1}\lambda_{j,j+1}=\varrho(\alpha)^{\lambda_{j,j+1}}
\end{align}
Therefore the braids $\varrho(\alpha)$ and $\varrho(\beta)$ are conjugated and by the induction hypothesis the braids $\varrho^*(\alpha)$ and $\varrho^*(\beta)$ are conjugated by the element from $S_n$.
Case 3.2.5. $i=s-1$. In this case we can consider that $k_s< s-1$ since the case when $k_s=s-1=i$ is already solved in the case 3.2.3.2.
Case 3.2.5.1. $\iota(\alpha)(s-1)=s$. From the case 3.1.5.1. we have
$$\alpha_1=\eta v_1\dots v_{n-2} w_1\dots w_{n-1}\rho_{n-1}B_{k_{n},n-1},$$
$$\varrho(\alpha)=\eta v_1\dots v_{n-2} (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1},$$
$$\varrho^{2}({\alpha})=\eta (v_1\dots v_{n-3})^{\rho_{n-2}} (w_1\dots w_{n-3})^{\rho_{n-1}\rho_{n-2}}B_{k_{n},n-2},$$
where the braid $\eta$ does not involve the strands $n-1,n$.
Let us count $\varrho(\beta)$.
Step 1. The maximal number which is not fixed by $\beta$ is equal to $s$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=k_s+n-s=k_n\neq n$.
Step 3. By Lemma <ref> we have $\lambda_{s-1,s}B_{1,n}^{n-s}=B_{1,n}^{n-s}\lambda_{n-1,n}$, therefore
\begin{align}
\notag\beta_1&=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}=B_{1,n}^{s-n}\lambda_{s-1,s}^{-1}\alpha\lambda_{s-1,s} B_{1,n}^{n-s}=\lambda_{n-1,n}^{-1}\alpha_1\lambda_{n-1,n}\\
\notag&=\lambda_{n-1,n}^{-1}\eta v_1\dots v_{n-2} w_1\dots w_{n-1}B_{k_{n},n}\lambda_{n-1,n}\\
\notag&=\eta v_1\dots v_{n-2} \lambda_{n-1,n}^{-1}w_1\dots w_{n-1}\lambda_{n-1,n}^{B_{k_{n},n}^{-1}}B_{k_{n},n}\\
\notag&=\eta v_1\dots v_{n-2} \lambda_{n-1,n}^{-1}w_1\dots w_{n-1}\lambda_{n-2,n-1}B_{k_{n},n}\\
\notag&=\eta v_1\dots v_{n-2}\lambda_{n-2,n-1} w_1\dots w_{n-2} \lambda_{n-1,n}^{-1}w_{n-1}B_{k_{n},n},
\end{align}
where the braid $\eta v_1\dots v_{n-2}\lambda_{n-2,n-1}$ does not involve the strand $n$.
Step 4. $\beta_2=\eta v_1\dots v_{n-2}\lambda_{n-2,n-1} w_1\dots w_{n-2} \rho_{n-1}B_{k_{n},n-1}$.
Step 5. $\varrho(\beta)=\eta v_1\dots v_{n-2}\lambda_{n-2,n-1} (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1}$.
Let us count $\varrho^2(\beta)$.
Step 1. Since the braid $\eta$ does not involve the strand $n-1$, then the image of $n-1$ under the permutation $\iota(\varrho(\beta))$ is equal to $\iota(B_{k_n,n-1})(n-1)=k_n\neq n-1$.
Step 2. $\varrho(\beta)_1=\varrho(\beta)$.
Step 3.The braid $\varrho(\alpha)_1$ has the following form.
\begin{align}
\notag\varrho(\beta)_1&=\varrho(\beta)=\eta v_1\dots v_{n-2}\lambda_{n-2,n-1} (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1}\\
\notag&=\eta v_1\dots v_{n-3} (w_1\dots w_{n-3})^{\rho_{n-1}}v_{n-2}\lambda_{n-2,n-1}w_{n-2}^{\rho_{n-1}}B_{k_{n},n-1}
\end{align}
Step 4. $\varrho(\beta)_2=\eta v_1\dots v_{n-3} (w_1\dots w_{n-3})^{\rho_{n-1}}{\rho_{n-2}}B_{k_{n},n-2}$
Step 5. $\varrho^2(\beta)=\eta v_1\dots v_{n-3} (w_1\dots w_{n-3})^{\rho_{n-1}\rho_{n-2}}B_{k_{n},n-2}=\varrho^2(\alpha)$.
By the induction hypothesis the braids $\varrho^*(\alpha)$ and $\varrho^*(\beta)$ are conjugated by the element from $S_n$.
Case 3.2.5.2. $\iota(\alpha)(s-1)=s-1$. In this case we have
\begin{align}
\notag\alpha_1&=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} w_1\dots w_{n-1}\rho_{n-1}B_{k_{n},n-1}\\
\notag\varrho(\alpha)&=\eta v_1\dots v_{n-2} B_{k_{n-1},n-1}(w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1},
\end{align}
where the braid $\eta$ does not involve the strands $n-1,n$ and $k_{n-1}\geq k_n$.
Since $\iota(\alpha)(s-1)=s-1$, then $\iota(\alpha_1)(n-1)=n-1$, therefore $k_{n-1}=n-2$ and $B_{k_{n-1},n-1}=\rho_{n-2}$.
\begin{align}
\notag\alpha_1&=\eta v_1\dots v_{n-2}\rho_{n-2} w_1\dots w_{n-1}\rho_{n-1}B_{k_{n},n-1}\\
\notag\varrho(\alpha)&=\eta v_1\dots v_{n-2} \rho_{n-2}(w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1},
\end{align}
Let us count the braid $\varrho(\beta)$.
Step 1. The maximal number which is not fixed by $\beta$ is equal to $s$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=k_s+n-s=k_n\neq n$.
Step 3. Similarly to the case 3.2.5.1. we have $\beta_1=\lambda_{n-1,n}^{-1}\alpha_1\lambda_{n-1,n}$ and therefore
\begin{align}
\notag\beta_1&=\lambda_{n-1,n}^{-1}\eta v_1\dots v_{n-2}\rho_{n-2} w_1\dots w_{n-1}B_{k_{n},n}\lambda_{n-1,n}\\
\notag&=\eta v_1\dots v_{n-2}\lambda_{n-1,n}^{-1}\rho_{n-2} w_1\dots w_{n-1}\rho_{n-1}\lambda_{n-1,n}^{B_{k_{n},n}^{-1}}B_{k_{n},n}\\
\notag&=\eta v_1\dots v_{n-2}\rho_{n-2}(\lambda_{n-1,n}^{-1})^{\rho_{n-2}} w_1\dots w_{n-1}\lambda_{n-2,n-1}B_{k_{n},n}\\
\notag&=\eta v_1\dots v_{n-2}\rho_{n-2}\lambda_{n-2,n-1}\lambda_{n-2,n}^{-1} w_1\dots w_{n-1}B_{k_{n},n},
\end{align}
where the braid $=\eta v_1\dots v_{n-2}\rho_{n-2}\lambda_{n-2,n-1}$ does not involve the strand $n$.
Step 4. $\beta_1=\eta v_1\dots v_{n-2}\rho_{n-2}\lambda_{n-2,n-1}\lambda_{n-2,n}^{-1} w_1\dots w_{n-2}\rho_{n-1}B_{k_{n},n-1}$.
Step 5. The braid $\varrho(\beta)$ follows.
\begin{align}\notag\varrho(\beta)&=\eta v_1\dots v_{n-2}\rho_{n-2}\lambda_{n-2,n-1}(\lambda_{n-2,n}^{-1} w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1}\\
\notag\varrho(\beta)&=\eta v_1\dots v_{n-2}\rho_{n-2}\underline{\lambda_{n-2,n-1}\lambda_{n-2,n-1}^{-1}} (w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1}=\varrho(\alpha).
\end{align}
By the induction hypothesis the braids $\varrho^*(\alpha)$ and $\varrho^*(\beta)$ are conjugated by the element from $S_n$.
Case 3.2.5.3. $\iota(\alpha)(s-1)\leq s-2$.
Case 3.2.5.3.1. $\iota(\alpha)(s-1)\geq \iota(\alpha)(s)+1$. From the case 3.1.5.3.1. we have
\begin{align}
\notag\alpha_1&=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} w_1\dots w_{n-1}\rho_{n-1}B_{k_{n},n-1}\\
\notag\varrho(\alpha)&=\eta v_1\dots v_{n-2} B_{k_{n-1},n-1}(w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1}\\
\notag\varrho^2(\alpha)&=\eta(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots
\notag& ~\cdot~(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}\rho_{n-2}} B_{k_{n-1}+1,n-2}
\end{align}
Where the braid $\eta$ does not involve the strands $n-1,n$ and $k_{n-1}\geq k_n$.
Let us count the braid $\varrho(\beta)=\varrho(\lambda_{n-1,n}^{-1}\beta\lambda_{n-1,n})$.
Step 1. The maximal number which is not fixed by $\beta$ is equal to $s$.
Step 2. We have $\beta_1=B_{1,n}^{s-n}\beta B_{1,n}^{n-s}$ and $\iota(\beta_1)(n)=k_s+n-s=k_n\neq n$.
Step 3. Similarly to the case 3.2.5.1. we have $\beta_1=\lambda_{n-1,n}^{-1}\alpha_1\lambda_{n-1,n}$ and therefore
\begin{align}
\notag\beta_1&=\lambda_{n-1,n}^{-1}\eta v_1\dots v_{n-2}B_{k_{n-1},n-1} w_1\dots w_{n-1}B_{k_{n},n}\lambda_{n-1,n}\\
\notag&=\eta v_1\dots v_{n-2}\lambda_{n-1,n}^{-1}B_{k_{n-1},n-1} w_1\dots w_{n-1}\lambda_{n-1,n}^{B_{k_{n},n}^{-1}}B_{k_{n},n}\\
\notag&=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1}(\lambda_{n-1,n}^{-1})^{B_{k_{n-1},n-1}} w_1\dots w_{n-1}\lambda_{n-2,n-1}B_{k_{n},n}\\
\notag&=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1}\lambda_{k_{n-1},n}^{-1} w_1\dots w_{n-1}\lambda_{n-2,n-1}B_{k_{n},n}\\
\notag&=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1}\lambda_{n-2,n-1}\lambda_{k_{n-1},n}^{-1} w_1\dots w_{n-1}B_{k_{n},n}
\end{align}
where the braid $\eta v_1\dots v_{n-2}B_{k_{n-1},n-1}\lambda_{n-2,n-1}$ does not involve the strand $n$.
Step 4. $\beta_2=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1}\lambda_{n-2,n-1}\lambda_{k_{n-1},n}^{-1} w_1\dots w_{n-2}\rho_{n-1}B_{k_{n},n-1}$.
Step 5. $\varrho(\beta)=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1}\lambda_{n-2,n-1}(\lambda_{k_{n-1},n}^{-1} w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1}$.
Let us count $\varrho^2(\beta)$.
Step 1 The maximal number which is not fixed by the braid $\varrho(\beta)$ is $n-1$ and $\iota(\varrho(\beta))(n-1)=k_{n-1}+1$.
Step 2 $\varrho(\beta)_1=\varrho(\beta)$.
Step 3 The braid $\varrho(\beta)_1$ can be rewritten
\begin{align}\notag\varrho(\alpha_1)&=\eta v_1\dots v_{n-2}B_{k_{n-1},n-1}\lambda_{n-2,n-1}(\lambda_{k_{n-1},n}^{-1} w_1\dots w_{n-2})^{\rho_{n-1}}B_{k_{n},n-1}\\
\notag&=\eta v_1\dots v_{n-2}\lambda_{n-2,n-1}^{B_{k_{n-1},n-1}^{-1}} (\lambda_{k_{n-1},n}^{-1}w_1\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_{n-1},n-1}B_{k_{n},n-1}\\
\notag&=\eta v_1\dots v_{n-2}\lambda_{n-3,n-2} (\lambda_{k_{n-1},n}^{-1}w_1\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_{n-1},n-1}B_{k_{n},n-1}
\end{align}
Since $k_{n-1}\geq k_n$, then
\begin{equation}\label{a3}
\end{equation}
and and therefore
\begin{align}\notag\varrho(\alpha_1)&=\eta v_1\dots v_{n-2}\lambda_{n-3,n-2} (\lambda_{k_{n-1},n}^{-1}w_1\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}}B_{k_n,n-2}B_{k_{n-1}+1,n-1}\\
\notag&=\eta\lambda_{n-3,n-2}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} \\
\notag &~\cdot~v_1\dots v_{n-2}(\lambda_{k_{n-1},n}^{-1}w_{k_{n-1}})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} B_{k_n,n-2}B_{k_{n-1}+1,n-1}\\
\notag&=\eta\lambda_{n-3,n-2}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} B_{k_n,n-2}\\
\notag &~\cdot~(v_1\dots v_{n-2})^{B_{k_n,n-2}}(\lambda_{k_{n-1},n}^{-1}w_{k_{n-1}})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}} B_{k_{n-1}+1,n-1}\\
\notag&=\eta\lambda_{n-3,n-2}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} B_{k_n,n-2}\\
\notag &~\cdot~(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}}(\lambda_{k_{n-1},n}^{-1}w_{k_{n-1}})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}} v_{n-3}^{B_{k_n,n-2}} B_{k_{n-1}+1,n-1},
\end{align}
where the braid $\eta\lambda_{n-3,n-2}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} B_{k_n,n-2}$ does not involve the strand $n-1$.
Step 4 The braid $\varrho(\beta)_2$ has the following form
\begin{align}
\notag\varrho(\beta)_2&=\eta\lambda_{n-3,n-2}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} B_{k_n,n-2}\\
\notag &~\cdot~(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}}(\lambda_{k_{n-1},n}^{-1}w_{k_{n-1}})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}} \rho_{n-2} B_{k_{n-1}+1,n-2},
\end{align}
Step 5Finally, the braid $\varrho^2(\alpha)$ follows
\begin{align}
\notag\varrho^2(\beta)&=\eta\lambda_{n-3,n-2}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}} B_{k_n,n-2}\\
\notag &~\cdot~(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}}(\lambda_{k_{n-1},n}^{-1}w_{k_{n-1}})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}\rho_{n-2}} B_{k_{n-1}+1,n-2}\\
\notag&=\eta B_{k_n,n-2}\lambda_{n-3,n-2}^{B_{k_n,n-2}}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}} \\
\notag &~\cdot~(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}}(\lambda_{k_{n-1},n}^{-1}w_{k_{n-1}})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}\rho_{n-2}} B_{k_{n-1}+1,n-2}\\
\notag&=\eta B_{k_n,n-2}\lambda_{n-2,k_n}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}} \\
\notag &~\cdot~(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}}\lambda_{n-2,k_n}^{-1}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}\rho_{n-2}} B_{k_{n-1}+1,n-2}\\
\notag&=\eta B_{k_n,n-2}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}} \\
\notag &~\cdot~(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}}\lambda_{n-2,k_n}\\
\notag&~\cdot~[\lambda_{n-2,k_n},(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}} \\
\notag &~\cdot~(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}}]
\lambda_{n-2,k_n}^{-1}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}\rho_{n-2}} B_{k_{n-1}+1,n-2}
\end{align}
\begin{align}
\notag~~~~~~~&=\eta B_{k_n,n-2}(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}} \\
\notag &~\cdot~(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}}\\
\notag&~\cdot~[\lambda_{n-2,k_n},(w_1\dots w_{k_{n-1}-1}w_{k_{n-1}+1}\dots w_{n-2})^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}} \\
\notag &~\cdot~(v_1\dots v_{n-4}v_{n-2})^{B_{k_n,n-2}\rho_{n-2}}]^{\lambda_{n-2,k_n}^{-1}}w_{k_{n-1}}^{\rho_{n-1}B_{k_{n-1},n-1}^{-1}B_{k_n,n-2}\rho_{n-2}} B_{k_{n-1}+1,n-2}
\end{align}
Therefore the braids $\varrho^2(\alpha)$ and $\varrho^2(\beta)$ are equal modulo $UVP_{n-2}^{\prime}$ and by the remark <ref> the braids $\varrho^*(\alpha)$ and $\varrho^*(\beta)$ are equal modulo $UVP_n^{\prime}$, therefore $\overline{\varrho^*(\alpha)}=\overline{\varrho^*(\beta)}$.
Case 3.2.5.3.2. $\iota(\alpha)(s-1)\leq \iota(\alpha)(s)$. This case literally repeats the case 3.2.5.3.1 using the fact that in this case $k_{n-1}<k_n$ and then using the equalities
instead of equality (<ref>).
Case 3.2.6. $i=s$. By Lemma <ref> we have $\lambda_{s,s+1}B_{1,n}^{n-s}=B_{1,n}^{n-s}\lambda_{n,1}$, therefore
$\beta_1=\lambda_{n,1}^{-1}\alpha_1\lambda_{n,1}.$ By the definition $\lambda_{1,n}=B_{2,n}\lambda_{1,2}B_{2,n}^{-1}$, therefore
where $\delta_1=B_{2,n}^{-1}\alpha_1B_{2,n}$. By the case 3.1 the braids $\overline{\varrho^*(\alpha_1)}$ and $\overline{\varrho^*(\delta_1)}$ are conjugated by the element from $S_n$.
Since $\alpha$ is not a pure braid, then $\alpha_1$ and $\delta_1$ are not pure braids. Therefore the maximal number which is not fixed by $\delta_1$ is greater then or equal to $2$. Therefore, by the cases 3.2.1-3.2.5 the braids $\overline{\varrho^*(\delta_1)}$ and $\overline{\varrho^*(\lambda_{1,2}^{-1}\delta_1\lambda_{1,2})}$ are conjugated by the element from $S_n$. Also by the case 3.1 the braids $\overline{\varrho^*(\lambda_{1,2}^{-1}\delta_1\lambda_{1,2})}$ and $\overline{\varrho^*(B_{2,n}\lambda_{1,2}^{-1}\delta_1\lambda_{1,2}B_{2,n}^{-1})}=\overline{\varrho^*(\beta_1)}$ are conjugated by the element from $S_n$, therefore the braids $\overline{\varrho^*(\alpha_1)}=\overline{\varrho^*(\alpha)}$ and $\overline{\varrho^*(\beta_1)}=\overline{\varrho^*(\beta)}$ are conjugated by the element from $S_n$.
Case 3.2.7. $i\geq s+1$. In this case $\beta_1=\lambda_{j,j+1}^{-1}\alpha_1\lambda_{j,j+1}$, where
Since the maximal number which is not fixed by $\alpha_1$ is equal to $n$, then $\overline{\varrho^*(\alpha_1)}$ and $\overline{\varrho^*(\beta_1)}$ are conjugated by the element from $S_n$ by the cases 3.2.1-3.2.5.
The proposition is proved.
The following main result of the paper follows immediately from Corollary <ref> and Proposition <ref>.
Let $\alpha$ and $\beta$ be unrestricted virtual braids. Then their closures $\widehat{\alpha}$ and $\widehat{\beta}$ are equivalent as fused links if and only if $\overline{\varrho^*(\alpha)}$ and $\overline{\varrho^*(\beta)}$ are conjugated by the element from $S_n<UVB_n$.
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We present the first smart phone tool that is capable of self-compilation, mutation and viral spreading.
Our autonomous app does not require a host computer to alter its functionality, change its appearance and lacks the normal necessity of a central app store to spread among hosts.
We pioneered survival skills for mobile software in order to overcome disrupted Internet access due to natural disasters and human made interference, like Internet kill switches or censored networks.
Internet kill switches have proven to be an effective tool to eradicate open Internet access and all forms of digital communication within an hour on a country-wide basis.
We present the first operational tool that is capable of surviving such digital eradication.
self-compilation, mutation and viral spreading.
§ INTRODUCTION
Natural disasters and human made disruptions can both be the cause of communication systems failure.
Whenever a natural disaster occurs communication infrastructure is often completely destroyed or affected in such a way that it becomes unusable for longer periods of time.
Terrible loss of life has been blamed directly on the breakdown of cell phone service in the stricken area after the earthquake in Haiti in 2011 <cit.>.
The infrastructure in the capital had collapsed with the buildings and remaining networks were out of range and quickly overloaded.
The landfall of hurricane Katrina in 2005 caused a communication blackout and large scale network outages lasting for multiple weeks <cit.>.
Besides the flooding, the continuing power outages disrupted Internet access for months.
An unstable electrical grid has caused multiple large scale blackouts in the United States in recent history <cit.>.
Without a stable electricity supply many forms of modern communication are useless, because they depend on infrastructure that has to be powered continuously to function without sufficient backup power.
Not just highly developed countries have to deal with a troubled energy supply.
Especially the weak grid in Africa hurts the economy of countries like Ghana, Malawi and Tanzania <cit.>.
Besides the natural causes and indirect human causes just mentioned, there are also multiple known cases of various intentional communication blackouts orchestrated on a state level.
For instance Cuba is barely connected to the Internet all, prompting its citizens to come up with offline Internet <cit.>.
The Great Firewall of China is the well known name of all the layers of censorship that the Chinese government enforces in all their territories <cit.>.
This effectively causes a communication blackout that affects all outbound connections from within the borders of China.
The government of Iran blocks almost 50% of the world’s top 500 most visited websites <cit.>.
In Turkey social media were completely blocked for almost a day due to a temporary broadcast ban following a deadly attack in the capital <cit.>.
These are just a few examples of how a single actor can cause communication systems to fail to function.
§.§ Problem Description
We aim to address the problem of robust and resilient mobile systems, specifically usable without exiting infrastructure and consisting of innocuous hardware & software; moving toward a censorship-free Internet: <cit.>
* The adversary can observe, block, delay, replay, and modify traffic on all underlying transport. Thus, the physical layer is insecure.
* The adversary has a limited ability to compromise smart phones or other participating devices. If a device is compromised, the adversary can access any information held in the deviceś volatile memory or persistent storage.
* The adversary cannot break standard cryptographic primitives, such as block ciphers and message-authentication codes.
§ RELATED WORK
Various projects attempted to provide at least a partial solution to the problem just described.
Beginning with the fact that any network which has a single point of failure is vulnerable.
For example a mobile network that covers an area with a single cell tower, or wherever in dense cities there are tall buildings blocking the signals of all but one tower.
Even redundant infrastructure becomes useless when the switch point is wiped out or otherwise disabled.
That is one of the reasons why Internet switches are critical points for wide spread outages.
Additionally such central points also have been used in Egypt in 2011 <cit.> and Syria in 2012 <cit.> to quickly terminate all Internet access in a country intentionally.
Encrypted tunneling and anonymous proxy technology is available <cit.> to circumvent some forms of blockades but uses existing infrastructure, thus does not solve the problems just mentioned.
The challenge is now to use the pervasive smart phone phenomenon to create a resilient alternative Internet infrastructure.
Mesh networks are a solution to this problem because every wireless device effectively becomes a cell tower and switch itself, thus providing connectivity service wherever it goes, as long as it stays within reach of any other mobile device connected to the mesh network; otherwise it will be fairly limited connectivity.
Now there is no single point of failure anymore as all devices have this cell tower capability and could therefore take over for each other.
This way the network is protected against damage to infrastructure, because that is not required anymore, as well as power outages or an unstable electricity supply, because wireless devices typically carry batteries themselves.
For example the power bank phone carrying a 10000mAh battery capable of charging other devices as well via USB <cit.> is still small enough to be carried on person, so it will survive a natural disaster together with its user.
Compare this to a current smart phone having a capacity of less than a third than that, around 3000mAh.
The phone is popular in Ghana, which is not surprising since the capital regularly suffers from blackouts lasting more than 36 hours.
Mesh networks can differ significantly in the way they are set up.
For instance they may grow organically without oversight or they may be deployed and extended in an organized manner by a central authority.
Meraki is an example of the latter kind <cit.>.
Having a central authority re-introduces the problem of human made failure of the network.
To overcome this problem various communities have formed around projects that are deploying DIY-mesh-networks in multiple cities around the world.
For example Guifi.net in Spain <cit.>, the Free Network Foundation <cit.> and Commotion <cit.> in America, FreiFunk <cit.> in Germany and the Serval project <cit.> in New Zealand.
All of these projects offer software to run on oneś own router or smart phone to join the mesh network, subsequently providing other wireless devices access via said router or phone, effectively extending the mesh network.
In addition to that the Freedom Network Foundation also offers a hardware design for the FreedomTower to improve performance of the network in the entire neighborhood, and even a lab in two separate data centers open for anyone to find out and evaluate the best configuration of hardware and software for connectivity on a larger scale.
The general aim of these projects is to provide a robust and resilient censorship free communication network anywhere the users are, unaffected by the lack of communication infrastructure.
Having the same general aim here are a few specific applications for smart phones using Bluetooth and WiFi to implement the mesh network:
* Open Garden: Internet Sharing <cit.>.
* FireChat <cit.> also from Open Garden; chat only.
* Bleep <cit.> from BitTorrent; supports chat and voice-calls.
* ShadowTalk <cit.> from PeerSafe; supports chat and voice-chat with time-lock.
* Serval Mesh <cit.> from the Serval Project; supports file transfers in addition to chat and voice-calls.
* Tuse <cit.> from Tuse; supports sharing the app p2p in addition to file transfers, chat and voice-calls.
* Twimight <cit.> for using Twitter offline until it finds an uplink to Internet.
All but the last two of these apps support public messaging and private messaging with end to end encryption.
All of them are free to use and available on the Google app store.
However that central app store is, by definition of it being central, vulnerable to the same problems as mentioned before.
Either a decentralized app store is required, like tsukiji <cit.>, or the possibility of downloading and installing the app without the use of an app store all together.
This process is called side loading and is possible on the Android platform.
However few apps are capable of virally spreading.
Apps that could benefit greatly from using such technology are the eyeWitness app <cit.>, from the Eye Witness project, because a tactic to safeguard a user from an oppressive entity would be to make sure multiple copies of eyewitness evidence are virally spread as fast and easy as possible.
Although multiple IDE apps are available for Android <cit.>, none of them can replicate themselves like a Quine and neither are they capable of autonomously mutating their source.
An earlier app from Delft University of Technology called DroidStealth <cit.> has very limited mutation capability, only the binary form and not the source code can be mutated, meaning that the functionality cannot be changed, only the app name and icon, however it can virally spread via Android Beam.
A polymorphic computer virus avoids detection by mutating itself <cit.>.
Each time a new host is infected the binary software is altered to look completely different, yet remains unchanged in functionality.
Polymorphic viruses are an established technique, dating back before 1997.
These mutating viruses do not have access to their own source code and lack an embedded compiler, making self-compilation more generic and likely extra powerful.
§ AUTONOMOUS APP DESIGN
The following design aspects are aimed to accomplish the criteria set out in the problem description: a robust and resilient mobile system.
Smart phones have multiple wireless technologies on board, like Bluetooth and WiFi, carry a battery and are very widespread all around the world, making it the perfect platform to use for a mesh network.
Therefore to be most accessible the software will need to run on the open source platform with the largest install base: Android.
Mobile service providers often modify the operating system and do not grant users administrative privileges on their own device for whatever reason.
Gaining root access to take these administrative privileges by force is possible, but not without the help of a development computer.
This means that the app must not require root access to the phone in order to be autonomous.
§.§ Viral Spreading
Making the app capable of outputting itself and sending it to another device eliminates the normal necessity of a central app store and makes it capable of spreading like a virus.
This bio-inspired mutation and spreading requires modifying the source code and the capability of running on a variety of host hardware, as described in the next subsection.
Figure <ref> shows viral spreading from within an offline region until it reaches an area with unrestricted Internet access.
Building upon existing trust networks, like family or business partners, it becomes evident that interconnection of local clusters is key.
1) P2P wireless transfer via NFC and Bluetooth to any number of devices. 2) Any device can mutate the app and distribute without limitation. 3) One strain of the app travels out of the offline region. 4) The surviving strain is connected to the rest of the world.
§.§ Self-compile
While virally spreading the app may encounter different platforms, like a new Android version or hardware architecture, so the app needs to be able to re-compile for such a new environment.
In addition the app needs to solve dependencies for new environments that may be unavailable at the time the app needs to re-compile so it should contain all build tools and libraries that may be required during the entire Android build process.
Usually Android apps are developed on a computer with a software development kit (SDK) and integrated development environment (IDE).
To make the app fully function without the need for such a system requires integrating the SDK and entire build chain into the app.
Effectively it requires an embedded app factory consisting of all the required build tools, chained together.
Various levels of self-compilation can be discerned: 1. the source code 2. the 3rd party libraries 3. the build tools.
Compiling the non-Java tools and C/C++ libraries would require the native development kit (NDK) in addition to the SDK.
Because it is possible for the user to user to add and remove any content in the installation package the app is capable of serving as an alternative communication network.
§.§ Mutate
To increase the survival skills of the app in adverse conditions it must be able to be modified immediately when needed on the device itself.
Crowdsourcing would be a transparent and decentralized way of improving the survival skills of the app.
Being able to integrate other pieces of code, resources and new or modified libraries into the app makes it incredible versatile and adjustable to any situation.
Periodic mutation would prevent a single signature of the app to be made in order to block it.
User interface for mutation.
The user has full access to and control over the source code, but needs to be aware of possible compiler injection by the running instance of the code, especially in case the app was received from an untrusted source.
Leveraging the power of reflection in code the app could indicate and perhaps look for new components itself that are available in the mesh network and become even more resilient through continuous self-modeling.
The mutation is aimed at creating generations of the app with increased fitness.
§.§ Innocuous Components
Using innocuous hardware and software is especially important in case of a dissent network <cit.>.
A casual search or monitoring may not pick up on an app that looks like a calculator or is generally inconspicuous.
Also separate pieces of code may be innocuous on their own, so it is only a matter of putting these together.
A game could for example be embedded with a special launch pattern to open the encrypted content within.
§.§.§ Anonymity vs. Trust
Removing usage tracks and signing the application with the default debug key makes it hard to trace back to an individual user.
In case of illegal embedded content this could provide author, publisher and reader anonymity, but not server, document and query anonymity <cit.>.
The trustworthiness however of any true anonymous source is unverifiable, so transitive trust is desirable.
§.§.§ Plausible Innocence
To be able to use IDE features if the user is not a software developer may be tricky.
However this depends on the many different use cases for non-dissent purposes of the app, so the usage is not incriminating in and of itself.
As technology only amplifies human intent <cit.> having IDE features on your phone does not create political movement.
Like a hammer it can be used for good or evil and will find its role in the social network of people that is already established.
To minimize use for harm various attack vectors, like staining, need to be considered.
§ IMPLEMENTATION
We fully implemented our autonomous app design.
The source code of the SelfCompileApp is available on GitHub <cit.>.
As described in the previous chapter the Android SDK and platform library are required for building an installable app package (.apk).
Figure <ref> shows the implemented tool chain that is required to let the app self-compile on the device.
The Android Open Source Project (AOSP) made most of the build tools source code available online <cit.>.
All tools except the Android Asset Packaging Tool (aapt) are implemented in Java, which is convenient since Android apps can be written in Java as well.
Implemented build chain.
§.§ Android Asset Packaging Tool
The Android Asset Packaging Tool compiles the app manifest and resources like images, layout and language .xml files into one binary file and generates R.java.
This file contains all the IDs of the compiled resources that can be referenced from the app source code.
Aapt then packages the binary resources and all files in the assets folder into a preliminary .apk file.
Since aapt is not implemented in Java it requires the NDK to build it for any CPU instruction set(s) you want to support, which is part of the extended Android build process <cit.>.
The app JavaIDEdroid showed this works for Android 4.4 and older <cit.> and proved to be compatible with Android 5.0 by dropping the resources for ultra high density screens because of the used older version of aapt.
All required native shared libraries are bundled into one native shared library file (.so) and included in the app installer package.
Because of a missing dependency CynogenMod was used for all tested devices running an Android API level higher than 18.
To remove the NDK dependency completely only aapt would need to be implemented in Java.
This allows for easy viral spreading to new hardware platforms and make the app fully capable of mutating all the dependencies and build tools, without the need for the GNU Compiler Collection (GCC) which TerminalIDE uses <cit.>.
§.§ Java Compiler
The Eclipse Compiler for Java (ECJ) is chosen because it is implemented in Java, so it can be easily included as a library in the app but also makes the compiler capable of compiling itself.
Currently all build tools are just included in the app as a library, but could be build from source just like the app source code.
Build in progress.
§.§ Dalvik Executable
After compiling the source to Java bytecode it needs to be converted into Dalvik bytecode because Android used to run Java in a process virtual machine (VM) called Dalvik.
In Android 5.0 this VM was replaced by an application runtime environment called Android Runtime (ART) which is backwards compatible and still uses the exact same Dalvik executable files (.dex).
This conversion process with the dx tool proved to require a huge amount of RAM, or the tool contains a serious memory leak.
Fortunately it is possible to convert the libraries separately so they are cached and only converted again if the hash of the .jar file has changed.
The source code is converted into one .dex file as well and subsequently merged with with the converted libraries.
Both the dx tool and DexMerger tool are implemented in Java and available from AOSP <cit.>.
§.§ Android App Package Builder
All compiled resources, native libraries, compiled files and asset files are combined by the ApkBuilder tool that also implemented in Java and available from AOSP <cit.>.
This includes the Android.jar platform library, required for compiling the Java source code, and the Java KeyStore, required for the last link of the build chain.
The ApkBuilder has the capability of signing the .apk itself, but does not support Java KeyStore, so a user would not be able to use or create a private key store without additional tools.
The uncompiled resources, manifest, source code, libraries and pre-dexed libraries are zipped and also added to the .apk.
§.§ Package Signer
As long as the same package name is used and the .apk is signed with the same certificate as a previous installation of the app it can be seamlessly updated.
Otherwise the mutated app will be installed alongside the other version, or fail to install if the package name is the same but the certificates do not match.
The tool ZipSigner <cit.> is used to sign the .apk file that can use any Java KeyStore, one of which containing the default Android debug key is included in the app.
As long as side loading is enabled on the Android device, which is necessary to be able to install the .apk file at all, the use of a debug key does not pose any problems.
§.§ Wireless Transfer
To spread the app trough the air the wireless technology of Android Beam is used that leverages the ease of use of near field communication (NFC) for easy connection set up of a Bluetooth large file transfer.
The user only needs to hold the backs of two devices together and tap the screen while the app is running to transfer it.
Bluetooth is then enabled on both devices, the devices are paired and Bluetooth gets disabled again after the transfer is complete, all completely automatic.
§ PERFORMANCE ANALYSIS
We measured both the time to self-compile the app into an installable package (.apk) and also measured the time to wireless transfer the .apk to different devices like in a typical viral spreading scenario.
Figure <ref> shows the build time for 10 consecutive runs on four different devices.
Time to self-compile the app into an installable package.
Very notable is the significant difference in speed between the Galaxy Nexus and the other devices.
Besides the different hardware capabilities, this can be partially explained by the difference in how the Dalvik bytecode is executed.
As described in the previous section, Android used to run Java in a process virtual machine (VM) called Dalvik that performed just-in-time (JIT) compilation.
In Android 5.0 this VM was replaced by an application runtime environment called Android Runtime (ART) that performs ahead-of-time (AOT) compilation.
Also very notable was the apparent thermal throttling of the Nexus 5 device.
After a period of letting it cool down the results returned to the level of the first couple measurements.
The wireless transfer speed of four different Android Beam and Bluetooth large file transfer enabled devices was tested by sending over the .apk of 30.1MB.
For each pair of the four devices the averages of 3 consecutive transfers are shown in table <ref>.
Two cells are left blank because only one Nexus 5 and one Nexus 10 device were at our disposal.
Notable is the slow transfer speed of only the Nexus 10.
From:To: Galaxy Nexus Nexus 5 Nexus 6 Nexus 10
Galaxy Nexus 227 221 209 419
Nexus 5 211 149 360
Nexus 6 198 147 139 357
Nexus 10 409 400 359
Average transfer time in seconds of 3 consecutive transfers per device pair.
§ CONCLUSION & FUTURE WORK
Our work shows that it is possible for an Android app to be autonomous in recompiling, mutating and virally spreading over multiple hosts all without the need for a host computer, root permissions and an app store.
Therefore our work provides a robust and resilient network and platform to overcome disruptions due to natural disasters or made made interference.
Since the user can add and remove any content in the app a social media experience does no longer have to be hampered by Internet kill switches as it is now.
The app currently can spread among Android devices and as a next step could be considered cross-compiling for the Android, iOS and Windows Phone platforms.
Continuous self-modeling through code reflection and higher levels of self-compilation, by recompiling the 3rd party libraries and build tools themselves, are left for future work.
A point for consideration is the minimization of the use for harm of the app, and the risk for harm by use of the app.
Various attack vectors could be considered to improve the fitness of the app and lower the risk of usage in a dissent network use case.
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1511.00329
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In order to better manage the premiums and encourage safe driving, many commercial insurance companies (e.g., Geico, Progressive) are providing options for their customers to install sensors on their vehicles which collect individual vehicle's traveling data. The driver's insurance is linked to his/her driving behavior. At the other end, through analyzing the historical traveling data from a large number of vehicles, the insurance company could build a classifier to predict a new driver's driving style: aggressive or defensive. However, collection of such vehicle traveling data explicitly breaches the drivers' personal privacy. To tackle such privacy concerns, this paper presents a privacy-preserving driving style recognition technique to securely predict aggressive and defensive drivers for the insurance company without compromising the privacy of all the participating parties. The insurance company cannot learn any private information from the vehicles, and vice-versa. Finally, the effectiveness and efficiency of the privacy-preserving driving style recognition technique are validated with experimental results.
§ INTRODUCTION
One of the engineering innovations that will have a transformative impact on the society over the next few decades is the Connected Vehicles initiative <cit.>. The connectivity is enabled through the use of wireless communication over a dedicated spectrum to create local ad hoc networks of vehicles that are then able to communicate with other vehicles in its neighborhood, as well as with traffic infrastructure. The communication network provides opportunity for mobile devices or inertial sensors mounted in the vehicles to collect data regarding vehicle travel, driver behavior, and location data <cit.>. This data can be processed for implementing automatic collision avoidance, optimizing traffic in real time, and planning for infrastructure needs. While, these data sets can improve safety, reduce emissions, and reduce driver wait time, they can also be used to uncover sensitive personal information. For instance, location data can be used in criminal investigations, driver behavior can be used to determine fault in accidents, and a person's lifestyle choices can be revealed by processing the data with mapping software. The privacy challenge needs to be addressed with a careful balance between the utility of the collected data and the protection of personal information as well as corporate proprietary information.
The insurance industry can leverage vehicle travel data to determine their premiums by correlating driver behavior and accidents. For instance, vehicle data can be used to create a classification system that can classify drivers as defensive or aggressive <cit.>. In order to better manage their premiums and also encourage safe driving, many commercial insurance companies (e.g., GEICO and Progressive) are providing options for customers to install sensors on their vehicles that can collect the vehicle's operational parameters (such as braking, acceleration, speed, etc.) <cit.>. Their insurance is linked to their driving behavior. At the other end, through analyzing the historical travel data from a large number of vehicles, the insurance company could build a classifier to determine the thresholds for aggressive vs. defensive driving. This would allow the companies to better understand the risks associated with each of their drivers and be able to tailor premiums based on the risks. Insurance companies could also give drivers “report cards” to help drivers better understand their own driving habits. However, collection of the vehicle traveling data to make this happen explicitly breaches the drivers' personal privacy. For instance, the accelerating & braking data, trips, turning behaviors, risky driving hours, and even the geographical locations the driver can be inferred from the data violating their personal privacy. The goal of this research is to securely build a classifier that accurately predicts any given driver's driving style (aggressive or defensive) without compromising any participating party's privacy.
From the perspective of both business and the public sector, driver behavior or vehicle traveling data, in aggregate, has a variety of uses, including the ability to provide a classification, such as defensive or aggressive driving style, as well as the ability to predict an individual's driving style based on criteria obtained from the analysis of such data (viz. classification) <cit.>. For instance, in order to ensure safe driving, many commercial insurance companies (e.g., Geico, Progressive) have begun to request their customers to install sensors on their vehicles which periodically collect individual vehicle's traveling data <cit.>. If the customer drives defensively, a rewarding discounted premium rate of the insurance would be offered to them. To this end, the insurance company normally analyzes a large number of vehicles' historical traveling data and builds a classifier that can recognize any new driver's driving style: aggressive or defensive.
However, such ubiquitously collected vehicle traveling data explicitly breaches the drivers' personal privacy, such as the accelerating & braking data, trips, turning behaviors, risky driving hours, and even the geographical locations the driver has visited. Is it possible to build a classifier that accurately predicts any given driver's driving style without accessing all the drivers' sensitive vehicle traveling data?
There are two phases in driving style recognition: (1) building a classifier with vehicles' historical traveling data; (2) predicting the driving style for new drivers using the classifier.
* In phase (1), some drivers' historical vehicle traveling data
are analyzed to train the classifier. As part of this process, each record has a class label applied to it, the characteristics of an aggressively labeled driver would be based on the records of individuals who had received speeding tickets or been involved in an accident, while a defensive driver would be an individual who had not been in such situations.
Each driver privately holds a record of his/her vehicle traveling data with the attributes such as average acceleration ($m/s^2$), average deceleration in the braking events ($m/s^2$), average turning (degrees), # of risky driving hours, and # of trips <cit.>. In reality, both of the driver and insurance company know the driver's class label in the training data from the traffic tickets, reported accidents, etc. Therefore, such multiparty classifier training process has a hybrid scenario of data partitions: first, all the drivers' records are horizontally partitioned – each driver privately holds a record, including the class label <cit.>; second, the overall data (attributes) is also vertically partitioned <cit.> into two shares – the insurance company holds a share (the attribute class label) while all the drivers jointly holds both shares, including the class label. Finally, the output of phase (1) is a trained classifier, privately held by the insurance company.
* In phase (2), the insurance company privately holds its classifier whereas the new driver privately holds his/her vehicle traveling data. Since the two parties privately own different attributes, it is a vertical partition case in each individual driving style recognition. Then, they jointly predict the class label (aggressive or defensive) with their private inputs.
Since the data partitions in the previous two phases are mixed with both horizontal and vertical partitions, the prior works on privacy-preserving classification <cit.> are not directly applicable to this research problem.
§.§ Research Contributions
In this paper, we develop privacy-preserving techniques for two phases of driving style recognition based on decision tree induction <cit.>. More specifically, in phase (1), each driver only knows its vehicle traveling data (a record) and its class label; the insurance company only knows all the drivers' class labels, and the final output – a decision tree. In phase (2), each new driver only knows its vehicle traveling data (a record) and learns nothing but the driving style recognition result; the insurance company only knows its input (the decision tree) and also learns nothing but the driving style recognition result. Thus, the main contributions are summarized as below:
* We propose two secure communication protocols under secure multiparty computation (SMC) <cit.> to implement the two phases of driving style recognition (based on decision trees) for the participating parties without private information disclosure.
* We analyze the privacy risks for all the participating parties in the secure communication protocols for both classifier training and driving style prediction.
* We experimentally validate the performance of our proposed approaches on the synthetic datasets generated following the format of data collected in <cit.>.
The remainder of this paper is organized as follows. We first briefly review the related work in Section <ref>. Then, we present the algorithms and demonstrate the experimental results in Sections <ref> and <ref> respectively. Finally, we conclude this paper and discuss the future work in Section
§ RELATED WORK
Vehicle traveling data has been commonly collected for analysis in Intelligent Transportation Systems (ITS). Ly et al. <cit.> demonstrated a methodology to collect vehicle data (e.g., accelerating, braking, turning) using inertial sensors. Hull et al. <cit.> collected the vehicle data with his CarTel system. There has also been expansive work into adaptive cruise control, which uses prediction algorithms to adapt to the road based on curve patterns <cit.>. Moreover, Rass et al. <cit.> formulated a system to provide feedback on the driving habits. Also, other research has focused on driver modeling & evaluation <cit.> and maneuver recognition <cit.>. While there has been a variety of ventures into many ITS applications, only a surprising few of these tackled such arising privacy concerns, including: Hoh et al. <cit.> proposed using the virtual trip lines to monitor traffic conditions while preserving privacy. Checkoway et al. <cit.> examined the attack vectors for hackers to infiltrate vehicle through its Electronic Control Unit. Han et al. evaluated authentication methods for securely integrating mobile devices in vehicular networks <cit.>. To the best of our knowledge, privacy risks in driving style recognition have not been systematically studied.
Privacy-preserving schemes are generally developed based on data transformation and/or secure computation. The former one transforms the original data to a privacy-compliant format and minimizes the utility loss in the process of data transformation (e.g., k-anonymity <cit.>, differential privacy <cit.>). The latter makes two or more parties jointly compute a function possible without revealing private data to each other (formally defined as Secure Multiparty Computation <cit.>). The function can be as simple as sum or as complex as big data analysis/mining <cit.>. Several researchers have addressed the privacy concerns in other contexts, such as classification <cit.>, location based services <cit.>,
search engine queries <cit.>, scheduling <cit.>, transportation <cit.>, and smart grid <cit.>. Following a similar line of research, we develop a privacy-preserving driving style recognition technique that can analyze the vehicle traveling data (e.g., an insurance company) without breaching participating parties' privacy.
§ SECURE COMMUNICATION PROTOCOLS
In this paper, we assume that the adversaries are semi-honest. The semi-honest model in Secure Multiparty Computation (SMC) <cit.> defines that the adversaries are honest to follow the a given protocol, but are curious to infer private information from each other. Two secure communication protocols will be given for two phases of driving style recognition in Sections <ref> and <ref> respectively.
§.§ Phase (1): Privacy Preserving Classifier Training
In phase (1) of driving style recognition, a set of drivers and the insurance company jointly derive a decision tree based on the drivers' vehicle traveling data and class labels (aggressive or defensive). In this scenario, we have:
* Every driver's class label in the training data is known by both the driver and the insurance company. Since the “Aggressive” class label in the historical data can be derived from the “traffic tickets or accidents” which is indeed known by both parties in real world.
* All drivers and the insurance company know the name of every attribute in the dataset, such as Acceleration Events (#), Average Acceleration ($m/s^2$), Braking Events (#), and Average Braking ($m/s^2$). Insurance company initializes the attributes in the sensors which are also known to the drivers in real world.
* Every attribute has a threshold to divide the values into two categories: “less than the threshold” or “no less than the threshold”. For example, the average deceleration when braking can be less than a threshold $4.9 m/s^2$ or no less than a threshold. Such split is used to determine the branches of a node on the decision tree. We assume that every attribute's threshold is known to all the drivers and the insurance company (e.g., thresholds of speed/mileage/acceleration/braking can be available as public information).
Note that our privacy-preserving decision tree training is extended from the ID3 Algorithm <cit.>, which iteratively finds the best attribute to split values based on its threshold (as the current node of the tree) by comparing the entropy or information gain of all the remaining attributes in the classification results. In this paper, we choose the entropy as the measure of uncertainty in the threshold based split $H=-\sum_{x\in X}p(x)log p(x)$ where $X$ is the set of classes (“Aggressive” or “Defensive”) and $p(x)$ is the proportion of the number of elements in class $x$ to the number of elements in all the data. Therefore, in a distributed manner, each driver/vehicle owns a record and they should securely sum their shares for every $p(x)$. In this section, we first present the Secure Sum algorithm which is iteratively invoked by the protocol of privacy-preserving classifier training.
§.§.§ Secure Sum
The secure sum algorithm is developed using Homomorphic Cryptosystem (e.g., Paillier <cit.>). It begins by having the insurance company $I$ generate a key pair: a public key $pk$ and a private key $sk$. The insurance company (party $I$) then sends the public key $pk$ to all $m$ drivers, denoted as $D_1,\dots, D_m$. $D_1$ then encrypts its share and passes along the encrypted data to the next driver (w.l.o.g, say $D_2$). $D_2$ then computes their encrypted sum to the previous number and this is passed through all the $m$ drivers in the circuit. After this, the encrypted sum is passed back to the insurance company who decrypts it with the private key $sk$ to get the sum. As shown in Algorithm <ref>, all the parties' data remains private while only allowing the insurance company to obtain the sum.
$m$ drivers' share of $p(x)$: $p(x)_1,\dots, p(x)_m$
insurance company $I$ learns $p(x)=\sum_{j=1}^mp(x)_j$
$I$ generates a pair of public-private key $(pk, sk)$ and sends the public key $pk$ to $P_1,\dots,P_n$
$i=1,\dots, m$
$D_i$ encrypts $p(x)_i$ using $pk$ to get $E[p(x)_i]$, and computes $E[\sum_{j=1}^ip(x)_j]=\prod_{j=1}^{i-1}E[p(x)_j]*E[p(x)_i]$ (ensured by Homomorphic Property <cit.>)
$D_i$ sends $E[\sum_{j=1}^ip(x)_j]$ to the next party $P_{i+1}$ (if $i=m$, the next party is $I$)
$I$ decrypts $E[\sum_{j=1}^mp(x)_j]$ with its private key $sk$ to obtain $\sum_{j=1}^mp(x)_j$
Secure Sum
§.§.§ Secure Communication Protocol
In the secure communication protocol for classifier training, the insurance company $I$ repeatedly finds the best attribute (that has the smallest entropy $H$; viz. lowest uncertainty) as the current node during the construction of the tree.
Note that the key value $p(x)$ in the entropy $H=-\sum_{\forall x\in X}p(x)\log p(x)$ is split into $m$ shares $p(x)_1,\dots, p(x)_m$, held by $m$ vehicles respectively. In each iteration, each of the remaining attributes' $p(x)$ is securely summed by all $m$ vehicles and the insurance company $I$ (Algorithm <ref>). This use of the secure sum ensures that $I$ is only ever able to learn the $p(x)$ of each attribute while preserving the privacy of each of the $m$ vehicles.
In turn, the only information that is learned by each of $m$ vehicles is the $pk$ that is sent by the insurance company $I$. The purpose of this approach is to provide a mechanism to securely sum the data needed to compute the entropy values for each attribute while obfuscating all vehicle identifying characteristics in $p(x)_1,\dots,p(x)_m$. In addition, the use of the secure sum ensures that no other parties will be able to uncover the identity of any vehicle or the vehicle's collected data.
In the final phase of this algorithm, $I$ locally computes the entropy $H$ of all the remaining attributes, and selects the attribute with the smallest entropy value as the current node. This process is performed iteratively until all the leaf nodes of the decision tree have $p(x)=1$. The details of the secure communication protocol is given in Algorithm <ref>
Hong's comments: you should describe it in a distributed manner...Now one party does what, another party does what.... $p(x)$ should be defined here (The proportion
of the number of elements in class x to the number of elements in the dataset). $p(x)$ is split into $n$ shares. In every iteration, secure sum them, then party $I$ gets a securely summed result.
compute entropy $H=-\sum_{i=1}p(x)log p(x)$ and information gain $IG=H-\sum_{i=1}^n p(x)\log p(x)$ for all the remaining attributes and find the best one
based on the smallest entropy or largest information gain...remember that only exactly one measure (not both) is required...we choose information gain... entropy should be
computed before computing information gain
$m$ is the total number of drivers/vehicles
Decision tree $T$
existing a leaf node's $p(x)!=1$
$i = 1, \dots, m$
Driver $D_i$ computes the share of $p(x)$ for all the remaining attributes
Securely sum shares of $p(x)$ (Algorithm <ref>) for all the remaining attributes (only party $I$ knows that)
$I$ computes entropy $H=-\sum_{i=1}p(x)log p(x)$ for all the remaining attributes
$I$ selects the best attribute (smallest entropy) as the root or leaf node (leaf node is added along the branch of the tree with $p(x)!=1$)
Privacy Preserving Classifier Training
The final product of this algorithm consists of a decision tree which contains a set of the attributes from the dataset. This tree outlines the pathways which represent determinable outcomes based on any given driver's collection of vehicle traveling data. This tree will be used
to help securely predict a given new driver as either aggressive or defensive.
Please discuss the following in details: while securely training the decision tree, insurance company $I$ only learns every attribute's $p(x)$ in each iteration. All $n$ vehicle only learns the public key $pk$ generated by the insurance company $I$. No private information is revealed to other parties.
§.§ Phase (2): Privacy-preserving Driving Style Recognition
The insurance company has the means to classify driving style/behavior for given drivers, but has an equal interest in keeping its decision tree $T$ private.
To this end, we develop another secure communication protocol to predict the driving styles without sharing information between the insurance company and the new driver, detailed below.
is defined as a path in the decision tree $T$ that leads to the class of aggressive driving.
Letting $|T|$ be the number of aggressive paths in $T$, with all the thresholds of the attributes in $T$, each aggressive path can be represented by an $n$-digit binary vector:
\begin{equation}
\forall i\in [1,|T|], \vec{c_{i}}=[c_{i1},c_{i2},\dots, c_{in}]
\end{equation}
where $\forall c_{ij}\in \{0,1\}$. Note that, in the decision tree $T$, $c_{ij}=1$ means the child value of the $j^{th}$ attribute (out of all $n$ attributes in total) along the aggressive path $\vec{c_i}$ exceeds the threshold value; otherwise, $c_{ij}=0$. For instance, in Figure <ref>, there are six attributes in total used for training decision tree, and four of them are utilized to build the tree $T$ (as shown in Figure <ref>). Two paths in $T$: “# of acceleration events $< 110$ $\longrightarrow$ # of braking events $\geq 150$ $\longrightarrow$ average braking ($m/s^2$) $\geq 4.9 m/s^2$” and “# of acceleration Events $\geq 110$ $\longrightarrow$ high risk driving hours $\geq 80$” can lead to “Aggressive”. Then, they can be represented as two binary vectors $(0,0,1,1,0,0)$ and $(1,0,0,0,0,1)$ respectively. Other paths in $T$ are simply considered as “Defensive Paths” which can be also represented as $n$-digit binary vector in a similar way.
[Decision Tree]
[Aggressive and Defensive Paths]
[Optional caption for list
of figures]An Example of Decision Tree and Aggressive/Defensive Paths
The insurance company $I$ now owns the decision tree $T$, the number of aggressive paths $|T|$, as well as the path(s) that identify aggressive driving behavior $\vec{c_{|T|}}$. Since the insurance company privately possesses such information, the aggressive paths must be encrypted for computation. Specifically, $I$ generates a public/private key pair $(pk,sk)$ based on the Paillier cryptosystem <cit.>. $I$ encrypts the aggressive paths from the decision tree and the total number of aggressive paths $|T|$ as well as the inner products of all the aggressive paths (which is the total number of “1” in each binary vector) using the public key $pk$ such that: $E(\vec{c_1})$, …, $E(\vec{c_{|T|}})$ and $E(\vec{c_1} \cdot \vec{c_1})$, …, $E(\vec{c_{|T|}} \cdot \vec{c_{|T|}})$ are both then transmitted along with $pk$ to the new driver/vehicle $D$.
At the other end, similar to the aggressive/defensive paths, the new driver $D$ privately holds a:
is an $n$-digit binary vector: $\vec{v}=[v_1,v_2,\dots, v_n]$ with $0$ representing the attributes with values below the threshold, and otherwise $1$.
After receiving $pk$ and the ciphertexts from the insurance company $I$, the driver/vehicle $D$ then securely computes the following scalar products with the ciphertexts and its vector $\vec{v}$:
\begin{equation}
\forall i\in [1,|T|], E(\vec{c_i}\cdot \vec{v}) =
\label{eq:scalar}
\end{equation}
Then, driver/vehicle $D$ encrypts $\vec{v}$ and computes:
\begin{equation}
\forall i\in[1,|T|], E(\vec{c_i}\cdot \vec{v}-\vec{c_i}\cdot \vec{c_i}) = \frac{E(\vec{c_i}\cdot \vec{v})}{E(\vec{c_i}\cdot \vec{c_i})}
\label{eq:onepath}
\end{equation}
If any of $\forall i\in[1,|T|], \vec{c_i}\cdot \vec{v}-\vec{c_i}\cdot \vec{c_i}$ equals $0$, the vehicle traveling vector $\vec{v}$ would match the corresponding aggressive path, and the driver $D$ is predicted as an aggressive driver. If all of $\forall i\in[1,|T|], \vec{c_i}\cdot \vec{v}-\vec{c_i}\cdot \vec{c_i}$ are non zero, the vehicle traveling vector $\vec{v}$ would not match any aggressive path, and the driver $D$ can be predicted as a defensive driver. To minimize information disclosure, the driver permutes all the ciphertexts $\forall i\in[1,|T|], E(\vec{c_i}\cdot \vec{v}-\vec{c_i}\cdot \vec{c_i})$ and send them to the insurance company $I$ one by one, and $I$ decrypts a ciphertext immediately. As long as a $0$ is found, conclude $D$ as an aggressive driver and terminate the protocol (no more ciphertext will be sent). If no $0$ is found after examining all $|T|$ results, then conclude $D$ as an defensive driver. $I$ can share the classification result to $D$ if necessary.
Also, needs more details for describing why one of $(c_1\cdot v-c_1\cdot c_1)$, …, $(c_{|T|}\cdot v-c_{|T|}\cdot c_{|T|})=0$ leads to aggressive
...and all non zero leads to defensive. Describes why number of “1” in the aggressive path equals the scalar product $c_1\cdot v$ make $v$ aggressive.You can
use an example to illustrate it
Insurance company $I$ and a new driver $D$;
$D$'s vehicle traveling vector $\vec{v}$;
$T$ represents the complete decision tree;
The number of aggressive paths $|T|$;
All the aggressive paths $\vec{c_1},\dots, \vec{c_{|T|}}$
The new driver is aggressive or not
A random nonce is generated for every single encryption
Party $I$ generates a public/private key pair based on Paillier Cryptosystem $(pk, sk)$
$I$ encrypts $\vec{c_1},\dots, \vec{c_{|T|}}$, $|T|$, and inner products $\vec{c_1}\cdot \vec{c_1}$, …, $\vec{c_{|T|}}\cdot \vec{c_{|T|}}$ using $pk$ to obtain
$E(\vec{c_1}),\dots,E(\vec{c_{|T|}})$, $E(\vec{c_1}\cdot \vec{c_1})$, …, $E(\vec{c_{|T|}}\cdot \vec{c_{|T|}})$, and sends the ciphertexts and $pk$ to the driver $D$
$D$ encrypts $\vec{v}$ and computes the ciphertexts of $|T|$ scalar products $E(\vec{c_1}\cdot \vec{v})$, …, $E(\vec{c_{|T|}}\cdot \vec{v})$ using Equation <ref>
$V$ computes $E(\vec{c_1}\cdot \vec{v}- \vec{c_1}\cdot \vec{c_1})$, …, $E(\vec{c_{|T|}}\cdot \vec{v}-\vec{c_{|T|}}\cdot \vec{c_{|T|}})$ using Equation <ref> and permutes them
$i=1,\dots, |T|$
$D$ sends the permuted ciphertext $E(\vec{c_i}\cdot \vec{v}- \vec{c_i}\cdot \vec{c_i})$ to $I$
$I$ decrypts the current $E(\vec{c_i}\cdot \vec{v}- \vec{c_i}\cdot \vec{c_i})$ using its private key $sk$ to get $\vec{c_i}\cdot \vec{v}- \vec{c_i}\cdot \vec{c_i}$
$\vec{c_i}\cdot \vec{v}- \vec{c_i}\cdot \vec{c_i}=0$
$D$ is an aggressive driver and terminate the algorithm
$\forall i\in[1,|T|], \vec{c_i}\cdot \vec{v}- \vec{c_i}\cdot \vec{c_i}\ne 0$
$D$ is a defensive driver
Privacy Preserving Driving Style Recognition
Upon completion, each driver has the ability to access his or her computed rating of either aggressive or defensive. $I$ has sole possession of the decision tree $T$ developed from training data and is the only party which is able to view all of the pathways which lead to an aggressive classification. The driver $D$, on the other hand, is the only party able to access $\vec{v}$, keeping specific driving behavioral data private from $I$. Ultimately, both parties will have access to the computed classification result. However, the insurance company can only infer some trivial information from $D$ such as how many values in $\vec{v}$ has met or exceeded the corresponding attributes' threshold.
§ EXPERIMENTAL RESULTS
We implemented application of privacy-preserving driving style recognition in Java on a PC with AMD FX-4350 4.55 GHZ CPU and 16G RAM.
Synthetic datasets are generated falling into a similar value range as <cit.>.
7500 drivers' vehicle traveling data are generated for training classifier (with class labels in the training data) while 2500 drivers' traveling data are generated for predicting the driving style by the classifier (without class labels in the dataset). These records
simulated driving activity over a 6-month period and featured 9 attributes:
* total number of trips taken
* total mileage driven
* the number of acceleration events
* the average amount of acceleration
* the number of braking events
* the average deceleration when braking
* the average number of degrees turned
* the total number of hard braking events
* the hours driven in the highest risk time (0 to 4 AM)
The cryptographic keys are generated with lengths of 512 and 1024-bit using Paillier Homomorphic Cryptosystem <cit.>.
For examining the computational costs, we tested the overall runtime of the protocols including encryption, computation and decryption. For examining the communication overheads, we tested the overall bandwidth consumption, which is equivalent to the overall size of the ciphertexts and plaintexts to be transmitted among all the distributed parties in the protocols. Due to the novelty of the data partition scenario and protocols devised for driving style prediction, there are no available benchmarks to compare against.
§.§ Classifier Training
[Computational Cost]
[Communication Overheads]
[Optional caption for list
of figures]Privacy Preserving Driving Style Classifier Training (Algorithm <ref>)
Algorithm <ref> securely trains a decision tree out of the privately held distributed records.
We conducted a group of experiments for classifier training (Algorithm <ref>) using $1000$, $2000$, $3000$, $4000$ and $5000$ drivers' traveling data featuring $9$ attributes. In each group of experiments, we tested the runtime for the encryption and decryption as well as the size of all the ciphertexts.
As shown in Figure <ref> and <ref>, both the computational cost and the communication overheads present a linear increase trend as the number of vehicles increases.
§.§ Driving Style Recognition
[Computational Cost]
[Communication Overheads]
[Optional caption for list
of figures]Privacy Preserving Driving Style Recognition (Algorithm <ref>)
Algorithm <ref> securely predicts the class for new individual drivers. Since the algorithm runs driving style recognition for all the drivers individually, we tested its computational costs and communication overheads again for a predicting a single driver's class. A group of experiments is conducted with a varying number of aggressive paths in the decision tree: $|T|=1,\dots,10$. As shown in Figure <ref> and <ref>, the costs increase slowly as the number of aggressive paths $|T|$ increases.
§ CONCLUSION AND FUTURE WORK
In this paper, we have developed two secure communication protocols to tackle the privacy concerns in the two phases of driving style recognition among the insurance company and various vehicles. Participating parties can jointly train a decision tree based on the vehicles' historical traveling data without compromising their privacy. The insurance company can also use its decision tree to securely predict the driving style (aggressive or defensive) of any given driver with limited disclosure. We have also experimentally validated the performance of our proposed secure communication protocols.
In the future, we have several directions to extend this work. First, we assume a semi-honest adversarial model in this paper such that every participant will follow the outlined secure communication protocols.
In the real world, one or multiple parties may become more malicious to corrupt the protocol for additional payoff, or even collude with each other to breach privacy or jeopardize the utility of the protocol. We will explore efficient solutions to address the security and privacy concerns for multiparty driving style recognition in malicious adversarial model. Second, maybe more than one entities (e.g., multiple insurance companies and police department) would like to collaboratively predict the driving style of the drivers with their private inputs. Introducing more parties into this problem will influence the data partition scenario, and then the required secure communication protocols might be thoroughly different from the current ones. Third, in the future, vehicles' traveling data used for driving style recognition might be in real-time format rather than the historical aggregated format. In such scenario, the challenges on efficiency and bandwidth should be resolved.
* Malicious model
One of the assumptions acted upon in this paper is that every participant in this data
exchange is invested in the privacy preservation aspect and will follow the outlined
protocols, i.e. the semi-honest model. Should one of the parties instead become
interested in divining one or more data aspects that the party is not otherwise privy
to may jeopardize the utility of the protocol. Investigation into the utility of stronger
encryption or other alternatives should be examined with an eye toward maintaining
For instance, in the protocol outlined in this paper, the decision tree paths associated with aggressive
driving are encrypted and shared with all vehicles connected to the insurance company in order to
determine whether or not the driver's behavior can be classified as aggressive. While the encryption
provides a level of security, even greater security may be achieved through not sharing the
decision tree pathways and instead performing the computations remotely on the insurance
company's hardware, removing the possibility that a user or cluster of users can identify the
classification criteria.
* Multiple insurance companies with multiple drivers
The introduction of multiple parties will change the many-to-one, drivers to rating operation, topology
to a manytomany
increase the amount of computation necessary to ensure privacy protection. Furthermore, there will
be additional computational overhead for bridging additional data sources, requiring a deeper look into
the efficiency of this approach.
* Availability of real-time data
The inclusion of granular, street level data from data capturing programas will highlight current methods of data
collection along with privacy preservation techniques, and will provide run-time benchmarks against which
future methods may be tested.
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1511.00353
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We consider online power control for an energy harvesting system with random i.i.d. energy arrivals and a finite size battery. We propose a simple online power control policy for this channel that requires minimal information regarding the distribution of the energy arrivals and prove that it is universally near-optimal for all parameter values. In particular, the policy depends on the distribution of the energy arrival process only through its mean and it achieves the optimal long-term average throughput of the channel within both constant additive and multiplicative gaps. Existing heuristics for online power control fail to achieve such universal performance. This result also allows us to approximate the long-term average throughput of the system with a simple formula, which sheds some light on the qualitative behavior of the throughput, namely how it depends on the distribution of the energy arrivals and the size of the battery.
§ INTRODUCTION
Recent advances in energy harvesting technologies enable
wireless devices to harvest the energy they need for communication from the natural
resources in their environment. This development opens
the exciting possibility to build wireless networks that are
self-powered, self-sustainable and which have lifetimes limited
by their hardware and not the size of their batteries.
Communication with such wireless devices requires the design of good power control policies that can maximize throughput under random energy availability. In particular, available energy should not be consumed
too fast, or transmission can be interrupted in the future due to an
energy outage; on the other hand, if the energy consumption
is too slow, it can result in the wasting of the harvested
energy and missed recharging opportunities in the future due
to an overflow in the battery capacity.
The problem of power control for energy harvesting communication has received significant interest in the recent literature <cit.>.
In the offline case, when future energy arrivals are known ahead of time, the problem has an explicit solution <cit.>. The optimal policy keeps energy consumption as constant as possible over time while ensuring no energy wasting due to an overflow in the battery capacity. The more interesting case is the online scenario where future energy arrivals are random and unknown. In this case, the problem can be modeled as a Markov Decision Process (MDP) and solved numerically using dynamic programming <cit.>. However, this numerical approach has several shortcomings. First, although there exist numerical methods to find a solution which is arbitrarily close to optimal, such
as value iteration and policy iteration, these methods rely
on quantization of the state space and the action space.
Specifically, the computational load of each iteration grows
as the cube of the number of quantized states and/or actions and may not be suitable for sensor nodes with limited computational capabilities.
Second, the solution depends on the exact model for the statistical distribution of the
energy arrival process; this may be hard to obtain in practical
scenarios and may require recomputing the dynamic programming solution periodically to track changes in the harvesting process. Finally, the numerical solution does not provide much insight on the structure of the optimal online power control policy and the
qualitative behavior of the resultant throughput, namely how it
varies with the parameters of the problem. This kind of insight
can be critical for design considerations, such as choosing the
size of the battery to employ at the transmitter.
Due to these limitations of the numerical approach, there has been significant effort in the recent literature to develop simple heuristic online policies. These policies come either with no guarantees or only asymptotic guarantees on optimality <cit.>. For example, two natural heuristic policies, the variations of which have been widely studied in the literature, are the greedy policy and what we will refer to as the constant policy. In the greedy policy, the transmitter instantaneously uses all the harvested energy in each time slot. This greedy approach ensures that energy is never wasted due to an overflow in the battery capacity and it naturally becomes optimal in the limit when the harvested energy (or SNR) goes to zero, since in the low-SNR limit the capacity becomes proportional to the total energy transferred to the receiver.
The constant policy on the other hand aims to keep power allocation as constant as possible over time, for example by always allocating energy equal to the mean $\mu$ of the energy harvesting process as long as there is sufficient energy in the battery.
It is easy to see that this policy becomes optimal in the limit when the battery size goes to infinity, since an infinite battery can allow to average out the randomness in the harvesting process and therefore allocate energy equal to $\mu$ in almost all time-slots. However, such asymptotic results do not provide any insights on the gap to optimality for given finite parameter values. In particular, given a certain distribution of the energy harvesting process and a given finite size of the battery, these two policies can be arbitrarily away from optimality and it is not even clear which one of these two structurally very different policies would be a better choice for the given problem.
The goal of this paper is to address this problem. Instead of seeking policies that become asymptotically optimal in one limit or the other, we look for policies that are provably close to optimal across all parameter regimes and any distribution of the energy arrivals. In particular, we seek policies that achieve the optimal long-term average throughput of the system simultaneously within a constant multiplicative factor and a constant additive gap for all parameter values and distributions of the energy arrivals. This more stringent requirement ensures that these policies are universally near-optimal in the sense that their gap to optimality remains bounded across all parameter ranges. It would be moreover desirable for these policies to have minimal dependence on the distribution of the energy arrivals, for example depend only on the mean energy arrival rate. This would enable one to apply them to any given problem with arbitrary parameter values, without even knowing the exact distribution of the energy arrivals, while she/he would be assured to achieve a performance that is very close to the one achieved by an optimal policy specifically optimized for the given problem, in particular the exact distribution of the energy arrivals. While the two policies discussed above can be applied with minimal knowledge of the energy arrival distribution (the greedy policy does not require any information about the distribution while the constant policy is based only on the knowledge of the mean), it is easy to show that neither of them is universally near-optimal; there are parameter regimes and distributions for which their gap to optimality grows unboundedly.
The main result of this paper is to propose a simple novel online power control policy which depends on the harvesting process only through its mean: at each time-slot, the policy uses a constant fraction of the energy available in the battery where the fraction is chosen as the ratio of the mean of the energy arrival distribution and the battery size. This policy is structurally very different from the two policies discussed above and may appear a priori counter-intuitive. We show that it is naturally motivated by the case where the energy arrival process is i.i.d. Bernoulli, in which case the optimal online power control policy can be explicitly characterized. We then establish the near-optimality of this policy for any i.i.d. harvesting process and any size of the battery. In particular, we show that this policy achieves the optimal long-term average throughput of the system simultaneously within a constant multiplicative factor and a constant additive gap for all parameter values. This implies that this policy can be applied under any i.i.d. harvesting process, without even knowing the statistical distribution of the energy arrivals. The multiplicative and additive approximations guarantee that it will perform close to the best strategy optimized for the exact distribution of the energy arrivals across all parameter values (both in the high- and the low-SNR regimes). The main ingredient of our proof is to show that for the proposed policy Bernoulli is the worst case distribution for the energy arrivals. Therefore, the performance of the scheme under a Bernoulli distribution provides a lower bound on its performance under any i.i.d. process. In this sense, our policy can be thought of as building on the insights from the worst-case scenario, hence performs well in the worst-case sense, while previous heuristics can be thought of as building on the insights from the best case scenario, i.e. when energy arrivals are constant equal to $\mu$.
This result also leads to a simple approximation of the optimal long-term average throughput of an AWGN energy harvesting channel. In particular, we show that within a constant gap, the average throughput is given by
\[
\Theta\approx\frac{1}{2}\log\left(1+\mathbb{E}[\min\{E_t,\bar{B}\}]\right),
\]
where $E_t$ denotes the energy arrival process and $\bar{B}$ is the battery capacity. This shows that a battery large enough to capture the maximal energy arrival over a single time-slot is sufficient to approximately achieve the AWGN capacity.
§ SYSTEM MODEL
We begin by introducing the notation used throughout the paper.
Let $\mathbb{E}[\,\cdot\,]$ denote expectation.
For $m\leq n$, denote $X_m^n=(X_m,X_{m+1},\ldots,X_{n-1},X_n)$ and $X^n=X_1^n$.
All logarithms are to base 2, and $\ln$ will denote log to base $e$.
We consider a point-to-point single user channel with additive white Gaussian noise (AWGN). We assume a quasi-static fading channel, in which the channel coefficient remains constant throughout the entire transmission time.
The system operation is slotted, i.e. time is discrete ($t=1,2,\ldots$).
At time $t$, the received signal is $y_t=\sqrt{\gamma}x_t+z_t$, where $x_t$ is the transmitted signal, $\gamma$ is the channel coefficient or SNR, and $z_t$ is unit-variance zero-mean white Gaussian noise.
The transmitter is equipped with a battery of finite capacity $\bar{B}$.
Let $E_t$ be energy harvested at time $t$, which is assumed to be a non-negative i.i.d. process with $\mathbb{E}[E_t]>0$.
We assume the energy arrival process is known causally at the transmitter.
A power control policy for an energy harvesting system is a sequence of mappings from energy arrivals to a non-negative number, which will denote a level of instantaneous power.
In this work, we will focus on online policies.
An online policy $\mathbf{g}=\{g_t\}_{t=1}^{\infty}$ is a sequence of mappings
\begin{equation}\label{eq:online}
\end{equation}
By allocating power $g_t$ at time $t$, the resultant instantaneous rate is $r_t=\frac{1}{2}\log(1+\gamma g_t)$.
Let $b_t$ be the amount of energy available in the battery at the beginning of time slot $t$.
An admissible policy $\mathbf{g}$ is such that satisfies the following constraints for every possible harvesting sequence $\{E_t\}_{t=1}^{\infty}$:
\begin{align}
&0\leq g_t\leq b_t&&,t=1,2,\ldots,\label{eq:power}\\
\end{align}
where we assume $b_1=\bar{B}$ without loss of generality.
For a given policy $\mathbf{g}$, we define the $n$-horizon expected throughput to be:
\begin{equation}
\T_n(\mathbf{g})=\frac{1}{n}\mathbb{E}\left[
\sum_{t=1}^{n}\frac{1}{2}\log(1+\gamma g_t(E^t))\right],
\label{eq:def_throughput}
\end{equation}
where the expectation is over the energy arrivals $E_1,\ldots,E_n$.
Finally, our goal is to characterize the optimal online power control policy and the resultant long-term average throughput:
\begin{align}
\Conline
&=\sup_{\mathbf{g}\text{ admissible}}
\liminf_{n\to\infty}\T_n(\mathbf{g}).
\label{eq:online_opt}
\end{align}
§ PRELIMINARY DISCUSSION
While the optimal offline power control policy has been explicitly characterized in <cit.>, in which the energy arrival sequence $E^n$ is assumed to be known ahead of time, there is limited understanding regarding the structure of the optimal online power control policy and the resultant long-term average throughput.
It is easily observed that this is an MDP, with the state being the battery level $b_t$, the action $g_t$ allowed to take values in the interval $[0,b_t]$, and the disturbance $E_{t+1}$. The state dynamics are given by
\begin{equation}
\label{eq:DP_state_dynamics}
\end{equation}
and the stage reward is $r_t=\frac{1}{2}\log(1+\gamma g_t)$.
It then follows by a well-known result in MDPs <cit.> that the optimal policy is Markovian, i.e. it depends only on the current state: $g_t^\star(E^t)=g_t^\star(b_t)$.
If the policy depends only on the current state and it is time-invariant, i.e. $g_t^\star(E^t)=g^\star(b_t)$, we say it is stationary.
The optimal policy can be found by means of dynamic programming, which involves solving the Bellman equation:
If there exists a scalar $\lambda\in\mathbb{R}_+$ and a bounded function $h:[0,\bar{B}]\to\mathbb{R}_+$ that satisfy
\begin{align}
&\lambda+h(b)=\sup_{0\leq g\leq b}\big\{
\tfrac{1}{2}\log(1+\gamma g)\nonumber\\*
\big\}
\label{eq:Bellman}
\end{align}
for all $0\leq b\leq\bar{B}$,
then the optimal throughput is $\Conline=\lambda$.
Furthermore, if $g^\star(b)$ attains the supremum in (<ref>) then the optimal policy is stationary and is given by $g_t^\star(E^t)=g^\star(b_t(E^t))$.
The functional equation (<ref>) is hard to solve explicitly, and requires an exact model for the statistical distribution of the energy arrivals $E_t$, which may be hard to obtain in practical scenarios. The equation can be solved numerically using value iteration <cit.>, but this can be computationally demanding and the numerical solution cannot provide insight as to the structure of the optimal policy and the qualitative behavior of the optimal throughput, namely how it varies with the parameters of the problem.
In the sequel, we propose an explicit online power control policy and show that it is within a constant gap to optimality for all i.i.d. harvesting processes. This policy depends on the harvesting process only through its mean, and does not depend on the channel gain whereas the optimal solution may depend on $\gamma$. It also leads to a simple and insightful approximation of the achievable throughput. We first discuss a special case in which the optimal online solution can be explicitly found. This inspires the approximately optimal power control policy for general i.i.d. energy harvesting processes.
§ BERNOULLI ENERGY ARRIVALS
Assume the energy arrivals $E_t$ are i.i.d. Bernoulli random variables (RVs):
\begin{equation}\label{eq:Bernoulli_Et}
\begin{cases}
\bar{B}&\text{w.p. }p\\
0&\text{w.p. }1-p,
\end{cases}
\end{equation}
i.e. at each time $t$ either the battery is fully charged to $\bar{B}$ with probability $p$ or no energy is harvested at all with probability $1-p$.
This simple case was studied extensively in <cit.>, and was shown there to be solved exactly. Specifically, we have the following Theorem, which we prove in Appendix <ref>.
Let the energy harvesting process be defined by (<ref>).
Let $j_t(E^t)$ be the time of the last energy arrival, i.e.
\[
j_t(E^t) = \{\sup\ \tau\leq t:\ E_{\tau}=\bar{B}\}.
\]
Then the optimal policy is given by
\begin{equation}
\frac{1}{\gamma}\left(\frac{\tilde{N}+\gamma\bar{B}}{1-(1-p)^{\tilde{N}}}p(1-p)^{t-j_t}-1\right)
0 &,t-j_t\geq\tilde{N}
\end{cases}
\label{eq:Bernoulli_gt_solution}
\end{equation}
where $\tilde{N}$ is the smallest positive integer satisfying
\[
1 > (1-p)^{\tilde{N}}[1+p(\gamma\bar{B}+\tilde{N})].
\]
It can be seen that this is a stationary policy, i.e. $g_t^\star$ can be written as a time-invariant function of $b_t$.
Roughly speaking, the energy is allocated only to the first $\tilde{N}$ time slots after each battery recharge and decays in an approximately exponential manner.
For the purpose of extending this policy to general i.i.d. processes in the next section, it is useful to simplify it to the following form by preserving its exponentially decaying structure:
\begin{equation}\label{Bernoulliform1}
\end{equation}
where $j_t$ is the time of the last energy arrival, as defined above. With this simplified policy, the amount of energy we allocate to each time slot decreases exactly exponentially with the time elapsed since the last battery recharge (or equivalently energy arrival).
Note that this is clearly an admissible strategy since
\sum_{k=j_t}^\infty\bar{B}p(1-p)^{k-j_t}=\bar{B},
i.e. the total energy we allocate until the next battery recharge can never exceed $\bar{B}$, the amount of energy initially available in the battery.
Another way to view this strategy is that we always use $p$ fraction of the remaining energy in the battery, i.e.
\begin{equation}\label{Bernoulliform2}
\end{equation}
where $b_t$ is the available energy in the battery given by
$b_t= (1 - p)^{t-j_t}\bar{B}$.
Hence, it is a stationary policy.
This simplified policy can be intuitively motivated as follows: for the Bernoulli
arrival process $E_t$, the inter-arrival time is a geometric random
variable with parameter $p$. Because the geometric
random variable is memoryless and has mean $1/p$,
at each time step the expected time to the next
energy arrival is $1/p$. Since $\log(\cdot)$ is a concave
function, uniform allocation of energy maximizes throughput, i.e. if the current energy level
in the battery is $b_t$ and we knew that the next battery recharge would be in exactly $m$ channel uses, allocating $b_t/m$ energy to each of the next $m$ channel uses would maximize throughput. For the online case of interest here, we can instead use the expected time to the next energy arrival: since at each time
step, the expected time to the next energy arrival is $1/p$, we
always allocate a fraction $p$ of the currently available energy in the battery.
Fig. <ref> illustrates this power control policy.
every node=[font=];
//ıin 0/1/1, 0.5/0.5/2, 1/0.25/3, 1.5/0.125/4, 2/1/1, 2.5/0.5/2, 3/0.25/3, 3.5/1/1, 4/0.5/2
[fill=cyan!!white,draw=black] (,0) – (+.5,0) – (+.5,) – (,) – cycle;
[above] at (+.25,) $g_{\i}$;
at (-.5,.7) $g_t$;
//̱/ı//in 0/2/B̅/1/100/100, 0.5/1/0/2/0/0, 1/0.5/0/3/0/100, 1.5/0.25/0/4/30/100, 2/2/B̅/5/100/100, 2.5/1/0/6/0/0, 3/0.5/0/7/40/100, 3.5/2/B̅/8/100/100, 4/1/0/9/0/40
[fill=green!!white,draw=black] (,) – (+.5,) – (+.5,+)̱ – (,+)̱ – cycle;
in 0,2,3.5
[->] (+.25,+2.8) node[above] $\bar{B}$ – (+.25,+2.4);
[above] at (-.5,+2.8) $E_t$;
at (-.5,+1) $b_t$;
The approximately optimal online power control policy for Bernoulli energy arrivals.
While this simplified policy is clearly suboptimal, it was shown in <cit.> to be at most within a gap of $0.97$ to optimality for all values of $\bar{B}$ and $p$. Here we improve upon this bound. Before we state this result, we present the following proposition which provides a simple upper bound on the achievable throughput for any i.i.d. harvesting process. The proof follows from Jensen's inequality, and can be found in e.g. <cit.>.
For completeness, we provide the proof in Appendix <ref>.
The optimal throughput under any i.i.d. harvesting process $E_t$ is bounded by
\[
\Conline
\leq\frac{1}{2}\log(1+\gamma\mu),
\]
where $\mu\triangleq \mathbb{E}[\min\{E_t,\bar{B}\}]$.
We next lower bound the performance of our simplified policy in terms of this upper bound.
Let $E_t$ be given by (<ref>) and consider the policy $\mathbf{g}$ given by (<ref>) (or equivalently (<ref>)).
Then the gap to optimality is bounded as follows:
\begin{equation}
\liminf_{n\to\infty}\T_n(\mathbf{g})\geq \frac{1}{2}\log(1+\gamma\mu)-0.72.
\label{eq:Bernoulli_lower_bound_final}
\end{equation}
See Section <ref> for the proof.
The two propositions above state that the simplified policy is always within 0.72 bits/channel use of optimality. Numerical evaluations show, however, that the real gap to optimality is much smaller than the one given in Proposition <ref>. In fact, Fig. <ref> shows that the throughput obtained by our simplified suboptimal policy is almost indistinguishable from the optimal throughput.
Optimal throughput and near-optimal throughput using the simplified Bernoulli policy, for $p=0.1$ and $\gamma=1$.
These are plotted along with the upper bound suggested in Proposition <ref>.
An additive bound is especially useful when $\bar{B}$ is very large; as seen in Fig. <ref>, the gap remains constant while the throughput grows unboundedly.
For small $\bar{B}$, however, the additive gap becomes useless as the optimal throughput itself falls below $0.72$.
Yet, it should be clear from the figure that as the throughput becomes small, the numerically evaluated additive gap also tends to zero.
We capture this fact in the following proposition where we provide a bound on the ratio between the throughput achieved by the simplified Bernoulli policy and the upper bound on the throughput.
For i.i.d. energy arrivals given by (<ref>) and the policy given by (<ref>), the ratio to optimality is bounded below as follows:
\[
\liminf_{n\to\infty}\T_n(\mathbf{g})\geq \frac{1}{2}\cdot\frac{1}{2}\log(1+\gamma\mu) .
\]
The proof is provided in Section <ref>.
The two propositions together imply that the simplified Bernoulli policy is good for all values of $\bar{B}$, i.e. across all SNR regimes.
§ APPROXIMATELY OPTIMAL POLICY FOR GENERAL I.I.D. ENERGY ARRIVAL PROCESSES
We now assume that $E_t$ is an i.i.d. process with an arbitrary distribution. As discussed in Section <ref>, finding the optimal solution for this general case is a hard problem.
In this section, we present a natural extension of the approximately optimal policy (<ref>) in the Bernoulli case and show that it is approximately optimal for all i.i.d. harvesting processes. The policy reduces to (<ref>) when the harvesting process is Bernoulli.
Before presenting the policy, observe that without loss of generality we can replace the random process $E_t$ with $\tilde{E}_t\triangleq\min\{E_t,\bar{B}\}$ without changing the system because of the store-and-use model we assume in (<ref>) and (<ref>). This is due to the fact that whenever an energy arrival $E_t$ is larger than $\bar{B}$, it will be clipped to at most $\bar{B}$. Denote $\mu=\mathbb{E}[\tilde{E}_t]$.
*The Fixed Fraction Policy
Let $q\triangleq \mu/\bar{B}$. Note that $\mu\in[0,\bar{B}]$ so $q\in[0,1]$.
We will use $q$ here instead of the parameter $p$ in the Bernoulli case. Notice that in that case, we also have $\mathbb{E}[E_t]=p\bar{B}$, hence this is a natural definition.
The Fixed Fraction Policy is defined as follows:
\[
g_t=q b_t.\qquad t=1,2,\ldots
\]
Inspired by (<ref>), at each time slot, this policy allocates a fraction $q$ of the currently available energy in the battery.
Clearly this is an admissible policy, since $q\leq 1$.
§.§ Main Result
The main result of this paper is that the Fixed Fraction Policy achieves the upper bound in Proposition <ref> within a constant additive gap and a constant multiplicative factor for any i.i.d. process.
We prove this result by showing that under this policy, the Bernoulli harvesting process yields the worst performance compared to all other i.i.d. processes with the same mean $\mu$.
This implies that the lower bounds obtained for the throughput achieved under Bernoulli energy arrivals apply also to any i.i.d. harvesting process with the same mean, giving the following theorem.
Let $E_t$ be an i.i.d. non-negative process with $\mu=\mathbb{E}[\min\{E_t,\bar{B}\}]$, and let $\mathbf{g}$ be the Fixed Fraction Policy.
Then, the throughput achieved by $\mathbf{g}$ is bounded by
\[
\liminf_{n\to\infty}\T_n(\mathbf{g})\geq\frac{1}{2}\log(1+\gamma\mu)-0.72,
\]
\[
\liminf_{n\to\infty}\T_n(\mathbf{g})\geq\frac{1}{2}\cdot\frac{1}{2}\log(1+\gamma\mu).
\]
The proof of this theorem is given in Section <ref>. The following approximation for the optimal throughput is an immediate corollary of the above theorem and proposition.
The optimal throughput under any i.i.d. energy harvesting process $E_t$ is bounded by
\[
\frac{1}{2}\log(1+\gamma\mu)-0.72\leq\Conline
\leq\frac{1}{2}\log(1+\gamma\mu),
\]
\[
\frac{1}{2}\leq\frac{\Conline}{\frac{1}{2}\log(1+\gamma\mu)}\leq 1,
\]
where $\mu\triangleq \mathbb{E}[\min\{E_t,\bar{B}\}]$.
This corollary gives a simple approximation of how the optimal throughput depends on the energy harvesting process $E_t$ and the battery size $\bar{B}$. This characterization identifies two fundamentally different operating regimes for this channel where the dependence of the average throughput on $E_t$ and $\bar{B}$ is qualitatively different. Assume that $E_t$ takes values in the interval $[0, \bar{E}]$. When $\bar{B}\geq \bar{E}$, we have
\begin{equation}\label{eq:mainres1}
\Conline\approx \frac{1}{2}\log\left(1+\gamma\mathbb{E}[E_t]\right)\qquad\text{bits/s/Hz},
\end{equation}
and the throughput is approximately equal to the capacity of an AWGN channel with an average power constraint equal to the average energy harvesting rate. This is surprising given that the transmitter is limited by the additional constraints (<ref>) and (<ref>), and at finite $\bar{B}$ this can lead to part of the harvested energy being wasted due to an overflow in the battery capacity. Note that in this large battery regime, the throughput depends only on the mean of the energy harvesting process – two energy harvesting profiles are equivalent as long as they provide the same energy on the average – and is also independent of the battery size $\bar{B}$. In particular, choosing $\bar{B}\approx \bar{E}$ is almost sufficient to achieve the throughput at infinite battery size.=-1
every node=[font=];
[->] (0,0) – (3,0);
[->] (0,0) – (0,2);
[below] at (0,0) $0$;
[above] at (0,2) $f_{\tilde{E}}(x)$;
[right] at (3,0) $x$;
[red,thick] (0,0.75) to[out=0,in=180] (0.75,1.25);
[red,thick] (0.75,1.25) to[out=0,in=135] (1.75,0);
[below] at (1.75,-0.1) $\bar{E}$;
(1.75,0) – (1.75,-0.1);
[dashed,thick] (2.5,0) – (2.5,2);
[below] at (2.5,-0.1) $\bar{B}$;
(2.5,0) – (2.5,-0.1);
every node=[font=];
[->] (0,0) – (3,0);
[->] (0,0) – (0,2);
[below] at (0,0) $0$;
[above] at (0,2) $f_{\tilde{E}}(x)$;
[right] at (3,0) $x$;
[red,thick] (0,0.75) to[out=0,in=180] (0.75,1.25);
[red,thick] (0.75,1.25) to[out=0,in=140] (1.1,1.12);
[below] at (1.1,-0.1) $\bar{B}$;
(1.1,-0.1) – (1.1,0);
[->,red,thick,>=latex] (1.1,0) – (1.1,1.5);
$\tilde{E}_t=\min\{E_t,\bar{B}\}$ in the two battery regimes.
When $\bar{B}\leq \bar{E}$, note that one can equivalently consider the distribution of $E_t$ to be that in Fig. <ref>-subfig:small_battery_regime: since every energy arrival with value $E_t\geq\bar{B}$ fully recharges the battery, this creates a point mass at $\bar{B}$ with value $\Pr(E_t\geq\bar{B})$. In this case, Corollary <ref> reveals that the throughput is approximately determined by the mean of this modified distribution. This can be interpreted as the small battery regime of the channel. In particular, in this regime the achievable throughput depends both on the shape of the distribution of $E_t$ and the value of $\bar{B}$.
§ NUMERICAL RESULTS
We compare the Fixed Fraction Policy to two additional policies that have been studied in the literature: The greedy policy <cit.>, i.e. $g_t=b_t$, and the “constant” policy <cit.>,
in which $g_t=\mu\cdot\mathbf{1}\{b_t\geq\mu\}$, where $\mu=\mathbb{E}[\min\{E_t,\bar{B}\}]$. The latter attempts to transmit at a constant power $\mu$, and if there is not enough energy in the battery, it simply waits until the battery is recharged again to a level at least $\mu$.[Reference <cit.> considers a refined version of this policy where the allocated energy is either $\mu+\delta$ or $\mu-\delta$ where $\delta=\beta\sigma^2\frac{\log\bar{B}}{\bar{B}}$ for some constant $\beta\geq2$ and $\sigma^2$ is the variance of the energy arrival distribution.
They show this policy is asymptotically optimal if $\bar{B}\to\infty$ and $\sigma^2$ is finite, in which case $\delta\to0$ and the strategy approaches the constant strategy. However, this policy is not applicable for all finite values of $\bar{B}$. For example, consider i.i.d. energy arrivals uniformly distributed on $[0,\bar{B}]$. Then $\sigma^2=\frac{1}{12}\bar{B}^2$, which yields $\delta=\frac{\beta}{12}\bar{B}\log\bar{B}$. Observe that for $\bar{B}<1$ we get $\delta<0$, and for $\bar{B}>e^{6/\beta}$ we get $\delta>\mu$, where $\mu=\bar{B}/2$. For $\beta=2$, which is the minimal value of $\beta$ according to <cit.>, the policy is only applicable for $1\leq\bar{B}\leq 20.1$.
Therefore we do not include this policy in our numerical evaluations.
All policies are compared to the optimal throughput $\Conline$ obtained by dynamic programming via value iteration (See Section <ref>).
Figures <ref>, <ref> and <ref> depict the
performance of these policies
when the energy distribution is Bernoulli with $p=0.1$, $p=0.5$ and $p=0.9$ respectively.
The plots are, from left to right: the achievable throughput, the additive gap to optimality, and the ratio between the throughput and the optimal one.
The figures show the absolute attainable throughput using these policies, as well as the gap to optimality as measured by $\Conline-\liminf_{n\to\infty}\T_n(\mathbf{g})$ and the ratio $\frac{\liminf_{n\to\infty}\T_n(\mathbf{g})}{\Conline}$ between the suboptimal policy and the optimal one, where $\mathbf{g}$ is any of the policies mentioned above.
Note that for the Fixed Fraction Policy, the gap to optimality remains small in all cases, while the gap to optimality grows unboundedly for the greedy and the constant policy as $\bar{B}$ increases. In particular, note that although the greedy policy is (not surprisingly) the best of all strategies at small $\bar{B}$ values, which correspond to low-SNR, its performance is not universally good across all SNR regimes. This can be also observed by noting that for Bernoulli arrivals the throughput achieved by the greedy policy is given by $p\frac{1}{2}\log(1+\bar{B})$ while the optimal throughput is $\frac{1}{2}\log(1+p\bar{B})$ within $0.72$ bits/channel use as shown in Proposition <ref>. Obviously, when $p$ is small and $\bar{B}$ is large the gap between the two expressions can be arbitrarily large. Also, note that as $p$ increases, the gap to optimality decreases for all strategies. This is not surprising as when $p\rightarrow 1$, the Bernoulli distribution approaches a constant equal to $\mu=\bar{B}$. In this trivial case, all three policies reduce to the optimal policy that always allocates energy equal to $\mu$. However note that even with $p=0.9$, the fixed fraction policy is still able to provide gain over the other strategies.
Plots of the Fixed Fraction, Greedy, and Constant policies, for $\gamma=1$ and $E_t\sim \text{Bernoulli}(0.1)$, where $E_t\in\{0,\bar{B}\}$.
Plots of the Fixed Fraction, Greedy, and Constant policies, for $\gamma=1$ and $E_t\sim \text{Bernoulli}(0.5)$, where $E_t\in\{0,\bar{B}\}$.
Plots of the Fixed Fraction, Greedy, and Constant policies, for $\gamma=1$ and $E_t\sim \text{Bernoulli}(0.9)$, where $E_t\in\{0,\bar{B}\}$.
Plots of the Fixed Fraction, Greedy, and Constant policies, for $\gamma=1$ and $E_t\sim \textrm{Unif}[0,\bar{B}]$.
Plots of the Fixed Fraction, Greedy, and Constant policies, for $\gamma=1$ and $E_t\sim \text{Exp}(\frac{1}{0.1\cdot\bar{B}})$.
Figures <ref> and <ref> provide the corresponding plots for the uniform and the exponential distribution. The exponential distribution corresponds to an energy harvesting process arising from a Gaussian signal. Note that while the absolute gap to optimality for the three policies depend on the distribution, the general trends for the Bernoulli case prevail.
§ LOWER BOUNDING THE THROUGHPUT ACHIEVED BY THE FIXED FRACTION POLICY
In this section we will prove the main theorem of the paper, namely Theorem <ref>.
First, we will derive lower bounds on the throughput obtained by the Fixed Fraction Policy when the energy arrivals are i.i.d. Bernoulli. These are derived in Sections <ref> and <ref>, in the form of an additive gap and a multiplicative gap from optimality, respectively.
Next, in Section <ref> we will show that the throughput of the Fixed Fraction Policy under any i.i.d. energy arrival process is necessarily larger than under Bernoulli energy arrivals. This will imply that the lower bounds derived for the Bernoulli case, apply also to any harvesting process.
§.§ Additive Gap for Bernoulli Energy Arrivals: Proof of Proposition <ref>
Before moving forward to establish the approximate optimality of this power control policy, we provide a few definitions and results from renewal theory.
A stochastic process $\{X_t\}_{t=1}^{\infty}$ is called a non-delayed regenerative process if there exists a random time $\tau>0$ such that the process $\{X_{\tau+t}\}_{t=1}^{\infty}$ has the same distribution as $\{X_t\}_{t=1}^{\infty}$ and is independent of the past $(\tau,X^{\tau})$.
Observe that a regenerative process is composed of i.i.d. “cycles” or epochs, which have i.i.d. durations $\tau_1,\tau_2,\ldots$. At the beginning of each epoch, the process “regenerates” and all memory of the past is essentially erased.
The following lemma establishes an important time-average property of regenerative processes.
Let $\{X_t\}_{t=1}^{\infty}$, $X_t\in\mathcal{X}$, be a non-delayed regenerative process with associated epoch duration $\tau$, and let $f:\mathcal{X}\to\mathbb{R}$.
If $\mathbb{E}\tau<\infty$ and $\mathbb{E}[\sum_{t=1}^{\tau}|f(X_t)|]<\infty$ then:
\[
\lim_{n\to\infty}\frac{1}{n}\sum_{t=1}^{n}f(X_t)=\frac{1}{\mathbb{E}\tau}\mathbb{E}\left[\sum_{t=1}^{\tau}f(X_t)\right]
\quad\text{a.s.}
\]
This is an immediate consequence of Theorem 3.1 in <cit.> or of the renewal reward theorem <cit.>.
We now return to our throughput optimization problem with Bernoulli energy arrivals.
Denote by $L$ the random time between two consecutive energy arrivals, or the length of an epoch. Evidently, $L\sim\mathrm{Geometric}(p)$. That is,
\[
\Pr(L=k)=p(1-p)^{k-1}
\qquad,k=1,2,\ldots
\]
Recall that we assume, without loss of generality, that $b_1=\bar{B}$
(It is shown in Appendix <ref> that
the initial battery level is irrelevant to the long-term average throughput
– this follows from the fact that we can always wait until the battery recharges to $\bar{B}$ before starting transmission, with a vanishing penalty to the average throughput).
Equipped with Lemma <ref>, we consider the Fixed Fraction Policy $\mathbf{g}$ for the Bernoulli case (see Section <ref>). Note that in this case this policy reduces to (<ref>) (or equivalently (<ref>)).
Observe that $g_t(E^t)$ is a non-delayed regenerative process with epoch duration $L$.
We apply Lemma <ref> with $f(x)=\frac{1}{2}\log(1+x)$.
Note that $\mathbb{E}L=1/p<\infty$ and
\[\mathbb{E}\Big[\sum_{t=1}^{L}|\tfrac{1}{2}\log(1+g_t(E^t)|\Big]\leq\mathbb{E}[L\cdot\tfrac{1}{2}\log(1+\bar{B})]<\infty,\]
so the conditions of the lemma are satisfied. We obtain
\begin{align}
\label{eq:SLLN_power_control_policy}
\end{align}
We proceed to lower bound the average throughput obtained by our suggested power control policy:
\begin{align}
\hspace{1em}\lefteqn{\hspace{-1em}\liminf_{n\to\infty}\T_n(\mathbf{g})}\nonumber\\*
&=\liminf_{n\to\infty}\frac{1}{n}\sum_{t=1}^n \mathbb{E}\left[\frac{1}{2}\log (1+\gamma g_t(E^t))\right]\nonumber\\*
&\overset{\text{(i)}}{\geq}\mathbb{E}\left[\liminf_{n\to\infty}\frac{1}{n}\sum_{t=1}^{n}\frac{1}{2}\log(1+\gamma g_t(E^t))\right]\nonumber\\
&\overset{\text{(ii)}}{=}\mathbb{E}\left[\frac{1}{\mathbb{E}L}\mathbb{E}\left[\sum_{t=1}^{L}\frac{1}{2}\log(1+\gamma g_t(E^t))\right]\right]\nonumber\\
&=\frac{1}{\mathbb{E}L}\mathbb{E}\left[\sum_{t=1}^{L}\frac{1}{2}\log(1+\gamma g_t(E^t))\right]\nonumber\\
&\overset{\text{(iii)}}{=}\frac{1}{\mathbb{E}L}\mathbb{E}\left[\sum_{i=1}^{L}\frac{1}{2}\log(1+\gamma \bar{B}p(1-p)^{i-1})\right]\label{eq:Bernoulli_scheme_exact_TP}\\
&\overset{\text{(iv)}}{\geq}\frac{1}{\mathbb{E}L}\mathbb{E}\left[\sum_{i=1}^{L}\left(\frac{1}{2}\log(1+\gamma p\bar{B})+(i-1)\frac{1}{2}\log(1-p)\right)\right]\nonumber\\
&=\frac{1}{\mathbb{E}L}\mathbb{E}\left[L\frac{1}{2}\log(1+\gamma p\bar{B})+\frac{L(L-1)}{2}\frac{1}{2}\log(1-p)\right]\nonumber\\
&=\frac{1}{2}\log(1+\gamma p\bar{B})-\frac{1}{4}\left(\frac{\mathbb{E}[L^2]}{\mathbb{E}L}-1\right)\log\left(\frac{1}{1-p}\right)\nonumber\\
&\overset{\text{(v)}}{=}\frac{1}{2}\log(1+\gamma p\bar{B})-\frac{1-p}{2p}\log\left(\frac{1}{1-p}\right),
\label{eq:Bernoulli_first_lower_bound}
\end{align}
where (i) is by Fatou's lemma <cit.>; (ii) is due to (<ref>); (iii) is by definition of the Fixed Fraction policy; (iv) is due to the inequality $\log(1+\alpha x)\geq\log(1+x)+\log\alpha$ for $0<\alpha\leq 1$;
and (v) is because $L\sim\mathrm{Geometric}(p)$.
The second term in (<ref>) achieves its maximum when $p\to0$, in which case it is given by $\frac{1}{2\ln 2}\approx 0.72$. We conclude that for Bernoulli energy arrivals:
\begin{equation}
\label{eq:Bernoulli_lower_bound}
\liminf_{n\to\infty}\T_n(\mathbf{g})
\geq\frac{1}{2}\log(1+\gamma \mu)-\frac{1}{2\ln 2},
\end{equation}
where $\mu=p\bar{B}$ is the average energy arrival rate of the Bernoulli process.
§.§ Multiplicative Gap for Bernoulli Energy Arrivals: Proof of Proposition <ref>
We start from (<ref>), which was derived in the previous section:
\begin{align}
\hspace{2em}\lefteqn{\hspace{-2em}\liminf_{n\to\infty}\T_n(\mathbf{g})}\nonumber\\*
&\geq\frac{1}{\mathbb{E}L}\mathbb{E}\left[\sum_{i=1}^{L}\frac{1}{2}\log(1+\gamma \bar{B}p(1-p)^{i-1})\right]\nonumber\\
\left[\sum_{i=1}^{L}(1-p)^{i-1}\frac{1}{2}
\log(1+\gamma\bar{B}p)\right]\nonumber\\
\sum_{i=1}^{k}(1-p)^{i-1}\frac{1}{2}
\log(1+\gamma\bar{B}p)\nonumber\\
\frac{1}{2}\log(1+\gamma\bar{B}p)\nonumber\\
\log(1+\gamma\bar{B}p)\nonumber\\
\frac{1}{2}\log(1+\gamma\mu),\label{eq:Bernoulli_lower_bound_mult}
\end{align}
where (i) is by the inequality $\log(1+\alpha x)\geq \alpha\log(1+x)$ for $0\leq\alpha\leq 1$;
(ii) is because $L\sim\text{Geometric}(p)$;
and (iii) is because $0\leq p\leq1$ and $\mu=p\bar{B}$.
§.§ General i.i.d Energy Harvesting Processes: Proof of <ref>
We will now use the result of the previous section to lower bound the throughput of the Fixed Fraction Policy for general i.i.d. energy harvesting processes.
We will show that under all distributions of (clipped) energy arrivals $\tilde{E}_t=\min\{E_t,\bar{B}\}$ with mean $\mu=\mathbb{E}[\tilde{E}_t]$, the lowest throughput is obtained when $\tilde{E}_t$ is Bernoulli, taking the values $0$ or $\bar{B}$.
We begin with a few notations and definitions.
Recall that the Fixed Fraction Policy is given by $g_t=qb_t$, where $q=\mu/\bar{B}$.
Under this policy:
\begin{align*}
\end{align*}
where, as in the previous section, we assume $b_1=\bar{B}$.
In what follows, we consider the performance of this policy under different distributions for the energy arrivals and different initial battery levels.
Therefore, with a slight abuse of notation, we define the expected $n$-horizon throughput for initial battery level $x\in[0,\bar{B}]$ and i.i.d. energy arrivals distributed according to the distribution of $E_1$:
\begin{align*}
\T_n(\mathbf{g},E_1,x)\triangleq
\frac{1}{n}\sum_{t=1}^{n}\mathbb{E}[\tfrac{1}{2}\log(1+\gamma qb_t)
\end{align*}
Note that the long term average throughput under i.i.d. energy arrivals with distribution $\tilde{E}_1$ is given by $\liminf_{n\to\infty}\T_n(\mathbf{g},\tilde{E}_1,\bar{B})$.
Let $\hat{E}_t$ be i.i.d. Bernoulli RVs, specifically $\hat{E}_t\in\{0,\bar{B}\}$ and $\Pr(\hat{E}_t=\bar{B})=q$. Note that
\[\mathbb{E}[\hat{E}_t]=\mathbb{E}[\tilde{E}_t]=\mathbb{E}[\min\{E_t,\bar{B}\}]=\mu.\]
In the following proposition, we claim that the $n$-horizon expected throughput for any distribution of i.i.d. energy arrivals is always better than the throughput obtained for i.i.d. Bernoulli energy arrivals with the same mean, for any $n$ and any initial battery level $x$.
For any $x\in[0,\bar{B}]$ and any integer $n\geq1$:
\[
\T_n(\mathbf{g},\tilde{E}_1,x)\geq\T_n(\mathbf{g},\hat{E}_1,x).
\]
Before proving this proposition, we state the following lemma.
Let $f(z)$ be concave on the interval $[0,\bar{B}]$, and let $Z$ be a RV confined to the same interval, i.e. $0\leq Z\leq \bar{B}$.
Let $\hat{Z}\in\{0,\bar{B}\}$ be a Bernoulli RV with $\Pr(\hat{Z}=\bar{B})=\mathbb{E}Z/\bar{B}$.
\[
\mathbb{E}[f(Z)]\geq \mathbb{E}[f(\hat{Z})].
\]
By concavity, for any $z\in[0,\bar{B}]$:
\[
\]
Setting $z=Z$ and taking expectation yields
\begin{align*}
\mathbb{E}[f(Z)]&\geq\frac{\mathbb{E}Z}{\bar{B}}f(\bar{B})
\tag*{\qedhere}
\end{align*}
We will give a proof by induction.
Clearly for $n=1$ we have
\[
\T_1(\mathbf{g},\tilde{E}_1,x)=\T_1(\mathbf{g},\hat{E}_1,x)
=\tfrac{1}{2}\log(1+\gamma qx).
\]
Observe that this is a non-decreasing concave function of $x$.
This will in fact be true for every $\T_n(\mathbf{g},\hat{E}_1,x)$, $n\geq 1$, and we will use this in the induction step.
Assume that $\T_{n-1}(\mathbf{g},\tilde{E}_1,x)\geq\T_{n-1}(\mathbf{g},\hat{E}_1,x)$ for all $x\in[0,\bar{B}]$, and also that $\T_{n-1}(\mathbf{g},\hat{E}_1,x)$ is monotonic non-decreasing and concave in $x$.
For the induction step, observe that:
\begin{align*}
&=\tfrac{1}{2}\log(1+\gamma qx)
\end{align*}
where the expectation is over the RV
This is due to the process $b_t$ being a time-homogeneous Markov chain.
By the induction hypothesis, we have:
\begin{equation}
\geq\tfrac{1}{2}\log(1+\gamma qx)
\label{eq:induction_step}
\end{equation}
where still $b_2=\min\{(1-q)x+\tilde{E}_2,\bar{B}\}$.
\begin{align*}
\hspace{1ex}\lefteqn{\hspace{-1ex}\T_{n-1}(\mathbf{g},\hat{E}_1,b_2)}\nonumber\\*
\T_{n-1}(\mathbf{g},\hat{E}_1,\bar{B})\big\},
\end{align*}
where the second equality is because $\T_{n-1}(\mathbf{g},\hat{E}_1,{}\cdot{})$ is non-decreasing, due to the induction hypothesis.
Next, we claim that the function $f_1(z)\triangleq\T_{n-1}(\mathbf{g},\hat{E}_1,(1-q)x+z)$ is concave. This is true again by the induction hypothesis that $\T_{n-1}(\mathbf{g},\hat{E}_1,{}\cdot{})$ is concave.
Therefore, since $\T_{n-1}(\mathbf{g},\hat{E}_1,\bar{B})$ is simply a constant, the function $f_2(z)\triangleq\T_{n-1}(\mathbf{g},\hat{E}_1,\min\{(1-q)x+z,\bar{B}\})$ is a minimum of two concave functions, hence it is itself concave.
We can now apply Lemma <ref> to obtain:
\begin{align*}
\mathbb{E}[\T_{n-1}(\mathbf{g},\hat{E}_1,b_2)]
&= \mathbb{E}[\T_{n-1}(\mathbf{g},\hat{E}_1,\hat{b}_2)],
\end{align*}
where $\hat{b}_2\triangleq\min\{(1-q)x+\hat{E}_2,\bar{B}\}$.
Substituting this into (<ref>):
\begin{align*}
&\geq\tfrac{1}{2}\log(1+\gamma qx)
\end{align*}
It is left to verify that $\T_{n}(\mathbf{g},\hat{E}_1,x)$ is concave and non-decreasing in $x$.
Writing it explicitly:
\begin{align*}
&=\tfrac{1}{2}\log(1+\gamma qx)
\end{align*}
we see that it is a sum of non-decreasing concave functions of $x$, hence it is a non-decreasing concave function of $x$.
As an immediate result of Proposition <ref>, we obtain
\begin{equation}
\liminf_{n\to\infty}\T_n(\mathbf{g},\tilde{E}_1,\bar{B})
\geq\liminf_{n\to\infty}\T_n(\mathbf{g},\hat{E}_1,\bar{B}).
\label{eq:bernoulli_is_worst}
\end{equation}
Now we can apply the results of the previous section.
From (<ref>) we have
\begin{align*}
\liminf_{n\to\infty}\T_n(\mathbf{g},\hat{E}_1,\bar{B})
&\geq\frac{1}{2}\log(1+\gamma q\bar{B})-\frac{1}{\ln2}\\*
\end{align*}
and from (<ref>) we have
\begin{align*}
\liminf_{n\to\infty}\T_n(\mathbf{g},\hat{E}_1,\bar{B})
&\geq\frac{1}{2}\cdot\frac{1}{2}\log(1+\gamma q\bar{B})\\
\end{align*}
Substituting in (<ref>) completes the proof of Theorem <ref>.
§ CONCLUSION
We proposed a simple online power control policy for energy harvesting channels and proved that it is within constant additive and multiplicative gaps to the AWGN capacity for any i.i.d. harvesting process and any battery size. This allowed us to develop a simple and insightful approximation for the optimal throughput. While optimal power control in the offline case and the online case with infinite battery size have been characterized in the previous literature, the strategies developed for the online case have been mostly heuristic with no or only asymptotic guarantees on optimality. We believe the approximation approach we propose in this paper can be fruitful in developing further insights on online power control under various assumptions, such as processing cost and battery non-idealities as well as multi-user settings, the rigorous treatment of which have been so far mostly limited to either the offline case or the case with infinite battery. For example, in <cit.> this approach is extended to derive a universally near-optimal power control policy for the multiple-access channel. It is shown that in an energy-harvesting MAC the users can achieve a symmetric capacity equal to the AWGN capacity as $K\to\infty$.
Additionally, an approximation of the optimal throughput can be used to derive bounds on the information-theoretic capacity of the energy harvesting channel, as done in <cit.>.
An important step in our proof was to show that i.i.d. Bernoulli energy arrivals constitute the worst case for our proposed policy among all i.i.d. energy arrival processes with the same mean, i.e. the throughput achieved by our proposed policy is smallest when the process is Bernoulli. Whether i.i.d. Bernoulli energy arrivals are also the worst case in terms of the optimal throughput is an interesting question.
§ UPPER BOUND ON THE OPTIMAL THROUGHPUT
Recall $\tilde{E}_t=\min\{E_t,\bar{B}\}$ and $\mu=\mathbb{E}[\tilde{E}_t]$.
For any $n$ and any policy $\mathbf{g}$, we have:
\begin{align*}
\T_n(\mathbf{g})
\frac{1}{n}\sum_{t=1}^n \mathbb{E}
\left[\frac{1}{2}\log
\big(1+\gamma g_t(\tilde{E}^n)\big)\right]\\
\frac{1}{2}\log\left(1+\frac{\gamma}{n}
\mathbb{E}\big[\sum_{t=1}^{n}
\frac{1}{2}\log\left(1+ \frac{\gamma}{n}\mathbb{E}
\big[\bar{B}+\sum_{t=2}^{n} \tilde{E}_t\big]
\right)\\
&=\frac{1}{2}\log\left(1+ \frac{\gamma}{n}\bar{B}+\gamma\tfrac{n-1}{n}\mu\right)
\end{align*}
where (i) is by concavity of $\log$; (ii) follows from the fact that the total allocated energy up to time $n$, can not exceed the total energy that arrives up to time $n$ plus the energy initially available in the battery, which cannot be more than $\bar{B}$:
\[
\sum_{t=1}^{n} g_t\leq \bar{B}+\sum_{t=2}^{n} \tilde{E}_t.
\]
The last term tends to $\frac{1}{2}\log(1+\gamma\mu)$ as $n\to\infty$. Note that this is true for any energy arrival process $E_t$. We therefore have:
\begin{equation*}
\Conline\leq\frac{1}{2}\log(1+\gamma\mu),
\end{equation*}
where $\mu=\mathbb{E}[\min\{E_t,\bar{B}\}]$.
§ THROUGHPUT DOES NOT DEPEND ON INITIAL BATTERY STATE
We state and prove the following proposition:
Let $\mathbf{g}$ be a policy which achieves throughput
\liminf_{n\to\infty}\T_n(\mathbf{g})
when the initial battery level is $b_1=\bar{B}$.
Then for every $\epsilon>0$ there exists a policy $\tilde{\mathbf{g}}$ which achieves throughput
\[
\liminf_{n\to\infty}\T_n(\tilde{\mathbf{g}})
\geq \liminf_{n\to\infty}\T_n(\mathbf{g})-\epsilon,
\]
when the initial battery level is $b_1=0$.
Since a policy which is admissible for $b_1=0$ is also admissible for any $b_1\in[0,\bar{B}]$,
this implies that we can compute the throughput $\Conline$ for any initial battery level, say $\bar{B}$, regardless of the actual battery level of interest $b_1$.
Fix $\ell\geq1$. Consider the following online power control policy $\tilde{\mathbf{g}}$ for initial battery level $b_1=0$: Transmit zeros ($g_t=0$) for the first $\ell$ time slots. This will allow the battery to completely recharge to $\bar{B}$ with high probability. Then, if $b_{\ell}=\bar{B}$, transmit the policy $\mathbf{g}$ (note that $b_\ell=\bar{B}$ and $g_\ell=0$ imply $b_{\ell+1}=\bar{B}$). Otherwise, transmit zeros (i.e. give up on the entire transmission).
More precisely, define the new policy as follows, for $t=1,2,\ldots$:
\[
\tilde{g}_t(E^t)=\begin{cases}
0&,1\leq t\leq \ell\\
1_{\{b_{\ell}=\bar{B}\}}\cdot {g}_{t-\ell}(E_{\ell+1}^{t})
&,\ell+1\leq t
\end{cases}
\]
where $1_{\{\cdot\}}$ is the indicator function.
Observe that $b_{\ell}$ is a deterministic function of $E^{\ell}$, which is given by
We have for any $n>\ell$:
\begin{align}
\T_n(\tilde{\mathbf{g}})
\label{eq:T0_geq_plTBbar}
\end{align}
where (i) is because $b_\ell$ depends only on $E^\ell$, and $E_t$ is i.i.d.;
and (ii) is because $E_t$ is i.i.d.
Since $\mathbb{E}[E_t]>0$, we can lower-bound the probability of recharging the battery using the law of large numbers:
\begin{align*}
\Pr\{b_\ell=\bar{B}\}
&\geq 1 - \epsilon_\ell,
\end{align*}
where $\epsilon_\ell\to0$ as $\ell\to\infty$.
Substituting this in (<ref>) and taking $n\to\infty$ yields
\[
\liminf_{n\to\infty}\T_n(\tilde{\mathbf{g}})
\geq(1-\epsilon_\ell)\liminf_{n\to\infty}\T_n(\mathbf{g}).
\]
By choosing $\ell$ large enough, we can approach the throughput of $\mathbf{g}$ arbitrarily close for any initial battery level.
§ OPTIMAL THROUGHPUT FOR BERNOULLI ENERGY ARRIVALS: PROOF OF THEOREM <REF>
Recall Proposition <ref> and the preceding discussion.
It can be argued <cit.> that the there exists an stationary policy, i.e. there exists a function $g^\star(b)$, satisfying $0\leq g^\star(b)\leq b$ for $0\leq b\leq\bar{B}$, s.t. the optimal throughput is given by
\[
\Conline=\liminf_{n\to\infty}\frac{1}{n}\sum_{t=1}^{n}\mathbb{E}[\tfrac{1}{2}
\log(1+\gamma g^\star(b_t))].
\]
Under such a stationary policy, the battery state $b_t$ is a regenerative process (see Definition <ref>). The regeneration times $\{T(n)\}_{n=0}^{\infty}$ are the energy arrivals, i.e. $E_{T(n)}=\bar{B}$, which implies $b_{T(n)}=\bar{B}$.
Additionally, we assume $b_1=\bar{B}$ (See Appendix <ref>),
which implies the process is non-delayed (i.e. the first regeneration time is $T(0)=1$).
Applying Lemma <ref> in Section <ref>, we obtain:
\begin{align*}
\Conline&=\frac{1}{\mathbb{E}L}\mathbb{E}\left[\sum_{t=1}^{L}
\tfrac{1}{2}\log(1+\gamma g^\star(b_t))\right],
\end{align*}
where $L=T(1)-T(0)$ is a $\text{Geometric}(p)$ RV, which follows from the fact that $E_t$ are i.i.d. $\text{Bernoulli}(p)$.
Observe that for $2\leq t\leq L$ there are no energy arrivals. Hence, we have the following deterministic recursive relation:
\begin{equation}
\begin{aligned}
\end{aligned}
\label{eq:bt_recursive_relation}
\end{equation}
Since $L$ can take any positive integer, this defines a sequence $\{\mathcal{E}^\star_i\}_{i=1}^{\infty}$ such that $g^\star(b_i)=\mathcal{E}^\star_i$.
We can therefore write
\begin{align*}
\Conline&=\frac{1}{\mathbb{E}L}\mathbb{E}\left[
\sum_{i=1}^{L}\tfrac{1}{2}\log(1+
\gamma\mathcal{E}^\star_i)\right]\\
\log(1+\gamma\mathcal{E}^\star_i)\\
% \tfrac{1}{2}\log(1+\gamma\mathcal{E}^\star_i)\\
\end{align*}
Moreover, by the constraint $g^\star(b_t)\leq b_t$ and the recursive relation (<ref>), we must have $\sum_{i=1}^{\infty}\mathcal{E}^\star_i\leq\bar{B}$, in addition to ${\mathcal{E}^\star_i\geq0}$ for all $i\geq1$.
To find $\{\mathcal{E}_i^\star\}_{i=1}^{\infty}$ we need to solve the following infinite-dimensional optimization problem:
\begin{equation}
\begin{aligned}
\text{maximize}
\text{subject to}&\qquad
\mathcal{E}_i\geq0,\quad i=1,2,\ldots,\\
&\qquad\sum_{i=1}^{\infty}\mathcal{E}_i\leq \bar{B}.
\end{aligned}
\label{eq:def_Cinf}
\end{equation}
Let $\{\mathcal{E}_i^\star\}_{i=1}^{\infty}$ and $\Theta$ be the optimal sequence and optimal objective, respectively, of (<ref>).
We will show that (<ref>) can be solved by the limit as $N\to\infty$ of the following $N$-dimensional optimization problem:
\begin{equation}
\begin{aligned}
\text{maximize}
\text{subject to}&\qquad
\mathcal{E}_i\geq0,\quad i=1,2,\ldots,N,\\
&\qquad\sum_{i=1}^{N}\mathcal{E}_i\leq \bar{B}.
\end{aligned}
\label{eq:def_Cbar}
\end{equation}
Denote by $\Theta_N$ the optimal objective of (<ref>).
Clearly $\Theta_N$ is non-decreasing and $\Theta_N\leq\Conline$.
Observe that the first $N$ values of the infinite-dimensional solution, $\{\mathcal{E}_i^\star\}_{i=1}^{N}$, are a feasible solution for (<ref>). Therefore,
\begin{align*}
\Theta_N&\geq \sum_{i=1}^{N}p(1-p)^{i-1}\tfrac{1}{2}\log(1+\gamma\mathcal{E}_i^\star)\\
\log(1+\gamma\mathcal{E}_i^\star)\\
\log(1+\gamma\bar{B})\\
\end{align*}
where $(\ast)$ is because $\mathcal{E}_t^\star\leq\bar{B}$.
Along with the inequality $\Theta_N\leq\Theta$, this implies
\[
\Theta=\lim_{N\to\infty}\Theta_N.
\]
We continue with the explicit solution of (<ref>).
Writing the problem in standard form and using KKT conditions, we have for $i=1,\ldots,N$:
\begin{equation}
\label{eq:KKT_equation}
\end{equation}
with $\lambda_i,\tilde{\lambda}\geq0$ and the complementary slackness conditions:
$\lambda_i\mathcal{E}_i=0$ and $\tilde{\lambda}(\sum_{i=1}^{N}\mathcal{E}_i-\bar{B})=0$.
To obtain the non-zero values of $\mathcal{E}_i$, we set $\lambda_i=0$ in (<ref>):
\begin{equation}\label{eq:Esolution}
\mathcal{E}_i=\frac{p(1-p)^{i-1}}{2\tilde{\lambda}}-
\frac{1}{\gamma}.
\end{equation}
Since $\mathcal{E}_i\geq0$, this implies $\tilde{\lambda}\leq\frac{\gamma}{2}p(1-p)^{i-1}$ for all $i$ for which $\mathcal{E}_i>0$.
Since the RHS is a decreasing function of $i$, there exists an integer $\tilde{N}$ such that $\mathcal{E}_i>0$ for if $i\leq\tilde{N}$ and $\mathcal{E}_i=0$ otherwise.
Therefore $\tilde{N}$ is the largest integer satisfying $\tilde{N}\leq N$ and
\begin{equation}
\tilde{\lambda}\leq\frac{\gamma}{2}p(1-p)^{\tilde{N}-1}.
\label{eq:Ntilde_bound}
\end{equation}
Next, consider the total energy constraint $\sum_{i=1}^{N}\mathcal{E}_i\leq\bar{B}$.
Since increasing $\mathcal{E}_i$ for any $i$ will only increase the objective, this constraint must hold with equality:
\begin{align*}
\bar{B}&=\sum_{i=1}^{N}\mathcal{E}_i\\
\left(\frac{p(1-p)^{i-1}}
\end{align*}
This yields:
\begin{equation}\label{eq:lambda_tilde}
\tilde{\lambda}=\frac{1-(1-p)^{\tilde{N}}}
\end{equation}
Along with (<ref>), we deduce that $\tilde{N}$ is the largest integer satisfying $\tilde{N}\leq N$ and
\[
\frac{1-(1-p)^{\tilde{N}}}{2(\bar{B}+\tilde{N}/\gamma)}
\leq \frac{\gamma}{2}p(1-p)^{\tilde{N}},
\]
or equivalently
\[
1 \leq (1-p)^{\tilde{N}}[1+p(\gamma\bar{B}+\tilde{N})].
\]
Observe that for $N$ large enough, the optimal $\tilde{N}$ does not depend on $N$, and therefore $\Theta_N$ will not depend on $N$.
For such values of $N$, the optimal sequence $\{\mathcal{E}_i\}_{i=1}^{N}$ for (<ref>) is given by substituting (<ref>) in (<ref>) for $i=1,\ldots,\tilde{N}$, and $\mathcal{E}_i=0$ for $i>\tilde{N}$.
Since $\Theta_N$ does not depend on $N$ and $\Theta=\lim_{N\to\infty}\Theta_N$,
the optimal sequence for (<ref>) is the same.
This gives (<ref>).
|
1511.00295
|
Categorification of Dijkgraaf-Witten Theory]Categorification
of Dijkgraaf-Witten Theory
A. Sharma]Amit Sharma
School of Mathematics
University of Minnesota
Minneapolis, MN 55455, USA
A. A. Voronov]Alexander A. Voronov
School of Mathematics
University of Minnesota
Minneapolis, MN 55455, USA, and Kavli IPMU (WPI), UTIAS, The
University of Tokyo, Kashiwa, Chiba 277-8583, Japan
This work was supported by the World Premier International
Research Center Initiative (WPI), MEXT, Japan, and a
grant from the Simons Foundation (#282349 to A. V.).
The goal of the paper is to categorify Dijkgraaf-Witten (DW) theory,
aiming at providing foundation for a direct construction of DW
theory as an Extended Topological Quantum Field Theory. The main
tool is cohomology with coefficients in a Picard groupoid, namely
the Picard groupoid of hermitian lines.
§ INTRODUCTION
R. Dijkgraaf and E. Witten in <cit.> constructed a gauge theory
with a finite gauge group $G$ as a “toy model,” a tool for studying
more general gauge theories with compact gauge groups. Their goal was
to describe this theory, known as DW theory, as a Topological
Quantum Field Theory (TQFT), i.e., a functor on the category of
3-dimensional (3d) cobordisms to that of vector spaces, starting with
an action given by a cocycle $\alpha \in Z^3(G; U(1))$. Dijkgraaf and
Witten indicated that the vector space $\Phi(Y)$ corresponding to a
closed oriented 2d manifold $Y$ was closely related to the set
$\Hom(\pi_1 (Y), G)/G$ of equivalence classes of principal $G$-bundles
over $Y$ and that it could be constructed by cutting the surface $Y$
into pairs of pants, as $\Phi$ was expected to be a functor. The
linear map $\Phi(X): \del_- X \to \del_+ X$ corresponding to a 3d
oriented cobordism $X$ between closed manifolds $\del_- X$ and $\del_+
X$ depended on such choices as the choice of a map $\Hom (\pi_1 (X,
x_0), G) \to \Map(X, BG)$, the choice of a basepoint $x_0$, the choice
of a chain, via triangulation, representing the relative fundamental
cycle $[X] \in H_3(X, \del X; \Z)$, which was interpreted as “lattice
gauge theory.” One can say that, from the categorical point of view,
Dijkgraaf and Witten constructed a TQFT functor on a certain
subcategory of cobordisms decorated with appropriate extra structure
utilized in their constructions. They used an orbifold approach to
taking the homotopy quotient by $G$, that is to say, worked with the
$G$-set $\Hom(\pi_1(Y), G)$.
D. Freed and F. Quinn in <cit.> streamlined the construction
of the TQFT functor $\Phi$, so that $\Phi(X)$ would no longer depend
on the choice of a representative of the fundamental cycle $[X]$ and
thereby would produce a TQFT functor on the category of
cobordisms. They also generalized the construction to $n$-dimensional
cobordisms. Their main tool was to define pairings between cocycles in
$Z^{n+1} (Y, U(1))$ and cycles $Z_n (Y, \Z)$ and between $Z^{n+1} (X,
U(1))$ and cycles $Z_{n+1} (X, \del X; \Z)$, resembling but certainly
different from cap product, which would not even be defined because of
dimension considerations. Freed and Quinn introduced the idea of an
invariant section of a flat hermitian line bundle over a
groupoid. This is a particular case of the idea of the limit of a
functor, and in this context, is akin to taking a global section.
J. Lurie in <cit.> sketched a different construction of
Dijkgraaf-Witten theory. Rather than using the orbifold $\Hom
(\pi_1(Y), G)/G$, he modeled the set of equivalence classes of
principal $G$-bundles on the mapping space $\Map(Y,BG)$. Given a
cohomology class $\alpha \in H^{n+1}(BG; U(1))$ and a closed oriented
$n$-manifold $Y$, he used a “push-pull” construction $\pi_* \ev^*
\alpha \in H^1 (\Map (Y, BG); U(1))$ for the diagram
\[
\begin{CD}
Y \times \Map (Y,BG) @>\ev>> BG\\
\Map (Y, BG)
\end{CD}
\]
to obtain a hermitian line bundle $\L_Y$ over $Y$. Then he defined the
TQFT functor $\Phi$ on objects by taking the space
\[
\Phi(Y) := H^0(\Map (Y, BG), \L_Y)
\]
of global sections. He used ambidexterity, a natural
\[
H^0 (\Map (Y, BG), \L_Y) \xrightarrow{\sim} H_0 (\Map (Y, BG), \L_Y),
\]
to produce a linear map
\[
\Phi(X): \Phi (\del_- X) \to \Phi (\del_+ X),
\]
using push-pull again, now along the diagram
\[
\Map (\del_- X, BG) \xleftarrow{p_-} \Map (X, BG) \xrightarrow{p_+}
\Map (\del_+ X, BG).
\]
Lurie's construction deliberately avoided the following subtlety. The
hermitian line bundle $\L_Y$ is determined by the cohomology class
$\alpha$ only up to isomorphism. Starting with a cocycle $\alpha \in
Z^{n+1}(BG; U(1))$ would partially fix the problem, because the
resulting cocycle $\pi_* \ev^* \alpha \in Z^1(\Map (Y, BG); U(1))$ is
not quite the same as a hermitian line bundle: isomorphic, but
different hermitian line bundles may correspond to the same cocycle,
whereas the cocycle is determined by a hermitian line bundle only up
to condoundary. Moreover, the push-pull cocycle $\pi_* \ev^* \alpha$
will depend on the choice of a cycle representing the fundamental
class $[Y] \in H_n (Y; \Z)$.
In the current paper, we replace the coefficient group $U(1)$ with an
equivalent Picard groupoid, namely the Picard groupoid $\L$ of
hermitian lines, and notice that an object of $H^0(M, \L)$ is exactly
a flat hermitian line bundle over $M$, see Section <ref>.
The paper <cit.> attempted the construction of an Extended
Topological Quantum Field Theory (ETQFT), which is defined on
cobordisms with corners, rather than boundary, and a generalization of
the DW theory to the case of a compact group $G$. The construction
utilizes the Cobordism Hypothesis, which asserts that an ETQFT is
determined by its value on zero-dimensional manifolds. The
two-dimensional case of the cobordism hypothesis was proved by
C. J. Schommer-Pries in <cit.>, and the full version was proven by
Lurie in <cit.>. However, Freed, Hopkins, Lurie, and Teleman
emphasize the importance of a direct construction, which has not been
been done yet.
This paper arose from the authors' trying to find an approach to this
hypothetical direct construction of an ETQFT. In the process we have
realized that Freed and Quinn's pairing makes sense as a cohomological
operation, cap product, if the group $H^{n+1} (Y; U(1))$ is replaced
with cohomology $H^n (Y; \L)$ with coefficients in the Picard groupoid
$\L$ of hermitian line bundles. Categorifying the coefficients goes
along with lowering the cohomological degree, thus opening a way to
defining cap products as well as extending the TQFT to an ETQFT by
further categorification to higher Picard groupoids and higher gerbes.
Another novel feature of our approach is that we do not use
ambidexterity, but rather a transfer map in the context of cohomology
with coefficients in Picard groupoids. In principle, one can view the
transfer map as an avatar of ambidexterity, but it might be argued
that using an avatar is less demanding than engaging the full power of a
We thank Jim Stasheff for valuable comments on the first version of
the manuscript. A. V. gratefully acknowledges support from the
Simons Center for Geometry and Physics, Stony Brook University, and
the Graduate School of Mathematical Sciences, The University of
Tokyo, at which some of the research for this paper was performed.
§ SETUP
We will consider (flat) hermitian line gerbes over simplicial sets. To
deal with gerbes over manifolds and topological spaces, we will
associate simplicial sets to them in a standard way: by taking
singular simplices or the nerve of an open cover. Flat hermitian line
gerbes are analogous to more traditional gerbes over topological
spaces with the constant sheaf $U(1)$ as the band, whether given as
stacks of groupoids, via gluing (descent) data, or as higher bundles,
<cit.>. We will take the
liberty of omitting the adjective “flat” when referring to flat
hermitian line bundles and gerbes.
We will describe cohomology with coefficients in Picard groupoids over
simplicial sets and later apply this construction to cobordisms, which
are manifolds, rather than simplicial sets. This may be done by
working with the simplicial set of singular chains associated to the
cobordism or by using the nerve of a sufficiently fine open covering,
see examples in Section <ref>.
§.§ Cohomology with coefficients in Picard groupoids
A Picard groupoid is a symmetric monoidal groupoid in which
every object is invertible, up to isomorphism, with respect to the
tensor product, which, by a slight abuse of notation, we denote
$+$. More precisely, for each object $s$ of a Picard groupoid $\A$,
the functors $t \mapsto s + t$, and $t \mapsto t + s$ define
autoequivalences of $\A$ as a category. In this case, one can define a
functor $\A \to \A$, $s \mapsto -s$, and natural isomorphisms
\[
m= m_s: s + (-s) \to 0, \qquad n=n_s: (-s) + s \to 0
\]
such that $l_s (m_s + \id_s) = r_s (\id_s + n_s) \alpha_{s,-s,s}$ for
all objects $s$ of $\A$, where $0$ is the zero (also known as unit)
object of $\A$ and
\begin{gather}
\label{structure1}
\alpha_{s,t,u}: (s+t)+u \to s + (t+u) \qquad \text{and}\\
\label{structure2}
l_s: 0+s \to s, \quad r_s: s + 0 \to s
\end{gather}
are the natural transformations of the monoidal structure on $\A$. We
will assume that $-0 = 0$, $m_0 = r_0$, and $n_0 = l_0$. Another
structure natural transformation is a symmetry:
\[
\beta_{s,t}: s+t \to t+s,
\]
making $\A$ to be a symmetric monoidal category. Given a Picard
groupoid $\A$, let $\pi_0 (\A)$ denote the abelian group of its
connected components and $\pi_1 (\A)$ denote the abelian group of
automorphisms of the zero object.
A homomorphism between two Picard groupoids $\A$ and $\B$ is a
functor $F: \A \to \B$ and an assignment of a coherence
morphism which is an arrow of $\B$, $\phi^F_{s,t}:F(s) + F(t) \to
F(s+t)$, to every pair of objects $s,t \in \A$ which is natural in
both variables $s$ and $t$ such that the assignment respects the
symmetry natural transformations $\beta$ of $\A$ and $\B$ in the
following sense:
\[
F(\beta_{s,t}) \circ \phi^F_{s,t} =
\phi^F_{t,s} \circ \beta_{F(s),F(t)}
\]
and also respects the associativity in the following sense:
\[
\phi^F_{s,t+u} \circ (id_{F(s)} + \phi^F_{t,u})\circ \alpha^{\B}_{F(s),F(t),F(u)}
= F(\alpha^{\A}_{s,t,u}) \circ \phi^F_{s+t,u} \circ (\phi^F_{s,t} + id_u),
\]
for each triple of objects $s,t,u \in \A$ and where $\alpha^\A$
and $\alpha^\B$ are the associativity natural transformations of
$\A$ and $\B$ respectively.
A homomorphism between two Picard groupoids $F: \A \to \B$ will be
called a strict homomorphism if the coherence morphisms
$\phi^F_{s,t}$ are identities for all pairs $s,t \in \A$ and $F(0) =
two homomorphisms $F$ and $F': \A \to \B$, a monoidal natural
transformation from $F$ to $F'$ is a natural transformation
$\theta: F \Rightarrow F'$ which is compatible with the coherence
morphisms of both homomorphisms $F$ and $F'$ in the following
\[
\phi^{F'}_{s,t} \circ (\theta_s + \theta_t) = \theta_{s+t} \circ \phi^F_{s,t},
\]
for all pairs $s, t \in \A$.
Given any two Picard groupoids $\A$ and $\B$, the category whose
objects are all homomorphisms from $\A$ to $\B$ and whose morphisms
are monoidal natural transformations
between these homomorphisms has the structure of a Picard groupoid
which we denote by $[\A,\B]$, see <cit.> for a detailed proof of this
assertion. One can associate another Picard groupoid with $\A$ and $\B$
which we denote by $\A \otimes \B$, and which will be called the
tensor product. We will not recall its construction, which
is rather elaborate, see <cit.>, but mention that
the tensor product 2-functor is determined by an adjunction
\[
[\A, [\B, \C]] \xrightarrow{\sim} [ \A \otimes \B, \C]
\]
in the bicategory of Picard groupoids. This bicategory also has a unit
object $I$ for the monoidal structure. The bicategory of Picard
groupoids, not only has an internal hom as indicated above, but it has
the structure of a $\Pic$-category, see appendix
<ref> for a definition of a
$\Pic$-category. More precisely, Picard groupoids, homomorphisms
between Picard groupoids and monoidal natural transformations between
homomorphisms form a $\Pic$-category which we denote by
$\Pic$. Further, $\Pic$ is the archtype example of a
$\Pic$-category. Our point of view on $\Pic$ is that it is the analog
of the category of Abelian groups, $\mathbf{Ab}$, in the world of
The groupoid of lines, i.e., one-dimensional vector spaces, and
$G$-torsors for a given abelian group $G$ have natural structures of
Picard groupoids with respect to tensor products and the product of
torsors over $G$, respectively. We will later focus our attention on
the Picard groupoid $\L$ of hermitian lines, where the hermitian form
on the tensor product of hermitian lines is the tensor product of the
hermitian forms one each line.
Let $X_\bullet$ be a simplicial set and $\A$ be a Picard groupoid. We
will define cohomology $H^\bullet (X_\bullet, \A)$ of $X_\bullet$ with
values in $\A$, following <cit.> and
<cit.>. Similar cohomology may be defined for topological
spaces and, more generally, with coefficients in sheaves of Picard
Let us associate with $X_\bullet$ and $\A$ a cosimplicial Picard
groupoid, that is to say, a cosimplicial object in the category of
Picard groupoids, defined as the “mapping space” $\A^{X_\bullet} :=
\Map (X_\bullet, \A)$: for each $n \ge 0$, we define the Picard
groupoid $\A^{X_n}$ whose objects are maps $X_n \to \Ob \A$, morphisms
are maps $X_n \to \Mor \A$, and the tensor product and morphism
composition are defined “point-wise.” The cosimplicial structure is
comprised of homomorphisms of Picard groupoids:
\[
\xymatrix{ \A^{X_0} \ar@/_/[rr]|{d_0^*} \ar@/_/@<-1ex>[rr]_{d_1^*} &&
\A^{X_1} \ar@/_/[ll]|{s_0^*} \ar@/_/[rr]|{d_0^*} \ar@/_/@<-1ex>[rr]
\ar@/_/@<-2ex>[rr]_{d_2^*} && \A^{X_2} \ar@/_/[ll]|{s_0^*}
\ar@/_/@<-1ex>[ll]_{s_1^*} \ar@/_/[rr]|{d_0^*} \ar@/_/@<-1ex>[rr]
\ar@/_/@<-2ex>[rr] \ar@/_/@<-3ex>[rr]_{d_3^*} && \ar@/_/[ll]|{s_0^*}
\ar@/_/@<-1ex>[ll] \ar@/_/@<-2ex>[ll]_{s_2^*} \dots },
\]
where the coface and codegeneracy homomorphisms $d_i^*: \A^{X_n} \to
\A^{X_{n+1}}$ and $s_j^* : \A^{X_{n+1}} \to \A^{X_n}$ are obtained by
composition with the face maps $d_i: X_{n+1} \to X_n$ and degeneracy
maps $s_j: X_n \to X_{n+1}$ of the simplicial set $X_\bullet$,
Now, by taking alternating sums, we obtain a $($cochain$)$
complex of Picard groupoids:
\begin{equation*}
C^\bullet (X_\bullet, \A): \xymatrix{ 0 \ar[r] &
\A^{X_0} \ar[r]^d \rruppertwocell<10>^0{\omit}
& \A^{X_1} \ar[r]^d \rrlowertwocell<-10>_0{<3>\chi} & \A^{X_2}
\lltwocell<\omit>{<3>\chi} \ar[r]^d \rruppertwocell<10>^0{\omit} &
\A^{X_3} \ar[r]^d & \lltwocell<\omit>{<3>\chi} \dots }
\end{equation*}
with $d = \sum_{i=0}^{n+1} (-1)^i d_i^* : \A^{X_n} \to \A^{X_{n+1}}$
and a monoidal transformation $\chi: d^2 \Rightarrow 0$, obtained in a
unique way from the structure isomorphisms $\alpha$, $m$ and $n$. This
system of coboundary homomorphisms $d$ and monoidal transformations
$\chi$ is coherent, i.e., $\chi d = d \chi$ as 2-cells $d^3
\Rightarrow 0$. Let $\2ch$ denote the $\Pic$-category of complexes
of Picard groupoids. The objects of $\2ch$ are complexes of
Picard groupoids. A 1-morphism between $\A^{\bullet},
\B^{\bullet} \in Ob(\2ch)$, pictured below:
\begin{equation*}
\xymatrix{ \dots
\A^{n-1} \ar[d]_{f^{n-1}} \ar[r]^{d_{\A}} \rruppertwocell<10>^0{\omit}
& \A^{n} \ar[d]_{f^{n}} \ar[r]^{d_{\A}} \ar@{=>}_{\phi^{n}}(12,-5)*{};(9,-9)*{}
& \A^{n+1} \ar[d]^{f^{n+1}} \lltwocell<\omit>{<3>\chi_{\A}}
\ar@{=>}^{\phi^{n+1}}(29,-5)*{};(26,-9)*{} \dots
\\
\dots
\B^{n-1} \ar[r]_{d_{\B}} \rrlowertwocell<-10>_0{\omit}
& \B^{n} \ar[r]_{d_{\B}} & \B^{n+1} \lltwocell<\omit>{<-3>\chi_{\B}} \dots },
\end{equation*}
is a pair $F = (f, \phi)$, where $f$ is a sequence of homomorphisms
$f^{n}: \A^{n} \rightarrow \B^{n}$ and $\phi$ is a sequence of
monoidal natural transformations $\phi^{n}:f^{n}d_{\A} \Rightarrow
d_{\B}f^{n-1}$ in $\Pic$, satisfying the following coherence conditions
$\phi^{n+1}d_{\A} = d_{\B}\phi^{n}$ and $(f^{n+1}\chi_\A) \circ
(\phi^{n+1}d_\A) \circ (d_\B \phi^{n}) = \chi_\B f^{n-1}$.
A 2-morphism $(f, \phi)\Rightarrow(f', \phi')$ is a sequence
$\lbrace \gamma_n \rbrace_{n \in \mathbb{Z}}$, where $\gamma_n:f^n
\Rightarrow f'^n$ is a monoidal natural transformation, for all $n \in \mathbb{Z}$,
and the following coherence condition is satisfied:
$(\gamma_{n+1}d_\A) \circ \phi_n = \phi'_n \circ (d_\B \gamma_n)$.
It would be useful to describe an alternative, equivalent, notion of
a $2$-morphism in $\2ch$ which is a generalization of cochain homotopy
to the Picard groupoid context. In this notion, a $2$-morphism is
also a pair $H = (h, \psi)$, where $h^{n}: \A^{n} \rightarrow
\B^{n-1}$ and $\psi$ is a sequence of monoidal natural transformations
$\psi^{n}: d_{\B}h^{n} + f^{n} \Rightarrow f'^{n} + h^{n+1}d_{\A}$
satisfying an obvious coherence condition.
We leave the establishment of an equivalence between the two notions
of $2$-morphisms in $\2ch$ as an excercise for an interested reader.
The cohomology $H^\bullet (X_\bullet, \A)$ of $X_\bullet$ with
coefficients in a Picard groupoid $\A$ is defined as the cohomology of
the complex $(\A^{X_\bullet}, d, \chi)$ of Picard groupoids. The
cohomology of a complex of Picard groupoids may be defined as
follows. In principle, to define the $n$th cohomology $H^n (X_\bullet,
\A)$, we want to take the kernel $\Ker d$ of the homomorphism
$d:\A^{X_n} \to \A^{X_{n+1}}$ and then the cokernel of the
homomorphism $d': \A^{X_{n-1}} \to \Ker d$ induced by $d:\A^{X_{n-1}}
\to \A^{X_{n}}$, but these need to be defined in a suitable
categorified sense. In particular, the kernels, cokernels, and
cohomology will depend on two subsequent coboundary homomorphisms $d$
as well as $\chi$ and be Picard groupoids. The objects of the category
$\Ker (d,\chi)$ (of $n$-cocycles) are pairs $(a,\phi)$ in which
$a$ is an object of $\A^{X_n}$ and $\phi: da \to 0$ is a morphism in
$\A^{X_{n+1}}$ satisfying a cocycle condition:
\[
d (\phi) = \chi_a: d^2 (a) \to 0.
\]
A morphism $(a,\phi) \to (a',\phi')$ in $\Ker (d,\chi)$ is given by a
morphism $f: a \to a'$ in $\A^{X_n}$ such that $ \phi'\circ d (f) =
\phi$. The monoidal structure on $\Ker (d,\chi)$ is inherited from
that of $\A$. The kernel $\Ker (d,\chi)$ naturally participates in a
complex of Picard groupoids, as follows:
\begin{equation*}
\xymatrix{ \A^{X_{n-2}} \ar[r]^d \rruppertwocell<10>^0{\omit}
& \A^{X_{n-1}} \ar[r]^{d'} & \Ker (d,\chi) \lltwocell<\omit>{<3>\chi'}
\end{equation*}
The cohomology $H^n (X_\bullet, \A)$ is defined as the cokernel
$\Coker (d',\chi')$ in this complex. The cokernel $\Coker (d',\chi')$
is a Picard category whose objects are the same as those of $\Ker
(d,\chi)$, i.e., of the type $(a,\phi)$, where $a$ is an object of
$\A^{X_{n}}$ and $\phi: da \to 0$ is a morphism in $\A^{X_{n+1}}$
satisfying the cocycle condition above. A morphism $(a,\phi) \to
(a',\phi')$ in $\Coker (d',\chi')$ is given by an equivalence class of
pairs $(b,f)$, where $b$ is an object of $\A^{X_{n-1}}$ and $f:
(a,\phi) \to (d'b + a', \chi_b + \phi')$ is a morphism in $\Ker
(d,\chi)$. Two morphisms $(b,f)$ and $(b',f'): (a,\phi) \to
(a',\phi')$ are equivalent, if there is a pair $(c,g)$ with $c$
being an object of $\A^{X_{n-2}}$ and $g: b \to dc + b'$ a morphism in
$\A^{X_{n-1}}$ such that the following diagram commutes:
\[
\begin{CD}
a @>f>> d'b + a' @>{d'(g)+ \id}>> (d'dc
@Vf'VV & & @VV{\alpha}V \\
d'b' + a' @<<{l}< 0 + (d'b'+a') @<<{\chi'_c + \id + \id}< dd'c
+(d'b'+a') .
\end{CD}
\]
One can check that $ \pi_0 (H^n (X_\bullet, \A)) \cong \pi_1
(H^{n+1}(X_\bullet, \A))$.
The simplicial homology of a simplicial set $X_\bullet$ with
coefficients in a Picard groupoid $\A$ may be defined similarly by
looking at the simplicial Picard groupoid $\A X_\bullet$ whose
$n$-simplices are formal “linear combinations” $a_1 s_1 + \dots +
a_k s_k$ of pairwise distinct elements $s_1, \dots, s_k$ in $X_n$ with
coefficients $a_1, \dots, a_k$ in $\A$. Perhaps, a better way of
looking at $\A X_\bullet$ is to view it as $\A$-valued functions on
$X_\bullet$ with finite support and apply the same treatment to it as
that for $\A^{X_\bullet}$. In particular, summing up the face
homomorphisms gives rise to a chain complex of Picard groupoids
$C_\bullet (X_\bullet, \A)$, which determines the homology Picard
groupoids $H_n (X_\bullet, \A)$ for $n \ge 0$.
When $A$ is an abelian group, we will think of it as a discrete
Picard groupoid, denoted $A[0]$, with $A$ being the set of objects and
identities being the only morphisms, so as $\pi_0 (A[0]) = A$ and
$\pi_1 (A[0]) = 0$. Then the (co)homology with coefficients in the
Picard groupoid $A[0]$ will be related to the usual simplicial
(co)homology with coefficients in the group $A$ as follows:
\begin{eqnarray*}
\pi_0 H^\bullet (X_\bullet; A[0]) & = & H^\bullet (X_\bullet; A),\\
\pi_0 H_\bullet (X_\bullet; A[0]) & = & H_\bullet (X_\bullet; A).
\end{eqnarray*}
§.§ Relative cohomology
Let $\A \in \Pic$, let $X_{\bullet}$ be a simplicial set, let
$Y_{\bullet} \subset X_{\bullet}$ be a simplicial subset. There is an
inclusion map $Y_{\bullet} \hookrightarrow X_{\bullet}$ in that category
of simplicial sets. This inclusion induces a $1$-morphism
\[
i_{\bullet}:C_{\bullet}(Y_{\bullet}, \A) \hookrightarrow C_{\bullet}(X_{\bullet},
\A)
\]
in $\2ch$. We define relative homology
$H_{\bullet}(X_{\bullet}, Y_{\bullet}, \A)$ to be the homology of the
2-chain complex given by the cokernel of $i_{\bullet}$ in $\2ch$.
We call this 2-chain complex, given by the cokernel, a relative 2-chain
complex so $H_{\bullet}(X_{\bullet}, Y_{\bullet}, \A)$ is the homology of
the relative 2-chain complex $C_{\bullet}(X_{\bullet}, Y_{\bullet}, \A)$.
The $nth.$ degree of the relative
2-chain complex is the Picard groupoid given by the cokernel, in the
category of Picard groupoids, of the map
$i_{n}:C_{n}(Y_{\bullet}, \A) \hookrightarrow C_{n}(X_{\bullet}, \A)$.
Relative cohomology is defined similarly, $H^{\bullet}(X_{\bullet},
Y_{\bullet}, \A)$ is the cohomology of the relative 2-cochain complex
given by the cokernel of the following map, induced by the inclusion
$Y_{\bullet} \hookrightarrow X_{\bullet}$
\[
i^{\bullet}:C^{\bullet}(Y_{\bullet}, \A) \to C^{\bullet}(X_{\bullet},\A).
\]
The objects of $i^{\bullet}(C^{n}(Y_{\bullet}, \A))$ are those functions,
$X_n \to Ob(\A)$, which vanish outside of $Y_{n}$.
$C^{n}(X_{\bullet}, Y_{\bullet}, \A)$ is a Picard subgroupoid
of $C^{n}(X_{\bullet}, \A)$ whose objects are the same as those of
$C^{n}(X_{\bullet}, \A)$. A morphisms in $C^{n}(X_{\bullet},
Y_{\bullet}, \A)$ is a certain equivalence class of morphisms in
$C^{n}(X_{\bullet}, \A)$. The cokernel also gives a $1$-morphism, in $\2ch$,
$p^{\bullet}:C^{\bullet}(X_{\bullet}, \A) \to C^{\bullet}(X_{\bullet}, Y_{\bullet},
\A)$ and a $2$-morphism $\phi^{\bullet}: p^{\bullet} \circ i^{\bullet} \Rightarrow
0:C^{\bullet}(Y_{\bullet}, \A) \to C^{\bullet}(X_{\bullet}, Y_{\bullet}, \A)$,
where $0$ is the zero homomorphism. If $\alpha
\in Ob(C^{n}(Y_{\bullet}, \A))$ then the natural transformation
$\phi^n$ assigns to $\alpha$, a morphism $i(\alpha) \to 0$ in
$C^{n}(X_{\bullet}, Y_{\bullet}, \A)$. In other words those objects
of $C^{n}(Y_{\bullet}, \A)$ are isomorphic to the zero object in
$C^{n}(X_{\bullet}, Y_{\bullet}, \A)$.
§.§ Functoriality
The $n$th cohomology (and $n$th homology) defined above is a functor of
$\Pic$-categories, see appendix <ref>,
$H^{n}:\2ch \rightarrow \Pic$. Moreover, every $F \in
\Mor_{\2ch}(\A^{\bullet}, \B^{\bullet})$ determines a morphism
$H^{\bullet}(F) \in \Mor_{\2ch}( H^\bullet (\A^\bullet ),\newline H^{\bullet}
(\B^\bullet))$ on cohomology. This fact follows from properties of
relative kernels and cokernels; for a direct proof of this fact see
Note that the described cohomology and homology are (strictly)
functorial with respect to simplicial maps. If $\A$ is Picard
groupoind and $f: X_\bullet \to Y_\bullet$ is a simplicial map, then
we get a strict morphism between the corresponding cochain complexes
of Picard groupoids $f^*: C^\bullet (Y_\bullet, \A) \to C^\bullet
(X_\bullet, \A)$, which yields a strict morphism on cohomology $f^*:
H^n (Y_\bullet, \A) \to H^n (X_\bullet, \A)$ for $n \ge 0$. Moreover,
a simplicial homotopy between two simplicial maps induces a monoidal
natural transformation on cohomology, cf. <cit.> and <cit.> and the discussion of 2-morphisms in
$\2ch$ in Section <ref>. The same statements are true for
§.§ The long 2-exact sequence
We begin this subsection by recalling the notion of a short 2-exact
sequence of Picard groupoids. Here we will only recall this notion in
a subcategory of $\Pic$ which has the same objects as $\Pic$ and whose
morphisms are homomorphisms which preserve the unit of addition. For
the general case see <cit.>. A complex
\[
\xymatrix{
0 \ar[r] &\A \ar[r]_F \rruppertwocell<10>^0{\omit} &\C \ar[r]_G
&\B \lltwocell<\omit>{<3>\phi} \ar[r] &0
\\%\ar[ru]_G \ar@{=>}[u]_{\phi}
\]
is called a short $2$-exact sequence of Picard groupoids if the
unique morphism $\overline{G}: \Coker(F, \id_0) \to \B$ is full and
faithful and further, $\pi_1 (\Ker(F, \phi)) = 0$ and $\pi_0(\Coker(G,
\phi)) = 0$.
A 2-exact sequence of complexes of Picard groupoids is a diagram
\begin{equation}
\label{short-2-exact-complex}
\xymatrix{
0 \ar[r] &\A^{\bullet} \ar[r]_{F^\bullet} \rruppertwocell<10>^0{\omit} &\B^{\bullet}
\ar[r]_{G^\bullet} & \C^{\bullet} \lltwocell<\omit>{<3>\phi^\bullet} \ar[r] &0,
\\%\ar[ru]_G \ar@{=>}[u]_{\phi}
\end{equation}
where $F^\bullet$ and $G^\bullet$ are $1$-morphisms and $\phi^\bullet$
is a $2$-morphism in $\2ch$, such that in every degree, the above
diagram in $\2ch$, reduces to a short $2$-exact sequence of Picard
The following example of a short $2$-exact sequence is of particular
interest and would be referenced frequently.
Let $X_{\bullet}$ be a simplicial set and let $Y_{\bullet} \subset
X_{\bullet}$ be a simplicial subset. Then for any Picard groupoid
$\A$, there is a morphism $i:\C^{\bullet}(Y_{\bullet}; \A) \to
\C^{\bullet} (X_{\bullet}; \A)$ of (cochain) complexes of Picard
groupoids. This morphism determines a short $2$-exact sequence of
complexes of Picard groupoids:
\begin{equation}
\label{cohomology-short-2-exact-seq}
\xymatrix{ 0 \ar[r] &{\C^{\bullet}(X_{\bullet}, Y_{\bullet}; \A)}
\ar[r] \rruppertwocell<10>^0{\omit}
&{\C^{\bullet}(X_{\bullet}; \A)} \ar[r]_{i}
&{\C^{\bullet}(Y_{\bullet}; \A)} \lltwocell<\omit>{<3>\pi
\ \ } \ar[r] &0. \\%\ar[ru]_G \ar@{=>}[u]_{\phi}
\end{equation}
The inclusion of simplicial sets induces another morphism of (chain)
complexes of Picard groupoids, $i: \C_{\bullet}(Y_{\bullet}; \A) \to
\C_{\bullet} (X_{\bullet}; \A)$. This morphism determines a short
$2$-exact sequence of (chain) complexes of Picard groupoids:
\begin{equation}
\label{homology-short-2-exact-seq}
\xymatrix{ 0 \ar[r] &{\C_{\bullet}(Y_{\bullet}; \A)} \ar[r]_{i}
\rruppertwocell<10>^0{\omit} &{\C_{\bullet}(X_{\bullet}; \A)}
\ar[r] &{\C_{\bullet}(X_{\bullet}, Y_{\bullet}; \A)}
\lltwocell<\omit>{<3>\pi \ \ } \ar[r] &0.
\\%\ar[ru]_G \ar@{=>}[u]_{\phi}
\end{equation}
A short 2-exact sequence of complexes (<ref>)
has an associated long $2$-exact sequence of cohomology
\begin{equation}
\label{long-2-exact-sequence}
\xymatrix@C=15pt{ \dots \to
H^n(\A^{\bullet}) \ar[r]_{\;\;\;\;\;\; H^n(F)} \rruppertwocell<10>^0{\omit}
& H^n(\B^{\bullet}) \ar[r]^{H^n(G)} \rrlowertwocell<-10>_0{<3>\ \ \Sigma^n } & H^n(\C^{\bullet})
\lltwocell<\omit>{<3>H^n(\phi) \ \ \ \ \ } \ar[r]^{\partial^n} \rruppertwocell<10>^0{\omit} &
H^{n+1}(\A^{\bullet}) \ar[r]_{\!\!\!\!\! H^{n+1}(F)} & \lltwocell<\omit>{<3>\Psi^n \ \ } H^{n+1}(\B^{\bullet}) \to
\dots }
\end{equation}
We will briefly outline the construction of the 1-morphism
$\partial^n$ and the $2$-morphism $\Psi^n$ here. For a more
elaborate description of the various components of this long exact
sequence, we refer the interested reader to Section 4 of
<cit.>. For an outline we will refer to the following diagram:
\begin{equation*}
\xymatrix@C=22pt@!C{ \!\!\!\!\!\!\!\!\!\!\!\!\!\! \dots \to
\A^{n} \ar[d]_{d^n_{\A}} \ar[r]^{f^n} \rruppertwocell<10>^0{\omit}
& \B^{n} \ar[d]_{d^n_{\B}} \ar[r(0.8)]^{g^n} \ar@{<=}_{\lambda^{n}}(20,-5)*{}
;(16,-9)*{} & \;\;\;\;\;\;\;\;\;\;\; \C^{n} \ar[d]^{d^n_{\C}} \lltwocell<\omit>{<3>\!\! \phi^{n}}
\ar@{<=}^{\mu^{n}}(57,-5)*{};(53,-9)*{} \;\;\; \to \dots
\\
\!\!\!\!\!\!\!\!\! \dots \to
\A^{n+1} \ar[r]_{f^{n+1}} \rrlowertwocell<-10>_0{<2.7>
\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\phi^{n+1}}
& \B^{n+1} \ar[r(0.8)]_{g^{n+1}} & \;\;\;\;\;\;\;\;\;\; \C^{n+1} \to \dots }
\end{equation*}
Let $(C_n, c_n: d_{\C}^n(C_n) \to 0)$ be an object in $Ker(d_{\C}^n)$;
since $\pi_1(\Coker(g_n,\phi^{n})) = 0$, there is a $B_n \in \B^n$ and
$i: g^n(B_n) \to C_n$. Since the following pair
\[
(d_{\B}^n(B_n), c_n \circ d_{\C}^n(i) \circ \mu_n(B_n):g^{n+1}(d_{\B}^n(B_n)) \to d_{\C}^n(g^n(B_n))
\to d_{\C}^n(C_n) \to 0)
\]
is an object of $Ker(g^{n+1})$ and the factorization of
$f^{n+1}:\A^{n+1} \to \B^{n+1}$ through $Ker(g^{n+1})$ is an
equivalence, there are $A_{n+1} \in \A^{n+1}$ and $j: f^{n+1}(A_{n+1})
\to d^n_{\B}(B_n)$ such that
\[
g^{n+1}(j) \circ c_n \circ d_{\C}^n(i) \circ \mu_n(B_n) =
\phi^{n+1}(A_{n+1}).
\]
We now need an arrow $a_{n+1}:d^{n+1}_{\A} \to 0$. Since the
factorization ${f'}^{n+2}$ of $f^{n+2}$ through $Ker(g^{n+2})$ is an
equivalence of categories, it is enough to find an arrow
${f'}^{n+2}(d^{n+1}_{\A}(A_{n+1})) \to {f'}^{n+2}(0)$. This is given
by the following:
\[
\chi_{\B}(B_n) \circ d^{n+1}_\B(j) \circ \lambda^{n+1}(A_{n+1}): f^{n+2}
(d^{n+1}_\A(A_{n+1})) \to 0 \cong f^{n+2}(0).
\]
We put $\partial^{n}(C_n, c_n) := (A_{n+1}, a_{n+1})$. This is an
object of $H^{n+1}{\A^{\bullet}}$: the condition $d^{n+1}_\A(a_{n+1})
= \chi_\A (A_{n+1})$ can be easily checked by applying the faithful
functor $f^{n+3}$. The arrow function of the functor $\partial^n$ has
a much more elaborate description and moreover it is not used in the
construction of our theory. We will refer the interested reader to
Before we can describe a construction of the $2$-morphism $\Psi^n$, we
need another description of $H^{n}(\C^\bullet)$. Since $(f^n, \phi^n,
g^n)$ is a $2$-short exact sequence, $\C^n$ is equivalent to the
cokernel of $f^n$, we get the following alternative description of
$H^n(\C^\bullet)$. An object is a pair
\[
(B_n \in \B^n, [A_{n+1} \in \A_{n+1}, a_{n+1}: d^n_{\B}(B_n) \to
\]
where $[A_{n+1}, a_{n+1}] \in \Mor_{\Coker (f^{n+1},\id_0)}(d^n_{\B}(B_n), 0)$,
such that there exists an arrow $t^{n+2}:d^n_{\A}(A_{n+1}) \to 0$
making the following diagram commutative
\[
\begin{CD}
d^{n+1}_{\B}(d^n_{\B}(B_n)) @>{d^{n+1}_{\B}(a_{n+1})}>>
@V{\chi_\B(B_n)}VV @VV{(\lambda^{n+1})^{-1}(A_{n+1})}V \\
0 @<<{f^{n+2}(t^{n+2})}< f^{n+2}(d^{n+1}_{\A}(A_{n+1})) .
\end{CD}
\]
Note that $t^{n+2}$ is necessarily unique because $f^{n+2}$ is faithful.
Now we begin the construction of $\Psi^n$, given an object
\[
(B_n \in \B^n, [A_{n+1} \in \A^{n+1}, a_{n+1}: d^n_{\B}(B_n) \to
\]
in $H^n(C^\bullet)$, we apply $\partial^n$ and $H^{n+1}(f)$ and
obtain the following object of $H^{n+1}(\B^\bullet)$:
\[
(f^{n+1}(A_{n+1}), f^{n+2}(t^{n+2}) \circ \lambda^{-1}_{n+1}
(A_{n+1}):d^{n+1}_{\B}(f^{n+1}(A_{n+1})) \to f^{n+2}(d^{n+1}_{\A}
(A_{n+1})) \to 0).
\]
This object is naturally isomorphic to the unit of the addition
$0 \in H^{n+1}(\B^\bullet)$ via the following morphism which we
take as the definition of $\Psi^n$ on the object
$(B_n, [A_{n+1}, a_{n+1}])$
\[
\Psi^n(B_n, [A_{n+1}, a_{n+1}]) := [B_n \in \B^n, a_{n+1}^{-1}:
f^{n+1}(A_{n+1}) \to d_\B^n(B_n)].
\]
The following example describes the images under the morphism
$\partial^n$ and the natural transformation $\Psi^n$ of an object in
degree $n$ of the cohomology sequence associated to the $2$-short
exact sequence (<ref>).
Let $X$ be a compact finite-dimensional manifold with boundary
$\partial X$. We denote by $H_n(X, \partial X;\L)$ the $n$th
homology Picard groupoid of the chain complex
$C_{\bullet}(X_\bullet, \partial X_{\bullet}; \L)$. Let $(X_{n},
[X'_{n-1},x'_{n-1}]) \in \Ob \ H_n(X, \partial X;\L)$, where $X_n
\in \Ob \ C_n(X_\bullet; \L)$ and the morphism
$[X'_{n-1},x'_{n-1}]:d(X_n) \to 0$ in $\Mor \ C_{n-1}(X_\bullet;
\partial X_\bullet; \linebreak[0] \L)$ consists of an object
$X'_{n-1} \in C_{n-1} (\partial X_\bullet;\L)$ and a morphism
$x'_{n-1}:d^n(X_n) \to X'_n$ $\in \Mor \ C_{n-1}(X_\bullet;
\L)$. The coboundary $\partial_n (X_{n}, [X'_{n-1},x'_{n-1}])$ is
the pair $(X'_{n-1}, \linebreak[0] x'_{n-1}) \in \Ob \
H_{n-1}(\partial X;\L)$. The natural transformation $\Psi_n(X_{n},
[X'_{n-1},x'_{n-1}])$ is a morphism in $ \Hom_{H_{n-1}
(X;\L)}((X'_{n-1}, x'_{n-1}), 0)$ given by the equivalence class
$[X_n,(x'_{n-1})^{-1}]$. Thus every object in $H_n(X, \partial
X;\L)$ produces a morphism in $H_{n-1}(X ;\L)$.
§.§ The Cap Product
In this section we develop a cap product between cohomology with
coefficients in a Picard groupoid and homology with coefficients in
the Picard groupoid $\Z[0]$:
\[
\cap : H_\bullet (X_\bullet, \Z[0]) \otimes H^\bullet(X_\bullet, \A)
\rightarrow H_\bullet (X_\bullet, \A).
\]
In order to do that, we will define a chain map i.e a morphism in $\2ch$
\[
H_\bullet (X_\bullet, \Z[0]) \to [H^\bullet(X_\bullet, \A), H_\bullet (X_\bullet, \A)].
\]
We start by defining the following chain map
\begin{equation}
\label{chain-map-cap-product}
\cap^{ch}:C_\bullet (X_\bullet, \Z[0]) \to [C^{\bullet}(X_\bullet, \A) , C_\bullet (X_\bullet, \A)]
\end{equation}
where the right hand side is the chain complex $[C^\bullet (X_\bullet,
\A),C_\bullet (X_\bullet, \A)]_{\bullet}$ defined in Appendix 2. We
define the map in degree $p$ as follows: On objects the chain map is
given by
\[
\sigma_{q} \mapsto \underset{q \ge p}\prod F_{q},
\]
where $F_{q} \in \Mor(\Pic)$ is defined on objects by
\[
F_{q}: \alpha \mapsto \alpha(d_{f}^{p}(\sigma_{q}))d_{l}^{q-p}(\sigma_{q}),
\]
where $d_{f}^{p}$ and $d_{l}^{q-p}$ are the restrictions of $d:C_{q+1}
(X_\bullet, \Z[0]) \rightarrow C_{q} (X_\bullet, \Z[0]) $ to the
simplex determined by the first $p+1$ vertices and the last $q-p+1$
vertices of $\sigma_{q}$ respectively. On morphisms, $F_{q}$ given by
\[
\to \beta \rbrace \mapsto \lbrace \alpha(d_{f}^{p}(\sigma_{q}))d_{l}^{q-p}(\sigma_{q})
\to \beta(d_{f}^{p}(\sigma_{q}))d_{l}^{q-p}(\sigma_{q}) \rbrace
\]
The map on the right side is determined by the natural
transformation $\alpha \to \beta$.
This chain map induces a map on homology
\begin{equation}
H_{\bullet}(X_\bullet, \Z[0]) \to H_{\bullet}( [C^\bullet (X_\bullet, \A), C_\bullet (X_\bullet, \A)]).
\end{equation}
Composition with the following obvious morphism gives us the desired chain map
\begin{equation}
\label{homology-to-homology-of-internal-hom}
H_{\bullet}( [C^\bullet (X_\bullet, \A), C_\bullet (X_\bullet, \A)]) \to [H^\bullet (X_\bullet, \A), H_\bullet (X_\bullet, \A)].
\end{equation}
§.§ Relative cap product
We now construct a relative version of the cap product. The $2$-functor
$[C^{\bullet}(X_{\bullet}; \A), -]:\2ch \to \2ch$ and the chain map
(<ref>), determine a composite $1$-morphism
and a $2$-morphism $\phi_{\bullet}$ in $\2ch$
\[
\xymatrix@C=1pt@!C{
C_{\bullet}(Y_{\bullet}; \Z[0]) \ar[r]^{i_\bullet} \rruppertwocell<10>^0{\omit}
& C_{\bullet}(X_{\bullet}; \Z[0]) \ar[r]^{\cap^{ch}_{rel} \ \ \ \ \ \ } &[C^{\bullet}(X_{\bullet}; \A),
C_{\bullet}(X_{\bullet}, Y_{\bullet}; \A)] \lltwocell<\omit>{<3>\phi_{\bullet} \ \ }. \\
\]
In order to define the $1$-morphism $\cap^{ch}_{rel}$ and the $2$-morphism $\phi_\bullet$
in the above diagram, we need to define a
restriction of the chain map <ref>. The image of the restriction
of this chain map to $i_{\bullet}(C_{\bullet}(Y_{\bullet}; \Z[0]))$ is
contained in the $2$-(chain) complex $[C^{\bullet}(X_{\bullet}; \A),
C_{\bullet}(Y_{\bullet}; \A)]$, this determines the following commutative diagram
\[
\xymatrix@C=65pt@R=35pt{
i_\bullet(C_{\bullet}(Y_{\bullet}; \Z[0]))
\ar[r]^{\cap^{ch}|_{i_\bullet(C_{\bullet}(Y_{\bullet}; \Z[0]))} \ \ } \ar[rd]_{\cap^{ch}_{Y_\bullet}}
&[C^{\bullet}(X_{\bullet}; \A),C_{\bullet}(X_{\bullet}; \A)] \\
&[C^{\bullet}(X_{\bullet}; \A),C_{\bullet}(Y_{\bullet}; \A)] \ar[u]_{[C^{\bullet}(X_{\bullet}; \A),i_\bullet]}
\]
The following composite chain map will be called the restricted relative cap product chain map.
\begin{equation}
\label{restricted-cap-product}
\cap^{ch}_{res}:C_{\bullet}(Y_{\bullet}; \Z[0]) \overset{i_\bullet}
\to i_\bullet(C_{\bullet} (Y_{\bullet}; \Z[0])) \overset{\cap^{ch}_{Y_\bullet}}
\to [C^{\bullet}(X_{\bullet}; \A), C_{\bullet} (Y_{\bullet}; \A)]
\end{equation}
The $2$-morphism $\phi_{\bullet}$ is the composition $\cap^{ch}_{res}
\circ ([C^{\bullet}(X_{\bullet}; \A),\pi_{\bullet}^{\A})])$ as
described in the following diagram
\[
\xymatrix@C=10pt{
C_{\bullet}(Y_{\bullet}; \Z[0]) \rruppertwocell<12>^0{\omit}
\ar[d]_{\cap^{ch}_{res}} \ar[r]^{i_\bullet} &C_{\bullet}(X_{\bullet}; \Z[0])
\ar[d]_{\cap^{ch}} \ar[rd]_{\cap^{ch}_{rel}} \ar[r]^{p_\bullet} &C_{\bullet}(X_{\bullet}, Y_{\bullet};
\Z[0]) \ar@{=>}^{\ \ \lambda_\bullet}(72,-4)*{} ;(67,-6)*{} \ar@{-->}[d]^{u}
\lltwocell<\omit>{<3> \pi_{\bullet} }.\\
[C^{\bullet}(X_{\bullet}; \A), C_{\bullet}(Y_{\bullet}; \A] \ar[r] \rrlowertwocell
<-12>_0{<3> \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [C^{\bullet}(X_{\bullet}; \A),\pi_{\bullet}^{\A})] }
& [C^{\bullet}(X_{\bullet}; \A),C_{\bullet}(X_{\bullet}; \A) \ar[r]
&[C^{\bullet}(X_{\bullet}; \A), C_{\bullet}(X_{\bullet}, Y_{\bullet}; \A)] \\
% \lltwocell<\omit>{ }. \\
\]
where $\pi_{\bullet}^{\A}$ is the following $2$-morphism
\[
\xymatrix@C=26pt{
C_{\bullet}(Y_{\bullet}; \A) \ar[r]^{i_{\bullet}} \rruppertwocell<12>^0{\omit}
& C_{\bullet}(X_{\bullet}; \A) \ar[r]_{p^\A_\bullet } &
C_{\bullet}(X_{\bullet}, Y_{\bullet}; \A) \lltwocell<\omit>{<3>\pi_{\bullet}^{\A} \ \ }. \\
\]
The universality of the cokernel determines a unique pair consisting of a $1$-morphism in $\2ch$
\[
u:C_{\bullet}(X_{\bullet}, Y_{\bullet};\Z[0]) \to [C^{\bullet}(X_{\bullet}; \A),
C_{\bullet}(X_{\bullet}, Y_{\bullet}; \A)],
\]
and a $2$-morphism in $\2ch$, $\lambda_\bullet: u \circ p_\bullet \Rightarrow \cap^{ch}_{rel}$
such that the following diagram commutes
\[
\xymatrix@C=20pt{
u \circ p_\bullet \circ i_\bullet \ar@{=>}[r]^{\ \ \ \ \ \ u \cdot \pi_\bullet}
\ar@{=>}[d]_{\lambda_\bullet \cdot i_\bullet} & u \circ 0 \ar@{=>}[d] \\
\cap^{ch}_{rel} \circ i_\bullet \ar@{=>}[r]_{\phi_\bullet} &0
\]
For more details on the universality of this cokernel we refer the
interested reader to <cit.>. This unique $1$-morphism, $u$,
induces the following $1$-morphism on passing to homology
\begin{equation}
H_{\bullet}(X_\bullet, Y_\bullet, \Z[0]) \to H_{\bullet}( [C^\bullet (X_\bullet, \A),
C_\bullet (X_\bullet, Y_\bullet, \A)]).
\end{equation}
Composition with the following chain map
\begin{equation}
\label{Relative-homology-to-homology-of-internal-hom}
H_{\bullet}( [C^\bullet (X_\bullet, \A), C_\bullet (X_\bullet, Y_\bullet, \A)]) \to
[H^\bullet (X_\bullet, \A), H_\bullet (X_\bullet, Y_\bullet, \A)].
\end{equation}
and the adjointness of the tensor product gives us the desired chain map
\begin{equation}
\label{relative-cap-product}
\cap : H_\bullet (X_\bullet, Y_\bullet, \Z[0]) \otimes H^\bullet(X_\bullet, \A)
\rightarrow H_\bullet (X_\bullet, Y_\bullet, \A).
\end{equation}
which we will call the relative cap product.
As in the classical case, the boundary map in homology is natural with
respect to relative cap product, in a sense made precise below.
The following diagram of Picard groupoids is commutative up to
natural isomorphism for $p-1 \ge q \ge 0$:
\begin{equation*}
\xymatrix{
H_{p} (X_\bullet, Y_\bullet; \Z[0]) \otimes H^q(X_\bullet, \A)
\ar[r]^-{\cap} \ar[d]_{\del \otimes i^*}& H_{p-q} (X_\bullet, Y_\bullet; \L)
\ar[d]^{\partial} \ar@{=>}(30,-5)*{};(20,-9)*{}
\\
H_{p-1} (Y_\bullet; \Z[0]) \otimes H^q(Y_\bullet, \A) \ar[r]^-{\cap}
& H_{p-q-1} (Y_\bullet; \L)
\end{equation*}
§ HERMITIAN LINE GERBES
In this section we describe geometric objects which we call
$($flat$)$ hermitian line $n$-gerbes. Then we give an example
describing a over the simplicial set $BG$, where $G$ is a
(discrete) group. We move on to describe certain geometic objects over
a topological space $X$ which are classified by the cohomology of
$X$ with coefficients in $U(1)$ and which we call s over
$X$. We describe these in two ways. For the first description, we
define a category $\cu$, associated to an open cover of $X$ and show
that hermitian line $0$-cocycles on the simplicial set $N(\cu)$
represent (flat) s. Our second description is that a flat
can be represented by a functor from the first fundamental
groupoid of $X$ into the Picard groupoid of hermitian lines
$\L$. Finally, we move on to describe higher hermitian line gerbes
over $X$.
A hermitian line $n$-gerbe on a simplicial set $X_\bullet$ is
an an $n$-cocycle on the simplicial set $X_\bullet$ with values in $\L$
i.e. an object $K \in \Ob H^n (X_\bullet, \L)$ of degree $n$ cohomology
Picard groupoid of $X_\bullet$ with coefficients in the Picard groupoid
$\L$ of hermitian lines.
A $n$-gerbe should be properly defined as a $0$-cocycle with
coefficients in an appropriate Picard $(n+1)$-groupoid
but this would be out of scope of this paper.
The following example shows that a $2$-gerbe on $BG$, where
$G$ is a finite group, is exactly the same as a $2$-cocycle
with values in hermitian lines as defined in <cit.>.
Any group $G$ can be viewed as a category with a single object.
The simplicial set $BG$ is the nerve of this category.
An object of $H^2(BG;\L)$ consists of a pair $(\beta, \phi:d\beta \to 0)$,
where $\beta \in \Ob C^2(BG;\L)$ and $\phi$ is an arrow in $C^3(BG;\L)$
and $d$ is the differential of the $2$-complex $C^\bullet(BG;\L)$.
$\beta$ is a set function whose domain is the underlying set
of $G \times G$ and codomain is $\Ob \L$ i.e. it assigns to
each pair $(g_1, g_2) \in G \times G$ a hermitian line
$l_{g_2,g_1} \in \Ob \L$. The arrow $\phi$ gives, for
every triple $(g_1, g_2, g_3) \in G \times G \times G$,
a following isomorphism in $\L$
\[
t_{g_3, g_2, g_1}:l_{g_3, g_2} - l_{g_3,g_2g_1}
+l_{g_3g_2, g_1} - l_{g_2, g_1} \to \mathbb{C}.
\]
The morphism $d(\phi):d^2(\beta) \to 0$
gives, for every quadruple $(g_1, g_2, g_3, g_4) \in G \times G
\times G \times G$, the following isomorphism
\[
t_{g_4,g_3,g_2} - t_{g_4,g_3,g_{21}} + t_{g_4,g_{32},g_1}
-t_{g_{43,g_2,g_1}} + t_{g_3,g_2,g_1},
\]
which is the canonical isomorphism $\chi_\beta((g_1, g_2, g_3, g_4)):
d^2\beta((g_1, g_2, g_3, g_4)) \to \mathbb{C}$.
By the definition above, a flat on a simplicial set $X_\bullet$
is just an object of the Picard groupoid $H^0(X_\bullet; \L)$. Thus,
given a topological space $X$, we may look at s over $X$ in two
ways: associating simplicial sets $\Sing_\bullet X$ and $N(\cu)$ to
$X$, where $N(\cu)$ is the simplicial sets obtained by taking the
nerve of a category associated to the cover of $X$, $\cu$, which we
now define:
We define $\cu$ to be a category whose object set is
the collection $\mathfrak{U}_I = \lbrace U_i: i \in I \rbrace\ $,
which is a chosen open cover of the topological space $X$. If the
set $U_{i,j} \neq \emptyset$, then $\Hom_{\cu}(U_i, U_j)
= \lbrace U_{i,j} \rbrace$, otherwise the set
$\Hom_{\cu}(U_i, U_j) = \emptyset$. Composition in $\cu$
is defined as follows: $U_{i,j} \circ U_{j,k} := U_{i,j,k}
:= U_{i,j} \cap U_{j,k}$. $id_{U_{i}} := U_{ii}$. The source
of an arrow $U_{i,j}$ is $U_i$ and its target is $U_j$.
This leads to two interpretations of flat s: Definitions
<ref> and <ref> below.
A flat over $X$, $\mathcal{G}^0(\Lambda, \theta)$, is
defined by the following data
1. A function $\Lambda_0:(\Sing_\bullet X)_0 \to \Ob(\L)$,
i.e. an assignment of a hermitian line to each point of $X$.
2. A function $\Lambda_1:(\Sing_\bullet X)_1 \to \Mor(\L)$
which assigns to each $f \in (\Sing_\bullet X)_1$, a linear isometry
$\Lambda_1(f):\Lambda_0\partial_1(f) \to \Lambda_0\partial_0(f)$ in
$\L$ such that for all $f, g \in (\Sing_\bullet X)_1$ satisfying
$\partial_1(f) = \partial_0(g)$, $\Lambda_1(g \circ f) = \Lambda(g)
\circ \Lambda(f)$.
This data is subject to the following condition. For each
$n \ge 2$, there exists a function $\Lambda_n:(Sing_\bullet X)_n \to \Mor(\L)$
such that for all $\sigma_n \in (Sing_\bullet X)_n$,
$\Lambda_n(\sigma_n) = \Lambda_{n-1}(\partial_0\sigma_n) \circ
\Lambda_{n-1}(\partial_1\sigma_n) \circ \cdots \circ \Lambda_{n-1}
The above definition assigns a hermitian line to each point of $X$.
Further, two homotopic paths in $X$, relative to endpoints,
are assigned the same linear isometry. In other words the above data
is equivalent to defining a functor from the first fundamental groupoid
of the space $X$, $\Pi_1(X)$, to $\L$.
A flat over $X$, $\mathcal{G}^0(\Lambda, \theta)$, is
defined by the following data
1. A constant hermitian line bundle $\Lambda_i$
over every open set $U_{i}$ for all $i \in I$.
2. For each ordered pair of distinct indices
$(i, j) \in I \times I $, a constant, non-zero section
\[
\theta_{i,j} \in \Gamma(U_{i,j}; \Lambda_i \otimes
\Lambda_j)
\]
This data is subject to a cocycle condition, on $U_{i,j,k}$
which we denote by $\delta \theta \Rightarrow 0$.
The cocycle condition is that over any three fold
intersections $U_{i,j,k}$, we can tensor the three
sections of the coboundary to give a trivialization
of the following hermitian line bundle
\begin{equation*}
% \label{Delta-Lambda}
(\Lambda_i \otimes \Lambda_j) \bigotimes
(\Lambda_i \otimes \Lambda_k)^{-1} \bigotimes
(\Lambda_j \otimes \Lambda_k).
\end{equation*}
over $U_{i,j,k}$. Notice that the above hermitian line bundle
is canonically trivial, so the cocycle condition
is the requirement that the following
\[
\theta_{i,j} - \theta_{i,k} + \theta_{j,k}
\]
be the canonical section of this trivial hermitian line bundle
over $U_{i,j,k}$.
Each point $x \in X$ has a neighborhood $U_i$ such that the
hermitian line bundle $\Lambda_i$ is isomorphic to the
trivial hermitian line bundle $U_i \times \mathbb{C}$.
Further, the specification of constant, non-zero section
$\theta_{i,j}$ is the same as specifying a hermitian line
bundle isomorphism $g_{i,j}:\Lambda_i|_{U_{i,j}} \to \Lambda_j|_{U_{i,j}}$,
which restricts to the same linear isometry on every fiber.
These two observations along with the data in the definition
above are sufficient to construct a (flat) hermitian line bundle
over the space $X$.
Now we move on to define higher hermitian line gerbes. Our definition
of a closely follows the definition of a “1-gerb” developed
in <cit.>.
A over $X$, $\mathcal{G}^1(\Lambda, \theta)$, is defined
by the following data
1. A constant hermitian line bundle $\Lambda_i^j$
over the intersection $U_{i,j}$ for every ordered pair
$(i, j) \in I \times I$ and $i \ne j$, such that
$\Lambda_i^j$ and $\Lambda_j^i$ are dual to each other.
2. For each ordered triple of distinct indices
$(i, j, k) \in I \times I \times I$, a nowhere zero section
\[
\theta_{i,j,k} \in \Gamma(U_{i,j,k}; \Lambda_i^j \otimes
\Lambda_j^k \otimes \Lambda_k^i)
\]
such that the sections of reorderings of triples
$(i, j, k)$ are related in the natural way.
This data is subject to a cocycle condition, on $U_{i,j,k,l}$
which we denote by $\delta \theta \Rightarrow 0$.
The cocycle condition is that over any four fold
intersections $U_{i,j,k,l}$, we can tensor the four
sections of the coboundary to give a trivialization
of the following hermitian line bundle
\begin{equation}
\label{Delta-Lambda}
(\Lambda_i^j \otimes \Lambda_j^k \otimes \Lambda_k^i) \bigotimes
(\Lambda_i^j \otimes \Lambda_j^l \otimes \Lambda_l^i)^{-1} \bigotimes
(\Lambda_i^k \otimes \Lambda_k^l \otimes \Lambda_l^i) \bigotimes
(\Lambda_j^k \otimes \Lambda_k^l \otimes \Lambda_l^j)^{-1}.
\end{equation}
over $U_{i,j,k,l}$. Notice that the above hermitian line bundle
is canonically trivial, so the cocycle condition
is the requirement that the following
\[
\theta_{i,j,k} - \theta_{i,j,l} + \theta_{i, k, l}
\]
be the canonical section of this trivial hermitian line bundle
over $U_{i,j,k,l}$.
The tensor product of two s is obtained by tensoring line bundles
and sections in an obvious way.
Let $(\alpha, \phi) \in \Ob H^1(N(\cu;\L)$. To each $U_{i,j}$, the
cochain $\alpha$ assigns a hermitian line $l_i^j$ and the morphism
$\phi$ specifies a linear isometry for each $U_{i,j,k}$
\[
\phi(U_{i,j,k}):l_i^j - l_k^i + l_j^k \to \mathbb{C}.
\]
Equivalently, the specification of this linear isometry
is the specification of a constant function
$t_{i, j, k}:U_{i,j,k} \to U(1)$. In other words
\[
t_{i,j,k}(x) = \phi(U_{i,j,k}),
\]
$\forall x \in U_{i,j,k}$. Put constant
hermitian line bundles $\Lambda_i^j = U_{i,j}
\times l_i^j$ over each $U_{i,j}$. Then $t_{i,j,k}$
gives a trivialization of the coboundary line bundle
$\Lambda_i^j \otimes \Lambda_j^k \otimes \Lambda_k^i$.
We define the section
\[
\theta_{i,j,k}(x) = t_{i,j,k}^{-1}(x, e_1).
\]
The morphism
\[
dt_{i,j,k} = d\phi(U_{i,j,k,l}) = t_{i,j,k} -t_{i,j,l}
+ t_{i,k,l} - t_{j,k,l}
\]
gives a trivialization of the line bundle (<ref>) over
$U_{i,j,k,l}$. The following section corresponds to the above
trivialization of the the hermitian line bundle (<ref>)
\[
\theta_{i,j,k} - \theta_{i,j,l} + \theta_{i,k,l}
\]
Clearly this is the canonical section.
Conversely, a over $X$ defines a $1$-cocycle in $H^1(N(\cu;
\L))$. We leave the easy verification of this fact as an excercise
for the reader.
Let $\mathcal{G}^1(\Lambda, \theta)$ and $\mathcal{H}^1(\Upsilon,
\eta)$ be two s over $X$ and let $(g, \phi)$ and $(h, \psi)$
the two $1$-cocycles in $H^1(N(\cu;\L))$ determined by them, then
the the two gerbes $\mathcal{G}^1(\Lambda, \theta)$ and
$\mathcal{H}^1(\Upsilon, \eta)$ are equivalent if there exists a
morphism $(g, \phi) \to (h, \psi)$ in $H^1(N(\cu);\L)$.
If $\mathcal{G}^1(\Lambda, \theta)$ and $\mathcal{H}^1
(\Upsilon, \eta)$ are equivalent, then there are
hermitian line bundle isomorphisms
\[
\Lambda_i^j \cong \Upsilon_i^j,
\]
over each $U_{i,j}$, such that the isomorphisms induce
a mapping
\[
\theta_{i,j,k} \mapsto \eta_{i,j,k}.
\]
A $\mathcal{G}^1(\Lambda, \theta)$ is globally
trivialized by displaying a basis $\lambda_i^j$ for each line
bundle $\Lambda_i^j$ such that on each $U_{i,j,k}$, we can express
the sections on three fold intersections, in terms of coordinates
specifed specified by the data and the ring
$C^\infty(U_{i,j,k};U(1))$, as follows:
\[
\theta_{i,j,k} = 1(x) \lambda_i^j \otimes \lambda_j^k
\otimes \lambda_k^i,
\]
where $1(x) \in C^\infty(U_{i,j,k};U(1))$ is the constant
function which assigns to each point $x \in U_{i,j,k}$, the
identity of the group $U(1)$.
Let $\G$ be a globally trivial and $(\alpha, \phi)$ be the
determined by $\G$. Then for $U_{i,j,k}$
\[
\phi(U_{i,j,k}):l_i^j - l_j^k + l_k^i \to
\mathbb{C}
\]
is the canonical isomorphism.
A is trivial if it is equivalent to the zero
$1$-gerbe over $X$, which is the determined by the cocycle
$(0,id_0) \in H^1(N(\cu);\L)$.
The notion of a trivial hermitian line 1-gerbe can equivalently
be defined by a geometric entity called an object, which we
define next.
Given a $\mathcal{G}^1(\Lambda, \theta)$, an object
compatible with $\mathcal{G}^1$, denoted $\O(L,m)$ is specified by
the following data
1. Constant hermitian line bundles $L_i$ over each $U_i$;
2. Hermitian line bundle isomorphisms over each intersection
$U_{i, j}$
\[
m_i^j:L_i \cong \Lambda_i^j \otimes L_j;
\]
such that the composition on three fold intersection
\[
L_i \longrightarrow (\Lambda_i^j \otimes \Lambda_j^k \otimes
\Lambda_k^i) \otimes L_i
\]
is exactly
\[
(id \otimes m_k^i) \circ (id \otimes m_j^k)
\circ m_i^j \equiv \theta_{i,j,k} \otimes id.
\]
Here we are abusing notation by denoting the trivialization
determined by the section $\theta_{i,j,k}$ also by
$\theta_{i, j, k}$.
Let $\mathcal{G}^1$ be a over $X$ and let $(\alpha, \phi)$ be
the hermitian line $1$-cocycle determined by $\G^1$. Then
$\G^1$ has an object, $\O(L,m)$, compatible with it iff there is a
hermitian line $0$-chain $\beta \in \Ob C^0(N(\cu);\L)$ and a
morphism $f:(\alpha, \phi) \to (d\beta, \chi_\beta)$, in $Ker(d,
\chi)$, such that $(\beta, f)$ is a representative of a morphism
$[(\beta, f)]: (\alpha, \phi) \to (0, id_0)$ in $H^1(N(\cu);\L)$.
Let $\G^1$ be a trivial over $X$ as above. Then there exists a
morphism $[(\beta, f)]: (\alpha, \phi) \to (0, id_0)$ in
$H^1(N(\cu);\L)$. Choose a representative $(\beta, f)$ of this
morphism. Now we define the constant line bundle, $L_i$, over each
$U_i$ as follows: $L_i := U_i \times \beta(U_i)$. The linear
isometry $f(U_{i,j}):\alpha(U_{i,j}) \to (\beta(U_i) - \beta(U_j))$
determines a morphism of hermitian line bundles
\[
m_i^j:L_i \to \Lambda_i^j \otimes L_j
\]
over each $U_{i,j}$. The condition over three fold intersections, in
definition <ref>, follows from the equation $\chi_\beta \circ
d(f) = \phi$. Conversely, given an object compatible with a trivial
$\G^1$, one can define the isomorphism $[(\beta, f)]:(\alpha,
\phi) \to (0, id_0)$ in $H^1(N(\cu);\L)$.
Finally, we are ready to define a over $X$.
A over $X$, $\G^2(\G, \O, \theta)$, is defined by the
following data
1. A $\G_i^j$ over the intersection $U_{i,j}$ for every
ordered pair $(i, j) \in I \times I$ and $i \ne j$ such that
$\G_i^j$ and $\G_j^i$ are dual to each other.
2. For each ordered triple of distinct indices $(i, j, k) \in
I \times I \times I$, an object $\O_{i,j,k}$ compatible with the
coboundary gerbe
\[
\G_i^j \otimes \G_j^k \otimes \G_k^i
\]
such that the sections of reorderings of triples
$(i, j, k)$ are related in the natural way.
3. For each ordered quadruple of distinct indices
$(i, j, k, l) \in I \times I \times I \times I$,
trivializations $\theta_{i,j,k,l}$ of coboundaries
of objects
\[
\O_{i,j,k} \otimes \O_{i,j,l}^{-1} \otimes
\O_{i,k,l} \otimes \O_{j,k,l}^{-1}
\]
on $U_{i,j,k,l}$. Notice that each pair $(\O_{i,j,k}
\otimes \O_{i,j,l}^{-1})$ is a line bundle over
$U_{i,j,k,l}$ so asking for a trivialization
of the object is ligitimate.
This data is subject to a cocycle condition, on $U_{i,j,k,l,m}$
which we denote by $\delta \theta \Rightarrow 0$.
The cocycle condition is that over any five fold
intersections $U_{i,j,k,l,m}$, we can tensor the five
sections of the coboundary objects to give a trivialization
of the following hermitian object
\[
(\O_{i,j,k} \otimes \O_{i,j,l}^{-1} \otimes \O_{i,k,l}
\otimes \O_{j,k,l}^{-1}) \bigotimes
(\O_{i,j,k} \otimes \O_{i,j,m}^{-1} \otimes
\O_{i,k,m} \otimes \O_{j,k,m}^{-1})^{-1}
\]
\[
\bigotimes
(\O_{i,j,l} \otimes \O_{i,j,m}^{-1} \otimes \O_{i,l,m}
\otimes \O_{j,l,m}^{-1}) \bigotimes
(\O_{i,k,l} \otimes \O_{i,k,m}^{-1} \otimes \O_{i,l,m}
\otimes \O_{k,l,m}^{-1})^{-1}
\]
\[
\bigotimes
(\O_{j,k,l} \otimes \O_{j,k,m}^{-1} \otimes \O_{j,l,m}
\otimes \O_{k,l,m}^{-1})
\]
Notice that the above object is canonically trivial,
so the cocycle condition is that the following
\[
\theta_{i,j,k,l} - \theta_{i,j,k,m} + \theta_{i,j,l,m}
-\theta_{i,k,l,m} +\theta_{j,k,l,m}
\]
is the canonical section.
A hermitian line $2$-cocycle $(\alpha, \phi)$ represents a over
$X$. We outline a construction of a starting from the
$2$-cocycle $(\alpha, \phi)$. A , $\G^i_j(\Lambda, \theta)$ over
$U_{i,j}$ for every pair $(i,j) \in I \times I$, is determined by the
$2$-cocycle $(\alpha, \phi)$ as follows: Over each three-fold
intersection, $U_{i,j,k}$, a constant hermitian line bundle
$\Lambda_{i,j,k}$ is defined by $\Lambda_{i,j,k} := U_{i,j,k} \times
\alpha(U_{i,j,k})$. On every four-fold intersection $U_{i,j,k,l}$,
the section $\theta_{i,j,k,l}$ is determined by the linear isometry
\[
\phi(U_{i,j,k,l}):\alpha(U_{i,j,k}) - \alpha(U_{i,k,l})
+ \alpha(U_{i,j,l}) - \alpha(U_{j,k, l}) \to \mathbb{C}.
\]
This section satisfies the cocycle condition $\delta \theta
\Rightarrow 0$ over five-fold intersections, thus defining a over $U_{i,j}$. Notice that the coboundary $\G^i_j \otimes
\G^j_k \otimes \G^i_k$ over $U_{i,j,k}$, is trivial, therefore there
exists an object $\O_{i,j,k}$ compatible with this trivial coboundary
gerbe. This object $\O_{i,j,k}$ is specified by the $2$-chain $\alpha
\in C^2(N(\cu);\L)$. Over each four-fold intersection $U_{i,j,k,l}$,
a section $\theta_{i,j,k,l}$ of the coboundary Object $\O_{i,j,k}
\otimes \O_{i,j,l}^{-1} \otimes \O_{i,k,l} \otimes \O_{j,k,l}^{-1}$
is specified by the linear isometry $\phi(U_{i,j,k,l})$.
§ DIJKGRAAF-WITTEN THEORY
We would like to recover Dijkgraaf-Witten's construction
<cit.> of a TQFT. In principle, we follow their
construction, using Freed-Quinn's hermitian-line incarnation
<cit.>, and placing it further within the framework of
cohomology with coefficients in the Picard groupoid of hermitian
§.§ Hermitian line corresponding to a closed $n$-manifold
We start with an $n$-cocycle $\alpha$ which is an object of the Picard
groupoid $H^n (BG; \L)$. For each map $f: Y \to BG$ from a closed
$n$-manifold $Y$, we take the pullback $f^* \alpha$. Consider the cap
\[
\cap: H^n (Y; \L) \otimes H_n (Y; \Z[0]) \to H_0 (Y; \L),
\]
which is a morphism of Picard groupoids. If we substitute the given
cocycle $\alpha$ in the first factor, we will get a morphism
\begin{equation}
\label{capping-with-alpha}
f^* \alpha \cap - : H_n (Y; \Z[0]) \to H_0 (Y; \L).
\end{equation}
What we would like to do is to apply this morphism to the fundamental
cycle of $Y$. However, in the homology with coefficients in a Picard
groupoid, be it a discrete one, such as $\Z[0]$, no single object
represents the fundamental cycle canonically. It is rather a full
subgroupoid (not monoidal) $C_Y$ formed by all possible cycles
representing the fundamental cycle and connected by equivalence
classes of morphisms given by $n$-boundaries modulo
$(n+1)$-boundaries: a morphism $y \to y'$ is given by an $(n+1)$-chain
$x$ such that $y' = y + dx$; two morphisms $x: y \to y'$ and $x': y
\to y'$ are equivalent if there is an $(n+2)$-chain $w$ such that $x =
dw + x'$. Thus, we can restrict the above morphism
(<ref>) to this fundamental-cycle groupoid
$C_Y$ and get a functor
\[
f^* \alpha \cap - : C_Y \to H_0 (Y; \L).
\]
If we compose this functor with the degree map
\[
H_0 (Y; \L) \to \L
\]
which takes each linear combination $a_1 y_1 + \dots + a_k y_k$ of
points $y_1, \dots, y_k$ in $Y$ with coefficients $a_1, \dots, a_k$ in
$\L$ to the sum $a_1 + \dots + a_k$, which is an object in $\L$, we
obtain a functor
\begin{equation}
\label{cap-with-fundamental-cycle groupoid}
F: C_Y \to \L
\end{equation}
from the fundamental-cycle groupoid to the groupoid of hermitian
lines. Now we take the limit of this functor. The existence of the
limit is guaranteed by the following fact.
The functor
\[
F: C_Y \to \L,
\]
which represents the cap product of the cocycle $\alpha$ with the
fundamental-cycle groupoid $C_Y$, has a limit,
\[
\lim_{C_Y} F,
\]
in the category $\L$ of hermitian lines.
The limit of the functor $F$ may be realized by Freed-Quinn's
invariant-section construction: an invariant section is a
collection of elements in $\{s(y) \in F(y) \; | \; y \in \Ob C_Y\}$
such that for each morphism $x: y \to y'$ in $C_Y$, we have
$F(x)s(y) = s(y')$. The space of invariant sections is a hermitian
line, in other words, the limit of $F$ exists, if the functor has
no holonomy, i.e., $F(x) = \id$ for each automorphism
$x: y \to y$. This is indeed the case, due to the following
Being an object of $H^n (Y; \L)$, the cocycle $\alpha$ is
represented by a pair $(a,\phi)$, where $a$ is an object of $C^n (Y;
\L)$, i.e., a function $a: S^n(Y) \to \Ob \L$, and $\phi: da
\to 0$ is a morphism in $C^{n+1} (Y; \L)$, i.e., a function
$S^{n+1} (Y) \to \Mor \L$. The functor $F: C_Y \to \L$ acts in the
following way on objects and morphisms of the groupoid $C_Y$:
\[
F(y) = a(y) \qquad \text{for $y \in \Ob C_Y$},
\]
\[
F(x): a(y) \to a(y') \qquad \text{for $x \in \Mor C_Y$, $y' = y + dx$},
\]
is defined by $\phi(x): a(y') - a(y) = a (dx) = da(x) \to 0$ as a
composition of it with the structure natural transformations
(<ref>)-(<ref>) and their inverses.
Now suppose we have an automorphism $x: y \to y$, which in particular
means that we have a chain $x \in \Ob C_{n+1} (Y; \Z[0])$, such that
$dx = 0$. Since $H_{n+1} (Y; \Z[0])$ is trivial whenever $\dim Y = n$,
the cycle $x$ must be a boundary: $x = dw$ for some $w$. This renders
the equivalence class of the morphism $x$ to be trivial.
§.§ Linear isometry corresponding to an $(n+1)$-cobordism
Now let $X$ be a compact $n+1$-manifold with boundary $i: \partial X
= \partial X_- \coprod \partial X_+ \subset X$. As a starting point,
we use the same $n$-cocycle $\alpha$, which is an object of the Picard
groupoid $H^n (BG; \L)$. For any continuous function $f: X \to BG$, a
pullback of $\alpha$ along $f$ gives an $n$-cocycle $f^* \alpha$,
which is an object of the Picard groupoid $H^n (X; \L)$. Consider the
relative cap product
\[
\cap: H^n (X; \L) \otimes H_{n+1} (X, \partial X; \Z[0]) \to H_1
(X, \partial X; \L),
\]
which is a morphism of Picard groupoids. If we substitute $f^* \alpha$
in the first factor, we will get a functor
\begin{equation}
\label{relative-capping-with-alpha}
f^* \alpha \cap - : H_{n+1} (X, \partial X; \Z[0]) \to H_1 (X, \partial X; \L).
\end{equation}
As above, we restrict this functor to the relative
fundamental-cycle groupoid $C_{X, \partial X}$ which is the full
subgroupoid of $H_{n+1} (X, \partial X; \Z[0])$ whose objects are all
possible relative cycles representing the relative fundamental class
of $X$. The restriction gives us a functor
\[
f^* \alpha \cap - : C_{X, \partial X} \to H_1 (X, \partial X; \L).
\]
We compose this functor first with the $2$-morphism $\Psi_1$ from (the
chain version of) the long $2$-exact sequence
(<ref>) and then the degree map
\begin{equation}
\label{F-map}
\xymatrix{
C_{X, \partial X} \ar[r]^{f^* \alpha \cap - \ \ \ \ } & H_1 (X, \partial X; \L)
\ar[r]^{\ \ \partial_1} \rruppertwocell<10>^0{\omit}
& H_0 (\partial X; \L) \ar[r]^{H_0(i) \ \ } &
H_0 (X; \L) \lltwocell<\omit>{<3>\Psi_1 \ \ }
\overset{deg} \to \L
\end{equation}
This diagram gives us a $2$-morphism $t: F \Rightarrow 0$, where
$F: C_{X, \partial X} \to \L$ is the composite functor in the lower
row. Consider the following diagram:
\begin{equation}
\label{big_triangle}
\xymatrix{
(X, \del X; \Z[0]) \ar[r]^{\ \ \ f^* \alpha \cap - } \ar[d]^{\del}& H_1 (X, \del X; \L)
\ar[r]^{\ \ \partial_1} \rruppertwocell<10>^0{\omit}
& H_0 (\partial X; \L) \ar[r]^{H_0(i) \ \ } &
H_0 (X; \L) \lltwocell<\omit>{<3>\Psi_1 \ \ }
\overset{\deg} \to \L\\
H_{n} (\del X; \Z[0]) \ar[urr]_{f|_{\del X}^* \alpha \cap -} \ar@{=>}(9,-9)*{};(21,-4)*{}
\end{equation}
where the bottom 2-morphism is comes from
Proposition <ref>. When we restrict the boundary 1-morphism
$\del: H_{n+1} (X, \del X; \Z[0]) \to H_{n} (\del X; \Z[0])$ to the
full subcategory $C_{X, \partial X}$, we get the following commutative
\begin{equation*}
\xymatrix{
C_{X, \partial X} \ar[r] \ar[d]_{\del} \ar@{}[dr]|{\circlearrowright}& H_{n+1}
(X, \del X; \Z[0]) \ar[d]^{\del}\\
-C_{\partial_- X} \times C_{\partial_+ X} \ar[r] & H_{n} (\del X; \Z[0])
\end{equation*}
where $-C_{\partial_- X}$ is the negative fundamental-cycle groupoid
of $\del_- X$, the full subcategory in $H_{n} (\del X; \Z[0])$ made up
by representatives of the negative fundamental class of $\del_- X$ in
$H_{n} (\del_- X; \Z) \subset H_{n} (\del X; \Z)$. By stacking
together the last two diagrams, we obtain the following diagram:
\begin{equation}
\label{little_triangle}
\xymatrix{
C_{X, \partial X} \ar[r]^(.60){F} \ar[d]^{\del} \ruppertwocell<10>^0{\omit}
& \L \ltwocell<\omit>{<2.5>*!/^-2pt/{\labelstyle \Psi_F}}\\
-C_{\partial_- X} \times C_{\partial_+ X} \ar[ur]_(.58)*!/^-1pt/{\labelstyle F_- + F_+} \ar@{=>}(5,-7)*{};(9,-3)*{}_(0.7)*!/^3pt/{\labelstyle X_F}
\end{equation}
where $F_- := f|_{\del_- X}^* \alpha \cap -$ and $F_+ := f|_{\del_+
X}^* \alpha \cap -$ appended by $H_0 (i)$ and $\deg$ as in
Applying the limit functor, we get canonical morphisms
\[
-\lim_{C_{\del_- X}} F_- + \lim_{C_{\del_+ X}} F_+ \rightarrow
\lim_{-C_{\del_- X} \times C_{\del_+ X}} (F_-+F_+) \rightarrow
\lim_{C_{X,\del X}} F \to 0
\]
in $\L$, whence a morphism
\[
l_f: \lim_{C_{\del_- X}} F_- \rightarrow \lim_{C_{\del_+ X}} F_+,
\]
which translates into a canonical linear isometry between hermitian
§.§ The Dijkgraaf-Witten theory TQFT functor
Given a finite group $G$, for each $\alpha \in H^n(BG; \L)$,
we construct the Dijkgraaf-Witten theory TQFT functor,
\[
Z^\alpha: \mathbf{Cob}(n+1) \to \Vect,
\]
from the category $\mathbf{Cob}(n+1)$ of cobordisms to the category
$\Vect$ of complex vector spaces, using the ingredients developed in
the preceding sections. We first construct the values of the functor
on objects. Observe that for every $Y \in \Ob \ \mathbf{Cob}(n+1)$,
Proposition <ref> delivers a canonical hermitian line for
each $f \in \Map (Y, BG)$. We claim that these lines glue into a flat
hermitian line bundle over $\Map(Y, BG)$, or a local system
with values in $\L$, i.e., a functor
\[
\L_Y: \Pi_1\Map(Y, BG) \to \L
\]
from the fundamental groupoid of the mapping space $\Map(Y, BG)$ to
A morphism in $\Pi_1 \Map(Y, BG)$ is a homotopy class $[f]$ of a map
$f: Y \times I \to BG$. We can think of $Y \times I$ as the identity
cobordism between two copies of $Y$. Applying the construction of
Section <ref>, we get a morphism in $\L$,
\[
l_f: \lim_{C_Y} F_0 \rightarrow \lim_{C_Y} F_1.
\]
We define $\L_Y([f]) := l_f$. The cocycle $f^* \alpha$ does depend on
the representative $f$ of the homotopy class $[f]$, see
Section <ref>, however the difference disappears at the
homology level after applying the cap product with $f^* \alpha$ and
the “boundary homomorphism” $\del_1: H_1 (Y \times I, \del (Y \times
I); \L) \to H_0 (\del (Y \times I); \L)$ in (<ref>). Note
that $f|_{\del (Y \times I)}^* \alpha$ does not depend on the
representative of the homotopy class $[f]$, because the homotopy is
supposed to be relative to the boundary. Thus, the diagram
(<ref>) does not depend of the choice of a
representative of the homotopy class $[f]$, and the local system
$\L_Y$ is well defined.
One can view the construction of a local system $\L_y$ as
”integration of $\ev^* \alpha$ along fibers” of $\pi$ or a
construction of the push-pull in cohomology with values in Picard
groupoids along the following diagram:
\[
\begin{CD}
Y \times \Map (Y,BG) @>\ev>> BG\\
\Map (Y, BG),
\end{CD}
\]
\[
H^n(BG; \L) \xrightarrow{\ev^*} H^n (Y \times \Map (Y,BG); \L)
\xrightarrow{\pi_*} H^0 (\Map (Y, BG); \L),
\]
where $\pi_* \ev^* \alpha := \L_Y$, by definition, and we recall that
objects of $H^0(\Map (Y, BG); \newline \L)$ are identified with local
systems or $0$-gerbes, see Section <ref>.
For any $Y \in \Ob \ \mathbf{Cob}(n+1)$, we define the value $Z^\alpha
(Y)$ of the TQFT functor to be the space of global
sections of the local system $\L_Y$ over $\Map (Y, BG)$ constructed
\[
Z^\alpha(Y) := H^0 (\Map(Y, BG); \L_Y) := \lim \L_Y \in \Vect,
\]
where the limit is taken for a natural extension $\Pi_1 \Map(Y, BG)
\xrightarrow{\L_Y} \L \to \Vect$ of the functor $\L_Y$, denoted by the
same symbol. The limit exists, because the category $\Vect$ is
Now we construct the arrow function of the TQFT functor. This can also
be viewed as a construction of “fiberwise integral.” Let $X$ be an
$(n+1)$-dimensional cobordism from $\del_- X$ to $\del_+ X$. We get
two local systems $\L_{\del_- X}$ and $\L_{\del_+ X}$ over the mapping
spaces $\Map(\del_- X, BG)$ and $\Map(\del_+ X, BG)$,
respectively. Let $p_{\pm}: \Map(X, BG) \to \Map(\del_\pm X, BG)$
denote the natural restriction morphisms. We start with constructing a
morphism $\L_X: p^*_{-} \L_{\del_- X} \to p^*_{+} \L_{\del_+ X} $ of
local systems on $\Map(X, BG)$. i.e., a natural transformation
between functors $p^*_{-} \L_{\del_- X}$ and $p^*_{+} \L_{\del_+ X}:
\Pi_1(\Map(X, BG)) \to \L$. For each $f \in \Map(X, BG)$, by invoking
the construction of Section <ref> once again, we get two
functors $F_\pm: C_{\del_\pm X} \to \L$ and the following morphism
\[
l_f: \lim_{C_{\del_- X}} F_- \to \lim_{C_{\del_+ X}} F_+
\]
in $\L$. Note that the fiber of each pull-back local system $p^*_{\pm}
\L_{\del_\pm X}$ over $f \in \Map(X, \linebreak[0] BG)$ is by
definition the fiber of $\L_{\del_\pm X}$ over $p_\pm(f)$, and that
fiber is $\lim_{C_{\del_\pm X}} F_\pm$ by the construction of
Section <ref>. We define $\L_X(f)$ to be $l_f: p^*_{-}
\L_{\del_- X}|_f \to p^*_{+} \L_{\del_+ X}|_f $ on objects $f \in
\Map(X,BG)$ of $\Pi_1 (\Map (X,BG))$. A morphism $f \to g$ in the
fundamental groupoid $\Pi_1 (\Map (X,BG))$ is represented by a
homotopy $h \in \Map(X \times I, BG)$ between maps $f$ and $g \in \Map
(X, BG)$. To see that $\L_X$ consitutes a natural transformation, we
need to see that the diagram
\begin{equation}
\label{homotopy}
\begin{CD}
p^*_{-} \L_{\del_- X}|_f @>{l_f}>> p^*_{+} \L_{\del_+ X}|_f\\
@V{p^*_- l_{h|_{\del_- X \times I}}}VV @VV{p^*_+ l_{h|_{\del_- X \times I}}}V\\
p^*_{-} \L_{\del_- X}|_g @>{l_g}>> p^*_{+} \L_{\del_+ X}|_g
\end{CD}
\end{equation}
commutes. Indeed, the homotopy gives a morphism $H: f^* \alpha \to g^*
\alpha$ in the Picard groupoid $H^n (X; \L)$. Using the
bifunctoriality of the cap product, we get a 2-morphism $f^* \alpha
\cap - \Rightarrow g^* \alpha \cap - $ added to Diagram (<ref>),
resulting in a commutative triangle
\begin{equation*}
\xymatrix{
F \ar@{=>}[rr]^{\Psi_H} \ar@{=>}[dr]_{\Psi_F}& & G \ar@{=>}[dl]^{\Psi_G}\\
& 0
\end{equation*}
on top of the upper part of Diagram (<ref>) and,
similarly, a commutative square
\begin{equation*}
\xymatrix{
(F_- + F_+) \circ \del \ar@{=>}[r]^{\Psi_{\del H}} \ar@{=>}[d]_{X_F}
& (G_- + G_+) \circ \del \ar@{=>}[d]^{X_G}\\
F \ar@{=>}[r]^{\Psi_H} & G
\end{equation*}
on top of the lower part of Diagram (<ref>), with
$\Psi_{\del H}$ coming from the 2-morphism $f|_{\del X}^* \alpha \cap
- \Rightarrow g|_{\del X}^* \alpha \cap - $ added to the bottom
triangle in (<ref>). Passing to the limits, we see that
(<ref>) is commutative.
Now, after the morphism $\L_X: p^*_{-} \L_{\del_- X} \to p^*_{+}
\L_{\del_+ X} $ of local systems on $\Map(X, \linebreak[0] BG)$ is
constructed, we are ready to construct a linear map
\[
Z^\alpha (X): Z^\alpha (\del_- X) \to Z^\alpha (\del_+ X)
\]
\[
Z^\alpha (X): H^0 (\Map (\del_- X, BG); \L_{\del_- X}) \to H^0 (\Map
(\del_+ X, BG); \L_{\del_+ X}).
\]
The plan is to describe a push-pull along the diagram of spaces:
\[
\Map (\del_- X, BG) \xleftarrow{p_-} \Map (X, BG) \xrightarrow{p_+}
\Map (\del_+ X, BG).
\]
The pullback
\[
p_-^*: H^0 (\Map (\del_- X, BG); \L_{\del_- X}) \to H^0 (\Map (X, BG);
p_-^* \L_{\del_- X})
\]
is easy. So is an intermediate map:
\[
H^0( \L_X): H^0 (\Map (X, BG); p_-^* \L_{\del_- X}) \to H^0 (\Map (X,
BG); p_+^* \L_{\del_+ X}).
\]
The pushforward
\[
(p_+)_*: H^0 (\Map (X, BG); p_+^* \L_{\del_+ X}) \to H^0 (\Map (\del_+
X, BG); \L_{\del_+ X})
\]
is not straightforward, and its existence relies on the specifics of
the topology of mapping spaces to $BG$ for a finite group $G$.
Recall that the space $\Map(X,BG)$ may naturally be realized at the
classifying space for principal $G$-bundles over $X$. This leads to a
natural homotopy equivalence
\[
\Map (X, BG) \sim \coprod_{[P \to X]} B \Aut (P),
\]
where the disjoint union is taken over isomorphism classes $[P \to X]
\simeq \pi_0 \Map (X, \linebreak[0] BG)$ of principal $G$-bundles $P
\to X$. The map $p_+: \Map (X, BG) \to \Map (\del_+ X, BG)$ is
homotopy equivalent to the natural restriction map
\[
p'_+: \coprod_{[P \to X]} B \Aut (P) \to \coprod_{[P_+ \to \del_+ X]} B
\Aut (P_+),
\]
which is a finite covering map over each connected component $B \Aut
(P_+)$, sometimes with empty fiber.
We will define the pushforward
\[
(p_+)_*: H^0 (\Map (X, BG); p_+^* \L_{\del_+ X}) \to H^0 (\Map (\del_+
X, BG); \L_{\del_+ X})
\]
as a transfer map
\[
(p'_+)_*: H^0 \left( \coprod_{[P \to X]} B \Aut (P); (p'_+)^*
\L_{\del_+ X}\right) \to H^0 \left( \coprod_{[P_+ \to \del_+ X]} B
\Aut (P_+); \L_{\del_+ X} \right),
\]
which will be constructed using the definition of $H^0$ as a limit
over the fundamental groupoid. Indeed, for every path $\gamma_+$ in $B
\Aut (P_+)$, we take all its lifts to the component $B \Aut (P)$ over
$B \Aut (P_+)$, which is a finite, possibly zero, number. For each
such path $\gamma$, we have a linear isometry $(p'_+)^* \L_{\del_+ X}
(\gamma): (p'_+)^* \L_{\del_+ X}(\gamma(0)) \to (p'_+)^* \L_{\del_+
X}(\gamma(1))$, which, by definition of $(p'_+)^*$, is equal to the
isometry $\L_{\del_+ X} (\gamma_+): \L_{\del_+ X}(\gamma_+(0)) \to
\L_{\del_+ X}(\gamma_+(1))$. Since $H^0 \left( \coprod_{[P \to X]} B
\Aut (P); (p'_+)^* \L_{\del_+ X} \right)$ is a limit of the functor
$(p'_+)^* \L_{\del_+ X}$, we have a canonical commutative diagram of
linear maps:
\[
\xymatrix{
& H^0 \left( \coprod_{[P \to X]} B \Aut (P); (p'_+)^* \L_{\del_+ X} \right) \ar[dl] \ar[dr]\\
(p'_+)^* \L_{\del_+ X}(\gamma(0)) \ar[rr] \ar@{=}[d]& & (p'_+)^*
\L_{\del_+
X}(\gamma(1)) \ar@{=}[d]\\
\L_{\del_+ X}(\gamma_+(0)) \ar[rr] & & \L_{\del_+ X}(\gamma_+(1)).
\]
If, given $\gamma_+$, we add the linear maps $H^0 \left( \coprod_{[P
\to X]} B \Aut (P); (p'_+)^* \L_{\del_+ X} \right) \linebreak[3]
\to \linebreak[4] \L_{\del_+ X}(\gamma_+(0))$ over all possible
$\gamma$'s covering $\gamma_+$ and do the same for maps to $\L_{\del_+
X}(\gamma_+(1))$, we will get a commutative diagram
\[
\xymatrix{
& H^0 \left( \coprod_{[P \to X]} B \Aut (P); (p'_+)^* \L_{\del_+ X} \right) \ar[dl] \ar[dr]\\
\L_{\del_+ X}(\gamma_+(0)) \ar[rr] & & \L_{\del_+ X}(\gamma_+(1)).
\]
Since $H^0 \left( \coprod_{[P_+ \to \del_+ X]} B \Aut (P_+);
\L_{\del_+ X} \right)$ is a limit of the functor $\L_{\del_+ X}$, we
get a canonical linear map
\[
H^0 \left( \coprod_{[P \to X]} B \Aut (P); (p'_+)^* \L_{\del_+ X}
\right) \to H^0 \left( \coprod_{[P_+ \to \del_+ X]} B \Aut (P_+);
\L_{\del_+ X} \right),
\]
which we declare to be the transfer $(p'_+)_*$.
Finally, the TQFT functor
\[
Z^\alpha (X): H^0 (\Map (\del_- X, BG); \L_{\del_- X}) \to H^0 (\Map
(\del_+ X, BG); \L_{\del_+ X})
\]
is defined as the composition of $(p_+)_*$, $H^0(\L_X)$, and $p_-^*$.
The invariance of $Z^\alpha$ under diffeomorphisms $X' \to X''$ of
cobordisms is obvious, as a diffeomorphism induces an isomorphism of
simplicial sets $\Sing(X')$ and $\Sing(X'')$ representing the
cobordisms and leads to isomorphic diagrams (<ref>) and
(<ref>) in a strict sense, thus giving the same
isometry $l_f$ of hermitian lines in Section <ref>.
§ THE HOM $2$-CHAIN COMPLEX
In this section we define the Hom $2$ - chain complex and a
tensor product in $\mathbf{2Ch({\emph{SCG}}})$. We recall that
given any two Picard groupoids $\A, \B \in \Ob(\emph{SCG})$,
$\Hom_{\emph{SCG}}(\A, \B)$ inherits a Picard groupoid structure,
i.e. the category $\emph(SCG)$ is enriched over itself .
Let $A_{\bullet}, \B_{\bullet} \in
\Ob(\mathbf{2Ch({\emph{SCG}}}))$. Then,
$(\Hom_{\mathbf{2Ch({\emph{SCG}}})}(\A_{\bullet}, \B_{\bullet}), d,
\phi)$ is a
chain complex whose $nth$. degree is defined as follows:
$\Hom_{\mathbf{2Ch({\emph{SCG}}})}(\A_{\bullet}, \B_{\bullet})_{n} = \underset{p} \prod \Hom_{\emph{SCG}}(\A_{p},\B_{p+n})$.
The differential $d:\Hom_{\mathbf{2Ch({\emph{SCG}}})}(\A_{\bullet},
\B_{\bullet})_{n} \rightarrow
\Hom_{\mathbf{2Ch({\emph{SCG}}})}(\A_{\bullet}, \B_{\bullet})_{n-1}$
is given by $(df)_{p} = df_{p} +(-1)^{p+1}f_{p-1}d$ and a composition
of $2$ - morphisms $dd(f) \Rightarrow d^{2}f + fd^{2} \Rightarrow 0$,
where the first $2$ - morphism comes from the distributivity law on
each degree of the Hom complex which is the consequence of the
enrichment of $\emph{SCG}$ over itself and the second $2$ - morphism
in the composition is obvious. Similarly, we may define the Hom 2 -
chain complex of chain maps between a 2-cochain complex and a 2-chain
complex. Let $\C^{\bullet}$ be a 2-cochain complex of Picard
groupoids, let $\B_{\bullet} \in \Ob(\mathbf{2Ch({\emph{SCG}})})$,
then the 2-chain complex $([\C^{\bullet}, \B_{\bullet}], d, \phi)$ is
defined as the 2 - chain complex
$(\Hom_{\mathbf{2Ch({\emph{SCG}}})}(\C^{-\bullet}, \B_{\bullet}), d,
\phi)$, where $\C^{-\bullet} \in \Ob(\mathbf{2Ch({\emph{SCG}}}))$ is
the 2-chain complex obtained by negatively regrading $\C^{\bullet}$,
its degree $n$ is $[\C^{\bullet}, \B_{\bullet}]_{n} = \underset{p}
\prod \Hom_{\emph{SCG}}(\C^{-p},\B_{p+n})$. The tensor product of two
chain complexes could be defined similarly.
§ $\PIC$ CATEGORIES
In this appendix we give the definition of a mathematical structure
which is built on a bicategory but whose mapping categories have
the structure of Picard groupoids.
A $\Pic$-category $\C$ consists of the following data
* A small set, $\Ob(\C)$, whose elements will be called the
objects of $\C$.
* A function $\C(-,-):\Ob(\C) \times \Ob(\C) \to \Ob(\Pic)$,
where $\Ob(\Pic)$ is the set of all Picard groupoids.
* For each object $s \in \Ob(\C)$, a homomorphism $id_s:\ast
\to \C(s,s)$, where $\ast$ is the terminal Picard groupoid.
* For each triple of objects $s, t, u \in \Ob(\C)$, a composition
bifunctor $-\circ -:\C(t,u) \times \C(s,t) \to \C(s,u)$ which is subject
to the following conditions
* For each $h \in \Ob(\C(t,u))$, the functor
\[
h \circ -:\C(s,t) \to \C(s,u).
\]
is a homomorphism
* For each $g \in \Ob(\C(s,t))$, the functor
\[
- \circ g:\C(t,u) \to \C(s,u).
\]
is a homomorphism.
* For each triple of objects $s, t, u \in \Ob(\C)$ and each pair of
morphisms $g_1, g_2 \in \Ob(\C(s,t))$, a monoidal natural transformation
$\phi^-_{g_1,g_2}:-\circ g_1 + - \circ g_2 \Rightarrow - \circ g_1+g_2$,
where the homomorphism $-\circ g_1 + - \circ g_2:\C(t,u) \to \C(s,u)$ is
defined pointwise.
* For each triple of objects $s, t, u \in \Ob(\C)$ and each pair of
morphisms $h_1, h_2 \in \Ob(\C(t,u))$, a monoidal natural transformation
$\psi_-^{h_1,h_2}: h_1 \circ - + h_2 \circ - \Rightarrow h_1+h_2 \circ -$,
where the homomorphism $h_1 \circ - + h_2 \circ -:\C(s,t) \to \C(s,u)$ is
defined pointwise.
* For each quadruple of objects $s,t,u,v \in \Ob(\C)$, a natural
transformation, $\alpha$ called the associator,
between functors defined in the following diagram
\begin{equation*}
\xymatrix{
\C(u,v) \times \C(t,u) \times \C(s,t)
\ar[r]^{\ \ \ \ \ \ id \times -\circ-} \ar[d]_{-\circ- \times id} &
\C(u,v) \times \C(s,u) \ar[d]^{-\circ-} \ar@{}[dl]^{\alpha}|{\Leftarrow}\\
\C(t,v) \times \C(s,t) \ar[r] & \C(s,v).
\end{equation*}
and which is subject to the following conditions:
* For each pair $(g, h) \in \Ob(\C(t,u)) \times \Ob(\C(u,v))$,
the natural transformation $\alpha_{(h,g,-)}$ as in
the following diagram
\begin{equation*}
\xymatrix{
\C(s,t) \ar[r]_{g \circ -} \rruppertwocell<12>^{(h \circ g) \circ -}{\omit}
& \C(s,u) \ar[r]_{h \circ -} & \C(s,v) \lltwocell<\omit>{<3>\alpha_{(h,g,-)} \ \ \ \ \ \ \ \ }
\end{equation*}
is a monoidal natural transformation.
* For each pair $(f, h) \in \Ob(\C(s,t)) \times \Ob(\C(u,v))$,
the natural transformation $\alpha_{(h,-,f)}$ as in
the following diagram
\begin{equation*}
\xymatrix{
\C(t,u) \ar[r]^{h \circ-} \ar[d]_{-\circ f} &
\C(t,v) \ar[d]^{- \circ f} \ar@{}[dl]^{\alpha_{(f,-,h)}}|{\Leftarrow}\\
\C(s,u) \ar[r]_{h \circ-} & \C(s,v)
\end{equation*}
is a monoidal natural transformation.
* For each pair $(f, g) \in \Ob(\C(s,t)) \times \Ob(\C(t,u))$,
the natural transformation $\alpha_{(-,g,f)}$ as in
the following diagram
\begin{equation*}
\xymatrix{
\C(u,v) \ar[r]^{- \circ g} \rrlowertwocell<-12>_{- \circ (g \circ f)}{\omit}
& \C(t,v) \ar[r]^{- \circ f} & \C(s,v) \lltwocell<\omit>{<-3>\alpha_{(-,g,f)} \ \ \ \ \ \ \ \ }
\end{equation*}
is a monoidal natural transformation.
* For each pair of objects $s,t \in \Ob(\C)$, two monoidal
natural transformations
\begin{equation*}
\xymatrix{
\C(s,t) \ar@{=}[r] \rlowertwocell<-10>_{id_t \circ -}{\omit}
& \C(s,t) \ltwocell<\omit>{<-2>\lambda } &&
\C(s,t) \ar@{=}[r] \rlowertwocell<-10>_{- \circ id_s}{\omit}
& \C(s,t) \ltwocell<\omit>{<-2>\rho }
\end{equation*}
Let $\C$ and $\D$ be two $\Pic$-categories, A functor of
$\Pic$-categories $F:\C \to \D$ is a functor of bicategories
which respects the additional structure on the morphism categories
of $\C$ and $\D$. We will skip a precise definition of a functor
of $\Pic$-categories but an interested reader can define these functors
rigorously using our definition of $\Pic$-categories.
|
1511.00343
|
§ INTRODUCTION
Young open clusters provide useful test beds for the study of star formation processes because about 80 – 90% of young stars are
found in embedded clusters with more than 100 members <cit.>. Furthermore, the fundamental parameters of
clusters such as reddening, distance, and age can be properly constrained. These advantages allow us to derive a more reliable stellar
initial mass function (IMF) with which to investigate star formation processes.
Sh 2-254 – 258 is a famous star forming region (SFR) in the Gem OB1 association <cit.>. The main ionizing sources
of the HII regions are known to be one late-O and four early-B-type stars <cit.>. A number
of previous works found a few maser sources <cit.> as well as various molecular lines <cit.> in the region. Several sub-structures such as clumps and cores were also reported
from infrared (IR), sub-millimeter, millimeter, and centimeter observations <cit.>. These
observational properties commonly indicate that active star formation is in progress.
Finder chart and color composite image of the observed region in optical passbands (upper) and
near-infrared passbands (lower). Stars brighter than $V = 18$ mag and $K_S = 16$ mag are plotted in left-hand panels, respectively.
The size of the circles is proportional to the brightness of individual stars. The position of stars are relative to $\alpha = 06^{\mathrm{h}}
\ 12^{\mathrm{m}} \ 52^{\mathrm{s}}.1, \ \delta = +17^{\circ} \ 59' \ 16''.4$. Squares outlined by blue solid lines represent the field of view of
the Mont4K CCD camera. The color composite images were obtained from the Digital Sky Survey-2 and Two Micron All Sky Survey.
The position of two ionizing sources is marked by open circles in each image.
<cit.> took a census of young stellar objects (YSOs) using extensive near- to mid-IR imaging data. Most of the
young stars ($\sim 80$%) were found in several embedded sub-clusters. <cit.> continued their search for low-mass
YSOs in the quiescent phase, with a deep Chandra X-ray observation, and found that the total number of YSOs was consistent with
that expected from the Kroupa IMF <cit.> scaled to the number of ionizing sources. On the other hand, sequential
star formation scenarios within the SFR have been proposed <cit.>. Circumstantial
evidence, such as the age difference among HII regions, and the number ratio of YSOs at different evolutionary
stages, indicate that star formation activity propagated from Sh 2-255 and 257 into the molecular clouds behind the HII
bubbles. Hence, this SFR is one of the more interesting sites in which to study star and cluster formation processes.
The present work on the embedded young open clusters in the HII regions Sh 2-255 – 257 (hereinafter IC 2162)
is the sixth paper of the Sejong Open cluster Survey (SOS) project. <cit.> presented the
overview of the SOS project. Comprehensive studies of several open clusters IC 1848, NGC 1624, NGC 1893, NGC 1931,
and NGC 2353 were carried out as part of the project <cit.>. In this current work, we
revise the fundamental parameters of the SFR in a homogeneous manner, and constrain the IMF to study
star formation processes. The observational data we used are described in Section 2. In Section 3, we present several fundamental
parameters of IC 2162 obtained from photometric diagrams and discuss the reddening law toward the SFR.
The IMF is derived in Section 4, and the spectral energy distribution (SED) of pre-main sequence (PMS) members is investigated
in Section 5. Finally, the comprehensive results from this study are summarized in Section 6.
§ OBSERVATIONAL DATA
§.§ Optical Imaging Data
The observations of IC 2162 were made on 2013 February 5, using the Kuiper 61" telescope (f/13.5) of Steward Observatory
on Mt. Bigelow in Arizona, USA. Images were taken with the Mont4K CCD camera and 5 filters (Bessell $U$, Harris $BV$,
Arizona $I$, and H$\alpha$) in a $3 \times 3$ binning mode. The field of view (FOV) is about $9.^{\prime}7 \times 9.^{\prime}7$.
The target images comprise 12 frames that were taken in two sets of exposure times for each band (5 and 180 s $\times$ 2
in $I$, 5 and 180 s $\times 2$ in $V$, 7 and 300 s in $B$, 30 and 600 s in $U$, and 30 and 600 s in H$\alpha$). We also
observed several equatorial standard stars <cit.> at air masses of 1.2 – 2 on the same night in order to transform
the instrumental magnitudes to the standard magnitude and colors. Additional standard stars with extremely blue and red colors
in the Landolt standard star field Rubin 149 <cit.> were observed to determine the secondary extinction coefficients.
Atmospheric Extinction Coefficients and Photometric Zero points
Filter $k_1$ $k_2$ $\zeta$ (mag)
$I$ $0.045 \pm 0.008$ - $22.170 \pm 0.009$
$V$ $0.120 \pm 0.008$ - $23.560 \pm 0.007$
$B$ $0.232 \pm 0.008$ $0.023 \pm 0.002$ $23.548 \pm 0.006$
$U$ $0.444 \pm 0.018$ $0.031 \pm 0.005$ $22.069 \pm 0.008$
H$\alpha$ 0.085 - 19.565
All the pre-processing to remove the instrumental signals were carried out using the IRAF[Image Reduction
and Analysis Facility is developed and distributed by the National Optical Astronomy Observatories, which is operated
by the Association of Universities for Research in Astronomy under cooperative agreement with the National Science
Foundation.]/CCDRED packages. Simple aperture photometry was performed for the standard
stars with an aperture size of $14.^{\prime \prime}0$ (16.3 pixels). The primary and secondary atmospheric extinction
coefficients were determined from the photometric data of the standard stars using a weighted least-square method.
We present the coefficients and photometric zero points in Table <ref>. Point spread function (PSF) photometry of
stars in the target images was performed with a small fitting radius of one full width at half-maximum ($\leq 1.^{\prime \prime}0$)
using IRAF/DAOPHOT. Aperture photometry of bright, isolated stars with a photometric error smaller
than 0.01 mag in individual target images was obtained to correct for the aperture difference. The instrumental magnitudes
of stars in the target images were transformed to the standard magnitude and colors using the transformation relations
as described in Appendix of <cit.>. The finder chart for the stars brighter than $V = 18$ mag is
shown in the upper left-hand panel of Figure <ref>.
A total of 811 stars were detected from optical photometry. The completeness of our photometry was assessed from
the luminosity function of all observed stars. The luminosity function exhibits a single linear slope in the
magnitude range of $V = 13 - 19$ mag. If we assume that the linear slope is applicable down to the faint stars, the
turn-over magnitude gives the completeness limit. As a result, our photometry seems to be about 90% complete down
to $V = 19.3$ mag.
Optical spectra of four standard stars and two ionizing sources (from top to bottom) in the observed field of view.
The object name is denoted below each spectrum. Main spectral lines used in spectral classification are identified at the top of the
§.§ Optical Spectroscopic Data
The optical spectra of two main ionizing sources (ALS 19 and HD 253327) were obtained on 2015 March 10 with the
fiber-fed echelle spectrograph BOES (Bohyunsan Observatory Echelle Spectrograph – ) attached to the 1.8 m telescope
at Bohyunsan Optical Astronomy Observatory in Korea. A single frame for each target was taken with a 300 $\mu m$
fiber ($R = 30,000$), and the exposure time was 3600 seconds. The 3 $\times$ 3 binning mode allowed us to improve
the signal-to-noise ratio of the spectra. For the wavelength calibration, spectra of a ThAr lamp were also acquired
on the same night.
Pre-processing and extraction of spectra were made with the IRAF/ECHELLE package.
A sigma clipping method was used to minimize the influence of cosmic rays on the individual frames in a given order. We
normalized the spectra using the best solution found from a cubic spline interpolation and finally smoothed them by
a box size of 33. The spectra of ALS 19 and HD 253327 are shown in Figure <ref>. For comparison, the spectra
of standard stars [AE Aur (O9.5V), HD 36960 (B0.5V), HD 42401 (B2V), and $\eta$ Aur (B3V) – ]
observed with the same instrument on 2014 October 29 are also plotted in the same figure.
§.§ Archival Infrared Data
We transformed the CCD coordinates ($x_{\mathrm{CCD}}, y_{\mathrm{CCD}}$) of the optical photometric data
into celestial coordinates ($\Delta \alpha, \Delta \delta$) using the Two Micron All Sky Survey catalogue
(2MASS; ). Optical counterparts of near-IR sources in the 2MASS catalogue were searched
for with a matching radius of $1^{\prime \prime}$. A total of 361 optical counterparts were found.
<cit.> has made an extensive IR imaging survey across the entire molecular complex
incubating the HII regions Sh 2-254 – 258. This survey covers an area of $25^{\prime} \times
20^{\prime}$. Their catalogue includes the near-IR $JHK_S$ and Spitzer InfraRed Array Camera (IRAC) 4-band
photometry of 26,821 sources. The near-IR photometry in the IR source catalogue is reasonably tied to the 2MASS
photometric system within 0.03 mag. Only stars within the FOV of the optical imaging observations
were used in our analysis. A total of 3,426 sources were found within our FOV ($\sim 9.^{\prime}7 \times 9.^{\prime}7$),
of which 792 sources have optical counterparts within a matching radius of $1^{\prime \prime}$. We present the finder chart of these
stars in the lower left panel of Figure <ref>.
A post-BCD (basic calibrated and mosaiced) image of the Spitzer Multiband Imaging Photometer (MIPS)
24 $\mu$m image was taken from the data archive of the Spitzer Science Center (ObsID: 40005, PI G. Fazio). We carried
out PSF photometry for stars in the image using the IRAF/DAOPHOT with a fit radius of 2.4 pixel and a sky annulus of
20$^{\prime\prime}$ – 32$^{\prime\prime}$ (see ). The photometric zero point of 11.76 mag was calculated using the pixel
scale and the flux of a zeroth magnitude star as described in MIPS Handbook. Within our FOV, a total of 207 sources
were detected, of which 13 and 30 sources have counterparts in the optical and IR catalogues, respectively.
§ FUNDAMENTAL PARAMETERS
As seen in Figure <ref>, the finder charts and color composite images in the optical and near-IR passbands
exhibit completely different stellar distributions. This implies that the majority of young stars are embedded behind the HII
regions. Because only about a quarter of the stars were detected in the optical passbands, the canonical analysis based on the optical
photometric diagrams is limited. For this reason, the IR photometry of <cit.>, which is less sensitive to the effect of
extinction, is a powerful tool to probe embedded populations. However, the several visible stars are still very helpful to
determine fundamental parameters such as reddening and distance. In this section, we describe the identification of the
main ionizing sources, membership selection criteria, and the determination of reddening, distance, and age based on
the optical spectra and photometric diagrams as presented in Figure <ref> – <ref>.
Color-color diagrams of stars in IC 2162. The small dots (grey) represent all the stars. Other symbols
denote early-type members (black bold dots), Class I (magenta triangles), Class II (blue pluses), X-ray emission stars (large crosses),
X-ray emission candidates (small crosses), and H$\alpha$ emission stars or candidates (red open circles), respectively. The solid
line (black) in the left-hand panel exhibits the intrinsic color-color relation of <cit.>, and its reddened relation [$E(B-V) = 0.88$
mag] is shown by a dashed line (red). The solid line in the right-hand panel represents the empirical photospheric level of unreddened
main sequence stars, while the dashed and dotted lines are the lower limit of H$\alpha$ emission stars and H$\alpha$ emission candidates,
§.§ Spectral Types of Two Ionizing Sources
The influence of high-mass stars on the surrounding environment involves destructive and constructive processes. The
strong stellar wind and radiation pressure of high-mass stars can disperse their natal clouds, and thereby terminate
star formation. On the other hand, HII bubbles created by these stars can accumulate and compress
material as they expand into the molecular clouds. The condensed material can then form a new generation of
stars <cit.>. The high-mass stars can also drive the formation of the second generation of stars radiatively in
pre-existing clumps <cit.>. Therefore, the identification of the ionizing sources is essential for studying such
feedback of high-mass stars.
The brightest stars ALS 19 and HD 253327 (ALS 18) are known to be the main ionizing sources of IC 2162. In order to
classify the spectral type of these stars, we adopted the O and B-type star classification scheme of <cit.>.
The spectra of the stars in Figure <ref> contain several emission-like features that are the residuals of cosmic rays.
In the case of ALS 19, He II $\lambda$4200 and $\lambda$4542 are invisible in the spectrum, while
He II $\lambda$4686 absorption is clearly seen. The spectral type of this star is likely to be B0V; however, it is also
possible that ALS 19 is a late-O-type star (O9.5V or O9.7V) given the strength of He II $\lambda$4686
and the line ratio between Si IV $\lambda$4116 and He I $\lambda$4121. Since
the star was classified as Class II <cit.>, the spectrum of this young star shows a mixture of late-O and
early-B-type star characteristics.
Color-magnitude diagrams in the optical passbands. Left-hand panels: $V-I$ versus $V$ diagram. The location of
pre-main sequence stars is confined between the green dashed lines. Middle panels: $B-V$ versus $V$ diagram. Right-hand
panels: $U-B$ versus $V$ diagram. The solid lines represent the reddened zero-age main sequence relation of <cit.>. The
arrow denotes the reddening vector corresponding to $A_V = 3$ mag. The other symbols are the same as Figure <ref>.
He II $\lambda$4200 and $\lambda$4542 are also absent in the spectrum of HD 253327. Si III
$\lambda$4552 is invisible, while a weak He II 4686 absorption is authentically seen. Hence, the spectral type of
HD 253327 is likely to be B0V. The spectrum of this star is, indeed, similar to that of ALS 19. Our spectral classification is in a good
agreement with that of previous studies <cit.>.
§.§ Membership Selection
Early-type main sequence (MS) stars (putatively B-type stars) can be selected from the optical photometric diagrams
(Figure <ref> and <ref>) as they are bright in V and very blue in $U-B$. In addition, the reddening
and distance of the individual stars can be reliably determined because the intrinsic colors and absolute magnitude of such
stars have been well calibrated in the optical passbands. Probable early-type members are firstly selected from magnitude and
color cuts as $V \leq 15$ mag, $0.5 \leq B-V \leq 0.9$, $-0.6 \leq U-B \leq 0.5$, and $Q^{\prime} \leq -0.3$, where $Q^{\prime}
\equiv (U-B) - 0.72(B-V) - 0.025E(B-V)^2$ (Paper 0). We then removed several foreground late-type stars (probably F- or G-type)
restricting the reddening range to $E(B-V) > 0.8$ mag and color excess ratios. In addition, a few stars identified as YSOs
by <cit.> were also excluded from the MS member list. Only two stars were finally selected as the early-type
MS members of IC 2162.
We utilized H$\alpha$ photometry as a criterion to identify PMS stars in the SFR. A series of studies
demonstrated that H$\alpha$ photometry can effectively detect a number of low-mass PMS stars at the T Tauri stage in
young open clusters <cit.>.
In order to detect objects with an H$\alpha$ emission line, the H$\alpha$ index [$\equiv$ H$\alpha - (V + I)/2$] is used as the detection criterion <cit.>.
As shown in the right-hand panels of Figure <ref>, stars with an H$\alpha$ index smaller than the empirical photospheric level
(solid line) of normal MS stars by $-0.2$ (dashed line), or $-0.1$ mag (dotted line), was selected as H$\alpha$ emission stars and
candidates, respectively. We found 21 H$\alpha$ emission stars and six candidates. However, the H$\alpha$ emission star ID 659 ($V = 18.84$,
$V-I = 1.70$, $B-V = 1.45$, and $U-B = 1.07$) is likely an active late-type star in the field because its colors are similar to those of other field
stars. A total of 26 H$\alpha$ emission stars and candidates were selected as PMS members of IC 2162.
Excess emission at IR wavelengths, particularly the mid-IR, is a useful membership
selection criterion because a large fraction of PMS stars in young open clusters ($\leq$ 3 Myr) have been found to have warm
circumstellar disks or envelopes <cit.>. <cit.> identified 252 YSOs (87 Class I and 165 Class II) in the
HII regions Sh 2-254 – 258 using Spitzer/IRAC images. We used their YSO list and found
optical counterparts for 64 YSOs (11 Class I and 53 Class II) within our FOV. However, only 41 IR sources
(6 Class I and 35 Class II) were detected in the $V$ band.
PMS stars are also known as X-ray emitting objects <cit.>. <cit.>
made deep X-ray observations of these SFRs down to 0.5 $M_{\odot}$ with the Chandra X-ray observatory. The observations covered a $17^{\prime}
\times 17^{\prime}$ field, and detected a total of 364 X-ray sources. We used their X-ray source list to select
X-ray emitting PMS members. The optical counterparts of the X-ray sources and candidates were searched for with matching
radii of $1.^{\prime \prime}0$ and $1.^{\prime \prime}5$, respectively. We confirmed that 86 X-ray sources and three candidates were detected in the $V$ band.
Among them 73 X-ray sources and one candidate are associated with PMS stars, and the early-type MS star HD 253327
also turns out to be an X-ray source. The other 14 sources and candidates seem to be X-ray active field stars from their colors.
A total of 102 members were identified in the optical passbands. In addition, members of IC 2162 were independently selected from
the IR source catalogue of <cit.> in the same way. We found 216 members observed in the $J$ and $H$ bands (2 early-type
MS stars, 20 Class I, 61 Class II, 158 X-ray sources, 7 X-ray candidates, 19 H$\alpha$ emission stars, and 6 H$\alpha$ candidates).
The membership list from the IR source catalogue (216 stars) was compared with that from the optical data (102 stars). As a result,
100 stars were found in both lists, however the other two members were observed only in either the $J$ or $H$ band. These
membership lists were merged into a membership catalogue. The total number of members identified in this work is 218.
<cit.> estimated about 58 field interlopers scaling the number of contaminants in the FOV of the Chandra Carina
Complex Project <cit.> to that in the FOV of $17^{\prime} \times 17^{\prime}$. If we assume that the surface density
of the field interlopers is uniform across these SFRs, the number of the field interlopers in the FOV of $\sim 9.7^{\prime} \times
9.7^{\prime}$ is about 18. According to the X-ray source classification toward the Carina region <cit.>,
extragalactic sources are so faint that the contribution of these sources should be negligible in this study. Probable interlopers
in our FOV may therefore be foreground and background stars. However, as the stellar density in the direction of IC 2162 is much
lower than that of the Carina region located toward the tangential direction of the Sagittarius spiral arm, the
expected number of field interlopers with X-ray emission in our FOV may be much smaller than 18. As we identified
14 X-ray sources and candidates as field interlopers, the number of field interlopers in our member catalogue may be less than four.
Color excess ratios obtained from the early-type main sequence members. The solid line corresponds to $R_V = 3.1$.
The color excess ratios from optical to mid-infrared data suggest that the reddening law toward IC 2162
is normal.
§.§ Interstellar Extinction and the Reddening Law
Light from stars is obscured by the interstellar material distributed along the line of sight. Therefore, the
effect of extinction on the measured flux or magnitude should be corrected properly. The reddening of young open clusters
can be estimated by comparing the observed colors of early-type stars with their intrinsic colors in the ($U-B$, $B-V$) two color
diagram along the reddening slope (Paper 0). The reddening determined from two early-type MS members is about $E(B-V) = 0.88$
mag. In addition, the spectral type of two early-type PMS members ALS 19 (B0V) and 2MASS J06123651+1756548 (B0.9V)
is available from this work and <cit.>. The reddening of these stars obtained from the spectral type-color relation
(Table 5 in Paper 0) is about $E(B-V) = 1.41$ and 1.17 mag, respectively. The result is consistent with that of previous studies,
e.g. $E(B-V) =$ 0.88 – 1.16 mag <cit.>, 0.64 – 1.47 mag <cit.>, and 0.82 – 1.20 mag <cit.>. More
severe differential reddening is expected across IC 2162 because a large number of stars are embedded in the molecular
The ratio of total-to-selective extinction ($R_V$) is an essential diagnostic tool to investigate the extinction law toward SFRs or young
open clusters. The general interstellar medium (ISM) in the solar neighbourhood is known to have, on average, $R_V = 3.0$ – 3.1
<cit.>. On the other hand, $R_V$ may be larger than the normal value in some dusty SFRs <cit.>.
The extinction law depends on the size distribution of dust grains <cit.>. A large $R_V$ implies that the size of dust grains is,
on average, larger than that found in the general ISM.
We investigated the various color excess ratios of the early-type MS members to study the reddening law toward IC 2162. $R_V$
can be determined from the color excess ratios between two different colors <cit.>. The color excess $E(V-\lambda)$
(where $\lambda = I, J, H, K_S$, [3.6], [4.5], [5.8], and [8.0]) can be computed from the intrinsic color relations of Paper 0 and
Sung et al. (in preparation). Figure <ref> displays the color excess ratios of the early-type MS members.
The color excess ratios at different wavelengths are reasonably matched with the slope corresponding to $R_V = 3.1$ (solid lines in the figure).
This result is acceptable given the $R_V$ variation with the Galactic longitude <cit.>. The normal reddening law implies that the
dust evolution in the front side of the region had already progressed. However, this result may not represent the reddening law of the
embedded cluster.
Zero-age main sequence (ZAMS) fitting to the bright members in the $Q_{V \lambda}$-$Q^{\prime}$ diagrams. The
ZAMS relations of <cit.> are fitted to the lower ridge line of the members. The solid line (red) represents the
adopted distance modulus of 11.6 mag, equivalent to 2.1 kpc. The dashed lines indicate a 0.3 mag error in the ZAMS fitting.
The other symbols are the same as Figure <ref>.
§.§ Distance
The distance to an object is fundamental in determining its physical quantities. We determined the
distance of IC 2162 using the zero-age main sequence (ZAMS) fitting method. The ZAMS relations and reddening-independent indices
introduced in Paper 0 are adopted in the present work. Our ZAMS fitting procedure is based on $UBVIJHK_S$ multicolor photometry,
and therefore the distance can be determined consistently in the optical and near-IR passbands.
Figure <ref> shows $Q_{V \lambda}$-$Q^{\prime}$ diagrams of the bright members ($J < 12.5$). Since the luminosity of stars
can be affected by stellar evolution and binary effects, the ZAMS relations should be fitted to the lower ridge line of the MS
in the $Q_{V \lambda}$-$Q^{\prime}$ planes as shown in the figure. We adjusted the ZAMS relations above and below the faint members
at a given $Q^{\prime}$ index and adopted a distance modulus of 11.6 mag. The lower ridge line could be confined between the ZAMS
relations shifted from the adopted value by $\pm 0.3$ mag, and therefore the upper and lower envelopes are the uncertainty in the distance.
Our result ($2.1 \pm 0.3$ kpc) is in reasonable agreement with that of previous studies within the uncertainties, e.g. 1.9 kpc
<cit.>, 2.4 kpc <cit.>, and 2.5 kpc <cit.>.
§.§ Age
The Hertzsprung-Russell diagram (HRD) provides a comprehensive view of the evolutionary status of stellar system. The reddening of the individual
members was corrected for using the weighted mean reddening of four early-type MS and PMS members, where the weight is exponentially
decreased as the distance between the early-type stars and a given member increases. We converted the reddening-corrected color-magnitude
diagrams (CMDs) in the optical passbands to the HRD using several relations (; Paper 0). The effective temperature
$T_{\mathrm{eff}}$ of a star was basically estimated from the color-$T_{\mathrm{eff}}$ relations, and the spectral type-$T_{\mathrm{eff}}$ relation
was also used for the three early-type members ALS 19, HD 253327, and 2MASS J06123651+1756548. The weighted mean value of the temperatures was adopted
as the $T_{\mathrm{eff}}$ of four early-type MS and PMS members. Only the $V-I$ versus $T_{\mathrm{eff}}$ relation
<cit.> was used for the temperature scale of the PMS members to avoid the effect of excess emission due to accretion activities.
Bolometric correction values were estimated from the $T_{\mathrm{eff}}$ of the individual members using Table 5 of Paper 0.
We present the HRD of IC 2162 in Figure <ref>. The mass of the main ionizing sources in Sh 2-255 and 257 is larger than
$10 M_{\odot}$. It is difficult to estimate the age from the MS turn-off because the stars seem to be still at the MS or PMS stage. On
the other hand, the majority of the PMS members are evolving along Hayashi tracks, while only several members are approaching
the ZAMS along Henyey tracks. There are five PMS members near or below the ZAMS. These stars may have edge-on disks
<cit.>. The upper mass range of the PMS members appears to be as large as $15M_{\odot}$, and the PMS lifetime of high-mass stars
is very short. These facts imply that IC 2162 is very young. Using the evolutionary models for PMS stars <cit.>, we estimated the ages of the PMS members and found a median age of 1.3 Myr with an age spread of 3.3 Myr.
The age distribution is very similar to that obtained from the evolutionary models for half solar metallicity ($Z=0.01$). The age
of IC 2162 is probably about 1 Myr. However, as the majority of PMS members are still deeply embedded in the molecular clouds, the age of
these stars may be much younger.
§ TOTAL MASS AND THE INITIAL MASS FUNCTION
The masses of optically visible members are estimated by comparing their $T_{\mathrm{eff}}$ and $M_{\mathrm{bol}}$ with those
of the evolutionary tracks in the HRD. The evolutionary tracks of <cit.> were used for the early-type MS and PMS members,
while those of <cit.> were adopted for the low-mass PMS members. The solar metallicity was assumed for both models.
These theoretical evolutionary masses can be used as a good mass scale reference for those inferred from the near-IR data.
The masses of the members selected from the IR source catalogue were inferred from the $(J, J-H)$ diagram because the $J$ magnitude is
less affected by excess emission from a disk or envelope compared to the $K$ magnitude <cit.>. Figure <ref> shows the
near-IR CMD of IC 2162. The PMS members have a wide color range from $J-H = 0.5$ to 2.9. A few factors such as high differential
reddening, excess emission from disks and envelopes in the $H$ band, age spread, and photometric errors for faint stars may be responsible
for such a large color spread. Only stars brighter than $J = 16.5$ mag were used so as to minimize the inclusion of stars with large photometric
errors. In order to treat differential reddening, a model grid was constructed from several isochrones reddened by $E(J-H) = 0.25, 0.64, 0.96, 1.27$,
and 1.91 mag [equivalently $E(B-V) = 0.8, 2.0, 3.0, 4.0, 6.0$ mag],respectively.
We attempted to estimate the masses of the PMS members, especially those in the embedded sub-clusters, using the model grid and
compared the masses with those obtained from the optical data. We found the difference between the mass inferred from the
near-IR CMD and that from the HRD to be a strong function of the age of the adopted isochrones, due to the luminosity evolution
of PMS stars. For a given PMS star, the mass estimated from the reddened isochrone is larger for older isochrones. We found that
the difference in masses from the two different methods showed a minimum for a grid of 0.1 Myr isochrones.
On the other hand, the mass of PMS stars ($J > 11.5$ mag) bluer than $J-H = 1.59$ was obtained from the the grid of isochrones
with different ages (0.1, 0.5, 1, 2, 5, and 10 Myr) assuming the minimum reddening of $E(J-H) = 0.25$ mag, equivalent to $E(B-V) = 0.8$ mag.
We compared the masses of all the PMS members with those from the optical data, and confirmed that there was a systematic difference of
$-0.3$ to $0.4 M_{\odot}$ between them over the mass range of 0.2 to $1.6 M_{\odot}$. The systematic difference could be approximated by
a combination of two straight lines. The mass from the near-IR data was converted to the mass scale obtained from the optical data using
those relations. Finally, the masses of four early-type members and two members observed only in either $J$ or $H$ were obtained from
analysis of the optical data.
In order to examine the metallicity effect on the mass estimation, we also inferred the masses of the PMS stars
using the PMS star evolution models for the half solar metallicity. The mass difference from the two models ($Z = Z_{\odot}$
and $Z=1/2Z_{\odot}$) is about 0.13$M_{\odot}$. It was confirmed that this difference yields a negligible effect on the resultant IMF in the mass
range of 1 – 100$M_{\odot}$.
Hertzsprung-Russell diagram of IC 2162. A few isochrones (red solid lines) for the age of 0.5, 1, and 5 Myr are superimposed
on the diagram with the evolutionary tracks of several initial masses <cit.>, where the solar metallicity is assumed. The
other symbols are the same as Figure <ref>
Near-infrared color-magnitude diagram. The solid lines represent pre-main sequence star isochrones with different ages
(0.1, 0.5, 1, 2, 5, and 10 Myr) from <cit.>, where the model parameters are the same as seen in Figure <ref>. The isochrones are
reddened by $E(J-H) = 0.25, 0.64, 0.96, 1.27$, and 1.91 mag, respectively, after correction for the distance modulus of 11.6 mag. The
arrow denotes the reddening vector corresponding to $A_J = 3$ mag. The other symbols are the same as Figure <ref>.
We obtained the cluster mass of $169M_{\odot}$ from the sum of the masses of all the identified members. This is definitely a
lower limit because a number of sub-solar mass stars below the completeness limit were not considered. IC 2162 seems to be the smallest SFR
among the young open clusters studied by our research group, e.g., $> 510 M_{\odot}$ for NGC 1624 and NGC 1931 <cit.>,
$> 576 M_{\odot}$ for NGC 2264, $> 1,300 M_{\odot}$ for NGC 1893 <cit.>, $> 2,600 M_{\odot}$ for NGC 6231 <cit.>,
$> 7,400 M_{\odot}$ for Westerlund 2 <cit.>, and $> 50,000 M_{\odot}$ for Westerlund 1 <cit.>.
The IMF is, in general, expressed by the following relation <cit.>:
\begin{equation}
\xi (\log m) \equiv {N \over \Delta\log m \cdot S}
\end{equation}
where $N$, $\Delta\log m$, and $S$ represent the number of stars in a given mass bin, the size of a logarithmic mass bin, and
the area of the observed region, respectively. The size of the mass bin was set to be 0.4 to avoid uncertainties from small sample statistics.
We assumed that the upper limit of stellar mass to be $100 M_{\odot}$, and adopted a larger bin size of 1 for the highest mass bin ($10 < M/M_{\odot}
\le 100$) because the number of high-mass stars was insufficient to sample across various mass bins. We counted the number of stars
in each mass bin, and then divided the total by the size of the mass bin and the area of IC 2162 (the area of our FOV). In order to prevent
the binning effect, the IMF was re-derived for the same stars by shifting the mass bin by 0.2.
Figure <ref> shows the IMF of IC 2162. We also plotted the IMF of the young open cluster NGC 2264 <cit.> for comparison.
The shapes of the IMFs are similar to each other in the mass range from $1 M_{\odot}$ to the upper limit of stellar mass. The IMF is generally
quantified by its slope ($\Gamma$) to compare it with that of other SFRs. We computed the slope of the IMF using a linear least square
fitting method. The slope of the IMF is about $\Gamma = -1.6 \pm 0.1$. This result is consistent with that of NGC 2264, although the slope
is slightly steeper than that of the Salpeter/Kroupa IMF <cit.>. It implies that the underlying star formation processes in IC 2162
are optimized to produce low-mass stars rather than high-mass stars.
The initial mass function (IMF) of IC 2162. The IMF (open and filled circles) was derived using different binning of the same stars to
avoid the binning effect. The dashed line and shaded area represent the IMF of NGC 2264 and its uncertainty <cit.>, respectively.
The arrow indicates the the IMF below the completeness limit.
§ SPECTRAL ENERGY DISTRIBUTION ANALYSIS
The SED of the members was also analyzed using the SED fitting tool (called the SED fitter) of <cit.>. In
order to increase the number of samples, we included 43 stars with X-ray and mid-IR excess emission in the membership catalogue. A total of 261
members were used in this analysis. All the selected members were observed in more than 3 passbands, and their photometric errors were
better than 0.15 mag in the optical to near-IR passbands and 0.20 mag in the mid-IR passbands.
In order to minimize the degrees of freedom, we limited the distance to $2.1 \pm 0.3$ kpc based on the result of the ZAMS fitting above.
The total extinction $A_V$ was set to be 2.48 – 100 mag. The SED fitter suggests various models with $\chi ^2$ values for a given SED.
To select the most appropriate model among them, we followed the guidelines introduced by <cit.>. The guidelines recommended
the model that gives the smallest stellar mass out of the 10 suggested models if $\chi ^2 / \chi _{\mathrm{min}} ^2 \leq 2.0$.
If $\chi _{\mathrm{min}} ^2$ is less than 1.0, the mass-minimum model with $\chi ^2 < 2.0$ is adopted.
The model adopted from the SED fitter provides various physical quantities of the disks and envelopes as well as stellar parameters (mass and age).
The disk parameters such as mass, outer radius, and accretion rate evolve with time by a few orders of magnitude over 10 Myr. A strong
correlation between the disk accretion rate and disk mass was also found [$\log(dM_{\mathrm{disk}}/dt) = -4.89 (\pm 0.12) + 1.15 (\pm 0.03)
\log (M_{\mathrm{disk}}/M_{\star})$]. It appears that the envelope mass decreases dramatically after 1 Myr. While the angle of the cavity rises with time,
its density, in contrast, declines as a function of time. The circumstellar extinction is proportional to the envelope mass and shows a sharp
increase for stars with large envelope mass ($\log M_{\mathrm{env}}/M_{\star} > -3$). Stars with a massive envelope accrete much more
material than low-mass counterparts. These correlations between the results obtained from the SED fitter are very similar to those found by
We also compared the stellar mass and age inferred from the SED fitter with those from the analysis of the HRD. For
less reddened members, the masses obtained from the different methods are reasonably consistent, while the SED fitter
underestimated the ages of the PMS members. On the other hand, the SED fitter tends to overestimate the masses and ages of
highly reddened stars. This discrepancy has also been reported by <cit.>.
§ SUMMARY
IC 2162 is an active SFR in which very young sub-clusters are embedded. We present a multiwavelength
study of the SFR as part of the SOS project. This work provided homogeneous optical photometric data as well as a
comprehensive result for the young stellar population in IC 2162.
The main ionizing sources in Sh 2-255 and 257 are two B0 stars (ALS 19 and HD 253327). A total of 218
members were identified from the various photometric diagrams, the X-ray source list
<cit.>, and the YSO list <cit.>. It appears from the two early-type MS members that IC 2162 is reddened by at least
$E(B-V) = 0.8$ mag. A large portion of the color spread in the near-IR CMD may be attributed to the presence of severe differential
reddening, because a large number of stars are actually embedded in the molecular cloud behind the HII bubbles. The reddening law
toward IC 2162 was investigated with various color excess ratios and the ratio of total-to-selective extinction found to
be the normal value ($R_V = 3.1$). It implies that the dust evolution in the front side of the SFR has been completed.
We also revisited the distance to IC 2162 with the ZAMS fitting method and determined a distance of $2.1 \pm 0.3$ kpc.
The ages of the members were estimated from the HRD using several evolutionary models <cit.> for the solar metallicity. The median age
of the optically visible stars in IC 2162 was about 1.3 Myr, and an age spread of 3.3 Myr was found. We derived the IMF of IC 2162
by analyzing the HRD and the ($J, J-H$) diagram. The shape of the IMF is similar to that of the nearby young open cluster
NGC 2264 in the mass range $1 M_{\odot}$ to the upper limit of stellar mass. The slope of the IMF was $\Gamma = -1.6 \pm 0.1$,
which is slightly steeper than that of Salpeter/Kroupa IMF. This result indicates that it was low-mass star formation that mostly took place
throughout IC 2162. The lower limit on the cluster mass ($> 169 M_{\odot}$) was also constrained from the sum of the masses of all the identified members.
The SEDs of the members were also analyzed with the SED fitter <cit.>. The results included the disk and envelope parameters
of PMS stars as well as stellar parameters such as age and mass. The properties of the disk and envelope were investigated as a function
of stellar age or mass, respectively. The discrepancy between the results from the SED fitter and the analysis based on the
HRD was also pointed out.
This work was partly supported by a National Research Foundation of Korea grant funded
by the Korean Government (Grant No. 20120005318) and partly supported by the Korea Astronomy
and Space Science Institute (Grant No. 2015183014).
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|
1511.00518
|
Finnish Meteorological Institute, POB-503, FI-00101, Helsinki, Finland
E-sail Boltzmann PIC simulation
P. Janhunen
Pekka Janhunen
The solar wind electric sail (E-sail) is a planned in-space propulsion
device that uses the natural solar wind momentum flux for spacecraft
propulsion with the help of long, charged, centrifugally stretched
tethers. The problem of accurately predicting the E-sail thrust is
still somewhat open, however, due to a possible electron population
trapped by the tether. Here we develop a new type of particle-in-cell
(PIC) simulation for predicting E-sail thrust. In the new simulation,
electrons are modelled as a fluid, hence resembling hybrid simulation,
but in contrast to normal hybrid simulation, the Poisson equation is
used as in normal PIC to calculate the self-consistent electrostatic
field. For electron-repulsive parts of the potential, the Boltzmann
relation is used. For electron-attractive parts of the potential we
employ a power law which contains a parameter that can be used to
control the number of trapped electrons. We perform a set of runs
varying the parameter and select the one with the smallest number of
trapped electrons which still behaves in a physically meaningful way
in the sense of producing not more than one solar wind ion deflection
shock upstream of the tether. By this prescription we obtain thrust
per tether length values that are in line with earlier estimates,
although somewhat smaller. We conclude that the Boltzmann PIC
simulation is a new tool for simulating the E-sail thrust. This tool
enables us to calculate solutions rapidly and allows to easily study
different scenarios for trapped electrons.
The electric solar wind sail (electric sail, E-sail) is a planned
device for producing interplanetary spacecraft propulsion from the
natural solar wind by a set of centrifugally stretched thin metallic
tethers that are kept artificially at a high positive potential
<cit.>. In order to engineer E-sail devices in detail, one
should predict the magnitude of the Coulomb drag that the flowing
solar wind exerts on the charged tether. Thus far this problem has
been studied using particle-in-cell (PIC) simulations
<cit.>, Vlasov simulations <cit.> and other methods
<cit.>. However, a fully
satisfactory way of estimating E-sail thrust has not yet emerged,
except for negative polarity tethers <cit.>, which are
however more suitable to use in low Earth orbit (LEO) as a deorbiting
plasma brake device <cit.> than in the solar wind
<cit.>. Independent laboratory measurements of the width of the electron
sheath around a positively biased tether placed in streaming plasma
were made by <cit.>. The laboratory results were
produced in a plasma mimicking conditions in LEO and they are in good
agreement with PIC thrust predictions <cit.>.
In this paper we develop a new variant of the PIC simulation of
<cit.> which treats electrons as a fluid obeying the
Boltzmann relation, with two additional tricks to be detailed
below. The motivation is to have a code which is relatively fast and
easy to run and which makes it possible for the user to control the
assumed amount of trapped electrons in a simple way. We present the
simulation code and use it to derive a thrust estimate in a case which
has relevance to the E-sail. Making more comprehensive set of runs
with different voltages and in different solar wind conditions is
outside of the scope of the paper.
§ PHYSICS OF TETHER COULOMB DRAG
The negative polarity Coulomb drag effect can be successfully
simulated by a PIC code <cit.>. The negative polarity effect
is less challenging to simulate than the positive polarity one,
because in the negative polarity case, trapped particles do not
form. Electrons are not trapped because they are repelled by the
tether's negative potential structure. Ions (protons) are also not
trapped, because in relevant cases the surrounding plasma flow is
supersonic so that in a coordinate frame where the tether is
stationary, ions enter the potential structure with significant
kinetic energy and hence cannot be easily trapped by the potential.
In the positive polarity case which is the subject of the present
paper, electrons may become trapped because the tether attracts them
and because the electron flow is subsonic so that there are some
electrons which enter the potential structure with nearly zero initial
energy. The issue of trapped electrons was identified by
<cit.> and simulations were later made where chaotisation of
trapped electron orbits when scattering at the end of the tether and
at the spacecraft was explicitly included in the PIC code, as well as
occasional removal of trapped electrons due to collisions with the
tether <cit.>. However, in that calculation the
timescale where trapped electrons were removed was made much faster
than in reality, in order to complete the simulation run in a
reasonable time. Basically, processes controlling the size of the
trapped electron population are poorly known, because we think that
they happen at longer timescales than what is accessible by PIC
Consider a 2-D positive potential structure which thus forms a
negative potential energy well (attractive structure) for
electrons. If the potential is stationary and hosts no trapped
electrons and if the ambient electron distribution is a non-drifting
Maxwellian and the plasma is tenuous enough to be collisionless, then
it can be shown using Liouville's theorem that the local electron
density can nowhere exceed the background plasma density
<cit.>. In fact, under the stated
conditions the electron density is typically significantly less than the
background density in most regions covered by the attractive potential
structure. Physically the result is due to the fact that although the
tether attracts electrons and therefore focusses them to move through its vicinity,
the electron density remains low because accelerating the electron
reduces the time that the electron spends inside the attractive potential
structure. Conservation of the electron's originally nonzero angular momentum also limits the minimum
distance that the electron can have from the tether.
That the electron density nowhere exceeds the background density leads
to an interesting dilemma (Éric Choinière, private communication)
that is associated with trapped electrons because the ion density, on
the other hand, must be higher than the background density inside at
least in some regions of the upstream side ion deflection shock. Then,
a significant positive charge density can build up at the upstream ion
deflection region, which spreads out the electron-attractive potential
structure and thus decreases the local electron density further which,
unless some process intervenes, makes the charge imbalance worse. What
exactly happens remains not well known, but conceivably, instabilities
which are able to scatter electrons might develop and take place until
enough trapped electrons have been formed so that proper charge
neutralisation in the ion deflection shock region has been restored. A
simple approach to model this kind of general behaviour may be to
reduce the number of trapped electrons in the simulation until the
result becomes nonphysical in some way or another and postulate that
the most realistic run is the physical run with least amount of
trapped electrons. This is what the simulation presented next attempts
to accomplish.
§ SIMULATION CODE
We use a special version of electrostatic PIC where electrons are not
modelled as particles, but as a fluid whose local density $n_e(x,y)$
is an assumed function of the local potential $V(x,y)$. When the
potential repels electrons (i.e. when $V(x,y)<0$), the Boltzmann
relation is assumed,
\begin{equation}
n_e(x,y) = \nzero \exp\left[\frac{eV(x,y)}{T_e}\right],\quad \mathrm{when}\ V(x,y)<0
\label{eq:repulsive}
\end{equation}
where $\nzero$ is the background plasma density and $T_e=10$ eV is the
background electron temperature in energy units. The Boltzmann
relation is supposedly a good approximation for the density of
particles that try to penetrate thermally into a repulsive potential
For attractive parts of the potential, the electron density depends on
the number and distribution function of trapped electrons and thus
extra assumptions are needed. That one needs such assumptions is
natural because we are not modelling electron dynamics using first
principles. We want to have a functional relationship which contains a
parameter that can be used to tune the number of trapped electrons in
a simple way.
In this paper we use the following functional form for electron
density in attractive parts of the potential:
\begin{equation}
n_e(x,y) = \nzero \left[1+\frac{e V(x,y)}{T_e}\right]^\nu
\quad\mathrm{when}\ V(x,y)\ge 0
\label{eq:attractive}
\end{equation}
where $\nu>0$ is the trapped electron control parameter.
In Figure <ref> we show how the local electron density
depends on the local potential according to Eqs. (<ref>)
and (<ref>).
The code that we use in this paper is a descendant of a PIC code that
we used earlier <cit.>. It is an
electrostatic code with linear particle weighting and additional
grid-level smoothing which is implemented in the Fourier domain. The
code supports 2-D and 3-D simulations and optionally includes a
constant external magnetic field. The Poisson solver is implemented by
fast Fourier transform and the code is parallelised with OpenMP
threads and the Message Passing Interface (MPI) library. The code is
also vectorised, so that for a processor with 256-bit wide AVX2
instruction set, each processor core calculates eight particle updates
simultaneously. For particle coordinates and velocities, 32-bit
floating point accuracy is used. To avoid loss of accuracy, particle
coordinates are expressed relative to the centroid of the grid cell
where the particle resides. Particles are stored in lists belonging to
grid cells which enables fast accumulation of the charge density (no
irregular memory addressing patterns needed) and enables good data
locality and cache friendliness. When a particle crosses a grid cell
boundary, it is removed from the old list and entered into a new one.
An additional trick is needed to avoid solutions that look unphysical
and oscillate drastically in time. Namely, the electron density must not
react instantaneously to changed electric potential, but with some
delay. If $n_e^*(x,y)$ is the preliminary instantaneous electron
density computed by Eqs. (<ref>) and
(<ref>), then we propagate the real electron density
$n_e(x,y)$ by the differential equation
\begin{equation}
\frac{\partial n_e}{\partial t} = \frac{n_e^*-n_e}{\tau}
\label{eq:diffeq}
\end{equation}
where $\tau$ is a time constant. The solution $n_e(t)$ of
(<ref>) is effectively a low-pass filtered version of
$n_e^*(t)$ where frequencies above $\sim 1/\tau$ are filtered out. We
have tested different values of $\tau$ and found that in order to
produce good results (in the sense of avoiding unphysical
oscillations), $\tau$ should be proportional to the ion plasma
oscillation period, because shorter timescale phenomena are not
included in a model with electron fluid. Specifically, we use
\begin{equation}
\tau = C\,\frac{1}{\omega_{pi}},\quad C=1.5
\label{eq:tau}
\end{equation}
where $\omega_{pi}$ is the ion plasma frequency
\begin{equation}
\omega_{pi} = \sqrt{\frac{\nzero e^2}{\epsilonzero m_i}}.
\label{eq:omegapi}
\end{equation}
By numerical experimentation, $C\le 1$ produces unphysical
oscillations which disappear if $C>1$. We use $C=1.5$ to have some
extra margin of safety. On the other hand there is no reason to
increase $C$ further.
In summary, the code pushes ions forward by their equations of motion
(in this paper with $\mathbf{B}$=0),
\begin{eqnarray}
\label{eq:eqs_of_motion}
\frac{d \mathbf{v}_i(t)}{dt} &=& \MARKIII{\frac{e}{m_i}} \left[\mathbf{E}(\mathbf{x}_i(t))
+ \mathbf{v}_i(t)\times\mathbf{B}\right]\nonumber \\
\frac{d \mathbf{x}_i(t)}{dt} &=& \mathbf{v}_i(t),
\end{eqnarray}
accumulates ion density $n_i$ from the ion positions $\{\mathbf{x}_i(t)\}$
\begin{equation}
n_i({\bf x},t) = \sum_i S(\mathbf{x}-\mathbf{x}_i(t))
\end{equation}
where $S$ is the linear area weighing function
<cit.>. Then the code applies Fourier domain filtering
to smooth the ion density spatially, calculates the provisional electron density
$n_e^*$ by Eqs. (<ref>) and (<ref>),
updates the electron density $n_e$ by Eq. (<ref>), computes
the charge density $\rho_e = e(n_i-n_e)$, solves the Poisson equation
\begin{equation}
-\nabla^2\phi(\mathbf{x}) = \frac{1}{\epsilonzero}\,\rho_e(\mathbf{x}),
\end{equation}
computes the gridded electric field $\mathbf{E}(\mathbf{x})$ from $\phi(\mathbf{x})$ by finite
differencing and repeats the process for the next timestep.
The particle equations of motion (<ref>) are
discretised in time using the second order accurate leapfrog method as
is common in momentum conservative explicit PIC simulations <cit.>. Equation (<ref>) is
discretised by forward Euler method. This is equivalent to linearly
mixing the portion $\Delta t/\tau$ of $n_e^*$ into $n_e$ at each
timestep, where $\Delta t$ is the length of the timestep. Table
<ref> summarises the simulation parameters.
The timestep $\Delta t$ in our code is $\sim 50$ times longer than
what it would be in a full electron-ion PIC code using the same grid
resolution. In addition to that, it seems that the code can produce
useful results with a smaller number of ions per grid cell than a full
PIC code so that the improvement factor in practical computational
efficiency is probably several hundred. The latter property
is probably due to averaging over timescale $\tau\approx 210\,\Delta t$
which effectively reduces the numerical noise.
§ RESULTS
First we made a family of runs at low resolution, varying the trapped
electron parameter $\nu$ between 0.16 and 0.08. The grid spacing was
10 m and the box side length 1.92 km so that the grid was $192\times
192$. The timestep was 4 $\mu$s, corresponding to 2500 km/s maximum
signal speed, resulting in Courant-Friedrichs-Lewy (CFL) parameter of
0.16 with respect to the used solar wind speed of 400 km/s (Table
<ref>). The solar wind plasma density is 7.3 cm$^{-3}$
and for simplicity, no interplanetary magnetic field is used. The ion
and electron temperature is 10 eV. Pure proton plasma is used as the
solar wind. The plasma parameters correspond to average solar wind
properties at 1 au distance from the sun.
The real solar wind also has typically 10 nT magnetic field. However,
the resulting background electron Larmor radius of $\sim$1 km is much
larger than the diameter of the electron sheath $\sim$100-200 m. Thus
the magnetic field is expected to play only a minor role as far as the
magnitude of the E-sail thrust is concerned, so that assuming zero
magnetic field is not a bad first approximation.
For each run we determined the thrust by two complementary methods,
“Mom” and “Coul”. In method Mom we keep track of the momentum $x$
component of all particles entering and leaving the simulation box and
deduce that the momentum which is left in the box is the one which
gets transferred to the tether and the spacecraft. In method Coul we
evaluate the negative gradient of the plasma potential at the tether's
location and use Coulomb's law, $dF/dz=\lambda E$ where $dF/dz$ is the
thrust per tether length, $\lambda$ is the tether's linear charge
density and $E$ is the electric field caused by the plasma and
evaluated at the tether. The line charge $\lambda$ is given as
an input parameter (in the runs performed, $\lambda=8.64\cdot10^{-8}$
As/m), and the corresponding tether voltage $\Vzero$ is
calculated from the run results after the end of the run by summing
the tether's vacuum potential and the potential created by the
plasma. Because the plasma potential varies somewhat from run to run,
the resulting tether voltage $\Vzero$ is also slightly different in
each run. For each run, we made a qualitative visual check of the
result to determine if the solution was physically reasonable in the
sense that it contained not more than one ion deflection shock.
We then made an identical family of runs with halved grid spacing of 5
m. The computing time is about 8 times longer than in the
low resolution set. Results of both families of runs are summarised in
Table <ref>.
Run $\nu=0.12$ is selected as the baseline because our visual
inspection indicates that $\nu=0.12$ is the smallest value of $\nu$
for which we get not more than one ion deflection shock. Figure
<ref> shows the electron and ion density and the total
potential in the baseline $\nu=0.12$ run as well as in another run
where $\nu=0.08$ and which exhibits two ion shocks. In addition, in
cases where two shocks emerge in the solution, the shocks also tend to
slowly propagate upstream so that full convergence is not reached. The
high resolution run data are used, but for viewing speed convenience
the spatial grid resolution has been halved by neighbour averaging
before plotting in Fig. <ref>. The solar wind arrives from
the right in Fig. <ref>.
In Fig. <ref> one clearly sees that the run $\nu=0.12$ looks
physically meaningful and well-behaved, while run $\nu=0.08$ exhibits
two shocks and is thus physically not acceptable. Similar visual
judgement can be made also for the other performed runs, and
information presented for each run in Table <ref> was
compiled in this way.
Figure <ref> shows the thrust history of the $\nu=0.12$ run
(high resolution version). After natural initial transients have died
out, the Coul and Mom methods are in good agreement with each other
regarding the thrust per length produced by the E-sail. By
construction, the Mom method reacts to changes slower than Coul, and
during the first $\sim 10$ ms both methods give unreliable results.
[Discussion and conclusions]
Using Boltzmann electrons in PIC simulation with two additional tricks
enables us to rapidly compute reasonable looking solutions for the
positive polarity Coulomb drag problem, applicable to the E-sail in
the solar wind. The first trick is that while the Boltzmann relation
(<ref>) is a natural choice for the electron density in
repulsive parts of the potential, in the attractive parts we use an a
power law relationship (<ref>) containing one free
parameter that can be used to control the amount of trapped
electrons. The second trick is that to avoid spurious temporal
oscillations, the electron density is not updated instantaneously, but
frequencies higher than $\sim \omega_{pi}$ are filtered out by
Eq. (<ref>).
The calculated solutions depend on parameter $\nu$ that must be
specified by the user and that parametrises the number of trapped
electrons. Once $\nu$ is fixed, the simulation produces a prediction
for the thrust per tether length. We calculated the thrust per length
by two complementary methods and found good mutual agreement. For too
small values of $\nu$ the result looks unphysical, however, because it
contains two ion deflection shocks. Our strategy is that by finding
the smallest value of $\nu$ which gives a physically reasonable single
shock solution, we obtain a solution which is expectedly the best
approximation to reality within this framework.
For voltage $\Vzero$ of 17.7 kV and nominal solar wind at 1 au
(density 7.3 cm$^{-3}$, speed 400 km/s), the thrust predicted by the
framework is $\sim 320$ nN/m. In our earlier paper <cit.> we arrived at
estimate $500$ nN/m at $\Vzero=20$ kV if
there are no trapped electrons, 400 nN/m if trapped density equals
background density, and 320 nN/m if the average trapped density in
the sheath is four times
higher than background. If one
assumes that the thrust is approximately linearly proportional to
$\Vzero$, after scaling to 20 kV the new estimate becomes $\sim 360$ nN/m.
The result is thus in quantitative agreement with <cit.>
when one takes into account that the average trapped density
is $\sim 1.5$ in the new simulation. In summary, if one
assumes that the number of trapped electrons increases until the
solution has only one ion deflection shock, then one ends up with
$\sim 360$ nN/m thrust estimate for 20 kV and average 1 au solar wind.
One must be somewhat cautious when interpreting the numerical result,
because the electron Boltzmann simulation in any case contains less
physics than the earlier full PIC simulations, and, for example, the
particular functional form (<ref>) used to describe the
electron density in attractive parts of the potential was chosen
mostly by mathematical convenience. In any case, the Boltzmann
simulation is a new addition in the toolbox of the E-sail thrust
analyst. Its benefits are fast computation compared to traditional PIC
and easy possibility to test different assumptions concerning trapped
The author thanks Petri Toivanen for reading the manuscript and making
a number of useful suggestions, and Éric Choinière for teaching the
trapped electron dilemma to the author.
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Sanchez-Torres, A.: Propulsive force in an electric solar sail,
Contrib. Plasma Phys., 54, 314–319, 2014.
[Sanmartín et al.(2008)]SanmartinEtAl2008
Sanmartín, J.R., Choinière, E., Gilchrist, B.E., Ferry, J.-B. and
Martínez-Sánchez, M.: Bare-tether sheath and current: comparison
of asymptotic theory and kinetic simulations in stationary plasma,
IEEE Trans. Plasma Phys., 36, 2851–2858, 2008.
[Siguier et al.(2013)]SiguierEtAl2013
Siguier, J.-M., P. Sarrailh, J.-F. Roussel, V. Inguimbert, G. Murat
and J. SanMartin: Drifting plasma collection by a positive biased
tether wire in LEO-like plasma conditions: current measurement and
plasma diagnostics, IEEE Trans. Plasma Sci., 41, 3380–3386, 2013.
Local relative electron density $n_e(x,y)/\nzero$ as function of
local potential $V(x,y)$ for $\nu=0.12$, by Eqs. (<ref>)
and (<ref>). The electron temperature is $T_e=10$ eV.
Physically realistic run (left, $\nu=0.12$)
and non-physical run (right, $\nu=0.08$) which exhibits double ion
deflection shock. The panels are electron density (top), ion density (middle)
and electric potential (bottom). Densities are relative to
$\nzero$. White dot shows tether position at the origin.
Thrust history in $\nu=0.12$ run: method Mom (black) and method
Coul (red).
Simulation parameters (high resolution runs). For
definition of tether electric radius $r_w^*$, see <cit.>.
Parameter Symbol Value
Grid size $384 \times 384$
Grid spacing $\Delta x$ $5$ m
$X$ domain -1.44 .. 0.48 km
$Y$ domain -0.96 .. 0.96 km
Timestep $\Delta t$ $2 \mu$s
Run duration $t_{\rm max}$ $60$ ms
Number of timesteps 30000
Ions per cell $N_0$ 100 (in plasma stream)
Number of particles $\sim 15\cdot 10^6$
Plasma density $\nzero$ $7.3$ cm$^{-3}$
Ion mass $m_i$ 1 amu (protons)
Plasma drift $v$ 400 km/s
Electron temperature $T_e$ 10 eV
Ion temperature $T_i$ 10 eV
Electron Debye length $\lambda_\mathrm{De}$ 8.7 m
Tether voltage $\Vzero$ $\sim$17.7 kV
Tether electric radius $r_w^{*}$ 1 mm
Line charge density $\lambda$ $8.64\cdot 10^{-8}$ As/m
Performed runs. Thrust per tether length in nN/m by Mom and
Coul methods, NS=number of shocks. The physically preferred run $\nu=0.12$ is
singled out by horizontal lines.
3cLow resolution runs 3cHigh resolution runs 2l
$\nu$ Mom Coul $\Vzero$/kV Mom Coul $\Vzero$/kV NS Remarks
0.16 301.1 319.0 17.601 303.6 319.1 17.578 1
0.15 311.5 336.6 17.615 315.6 331.3 17.627 1
0.14 322.3 348.6 17.667 327.3 343.7 17.675 1
0.13 328.1 352.3 17.733 330.1 347.9 17.709 1 Thrust local max
0.12 312.7 340.6 17.745 313.7 332.8 17.725 1 Smallest $\nu$ with NS=1
0.11 296.0 328.5 17.751 300.7 319.8 17.749 1+
0.10 298.7 316.2 17.801 311.7 313.2 17.787 2
0.09 356.5 312.0 17.841 380.9 311.0 17.825 2 Mom$>$Coul
0.08 498.1 308.5 17.881 528.0 307.2 17.868 2 Mom$\gg$Coul
|
1511.00310
|
In the Integer Quadratic Programming problem input is an $n \times n$ integer matrix $Q$, an $m \times n$ integer matrix $A$ and an $m$-dimensional integer vector $b$. The task is to find a vector $x \in \mathbb{Z}^n$ minimizing $x^TQx$, subject to $Ax \leq b$. We give a fixed parameter tractable algorithm for Integer Quadratic Programming parameterized by $n + \alpha$. Here $\alpha$ is the largest absolute value of an entry of $Q$ and $A$. As an application of our main result we show that Optimal Linear Arrangement is fixed parameter tractable parameterized by the size of the smallest vertex cover of the input graph. This resolves an open problem from the recent monograph by Downey and Fellows.
§ INTRODUCTION
While Linear Programming is famously polynomial time solvable <cit.>, most generalizations are not. In particular, requiring the variables to take integer values gives us the Integer Linear Programming problem, which is easily seen to be NP-hard. On the other hand, integer linear programs (ILPs) with few variables can be solved efficiently. The celebrated algorithm of Lenstra <cit.> solves ILPs with $n$ variables in time $f(n)L^{O(1)}$ where $f$ is a (doubly exponential) function depending only on $n$ and $L$ is the total number of bits required to encode the input integer linear program. In terms of parameterized complexity this means that Integer Linear Programming is fixed parameter tractable (FPT) when parameterized by the number $n$ of variables to the input ILP. In parameterized complexity input instances come with a parameter $k$, and an algorithm is called fixed parameter tractable if it solves instances of size $L$ with parameter $k$ in time $f(k)L^{O(1)}$ for some function $f$ depending only on $k$. For an introduction to parameterized complexity we refer to the recent monograph of Downey and Fellows <cit.>, as well as the textbook by Cygan et al. <cit.>.
Following the algorithm of Lenstra <cit.> there has been a significant amount of research into parameterized algorithms for Integer Linear Programming, as well as generalizations of the problem to (quasi) convex optimization. Highlights include the algorithms for Integer Linear Programming with improved dependence on $n$ by Kannan <cit.>, Clarkson <cit.> and Frank and Tardos <cit.> and generalizations to $N$-fold integer programming due to Hemmecke et al. <cit.>, see also the book by Onn <cit.>.
Heinz <cit.> generalized the FPT algorithm of Lenstra to quasi-convex polynomial optimization. More concretely, the algorithm of Heinz finds an integer assignment to variables $x_1, \ldots, x_n$ minimizing $f(x_1, \ldots x_n)$ subject to the constraints $g_i(x_1, \ldots, x_n) \leq 0$ for $1 \leq i \leq m$, where $f$ and $g_1, \ldots, g_m$ are quasi-convex polynomials of degree $d \geq 2$. Here a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is quasi-convex if for every real $\lambda$ the set $\{x \in \mathbb{R}^n : f(x) \leq \lambda\}$ is convex. The algorithm has running time $L^{O(1)}n^{O(dn)}2^{O(n^3)}$, that is, it is fixed parameter tractable in the dimension $n$ and the degree $d$ of the input polynomials. Khachiyan and Porkolab <cit.> gave an even more general algorithm that covers the case of minimization of convex polynomials over the integer points in convex semialgebraic sets given by arbitrary (not necessarily quasi-convex) polynomials, see <cit.> for more details.
On the other hand, generalizations of Integer Linear Programming to optimization of possibly non-convex functions over possibly non-convex domains quickly become computationally intractable in the strongest possible sense. In fact, solving a system of quadratic equations over $232$ integer valued variables is undecidable <cit.>, and the same holds for finding an integer root of a single multi-variate polynomial of degree $4$ <cit.>.
Nevertheless, there are interesting special cases of non-convex (integer) mathematical programming for which algorithms are known to exist, and it is an under-explored research direction to investigate the parameterized complexity of these problems.
Perhaps the simplest such generalization is the Integer Quadratic Programming problem. Here the input is a $n \times n$ integer matrix $Q$, an $m \times n$ integer matrix $A$ and an $m$-dimensional integer vector $b$. The task is to find a vector $x \in \mathbb{Z}^n$ minimizing $x^TQx$, subject to $Ax \leq b$. Thus, in this problem, the domain is convex, but the objective function might not be.
It is a major open problem whether there exists a polynomial time algorithm for Integer Quadratic Programming with a constant number of variables.
Indeed, until quite recently the problem was not even known to be in NP <cit.>, and the first polynomial time algorithm for Integer Quadratic Programming in two variables was given by Del Pia and Weismantel <cit.> in 2014.
In this paper we take a more modest approach to Integer Quadratic Programming, and consider the problem when parameterized by the number $n$ of variables and the largest absolute value $\alpha$ of the entries in the matrices $Q$ and $A$. Our main result is an algorithm for Integer Quadratic Programming with running time $f(n,\alpha)L^{O(1)}$, demonstrating that the problem is fixed parameter tractable when parameterized by the number of variables and the largest coefficient appearing in the objective function and in the constraints.
On one hand Integer Quadratic Programming is a more general problem than Integer Linear Programming. On the other hand the parameterization by variables and coefficients is a much stronger parameterization than parameterizing just by the number $n$ of variables. By making the largest entry $\alpha$ of $Q$ and $A$ a parameter we allow the running time of our algorithms to depend in arbitrary ways on essentially all of the input. The only reason that designing an FPT algorithm for this parameterization is non-trivial is that the entries in the vector $b$ may be arbitrarily large compared to the parameters $n$ and $\alpha$. This makes the number of possible assignments to the variables much too large to enumerate all assignments by brute force. Indeed, despite being quite restricted our algorithm for Integer Quadratic Programming allows us to show fixed parameter tractability of a problem whose parameterized complexity was unknown prior to this work. More concretely we use the new algorithm for Integer Quadratic Programming to prove that Optimal Linear Arrangement parameterized by the size of the smallest vertex cover of the input graph is fixed parameter tractable.
In the Optimal Linear Arrangement problem we are given as input an undirected graph $G$ on $n$ vertices. The task is to find a permutation $\sigma : V(G) \rightarrow \{1, \ldots, n\}$ minimizing the cost of $\sigma$. Here the cost of a permutation $\sigma$ is $val(\sigma,G) = \sum_{uv \in E(G)} |\sigma(u)-\sigma(v)|$. The problem was shown to be NP-complete already in the 70's <cit.>, admits a factor $O(\sqrt{\log n}\log\log n)$ approximation algorithm <cit.>, but no admits no polynomial time approximation scheme, assuming plausible complexity-theoretic assumptions <cit.>.
We consider Optimal Linear Arrangement parameterized by the size of the smallest vertex cover of the input graph $G$. A vertex cover of a graph $G$ is a vertex set $C$ such that every edge in $G$ has at least one endpoint in $C$. When Optimal Linear Arrangement is parameterized by the vertex cover number of the input graph, an integer parameter $k$ is also given as input together with $G$ and $n$. An FPT algorithm is allowed to run in time $f(k)n^{O(1)}$ and only has to provide an optimal layout $\sigma$ of $G$ if there exists a vertex cover in $G$ of size at most $k$. We remark that one can compute a vertex cover of size $k$, if it exists, in time $O(1.2748^k + n^{O(1)})$ <cit.>. Hence, when designing an algorithm for Optimal Linear Arrangement parameterized by vertex cover we may just as well assume that a vertex cover $C$ of $G$ of size at most $k$ is given as input.
The parameterized complexity of Optimal Linear Arrangement parameterized by vertex cover was first posed as an open problem by Fellows et al. <cit.>. Fellows et al. <cit.> showed that a number of well-studied graph layout problems, such as Bandwidth and Cutwidth can be shown to be FPT when parameterized by vertex cover, by reducing the problems to Integer Linear Programming parameterized by the number of variables. For the most natural formalization of
Optimal Linear Arrangement as an integer program the objective function is quadratic (and not necessarily convex), and therefore the above approach fails.
Motivated by the lack of progress on this problem, Fellows et al. <cit.> recently showed an FPT approximation scheme for Optimal Linear Arrangement parameterized by vertex cover. In partiular they gave an algorithm that given as input a graph $G$ with a vertex cover of size at most $k$ and a rational $\epsilon > 0$, produces in time $f(k,\epsilon)n^{O(1)}$ a layout $\sigma$ with cost at most a factor $(1+\epsilon)$ larger than the optimum. Fellows et al. <cit.> re-state the parameterized complexity of Optimal Linear Arrangement parameterized by vertex cover as an open problem. Finally, the problem was re-stated as an open problem in the recent monograph of Downey and Fellows <cit.>. Interestingly, Downey and Fellows motivate the study of this problem as follows.
“Our enthusiasm for this concrete problem is based on its connection to Integer Linear Programming. The problem above is easily reducible to a restricted form of Integer Quadratic Programming which may well be FPT”.
We give an FPT algorithm for Optimal Linear Arrangement parameterized by vertex cover, resolving the open problem of <cit.>. Our algorithm for Optimal Linear Arrangement works by directly applying the new algorithm for Integer Quadratic Programming to the most natural formulation of Optimal Linear Arrangement on graphs with a small vertex cover as an integer quadratic program, confirming the intuition of Downey and Fellows <cit.>.
Preliminaries and notation.
In Section <ref> lower case letters denote vectors and scalars, while upper case letters denote matrices. All vectors are column vectors. For an integer $p \geq 2$, the $\ell_p$ norm of an $n$-dimensional vector $v = [v_1, v_2, \ldots, v_n]$ is denoted by $|v|_p$ and is defined to be $|v|_p = (v_1^p + v_2^p + \ldots + v_n^p)^{1/p}$. The $\ell_1$ norm of $v$ is $|v|_1 = |v_1| + |v_2| + \ldots + |v_n|$, while the $\ell_\infty$ norm of $v$ is $|v|_\infty = \max(|v_1|, |v_2|, \ldots,|v_n|)$.
§ ALGORITHM FOR INTEGER QUADRATIC PROGRAMMING
We consider the following problem, called Integer Quadratic Programming. Input consists of an $n \times n$ integer symmetric matrix $Q$, an $m \times n$ integer matrix $A$ and $m$-dimensional integer vector $b$. The task is to find an optimal solution $x^\star$ to the following optimization problem.
\begin{eqnarray}\label{eqn:prob1}
\nonumber \mbox{Minimize } x^TQx \\
\mbox{ subject to:~~} Ax \leq b \\
\nonumber x \in \mathbb{Z}^n.
\end{eqnarray}
A vector $x \in \mathbb{Z}^n$ that satisfies the constraints $Ax \leq b$ is called a feasible solution to the IQP (<ref>). Given an input on the form (<ref>) there are three possible scenarios. A possible scenario is that there are no feasible solutions, in which case this is what an algorithm for Integer Quadratic Programming should report. Another possibility is that for every integer $\beta$ there exists some feasible solution $x$ such that $x^TQx \leq \beta$. In that case the algorithm should report that the IQP is unbounded. Finally, it could be that there exist feasible solutions, and that the minimum value of $x^TQx$ over the set of all feasible $x$ is well defined. This is the most interesting case, and in this case the algorithm should output a feasible $x$ such that $x^TQx$ is minimized.
Note that the requirement that $Q$ is symmetric can easily be avoided by replacing $Q$ by $Q + Q^T$. This operation does not change the set of optimal solutions, since it multiplies the objective function value of every solution by $2$. We will denote by $a_i^T$ the $i$'th row of the matrix $A$, and by $b_i$ the $i$'th entry of the vector $b$. Thus, $Ax \leq b$ means that $a_i^Tx \leq b_i$ for all $i$. The maximum absolute value of an entry of $A$ and $Q$ is denoted by $\alpha$. Using a pair of inequalities one can encode equality constraints. It is useful to rewrite the IQP (<ref>) to separate out the equality constraints explicitely, obtaining the following equivalent form.
\begin{eqnarray}\label{eqn:prob}
\nonumber \mbox{Minimize } x^TQx \\
\mbox{ subject to:~~} Ax \leq b \\
\nonumber Cx = d \\
\nonumber x \in \mathbb{Z}^n.
\end{eqnarray}
Here $C$ is an integer matrix and $d$ is an integer vector. If input is given on the form (<ref>), then we still use $\alpha$ to denote the maximum value of an entry of $A$ and $Q$. The IQP (<ref>) could be generalized by changing the objective function from $x^TQx$ to $x^TQx + q^Tx$ for some n-dimensional vector $q$ also given as input. This generalization can be incorporated in the original formulation (<ref>) at the cost of introducing a new variable $\hat{x}$, adding the constraint $\hat{x} = 1$ to the system $Cx = d$ and adding $[0,q]$ as the row corresponding to the new variable $\hat{x}$ in $Q$.
We will denote by $\Delta$ the maximum absolute value of the determinant of a square submatrix of $C$. We may assume without loss of generality that the rows of $C$ are linearly independent; otherwise we may in polynomial time either conclude that the IQP has no feasible solutions, or remove one of the equality constraints in the system $Cx = d$ without changing the set of feasible solutions. Thus $C$ has at most $n$ rows. If the IQP (<ref>) is obtained from (<ref>) by replacing constraints $a_i^Tx \leq b_i$, $-a_i^Tx \leq -b_i$ with $a_i^Tx = b_i$, the maximum entry of $C$ is also upper bounded by $\alpha$ and then we have $\Delta \leq n! \cdot \alpha^n$. The next simple observation shows that we can in polynomial time reduce the number of constraints to a function of $n$ and $\alpha$.
There is a polynomial time algorithm that given as input the matrix $A$ and vector $b$ outputs an $m' \times n$ submatrix $A'$
of $A$ and vector $b'$ such that $m' \leq (2\alpha+1)^n$ and for every $x \in \mathbb{Z}^n$, $Ax \leq b$ if and only if $A'x \leq b'$.
Suppose $A$ has more than $(2\alpha+1)^n$ rows. Then, by the pigeon hole principle the system $Ax \leq b$ has two rows $a_i^Tx \leq b_i$ and $a_j^Tx \leq b_j$ where $i \neq j$ but $a_i = a_j$. Without loss of generality $b_i \leq b_j$, and then any $x \in \mathbb{Z}^n$ such that $a_i^Tx \leq b_i$ satisfies $a_j^Tx \leq b_j$. Thus we can safely remove the inequality $a_j^Tx \leq b_j$ from the system, and the lemma follows.
In the following we will assume that the input is on the form (<ref>). We will give an algorithm that runs in time $f(n,m,\alpha,\Delta)\cdot L^{O(1)}$, where $L$ is the length of the bit-representation of the input instance. Since we can reduce the input using Lemma <ref> first and $\Delta$ is upper bounded in terms of $n$ and $\alpha$ this will yield an FPT algorithm for Integer Quadratic Programming parameterized by $n$ and $\alpha$.
Let $r$ be the dimension of the nullspace of $C$. Using Cramer's rule (see <cit.>) we can in polynomial time compute a basis $y_1, \ldots y_r$ for the nullspace of $C$, such that each $y_i$ is an integer vector and $|y_i|_\infty \leq \Delta^2$. We let $Y$ be the $n \times r$ matrix whose columns are the vectors $y_1, \ldots y_r$. We will abuse notation and write $y_i \in Y$ to denote that we chose the $i$'th column $y_i$ of $Y$. We will say that a feasible solution $x$ is deep if $x + y_i$ and $x - y_i$ are feasible solutions for all $y_i \in Y$. A feasible solution that is not deep is called shallow.
Let $x$ be a shallow feasible solution to (<ref>). Then there exists a row $a_j^T$ of $A$ and integer $b_j'$ such that $a_j^Tx = b_j'$ and $b_j' \in \{b_j - \alpha \cdot n \cdot \Delta^2, \ldots, b_j\}$. Further, $a_j^T$ is linearly independent from the rows of $C$.
We prove the statement for $y_i \in Y$ such that $x + y_i$ is not a feasible solution to (<ref>). Then there exists a row $a_j^T$ of $A$ such that $a_j^T(x + y_i) > b_j$, and thus
$$b_j - a_j^Ty_i < a_j^Tx \leq b_j.$$
Thus $a_j^Tx = b_j'$ for $b_j' \in \{b_j - \alpha \cdot n \cdot |y_i|_\infty, b_j\}$. Since $|y_i|_\infty \leq \Delta^2$ we have that $b_j' \in \{b_j - \alpha \cdot n \cdot \Delta^2, b_j\}$.
We now show that $a_j^T$ is linearly independent from the rows of $C$. Suppose not, then there exists a coefficient vector $\lambda$ such that $\lambda^TC = a_j^T$. But then
$$a_j^T(x+y_i) = \lambda^TC(x + y_i) = \lambda^TCx + \lambda^TCy_i = \lambda^TCx = a_j^Tx \leq b_j,$$
which contradicts that $a_j^T(x + y_i) > b_j$. We conclude that $a_j^T$ is linearly independent from the rows of $C$. The proof for the case when $x - y_i$ is not a feasible solution to (<ref>) is symmetric.
Lemma <ref> suggests the following branching strategy: either all optimal solutions are deep or Lemma <ref> applies to some shallow optimal solution $x^\star$. In the latter case the algorithm can branch on the choice of row $a_j^T$ and $b_j'$ and add the equation $a_j^Tx = b_j'$ to the set of constraints. This decreases the dimension of the nullspace of $C$ by $1$. We are left with handling the case when all optimal solutions are deep.
For any pair of vectors $x$, $y \in \mathbb{R}^n$ and symmetric matrix $Q \in \mathbb{R}^{n \times n}$, the following are equivalent.
* $(x+y)^TQ(x+y) \geq x^TQx$ and $(x-y)^TQ(x-y) \geq x^TQx$,
* $- y^TQy \leq 2x^TQy \leq y^TQy.$
Expanding the inequalities of $(\ref{itm:noLamb})$ yields
$$x^TQx + 2x^TQy + y^TQy \geq x^TQx,$$
$$x^TQx - 2x^TQy + y^TQy \geq x^TQx.$$
Cancelling the $x^TQx$ terms and re-organizing yields $2x^TQy \geq - y^TQy$ and $2x^TQy \leq y^TQy$. Since the left hand side of the inequalities is the same we can combine the two inequalities in a single inequality,
$$- y^TQy \leq 2x^TQy \leq y^TQy,$$
completing the proof. Note that all the manipulations we did on the inequalities are reversible, thus the above argument does indeed prove equivalence and not only the forward direction $(\ref{itm:noLamb}) \rightarrow (\ref{itm:absVal})$.
Lemma <ref> suggests a branching strategy to find a deep solution $x^\star$: pick a vector $y_i$ in $Y$ such that $y_i^TQ$ is linearly independent of the rows of $C$, guess the value $z$ of $2(x^\star)^TQy_i$ and add the linear equation $2(x^\star)^TQy_i = z$ to the set of constraints. In each branch the dimension of the nullspace of $C$ decreases by $1$. Thus we are left with the case that all solutions are deep and there is no $y_i$ in $Y$ such that $y_i^TQ$ is linearly independent of the rows of $C$. We now handle this case.
For any deep optimal solution $x^\star$ of the IQP (<ref>) and $y_i \in Y$ such that $y_i^TQ$ is linearly dependent of the rows of $C$, the vectors $x^\star + y_i$ and $x^\star - y_i$ are also optimal solutions.
We prove the statement for $x^\star + y_i$. Since $x^\star$ is deep it follows that $x^\star + y_i$ is feasible, and it remains to lower bound the objective function value of $x^\star + y_i$. Since $y_i^TQ$ is linearly dependent of the rows of $C$ there exists a coefficient vector $\lambda^T$ such that $y_i^TQ = \lambda^TC$. Therefore,
\begin{eqnarray*}
2(x^\star + y_i)^TQy_i & = & 2y_i^TQ(x^\star+y_i) = 2\lambda^TC(x^\star+y_i) \\
& = & 2\lambda^TCx^\star+2\lambda^TCy_i = 2\lambda^TCx^\star = 2y_i^TQx^\star = 2(x^\star)^TQy_i
\end{eqnarray*}
Since $x^\star$ is deep, it follows that both $x^\star + y_i$ and $x^\star - y_i$ are feasible and therefore cannot have a higher value of the objective function than $x^\star$. Hence, by Lemma <ref> we have that $- y_i^TQy_i \leq 2(x^\star)^TQy_i \leq y_i^TQy_i$. Since $2(x^\star + y_i)^TQy_i = 2(x^\star)^TQy_i$, we have that
$$- y_i^TQy_i \leq 2(x^\star + y_i)^TQy_i \leq y_i^TQy_i.$$
Hence, Lemma <ref> applied to $(x^\star + y_i)$ implies that
$$(x^\star)^TQ x^\star = (x^\star+y_i-y_i)^TQ (x^\star+y_i-y_i) \geq (x^\star+y_i)^TQ(x^\star+y_i).$$
This means that the objective function value of $x^\star+y_i$ is at most that of $x^\star$, hence $x^\star+y_i$ is optimal. The proof for $x^\star - y_i$ is symmetric.
Lemma <ref> immediately implies the following corollary.
Suppose the IQP (<ref>) has an optimal solution, all optimal solutions to (<ref>) are deep, and for every $y_i \in Y$, $y_i^TQ$ is linearly dependent of the rows of $C$. Then, for every optimal solution $x^\star$ and integer vector $\lambda \in \mathbb{Z}^r$, $x^\star + Y\lambda$ is also an optimal solution of (<ref>).
Since $x^\star$ is optimal and deep, and for every $y_i \in Y$, $y_i^TQ$ is linearly dependent of the rows of $C$, it follows from Lemma <ref> that for every $y_i \in Y$, $x^\star + y_i$ and $x^\star - y_i$ are also optimal solutions of (<ref>). Since all optimal solutions are deep, $x^\star + y_i$ and $x^\star - y_i$ are deep. The statement of the corollary now follows by induction on $|\lambda|_1$.
We are now ready to state the main structural lemma underlying the algorithm for Integer Quadratic Programming.
For any Integer Quadratic Program of the form (<ref>) that has an optimal solution and any $x_0$ such that $Cx_0 = d$, there exists an optimal solution $x^\star$ such that at least one of the following three cases holds.
* There exists a row $a_j^T$ of $A$ and integer $b_j' \in \{b_j - \alpha \cdot n \cdot \Delta^2, \ldots, b_j\}$ such that $a_j^Tx^\star = b_j'$, and $a_j^T$ is linearly independent from the rows of $C$.
* There exists a $y_i \in Y$ such that $y_i^TQ$ is linearly independent of the rows of $C$ and
$2y_i^TQx^\star = z$ for $z \in \{-n^2\Delta^4\alpha, \ldots, n^2\Delta^4\alpha\}$.
* $|x^\star-x_0|_1 \leq \Delta^2 \cdot n$.
Suppose the integer quadratic program (<ref>) has a shallow optimal solution $x^\star$. Then, by Lemma <ref> case <ref> applies. In the remainder of the proof we assume that all optimal solutions are deep. Suppose now that there is a $y_i \in Y$ such that $y_i^TQ$ is linearly independent of the rows of $C$. Then, since $x^\star$ is a deep optimal solution, both $x^\star + y_i$ and $x^\star - y_i$ are feasible solutions, so
$(x^\star+y_i)^TQ(x^\star+y_i) \geq (x^\star)^TQx^\star$ and $(x^\star-y_i)^TQ(x^\star-y_i) \geq (x^\star)^TQx^\star$. Thus, Lemma <ref> implies that $2y_i^TQx^\star = z$ for $z \in \{-y_i^TQy_i, \ldots, y_i^TQy_i\}$.
Furthermore, $y_i^TQy_i$ is the sum of $n^2$ terms where each term a product of an element of $y_i$ (and thus at most $\Delta^2$), another element of $y_i$, and an element of $Q$. Thus $z \in \{-n^2\Delta^4\alpha, \ldots, n^2\Delta^4\alpha\}$ and therefore case <ref> applies.
Finally, suppose that all $y_i \in Y$ are linearly dependent of the rows of $C$. Let $\hat{x}$ be an arbitrarily chosen optimal solution to (<ref>). Since $C(x_0-\hat{x}) = 0$ and $Y$ forms a basis for the nullspace of $C$ there is a coefficient vector $\lambda \in \mathbb{R}^r$ such that $x_0 = \hat{x} + Y\lambda$. Define $\tilde{\lambda}$ from $\lambda$ by rounding each entry down to the nearest integer. In other words, for every $i$ we set $\tilde{\lambda}_i = \lfloor \lambda_i \rfloor$. Set $x^\star = \hat{x} + Y\tilde{\lambda}$. By Corollary <ref> we have that $x^\star$ is an optimal solution to (<ref>). But $|x^\star-x_0|_1 = |Y(\tilde{\lambda}-\lambda)|_1 \leq (\max_i |y_i|_1) \cdot n \leq \Delta(C)^2 \cdot n$, concluding the proof.
There exists an algorithm that given an instance of Integer Quadratic Programming, runs in time $f(n,\alpha)L^{O(1)}$, and outputs a vector $x \in \mathbb{Z}^n$. If the input IQP has a feasible solution then $x$ is feasible, and if the input IQP is not unbounded, then $x$ is an optimal solution.
We assume that input is given on the form (<ref>). The algorithm starts by reducing the input system according to Lemma <ref>. After this preliminary step the number of constraints $m$ in the IQP is upper bounded by $(2\alpha + 1)^n$. We give a recursive algorithm, based on Lemma <ref>. The algorithm begins by computing in polynomial time a basis $Y = {y_1, \ldots, y_r}$ for the nullspace of $C$, as described in the beginning of Section <ref>. In particular $Y$ is a matrix of integers, and for every $i$, $|y_i|_\infty \leq \Delta^2$.
If the dimension of the nullspace of $C$ is $0$ the algorithm solves the system $Cx = d$ of linear equations in polynomial time. Let $x^\star$ be the (unique) solution to this linear system. If $x^\star$ is not an integral vector, or $Ax^\star \leq b$ does not hold the algorithm reports that the input IQP has no feasible solution. Otherwise it returns $x^\star$ as the optimum.
If $C$ is not full-dimensional, i.e the dimension of the nullspace of $C$ is at least $1$, the algorithm proceeds as follows. For each row $a_j^T$ of $A$ and integer $b_j' \in \{b_j - \alpha \cdot n \cdot \Delta^2, b_j\}$ such that $a_j^T$ is linearly independent from the rows of $C$, the algorithm calls itself recursively on the same instance, but with the equation $a_j^Tx = b_j'$ added to the system $Cx = d$. Furthermore, for each $y_i \in Y$ such that $y_i^TQ$ is linearly independent of the rows of $C$ and every integer $z \in \{-n^2\Delta^4\alpha, \ldots, n^2\Delta^4\alpha\}$ the algorithm calls itself recursively on the same instance, but with the equation $2y_i^TQx = z$ added to the system $Cx = d$. Finally the algorithm computes an arbitrary (not necessarily integral) solution $x_0$ of the system $Cx = d$, and checks all (integral) vectors within $\ell_1$ distance at most $\Delta^2 \cdot n$ from $x_0$. The algorithm returns the feasible solution with the smallest objective function value among the ones found in any of the recursive calls, and the search around $x_0$.
In the recursive calls, when we add a linear equation to the system $Cx = d$ we extend the matrix $C$ and vector $d$ to incorporate this equation. The algorithm terminates, as in each recursive call the dimension of the nullspace of $C$ is decreased by $1$. Further, any feasible solution found in any of the recursive calls is feasible for the original system. Thus, if the algorithm reports a solution then it is feasible. To see that the algorithm reports an optimal solution, consider an optimal solution $x^\star$ satisfying the conditions of Lemma <ref> applied to the quadratic integer program (<ref>) and vector $x_0$. Either $x^\star$ will be found in the search around $x_0$, or $x^\star$ satisfies the linear constraint added in at least one of the recursive calls. In the latter case $x^\star$ is an optimal solution to the integer quadratic program of the recursive call, and in this call the algorithm will find a solution with the same objective function value. This concludes the proof of correctness.
We now analyze the running time of the algorithm. First, consider the time it takes to search all integral vectors within $\ell_1$ distance at most $\Delta^2 \cdot n$ from $x_0$. It is easy to see that there are at most $3^{\Delta^2 \cdot n + n}$ such vectors. For the running time analysis only we will treat this search as at most $3^{\Delta^2 \cdot n + n}$ recursive calls to instances where the dimension of the nullspace of $C$ is $0$. Then the running time in each recursive call is polynomial, and it is sufficient to upper bound the number of leaves in the recursion tree of the algorithm.
We bound the number of leaves of the recursion tree as a function of $n$ – the number of variables, $m$ – the number of rows in $A$, $\alpha$ – the maximum value of an entry in $A$ or $Q$, $\Delta$ – the maximum absolute value of the determinant of a square submatrix of $C$, and $r$ – the dimension of the nullspace of $C$. Notice that the algorithm never changes $Q$ or $A$, and that the number of variables remains the same throughout the execution of the algorithm. Thus $n$, $m$ and $\alpha$ do not change throughout the execution. For a fixed value of $n$, $m$ and $\alpha$, we let $T(r, \Delta)$ be the maximum number of leaves in the recursion tree of the algorithm when called on an instance with the given value of $n$, $m$, $\alpha$, $r$ and $\Delta$.
In each recursive call the algorithm adds a new row to the matrix $C$. Let $C'$ be the new matrix after the addition of this row, $r'$ be the dimension of the nullspace of $C$ and $\Delta'$ be the maximum value of a determinant of a square submatrix of $C'$. Since the new added row is linearly independent of the rows of $C$ it follows that $r' = r - 1$ in each of the recursive calls arising from case <ref> and case <ref> of Lemma <ref>. The remaining recursive calls are to leaves of the recursion tree.
When the algorithm explores case <ref>, it guesses a row $a_j$, for which there are $m$ possibilities, and a value for $b_j'$, for which there are $\alpha \cdot n \cdot \Delta^2$ possibilities. This generates $m \cdot \alpha \cdot n \cdot \Delta^2$ recursive calls. In each of these recursive calls $a_j$ is the new row of $C'$, and so, by the cofactor expansion of the determinant <cit.>, $\Delta' \leq n\alpha\Delta$.
When the algorithm explores case <ref>, it guesses a vector $y_i \in Y$, and there are at most $n$ possibilities for $y_i$. For each of these possibilities the algorithm guesses a value for $z$, for which there are $2n^2\Delta^4\alpha$ possible choices. Thus this generates $2n^3\Delta^4\alpha$ recursive calls. In each of the recursive calls the algorithm makes a new matrix $C'$ from $C$ by adding the new row $2y_i^TQ$.
We have that $|y_i|_\infty \leq \Delta^2$. Thus, $|2y_i^TQ|_\infty \leq n \cdot \Delta^2 \cdot \alpha$, and the cofactor expansion of the determinant <cit.> applied to the new row yields $\Delta' \leq n^2 \Delta^3 \cdot \alpha$, where $\Delta'$ is the maximum value of a determinant of a square submatrix of $C'$. It follows that the number of leaves of the recursion tree is gouverned by the following recurrence.
\begin{align*}
T(r, \Delta) \leq m \cdot \alpha \cdot n \cdot \Delta^2 \cdot T(r-1, n\alpha\Delta) + 2n^3 \cdot \Delta^4 \cdot \alpha \cdot T(r-1, n^2 \Delta^3 \alpha) + 3^{(\Delta^2 + 1) \cdot n} \\
\leq \alpha \cdot m \cdot n^3 \cdot \Delta^4 \cdot T(r-1, n^2 \Delta^3 \alpha) + 3^{(\Delta^2 + 1) \cdot n}
\end{align*}
The above recurrence is clearly upper bounded by a function of $n$, $m$, $\Delta$ and $\alpha$. Since $m$ is upper bounded by $(2\alpha+1)^n$ from Lemma <ref>, the theorem follows.
§.§ Detecting Unbounded IQPs
Theorem <ref> allows us to solve bounded IQPs, and is sufficient for the application to Optimal Linear Arrangement. However, it is somewhat unsatisfactory that the algorithm of Theorem <ref> is unable to detect whether the input IQP is bounded or not. Next we resolve this issue. Towards this, we inspect the algorithm of Theorem <ref>.
For purely notational reasons we will consider the algorithm of Theorem <ref> when run on an instance on the form (<ref>). The first step of the algorithm is to put the the IQP on the form (<ref>), and then proceed as described in the proof of Theorem <ref>.
The algorithm is recursive, and the only variables that change from one recursive call to the next are the matrix $C$ and the vector $d$. Furthermore, when making a recursive call, the new matrix $C'$ is computed from $C$ by either adding the row $a_j^T$ to $C$ or adding the row $2y_i^TQ$ to $C$. The vector $y_i^T$ is a vector from the basis $Y$ for the nullspace of $C$. In other words $C'$ depends only on $Q$, $A$, $C$ and $i$, and is independent of $b$ and $d$. Furthermore, the recursion stops when $C$ has full column rank. Thus, the family ${\cal C}$ of matrices $C$ that the algorithm of Theorem <ref> ever generates depends only on the input matrices $Q$ and $A$ (and not on the input vector $b$).
At this point we remark that the only reason we assumed input was on the form (<ref>) rather than (<ref>) was to avoid the confusing sentence “Thus, the family ${\cal C}$ of matrices $C$ that the algorithm of Theorem <ref> ever generates depends only on the input matrices $Q$ and $A$, and $C$,” where the meaning of the matrix $C$ is overloaded.
Let $\hat{\Delta}$ be the maximum absolute value of the determinant of a square submatrix of a matrix $C \in {\cal C}$ ever generated by the algorithm. In other words, $\hat{\Delta}$ is the maximum value of the variable $\Delta$ throughout the execution of the algorithm. Because $\Delta$ only depends on $C$, it follows that $\hat{\Delta}$ only depends on ${\cal C}$, and therefore $\hat{\Delta}$ is a function of the input matrices $A$ and $Q$.
We now discuss all the different vectors $d$ ever generated by the algorithm. In each recursive call, the algorithm adds a new entry to the vector $d$, this entry is either from the set $\{b_j - \alpha \cdot n \cdot \hat{\Delta}^2, b_j\}$ or from the set $\{-n^2\hat{\Delta}^4\alpha, \ldots, n^2 \hat{\Delta}^4\alpha\}$. Thus, any vector $d$ ever generated by the algorithm is at $\ell_\infty$ distance at most $n^2 \hat{\Delta}^4\alpha$ from some vector whose entries are either $0$ or equal to $b_j$ for some $j \leq m$. Given the vector $b$ and the integer $n$, we define the vector set ${\cal D}(b,n)$ be the set of all integer vectors in at most $n$ dimensions with entries either $0$ or equal to $b_j$ for some $j \leq m$. Observe that $|{\cal D}(b,n)| \leq (m+1)^{n}$. We have that every vector $d$ ever generated by the algorithm is at $\ell_\infty$ distance at most $n^2 \hat{\Delta}^4\alpha$ from some vector in ${\cal D}(b,n)$.
The algorithm of Theorem <ref> generates potential solutions $x$ to the input IQP by finding a (not necessarily integral) solution $x_0$ to the linear system $Cx = d$, and then lists integral vectors within $\ell_1$-distance at most $\hat{\Delta}^2 \cdot n$ from $x_0$. From Theorem <ref> it follows that if the input IQP is feasible and bounded, then one of the listed vectors $x$ is in fact an optimum solution to the IQP. The above discussion proves the following lemma.
Given an $n \times n$ integer matrix $Q$, and an $m \times n$ integer matrix $A$, let ${\cal C}$ be the set of matrices $C$ generated by the algorithm of Theorem <ref> when run on the IQP
\begin{eqnarray*}
\mbox{Minimize } x^TQx \\
\mbox{ subject to:~~} Ax \leq 0 \\
x \in \mathbb{Z}^n,
\end{eqnarray*}
and let $\hat{\Delta}$ be the maximum absolute value of the determinant of a square submatrix of a matrix $C \in {\cal C}$. Then, for any $m$-dimensional integer vector $b$ such that the IQP (<ref>) is feasible and bounded, there exists a $C \in {\cal C}$, a vector $d_0 \in {\cal D}(b,n)$, and an integer vector $d$ at $\ell_\infty$ distance at most $n^2 \hat{\Delta}^4\alpha$ from $d_0$ such that the following is satisfied. For any $x_0$ such that $Cx_0 = d$, there exists an integer vector $x^*$ at $\ell_1$ distance at most $\hat{\Delta}^2 \cdot n$ from $x_0$ such that $x^*$ is an optimal solution to the IQP (<ref>).
The algorithm in Theorem <ref> only adds a row to the matrix $C$ if this row is linearly independent from the rows of $C$. Hence all matrices in ${\cal C}$ have full row rank, and therefore they have right inverses. Specifically, for each $C \in {\cal C}$, we define $C_{right}^{-1} = C^T(CC^T)^{-1}$. It follows that $C C_{right}^{-1} = I$ and that therefore, $x_0 = C_{right}^{-1}d$ is a solution to the system $Cx = d$ for any vector $d$. Note that $C_{right}^{-1}$ is not necessarly an integer matrix, however Cramer's rule <cit.> shows that $C_{right}^{-1}$ is a matrix with rational entries with common denominator $\det(CC^T)$. This leads to the following lemma.
There exists an algorithm that given an $n \times n$ integer matrix $Q$, and an $m \times n$ integer matrix $A$ outputs a set ${\cal C}^{-1}$ of rational matrices, and a set ${\cal V}$ of rational vectors with the following property. For any $m$-dimensional integer vector $b$ such that the IQP (<ref>) defined by $Q$, $A$ and $b$ is feasible and bounded, there exists a matrix $C^{-1}_{right} \in {\cal C}$ a vector $v \in {\cal V}$ and a vector $d_0 \in {\cal D}(b,n)$, such that
$x^* = C^{-1}_{right}d_0 + v$ is an optimal solution to the IQP (<ref>).
The algorithm starts by applying the algorithm of Lemma <ref> to obtain a set ${\cal C}$ of matrices and the integer $\hat{\Delta}$. The algorithm then computes the set ${\cal C}^{-1}$ of matrices, defined as ${\cal C}^{-1} = \{C_{right}^{-1} ~:~ C \in {\cal C}\}$.
Next the algorithm constructs the set ${\cal V}$ of vectors as follows. For every matrix $C_{right}^{-1} \in {\cal C}^{-1}$, we have that $C_{right}^{-1} = C^T(CC^T)^{-1}$ for some $C \in {\cal C}$. For every integer vector $v_0$ with $|v_0|_\infty \leq n^2 \hat{\Delta}^4\alpha$, and every rational vector $v_1$ with $|v_1|_1 \leq \hat{\Delta}^2 \cdot n$ and denominator $\det(CC^T)$ in every entry, the algorithm adds $C_{right}^{-1}v_0 + v_1$ to ${\cal V}$. It remains to prove that ${\cal C}^{-1}$ and ${\cal V}$ satisfy the statement of the lemma.
Let $b$ be an m-dimensional integer vector such that the IQP (<ref>) defined by $Q$, $A$ and $b$ is feasible and bounded. By Lemma <ref> we have that there exists $C \in {\cal C}$, a vector $d_0 \in {\cal D}(b,n)$, and a vector $d$ at $\ell_\infty$ distance at most $n^2 \hat{\Delta}^4\alpha$ from $d_0$ such that the following is satisfied. For any $x_0$ such that $Cx_0 = d$, there exists an integer vector $x^*$ at $\ell_1$ distance at most $\hat{\Delta}^2 \cdot n$ from $x_0$ such that $x^*$ is an optimal solution to the IQP (<ref>).
Let $C \in {\cal C}$, $d_0 \in {\cal D}(b,n)$, and $d$ be as guaranteed by Lemma <ref>, and set $v_0 = d - d_0$. We have that $|v_0|_\infty \leq n^2 \hat{\Delta}^4\alpha$. Let $C^{-1}_{right}$ be the right inverse of $C$ in ${\cal C}^{-1}$, and let $x_0 = C^{-1}_{right}d = C^{-1}_{right}d_0 + C^{-1}_{right}v_0$. We have that ${\cal C}x_0 = d$.
Thus, there exists an integer vector $x^*$ at $\ell_1$ distance at most $\hat{\Delta}^2 \cdot n$ from $x_0$ such that $x^*$ is an optimal solution to the IQP (<ref>). Let $v_1 = x^* - x_0$, we have that $|v_1|_1 \leq \hat{\Delta}^2 \cdot n$. Further, $x^*$ is an integer vector, while $x_0 = C^{-1}_{right}d$ is a rational vector whose entries all have denominator $\det(CC^T)$. It follows that all entries of $v_1$ have denominator $\det(CC^T)$. Thus $v = C^{-1}_{right}v_0 + v_1 \in {\cal V}$ and $C^{-1}_{right}d_0 + v$ is an optimal solution to the IQP (<ref>), completing the proof.
Armed with Lemma <ref> we are ready to prove the main result of this section.
There exists an algorithm that given an instance of Integer Quadratic Programming, runs in time $f(n,\alpha)L^{O(1)}$, and determines whether the instance is infeasible, feasible and unbounded, or feasible and bounded. If the instance is feasible and bounded the algorithm outputs an optimal solution.
The algorithm of Theorem <ref> is sufficient to determine whether the input instance is feasible, and to find an optimal solution if the instance is feasible and bounded. Thus, to complete the proof it is sufficient to give an algorithm that determines whether the input IQP is unbounded. We will assume that the input IQP is given on the form (<ref>).
Suppose now that the input IQP is unbounded. For a positive integer $\lambda$, consider adding the linear constraints
$Ix \leq \lambda{\bf 1}$, and $-Ix \leq \lambda{\bf 1}$ to the IQP. Here 1 is the $n$-dimensional all-ones vector. In other words, we consider the IQP where the goal is to minimize $x^TQx$, subject to $A'x \leq b'$ where $A'$ is obtained from $A$ by adding $2n$ new rows containing $I$ and $-I$, and $b'$ is obtained from $b$ by adding $2n$ new entries with value $\lambda$. There exists a $\lambda_0$ such that for every $\lambda \geq \lambda_0$ this IQP is feasible. Further, for every $\lambda$ the resulting IQP is bounded. Note that the matrix $A'$ does not depend on $\lambda$.
We now apply Lemma <ref> on $Q$ and $A'$, and obtain a set ${\cal C}^{-1}$ of rational matrices, and a set ${\cal V}$ of rational vectors. We have that for every $\lambda \geq \lambda_0$, there exists a matrix $C^{-1}_{right} \in {\cal C}$ a $v \in {\cal V}$ and a $d_0 \in {\cal D}(b',n)$, such that $x^* = C^{-1}_{right}d_0 + v$ is an optimal solution to the IQP defined by $Q$, $A'$ and $b'$. Furthermore, the entries of $b'$ are either entries of $b$, $0$ or equal to $\lambda$. Hence $d_0 = d_b + \lambda d_1$ for $d_b \in {\cal D}(b,n)$ and $d_1 \in {\cal D}({\bf 1},n)$.
We can conclude that there exists a matrix $C^{-1}_{right} \in {\cal C}$ a vector $v \in {\cal V}$, a vector $d_b \in {\cal D}(b,n)$ and a vector $d_1 \in {\cal D}({\bf 1},n)$, such that
$$C^{-1}_{right}(d_b + \lambda d_1) + v = \lambda \cdot (C^{-1}_{right} d_1) + (C^{-1}_{right} d_b + v)$$
is an optimal solution to the IQP defined by $Q$, $A'$ and $b'$.
Thus, the input IQP is unbounded if and only if there exists a choice for $C^{-1}_{right} \in {\cal C}$, $v \in {\cal V}$, $d_b \in {\cal D}(b,n)$ and $d_1 \in {\cal D}({\bf 1},n)$, such that following univariate quadratic program with integer variable $\lambda$ is unbounded.
\begin{eqnarray*}
\mbox{Minimize } x^TQx \\
\mbox{ subject to:~~} Ax \leq b \\
x =\lambda \cdot (C^{-1}_{right} d_1) + (C^{-1}_{right} d_b + v) \\
\lambda \in \mathbb{Z}.
\end{eqnarray*}
Hence, to determine whether the input IQP is unbounded, it is sufficient to iterate over all choices of $C^{-1}_{right} \in {\cal C}$, $v \in {\cal V}$, $d_b \in {\cal D}(b,n)$ and $d_1 \in {\cal D}({\bf 1},n)$, and determine whether the resulting univariate quadratic program is unbounded. Since the number of such choices is upper bounded by a function of $Q$ and $A$, and univariate (both integer and rational) quadratic programming is trivially decidable, the theorem follows.
§ OPTIMAL LINEAR ARRANGEMENT PARAMETERIZED BY VERTEX COVER
We assume that a vertex cover $C$ of $G$ of size at most $k$ is given as input. The remaining set of vertices $I = V(G) - C$ forms an independent set. Furthermore, $I$ can be partitioned into at most $2^k$ sets as follows: for each subset $S$ of $C$ we define $I_S = \{v \in I ~:~ N(v) = S\}$. For every vertex $v \in I$ we will refer to $N(v)$ as the type of $v$, clearly there are at most $2^k$ different types.
Let $C = \{c_1, c_2, \ldots, c_k\}$. By trying all $k!$ permutations of $C$ we may assume that the optimal solution $\sigma$ satisfies $\sigma(c_i) < \sigma(c_{i+1})$ for every $1 \leq i \leq k-1$. For every $i$ between $1$ and $k-1$ we define the $i$'th gap of $\sigma$ to be the set $B_i$ of vertices appearing between $c_i$ and $c_{i+1}$ according to $\sigma$. The $0$'th gap $B_0$ is the set of all vertices appearing before $c_1$, and the $k$'th gap $B_k$ is the set of vertices appearing after $c_k$. We will also refer to $B_i$ as “gap $i$” or “gap number $i$”. For every gap $B_i$ and type $S \subseteq C$ of vertices we denote by $I_S^i$ the set $B_i \cap I_S$ of vertices of type $S$ appearing in gap $i$.
We say that an ordering $\sigma$ is homogenous if, for every gap $B_i$ and every type $S \subseteq C$ the vertices of $I_S^i$ appear consecutively in $\sigma$. Informally this means that inside the same gap the vertices from different sets $I_{S}$ and $I_{S'}$ “don't mix”. Fellows et al. <cit.> show that there always exists an optimal solution that is homegenous.
There exists a homogenous optimal linear arrangement of $G$.
For every vertex $v$ we define the force of $v$ with respect to $\sigma$ to be
$$\delta(v) = |\{u \in N(v) ~:~ \sigma(u) > \sigma(v)\}| - |\{u \in N(v) ~:~ \sigma(u) < \sigma(v)\}|.$$
Notice that two vertices of the same type in the same gap have the same force. Fellows et al. <cit.> in the proof of Lemma <ref> show that there exists an optimal solution that is homogenous, and where inside every gap, the vertices are ordered from left to right in non-decreasing order by their force. We will call such an ordering solution super-homogenous. As already noted, the existence of a super-homogenous optimal linear arrangement $\sigma$ follows from the proof of Lemma <ref> by Fellows et al. <cit.>.
There exists a super-homogenous optimal linear arrangement of $G$.
Notice that a super-homogenous linear arrangement $\sigma$ is completely defined (up to swapping positions of vertices of the same type) by specifying for each $i$ and each type $S$ the size $|I_S^i|$. For each gap $i$ and each type $S$ we introduce a variable $x_S^i \in \mathbb{Z}$ representing $|I_S^i|$. Clearly the variables $x_S^i$ need to satisfy
\begin{align}\label{constr:nonneg}\forall i \leq k, \forall S \subseteq C ~~~~ x_S^i \geq 0\end{align}
\begin{align}\label{constr:sumvert}\forall S \subseteq C ~~~~ \sum_{i=0}^k x_S^i = |I_S|.\end{align}
On the other hand, every assignment to the variables satisfying these (linear) constraints corresponds to a super-homogenous linear arrangement $\sigma$ of $G$ with $|I_S^i| = x_S^i$ for every type $S$ and gap $i$.
We now analyze the cost of $\sigma$ as a function of the variables. The goal is to show that $val(\sigma, G)$ is a quadratic function of the variables with coefficients that are bounded from above by a function of $k$. The coefficients of this quadratic function are not integral, but half-integral, namely integer multiples of $\frac{1}{2}$. The analysis below is somewhat tedious, but quite straightforward. For the analysis it is helpful to re-write $val(\sigma,G)$. For a fixed ordering $\sigma$ of the vertices we say that an edge $uv$ flies over the vertex $w$ if
$$\min(\sigma(u), \sigma(v)) < \sigma(w) < \max(\sigma(u), \sigma(v)).$$
We define the “fly over” relation $\sim$ for edges and vertices, i.e $uv \sim w$ means that $uv$ flies over $w$. Since an edge $uv$ with $\sigma(u) < \sigma(v)$ flies over the $\sigma(v)-\sigma(u)-1$ vertices appearing between $\sigma(u)$ and $\sigma(v)$ it follows that
$$val(\sigma,G) = |E(G)| + \sum_{uv \in E(G)} \sum_{\substack{w \in V(G) \\ uv \sim w}} 1.$$
We partition the set of edges of $G$ into several subsets as follows. The set $E_C$ is the set of all edges with both endpoints in $C$. For every gap $i$ with $i \in \{0,\ldots,k\}$, every $j \in \{1,\ldots,k\}$ and every $S \subseteq C$ we denote by $E_{i,j}^S$ the set of edges whose one endpoint is in $I_S^i$ and the other is $c_j$. Notice that $|E_{i,j}^S|$ is either equal to $x_S^i$ or to $0$ depending on whether vertices of type $S$ are adjacent to $c_j$ or not. We have that
\begin{align}\label{eqn:objfunc}
val(\sigma,G) = |E(G)| + \sum_{c_ic_j \in E_C} \sum_{\substack{w \in V(G) \\ c_ic_j \sim w}} 1 + \sum_{i,j,S} \sum_{uc_j \in E_{i,j}^S} \sum_{\substack{w \in V(G) \\ uc_j \sim w}} 1.
\end{align}
Further, for each edge $c_ic_j \in E_C$ (with $i < j$) we have that
$$\sum_{\substack{w \in V(G) \\ c_ic_j \sim w}} 1 = j-i-1 + \sum_{p=i}^{j-1} \sum_{S \subseteq C} x_S^p.$$
In other words, the first double sum of Equation <ref> is a linear function of the variables. Since $|E_C| \leq {k \choose 2}$ the coefficients of this linear function are integers upper bounded by ${k \choose 2}$.
We now turn to analyzing the second part of Equation <ref>. We split the triple sum in three parts as follows.
\begin{align}
\nonumber \sum_{i,j,S} \sum_{uc_j \in E_{i,j}^S} \sum_{\substack{w \in V(G) \\ uc_j \sim w}} 1 \\
= \sum_{i,j,S} \left( \sum_{uc_j \in E_{i,j}^S} \sum_{\substack{w \in C \\ uc_j \sim w}} 1 + \sum_{uc_j \in E_{i,j}^S} \sum_{\substack{w \in I_S^i \\ uc_j \sim w}} 1 + \sum_{uc_j \in E_{i,j}^S} \sum_{\substack{w \in I - I_S^i \\ uc_j \sim w}} 1\right)\label{eqn:tripleSumExpand}
\end{align}
For any fixed $i$, $j$ and $S$, any edge $uc_j \in E_{i,j}^S$ the number of vertices $w \in C$ such that $uc_j \sim w$ depends solely on $i$ and $j$. It follows that
$$ \sum_{uc_j \in E_{i,j}^S} \sum_{\substack{w \in C \\ uc_j \sim w}} 1 = f(i,j) \cdot x_S^i$$
for some function $f$, which is upper bounded by $k$ (since $|C| = k$).
Consider a pair of vertices $u$, $w$ in $I_S^i$ and a vertex $c_j \in C$ such that vertices of $u$'s and $w$'s type are adjacent to $c_j$. Either the edge $uc_j$ flies over $w$ or the edge $wc_j$ flies over $u$, but both of these events never happen simulataneously. Therefore,
$$\sum_{uc_j \in E_{i,j}^S} \sum_{\substack{w \in I_S^i \\ uc_j \sim w}} 1 = {x_S^i \choose 2} = \frac{(x_S^i)^2}{2} - \frac{x_S^i}{2}$$
In other words, this sum is a quadratic function of the variables with coefficients $\frac{1}{2}$ and $-\frac{1}{2}$. Further, if vertices in $I_S$ are not adjacent to $c_j$ this sum is $0$.
For the last double sum in Equation <ref> consider an edge $uc_j \in E_{i,j}^S$ and vertex $v \in I_{S'}^{i'}$ such that $S' \neq S$ or $i' \neq i$. If $uc_j$ flies over $v$ then all the edges in $E_{i,j}^S$ fly over all the vertices in $I_{S'}^{i'}$. Let $g(i,j,S,i',S')$ be a function that returns $1$ if vertices in $I_S$ are adjacent to $c_j$ and all the edges in $E_{i,j}^S$ fly over all the vertices in $I_{S'}^{i'}$. Otherwise $g(i,j,S,i',S')$ returns $0$. It follows that
$$\sum_{uc_j \in E_{i,j}^S} \sum_{\substack{w \in I - I_S^i \\ uc_j \sim w}} 1 = x_S^i \cdot \sum_{(i',S') \neq (i,S)} g(i,j,S,i',S') x_{S'}^{i'}.$$
In other words, this sum is a quadratic function of the variables with $0$ and $1$ as coefficients.
The outer sum of Equation <ref> goes over all $2^k$ choices for $S$, $k+1$ choices for $i$ and $k$ choices for $j$. Since the sum of quadratic functions is a quadratic function, this concludes the analysis and proves the following lemma.
$val(\sigma, G)$ is a quadratic function of the variables $\{x_S^i\}$ with half-integral coefficients between $-2^kk^2$ and $2^kk^2$. Furthermore, there is a a polynomial time algorithm that given $G$ computes the coefficients.
For each permutation $c_1, \ldots, c_k$ of $C$ we can make an integer quadratic program for finding the best super-homegenous solution to Optimal Linear Arrangement which places the vertices of $C$ in the order $c_1, \ldots, c_k$ from left to right. The quadratic program has variable set $\{x_S^i\}$ and constraints as in Equations <ref> and <ref>. The objective function is the one given by Lemma <ref>, but with every coefficient multiplied by $2$. This does not change the set of optimal solutions and makes all the coefficients integral. This quadratic program has at most $2^k \cdot (k+1)$ variables, $2^k \cdot (k+2)$ constraints, and all coefficients are between $-2^{k+1}k^2$ and $2^{k+2}k^2$. Furthermore, since the domain of all variables is bounded the IQP is bounded as well. Thus we can apply Theorem <ref> to solve each such IQP in time $f(k) \cdot n$. This proves the main result of this section.
Optimal Linear Arrangement parameterized by vertex cover is fixed parameter tractable.
§ CONCLUSIONS
We have shown that Integer Quadratic Programming is fixed parameter tractable when parameterized by the number $n$ of variables in the IQP and the maximum absolute value $\alpha$ of the coefficients of the objective function and the constraints.
We used the algorithm for Integer Quadratic Programming to give the first FPT algorithm for Optimal Linear Arrangement parameterized by the size of the smallest vertex cover of the input graph.
We hope that this work opens the gates for further research on the parameterized complexity of non-linear and non-convex optimization problems. There are open problems abound. For example, is Integer Quadratic Programming fixed parameter tractable when parameterized just by the number of variables? What about the parameterization by $n+m$, the number of variables plus the number of constraints?
It is also interesting to investigate the parameterized complexity of Quadratic Programming, i.e. with real-valued variables rather than integer variables.
Finally, there is no reason to stop at quadratic functions. In particular, investigating the parameterized complexity of special cases of (integer) mathematical programming with degree-bounded polynomials in the objective function and constraints looks like an exciting research direction. Of course, many of these problems are undecidable <cit.>, but for the questions that are decidable, parameterized complexity might well be the right framework to study efficient algorithms.
|
1511.00297
|
Kernel-Penalized Regression
Fred Hutchinson Cancer Research Centerm1 and University of Washingtonm2
The analysis of human microbiome data is often based on dimension-reduced graphical
displays and clusterings derived from vectors of microbial abundances in each sample.
Common to these ordination methods is the use of biologically motivated definitions of
similarity. Principal coordinate analysis, in particular, is often performed using
ecologically defined distances, allowing analyses to incorporate context-dependent,
non-Euclidean structure. In this paper, we go beyond dimension-reduced ordination methods and
describe a framework of high-dimensional regression models that extends these
distance-based methods. In particular, we use kernel-based methods to show how to
incorporate a variety of extrinsic information, such as phylogeny, into penalized
regression models that estimate taxon-specific associations with a phenotype or
clinical outcome. Further, we show how this regression framework can be used to address
the compositional nature of multivariate predictors comprised of relative abundances;
that is, vectors whose entries sum to a constant. We illustrate this approach with
several simulations using data from two recent studies on gut and vaginal microbiomes.
We conclude with an application to our own data, where we also incorporate a
significance test for the estimated coefficients that represent associations between
microbial abundance and a percent fat.
compositional data
distance-based analysis
kernel methods
microbial community data
penalized regression
§ INTRODUCTION
A common tool in the analysis of data from microbiome studies is a scatterplot of
dimension-reduced microbial abundance vectors. This is a display of the samples' beta
diversity which, in ecology, refers to differences among various habitats. When applied
to human studies, beta diversity describes the variation in microbial community structure
across sampling units (e.g., human subjects): a beta diversity plot displays the $n$
sampling units with respect to the principal coordinates of their microbial abundance
vectors, each consisting of measures on the $p$ taxa observed in the study; see, e.g.,
<cit.>. This principal coordinates
analysis (PCoA; or multidimensional scaling, MDS) begins with an $n\times n$ matrix of
pairwise dissimilarities between abundance vectors. The choice of dissimilarity measure
may greatly influence the biological interpretation <cit.>.
Euclidean distance is rarely used.
Dissimilarity measures that account for phylogenetic relationships among the taxa are
assumed to enhance statistical analyses — for instance, to improve the power of
statistical tests — because they incorporate the degree of divergence between sequences
<cit.> and do not ignore “the correlation between evolutionary and ecological
similarity" <cit.>. The UniFrac distance <cit.>, in
particular, is based on the premise that taxa which share a large fraction of the
phylogenetic tree should be viewed as more similar than those sharing a small fraction of
the tree. In the unweighted version of UniFrac, each taxon is quantified merely by its
presence or absence; the distance between a pair of samples is based on the number of
branches in the tree shared by both. Figure <ref>(a) is a beta
diversity plot of $n=100$ human microbial abundance vectors with $p=149$ taxa based on
data from <cit.>. The 2-dimensional coordinates of the samples are
displayed with respect to the unweighted UniFrac distance, and each sample is colored
according to the age of the subject.
PCoA plots of data from <cit.>. (a): PCoA plot with respect to unweighted UniFrac distance, colored according to log(age) of subject.
(b): PCoA plot with respect to unweighted UniFrac distance, colored according to
$y_{\mbox \footnotesize True}$ from the model in Eq. (<ref>) with $\epsilon=0$.
Dissimilarity measures in microbiome studies are many and varied, with a rich collection
that, like UniFrac, exploit the phylogentic structure: <cit.> generalize
UniFrac by reweighting rare and abundant lineages; double principal coordinate analysis
(DPCoA) <cit.>, as shown by <cit.>, generalizes PCA by
incorporating the covariance that would arise if the data was created by a process
modeled by the tree; the edge PCA method of <cit.> incorporates taxon
abundance information at all nodes in a phylogenetic tree, rather than just the leaves of
the tree, and <cit.> formalize the mathematical interpretation of
UniFrac as just one example within a large family of Wasserstein (or earth mover's)
metrics. A wide variety of non-phylogenetic dissimilarities are also in common use, such
as Bray-Curtis <cit.> and Jenson-Shannon <cit.>, among others.
While PCoA plots provide valuable graphical insight into the relationships among
microbial profiles and an outcome or phenotype, they do not quantify this
association. More importantly, the (sets of) taxa associated with the outcome — and the
magnitude or statistical significance of such associations — are not ascertained from a
PCoA plot; once a matrix of (dis)similarities between samples is formed, it is not clear
how to identify individual taxa that are associated with an outcome. Specifically, given
a PCoA plot as in Figure <ref>(a), with structure imposed by the
chosen dissimilarity matrix (e.g., unweighted UniFrac) and with associations implied by a
class label or continuous outcome (e.g., age), how does one estimate which taxa or
subcommunities are associated with this outcome? We address this question by formulating
multivariate regression models that are constrained by the structure of the
(dis)similarity matrix. This is made possible by exploiting an equivalence between a
taxon-based (primal space) and sample-based (dual space) formulation of our penalized
regression models. While exploiting such an equivalence is straightforward in the special
case of ridge regression (with purely Euclidean structure), it becomes complicated when
more general distance measures are used. To this end, we show how a little-used
regularization scheme by <cit.> provides a dual-space regression
coefficient estimate that naturally connects to primal-space coefficients. Because a
dissimilarity matrix can be used to construct a similarity matrix (as commonly done in
classical MDS <cit.>), we work with kernels, rather than distances,
and allow for general kernels, including those constructed from a nonlinear feature map.
In addition to complications stemming from more general distances, the analysis of
microbiome data is also complicated by the compositional nature of the data itself. More
specifically, taxon measures typically represent relative, rather than absolute,
abundances. The $p$-variate relative abundance vectors are thus compositional in
that they are constrained to a simplex within $\bbR^p$; such data do not reside in a
Euclidean vector space <cit.>. Consequently, spurious correlations arise
and standard multiple regression models fail.
Our proposed KPR framework, however, addresses this: the centered log (CLR) transform of
the relative abundance vectors first removes the vectors from the simplex, then the
estimation process is constrained using a penalization term defined by Aitchison's
variation matrix. This approach takes a different perspective from the recent proposal of
<cit.> which forces the estimated coefficient vector to reside in the simplex.
Given that the CLR transforms the compositional vectors to Euclidean space and that the
units of the Aitchison variation matrix are the same as the CLR transformed data
<cit.>, our constraint seems more suitable for the geometry of the
In summary, we describe a family of high-dimensional regression problems in
Section <ref>, which are designed to incorporate the assumptions that are
tacitly implied by various exploratory and graphically-focused PCoA plots common in
microbiome studies. We show how phylogenetic and other structure can be incorporated via
kernel penalized regression in either the primal ($p$-dimensional) feature space or the
dual ($n$-dimensional) samples space; see Sections <ref> and
<ref>. Finally, our proposed framework leads to an approach, described in
Section <ref>, for addressing well-known problems that arise from applying
standard (Euclidean-based) statistical models to compositional data.
Section <ref> illustrates the proposed framework with simulations based on
publicly available data, while Section <ref> presents an application to our
recent microbiome study of premenopausal women. In this analysis, we obtain estimates of
associations between microbial species and percent fat measured in premenopausal women,
and also provide inference for these estimates by applying a recent significance test
<cit.> in our kernel-penalized regression (KPR) framework.
§ KERNEL PENALIZED REGRESSION FOR MICROBIOME DATA
We describe a family of multiple regression problems aimed at incorporating assumptions
that are implicit in PCoA plots common in microbiome studies.
We begin in Section <ref> by establishing notation and concepts from
existing dimension-reduction (ordination) methods with the goal of extending them to
non-truncated (penalized) regression models.
Section <ref> extends PCoA and PCR to penalized regression models in the
primal space in a manner that incorporate structures implicit in recent microbiome
Section <ref> extends kernel ridge regression to general (non-$L^2$) structure
and the use of two kernels. This extension exploits a dual-space regularization
scheme of Franklin <cit.>.
Section <ref> describes how our proposed framework can be applied to
formulate a penalized regression model that accounts for the structure of compositional
We denote by $y_i$, $i=1,.., n$, a real-valued quantified trait, and by $x_i =
[x_{i1},..., x_{ip}]'$ a $p$-dimensional vector of microbial abundance values measured
for each of $n$ subjects. Denote by $X$ the $n\times p$ sample-by-taxon matrix whose
$i$th row is $x_i'$. We assume throughout that the columns of $X$ are mean centered. For
now, we assume that the abundance values are appropriately normalized/transformed and
postpone the treatment of compositional data to Section <ref>. The
transpose of a matrix $A$ is denoted by $A'$ and the Frobenius norm is denoted as
$\|A\|_F$. The Euclidean norm of a vector $x\in\bbR^p$ is denoted $\|x\|_{\bbR^p}$,
$\|x\|_{2}$ or simply $\|x\|$.
§.§ Background for PCoA and principal component regression
Consider first the Euclidean PCoA, which is obtained from the eigenvectors of the
kernel matrix $K_I:=XX'$ of inner products $K_{ij}=\langle x_i,x_j\rangle$ between
samples. Let $\calJ$ be the centering matrix, $\calJ = I -\frac{1}{n}
\boldsymbol{1}\boldsymbol{1}'$, where $\boldsymbol{1}$ is the $n\times 1$ vector of ones.
Then, it can be seen that $XX' = -\frac{1}{2}\calJ\Delta^E\calJ$, where $\Delta^E$ is the
$n\times n$ matrix of squared Euclidean distances between samples:
$\Delta^E_{i,j}=\|x_i-x_j\|_{\bbR^p}^2$. The relationship between a kernel and a distance
matrix $\Delta$ is more general. In particular, if $\Delta$ is any $n\times n$ symmetric
matrix of squared dissimilarities between vectors in $\bbR^p$ then $H
=-\frac{1}{2}\calJ\Delta\calJ$ serves as a kernel matrix summarizing similarities; see,
e.g., <cit.>. A particular case involves a $p\times p$
symmetric, positive definite matrix $Q$ that defines an inner product $\langle
x_i,x_j\rangle_Q=x_i'Qx_j$ on $\bbR^p$. If $\Delta^Q$ denotes the matrix of squared
distances, $\Delta^Q_{i,j}=\|x_i-x_j\|_{Q}^2 = \langle x_i-x_j, x_i-x_j\rangle_Q$,
defined with respect to this inner product, then $XQX'=-\frac{1}{2}\calJ\Delta^Q\calJ$ is
also a similarity kernel for the $n$ samples. We will denote this kernel by $K_Q=XQX'$.
Similarly, one may start with a matrix $\Delta^U$ of squared distances defined by a
tree-based UniFrac dissimilarity <cit.>, and define a similarity kernel by
In graphical displays, two or three coordinates are typically used to explore the
relationship between samples. Let $K = US^2U'$ be the eigen-decomposition of any
similarity kernel, $K$, where $U$ is the matrix whose columns are eigenvectors and
$S^2=\diag\{\sigma_j^2\}$ is the diagonal matrix of eigenvalues. The two-dimensional PCoA
plot is then the collection of points $\{\eta_{i1}, \eta_{i2}\}_{i=1}^n :=\{(\sigma_1
U_{i1}, \sigma_2 U_{i2})\}_{i=1}^n$; i.e., a plot of the points represented by the first
two columns of the matrix $US$. These points are often colored according to a grouping
label or continuous value, $\{y_i\}_{i=1}^n$, to graphically explore the existence of an
association between the outcome $y$ and the sample profiles summarized by the first few
columns of $US$. So, a PCoA plot is a graphical depiction of a two-component regression
model of association:
\begin{equation}\label{eq:yreduced}
y_i = \gamma_1\eta_{i1} + \gamma_2\eta_{i2}+\epsilon, \quad i=1,\ldots,n,
\end{equation}
where $\eta_1$ and $\eta_2$ are the first two PCoA axes. Ordinary principal component
regression (PCR) corresponds to the case that $\eta_{1}$ and $\eta_{2}$ come from the
Euclidean kernel $K_I=XX'$. On the other hand, the configuration of points in
Figure <ref>(b) correspond to the first two eigenvectors of the
kernel defined by an unweighted UniFrac distance matrix $\Delta^U$, and colors of
individual points correspond to the values of $y$ from eq. (<ref>) with
$\epsilon = 0$.
Let $A_{(k)}$ denote the first $k$ columns of a matrix $A$, or its first $k$ rows and
columns if $A$ is diagonal. Then, using the singular value decomposition (SVD), $X=USV'$,
if we express the dimension-reduced approximation of $X$ as $\breve{X} :=
U_{(2)}S_{(2)}V_{(2)}'$, then eq. (<ref>) can be written as
\begin{equation}\label{eq:gammaPCR}
\begin{aligned}
y &= \gamma_1\eta_{1} + \gamma_2\eta_{2}+\epsilon \\
&= U_{(2)}S_{(2)}\,\gamma + \epsilon \\
&= \breve{X}V_{(2)}\,\gamma + \epsilon,
\end{aligned}
\end{equation}
where $\gamma =[\gamma_1 \ \gamma_2]'$. Here, $\breve{X}V_{(2)}=U_{(2)}S_{(2)}$, and
$\Range(\breve{X}')=\Range(V_{(2)})$. Therefore, assuming a coefficient vector $\beta$ of
the form $\beta=\breve{X'}\gamma$, the model $y = \breve{X}V_{(2)}\,\gamma + \epsilon$
can be written as $y=\breve{X}\beta+\epsilon$. So inherent in a Euclidean PCoA plot is an
implicit coefficient vector, $\beta$, which models a linear association between $y$ and
$\breve{X}$. Using the SVD of $X$ in (<ref>), the PCR estimate of
$\beta\in\bbR^p$ is expressed as
\begin{equation}\label{eq:betaPCR}
\hat\beta_{\mbox{\tiny PCR}} = (\breve{X}'\breve{X})^{\dag}\breve{X}'y
= V_{(2)}S_{(2)}^{-1}U_{(2)}'y
= \sum_{k=1}^2\frac{1}{\sigma_k}u_k'y\,v_k,
\end{equation}
where $\dag$ denotes the Moore-Penrose inverse.
§.§ Penalized regression and DPCoA
An alternative to a Euclidean PCR is the ordinary ridge regression <cit.>,
\begin{equation}\label{eq:betaRidgeSVE}
\hat\beta_{\mbox{\tiny ridge}} = (X'X+\lambda I)^{-1}X'y
= \sum_{k=1}^n\left(\frac{\sigma_k^2}{\sigma_k^2+\lambda^2}\right)\frac{1}{\sigma_k}u_k'y\,v_k,
\end{equation}
in which the terms are re-weighted instead of being truncated, as in
$\hat\beta_{\mbox{\tiny PCR}}$. The estimate in (<ref>) is the solution
of the penalized least squares regression problem, $\hat\beta_{\mbox{\tiny ridge}} =
\argmin_{\beta}\left\{\|y-X\beta\|^2 + \lambda\|\beta\|^2\right\}$, where here and
throughtout $\lambda$ is a tuning parameter that controls the amount of shrinkage or size
of $\beta$ in the penalty term. Here the penalty is simply the Euclidean (or $\ell^2$)
norm on $\bbR^p$, but a wide range of penalty terms have been proposed to replace or
extend this particular form of regularization; see <cit.> for a review of
the most established methods. These methods, such as the lasso, elastic net or SCAD do
not incorporate any extrinsic information, but a variety of other penalization methods
have been proposed which aim to do this. For instance, <cit.> uses a
tree-guided penalty <cit.> to incorporate such structure into a penalized
logistic regression framework to encourage similar coefficients among taxa according to
their relationships in the phylogenetic tree. <cit.> study the solution
path for computing a “generalized lasso" estimate in which an $\ell^2$ penalty is
replaced with an $\ell^1$ penalty applied to a linear transformation of the features,
$\lambda\|L\beta\|_1$. Within the context of genetic networks, <cit.>
accounted for network structure by augmenting the $\ell^1$ penalty with a second penalty
of the form $\lambda_2\|\beta\|_{\mathcal{L}}^2 = \beta'\mathcal{L}\beta$, where
$\mathcal{L}$ denotes the graph Laplacian matrix corresponding to pre-defined connections
between genes in a pathway.
For now, we consider a positive definite $p\times p$ matrix $Q$ with a Cholesky
decomposition $Q=LL'$, and a penalty term of the form $\|L^{-1}\beta\|^2 =
\|\beta\|^2_{Q^{-1}}=\beta'Q^{-1}\beta$. The generalized ridge (or Tikhonov
regularization <cit.>) estimate with respect to $Q$ is then defined as
\begin{equation}\label{eq:hatBetaQ}
\begin{aligned}
\hat\beta_Q % = \argmin_{\beta}\{ \|y-X\beta\|_{\bbR^n}^2 + \alpha\|L\beta\|_{\bbR^p}^2\}
& =\argmin_{\beta}\{ \|y-X\beta\|^2 + \lambda\|\beta\|_{Q^{-1}}^2\}
= (X'X + \lambda Q^{-1})^{-1}X'y \\
& = \sum_{k=1}^{n} \left(\frac{\sigma_k^2}{\sigma_k^2+\lambda \mu_k^2}\right) \frac{1}{\sigma_k} u_k'y \, v_k,
\end{aligned}
\end{equation}
This estimate takes the same form as (<ref>) but now the vectors $u_k$
and $v_k$ arise from the SVD of $XL=USV'$.
As an aside, it is worth noting that if $A$ denotes any matrix with $p$ columns, the
structure of an estimate $\hat\beta_A$ from a penalty term of the form $\|A\beta\|_{2}^2$
is determined by the joint eigenstructure of the pair $(X,A)$ via the generalized
singular value decomposition.[We refer here to the generalized singular value
decomposition (GSVD) of <cit.>, a simultaneous diagonlization of two
matrices. A different SVD generalization <cit.> imposes constraints on
left and right singular vectors of a matrix.] In particular, the basis expansion of
$\hat\beta_Q$ in (<ref>) is given in terms of the generalized singular
vectors of $(X,L^{-1})$. Although the ridge estimate (with $Q=I_p$) is biased, an
informed choice of penalty term can, in fact, reduce the bias <cit.>.
Now consider the context of phylogentic information and let $\delta$ represent the matrix
of squared patristic distances between pairs of taxa — i.e., the sum of branch lengths
between each pair of taxa on the leaves of a phylogenetic tree. Set
$Q=-\frac{1}{2}\calJ\delta\calJ$, a matrix of similarities between taxa. Double
principal coordinate (DPCoA) analysis was proposed as a multi-step procedure by
<cit.> to provide an alternative to ordinary PCoA by incorporating both the
structure among samples and the structure implied by species distribution among
subcommunities; i.e., the phylogeny as summarized by $Q$. <cit.> clarified
the original multi-step DPCoA procedure and showed how it can be more simply understood
as a generalized PCA (gPCA) in which one obtains the new coordinates from the
eigenvectors of $K_Q=XQX'$.
Note that when $Q$ is the identity matrix ($Q = I_p$), DPCoA reduces to PCA/MDS. As
emphasized in <cit.>, the use of a non-identity $Q$ matrix incorporates
structure from known relationships between the $p$ taxa by exploiting a matrix
representation of phylogenetic relationships, thus providing a model for covariance
If we let $Q=LL'$ be a Cholesky decomposition of $Q$ and set $Z:=XL$, then the kernel
$K_Q$ has an SVD of the form $XQX'=VS^2V'$. This leads to a two-dimensional regression
estimate that takes the same the form as $\hat\beta_{\mbox{\tiny PCR}}$ in
(<ref>). Indeed, we can recover a primal space estimate in terms of singular
vectors as
\begin{equation}\label{eq:betaDPCR}
\hat\beta_{\mbox{\tiny DPCR}} := V_{(2)}S_{(2)}^{-1}U_{(2)}'y =
\sum_{k=1}^2\frac{1}{\sigma_k}u_k'y\,v_k.
\end{equation}
That is, implicit in a DPCoA plot is a coefficient vector, $\hat\beta_{\mbox{\tiny
DPCR}}$ which models a two-dimensional linear association between $y$ and $Z=XL$ in the
same way that $\hat\beta_{\mbox{\tiny PCR}}$ represents a two-dimensional linear
association between $y$ and $X$. Further, $XQX'=(XL)(XL)'$ and so $U$, $S$ and $V$ in
(<ref>) are the same as those in the penalized (non-truncated) estimate,
$\hat\beta_{Q}$, in (<ref>).
§.§ Kernel-based regression with two kernels
In addition to similarities among taxa, as in $Q$, it is often of interest to incorporate
similarities among samples as derived, for instance, from UniFrac distances:
$H=-\frac{1}{2}\calJ\Delta^U\calJ$. The symmetric positive definite $n\times n$ kernel
$H$ defines a new inner product on $\bbR^n$ given by $\langle u,w\rangle_H = u'Hw$, with
the corresponding norm $\|u\|^2_{H}=\langle u,u\rangle_H$. If we consider both a general
kernel, $H$, and a DPCoA kernel $K_Q=XQX'$, the generalized ridge estimate $\hat\beta_Q$
in (<ref>) can be extended to
\begin{equation}\label{eq:hatBetaQH}
\begin{aligned}
\hat\beta_{Q,H} & :=\argmin_{\beta\in\bbR^{p}}\left\{ \|y-X\beta\|_{H}^2 + \lambda\|\beta\|_{Q^{-1}}^2\right\}\\
& = (X'HX + \lambda Q^{-1})^{-1}X'Hy.
\end{aligned}
\end{equation}
In this section, we show that the estimate in (<ref>) is directly defined
based on the generalized eigenvectors of the two kernels $Q$ and $H$. Before proceeding
to the general case, let us examine the special case of ridge regression. In this case,
$H = I_n$ and $Q=I_p$. It is well known that ridge estimates can be obtained by solving
an equivalent optimization problem in the dual space $\bbR^n$, known as kernel
ridge regression <cit.>. Specifically, taking $K_I=XX'$, the ridge
estimate in (<ref>) can be obtained as $\hat\beta_{\mbox{\tiny
ridge}}=X'\hat\gamma_{\mbox{\tiny kernel ridge}}$, where
\begin{equation}\label{eq:KernelRidge}
\begin{aligned}
\hat\gamma_{\mbox{\tiny kernel ridge}}& =(K_I+\lambda I)^{-1}y = (K_I^2+\lambda K_I)^{-1}K_Iy\\
&= \argmin_{\gamma\in\bbR^{n}}\{ \|y-K_I\gamma\|^2 + \lambda\|\gamma\|^2_{K_I}\}.
\end{aligned}
\end{equation}
In the case of ridge, the connection between the dual- and primal-space estimates,
$\hat\gamma_{\mbox{\tiny kernel ridge}}$ and $\hat\beta_{\mbox{\tiny ridge}}$, relies on
the form $K_I=XX'$. Unfortunately, it is less clear how to extend this connection to a
general kernel (e.g., UniFrac or polynomial). One way to incorporate a general kernel $K$
and a second kernel $H$ in (<ref>) is to define the penalty in terms of
$H$ as
\begin{equation}\label{eq:K2Hinv}
\hat\gamma_{*} = (K^2+\lambda H^{-1})^{-1}Ky = \argmin_{\gamma}\{ \|y-K\gamma\|^2 + \lambda\|\gamma\|_{H^{-1}}^2\},
\end{equation}
which is exactly the Tikhonov regularization, but in the dual space; compare
eq. (<ref>). However, $\hat\gamma_{*}\in\bbR^n$ has no obvious connection to
a penalized estimate of $\beta\in\bbR^p$ and cannot be used to obtain a penalized
regression estimate in the primal space, even if $K=K_I=XX'$.
To bridge this gap, we instead apply the Franklin regularization scheme
<cit.>, a little-used alternative to Tikhonov regularization. More
specifically, for any kernels $K$ and $H$, we define the dual estimate
\begin{equation}\label{eq:Franklin}
\hat\gamma_{H,K} := (K+\lambda H^{-1})^{-1}y = \argmin_{\gamma\in\bbR^{n}}\{ \|y-K\gamma\|_{K^{-1}}^2 + \lambda\|\gamma\|_{H^{-1}}^2\}
\end{equation}
Comparing (<ref>) and (<ref>), one sees that the analytic form of
(<ref>) involves just $K$ rather than $K^2=K'K$. As shown in
Proposition <ref>, this subtle difference is a key for relating
$\hat\gamma_{H,K}$ and its primal-space counterpart. Before presenting the main result of
this section, we provide several equivalent forms of $\hat\gamma_{H,K}$.
\begin{equation}\label{eq:Franklin2}
\begin{aligned}
%(K+\lambda H^{-1})^{-1}y &
\hat\gamma_{H,K} &=(K+\lambda H^{-1})^{-1}y\\
&= \argmin_{\gamma\in\bbR^{n}}\{ \|y-K\gamma\|_{K^{-1}}^2 + \lambda\|\gamma\|_{H^{-1}}^2\}\\
&=\argmin_{\gamma\in\bbR^{n}}\{\|y-K\gamma\|_{H}^2+ \lambda \|\gamma\|_{K}^2\}\\
&=(HK+\lambda I)^{-1}Hy\\
&=H(KH+\lambda I)^{-1}y.
\end{aligned}
\end{equation}
In Proposition <ref>, we also refer to the special case corresponding to the
DPCoA ordination. As before, let $Z=XL$ so that $K_Q=XQX'=XLL'X'=ZZ'$. Taking $H=I$, the
dual-space estimate in (<ref>) is $\hat\gamma_{I,K_Q} = (K_Q+\lambda
I)^{-1}y$, and so the corresponding primal-space estimate is $\hat\beta\equiv
Z'\hat\gamma_{I,K_Q}$. Since this estimate arises from the DPCoA kernel, so we make the
following definition.
A primal space DPCoA estimate is of the form $\hat\beta_{\mbox{\tiny
DPCoA}}=Z'\hat\gamma_{I,K_Q}=L'X'(XQX'+\lambda I)^{-1}y$.
The next proposition collects several properties that emphasize the roles of $H$ and $K$
in our penalized regression framework. In particular, we show that the primal space
estimate $\hat\beta_{Q,H}$ can be recovered in terms of two kernels, $H$ and $K_Q$.
Let $H$ and $K$ be any two kernels constructed using the rows of $X$ in the regression
model $y=X\beta+\epsilon$. Then,
* $\hat\gamma_{H,K}$ is a linear combination of the eigenvectors of the matrix
product $HK$.
* For any kernel $H$ and DPCoA kernel $K_Q=XQX'$, then the primal- and dual-space
estimates in (<ref>) and (<ref>), respectively, are
related as: $\hat\beta_{Q,H} = QX'\hat\gamma_{H,K_Q}$.
* For $H=I$ and $Q=LL'$, the generalized ridge and DPCoA estimates are related as
$\hat\beta_{Q} = QX'(K_Q+\lambda I_n)^{-1}y = L\hat\beta_{\mbox{\tiny DPCoA}}$.
These relationships are based on several basic linear algebraic
properties. In particular, we make use of the following identities:
\begin{equation}\label{eq:XQHidentities}
\begin{aligned}
&(X'HX + \lambda Q^{-1})^{-1}X'H =(QX'HX+\lambda I_p)^{-1}QX'H \\
& = Q(X'HXQ+\lambda I_p)^{-1}X'H
= QX'(XQX'+\lambda H^{-1})^{-1} \\
& = QX'H (XQX'H+\lambda I_n)^{-1}.
\end{aligned}
\end{equation}
* Let $w_1,...,w_n$ denote the columns of the matrix $W$ satisfying $W'KW =
\diag\{\sigma_1,...,\sigma_n\}$ and $W'H^{-1}W = \diag\{\mu_1,...,\mu_n\}$. That
is, $\{w_k\}$ is a basis with respect to which a simultaneous diagonalization of
$K$ and $H^{-1}$ is obtained (see <cit.>). These
are the generalized eigenvectors of the pair $(K,H^{-1})$. Then
\begin{equation}\label{eq:gammaH_GSV}
\hat\gamma_{H,K} = \sum_{k=1}^n \left(\frac{\sigma_k}{\sigma_k+\lambda\mu_k}\right) %% CHECK the \sigma_k^2 numerator in [Hansen]
\frac{w_k'y}{\sigma_k} w_k.
\end{equation}
Since $H$ is invertible, the $w_k$'s are also eigenvectors of $HK$
* Using the identities in (<ref>), the estimate in
(<ref>) can be expressed as
\begin{equation}\label{eq:hatBetaQH2}
\begin{aligned}
\hat\beta_{Q,H} & = Q(X'HXQ+\lambda I_p)^{-1}X'Hy \\
& = QX'(XQX'+\lambda H^{-1})^{-1}y \\
& = QX'(K_Q+\lambda H^{-1})^{-1}y = QX'\hat\gamma_{H,K_Q}.
\end{aligned}
\end{equation}
* Setting $H=I$ in the first line of the above equalities gives $\hat\beta_{Q} =
QX'(XQX'+\lambda I_n)^{-1}y$. Using $Q=LL'$ gives
$\hat\beta_{Q}=L\,L'X'(K_Q+\lambda I_n)^{-1}y = L\,Z'(ZZ'+\lambda I)^{-1}y=
L\hat\beta_{\mbox{\tiny DPCoA}}$.
* Types of similarity kernels. In general, a sufficient condition for a
matrix $K$ to be a similarity kernel is that it is induced by a feature map
$\phi\colon \bbR^p \to \mathcal{K}$. More specifically, the $i,j$ entry of $K$ is
defined as the inner product of the observations $x_i\in \Bbb{R}^p$ with respect to
their transformed versions $K_{ij}=\langle\phi(x_i),\phi(x_j)\rangle$ in the new
inner product space, $(\mathcal{K},\langle\cdot,\cdot\rangle)$. Examples include
$K_I=XX'$ or $K_Q=XQX'$, where $\mathcal{K}$ is $\bbR^p$ with inner product
$\langle\cdot,\cdot\rangle_Q$ (as in DPCoA). It is this quadratic form that we
require for $K_Q$ in Proposition <ref>(2)–(3); see <cit.>
for genomic applications of this form. On the other hand, $H$ can be any symmetric
positive semi-definite matrix. Here, we are more interested in
biologically-motivated kernels, such as UniFrac or DPCoA, than
mathematically-derived ones, such as those constructed from polynomials or radial
basis functions <cit.>.
* Co-informative kernels and the HSIC. Any kernels $K$ and $H$ may be used
in (<ref>) and (<ref>), but to be useful in this
framework, we assume that they are “co-informative" in the sense that they exhibit
a shared eigenstructure; for instance, both should be informative for classifying
samples. This concept is illustrated in the simulation of Section <ref>
and Figure <ref>. The co-informativeness can be made precise using the
Hilbert-Schmidt information criteria (HSIC) <cit.>
or its relatives the distance covariance <cit.>
or RV statistic <cit.>.
<cit.> provide a nice review of these and related kernel-based
tests. The HSIC provides a test for the statistical dependence of two data sets,
$X_1$ ($n\times p$) and $X_2$ ($n\times q$), and is based on the eigen-spectrum of
covariance operators defined by kernels defined by $X_1$ and $X_2$, respectively.
For two kernels $K$ and $H$, the empirical HSIC is simply $\trace(HK)$. The HSIC is
thus of particular interest in item (1) of Proposition <ref>, which shows
how two co-informative kernels may be used to obtain a penalized estimate
* Linear mixed models and KPR. As an alternative to the regularization
framework presented here, it may be useful to consider a kernel as a generalized
covariance among either the $p$ variables (using $Q$) or $n$ subjects (using $H$)
<cit.>. This alternative representation can be made
precise using the linear mixed model (LMM) framework <cit.>.
Specifically, recall from equations (<ref>) and (<ref>)
\begin{equation*}
\begin{aligned}
\hat\beta_{Q,H} & = \argmin_{\beta\in\bbR^{p}}\{ \|y-X\beta\|_{H}^2 + \lambda\|\beta\|_{Q^{-1}}^2\}\\
%& = (X'HX + \lambda Q^{-1})^{-1}X'Hy\\
%& = QX'(XQX'+\lambda H^{-1})^{-1} \\
& = QX'(K_Q+\lambda H^{-1})^{-1}y \\
& = QX'\argmin_{\gamma\in\bbR^{n}}\{ \|y-K_Q\gamma\|_{H}^2 + \lambda\|\gamma\|_{K_Q}^2\}
= QX'\hat\gamma_{H,K_Q}.
\end{aligned}
\end{equation*}
These regression estimates are compatible with $\beta\sim N(0,\sigma_b^2 Q)$, $\epsilon\sim N(0,\sigma_e^2 H^{-1})$ and $var(y) = (\tau K_Q + \lambda H^{-1})^{-1}$.
And the estimate $\hat\gamma_{H,K_Q}$ is compatible with $\gamma\sim N(0,\sigma_a^2
K_Q^{-1})$ and $\epsilon\sim N(0,\sigma_e^2 H^{-1})$. With regard to the latter, a
genetic similarity between subjects (e.g., kinship) is often used for grouping subjects
and several authors have proposed this form of kernel for testing the (global) genetic
association with a trait or phenotype, $y$; see, e.g., <cit.>.
In particular, these methods use the LMM framework to motivate and define a “kernel
association test". The variance score statistic for testing the null hypothesis of no
association between $y$ and $X$ ($H_0: \beta=0$) is, using our notation above,
$\mathcal{T} := \|y\|_{H^{1/2}K_QH^{1/2}}^2$. The kernel association testing framework
has been applied to microbiome data using a single kernel at a time derived from
Unifrac <cit.>, but this is a test for whether $\beta\ne 0$ and, unlike
our KPR framework, provides no insight about which taxa, as represented by coordinates
of $\beta$, are associated with $y$.
§.§ Regression with compositional data
Data from 16S rRNA gene sequencing methods are random counts of the molecules in each
sample. The number of sequence reads assigned to a taxon contains no information about
the actual number of molecules in the sample; the total number of reads observed in
two samples can vary by several orders of magnitude. Hence, only relative amounts
can be investigated. Common approaches for normalizing these data include converting them
to proportions (relative percent) or subsampling the sequences to create equal library
sizes for each sample (rarefying). These data are “compositional" in the sense that the
microbial abundances represent a proportion of a constant total. It is well known,
however, that compositional measures can result in spurious correlations among taxa
<cit.>, an effect that can be quite extreme
when there are a few dominant taxa.
Compositional data reside on the simplex $\bbS^{p-1}$ of unit-sum vectors in
$\bbR^p$ and so standard multivariate methods do not apply
<cit.>. In particular, because
these data do not naturally reside in a Euclidean vector space, standard regression
models based on Euclidean covariance measures are inappropriate. However, ordinary
least-squares and ridge regression estimates are of the form $\hat\beta= ( X'X + \lambda
I)^{-1} X' y$ (with $\lambda = 0$ and $\lambda > 0$, respectively). Thus, these estimates
depend on the empirical covariance structure, $X'X$, among taxa, which may include
spurious correlations. Similarly, <cit.> points out that a naïve application
of lasso regression is not expected to perform well due to the compositional nature of
the covariates. He addresses this issue by applying a lasso regression model to the
log-ratio abundances and imposing an additional constant-sum constraint on the
coefficient vector, $\beta$.
We next show that the generality of KPR for handling non-Euclidean structures can be used
to address the compositional nature of microbiome data. In particular, we propose an
approach that uses the centered log-ratio transformation of the compositional vectors and
an estimate of covariance among the log taxa counts that is obtained via Aitchison's
variance matrix <cit.>.
Let $X$ be the $n\times p$ sample-by-taxon matrix whose rows are relative percent
(compositional) vectors $\{x_i\}_{i=1}^n\subset\bbS^{p-1}$. The columns of $X$ will be
denoted by $x^k$, corresponding to $k=1,...,p$ taxa. Let $g(z) =\left(\Pi_{k=1}^p
z^k\right)^{1/p}$ be the geometric mean of a row vector, $z$, and denote the centered
log-ratio (CLR) transform of $x_i$ by $\tilde x_i=\clr(x_i):=
\left[\log\frac{x_i^1}{g(x_i)},\ldots,\log\frac{x_i^p}{g(x_i)}\right]$. In what follows
we denote the matrix of CLR vectors by $\tilde X$, and use the normalized variation
matrix $T$, of $X$, as defined by <cit.>: $T_{k,\ell}=
\var\left(\frac{1}{\sqrt{2}}\log\frac{x^k}{x^\ell}\right)$. $T$ is a symmetric
dissimilarity matrix with zeros on the diagonal and entries that have squared Aitchison
distance units: the Aitchison norm of a vector $x\in\bbS^{p-1}$ is defined as $\|x\|_a^2
= \frac{1}{2p}\sum_{k,\ell}\left(\log \frac{x^k}{x^\ell} \right)^2$. In fact, $\|x\|_a^2
= \|\clr(x)\|^2$. One can show that $T$ is related to the covariance matrix, $C$ of the
log of the true unobserved taxa counts via $T = v\boldsymbol{1}' + \boldsymbol{1}v' - 2C$
<cit.>. Consequently, $C = -\frac{1}{2}\mathcal{J}T\mathcal{J}$, and we can use
$C$ in place of $Q$ in eq. (<ref>) to obtain
\begin{equation}\label{eq:hatBetaC}
\tilde\beta_C =\argmin_{\beta}\left\{ \|y-\tilde{X}\beta\|_{\bbR^n}^2 + \lambda\|\beta\|_{C^{-1}}^2\right\}.
\end{equation}
As a comparison, we observe that <cit.> proposed a constrained regression
\begin{equation}\label{eq:LiConstrainReg}
\E(y_i) = \beta_1\log x_i^1 +\dots + \beta_p\log x_i^p \quad \text{ subject to } \sum_{j=1}^p\beta_j=0,
\end{equation}
augmented with a lasso penalty to obtain an estimate of the form
\begin{equation*}
\argmin_{\beta} \left\{\frac{1}{2n} \| y- \sum_{j} \log(x^j) \beta_j \|_{\bbR^n}^2 + \lambda \sum_j | \beta_j | \right\} \quad \mbox{ subject to } \sum_{j=1}^p\beta_j=0.
\end{equation*}
The zero-sum constraint on $\beta$ was emphasized for interpretability advantages over
the standard lasso estimate. Temporarily denoting $\beta_p=-\sum_{j = 1}^{p-1}\beta_j$,
we see that (<ref>) is equivalent to
\begin{align*}
\E(y_i) &=\beta_1\log\frac{x_i^1}{x_i^p} + \beta_2\log\frac{x_i^2}{x_i^p} + \dots + \beta_{p-1}\log\frac{x_i^{p-1}}{x_i^p} \\
&=\beta_1\log x_i^1 + \beta_2\log x_i^2 + \dots + \beta_{p-1}\log x_i^{p-1} -\sum_{j = 1}^{p-1}\beta_j \cdot\log x_i^p.
\end{align*}
Since $\sum_{j = 1}^{p}\beta_j=0$, this can be rewritten as
\begin{align*}
\E(y_i) &= \beta_1\log x_i^1 +\dots +\beta_p \log x_i^p - (\sum_{j = 1}^p \beta_j) \log g(x_i)\\
&=\beta_1\log \frac{x_i^1}{g(x_i)} + \dots + \beta_p\log\frac{x_i^p}{g(x_i)} \qquad \text{ subject to } \sum_{j=1}^p\beta_j=0.
\end{align*}
Therefore, Li's proposal of regression on log-ratio abundances is equivalent to
regression on the CLR-transformed data $\tilde{X}$ provided a zero-sum constraint is
imposed on $\beta$. In contrast, however, our formulation does not explicitly impose a
constant-sum constraint. In fact, this constraint is not needed because the CLR transform
removes the analysis from the simplex to allow an analysis in Euclidean vector space
algebra <cit.>. Our model instead incorporates the appropriate
covariance structure for the CLR transformation, $C$.
As a final observation, we note that a positive-definite $C$ in (<ref>), or
more generally $Q$ in (<ref>), can be decomposed as a sum $Q=I+\tilde Q$ of
the identity plus a positive semi-definite singular matrix $\tilde Q$. The identity term
constrains $\sum_{j=1}^p\beta^2_j$ to be small while, overall, $\tilde Q$ encourages
extrinsic structure (e.g., smoothness). One may also control the size of
$\sum_{j=1}^p\beta_j^2$ by adding or subtracting values in the diagonal entries of $Q$.
This idea is similar to that of “Grace-ridge" in <cit.> where, in
addition to the penalty induced by $Q$, the authors propose to further impose a
ridge-type penalty in the objective. We apply the significance testing framework of
<cit.> in Section <ref>.
§ NUMERICAL EXPERIMENTS
To illustrate the proposed framework, we perform several data-driven simulations using
publicly available microbiome data. We consider three scenarios from the literature that
exploit extrinsic structure from a phylogenetic tree, including DPCoA, UniFrac and edge
PCA. To achieve realistic simulations, we simulate “true" signals of the type implied by
each of these methods in order to create benchmarks for performance evaluation. Our
emphasis is on formalizing the role that such structure plays in penalized regression
when modeling associations between the multivariate data, $X$, and a response variable,
$y$. Since $y$ is directly simulated from $X$ in these settings, the compositional nature
of the data discussed in Section <ref> does not affect the simulation
results. We will return to this topic when analyzing the relative abundance data in
Section <ref>.
The numerical experiments in this section are motivated by the relationship between the
PCoA plots and PCR described in Section <ref> and
Figure <ref>(b). This connection can be generalized to a number of
other commonly-used graphical representations in the microbiome literature. For instance,
any two-dimensional DPCoA plot involves an implicit coefficient vector, $\beta$, of
associations between $y$ and $X$.
Throughout this section, we compare the performance of KPR with ridge regression and
lasso. Ridge regression provides a direct extension of ordinary least squares and thus is
a natural benchmark for comparing various KPR estimates. Lasso, which gives sparse
estimates, is used as a benchmark in settings where the true $\beta$ is sparsely
non-zero. The choice of competing methods is limited by our emphasis on estimating
$\beta$, rather than predicting the outcome $y$. Indeed, most kernel methods focus on
prediction which renders them inappropriate for comparison.
In all simulation experiments, the tuning parameters for KPR, ridge and lasso are chosen
using 10-fold cross-validation. Specifically, to compare the prediction performance of
KPR, ridge and lasso, we choose the tuning parameters that minimize squared test error in
held out cross validation samples (CV min). On the other hand, the task of estimation
usually requires more smoothing than prediction <cit.>. Therefore, when
examining the estimation performances of KPR, ridge and lasso, we use the largest tuning
parameters such that the squared test errors are within one standard error of the minimum
squared test error (CV 1se), as suggested in <cit.>. For comparison, we
also consider the tuning parameters corresponding to the minimum squared test error for
ridge and lasso.
§.§ Regression and DPCoA
In our first example, we compare the estimation and prediction performances of KPR, ridge
and lasso using the data depicted in Figure <ref>. The rows of $X$
represent relative abundances of $p = 149$ taxa from $n=100$ subjects in a study by
<cit.>. The outcome $y$ is log-transformed age of each subject. For KPR,
we use $K_Q=XQX'$ and $H=I$, where $Q=-\frac{1}{2}\calJ\delta\calJ$ is a matrix of
similarities between taxa obtained from the matrix of squared patristic distances,
$\delta$. Motivated by DPCoA plots, we assume the underlying “true” response
$y_{\mbox{\tiny True}}$ is generated from the first two eigenvectors of $K_Q$. Let $L$ be
the Cholesky factor of $Q$, i.e., $Q=LL'$, and let $XL=U^{L}S^{L}(V^{L})'$. Recall that
$A_{(k)}$ denotes the first $k$ columns of matrix $A$, or its first $k$ rows and columns
if $A$ is diagonal. Motivated by (<ref>), we let
\begin{equation}\label{eq:truebetaDPCoA}
\beta_{\mbox{\tiny True}}= s\left(V^{L}_{(2)}(S^{L}_{(2)})^{-1}(U^{L}_{(2)})'y, \tau \right),
\end{equation}
where, $s(\cdot,\tau)$ is the hard-thresholding operator, i.e., $s(x, \tau) =
x\cdot1(|x|>\tau)$. The threshold $\tau\geq 0$ is set to achieve various levels of
sparsity: $\|\beta_{\mbox{\tiny True}}\|_0\in\{\lfloor0.2p\rfloor, \lfloor0.6p\rfloor,
p\}$. After generating $\beta_{\mbox{\tiny True}}$, we simulate
\begin{equation*}
y_{\mbox{\tiny True}}=U^L_{(2)}S^L_{(2)}(V^L_{(2)})'\beta_{\mbox{\tiny True}}.
\end{equation*}
The simulation is repeated 500 times, each with a different $\epsilon\sim N_n(0,
\sigma^2_\epsilon I_n)$ in $y_{obs} = y_{\mbox{\tiny True}} + \epsilon$. Further,
$\sigma^2_\epsilon$ is set to achieve $R^2 = \var(y_{\mbox{\tiny True}}) /
(\var(y_{\mbox{\tiny True}}) + \sigma^2_\epsilon)\in\{0.1,0.2,...,0.9\}$. In each
repetition, we estimate $\hat\beta_{\mbox{\tiny DPCoA}}$ from $y_{obs}$ according to
definition <ref>. To make the simulation more realistic, we do not assume
we always observe the $Q$ matrix used to simulate $\beta_{\mbox{\tiny True}}$ and
$y_{\mbox{\tiny True}}$. Rather, to estimate $\hat\beta_{\mbox{\tiny DPCoA}}$, we use
$Q_{obs}$, which is obtained by adding random Gaussian noise to $Q$. Eigenvalues of
$Q_{obs}$ are adjusted to be equal to the eigenvalues of $Q$.
The amount of Gaussian noise added to the entries of $Q_{obs}$ is empirically determined
to achieve $\|Q-Q_{obs}\|_F/\|Q\|_F\in\{0, 0.25, 0.5\}$. As a comparison, we estimate
$\hat\beta_{\mbox{\tiny Ridge}}$ and $\hat\beta_{\mbox{\tiny Lasso}}$ using only $X$ and
$y_{obs}$, without incorporating $Q$. From the estimated coefficients, we compute $\hat
y_{\mbox{\tiny DPCoA}}=XL_{obs}\hat\beta_{\mbox{\tiny DPCoA}}$, $\hat y_{\mbox{\tiny
Ridge}}=X\hat\beta_{\mbox{\tiny Ridge}}$ and $\hat y_{\mbox{\tiny
Lasso}}=X\hat\beta_{\mbox{\tiny Lasso}}$.
The performance metrics are the prediction sum of squared error (PSSE) from
$y_{\mbox{\tiny True}}$ and estimation sum squared error (ESSE) from $\beta_{\mbox{\tiny
Estimation sum squared error (ESSE: left panels) and prediction
sum squared errors (PSSE: right panels) of KPR (red), ridge regression (black) and lasso (blue),
and their 95% confidence bands. We consider three sparsity settings for
$\beta_{\mbox{\tiny True}}$, based on (<ref>): $\|\beta_{\mbox{\tiny True}}\|_0 = p$
in top panels; $\|\beta_{\mbox{\tiny True}}\|_0 = \lfloor0.6p\rfloor$ in center panels, and
$\|\beta_{\mbox{\tiny True}}\|_0 = \lfloor0.2p\rfloor$ in bottom panels.
For ridge and lasso, tuning parameters that produce the smallest cross-validated squared test
error (CV min), and the largest tuning parameters such that the cross-validated squared test
errors are within one standard error of the minimum cross-validated squared test error (CV 1se)
are considered. For KPR, we consider $\|Q-Q_{obs}\|_F/\|Q\|_F = 0$ (no $Q$ error), $0.25$
(small $Q$ error) and $0.5$ (large $Q$ error).
Figure <ref> shows the estimation and prediction performance of KPR, ridge
and lasso. KPR significantly outperforms both ridge regression and lasso for both
prediction and estimation in all settings. As expected, the performance of ridge and
lasso for estimation improve when using a larger tuning parameter. On the other
hand, neither mis-specification of $Q$ nor sparsity of $\beta_{\mbox{\tiny True}}$ seems
to substantially impact the relative performance of the three methods. This may be due to
the fact that KPR estimates the correct target $\beta_{\mbox{\tiny True}}$, even with
mis-specified $Q$, whereas ridge regression and lasso estimate the wrong target.
§.§ Regression and PCoA with respect to a UniFrac kernel
In the case of PCoA with respect to a UniFrac matrix $\Delta^U$ of squared
dissimilarities, the graphical displays are based on the eigen-decomposition of
$H=-\frac{1}{2}\calJ \Delta^U\calJ$. That is, for $H = U^H (S^H)^2 (U^H)' \approx
U^{H}_{(2)}(S^{H}_{(2)})^2(U^{H}_{(2)})'$, the $n$ samples are represented in two
dimensions by the columns of $U^{H}_{(2)}S^{H}_{(2)}$; this results in points
$\{\eta_{i1}, \eta_{i2}\}_{i=1}^n :=\{(\sigma_1 U^{H}_{i1}, \sigma_2
U^{H}_{i2})\}_{i=1}^n$, as plotted in Figure <ref>. When the points
are colored according to a response variable, $\{y_i\}_{i=1}^n$, the implied regression
model is
\begin{equation}\label{eq:KPCR-model}
\begin{aligned}
y &= \gamma_1\eta_{1} + \gamma_2\eta_{2}+\epsilon \\
&= U^H_{(2)}S^H_{(2)}\,\gamma + \epsilon.
% &= XV^H_{(2)}\,\gamma + \epsilon.
\end{aligned}
\end{equation}
However, in contrast to PCR in eq. (<ref>), where $US=XV$, it is not obvious
how to connect $\gamma$ directly to the $p$-coordinates corresponding to the $p$ columns
of $X$.
Here, we exploit the joint eigenstructure of kernels $K_I$ and $H$ (see
(<ref>)) by proceeding as in (<ref>) to obtain the estimate
$\hat\beta_{H}=X'\hat\gamma$ as in (<ref>), with $Q=I$.
In this example, we use the same data as in Section <ref>. For KPR, we use
$K=XX'$ and obtain $H=-\frac{1}{2}\calJ \Delta^U\calJ$ using the UniFrac distance matrix.
We simulate $\gamma_{\mbox{\tiny True}}$ and $y_{\mbox{\tiny True}}$ from the first two
eigenvectors of $H$, as in (<ref>):
\begin{align*}
\gamma_{\mbox{\tiny True}}&=\left((U^{H}_{(2)})'(S^{H}_{(2)})^2U^{H}_{(2)}\right)^{-1}S^{H}_{(2)}(U^{H}_{(2)})'y \\
y_{\mbox{\tiny True}}&=U^{H}_{(2)}S^{H}_{(2)}\gamma_{\mbox{\tiny True}}.
\end{align*}
This bivariate regression is illustrated in Figure <ref>(b).
The simulation is repeated 500 times, each with a different $\epsilon\sim N_n(0,
\sigma^2_\epsilon I_n)$ to produce various values of $R^2\in\{0.1,0.2,\ldots,0.9\}$. We
compute $\hat y_{\mbox{\tiny KPR}}=K\hat\gamma_{\mbox{\tiny KPR}}$, where
$\hat\gamma_{\mbox{\tiny KPR}}$ is estimated using (<ref>). Similar to the
last example, we do not assume we always observe the $H$ matrix that is used to generate
$\gamma_{\mbox{\tiny True}}$ and $y_{\mbox{\tiny True}}$; rather, we use a noisy version,
$H_{obs}$, of $H$ in KPR with $\|H-H_{obs}\|_F/\|H\|_F\in\{0, 0.25, 0.5\}$.
Since there is no meaningful way to simulate $\beta_{\mbox{\tiny True}}$, we do not
compare the methods based on their estimation performances, and only consider prediction.
For all three methods, we find the tuning parameters that minimize the cross-validated
$H_{obs}$-weighted squared test error. While the use of $H$ in tuning ridge and lasso
penalties deviates from the common practice, it results in improved performances, given
the important role of $H$ in this simulation. The $H$ matrix also defines the valid
distance in this example. Thus, to evaluate the prediction performances of various
methods, we use the $H$-weighted prediction sum of squared error (HPSSE), $\|\hat
y-y_{\mbox{\tiny True}}\|^2_H$.
Figure <ref> shows that KPR consistently outperforms ridge regression and
lasso in prediction, even with a reasonable amount of misspecification of $H$. This may
be due to the fact that, with the incorporation of the $H$ matrix, KPR estimates the
correct target whereas ridge and lasso do not.
$H$-weighted prediction sum of squared error (HPSSE) of KPR (red), ridge (black) and lasso (blue), with 95% confidence bands. For KPR, we consider
$\|H-H_{obs}\|_F/\|H\|_F = 0$ (no $H$ Error), $0.25$ (small $H$ Error) and $0.5$ (large $H$ Error).
Analysis of bacterial vaginosis data from <cit.>.
(a): representation of the samples in the space of the first two PC's
of the edge-matrix kernel $H=EE'$. The color of each point corresponds to the pH
level of the sample;
(b): heatmap of edge-matrix kernel used to generate the plot in (a);
(c): two-dimensional PCA plot based on the genus-level relative abundances;
(d): heatmap of the genus-abundance kernel $K=XX'$ used to create the plot in (c).
In (b) and (d), subjects are ordered by the pH values.
§.§ Regression and PCoA using an edge-matrix kernel
In this section, simulations are based on data from a study of bacterial vaginosis (BV)
by <cit.> in which 16S rRNA gene samples were collected using vaginal
swabs from $n=220$ women with and without BV. Here, the outcome $y$ represents pH
measured from vaginal fluid of each subject and we consider the association of $y$ with
genus-level taxa. In this example, we use the $p=62$ genera that exhibit non-zero
sequence counts in at least 20% of the subjects. So here, $X$ represents $220 \times
62$ abundances in a sample-by-genus matrix, and we use a kernel $K=XX'$. Additionally,
however, we define a second kernel $H=EE'$ based on the “edge mass difference matrix",
$E$, originally introduced by <cit.>. If the full phylogenetic tree has $q$
edges, each sample can be represented by a vector indexed by all $q$ edges, the
$e^{\text{th}}$ coordinate of which quantifies the difference between the fraction of
sequence reads on either side of the edge; i.e., the fraction of reads observed on the
root side of the tree minus the fraction of reads on the non-root side. We refer
to <cit.> for details and a discussion of “edge PCA", which refers to PCA
applied to the $n\times q$ matrix $E$. Note, in particular, that abundances from
every taxon level in the tree contribute to a similarity between subjects as
opposed to abundances at a single taxon level, which is used in UniFrac or DPCoA.
In summary, $X$ represents $p=62$ genus-level abundances while $E$ is based on all
$q=1770$ edges in the original phylogenetic tree. Figure <ref>(a) shows a PCA
plot of the 220 subjects in which their similarity is defined using the edge kernel
$H=EE'$; the color of each dot represents the subject's pH. Figure <ref>(b) is
a heatmap of the kernel $H$ used to create Figure <ref>(a). The columns and rows
of $H$ represent similarities between samples based on the edge mass difference matrix,
ordered by subject pH measurement. Similarly, Figure <ref>(c) is a PCA plot
based on similarities defined using the genus-level abundance kernel, $K=XX'$.
Figure <ref>(d) is a heatmap of the kernel $K$ used to create
Figure <ref>(c), and subjects are again ordered by pH. These figures illustrate
how two different measures of similarity (two separate kernels) may be co-informative in
the sense that they both provide information about grouping of subjects' microbiota in
relation to their pH. It is thus natural to expect that incorporating information from
both $H$ and $K$ within the KPR framework may result in improved estimates of association
between $y=\mbox{pH}$ and the taxon abundances.
For the simulation, we define a “true" association between pH and the genus-level taxa
in $X$ using the 2-dimensional PCR model in eq. (<ref>) and
(<ref>). Specifically, we use the apparent association between $y=\mbox{pH}$
and genus-level abundances in Figure <ref>(c) to construct a “true" coefficient
vector $\beta_{\mbox{\tiny True}}$ as follows. Using the SVD of $X$, $X=USV'$, set
\begin{equation*}
y_{\mbox{\tiny True}}=U_{(2)}S_{(2)}\left(U'_{(2)}S_{(2)}^2U_{(2)}\right)^{-1}S_{(2)}U'_{(2)}y;
\end{equation*}
then project $y_{\mbox{\tiny True}}$ onto the space spanned by the first two singular vectors
\begin{equation*}
\beta_{\mbox{\tiny True}}=V_{(2)}S_{(2)}^{-1}U_{(2)}'y_{\mbox{\tiny True}}.
\end{equation*}
Taking $H=EE'$ in a KPR model of the form (<ref>), we compare the resulting
estimate of $\beta$ with ridge and lasso estimates. (Note that $y_{\mbox{\tiny True}}$
and $\beta_{\mbox{\tiny True}}$ are not informed by $E$.) The simulation is repeated 500
times, each with a different $\epsilon\sim N_n(0, \sigma^2_\epsilon I_n)$ to produce
various values of $R^2\in\{0.1,0.2,...,0.9\}$. The performance metrics are the estimation
sum squared error (ESSE) the $H$-weighted prediction sum squared error (HPSSE) as in the
previous section. In this numerical example, we do not assume we always observe the true
$H$ matrix; rather, we use a noisy version, $H_{obs}$, of $H$ in KPR with
$\|H-H_{obs}\|_F/\|H\|_F\in\{0, 0.25, 0.5\}$.
For all three methods, tuning parameter values are chosen to minimize the sum of squared
test error weighted by $H_{obs}$. As in the simulation for DPCoA, we also allow for using
the largest tuning parameters such that the squared test error weighted by $H$ is within
one standard error of the minimum squared test error.
Figure <ref> shows that KPR significantly outperforms ridge and lasso in both
prediction and estimation. Note that even though $H$ is not used to simulate the true
association, the use of edge kernel in KPR enhances the performance of both estimation
and prediction, as long as $H$ is not severely misspecified. Once again, the performance
of estimates from ridge and lasso improve when using a larger tuning parameter (CV
In silico evaluation of using tree-based edge information in regression models. Estimation sum squared error (ESSE) and $H$-weighted prediction sum squared error (HPSSE) of KPR (red), ridge
regression (black) and lasso (blue), with the 95% confidence bands. For KPR, we consider
$\|H-H_{obs}\|_F/\|H\|_F = 0$ (no $H$ error), $0.25$ (small $H$ error) and $0.5$ (large
$H$ error).
§ APPLICATION TO AN OBSERVATIONAL STUDY
We apply our kernel-penalized regression framework to data from 16S rRNA gene collected
in a study of premenopausal women <cit.>. This study investigated aspects
of gut microbial communities in stool samples from premenopausal women
using 454 pyrosequencing of the 16S rRNA gene. The abundances of 127 species were zero
for more than 90% of the subjects and were removed from our analysis. The data set we
consider consists of $p=128$ species sampled from $n=102$ women.
To make the measurements comparable between subjects, the species abundances were scaled
by the total number of sequences measured in each sample. This scaling produces
compositional data (the relative abundances in each sample sum to 1) which introduces
analytical complications. In particular, regression analysis using compositional
covariates must somehow account for their unit sum constraint <cit.>.
For this reason, we apply the CLR transformation to the relative abundance values and use
this transformed data $\tilde{X}$ as the matrix of predictors in the KPR model.
Additionally, using Aitchison's variation matrix <cit.>, $T$, we obtain
the covariance matrix, $C$, as described prior to eq. (<ref>). As $C$
provides more accurate information on the covariance among the true abundances than does
the empirical covariance matrix from relative abundances, $X$, or their CLR transform,
$\tilde{X}$, we use $C$ in place of $Q$ in (<ref>).
In this example, we examine the effect of using the CLR transformed data $\tilde{X}$ and
covariance $C$ as in (<ref>) and fit penalized regression models with the
goal of estimating $\tilde\beta_C$ in (<ref>) for the purpose of identifying
specific species that may be associated with percent fat in the cohort described above.
To this end, we apply a recently developed significance testing procedure to three
high-dimensional models in order to identify species exhibiting evidence of association
with subjects' adiposity. This significance test for graph-constrained estimation,
called Grace <cit.>, provides a means to assign significance to
estimates from penalized regression models that incorporate structure of the type
provided by $Q$ in (<ref>) (or $C$ in (<ref>)). The method
asymptotically controls the type-I error rate regardless of the choice of $Q$. The
special case with $Q=I$ provides a significance test for ordinary ridge regression. In
each application of the Grace test, tuning parameters are selected based on the smallest
squared test error using 10-fold cross validation. Following <cit.>, the
assumed sparsity parameter is set to be $\xi=0.05$.
The tuning parameter for the initial estimator is set to be $\lambda_{init}=4\hat\sigma_\epsilon\sqrt{3\log p/n}$, where
$\hat\sigma_\epsilon$ is the estimated standard deviation of the random error $\epsilon$,
using the scaled Lasso <cit.>. To assess significance for the sparse
models using lasso, we apply the recently proposed significance test for lasso
regressions based on low-dimensional projection estimator (LDPE) <cit.>, which provides an asymptotically valid test for lasso-penalized
regression estimates.
Species found to be associated with percent fat (in increasing order of p-values) at
different significant levels using: KPR with centered log-ratio transformed abundances (CLR) ;
ridge and lasso regression with centered log-ratio transformed abundances; and ridge and
lasso regression with untransformed relative abundances (rel%).
$p<0.01$ $p<0.005$ FDR $<0.1$
KPR $+$ CLR Bacteroides, Anaerovorax, Acidaminococcus, Blautia, Dethiosulfatibacter,Asaccharobacter, Turicibacter, Lebetimonas, Streptobacillus, Anoxynatronum
Bacteroides, Turicibacter, Acidaminococcus,Dethiosulfatibacter
Ridge $+$ CLR (none) (none) (none)
Ridge $+$ rel% Catonella, Dethiosulfatibacter (none) (none)
Lasso $+$ CLR Roseburia (none) (none)
Lasso $+$ rel% Dethiosulfatibacter, Micropruina Dethiosulfatibacter (none)
We report on five regression estimation methods for which the significance of regression
coefficients can be evaluated using existing high-dimensional testing methods. Two are
obtained using the relative abundances, $X$, with respect to: (i) an ordinary ridge
penalty and (ii) a lasso penalty. Three are obtained using the CLR transformed
abundances, $\tilde{X}$, with respect to: (iii) an ordinary ridge penalty, (iv) a lasso
penalty, and (v) the KPR estimate in (<ref>). None of these methods result
in any species associated with the outcome of percent fat when controlled for false
discovery rate (FDR) at 0.1 using the Benjamini-Yekutieli procedure <cit.>.
However, when using a cut-off of $p=0.01$, the KPR estimate (<ref>) results
in ten species. With a cut-off of $p=0.005$, KPR results in four species. Ordinary ridge
regressions using the CLR-transformed vectors find no associations at a cut-off of
$p=0.01$, whereas using the relative abundances, ridge finds two species at the $p=0.01$
cut-off and none at $p=0.005$. Lasso regression with the CLR-transformed vectors
identifies one specie at the $p=0.01$ cut-off and none at $p=0.005$ cut-off. When using
the relative abundances, lasso identifies two species as significant at the $p=0.01$
cut-off and one at the $p=0.005$ cutoff. See Table <ref> for the list of
identified species.
§ DISCUSSION
We have formulated a family of regression models that naturally extends the
dimension-reduced graphical explorations common to microbiome studies. In this sense, we
have simply re-focused the role of the eigen-structures used in ordination methods toward
exploiting this structure in penalized regression models. The large family of models
developed here provides a supervised statistical learning counterpart to the unsupervised
methods of principal coordinate analysis (PCoA).
A primary motivations for PCoA graphical displays is the ability to incorporate
biologically-inclined measures of (dis)similarity. The popular use of UniFrac, for
instance, is motivated by the desire to impose phylogeny into the analysis. These
dissimilarities have also been used for rigorous statistical testing in the context of
Anderson's nonparametric MANOVA <cit.> or the closely-related kernel
machine regression score test <cit.> for global
association of a multivariate predictor with an outcome. However, the use of UniFrac and
other non-Euclidean distances make it difficult to identify specific associations between
the microbial abundance profiles and a phenotype; indeed, none of these analyses proceed
to estimate the individual associations. In addition to ordination displays and global
tests for associations, a variety of machine learning approaches have emphasized on
models that predict a response. In contrast, we focus on estimating the
coefficient vector, which is a key aspect of any approach used to draw scientific
conclusions based on the association of microbial communities with an outcome or
An interesting feature of the proposed kernel-penalized regression framework is its
ability to sidestep some of the problems inherent in compositional data analysis.
Indeed, as emphasized by <cit.> regression analysis with compositional
covariates must somehow acknowledge their unit-sum constraint and spurious correlations.
Our approach, which differs somewhat from that of <cit.>, may also be viewed as
a penalized version of the low-dimensional linear model for compositions by
<cit.>, who use the isometric log-ratio (ILR) coordinates.
We note that ILR coordinates arise from the SVD of mean-centered CLR-transformed data,
$\tilde{X}$ (see <cit.>), which is also used in our model. However,
to estimate $\beta\in\bbR^p$, we used instead a regularization framework; our penalty in
Section <ref> arises from Aithison's total variation matrix whose singular
values are the total variances of ILR components. Moreover, the proposed framework also
allows us to use existing inference frameworks for high-dimensional regression, and in
particular the Grace test <cit.>, to assess the significance of
estimated regression coefficients.
|
1511.00340
|
Heinrich-Heine Universität Düsseldorf,
Universitätsstrasse 1,
D-40225 Düsseldorf, Germany
Heinrich-Heine Universität Düsseldorf,
Universitätsstrasse 1,
D-40225 Düsseldorf, Germany
Departamento de Física, Facultad de Ciencias,
Universidad Nacional Autónoma de México,
Ciudad Universitaria, México D.F. 04510, Mexico
We study the structure of quasiperiodic Lorentz gases, i.e., particles bouncing elastically off fixed obstacles arranged in quasiperiodic lattices. By employing a construction to embed such structures into a higher-dimensional periodic hyperlattice, we give a simple and efficient algorithm for numerical simulation of the dynamics of these systems.
This same construction shows that quasiperiodic Lorentz gases generically exhibit a regime with infinite horizon, that is, empty channels through which the particles move without colliding, when the obstacles are small enough; in this case, the distribution of free paths is asymptotically a power law with exponent -3, as expected from infinite-horizon periodic Lorentz gases.
For the critical radius at which these channels disappear, however, a new regime with locally-finite horizon arises, where this distribution has an unexpected exponent of -5,
previously observed only in a Lorentz gas formed by superposing three incommensurable periodic lattices in the Boltzmann-Grad limit where the radius of the obstacles tends to zero.
61.44.Br, 66.30.je, 05.60.Cd, 05.45.Pq
§ INTRODUCTION
The Lorentz gas (LG) model was proposed by
Lorentz <cit.> as a model of a completely ionized gas to study the conductivity of metals.
During the last decades, Lorentz gases have became popular among mathematicians, as key models in probability theory and dynamical systems <cit.>. At the same time, numerous works in physics use modified versions of the Lorentz gas to study dynamical and statistical properties of systems
with periodic <cit.> and random <cit.> distributions of obstacles in one <cit.>, two <cit.>, three <cit.>, and higher dimensions <cit.>.
It has been shown heuristically, numerically <cit.>, and rigorously <cit.> that $m$-dimensional Lorentz gases generically present weak super diffusion if there are channels of dimension $m-1$ (also called principal horizons <cit.>) in which particles can move freely for infinite time i.e., the mean squared displacement $\langle \Delta x(t)^2 \rangle = \langle x(t)^2 \rangle - \langle{x(t)}\rangle^2 \sim t \log(t)$, where $\langle X \rangle$ is the ensemble average of $X$. This situation is generic for periodic Lorentz gases <cit.> and it has been suggested for quasiperiodic Lorentz gases <cit.>; however, the proof of a generic situation in quasiperiodic systems is still missing.
On the other hand, in solid state physics the exploration of aperiodic structures, such as quasicrystals, has become increasingly important, since these systems exhibit a number of surprising effects, such as phasons <cit.>.
Quasicrystals are structures with long-range order, but no
translational symmetry <cit.>, first found experimentally by Shechtman in a metalic alloy with a diffraction pattern with 10-fold symmetry <cit.>.
Since their discovery, quasicrystals have been produced with many different materials <cit.>.
Quasiperiodic arrays have also been found in other contexts, e.g.,
liquid quasicrystals <cit.>, auto-assemblies of
nanoparticles <cit.>, virus colonies <cit.>, and photonic quasicrystals <cit.>.
Furthermore, quasicrystals have been observed in nature <cit.>. In simulations,
quasicrystalline structures have been found as cluster quasicrystals <cit.> or in hard tetrahedral systems <cit.>, where a first-order phase transition was observed, as confirmed in experiments <cit.>.
In spite of the relevance of quasiperiodic systems, quasiperiodic Lorentz gases have only recently been investigated <cit.>, having previously been
proposed as an open problem in the theory of dispersing billiards <cit.>.
In particular, the distribution of free paths has been studied in the Boltzmann-Grad limit <cit.>, where the radius of the obstacles tends to zero. In this limit, it has been proved that the distribution should be similar to the periodic case <cit.>, i.e., the probability density of free paths of length $\ell$ should decay asymptotically as a power law $\ell^{-\alpha}$, with exponent $\alpha=3$ <cit.>.
(Color online) Periodic and quasiperiodic arrays of scatters.
(a) A channel in a square lattice of obstacles.
(b) Different channels (thin red (strong gray) lines) in the Penrose Lorentz gas. At the bottom are shown trajectories in the direction of a horizontal channel.
(c) Combining two Penrose Lorentz gases, the second rotated by an angle $\frac{\pi}{20}$, blocks the channels. At the bottom some trajectories are shown in a direction between two of the channels of the two Penrose arrays.
(d) Quasiperiodic array that results from projecting a 3D lattice into a 2D subspace. At the bottom we show a channel and some trajectories in the direction of the channel.
Consider a periodic Lorentz gas and a particle that moves outside a channel, but close to the direction of this channel. In general, the length of the free path of this particle will be bounded by the lattice spacing; see fig. <ref>(a)). When channels are blocked in a periodic Lorentz gas, the distribution of free path lengths becomes bounded above. It is not a priori clear if the same will hold for quasiperiodic lattices; see fig. <ref>(b–d). In fact, we show in this paper that quasiperiodic Lorentz gases generically have a regime with so-called locally-finite horizon, where the width of the largest channel is zero, but there is no upper bound on the free path length <cit.>. This situation occurs only precisely at a critical radius $r = r_c$, defined such that channels are present when $r < r_c$, and absent when $r > r_c$, i.e. in the limit when the width of the widest channel tends to $0$.
As mentioned above, in an $m$-dimensional Lorentz gas, if there are $(m-1)$-dimensional channels present, it is expected that the distribution of free path lengths is a power law with exponent $\alpha=3$ <cit.>, and that the diffusion has a logarithmic correction to the mean square displacement.
One of the most studied examples of a system with locally-finite horizon is the random Lorentz gas, in which the obstacles are often distributed with positions following a Poisson distribution (if overlapping is allowed). In this case, the distribution of the length of the free paths seems to be
exponential, at least in the Boltzmann-Grad limit <cit.>. It is natural to ask what distribution is found in quasiperiodic Lorentz gases.
There are many other possible random distribution of obstacles, for example, when the scatterers are of finite size, they are often considered to be non-overlapping and hence not Poisson. We are not aware of any numerical investigations of the free path distribution for this case at a high density, but in Ref. <cit.> this has been calculated for both overlapping and non-overlapping obstacles in the low density limit. In both cases the distribution of free path length has an exponential tail.
This paper is organized as follows: In Section <ref>, we define a quasiperiodic Lorentz gas and we summarize the procedure to embed a quasiperiodic potential into a
higher-dimensional periodic potential. In Section <ref> we define finite, infinite, and
locally-finite horizons in Lorentz gases and we prove the generic existence of channels for quasiperiodic Lorentz gases. We then show that quasiperiodic Lorentz gases have a locally-finite horizon for $r = r_c$. In Section <ref>, we measure numerically the free path length distribution, obtaining, for the locally-finite regime, a power law with an unexpected exponent $\alpha=5$, which we confirm with heuristic arguments. We finish with conclusions in Section <ref>.
§ MODEL AND METHODS
§.§ Lorentz gas
A Lorentz gas (LG) consists of an ensemble of non-interacting point particles moving in an array of fixed obstacles, usually spheres, placed at the vertices of a lattice in $\RR^{m}$. Each particle undergoes free motion until it collides with a scatter, and is then reflected elastically. If the lattice is quasiperiodic, then the Lorentz gas is also called quasiperiodic. There are several methods to produce quasiperiodic arrays, not all of which producing the same tiling <cit.>; one of the most popular is the projection method <cit.>. Reversing this method, it is possible to simulate and analyze the dynamics of a quasiperiodic LG as a periodic billiard, as two of the current authors previously showed <cit.>. Throughout this paper, we take the interaction potential to be that of the hard sphere.
One of the main interests in studying a Lorentz gas consists of measuring its diffusivity, i.e., how fast particles disperse through the system, characterized by the variance, or mean squared displacement, $\langle \Delta x(t)^2 \rangle$, of an initial cloud of particles as a function of time, $t$, where the ensemble average is defined by averaging with respect to the uniform measure over the unit cell in the higher-dimensional periodic system.
It has been shown, numerically <cit.> and analytically <cit.>, that the periodic version of these models can exhibit weak super-diffusion:
\begin{equation}
\langle {\Delta x^2}(t) \rangle \sim D t \log(t/\tau),
\label{eq: superdifusion}
\end{equation}
where $D$, the super-diffusion coefficient,
is a constant (for a given system) that depends on the geometry of the lattice on which the obstacles are positioned and the obstacle radius, and $\tau$ is the average time that a particle stays in a cell of a given size.
This occurs in the presence of the highest possible dimension of channels in the structure, i.e., subspaces of the system in $\mathbb{R}^m$, such that the dimension of the set of infinite directions of the channel is $m-1$, and which are devoid of obstacles. This happens generically in periodic LGs if the obstacles are small enough; however, it does not happen in random LGs.
Simulations of the mean square displacement in quasiperiodic LGs close to the locally-finite horizon regime suggest that such systems have normal diffusion <cit.>. Nonetheless, the simulation data may not be sufficient; in particular, calculating logarithmic corrections numerically is subtle <cit.>.
As an alternative, we examine the distribution of free path lengths obtained with a locally-finite horizon. It is expected that a power law with exponent $\alpha=3$ for this distribution corresponds to weak super-diffusion (logarithmic correction), while a smaller exponent instead corresponds to normal diffusion <cit.>.
§.§ Periodization of quasiperiodic potentials
We summarize the method for constructing finite-range quasiperiodic potentials introduced in <cit.>. The main idea is to produce an $n$-dimensional periodic potential, with $n > m$, based on the projection method <cit.>, with non-interacting classical particles moving inside. The initial conditions are constrained such that the dynamics of particles in the periodic potential will reproduce the dynamics in the $m$-dimensional quasiperiodic potential: the initial velocities must lie in the $m$-dimensional physical subspace $E$ onto which points of the higher-dimensional periodic lattice are projected. $E \subset \RR^n$ is a totally irrational subspace, i.e., such that $E \cap \ZZ^n = \{0\}$.
(a) (b)
(Color online) (a) Projection method used to produce the Fibonacci chain. $E$ is the physical space, where the particles are projected to produce the Fibonacci chain. (b) Periodization of the Fibonacci chain; particles can move in only two directions.
Fig. <ref>(a) shows, as an example of the projection method, the quasiperiodic Fibonacci chain, constructed by projecting a 2-dimensional periodic lattice onto a 1-dimensional totally irrational line $E$, i.e., a line with irrational slope. Figure <ref>(b) shows a 2-dimensional periodic potential, which represents the Fibonacci chain if the particles are constrained to move only parallel to $E$.
In general, we can construct periodic potentials in higher dimension that are equivalent to a two- or three-dimensional quasiperiodic array in $E$. For example, the Penrose tiling can be embedded in a 5-dimensional periodic potential or the icosahedral array can be embedded in a 6-dimensional periodic potential <cit.>. To do so, we proceed as follows:
* Take a unit hypercube $C$ of dimension $n$, corresponding to a Voronoi cell of the periodic lattice.
* Translate the space $E$ so that it passes through the center $c$ of the hypercube $C$, which we take as the origin.
* Project $C$ onto $E_\perp$, the subspace orthogonal to $E$ (of dimension $n-m$) that also passes through $c$. Call the resulting projected object $W$.
* Apply periodic boundary conditions to $W$, by translating those parts of $W$ that lie outside $C$, to produce a “periodized” object $K$ inside $C$.
* Apply the $m$-dimensional potential (for example, the hard-sphere potential) in the direction of the hyperplane $E$, using $K$ as the axis of the potential. In the orthogonal direction, the potential is $0$.
We call this procedure to embed a quasiperiodic system into a higher-dimensional periodic one the periodization of the system.
§ HORIZONS IN QUASIPERIODIC LORENTZ GASES
In this section, we define finite, infinite and locally-finite horizons, and we prove the generic existence of channels in quasiperiodic Lorentz gases, as well as the locally-finite horizon regime.
§.§ Generic existence of channels in quasiperiodic Lorentz gases
The construction described in the previous section was originally designed to allow efficient numerical simulation of quasiperiodic Lorentz gases. Nonetheless, it also provides a powerful tool to analyze the geometric structure of these systems
<cit.>: here we use it to prove the generic existence of channels in quasiperiodic LGs; specific cases were studied in <cit.>.
The periodized model is a periodic LG in a higher dimension, in which the obstacles are no longer spheres, but are now $n$-dimensional cylinders (together with the constraint mentioned above on the initial velocities of the particles).
If the radius of the obstacles is small enough, then we expect that there will be channels of dimension $n-1$ in the $n$-dimensional periodic LG.
We need only prove that these channels are not all parallel to the plane $E$;
if so, then there are channels of dimension $m-1$ in the quasiperiodic LG, since the intersection of a subspace of dimension $n-1$ with a subspace of dimension $m \leq n$ is generically a subspace of dimension $m-1$; for example, the intersection of two planes in 3D is generically a line.
To show the existence of these
$(n-1)$-dimensional channels that are not parallel to $E$, note that if a face of $C$, which is an $(n-1)$-dimensional hyperplane, does not touch the obstacle, then there will be a channel with this property, since the plane $E$ is totally irrational, so that it cannot be parallel to any face of $C$. Thus, the intersection of the plane $E$ (of dimension $m$) and this face will produce a subspace of $E$ with dimension $m-1$, without any obstacle; that is exactly the definition of channel. This happens generically since $K$ has the same dimension as the orthogonal space to $E$, namely $n-m < n$, and its length is bounded by the length of the hypercube such that the intersection between the hypercube and $W$ intersects the same number of faces as $K$; see
Figs. <ref>(a) and <ref>(1). In this case, it is not possible that $K$ intersects all the faces; indeed, there are exactly $2m$ faces that it does not intersect, $m$ of them orthogonal, giving exactly $m$ channels if the obstacle is small enough.
Therefore, we expect weak super-diffusion if the obstacles are small enough, which agrees with numerical results founded in <cit.>. However, the numerical results are not completely convincing, especially when the obstacles are very small. The logarithmic correction to the mean square displacement is difficult to observe numerically even in periodic systems <cit.>. This problem persists in the quasiperiodic case, but is even more challenging, since there is an additional effect that results in slow convergence to the logarithmic correction, as we will see in the following.
(Color online) Trajectory of a particle constrained to move in plane parallel to one of the faces of the cube, in the case that the plane $E$ is totally irrational. The trajectory of the particle inside a channel represented in (a) fills densely the face of the cube, represented in (1) as a gray background with part of the trajectory highlighted. If the channels become blocked (b), the trajectory can still be dense in the face of the cube (3) or becomes finite if the particles moves in a plane parallel to the face of the cube (2), since it collides with the part of the ellipse (shown in black) that is the intersection of the cylinder with this plane. The thick (red) arrows on figure (b) show the intersection between the planes. The bottom arrow shows only a point while the other shows a larger area of intersection.
§.§ Locally-finite horizon
Periodic Lorentz gases can be classified into systems with finite or infinite horizon, according to whether the free path length is bounded above or can be infinite.
In contrast, random Lorentz gases are typically in the locally-finite regime, in which the probability to have unbounded free paths is $0$, but the length of free paths may be arbitrary large <cit.>. Furthermore, if we fix an arbitrary direction, the free path in that direction is still bounded, with probability $1$. Thus, we will say that a system has locally-finite horizon if the free path length is not bounded above, but for any fixed direction the probability of choosing a point with an unbounded trajectory in that direction is $0$. We will show here that quasiperiodic Lorentz gases can exhibit all three of these regimes: finite horizon, locally-finite horizon and infinite horizon, depending on the size of the obstacles.
Suppose that the plane $E$ of dimension $m$ in the periodization of a quasiperiodic array is totally rational, i.e., $E \cap \ZZ^n$ is a lattice of rank $m$, rather than totally irrational. In this case, the periodized system represents a periodic Lorentz gas, rather than a quasiperiodic one. Note that a trajectory in this system will not fill densely the available volume. For example, consider a particle with initial velocity and position in the intersection of the plane $E$ and a plane $F$ that contains one of the faces of the cube that is not touched by the obstacle if the obstacle is small enough, i.e., a channel. Then, as shown in fig. <ref>, the whole trajectory will be represented by a finite number of segments. If we increase the radius of the obstacle, at some moment the obstacle will intersect this face. The intersection of the obstacle and the plane $F$ will produce an $(n-1)$-dimensional object. If we continue growing the obstacle, this $(n-1)$-dimensional object will continue
growing until it intersects the trajectory. At this point, the channel becomes blocked, and the free path becomes finite and bounded. That is, the horizon changes from an infinite to a finite horizon, without passing through a locally-finite horizon.
The situation is similar if, instead of growing the obstacle, we continuously displace the plane $F$ to the plane $F_{\lambda}$, keeping it parallel to the face of the cube, where
$\lambda$ denotes the distance between the plane and the face of the cube. We can identify the region where particles with velocity $v$ in the direction of the intersection of the planes $F_{\lambda}$ and the plane $E$ can move freely by seeing where the intersection of the plane $F_{\lambda}$ and the obstacle does not intersect the trajectory of the particles constrained to move in this plane. This region is the channel with direction $v$.
(Color online) Trajectory of a particle constrained to move in a plane parallel to one of the faces of the cube, in the case that the plane $E$ is totally rational: (a) in the presence of a channel of width $\Delta$; (b) when the channels are blocked. Below are shown trajectories of particles moving in a face of the cube. (1) A particle in a channel; (2) and (3) show two cases where the particle touches tangentially the obstacle; (4) a particle that collides with an obstacle. The cases (3) and (4) are contained in the plane marked in a dark colour in figure (b), while figure (4) is contained in one of the faces of the cube.
Now, consider the quasiperiodic Lorentz gas, i.e., when the plane $E$ is totally irrational. Following the same procedure, we produce first a trajectory that fills densely the face of the cube. If the intersection between the plane $F$ and the obstacle is non-empty, the trajectory will be finite; see Fig. <ref>. This time, however, the length of the trajectory depends on the size of the $(n-1)$-dimensional object produced by this intersection. Consider the limiting case in which the obstacle is of the critical size such that it exactly touches the plane $F$, so that the intersection between the plane $F$ and the obstacle consists of exactly one point on the plane $F$. Then the channel associated to this face has measure $0$, and the length of the free paths constrained to move in the planes $F_\lambda$ depend on $\lambda$, with no upper bound. Thus, in the case when all channels become blocked and at least one of them is in this limiting case, the system has a locally-finite horizon.
This can explain why it is more difficult to see numerically the behavior of the mean square displacement of the form $D t \log(t/\tau)$ in quasiperiodic Lorentz gases than in the periodic case. The coefficient $D$ depends on the width of the channel <cit.>, but this width is, in some sense, a decreasing function of time $t$ for quasiperiodic systems: for any time $t$, there is a volume of the billiard (width of the channel), that we call an “effective channel”, in which particles with velocities in the direction of a channel can have free paths during a time of at least order $t$, but where there is no real channel. Thus, the convergence to $D t \log(t/\tau)$ is even slower than in the periodic case, because of the variation in width of these effective channels.
§ DISTRIBUTION OF FREE PATH LENGTHS
We performed numerical simulations of the distribution of free path lengths for a 2D quasiperiodic Lorentz gas obtained using the construction in
Section <ref>. These were performed for different radii, including the limit case $r=r_c \sim 0.309$, i.e., a Lorentz gas with locally-finite horizon, and radii $r=0.03$ and $r=0.36$. We used $10^7$ initial conditions, distributed homogeneously in a unit cell in the periodized system and velocities
with unit speed distributed with uniform directions parallel
to the subspace $E$. The orthogonal space $\Eperp$ was taken aligned along the unit vector $(\frac{1}{\phi +2},\frac{\phi}{\phi +2}, \frac{\phi}{\sqrt{\phi +2}})$, where $\phi= \frac{1+\sqrt{5}}{2}$ is the golden ratio.
We have also performed simulations on Sinai Billiard, where we measure the maximum free path length for a fixed direction (slope), as a function of the radius of the obstacle. The simulations were performed with $10^6$ trajectories, with the following slopes: $\frac{1}{\phi \sqrt{\phi+1}}$ (the slope in one of the faces of the unit cell used in the simulations), $\pi$, $\sqrt{2}$, the Liouville constant $\sim 0.110001000000000000000001$ and 10000 random slopes.
§.§ Simulation results
The free path distributions obtained from simulations are shown in Fig. <ref>. The distribution for $r = r_c$ is well-fitted by a power law with exponent $\alpha=5$, which corresponds to normal diffusion for this case. However, the resolution is not enough good to be conclusive and the calculation is computationally demanding. Nevertheless, we now give arguments that confirm that the distribution should be a power law with this exponent.
(a) (b)
(Color online) (a) Free path length distribution for radii $r=0.03$ (red (upper) line); $r=0.30$ (green (light gray)); and $r=0.36$ (blue (bottom)). (b) The corresponding cumulative distribution functions with the same color (gray scale) code.
§.§ Heuristic argument
Consider a simplification of the periodic case, where a 1D channel corresponds to a system with two parallel lines parallel to the $x$-axis; see Figure <ref>(a). In order to calculate the distribution of the length of the first free path, we need to count all the possible initial directions and positions with respect to the channel. If the channel has a width $\Delta$, due to the symmetry of the system, for the position it is enough to consider all possible initial conditions $(x,y)$ with $x$ an arbitrary constant, and $y \in (0,\Delta/2]$. By symmetry, it is enough to consider angles from $0$ to $\pi/2$ with respect to the $x$-axis; indeed, we are interested in angles close to 0, since our goal is to obtain the distribution for long paths. In this case, as is shown in
Fig. <ref>(a), the length of the free path is $l=\epsilon / \sin(\theta)$, so that $\epsilon= l \cdot \sin(\theta)$. Thus, for angles close to $0$, $\epsilon \sim l \cdot \theta$, the probability $p(l)$ to have a trajectory with length $l$ is
\begin{equation}
p(l) \sim \frac{1}{\theta_m \epsilon_m}\int_{0}^{\theta_m} \int_{0}^{\epsilon_m} \delta (l-\frac{\epsilon}{\theta})d\epsilon d\theta,
\end{equation}
where $\theta_m$ and $\epsilon_m$ are finite constants. Making the change of variables $\xi=\epsilon/\theta$, we obtain
\begin{equation}
p(l) \sim \frac{1}{\theta_m \epsilon_m}\int_{0}^{\theta_m} \int_{0}^{\frac{\epsilon_m}{\theta}} \delta (l-\xi)\theta d\xi d\theta.
\end{equation}
Using the fact that $\frac{d H(x)}{dx}=\delta(x)$, where $H(x)$ is the Heaviside step function and $\delta(x)$ is the Dirac delta, we obtain
\begin{equation}
\int_{0}^{\theta_m} \int_{0}^{\frac{\epsilon_m}{\theta}} \delta (l-\xi)\theta d\xi d\theta=\int_{0}^{\theta_m} H(\frac{\epsilon_m}{\theta}-l)\theta d\theta.
\end{equation}
Since we assume $\epsilon/l \sim \theta$ and $\epsilon < \epsilon_m$, we have $\epsilon_m / l > \theta$, giving the approximation
\begin{equation}
p(l) \sim \int_{0}^{\frac{\epsilon_m} {l}} H(\frac{\epsilon_m}{l}-\theta)\theta d\theta= \int_{0}^{\frac{\epsilon_m} {l}} \theta d\theta \sim \frac{1}{l^2}.
\end{equation}
(Color online)(a) Sketch of the geometry to calculate the free path length distribution inside a channel (as in the periodic Lorentz gases). $\epsilon$ measures the distance of the particle from the edge of the channel;
$\Delta$ is the width of the channel. (b) and (c) show how is defined $s$. (d) Another view of the periodized model of figure (b), showing the trajectory of two particles, one with velocity parallel to the direction of the channel (in black), an the other slightly deviated (in red (gray)). This trajectories are also shown in figure (e), where is sketched the geometry to calculate the free path length distribution in the locally finite horizon regime for a 2D quasiperiodic Lorentz gas.
In this procedure, we have not considered free paths outside the channel, since in the periodic case the free paths outside a channel are bounded, so the only important contribution to the free path length distribution for long paths is due to the particles inside a channel. However, for the quasiperiodic LG, we have seen that there may be an important contribution from particles outside the channels.
To analyze this contribution, consider the limiting case, in which the obstacles have the critical radius $r = r_c$. In this case, the only important contribution to the free paths are outside the channels, and changes in position become relevant: for each position, there is a different maximum length of free paths. Placing the particle at a perpendicular distance $\epsilon$ from the channel is equivalent to moving the plane $F$ to the plane $F_\epsilon$. Thus, the intersection of the obstacle with the plane will increase from a point to part of an ellipse; see Figs. <ref>(2) and <ref>(b,c). This ellipse has an effective width of $s=a \epsilon$, with $a$ a constant. Thus, the question becomes: how does the maximum free path length of a particle in a unit square with periodic boundary conditions depend on the radius of the obstacle?
Figure <ref> shows the results of numerical simulations, performed using an efficient algorithm that will be published elsewhere <cit.>, that are well approximated by the function $L(s)= C(s)/s \sim 1/\epsilon$, where $C(s)$ is a bounded function. In order to present these results more clearly, we have included only three slopes. However, we have performed simulations for 10004 different slopes, including 10000 random slopes, $\frac{1}{\phi \sqrt{\phi+1}}$, $\pi$, $\sqrt{2}$, and the Liouville constant. In any case, $C(s)$ seems to bounded below by the constant $1/2$, while the upper bound depends on the slope; the largest that we have found is for slope $\pi$, and it is around $37$. Thus, we conjecture that $L(s)\sim C(s)/s$, where $C(s)$ is a function bounded below by $1/2$ and bounded above with an upper bound depending on the slope.
On the other hand, assuming small angles, close to the channel and using trigonometry we obtain that $l \cos(\theta) \sim l (1-\theta ^2) \sim a/(\epsilon+l\sin(\theta)) \sim a/(\epsilon+l \theta)$; see fig. <ref>(d,e). Taking approximations of $\sin(\theta)$ and $\cos(\theta)$ up to second order, we can approximate $l \sim c/\epsilon_{eff}=c/(\epsilon+l \theta)$, with $c$ a constant.
Using this approximation, and the same procedure as for the periodic case, we obtain
\begin{equation}
p(l) \sim \frac{1}{\theta_m \epsilon_m} \int_{0}^{\theta_m} \int_{0}^{\epsilon_m} \delta(l-\frac{c}{\epsilon+l \theta}) d\theta d\epsilon.
\end{equation}
Now we substitute $\xi=c/(\epsilon+l\theta)$, so that $\epsilon=c/\xi-l\theta$, which gives
\begin{equation}
p(l) \sim \frac{1}{\theta_m \epsilon_m} \int_{0}^{\theta_m} H(l-\frac{c}{\epsilon_m+l \theta})H(\frac{c}{l \theta}-l) \frac{1}{l^2} d \theta.
\end{equation}
Assuming $l \gg 1$, we have $1/l-\epsilon_m<0$ and $\theta>1/l (1/l -\epsilon_m)$ or $l-\frac{c}{\epsilon_m+l\theta}>0$. On the other hand, $c/(l \theta)-l>0$ if and only if $\theta<c/l^2$, so the integral becomes
\begin{equation}
p(l) \sim \int_{0}^{\frac{c}{l^2}}\frac{1}{l^2} d l=\frac{1}{l^4}.
\end{equation}
In both cases (in the channel and outside the channel), we have computed the distribution of the free path lengths for the first collisions. However, since the distribution of the angles is different for the second collision, i.e., the free paths between two obstacles (rather than starting from a random initial condition in a cell), we still need to calculate the distribution for this case. To do so, assume a distribution of free paths equal to $\rho(l)$, a continuous function of $l$. Since the system is ergodic (this has been proved for periodic systems <cit.>, and we assume that it also holds for quasiperiodic ones), we run a simulation with $n$ particles, and stop it at an arbitrary time $t$. The distribution of angles and positions should be homogeneous. So, the free path length distribution at this point should be the same as the first free path length distribution. We ask for this distribution as a function of $\rho(l)$. This problem is equivalent to have a $\rho(l)$ distribution of segments
of length $l$ on a line, and to choose a point randomly on this line. The probability density that the point particle has a free path of length $l$ if it always moves in a positive direction is $p(l)=\rho(l) \cdot l/2$, so
\begin{equation}
\rho(l)=2\frac{p(l)}{l}.
\end{equation}
Thus, our calculations give $\rho(l) \sim l^{-3}$ and $\rho \sim l^{-5}$ for the distribution of free path lengths inside the channel and outside the channel, respectively. This implies that the distribution should be dominated by $\rho(l) \sim l^{-3}$ when there is an infinite channel, but if the width of the channel $\Delta$ tends to $0$, then we expect the distribution to be $\rho(l) \sim l^{-5}$. Both results are in good agreement with our numerical simulations, shown in Fig. <ref>.
(Color online) Measurement of the maximum free path length as a function of the radius of an obstacle for fixed direction (slope) in a 2D Lorentz gas with the square arrangement. The employed slopes are: $\frac{1}{\phi \sqrt{\phi+1}}$, $\sqrt{2}$, $\pi$ and the Liouville constant $\sim [0.110001000000000000000001]$
However, this situation does not always occur, because the window to produce the quasiperiodic arrangement is not always totally irrational, even if the space $E$ is. For example, for the Penrose Lorentz gas, this situation does not happen, since the window in one direction forms a rational angle with the channels. In this case, the behavior is similar to that in periodic Lorentz gases. An example of a real quasicrystal arrangement which exhibits this behavior is presented in Ref. <cit.>.
Simulations of this system are too slow to study the free path length distribution, but we have computed the directions of the channels using the method of Ref. <cit.>, and we find that these channels are totally irrational with respect to the window.
§ CONCLUSIONS
We have studied the
properties of Lorentz gases where obstacles are arranged according to quasicrystalline symmetry,
providing a proof of the generic existence of channels for quasiperiodic Lorentz gases in $n$ dimensions for small enough obstacles. This was conjectured in Ref. <cit.>, but until now the proof was missing. Furthermore, we have given a method to identify these channels and measure their volumes, which is closely related to the super-diffusion coefficient.
We have also proved the existence of a locally-finite horizon regime for quasiperiodic Lorentz gases. This regime occurs at a critical radius $r_c >0$ of obstacles, when the volume of the channels tends to $0$. With this, we have shown that quasiperiodic arrays of obstacles can exhibit three different regimes, finite, infinite and
locally finite, thus exhibiting richer behaviour than that found in periodic systems.
In addition, we have performed numerical simulations and heuristic calculations showing that the free path length distribution in the
locally-finite regime for a 2D quasiperiodic array is asymptotically a power law with exponent $-5$, instead of $-3$ as in the infinite horizon regime. This allows us to deduce that diffusion in the locally-finite regime is normal, in agreement with the numerical results of
Ref. <cit.>.
We remark that a similar situation has been found for the Boltzmann-Grad limit for two overlapping periodic lattices of obstacles with non-commensurate directions <cit.>.
These results suggest that surprising behaviors can be found in quasiperiodic systems, where not only the mean square displacement, but also the mean free path length is relevant, such as in the calculation of band gaps in photonic crystals <cit.> recently studied with Lorentz gas models <cit.>, or thermal and electrical conductivity <cit.> of quasicrystals.
§ ACKNOWLEDGEMENTS
We thank Domokos Szász and Jürgen Horbach for useful discussions about the distributions of free path lengths. ASK and MS received support from the DFG withing the Emmy Noether program (grant Schm 2657/2). DPS received financial support from CONACYT Grant CB-101246 and DGAPA-UNAM PAPIIT Grants IN116212 and IN117214.
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1511.00593
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Релятивіська дзиґа]РЕЛЯТИВІСЬКА ДЗИҐА В ДИНАМІЦІ ОСТРОГРАДСЬКОГО
Р. Мацюк]
Роман МАЦЮК
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1511.00006
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1Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112; dylan.gregersen@utah.edu, aseth@astro.utah.edu
2Department of Astronomy, Box 351580, University of Washington, Seattle, WA 98195
3McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
4Osservatorio Astronomico di Padova - INAF, Vicolo dell'Osservatori 5, I-35122 Padova, Italy
5Minnesota Institute for Astrophysics, University of Minnesota, Minneapolis, MN 55455, USA
6Department of Astronomy, University of Michigan, 500 Church Street, Ann Arbor, MI 48109, USA
7Raytheon, 1151 E. Hermans Road, Tucson, AZ 85706
8MPIA, Koenigstuhl 17, 69117 Heidelberg, Germany
9UCO/Lick Observatory, University of California at Santa Cruz, 1156 High Street, Santa Cruz, CA 95064
10Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218
11Space Telescope Science Institute, Baltimore, MD 21218
12MPA, Garching, Germany
13NOAO, Tucson, AZ 85719, USA
We present a study of spatial variations in the metallicity of old red giant branch stars in the Andromeda galaxy. Photometric metallicity estimates are derived by interpolating isochrones for over seven million stars in the Panchromatic Hubble Andromeda Treasury (PHAT) survey. This is the first systematic study of stellar metallicities over the inner 20 kpc of Andromeda's galactic disk. We see a clear metallicity gradient of $-0.020\pm0.004$ dex/kpc from $\sim4-20$ kpc assuming a constant RGB age. This metallicity gradient is derived after correcting for the effects of photometric bias and completeness and dust extinction and is quite insensitive to these effects. The unknown age gradient in M31's disk creates the dominant systematic uncertainty in our derived metallicity gradient. However, spectroscopic analyses of galaxies similar to M31 show that they typically have small age gradients that make this systematic error comparable to the 1$\sigma$ error on our metallicity gradient measurement. In addition to the metallicity gradient, we observe an asymmetric local enhancement in metallicity at radii of 3-6 kpc that appears to be associated with Andromeda's elongated bar. This same region also appears to have an enhanced stellar density and velocity dispersion.
§ INTRODUCTION
Spiral disks form as a result of complex interactions of star formation, accretion, and gas. These processes are recorded in a galaxy's stellar component. The stellar metallicity, in particular, encodes information on a galaxy's overall chemical enrichment resulting from its history of star formation, gas accretion, and gas outflows.
Spiral galaxies are commonly thought to form “inside out” with the inner
regions forming earliest <cit.>. This effect has been
observed by measuring a decreasing age with increasing radius in some nearby
galaxies <cit.>. A metallicity gradient
is a natural consequence of inside out growth. Observationally, present-day
metallicity gradients can be measured using HII regions or the atmospheres of young stars. Studies of the present-day metallicity gradients in the Milky Way and other nearby spirals do show lower metallicities at larger radii <cit.>.
To measure metallicity gradients not at the present-day, but in the
past, one must analyze stellar metallicities as a function of age. The
simultaneous measurement of age and metallicity is non-trivial, but
can be done using either the observations of resolved stars or using
integrated spectra. Recently, the CALIFA survey <cit.> used integral field
spectroscopic data to study both the present-day and the mean stellar
metallicity gradients in a large sample of galaxies <cit.>. The mean mass-weighted stellar metallicity gradient found among the CALIFA galaxies is relatively shallow, $-0.05$ dex/r$_e$, where r$_e$ is the disk effective radius <cit.>. The galaxies in this CALIFA subsample are very similar to M31 in Hubble type and luminosity. Similar results based on photometric modeling were also found by <cit.> for spiral galaxies in the Virgo cluster. In both studies, there is no clear dependence on Hubble type for spiral galaxies. In the Milky Way, the best gradients utilize measurements of 1000s of individual stars from large surveys (e.g. SDSS) and found a gradient of $\sim$-0.06 dex/kpc <cit.>.
In addition to the smooth gradients in metallicity, distinct metallicity structures are visible within galaxies. Metallicity substructure in the halos of galaxies have been linked to galactic merger events <cit.>. Within the disks of galaxies, structures with distinct metallicity signatures can arise due to bars, bulges and other dynamical structures <cit.>.
The ability to resolve individual stars all at the same distance makes Andromeda a unique object with which to study massive galaxy disks. Distance is the main source of uncertainty within the Milky Way and large-scale morphology observations are largely restricted to the solar neighborhood. In contrast, stars can be resolved with ease at a wide range of radii in Andromeda. The Panchromatic Hubble Andromeda Treasury <cit.> survey recently observed 117 million stars over a third of Andromeda, providing a unique database for studying Andromeda's stellar populations <cit.>.
In this paper, we present a detailed analysis of the metallicities of Andromeda's red giant branch (RGB) stars using PHAT photometry. Previous work on the RGB metallicities in Andromeda have focused on the halo with limited information in the disk of the galaxy <cit.>.
For this study, we assume the following parameters: The center of M31 is at RA=10.68458and Dec=1.2692<cit.>; M31 position angle and inclination angle are respectively 38and 74<cit.>; the distance modulus is 24.45$\pm$0.05 mag or 776$\pm$18 kpc (mean distance, see <cit.> for discussion). Due to the small uncertainty in M31's distance, and the primary dependence of the RGB metallicity on color and not luminosity, the distance uncertainty should have little effect on the analysis we present in this paper. The foreground extinction is $A_V = $ 0.17 mag <cit.>; M31's effective radius is 8.9$\pm$0.8 kpc <cit.>.
The paper is structured as follows: In Section <ref>, we explain our data sources, including the RGB star photometry and dust extinction measurements. In Section <ref>, we discuss our method for estimating metallicities from isochrone models and the uncertainties caused by photometric crowding and dust extinction. We present and discuss our results on Andromeda's metallicity gradient and a metal rich structure associated with the bar in Section <ref>.
§ DATA
§.§ RGB Photometry
The PHAT survey observed approximately 117 million stars with the Hubble Space Telescope (HST) in ultraviolet (F275W and F336W), optical (F475W and F814W), and infrared (F110W and F160W) filters using the WFC3/UVIS, ACS/WFC and WFC3/IR HST camera, respectively. The camera pointing angles divided the survey into 23 bricks each consisting of 18 overlapping IR-fields defined from the WFC3/IR camera footprint (for details see ).
Color-magnitude diagram (CMD) of the PHAT optical photometry. Data points and gray scale density contours show our data after photometric quality cuts. The plot bounds contain a total of $\sim$3.47 stars. The red clump is visible at $F475W_o-F814W_o \approx 1.8$ and $F814W_o \approx$ 24.3 mag. The black dashed line indicates the RGB selection box used for this paper (see Section <ref>). Color indicates the metallicity. The three color lines plot Padova PARSEC1.2s isochrones at $[M/H]=$-2.0, 0.0, and 0.6 dex (in violet, orange, and yellow, respectively) at a fiducial age of 4 Gyr. The black star marks an example from our data sample at a deprojected radius of 6.1 kpc whose interpolated metallicity estimate is 0.033 dex (see Section <ref>). The black arrow indicates 1 mag of dust extinction in $V$ Band. Extinction causes stars metallicities to be overestimated as discussed at length in Section <ref>.
Two versions of the photometry are publicly available. First, there is camera-by-camera photometry derived separately on the pairs of filters in each camera producing a UV, optical, and infrared catalog that are then merged together to form a single six-filter catalog [Camera-by-camera photometry can be downloaded at http://archive.stsci.edu/prepds/phat/datalist.html] <cit.>. Second, the “six-filter photometry” is derived across all filters simultaneously[Six filter photometry can be downloaded at http://archive.stsci.edu/missions/hlsp/phat/] <cit.>. In both versions, each star is parameterized by a position and magnitude, magnitude error, signal-to-noise ratio, crowding, and sharpness for each filter. Sharpness parameterizes the shape of each source relative to the point spread function; and, crowding parameterizes the brightness of nearby sources.
In this paper, we use the optical camera-by-camera photometry to derive RGB metallicity estimates. The primary reason for using the camera-by-camera photometry over the six filter photometry is the availability of extensive artificial star tests (also publicly available from MAST archive). These artificial star tests allow us to understand the photometric completeness and bias (see Section <ref>). The optical color of RGB stars also is more sensitive to the metallicity relative to the IR photometry, while the UV photometry suffers from significant incompleteness for even the brightest RGB stars. The smaller effect of dust extinction on the IR data makes it logical to include this data in our analysis, however, artificial stars are only available for the two-camera photometry in the IR, and this data is significantly shallower and crowding limited than the optical data <cit.>. The six filter photometry is significantly deeper in the IR due to the ability to use the optical data to determine the positions of stars, but it is not computationally feasible to run artificial star tests over the full PHAT footprint for the six band photometry <cit.>.
We create an optical only ($F475W$ and $F814W$) photometric catalog with uniform coverage from the individual ACS/WFC field images. This involves two primary steps: defining non-overlapping spatial regions, and filling in the chip gaps of each individual field. The boundary of each field is first defined using a grid of non-overlapping regions that roughly follow the individual WFC3/IR field footprints. We additionally crop Bricks 9 and 12, which overlap with neighboring bricks, because of orientation differences described in Section 3.5 of <cit.>. The PHAT tiling scheme allows us to use neighboring ACS/WFC fields to fill in the chip gap of each ACS/WFC field. Thus, our final catalog has uniform spatial coverage over the full PHAT survey region but does not make use of multiple overlapping observations. This same catalog creation process is also used to deal with chip gaps in the the six-filter photometry as described in <cit.>.
We apply quality cuts on our photometry. The cuts use signal-to-noise, sharpness, and crowding in the F475W and F814W filters for each star. We use the same parameterization which defines “gst” (good star) cuts from <cit.> by applying all of the following equations:
\begin{array}{rcl}
F475W\_SNR & \geq & 4.0 \\
F814W\_SNR & \geq & 4.0 \\
(F475W\_SHARP + F814W\_SHARP)^2 & \leq & 0.075 \\
F475W\_CROWD + F814W\_CROWD & \leq & 1.0\\
\end{array}
The variables correspond to the names given in the data files. For example, $F475W\_SNR$, $F475W\_SHARP$, and $F475W\_CROWD$ respectively correspond to the signal-to-noise, sharpness, and crowding parameters for the $F475W$ filter.
We correct the magnitudes in $F475W$ and $F814W$ for foreground extinction using $A_V = $0.17 mag <cit.>. This extinction in $A_V$ is converted to $F475W$ and $F814W$ extinctions by multiplying with $A_{\lambda}/A_V$ equal to 1.19119 and 0.60593, respectively[$A_{\lambda}/A_V$ taken from http://stev.oapd.inaf.it/cgi-bin/cmd]. The corrected photometry is indicated by a subscript zero (e.g. $F814W_o$).
In addition to the “gst” cuts, we require that $F814W_o \leq 23$ to remove stars with higher photometric bias and lower completeness. The foreground reddening correction is applied before this brightness cut. After applying the spatial and photometric cuts our sample contains $\sim$7.06 stars.
In Figure <ref>, we present the optical color-magnitude diagram after quality cuts and foreground extinction corrections. This color-magnitude diagram (CMD) shows the stellar density prior to the magnitude cut in gray scale contours. The red-clump and bump of early-AGB stars are both visible as peaks in the density below our selection region. The dashed area shows the selection region in which we determine RGB metallicities. In this paper, we estimate metallicities for RGB stars from their positions in the CMD. Isochrones at a range of metallicities are shown in Figure <ref>, showing that stars of different metallicities are clearly separated in color; we use this information to infer metallicities after considering a number of subtleties that can affect this measurement (Section <ref>).
§.§ Sources of Dust Extinction
Dust extinction has a significant impact on the observed photometry and thus our metallicity estimates. We discuss this in more detail in Section <ref>. In this section, we describe the source of our dust extinction measurements.
We use extinction measurements from <cit.> (hereafter ). uses a novel method to directly measure the dust extinction from the IR photometry (filters $F110W$ and $F160W$) of PHAT RGB stars. In the IR CMD, unreddened RGB stars form a tight locus that is insensitive to age and metallicity. Dust extinction causes RGB stars to redden off this locus and form a second, broad RGB. models the unreddened and reddened stars using three parameters: median extinction (), spread of extinctions (), and the fraction of reddened stars (). This latter parameter takes into account the geometry of the dust relative to the RGB stars; 1- stars are assumed to be in front of the dust, and thus unaffected by it. Maps of each parameter are created with 2” pixels across the PHAT survey region. Note that the central region of the PHAT survey (Brick 1) is not included in the dust maps due to extreme crowding.
In Figure <ref>, we present maps of the and used in this paper. We rebinned and down-sampled the data into spatial bins used in our final results (0.01squares). The width is approximately 0.3 everywhere and not shown because it has little bearing on interpreting the effect of extinction. In contrast, the plays a significant role. Increases in cause the reddening to be more pronounced at the same mean extinction values. For instance, in the case where $<0.5$ and high , the median metallicity remains unchanged as stars in front of the thin dust layer are unaffected while those behind are reddened beyond selection bounds. One feature visible in the map of is the increase towards the lower (northwest) edge of the survey. In this region a higher fraction of stars appear behind the dust because of the disk inclination.
Dust maps of M31. Top - the mean dust extinction map from which we use throughout this paper. The map shows that the dust extinction is highest in the 10 kpc ring region, with lower extinction values in other star-forming regions. Bottom - map of the fraction of RGB stars that are reddened, from . The parameter is important in understanding structure caused by dust; e.g., at a fixed $A_V$ increasing means more stars will be extincted. The final dust parameter, , is roughly 0.3 everywhere and is not shown here because its effect on our results is negligible.
§ PRINCIPLES OF PHOTOMETRIC METALLICITIES
In this section, we explain our method for deriving metallicity estimates and the sources of systematic and random uncertainty in these estimates. This section is organized as follows.
In Section <ref>, we discuss our choice of isochrone models, which we then use to interpolate the metallicity of individual stars based on their optical color and magnitude in Section <ref>. We then spatially bin the individual star metallicities to create a map of median metallicity. Because of the large number of stars, the random sampling uncertainties are negligible. We address systematic uncertainties in the median metallicity maps due to photometric bias and completeness (<ref>), dust extinction (<ref>), and the uncertainties in the absolute metallicity scale due to isochrone choices and age (<ref>). We correct the median metallicity map for the systematic effects of photometric bias, completeness and dust extinction and present results from this map in Section <ref>.
§.§ Isochrone Models and Age Assumptions
The intrinsic color of an RGB star depends primarily on its age and metallicity. Thus, given the photometry of a star and an assumed age, we can use isochrones to estimate a star's metallicity from its color and magnitude. When estimating individual star metallicities (Section <ref>), we break the age-metallicity degeneracy using assumptions about the age of M31's disk motivated by literature estimates. In this section, we discuss our choice of isochrone models and the possible impact of age on our metallicity estimates; this impact is further explored throughout the paper.
For isochrone models, we use Padova PARSEC1.2s isochrones <cit.>. At ages $>$1 Gyr, these isochrone models are available at metallicities ($Z$) between 0.0001 and 0.06 using $Z_{\sun}$=0.0152). We use the OBC bolometric corrections described in <cit.> to convert the isochrones to photometric brightnesses in our optical bands.
We test dependency on the isochrone models by comparing models from PARSEC to BaSTI <cit.> and find the later has a typical offset of approximately +0.3 dex. This offset drops to $\sim$0 for metal poor ($\lesssim-1.0$ dex) and metal rich ($\gtrsim0.3$ dex) stars. Because the majority of our stars have the typical offset, the resulting median metallicity map is only affected by a change in the absolute scale of the metallicity. As we discuss in Section <ref>, absolute metallicity changes do not affect our results.
The isochrone models for the bright RGB stars have a well known degeneracy between age and metallicity <cit.>. To break this degeneracy we adopt a single age motivated by the literature. We assume that this age is either uniform throughout the disk, or that it varies linearly with radius.
Current constraints on the age of M31's disk are relatively minimal. Spectroscopic measurements of the age by <cit.> show a mean age of $\sim$12 Gyr throughout the bulge, but this drops to $\sim$8 Gyr at their maximum radius of $\sim$2 kpc (500”) where the disk light starts to contribute significantly <cit.>. Only weak constraints on the ancient star formation history (SFH) can be obtained from the PHAT data in the inner disk, but measurements by Williams et al. ( in prep) suggest a mean SFH age of 10 Gyr throughout the outer disk. We can also get some insight into the age and age gradient present in M31's disk by examining similar galaxies. As noted in the introduction, the subsample of galaxies in the CALIFA survey analyzed by <cit.> includes a large number of galaxies similar to M31. Taking the 25 galaxies within one magnitude of M31 in $r$ band luminosity <cit.> and with Hubble type Sab to Sbc, we find that the minimum mass-weighted age in the disks of these galaxies is 3 Gyr, with typical values of $\sim$7 Gyr. These measurements were made over similar physical scales (0.5-2 $r_{eff}$) as the region we probe with the PHAT data. These old mass-weighted ages of spiral disks are also consistent with previous work by <cit.>.
While these previous studies suggest that the disk of M31 is old, the average age of stars observed along the upper RGB is not equivalent to the mass-weighted average age of a stellar system due to varying RGB lifetimes and the difference in masses of RGB stars with age. The average age of stars along the upper RGB is normally younger than the mass-weighted average stellar age. By simulating a CMD using the star formation history derived by Williams et al. ( in prep), we calculate the mean age of RGB stars to be $\sim$4 Gyr throughout the disk.
For the primary median metallicity maps we assume a flat fiducial age of 4 Gyr for our RGB stars. In Section <ref>, we show the metallicity gradient is not dependent on the exact fiducial age because the primary effect is a scaling of the absolute metallicity which leaves the gradient unchanged. In Section <ref>, we show that the median metallicity gradient can be changed if we assume a significant age gradient in the disk.
§.§ Estimating RGB Star Metallicity from Isochrones
We estimate individual star metallicities by interpolating isochrones at the distance of Andromeda ($m-M=24.45$) in the $F475W_o-F814W_o$ vs $F814W_o$ color-magnitude diagram. For interpolation, we use the LinearNDInterpolator function within the Python package scipy. This function finds the closest three isochrone points that create a triangle containing a star's color and magnitude using Delaunay triangulation. Then it calculates the star's metallicity by linearly interpolating the metallicity at those three nearest isochrone points. The isochrones used in this procedure are sampled at $\Delta Z = 0.0001$ over the full range of metallicities considered, so the interpolation in metallicity is very small. In Figure <ref>, we overplot the star metallicity on top of the stellar density contours in color and magnitude.
In Figure <ref>, we present the metallicity distribution function (MDF) of our RGB stars using an fiducial age of 4 Gyr. We additionally show the MDF generated from stars in low extinction regions. From this figure, we see that the dust has a small effect on the overall MDF of our sample.
The metallicity distribution function (MDF) of our RGB star sample using a flat fiducial age of 4 Gyr. The cyan MDF uses this paper's RGB selection. In purple, we show only RGB stars from low extinction regions within the PHAT survey ( $< 0.25$ mag; see Section <ref>). The median metallicity for both histograms is $\sim$-0.11 dex (shown as a vertical line). We note this MDF is not corrected for the systematic biases in the metallicity discussed in Section <ref> and <ref>, however, after correction, the MDFs look qualitatively very similar.
Our primary goal in this paper is to examine the spatial variation in the metallicity distribution across Andromeda's disk. We present a spatial map of the median metallicity of the RGB in the top panel of Figure <ref>. We derive the median metallicity in 0.01(135 pc, projected) square bins. Each bin contains between 100 and 19000 stars. The upper limit becomes 3300 after we exclude the inner regions ($\sim4$ kpc) because of completeness (discussed in Section <ref>). The typical bin contains $\sim$560 stars. In addition to the true variations in median RGB metallicity, this map shows features due to photometric bias, completeness, and dust. We attempt to correct these effects in subsequent sections.
Top: Raw map of the RGB median metallicity without any systematic corrections for dust extinction or photometric bias and completeness. Each spatial bin is 0.01(36 arcsec, 135 pc) square. Middle: Model for median metallicity changes due to the photometric bias and completeness. These systematics increase towards the central dense regions. The green contour shows the region outside which completeness effects are correctable (see Section <ref>). Bottom: Model for the expected changes in the median metallicity due to dust extinction. We interpret the increases and decreases in median metallicity in Section <ref>. For low extinction regions the dust causes an increase of $\sim$0.03 dex in the absolute metallicity and no significant spatial variations in the median metallicity. In our results (Figure <ref>), we subtract the two models from the raw median metallicity map to create a corrected version (presented in Figure <ref>). Note, all color bars have same range so relative differences can be assessed by eye.
§.§ Photometric Bias and Completeness Effects on Metallicity
Photometric bias and completeness affect a star's observed photometry and thus the inferred metallicity estimate. Artificial stars allow us to evaluate and ultimately correct for these two systematics. <cit.> describes the optical camera artificial star tests in detail. To summarize, $10^5$ artificial stars were distributed uniformly over each PHAT ACS/WCS field of view, resulting in a total of $\sim$47 stars over the full survey. The input $F475W-F814W$ colors and $F814W$ magnitudes for the artificial stars were split into two halves: the first half sample the full color and magnitude range of the data, while the other half sample the local observed color-magnitude diagram with an extrapolation to the faintest observed magnitudes. Each artificial star was inserted into and recovered from the data images independently using the DOLPHOT package <cit.>. The same photometric cuts described in Section <ref> were applied to the output photometry. We use these artificial stars to measure the photometric bias in our measurement in Section <ref>, and to model the completeness in Section <ref>. Then in Section <ref>, we describe how we apply corrections for bias and completeness to our photometric metallicity estimates.
§.§.§ Evaluating Photometric Bias
Unresolved and bright stars in crowded regions cause a bias in the measured photometry that becomes increasingly large in more crowded regions. This bias generally causes a star to appear brighter and with a color closer to the mean color of the overall galaxy due to blending with fainter undetected stars.
In Figure <ref>, we show the average color and magnitude bias in bins across the CMD for a region of M31 at radii between 4-5 kpc where the bias is significant. The bias of each individual artificial star is measured from the recovered magnitude minus the input magnitude. Because the star's recovered CMD position is changed from its original position, this photometric bias results in a bias in our metallicity estimates as well. Examining Figure <ref> shows stars within our RGB region are typically detected to be brighter and bluer than their intrinsic CMD position. Our metallicity estimates for these stars are thus biased lower than the true metallicity, with the effect being more severe for high metallicity stars which are closer to the $F475W$ detection limit due to their intrinsically redder colors.
In Section <ref>, we use the measured bias to partially correct for this effect. For computational efficiency, we create a binned data set of the average bias. We bin using 4 dimensions: two spatial dimensions with step size of 0.02(270 pc) in both projected major and minor axis of M31; and, two color-magnitude dimensions with step size of 0.45 mag in both dimensions (sizes for spatial and CMD bins seen in Figures <ref> and <ref> respectively). The result is a median of 3.34 stars input for every spatial pixel and 36 input stars for every 4D bin. In each bin with $>10$ recovered artificial stars, we calculate the mean photometric bias for each filter from all the artificial stars. Stars in 4D bins with $<10$ artificial stars receive no correction value.
Color-magnitude diagram (CMD) showing the mean completeness and bias. This figure is computed from artificial stars at radii of $\sim$2-4 kpc, and therefore highlights the region most affected by completeness and bias that we consider in our analysis (Section <ref>). Our RGB selection box is indicated as a dotted line. Each bin is 0.45 mag on each side. The completeness is the number of detected artificial stars divided by the number of those inserted in each CMD bin (Section <ref>). The arrows represent the magnitude and direction of bias for each bin (Section <ref>). The bias generally shifts stars brighter and closer to the median color of the galaxy. We correct our median metallicity determinations for completeness and bias as discussed in Section <ref>.
§.§.§ Evaluating our Photometric Completeness
The completeness is the number of detected stars divided by the total number of stars present at a given magnitude and color. Our completeness correlates with magnitude; at fainter magnitudes we detect a smaller fraction of stars and are thus less complete. Because we require stars to be detected in both filters, each filter contributes to the completeness; for blue stars the F814W magnitude determines the completeness (thus creating a horizontal completeness limit on our CMD), while for redder stars, the F475W magnitude limits the completeness, creating a diagonal completeness limit on our CMD. These completeness limits are visible in Figure <ref> where the darkest pixels represent $\sim0\%$ completeness. The faint F475W magnitudes of metal-rich giants causes these stars to be most affected by completeness, and thus we detect a smaller fraction of the most metal-rich stars.
We measure the completeness from the artificial star tests as follows. We define an artificial star as detected if it passes the photometry cuts defined in Section <ref> and has an absolute magnitude change less than 0.75 mag. The completeness is then simply the fraction of artificial stars detected in a particular spatial and CMD volume. We measure the completeness using the same 4D bins described previously in Section <ref>.
To correct for completeness, we divide the total number of artificial stars recovered and inserted at a given CMD position using the input magnitudes (i.e. after the bias correction, Section <ref>). This is somewhat different from the standard practice where completeness is calculated using the observed magnitudes to determine the detections and the input magnitudes to determine the total number of stars inserted at a given CMD position.
Due to the small number of stars in each individual 4D bin, the errors on the completeness in an individual bin are large. To reduce these uncertainties, we model the completeness at a given color as a function of $F814W_o$ magnitude using a sigmoid function (e.g. ):
\begin{equation} % sigmoid function
\label{eq:completeness-model}
C_{model}(F814W_o|a,b) = (1-exp(\frac{1}{a}\cdot{}F814W_o+b))^{-1}
\end{equation}
Our modeling process is a loop with five steps:
First, we select artificial stars within a 2-dimensional spatial bin. Second, we select artificial stars within a particular color bin and in the neighboring color bin(s). Using neighboring bins increases the number of stars used for each fit. Also as a result the fit parameters become covariant. This is beneficial because the covariance effectively smooths along the color dimension and makes the measurement more robust. Third, we bin the color-selected stars into independent $F814W_o$ magnitude bins. The completeness for each 2-dimensional color-magnitude bin is the fraction of detected to input artificial stars. An error in this measurement is also estimated. Fourth, we fit the completeness vs $F814W_o$ to Equation <ref> using $\chi^{2}$ minimization and determine $a$ and $b$ for that color bin. Fifth, we use the model to assign a fitted completeness to each 4D bin.
At the end of this process, we have robust measurements of our completeness binned in 4D. We show a spatial map of our completeness for RGB stars in Figure <ref>; this figure is discussed further in Section <ref>.
§.§.§ Applying Photometric Corrections
Using the results from <ref> and <ref> We correct our final median metallicity measurements for the photometric bias and completeness systematics. To apply this correction, we first correct the observed photometry for photometric bias. Then, when aggregating into the median metallicity, we correct for completeness by weighting each star by the inverse of the recovered completeness from Section <ref>. In this section, we further describe these corrections and the creation of a photometrically corrected median metallicity map.
To correct the observed star photometry for the photometric bias in each filter, we first bin the observed stars using the same 4D bins described in Section <ref>. Then, we select each spatial bin and consider only the mean bias values in the associated CMD bins. We interpolate the bias for each star in both filters using the CMD bin centers and mean bias values (as we did for the metallicity in Section <ref>). Finally, the observed star photometry is corrected for bias by subtracting interpolated bias from the observed photometry.
We use the bias corrected photometry to estimate a photometric completeness for each star. The process is similar to interpolating the bias values for each star, only we now use the 4D recovered completeness data from Section <ref>. We apply a minimum completeness limit of 0.05 so when weighting by completeness no star can dominate the calculations. This limit affects $<0.0002\%$ of our stars.
For every spatial bin in our final maps, we calculate a bias and completeness corrected median metallicity. We first re-derive metallicities for the stars using the bias corrected photometry. To calculate the median metallicity, we weigh each star by the inverse of its completeness and then take the 50th percentile of the stars' MDF.
The middle panel of Figure <ref> shows the effect of bias and completeness on the median metallicity map; specifically, it shows the median metallicity determined from the uncorrected photometry minus the median metallicity after we apply our bias and completeness correction. The bias and completeness correction is largest near the center where the density of stars is high and is negligible in the outer parts of the galaxy. We discuss the limits to these corrections in Section <ref>.
This correction model is not exactly accurate due to uncertainties in the artificial star tests caused by stochasticity and assumptions in the prior distribution of the stars. Nevertheless, the model provides a reasonable and robust correction to this known systematic.
§.§.§ Completeness Limit
Where the stellar density becomes high, photometric bias and completeness effects become large. When the completeness drops to near 0%, our bias and completeness corrections are no longer valid as they require the intrinsic distribution of stars to be at least partially detected. This limits how close to Andromeda's galactic center we can measure metallicities and limits our ability to detect the most metal-rich stars.
We use a minimum completeness metric to quantify exactly how close to the center our corrections are applicable. At each spatial position, we measure the completeness along the 90th percentile density contour of our RGB stars (shown as a black contour line on Figure <ref>) limited our RGB $F814W$ magnitude limit. We use the minimum completeness along the contour to create the map shown in Figure <ref>. For almost all spatial positions the minimum completeness is measured at the most red RGB tip. This means the completeness along our $F814W$ magnitude limit is greater than this minimum completeness metric at practically all spatial positions.
From the minimum completeness map (Figure <ref>), we define the limiting central region at the 50% contour which is $\sim$4 kpc. At this limit, the bias and completeness correction is $\sim$0.1 dex. We exclude data within from our final analysis. Beyond the limit, we find the same correction has a $\sim1\sigma$ effect on our metallicity graident (Section <ref>) and no bearing on our observed metal rich bar structure (Section <ref>). We conclude this limiting region to be a reasonable choice. In all subsequent figures, we include the results at all radii but shade the excluded central region.
Map of the minimum completeness for stars within our RGB selection box (see Figure <ref>). This metric is calculated by assigning a completeness to all the stars then taking the minimum from the 90th percentile—in CMD density—of our data. The solid lines indicate 50%, 75%, and 90% minimum completeness. In general, the stars most affected by completeness are the reddest, highest metallicity portion of the RGB. At the center of Andromeda these stars are completely undetectable. As the sample becomes more incomplete it becomes impossible to correct properly for the bias and completeness; we therefore only analyze the area with $>50$% completeness (see Section <ref>).
§.§ Effects of Dust Extinction
Dust extinction causes stars to appear dimmer and redder. For RGB stars, this causes us to overestimate the metallicity of reddened stars (see arrow in Figure <ref>). We use information on the dust distribution (see Section <ref>) to both identify high extinction regions and to model the effect of dust extinction on the median metallicity.
To identify regions where the extinction is likely to cause biased metallicity estimates, we multiply the mean extinction by the fraction of reddened stars () from within each spatial bin. We find $ > 0.25$ mag is a good limit for identifying regions where extinction significantly biases our results.
We measure the significance of dust extinction on our median metallicity measurements by forward modeling the effect of dust extinction on an unreddened fiducial CMD.
To forward model our data, we first need a realistic input CMD. We take this input CMD from unreddened regions in the survey. For the unreddened fiducial CMD, we use stars in the low extinction region ( $<0.25$ mag) at $\sim$ 14 kpc along the major axis (visible in Figure <ref>). Our final result should be insensitive to low level extinction present in our fiducial CMD because we measure the relative change in median metallicity of the sample. Our results are very similar using other low extinction regions within the survey (e.g. along the NE edge). The primary weakness of our approach is that we do not account for the stellar population differences (i.e. young stars) we expect between the dust-free and dusty regions of the galaxy; trying to account for this change would require full star formation history modeling and is beyond the scope of this paper. We expect that any young contaminants not included in our fiducial CMD would be brighter bluer stars reddened into our CMD region, and thus would likely reduce the median metallicity at moderate extinctions.
We apply reddening to this CMD to measure the resulting change in median metallicity. The extinction for each individual star in the CMD is drawn from the extinction probability function from using their maps of the best fit , and dust parameters (see Figure <ref>). The stars are binned into 0.01spatial bins which correspond to the bin size used in our median metallicity maps. In each bin, we take the associated dust parameters and determine a random extinction for each star. Then we apply that extinction to create a new reddened data set and re-evaluate the metallicity as described in Section <ref>. Finally, we measure the change in median metallicity between new reddened data and the original unreddened sample.
In Figure <ref>, we present the change in median metallicity as a function of for each spatial bin colored by . These two parameters dominate the predicted change in median metallicity estimates; changing has a $<0.005$ dex change in the median metallicity estimates. As previously noted, for a particular the change in median metallicity increases with increasing . Lines of constant (and constant =0.3) are also plotted on Figure <ref>. For these constant lines, the change in median metallicity first increases then decreases with increasing . The increases are easily understood as stars becoming reddened and appearing more metal rich. The decrease is caused by stars becoming so reddened they are moved beyond our RGB selection box and the most metal rich isochrone; the remaining unreddened stars (which are in front of the dust distribution) yield a very similar median metallicity to the input CMD.
The individual points in Figure <ref> correspond to individual spatial bins. We also present these data as a map of the change in median metallicity (lower panel of Figure <ref>). This map shows sharp changes due to dust especially along the upper (northeastern) edge of the survey and in the 10 kpc ring. Our final results avoid these regions by imposing a cut to remove these high extinction regions ($ > 0.25$ mag).
In the low extinction regions, the dust still has a low level effect. We apply a dust correction by subtracting the change in median metallicity due to dust. The changes in metallicity are small (typically $\sim0.03$ dex) and—more importantly—distributed randomly over the survey area so the result is only a systematic on the absolute metallicity scale and not on spatial variations. Systematics on the absolute metallicity scale do not heavily impact our results.
Simulated change in median metallicity ($\Delta$[M/H]) for each spatial bin in the data. The simulation uses a fiducial CMD taken from a low-dust region and applies reddening to individual stars using the dust parameters (,, and ) from the dust maps for every spatial bin. The solid lines show changes in median metallicity at constant values of 0.1, 0.2, 0.3, and 0.4. Along one of these solid lines (and assuming =0.3). The dust increases the median metallicity until $\approx$0.5 because the reddened stars receive an overestimated metallicity. The median metallicity decreases for high extinction when the reddened stars are moved beyond the edge of our RGB box and no longer affect the measurement. The points below the dashed line are low extinction regions with $< 0.25$. These are the regions used in our metallicity gradient measurement. Their $\Delta$[M/H] is $<0.12$ dex, with typical values of 0.03 dex. Details on the dust simulations are given in Section <ref>; a map of $\Delta$[M/H] is shown in the bottom panel of Figure <ref>.
§.§ Uncertainties in Absolute Metallicities
The absolute metallicities we determine have large uncertainties which we discuss in this section.
Age: Older populations have redder RGBs than younger populations of the same metallicity. Therefore, using older fiducial ages results in lower metallicity estimates. This effect is quite significant, with metallicity estimates varying by $\sim$0.4 dex when changing the age from 3 to 13 Gyr (we discuss this further in Section <ref>). Overall, this is the dominant uncertainty in the absolute metallicity. Because we are primarily interested in the metallicity gradient, an age gradient can significantly affect our result; this is discussed further in Section <ref>.
Isochrones: As noted in Section <ref>, use of the BaSTI isochrones results in a median metallicity that is $\sim$0.3 dex higher than the PARSEC 1.2s isochrones we adopt.
Dust: From our simulations in the previous section, we find the low level dust extinction present throughout the survey results in a slight overestimate of the intrinsic metallicity; however this bias is only $\sim$0.03 dex.
Photometric Bias and Completeness: Near the center, the bias and completeness correction raise our median metallicity by more than 0.1 dex. Unlike the effects above, this is primarily a bias in the relative metallicity between the center and outskirts, as the bias and completeness have no affect on the median metallicity in the outer disk regions.
Because of the large uncertainties in the absolute metallicities, we choose to focus on relative measurements for our results. Particularly, the metallicity gradient of Andromeda and metallicity substructure of the disk. After applying corrections for known systematics, we present the median metallicity map and results in the next section.
§ RESULTS: METALLICITY GRADIENT AND STRUCTURES IN ANDROMEDA
In this section, we analyze the radial and spatial variation of the median RGB metallicity. Our results are based on the spatial map of corrected median metallicities presented in Figure <ref>. Our results focus on two interesting features in the corrected median metallicity map. The first is an overall gradient towards lower metallicities at larger radii (Section <ref>). The second feature is a region of high metallicity at radii of 3-7 kpc lying just above (southeast of) the major axis, which we associate with an elongated bar (Section <ref>).
Top Panel – Map of the RGB median metallicity after corrections for dust extinction, photometric bias, and completeness. The shaded black regions indicate the crowded central region where our bias and completeness corrections fail in recovering the true metallicity distribution (Section <ref>). Included as a reference point, the black ellipse is a ring with with constant deprojected radius at 10 kpc. Our measurement of the metallicity gradient uses these corrected median metallicities (Section <ref>). Middle Panel – on top of the median metallicity map, we plot green contours at constant RGB density as inferred from the IR photometry (<cit.>, Seth et al. in prep). Additionally, a solid black line shows Andromeda's bar length and azimuthal offset measured by <cit.>. We observed a region of high median metallicity correlated with the RGB density and bar. In Section <ref>, we argue these high metallicities are likely a result of bar interaction. Bottom Panel – we overplotted the median metallicity map with the average reddening (). Every spatial bin below the lowest dust contour we define as low extinction ($ < 0.25$ mag). We base our results on these low extinction areas. This figure shows the high metallicity region we associate with Andromeda's bar is not caused by dust.
The median metallicity gradient in M31. To create this figure, the median metallicity map (Figure <ref>) was divided into annular bins with $b/a=0.275$; within these bins the mean of the map is shown as the solid line, while the shaded region shows the standard deviation of the pixels in the bin. High extinction regions ( $> 0.25$) were excluded. The red line shows the gradient including corrections for dust extinction and photometric bias and completeness, while in cyan the metallicities have only been corrected for dust extinction. The bias and completeness correction changes the gradient by $\sim$1$\sigma$, while the dust extinction correction changes the gradient by less than a 0.001$\sigma$. The black line and shaded region shows the best linear fit at deprojected radii $R > 4.45$ kpc ($\frac{1}{2} R_{eff}$; <cit.>) to the corrected metallicities, the best-fit slope is -0.020$\pm$0.004 dex/kpc. The black shaded region at $R \lesssim 4$ kpc where our RGB data becomes incomplete at high metallicities.
§.§ Radial Metallicity Gradient
We present the first measurement of the RGB metallicity gradient in the M31 disk from 4 to 20 kpc. We binned the median metallicity (shown in Figure <ref>) into elliptical apertures with a size of 1 kpc along the major axis, and assuming $b/a = 0.275$ <cit.> to account for the disk's inclination. We exclude the high extinction regions with $> 0.25$ to limit the effect of dust on our measurement; this limit is shown as the white contour in the bottom panel of Figure <ref>. After excluding these regions, we take the average and standard deviation of the median metallicity within each elliptical aperture to create the radial metallicity profile shown in Figure <ref>.
We fit a gradient to the measured metallicity profile at $R > 4.45$ kpc ($\frac{1}{2} R_{eff}$; <cit.>). This radial cut excludes the regions we distrust due to low completeness (shown as the shaded region in Figures <ref> and <ref>). We fit the mean and standard deviation of the metallicity in each radial bin using a Monte Carlo simulation with a linear fit. For each iteration, we calculate the best fit slope and intercept using least squares optimization. We repeat this and obtain a distribution of fitted slopes and intercepts; this procedure also allows us to examine the covariance of the slope and intercept. We show the best fit parameters and 1$\sigma$ contours in Figure <ref> for various fiducial ages including our primary result with 4 Gyr.
We measure Andromeda's metallicity gradient to be $-0.020\pm0.004$ dex/kpc. This gradient uses a 4 Gyr fiducial age (see Section <ref>, Fig <ref>) and our corrected metallicities. The error on the gradient is calculated using the Monte Carlo technique described above; bootstrapping errors are significantly smaller than this quoted value. The photometric bias, completeness and dust corrections have little impact on producing the observed gradient: the dust corrections change the gradient by $<1\%$, and the photometric bias and completeness corrections steepen the relation somewhat, but still agree at about the 1$\sigma$ level. We also measure a consistent metallicity gradient (within 0.25$\sigma$) when selecting stars just along the major axis which indicates the assumed inclination angle has little impact on the gradient. The intercept (central metallicity) of this fit is [M/H]$=0.11\pm0.05$; though caution should be taken in interpreting the absolute metallicity as explained in Section <ref>.
To facilitate comparisons with the literature, we also present our gradient with different units. In terms of Andromeda's half light radius (8.9$\pm$0.8 kpc; <cit.>), the gradient is $-0.181\pm0.040$ dex/$R_{eff}$. Using the radius of the isophote at a bolometric B-band of 25 mag ($21.5$ kpc; <cit.>), the gradient is $-0.43\pm0.09$ dex/R25.
§.§.§ The Effect of Changes to the Fiducial Age
As discussed in Section <ref>, the assumed age influences the estimated metallicity of our stars. We evaluate the sensitivity of our metallicity gradient results to changes in the fiducial age by repeating our analysis using isochrone models with ages of 2, 4, 6, 8, 10, and 12 Gyr. For each age, we calculate a distribution of metallicity gradients and intercepts; In Figure <ref>, we present the best fit and 1$\sigma$ contour of the distribution.
The metallicity gradient across all fiducial ages is consistent within 0.25$\sigma$. This is because the RGB of isochrones at different ages have similar shapes in the CMD, with the primary difference being that older ages have a lower metallicity at the same RGB color. This results in a significant change in the normalization of the metallicity gradient and the fitted intercept in metallicity, but little change in the slope. We find a change in the intercept of roughly $-0.037\pm0.006$ dex/Gyr.
This figure shows the best fit slope and intercept for the median metallicity gradient using fiducial ages of 2, 4, 6, 8, 10, and 12 Gyr. The line surrounding each point represents the 1$\sigma$ confidence after performing a Monte Carlo simulation to evaluate the slope and intercept. Our primary result presented in Section <ref> uses a fiducial age of 4 Gyr shown here as a diamond. The age gradient for all fiducial ages are within 0.25$\sigma$. Changing the fiducial age does have an effect on the absolute metallicity which we discuss in Section <ref>. This figure also demonstrates a covariance between the fitted slope and intercept of the metallicity gradient.
§.§.§ The Effect of a Radial Age Gradient
A gradient in the age has a significant impact on our metallicity gradient. We evaluate this impact using the following experiment; we fix the age of stars at the center of Andromeda to 4 Gyr. Then, we calculate the age for each star using its spatial position for age gradients of -0.35, -0.25, -0.1, 0.0, and 0.1 Gyr/kpc. We calculate the metallicities similar to in Section <ref> with age in addition to color and magnitude. In this application, the Delaunay triangulation finds the nearest four color, magnitude, and age isochrone points which contain a particular star's parameters and performs the linear interpolation of the metallicity between the points. Using these new metallicities, we recalculate the median metallicity map and derive the distribution of best fit metallicity gradients and intercepts as described in Section <ref>.
In Figure <ref>, we present the results of this age gradient experiment. As mentioned, as the age gradient becomes negative, the metallicity gradient flattens, while a positive age gradient steepens the gradient. For an age gradient of -0.35 Gyr/kpc the metallicity gradient is consistent with being flat. This figure can be combined with Figure <ref> to determine the inferred age gradient and intercept for a range of central ages and age gradients. For instance, if the mean age of stars in our RGB box is 8 Gyr in the center changing by -0.1 Gyr/kpc, then our measured metallicity gradient would be $\sim$-0.018 dex/kpc with a central metallicity of $\sim$-0.04 dex.
We expect any age gradient in M31 to be small. This is based on results from the PHAT data (Williams et al. in prep). Also, <cit.> find flat mass weighted age-gradients ($< 0.1 $dex/$r_{eff}$) for a majority of galaxies in the CALIFA sample, with the small age gradients being both positive and negative. This translates to an age gradient less than 0.1 Gyr/kpc using values for M31 and our fiducial age. However, resolved star formation histories of M33 and NGC 300 (lower mass, later-type disk galaxies than M31) have shown steeper negative age gradients, with younger ages at larger radii <cit.>. If such gradients exist in M31, the younger stars at larger radii will cause a flatter metallicity gradient than using a single fiducial age at all radii. However, resolved star formation histories of M33 and NGC 300 (lower mass, later-type disk galaxies than M31) have shown steeper negative age gradients, with younger ages at larger radii <cit.>. If such gradients exist in M31, the younger stars at larger radii will cause a flatter metallicity gradient than using a single fiducial age at all radii. If any new information becomes available, Figure <ref> can be used to update our metallicity gradient result based on the determined age gradient.
This figure shows the effects of an age gradient on the median metallicity. The best fit metallicity gradient and intercept were calculated as described in Section <ref>, but with a variation of stellar age with radius. The age gradients shown have slopes of -0.35, -0.25, -0.1, 0.0, and 0.1 Gyr/kpc with a central age of 10 Gyr. The metallicity gradient flattens as the age gradient becomes more negative with it being consistently 0 dex/kpc when the age gradient is -0.35 Gyr/kpc. Based on observations of external galaxies similar to Andromeda, we expect the age gradient to be $\lesssim\arrowvert0.1\arrowvert$ Gyr/kpc (see Section <ref>).
§.§.§ Comparisons to Literature
In this section, we compare our measured metallicity gradient to other metallicity gradient measurements in M31 and to metallicity gradients measured in other galaxies. We find some evidence that the metallicity gradient may become flatter at large radii, but does not appear to vary with age.
The metallicity gradient of the young population of Andromeda has been measured by <cit.> using HII region abundances over the same radial region covered by the PHAT data. They find a gradient of $-0.023\pm0.002$ dex/kpc. This gradient is consistent with our RGB metallicity gradient, suggesting that the age gradient has been stable in M31 for some time.
At larger radii, <cit.> used abundance measurements of planetary nebula to measure a gradient with deprojected radii of 18 to 43 kpc. These planetary nebulae have typical ages of 1-2 Gyr, thus probing slightly younger stars than the RGB stars probed here. They find a gradient of $-0.011\pm0.004$ dex/kpc, significantly flatter than our gradient. This may be evidence that the metallicity gradient flattens at larger radii. Another, more indirect approach to measuring metallicities is the stellar population modeling of integrated photometry of M31 from the ANDROIDS survey <cit.>. Their preliminary results seem to show a somewhat shallower gradient of $\sim$-0.01 dex/kpc out to $\sim$30 kpc, with a possible steepening at even larger radii.
These comparisons provide conflicting support for the expectations due to radial migration <cit.>. On one hand, the shallower metallicity gradient in old populations at large radii could be explained if an increasing fraction of stars at large radii come from smaller radii. However, radial migration also predicts that any metallicity gradient should get shallower over time, thus the similarity of the metallicity gradients in the young and old populations in the inner disk suggests that either the metallicity gradient was steeper in the past, or significant migration has not occurred in the inner disk.
Higher quality stellar population modeling can be done with spectra, but published results include only the central bulge region. Spectral index measurements of M31's bulge by <cit.> show the metallicity at radii between 100 pc and 2 kpc is flat and approximately solar. At smaller radii, the metallicity rises sharply, with [M/H]$\sim$0.5 in the center. At the outer most radii, these measurements include a substantial disk component. Our central metallicity extrapolated inwards to 2 kpc is roughly consistent with this measurement though the absolute metallicity scale in our measurement is sensitive to many systematics (Figure <ref> and Section <ref>).
The recent CALIFA survey observed $\sim600$ face-on spiral galaxies using wide-field integrated field units (IFU). Andromeda's size, morphology, and central velocity dispersion are typical among this sample. From the CALIFA sample, <cit.> selected 62 face on galaxies and measured their stellar metallicities and ages. The mean mass-weighted metallicity gradient of these galaxies is $-0.089\pm0.151$ dex/$R_{eff}$ or $-0.010\pm0.017$ dex/kpc when scaled using M31's effective radius ($R_{eff}=8.9$ kpc; <cit.>). Our metallicity gradient is slightly steeper than this mean metallicity gradient, but is well within the scatter observed of their galaxies.
Mass-weighted age and metallicity gradients measured for 62 CALIFA galaxies by <cit.>. Gray points indicate the full sample, while the 25 galaxies most similar to M31 (Hubble types Sab to Sbc and $-20.5 > M_r > -22.5$) are highlighted in black. Our M31 metallicity gradient (assuming no age gradient) is shown as the blue star. Dashed lines give the median of the full sample (gray), and the M31-like sample (black) along both axes.
The metallicity gradient in the Milky Way (MW) disk appears to be significantly steeper than the gradient in M31. <cit.> measure the metallicity gradient as a function of height (Z) from the disk for SEGUE Survey stars. For stars near the $|Z|=0$ they measure $\sim-0.06$ dex/kpc which drops to $\sim0$ dex/kpc for $|Z| > 1$ kpc (see Figure 8 of <cit.> for metallicity gradient vs Z height). Given that the disk mass is dominated by stars below this height, the MW gradient does appear to be steeper than the M31 gradient. A similarly steep mid-plane gradient is found in other survey data as well including APOGEE data <cit.>, RAVE data <cit.> and the Gaia-ESO survey <cit.>. The shallower gradient in M31 relative to the Milky Way is also discussed by <cit.>.
§.§ Metal Rich Bar Feature
We observe a region of high median metallicity likely associated with the end of the bar. In Figure <ref>, this non-axisymmetric region lies just above the major axis between radii of $\sim$3 and 7 kpc. This region contains the highest metallicities found anywhere in our maps. Examination of contours of RGB density in the middle panel of Figure <ref> shows a similar axial asymmetry. This asymmetry coincides with M31's bar as measured by <cit.> at 8above the major axis. <cit.> also noted an overdensity in AGB star counts near the end of the bar. In addition to the presence of a stellar overdensity, a peculiarly high velocity dispersion ($\sim$150 km/s) is observed at the outer edge of this region <cit.>. This high dispersion could be explained if this represents the end of a misaligned bar.
This high metallicity region is not caused by dust but represents a real change in the stellar population. As previously noted in Section <ref>, dust causes the median metallicity to appear more metal rich for moderate extinctions of $A_V\approx0.5$ mag. We observe the high metallicity region prior to the dust correction (Figure <ref>) and it becomes even clearer after the correction. Even more noteworthy, the high metallicity is seen even in the very low extinction regions (dust contours shown in Figure <ref>).
This higher median metallicity region may be expected if either: (1) gas collects at the bar end resulting in higher enrichment there <cit.>; or, (2) the bar scatters stars from the inner part of the galaxy which have higher metallicity. An additional possibility is there could be a shift in the RGB age in the bar region; however, to get a high metallicity region, the average age would need to drop in the bar relative to its surroundings. Regardless of the mechanism for metal enrichment, this metallicity adds further evidence for a very long bar in M31.
§ CONCLUSION
We present an analysis of the spatial variations in the metallicity of RGB stars in Andromeda's disk using the PHAT survey. This study is the first large study of RGB metallicities within the Andromeda disk and covers a radial region between 4 and 20 kpc. Metallicities are estimated for 7.06 RGB stars using their optical photometry and isochrone models. We create a map of median metallicity and correct this map for the effects of dust extinction and photometric bias and completeness. Our final results are based on the median metallicity map after correcting for these systematics (Figure <ref>). From the corrected median metallicity map we conclude:
* Over the inner disk PHAT survey ($\sim$4-20 kpc), we measure a clear gradient in the median metallicity. The measured gradient is [M/H]=$-0.020\pm0.004$ dex/kpc assuming a constant fiducial age of 4 Gyr (Section <ref>). This gradient has a less than 1 sigma sensitivity to systematic effects of dust extinction, photometric bias and completeness, and fiducial age changes. The derived metallicity gradient is sensitive to any age gradient that may be present in Andromeda, with negative age gradients resulting in shallower metallicity gradients. A steep age gradient of -0.35 Gyr/kpc would result in no metallicity gradient; however, spectroscopic observations of other similar galaxies typically show much shallower age gradients <cit.>.
* We measure an enhanced metallicity region offset from the major axis. We show this region is coincident with an overdensity of RGB stars and a region of enhanced dispersion <cit.>. In Section <ref>, we discuss how this likely is due to Andromeda's bar <cit.>.
We thank the anonymous referee for suggestions that improved the paper. This work was supported by the Space Telescope Science Institute through GO-12055. This research made use of codes in the SciPy stack[http://www.scipy.org/], as well as Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration, 2013)[http://www.astropy.org].
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|
1511.00580
|
Let $\Gamma$ be a cocompact discrete subgroup of $\pslc$ and denote by $\hyps$ the three dimensional upper half-space.
For a $p\in\hyps$,
we count the number of points in the orbit $\Gamma p$, according to their distance, $\arccosh X$, from
a totally geodesic hyperplane. The main term in $n$ dimensions was obtained by Herrmann
for any subset of a totally geodesic submanifold. We prove a pointwise error term of $O(X^{3/2})$
by extending the method of Huber and Chatzakos–Petridis to three dimensions.
By applying Chamizo's large sieve inequalities we obtain the conjectured
error term $O(X^{1+\epsilon})$ on average in the spatial aspect.
We prove a corresponding large sieve
inequality for the radial average and explain why it only improves on the pointwise bound by $1/6$.
§ INTRODUCTION
Let $\Gamma$ be a discrete group acting discontinuously on a hyperbolic space $\hyps$ and denote
the quotient space by $M=\hypmodg$.
The standard hyperbolic lattice point problem asks to count the number of points in the orbit $\Gamma p$ within a given distance from some fixed point $q\in\hyps$.
For example, in two dimensions the counting function is
\[N(z,w,X) = \#\Set{\gamma\in\Gamma\given 4u(\gamma z, w)+2\leq X},\]
where $u$ is the standard point-pair invariant on $\uph$ and $z,w\in\uph$, and it measures
the number of lattice points $\gamma z$ in a hyperbolic disc of radius $\arccosh(X/2)$
centered at $w$. This problem was first considered by e.g. Huber and Selberg. Selberg proved that for fixed $z,w\in\uph$,
\[N(z,w,X)=\sqrt{\pi}\sum_{s_{j}\in(1/2,1]}\frac{\Gamma(s_{j}-1/2)}{\Gamma(s_{j}+1)}u_{j}(z)\overline{u}_{j}(w)+E(z,w,X),\]
where the error term satisfies $E(z,w,X)=O(X^{2/3})$. The bound on the error term has
not been improved for any cofinite $\Gamma$ or any choice of points $z,w\in\uph$.
To find more evidence of the conjectured error term $E(z,w,X)=O(X^{1/2+\epsilon})$, it is useful to consider various averages.
For example, hill2005 look at the variance of the counting function in terms of the centre over the whole fundamental domain
of any cofinite $\Gamma$ in hyperbolic $n$-space.
For the case $\hyps=\uph$ and $\Gamma=\pslz$, with no eigenvalues $\lambda_{j}\leq 1/4$, their result is
\[\int_{\Gamma\backslash\uph}\abs[\bigg]{N(z,w,X)-\frac{\pi}{\vol(\Gamma\backslash\uph)}X}^{2}\d{\mu(w)}=O(X),\]
where $\mu(w)$ is the standard hyperbolic measure on $\uph$.
On the other hand, petridis2015 looked at a local average of $N(z,z,X)$ over $z$. Suppose that $f$ is smooth, non-negative, compactly supported
function on $M$. For $\Gamma=\pslz$ they proved that
\[\int_{\Gamma\backslash\uph}f(z)N(z,z,X)\d{\mu(z)}=\frac{\pi X}{\vol(\Gamma\backslash\uph)}\int_{\Gamma\backslash\uph}f(z)\d{\mu(z)}+O(X^{7/12+\epsilon}),\]
where the error term depends on $\epsilon$ and $f$ only.
This improves Selberg's bound halfway to the expected $1/2+\epsilon$. Their method requires
knowledge of the average rate of QUE for Maaß cusp forms on $M$ and other arithmetic information only available to groups similar to $\pslz$.
In 1996 chamizo21996 showed that it is possible to apply large sieve methods on $M$. As an application, he proved that by averaging
over a large number of radii, one gets the expected bound on the error term $E(z,w,X)$:
\begin{equation}\label{eq:chamizoresult}
\frac{1}{X}\int_{X}^{2X}\abs{E(z,w,x)}^{2}\d{x}=O(X\log^{2}X).
\end{equation}
Furthermore, Chamizo also proves a similar result for the second and fourth moments of discrete averages over sufficiently spaced centres, which leads to
\begin{equation}\label{eq:chamizoresult2}
\biggl(\int_{\Gamma\backslash\uph}\abs{E(z,w,X)}^{2m}\d{\mu(z)}\biggr)^{\frac{1}{2m}}\!\!=O(X^{1/2}\log X),
\end{equation}
for $m=1,2$.
Instead of measuring the distance between two points of $\hyps$, it is also possible to consider geodesic segments between various subspaces of $M$.
In two dimensions, huber1956 looked at geodesic segments between a point and a fixed closed geodesic $\ell$.
The geodesic $\ell$ corresponds to a hyperbolic conjugacy
class $\mathfrak{H}$, given by some power $\nu$ of a primitive hyperbolic element $g\in\Gamma$. For cocompact $\Gamma$,
Huber explained that counting
\begin{equation}\label{eq:huberdef}
N_{z}(T)=\#\Set{\gamma\in\mathfrak{H}\given d(z,\gamma z)\leq T}
\end{equation}
is equivalent to counting
the geodesic segments from $z$ to $\ell$ according to length. If $\Gamma$ has no small eigenvalues,
then Huber's main result in <cit.> says that
\begin{equation}\label{eq:huber}
\end{equation}
where $\mu$ is the length of the invariant geodesic corresponding to $\mathfrak{H}$, and $X=\frac{\sinh T/2}{\sinh \mu/2}$.
Independently, good1983 proved
a stronger error bound of $O(X^{2/3})$. Good's methods also extend to more general counting problems for cofinite groups $\Gamma$.
There is another interesting geometric interpretation that Huber gave for the counting problem in conjugacy classes.
After conjugation we may assume that the geodesic $\ell$
lies on the imaginary axis. Then the counting in $N_{z}(T)$ is equivalent to counting $\gamma z$ in the cosets $\gamma\in\Gamma/\langle g\rangle$, such that
$\gamma z$ lies inside the sector formed by the imaginary axis and some angle $\Theta$.
Chatzakos and Petridis <cit.> showed that it is possible to apply Chamizo's methods to obtain results analogous to (<ref>)
and (<ref>) for both cocompact and cofinite $\Gamma$. This was done by extending the method of Huber. Along the way they also
obtain a new proof of Good's error term $O(X^{2/3})$.
In $n$ dimensions, herrmann1962 investigated the number of geodesic segments from a point to any Jordan measurable subset $Y$ of a totally
geodesic submanifold $\mathcal{Y}\subset \hyps$.
Let $N(r,Y,\Gamma p)$ be the number of orthogonal geodesic segments from $\gamma p$, for any $\gamma\in\Gamma$, to $Y$ with length at most $r$.
For cocompact $\Gamma$, Herrmann proves that
\begin{equation}\label{eq:herrmann}
N(r, Y, \Gamma p)\sim \frac{2}{n-1}\frac{\pi^{(n-k)/2}}{\Gamma(\frac{n-k}{2})}\frac{\vol(Y)}{\vol(\Gamma\backslash\hyps)}\cosh^{n-1}r.
\end{equation}
His method is geometric, not depending on the action of the group $\Gamma$ on $\hyps$. He introduces an associated Dirichlet series and
studies its analytic continuation. It is difficult to prove strong error terms with this method.
We are interested in the error term of (<ref>) for $n=3$ and $k=n-1$. We study this for $Y=\mathcal{Y}$ by adapting the method
of Huber and Chatzakos–Petridis
to $\hyps=\uphs$ for cocompact $\Gamma\subset\pslc$.
For the rest of this paper we fix $\hyps=\uphs$.
We also focus solely on cocompact $\Gamma$, see Remark <ref> for a discussion on cofinite groups.
Let $\{u_{j}\}_{j\geq 0}$ be a complete orthonormal system of eigenfunctions of the Laplacian $\Delta$
on $M$ with eigenvalues $\lambda_{j}=s_{j}(2-s_{j})\geq 0$.
Let $\hypp$ be a totally geodesic hyperplane in $\hyps$ and define $v(p)=\arctan(x_{2}(p)/y(p))$.
We prove the following theorem.
Let $\Gamma$ be a cocompact discrete subgroup of $\pslc$. Set $H=\Gamma\cap\stab_{\pslc}(\hypp)$ and let
$\period_{j}$ be the period integral of $u_{j}$ over the fundamental domain of $H$ restricted to $\hypp$.
\begin{equation*}
N(p,X)=\#\Set{\gamma\in\cosets\given(\cos v(\gamma p))^{-1}\leq X}.
\end{equation*}
\[N(p,X) = M(p,X) + \error,\]
\begin{equation}\label{eq:mainterm}
M(p,X)= \frac{\vol(H\backslash\hypp)}{\vol(\hypmodg)}X^{2}+\sum_{1<s_{j}<2}\frac{2^{s_{j}-1}}{s_{j}}\period_{j}u_{j}(p)X^{s_{j}},
\end{equation}
\[\error = O(X^{3/2}).\]
Here we understand $\vol(H\backslash\hypp)$ as the hyperbolic area in two dimensions.
We also apply Chamizo's large sieve results in this case.
For the radial average, Chamizo only provides a large sieve inequality in two dimensions.
We generalise it to three dimensions and prove an improvement of $1/6$ on the pointwise
error term on average.
In <cit.> we show that the radial large sieve yields the same improvement for
the error term in the standard hyperbolic lattice point problem in three dimensions.
Due to structural reasons the improvements from the large sieve get worse for higher dimensions.
We expect that in $n$ dimensions it is possible to obtain the conjectured bound for the second moment in the
spatial aspect while for the radial aspect we get diminishing returns even for the mean square.
We summarise our results in the following theorems. The corresponding discrete
averages and more precise statements are given in Theorems <ref> and <ref>.
Let $\Gamma$ be a cocompact discrete subgroup of $\pslc$. Then, for $X>2$,
\[\frac{1}{X}\int_{X}^{2X}\abs{\error[x][p]}^{2}\d{x}\ll X^{2+2/3}\log X.\]
Let $\Gamma$ be a cocompact discrete subgroup of $\pslc$. Then, for $X>2$,
\[\int_{\hypmodg}\abs{\error[X][p]}^{2}\d{\mu(p)}\ll X^{2}\log^{2}X.\]
It is also possible to obtain the radial mean square (<ref>) in the standard two dimensional lattice point problem
by direct integration in the spectral expansion of the error term.
This is done for a smoothed error term by phillips1994. It is possible to deduce
the result of Chamizo from their computations <cit.>.
It would be interesting to see if this can be done in our problem and whether it improves on the above estimate coming from the large sieve.
The other possible case in three dimensions, $k=1$, is substantially harder with our method. Currently, the spectral
expansion of the automorphic function corresponding to $N(p,X)$ can be written in terms of $\period_{j}$ and
an explicit solution to an ordinary differential equation. For $k=1$ we can no longer solve the corresponding
eigenvalue equation as it remains a partial differential equation.
Geometrically, $k=1$ corresponds to counting in a cone, while $k=2$ is counting in a sector.
The majority of computations in this paper are more explicit than in two dimensions (cf. <cit.>).
This is because the Selberg transform, the spherical eigenfunctions and the special functions in the spectral expansion
of the counting function (<ref>) can all be expressed in an elementary form.
We expect that there is always such a distinction between even and odd dimensions.
Dynamical systems and ergodic methods have also been applied to study lattice point counting. Their advantage
is that the results generally apply to a larger set of manifolds. On the other hand, these methods fail to produce
finer results, such as strong error terms.
For example, parkkonen2015
extend the counting in conjugacy classes problem of Huber to higher dimensions for loxodromic, parabolic and elliptic
conjugacy classes for any discrete group of isometries $\Gamma$. See also <cit.> for a survey on a wider
variety of counting problems analogous to herrmann1962.
Moreover, eskin1993 obtain main terms for a variety of counting problems on affine symmetric spaces
defined by Lie groups. In particular, they give an alternate proof for the main term on homogenous affine varieties, which was also
proved by duke1993 through spectral methods.
§ THE GEOMETRIC SETUP OF THE PROBLEM
We refer to <cit.> for the basic definitions concerning $\hyps$.
We denote points $p\in\hyps$ by $p=z+yj=(x_{1},x_{2},y)$, where $z=x_{1}+ix_{2}\in\complex$ and $y>0$.
The action of $\gamma=\begin{psmallmatrix}a & b\\ c&d\end{psmallmatrix}\in\Gamma$ on $\hyps$ is given by
\[\gamma p=\left(\frac{(az+b)\overline{(cz+d)}+a\overline{c}y^{2}}{\norm{cp+d}^{2}},\frac{y}{\norm{cp+d}^{2}}\right),\]
where $\norm{p}^{2}=\abs{z}^{2}+y^{2}$ is the Euclidean norm of $p$. Equipped with the hyperbolic metric, the hyperbolic distance between
$p,\,p'\in\hyps$ is given by
\[\cosh d(p,p')=\delta(p,p'),\]
where $\delta$ is the standard point-pair invariant.
This gives rise to the volume element
\begin{equation}\label{eq:volume}
\end{equation}
and the hyperbolic Laplacian
\[\Delta=y^{2}\left(\pd[2]{x_{1}}+\pd[2]{x_{2}}+\pd[2]{y}\right)-y\pd{y}.\]
The totally geodesic hypersurfaces in $\hyps$ are the Euclidean hyperplanes and semispheres orthogonal to the complex plane.
Motivated by Huber, we define a new set of coordinates. Let
\begin{align*}
x &= x_{1}, &
u &= \log \sqrt{x_{2}^{2}+y^{2}}, &
v &= \arctan\frac{x_{2}}{y},
\end{align*}
and transform to $p=(x(p), u(p), v(p))$. We often write $(x, u, v)$ for the same point as a shorthand if
the point in question is clear. The effect of this change of coordinates on the metric and the Laplacian
is summarised in the next lemma.
With the $(x,u,v)$ coordinates defined as above, we have
\begin{align*}
ds^{2} &= \frac{dx^{2}}{e^{2u}\cos^{2}v}+\frac{du^{2}+dv^{2}}{\cos^{2}v},\\
d\mu(p) &= \frac{dx\,du\,dv}{e^{u}\cos^{3}v},
\end{align*}
\[\Delta = e^{2u}\cos^{2}v\pd[2]{x} + \cos^{2}v\left(\pd[2]{u}+\pd[2]{v}\right)-\cos^{2}v\pd{u}+\sin v\cos v\pd{v}.\]
The Jacobian of the transformation $(x_{1},x_{2},y)\mapsto(x,u,v)$ is
\begin{equation*}
\begin{pmatrix} 1 & & \\
& \sin v\,e^{u} & \cos v\,e^{u}\\
& \cos v\,e^{u} & -\sin v\,e^{u}
\end{pmatrix},
\end{equation*}
so that the hyperbolic metric tensor in these coordinates is
\begin{equation}\label{eq:tensor}
(g_{ij})=\frac{1}{\cos^{2}v}\begin{psmallmatrix} e^{-2u}\\
& 1\\
& & 1
\end{psmallmatrix}.
\end{equation}
For the Laplacian we get
\begin{align*}
\Delta &= e^{u}\cos^{3}v\left(\pd{x}\left(\frac{e^{u}}{\cos v}\pd{x}\right)+\pd{u}\left(\frac{1}{e^{u}\cos v}\pd{u}\right)+\pd{v}\left(\frac{1}{e^{u}\cos v}\pd{v}\right)\right),
\end{align*}
which simplifies to the required form.
[->, thick] (0, 0) – (0, 4) node (taxis) [above] $y$;
[->, thick] (30:-2) – (30:3) node (xaxis) [above] $x_{1}$;
[->, thick] (-2, 0) – (3, 0) node (yaxis) [right] $x_{2}$;
(30:-1.5) coordinate (p1) – ++(0, 3) coordinate (p2) – ++(30:3) coordinate (p3) – ++(0, -3) coordinate (p4);
[opacity=0.4, gray] (p1) – (p2) – (p3) – (p4) – cycle;
[dashed, thin] (30:-1.5) – ++(20:2) – ++(30:3) – ++(20:-2);
[->, thick] (0,0) – (20:1.75) node (uaxis) [right] $e^{u}$;
(0,0) ++(90:0.5) arc (90:20:0.5) node[midway, yshift=0.7em,xshift=0.2em] $v$;
The $(x,u,v)$ coordinates in $\hyps$.
Now, let $\hypp$ be a totally geodesic hyperplane in $\hyps$. After conjugation by an element of $\pslc$,
we may assume that $\hypp$ is given by the set $\Set{p\in\hyps\given v=0}$ (i.e. $x_{2}=0$).
Notice that in this case $v(p)$ measures the angle between $p$ and $\hypp$ as shown in Figure <ref>.
Let $p\in\hyps$. We denote the orthogonal projection (along geodesics) of $p$ onto $\hypp$ by $\proj{p}=(x(p),u(p),0)$.
Next we identify all the elements of $\pslc$ that stabilise the plane $\hypp$. Since we are no longer working
with a single geodesic, the stabiliser will be larger than in the two dimensional setting.
The stabiliser of $\hypp\subset\pslc$ is
\[\stab_{\pslc}(\hypp)=\pslr\bigcup\left(\begin{smallmatrix}i &\\ &-i\end{smallmatrix}\right)\pslr.\]
Denote the stabiliser by $A$.
Let $\gamma=\left(\begin{smallmatrix}a & b\\ c& d\end{smallmatrix}\right)\in A$. By definition
we have that $\gamma p\in\hypp$ for any $p\in\hypp$, that is
$x_{2}(p)=0$ implies that $x_{2}(\gamma p)=0$. Hence,
\begin{equation*}
x_{2}(\gamma p) = \frac{\im(a\overline{c}x_{1}^{2}+(a\overline{d}+b\overline{c})x_{1}+b\overline{d}+a\overline{c}y^{2})}{\norm{cp+d}^{2}}=0.
\end{equation*}
This needs to be true for any $x_{1}$ and $y$. Comparing coefficients we get
\begin{align}
\im(a\overline{c}) & = 0,\label{eq:trig1}\\
\im(a\overline{d}+b\overline{c}) &=0,\label{eq:trig2}\\
\im(b\overline{d}) &=0.\label{eq:trig3}
\end{align}
Now, write $a,b,c,d$ in polar form as
\[\begin{aligned}
a&=r_{a}e^{\theta_{a}i}, & b&=r_{b}e^{\theta_{b}i},\\
c&=r_{c}e^{\theta_{c}i}, & d&=r_{d}e^{\theta_{d}i}.
\end{aligned}\]
Solving the equations (<ref>), (<ref>) and (<ref>) tells us that
either $\theta_{a}$, $\theta_{b}$, $\theta_{c}$, $\theta_{d}\in\{0,\pi\}$ or
$\theta_{a}$, $\theta_{b}$, $\theta_{c}$, $\theta_{d}\in\{\frac{-\pi}{2},\frac{\pi}{2}\}$. In the first case the matrix is real and hence
gives $\pslr$. In the second case we have, after considering the determinant,
\[\gamma=\begin{pmatrix}i & \\ & -i\end{pmatrix}\begin{pmatrix}r_{a} & r_{b}\\ r_{c} & r_{d}\end{pmatrix}\quad\text{or}\quad\begin{pmatrix}r_{a} & r_{b}\\ r_{c} & r_{d}\end{pmatrix}\begin{pmatrix} i & \\ & -i\end{pmatrix}.\]
Now define $H=\stab_{\pslc}(\hypp)\cap\Gamma$. We can then write the counting function as
\[\widetilde{N}(p,\Theta)=\#\Set{\gamma\in\cosets\given\abs{v(\gamma p)}\leq \Theta},\]
where $\Theta\in(0,\pi/2)$.
If we set $(\cos \Theta)^{-1}=X$, then $\widetilde{N}$ takes on the following form (cf. huber1998)
\[\widetilde{N}(p,\Theta)=N(p,X)=\#\Set*{\gamma\in\cosets\given\frac{1}{\cos v(\gamma p)}\leq X}.\]
Counting in the sector is equivalent to counting orthogonal geodesic segments from $\gamma p$ to $\hypp$ according to length.
Hence, we can easily relate the main term (<ref>) to that of Herrmann's (<ref>). Given a point $p\in\hyps$,
the projection $p_{0}$ in the $(x_{1},x_{2},y)$-coordinates is given by $p_{0}=(x_{1},0,\sqrt{x_{2}^{2}+y^{2}})$. Then,
by the explicit formula for the point-pair invariant, we get
\[\delta(p,p_{0})=\frac{\sqrt{x_{2}^{2}+y^{2}}}{y}=\sec v.\]
For $N( p, X)$ we have $X=\sec\Theta$ so that the maximal distance we are counting
is $\arccosh\sec\Theta$. Substituting this into (<ref>) with $n=3$ and $k=2$ shows that the main terms agree.
[->] (0, 0) – (0, 4) node (taxis) [above] $y$;
[->] (30:-2) – (30:3.5) node (xaxis) [above] $x_{1}$;
[->] (-4, 0) – (4, 0) node (yaxis) [right] $x_{2}$;
(30:-1.5) coordinate (p1) – ++(0, 3) coordinate (p2) – ++(30:3) coordinate (p3) – ++(0, -3) coordinate (p4);
(90:0.7) coordinate (l2) +(0,2.3) coordinate (l1);
(30:0.5) ++(0,0.30) coordinate (l3) +(0,2.7) coordinate (l4);
[fill opacity=0.5, thick, fill=black] (l1) – (l2) .. controls ($ (l2) + (30:0.2) $) and ($ (l3) + (120:0.2) $) .. (l3) – (l4);
[thick] (0,0) ++(1.0,0) coordinate (m2) – +(1.7,0) coordinate (m1);
[thick] (l3) ++(-90:0.3) ++(0.3,0) coordinate (m3) – +(2.3,0) coordinate (m4);
[thick] (m2) .. controls ($(m2) +(120:0.2)$) and ($(m3) +(-30:0.2)$) .. (m3);
[thick] (0,0) ++(-0.7,0) coordinate (n2) – +(-1.9,0) coordinate (n1);
[thick] (l3) ++(-90:0.3) ++(-0.3,0) coordinate (n3) – +(-2.4,0) coordinate (n4);
[thick] (n2) .. controls ($(n2) +(30:0.2)$) and ($(n3) +(210:0.2)$) .. (n3);
[densely dashed] (n2) to[in=100,out=90,distance=26] (m2);
[densely dashed] (n3) to[in=100,out=90,distance=12] (m3);
[dashed] (n1) to[in=90,out=90,distance=113] (m1);
[dashed] (n4) to[in=90,out=90,distance=113] (m4);
[densely dashed] (n1) to (n4);
[densely dashed] (m1) to (m4);
Fundamental domain of $H$ in $\hyps$ with the part for $H\backslash\hypp$ highlighted (for $\Gamma=\pslzi$).
As we are working with invariance under $H$, we have to compute its fundamental domain,
$\fund[\hyps]{H}$. It has a particularly convenient description in the $(x,u,v)$-coordinates.
First, consider $H$ restricted to the plane $\hypp$ and denote the fundamental domain of $H$ in this space
by $S=\fund[\uph]{H}$.
We claim that $\fund[\hyps]{H}=F$, where $F$ is given by the union of rotated copies of $S$:
\[F=\bigcup_{\theta\in(-\frac{\pi}{2},\frac{\pi}{2})}S_{\theta},\]
and $S_{\theta}$ are defined by
\[S_{\theta}=\Set{p\in\hyps\given v(p)=\theta, \proj{p}\in S}.\]
This follows immediately from computing the action of $H$ on $\hyps$, which is seen to be independent of $v(p)$ in the $x$ and $u$-coordinates.
First, let $\iota=\begin{psmallmatrix}i & \\ & -i\end{psmallmatrix}$,
which acts on $p\in\hyps$ as a rotation by $\pi$ about the imaginary axis, $\iota p=\iota(z+yj)=-z+yj$,
so that
\begin{align*}
x(\iota p) &= -x(p),
& u(\iota p) & = u(p),
& v(\iota p) & = -v(p).
\end{align*}
On the other hand, for $\tau=\left(\begin{smallmatrix}a & b\\ c&d\end{smallmatrix}\right)\in\pslr$, we find that
\begin{align*}
x(\tau p)% &= \frac{ac\abs{z}^{2} +(ad+bc)x+bd+acy^{2}}{c^{2}\abs{z}^{2}+2cdx+d^{2}+c^{2}y^{2}}\\
&= \frac{ac\norm{p}^{2}+(ad+bc)x+bd}{c^{2}\norm{p}^{2}+2cdx+d^{2}}.
\end{align*}
And since $\norm{p}^{2}=x^{2}+e^{2u}$ it follows that $x(\tau p)$ does not depend on $v(p)$.
\begin{align*}
x_{2}(\tau p)& = \frac{x_{2}}{\norm{cp+d}^{2}},
& y(\tau p)&= \frac{y}{\norm{cp+d}^{2}},\\
\shortintertext{so that}
u(\tau p) &= \log\frac{\sqrt{x_{2}^{2}+y^{2}}}{\norm{cp+d}^{2}} & v(\tau p) &= v(p),\\
&= u(p)-\log\norm{cp+d}^{2}.\\
\end{align*}
It is easy to see that $\iota\tau\iota\in\slr$. Hence any $\gamma\in H$ can be written as $\iota\tau$, $\tau\iota$ or $\tau$.
The first consequence of the above calculations is that for $p\in\hyps$ and $\gamma\in H$ the action of the group and the
orthogonal projection to $\hypp$ commute, i.e. $\proj{(\gamma p)}=\gamma\proj{p}$.
Moreover, if
$v(p)=\theta$ then $v(\gamma p)=\pm\theta$ for any $\gamma\in H$. Thus, suppose that
$p,\gamma p\in S_{\pm\theta}$ for some $\theta\in(0,\pi/2)$ and $\gamma\in H$. It follows that
$\proj{p}, \gamma\proj{p}\in S$, which is a contradiction as $S$ is a fundamental domain for $H$ on the plane. This shows
that $\fund[\hyps]{H}\subseteq F$. Suppose, $\fund[\hyps]{H}\varsubsetneq F$. Then for some $\theta$ there is a point $p\in S_{\theta}$
and $\gamma\in H$ with $\gamma p\in F\setminus\fund[\hyps]{H}$. Projecting back to $S$ gives a contradiction.
For cocompact $\Gamma$ it is easy to see that $N(p,X)$ is uniformly bounded in $p$. On the other hand, for cofinite $\Gamma$
it is not true in general, although it is still possible to see that $N(p,X)$ is well-defined (finite for fixed $X$ and $p$).
For example, if $\Gamma=\pslzi$ then as $y(p)\tendsto\infty$, $N(p,X)$ becomes unbounded.
This introduces complications for the convergence of the corresponding automorphic form and is the main
reason for our restriction to cocompact groups. The problem lies in the fact that for non-compact $M$
the totally geodesic
surface can pass through the cusp. It should be possible to overcome this difficulty by restricting
the group $\Gamma$ appropriately.
§ SPECTRAL ANALYSIS
Let $\Gamma\subset\pslc$ be cocompact.
Define a function $A(f)$ on $\hyps$ by
\[A(f)(p) = \sum_{\gamma\in\cosets}f\left(\frac{1}{\cos^{2}v(\gamma p)}\right),\]
where $\func{f}{[1,\infty)}{\reals}$ has a compact support and finitely many discontinuities.
Then it is easy to see that $A(f)$ is automorphic, since $v(\gamma p)$ is constant
on the cosets. Since $A$ is sufficiently smooth (in $C^{2}(M)$), it follows by <cit.> that $A$ converges
pointwise to its spectral expansion.
Let $\{u_{j}\}_{j\geq 0}$ be a complete orthonormal system of automorphic eigenfunctions of $-\Delta$ with corresponding eigenvalues $\lambda_{j}$.
Since our problem differs from the standard lattice point counting problem, in that $A(f)$ does not define an automorphic kernel, we do not
have the usual expansion in terms of the Selberg transform of $f$. The correct substitute for this is the spectral expansion of $A(f)$ in term
of the $u_{j}$'s.
Let $a(f,t_{j})$ be the coefficients of the spectral expansion of $A(f)$ on $\hypmodg$ given by
\[a(f,t_{j}) = \langle A(f),u_{j}\rangle= \int_{\hypmodg}A(f)(p)\overline{u}_{j}(p)\d{\mu(p)}.\]
Then the spectral expansion of $A(f)$ in terms of the $u_{j}$'s is
\[A(f)(p) = \sum_{j} a(f,t_{j})u_{j}(p).\]
We now compute the coefficients $a(f,t_{j})$ explicitly in the manner of <cit.> and <cit.>.
Following this, we identify the special function that appears in the spectral expansion and prove some simple estimates on it.
For simplicity, consider $u_{j}$ instead of
We have
\begin{equation*}
a(f,t_{j}) = 2\period_{j} c(f,t_{j}),
\end{equation*}
\[\period_{j}=\int_{H\backslash\hypp}u_{j}(x,u,0)\frac{du\,dx}{e^{u}},\]
is a period-integral of $u_{j}$ over the fundamental domain of $H$ restricted to the plane $\hypp$.
\[c(f,t_{j})=\int_{0}^{\frac{\pi}{2}}f\left(\frac{1}{\cos^{2}v}\right)\frac{\xi_{\lambda_{j}}(v)}{\cos^{3}v}\d{v},\]
where $\xi_{\lambda}$ is the solution of the ordinary differential equation
\[\cos^{2}v\,\xi_{\lambda}''(v)+\sin v\cos v\,\xi_{\lambda}'(v)+\lambda\xi_{\lambda}(v)=0,\]
with the initial conditions
\[\xi_{\lambda}(0)=1,\quad\xi_{\lambda}'(0)=0.\]
In the following proofs we work with a fixed $\lambda$ and denote $\xi_{\lambda}$ by $\xi$.
Unfolding the spectral coefficients,
\begin{align*}
a(f,t_{j}) &= \int_{\hypmodg}\sum_{\gamma\in\cosets}f\left(\frac{1}{\cos^{2}v(\gamma p)}\right)u_{j}(p)\d{\mu(p)}\\
&= \int_{H\backslash\hyps}f\left(\frac{1}{\cos^{2}v}\right)u_{j}(x,u,v)\frac{dx\,du\,dv}{e^{u}\cos^{3}v}.
\end{align*}
We can express this in terms of the period integral as
\[a(f,t_{j})=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f\left(\frac{1}{\cos^{2}v}\right)\varphi_{j}(v)\frac{dv}{\cos^{3} v},\]
\[\varphi_{j}(v) = \int_{S_{v}} u_{j}(x,u,v)\frac{du\,dx}{e^{u}}.\]
It is immediate that $\varphi_{j}$ is even.
According to Lemma <ref>, in our new coordinates the eigenvalue equation becomes:
\[e^{2u}\cos^{2}v\pd[2][u_{j}]{x} + \cos^{2}v\left(\pd[2][u_{j}]{u}+\pd[2][u_{j}]{v}\right)-\cos^{2}v\pd[][u_{j}]{u}+\sin v\cos v\pd[][u_{j}]{v}+\lambda u_{j} = 0.\]
Now, dividing by $e^{u}$ and integrating over $S_{v}$ we get
\begin{multline}\label{eigeneq}
\cos^{2}v\int_{S_{v}}e^{2u}\pd[2][u_{j}]{x}\frac{dx\,du}{e^{u}}+\cos^{2}v\int_{S_{v}}\pd[2][u_{j}]{u}\frac{du\,dx}{e^{u}}+\cos^{2}v\pd[2][\varphi_{j}]{v}\\
-\cos^{2}v\int_{S_{v}}\pd{u}\int u_{j}\frac{dx\, du}{e^{u}}+\sin v\cos v\pd[][\varphi_{j}]{v}+\lambda\varphi_{j}(v)=0.
\end{multline}
Next, notice that the Laplacian on $S_{v}$ in the induced metric is exactly the restriction of $\Delta$ to $S_{v}$, that is,
$\Delta\restriction_{S_{v}}=\cos^{2}v \Delta_{S_{v}}$, where
\[\Delta_{S_{v}}=e^{2u}\pd[2]{x}+\pd[2]{u}-\pd{u}.\]
Hence, for a fixed $v$, (<ref>) becomes
\begin{equation}\label{eq:eigeneq2}
\cos^{2}v\int_{S_{v}}\Delta_{S_{v}}u_{j}\frac{dx\,du}{e^{u}}+\cos^{2}v\pd[2][\varphi_{j}]{v}+\sin v\cos v\pd[][\varphi_{j}]{v}+\lambda\varphi_{j}(v)=0.
\end{equation}
Denote the integral in (<ref>) by $I_{v}$. Then, by Stokes' theorem we have that
\[I_{v}=\int_{\partial S_{v}}\nabla u_{j}\cdot\vec{n}\d{\ell},\]
where $d\ell$ is the line element on $S_{v}$ and $\vec{n}$ is the unit normal vector (on the plane) to $S_{v}$. We wish to show that $I_{v}=0$.
To do this, recall some basic terminology for fundamental domains (in $\uph$) from beardon1983.
Let $\fund{}$ be a fundamental domain of a cofinite or cocompact Fuchsian group $G$.
Then $\fund{}$ is a convex hyperbolic polygon with finitely many sides.
A side of $\fund{}$ is a geodesic segment of the form $\overline{\fund{}}\cap g\overline{\fund{}}$ for any $g\in G$ with $g\neq I$.
A vertex of $\fund{}$ is a point of the form $\overline{\fund{}}\cap g\overline{\fund{}}\cap h\overline{\fund{}}$ for any $g\neq h\in G$ such that
$g,h\neq I$. If $\partial\fund{}$ contains an elliptic fixed point of $g\in G$ of order 2, then we consider the fixed point
as a vertex of $\fund{}$ and moreover $g$ identifies the adjacents sides with opposite orientation. In general, we can always find
a side-pairing for $\fund{}$, that is, for $i=1$, $\ldots$, $k$, there exist triples $(\Lambda_{i},\Psi_{i},g_{i})$ such that
$g_{i}\Lambda_{i}=\Psi_{i}$, and $g_{i}$ is the unique element in $G$ that does this, and that $\Lambda_{i}$ or $\Psi_{i}$ are not
paired with any other sides of $\fund{}$. Finally, we can always choose $\fund{}$ so that if we consider $\partial\fund{}$ as a contour
in $\hyps$, then the congruent sides occur with opposite orientation as segments of the contour <cit.>.
So, let $\Set{(\Lambda_{i},\Psi_{i},g_{i})\given i=1,\ldots, k}$ be a side-pairing of $S$. Then it immediately follows that
for any $S_{v}$ we get a corresponding side-pairing. Denote these by $\Set{(\Lambda_{i}^{v},\Psi_{i}^{v},g_{i})\given i=1,\ldots,k}$,
where $\proj{(\Lambda_{i}^{v})}=\Lambda_{i}$ and $\proj{(\Psi_{i}^{v})}=\Psi_{i}$. It follows that $I_{v}=0$ as the integral over $\Lambda_{i}^{v}$
is cancelled by the one over $\Psi_{i}^{v}$ since $\nabla u_{j}\cdot\vec{n}$ is invariant under $H$.
We are left with
\begin{equation}\label{eq:ode}
\cos^{2}v\,\varphi''(v)+\sin v\cos v\, \varphi'(v)+\lambda\varphi(v)=0,
\end{equation}
where $\varphi'(0)=0$, as $\varphi$ is even.
\[\omega(v)=\varphi(v)+\varphi(-v),\]
for $v\in(-\pi/2,\pi/2)$.
Hence, adding (<ref>) evaluated at $-v$ to itself yields
\begin{equation}\label{eq:fullode}
\cos^{2}v\,\omega''(v) + \sin v\cos v\,\omega'(v) + \lambda\omega(v)=0,
\end{equation}
with $\omega(0) = 2\period_{j}$ and $\omega'(0) = 0$.
Now, suppose that $\xi(v)$ is a solution to the second order homogenous linear ODE
\begin{equation}\label{ode}
\cos^{2}v\,\xi''(v)+\sin v\cos v\,\xi'(v)+\lambda\xi(v)=0,
\end{equation}
with initial conditions $\xi(0) = 1$ and $\xi'(0) = 0$.
Then we can write the full solution $\omega$ of (<ref>) as
\[\omega(v)=2\period_{j}\xi(v).\]
Therefore, the $a(f,t_{j})$'s can be written as
\begin{align*}
a(f,t_{j}) &=\left(\int_{0}^{\pi/2}+\int_{-\pi/2}^{0}\right)f\left(\frac{1}{\cos^{2}v}\right)\varphi(v)\frac{dv}{\cos^{3}v}\\
\end{align*}
We will also need some estimates on $\xi_{\lambda}$. Notice that the following lemma does not use the explicit
form of $\xi_{\lambda}$ (which we will compute later). This computation is analogous to <cit.>.
For all $v\in[0,\pi/2)$ we have
\begin{align}
\abs{\xi_{\lambda}(v)}&\leq 1,\label{eq:xibound1} \\
\xi_{\lambda}(v)&\geq 1-\frac{2+\lambda}{2}\tan^{2}v.\label{eq:xibound2}
\end{align}
Multiplying (<ref>) by $2\xi'(v)$, we can write
\[\cos^{2}v \left(\xi'(v)^{2}\right)'+2\sin v\cos v (\xi'(v))^{2}+\lambda(\xi(v)^{2})'=0.\]
Now, integrate over $[0,x]$ and use $\xi(0)=1$ and $\xi'(0)=0$
\[\lambda(1-\xi(x)^{2})=\cos^{2}x \xi'(x)^{2}+2\int_{0}^{x}\sin 2v\, \xi'(v)^{2}\d{v}\geq 0,\]
since $x\in[0,\pi/2)$ so that $\sin 2v$ is non-negative.
This proves the first part.
Now we can apply (<ref>) to get
\begin{align*}
\xi''(v) &\geq -\tan v \xi'(v)-\lambda\sec^{2}v.
\end{align*}
Integrating twice over $[0,x]$ and $[0,v]$ yields
\[\xi(v)-1\geq(2+\lambda)\log(\cos v)\geq \frac{-1}{2}(2+\lambda)\tan^{2}v.\]
\[\xi(v)\geq 1-\frac{2+\lambda}{2}\tan^{2}v.\]
With this, we have the following Hecke type bound for the mean square of the period integrals.
Let $\period_{j}$ be the period integral over $H\backslash\hypp$ of the automorphic form $u_{j}\in L^{2}(M)$. Then, for $T>1$
\[ \sum_{\abs{t_{j}}\leq T}\abs{\period_{j}}^{2}\ll T.\]
This is a surprising result in the sense that the order of growth is better than what we expect from the local Weyl law.
We suspect that the mean square should be bounded in all dimensions, cf. [Theorem 1]tsuzuki2009.
The proof is analogous to <cit.>.
Let $K=\sup_{p\in\hyps}N(p,X)$. This is well-defined as $N(p,X)$ is uniformly bounded.
\[\int_{\hypmodg}(A(f)(p))^{2}\d{\mu(p)}\leq K\int_{\hypmodg}A(f)(p)\d{\mu(p)}.\]
Also, define
\[\tan\theta = \sqrt{\frac{2}{y+2}}.\]
\[a(f,t_{j})=2\period_{j}\int_{0}^{\theta}\frac{\xi_{\lambda_{j}}(v)}{\cos^{3} v}\d{v}\]
with $\xi_{0}(v)=1$. In particular, the coefficient for the zero eigenvalues (so $t_{j}=i$) gives
\begin{align*}
a(f, t_{0}) &= 2 \period_{0}\int_{0}^{\theta}\cos^{-3}v\d{v}\\
&= 2 \period_{0}\left(\frac{1}{\sqrt{2}}\frac{\sqrt{4+y}}{y+2}+\frac{1}{2}\log\abs*{\frac{\sqrt{2}+\sqrt{4+y}}{\sqrt{y+2}}}\right)\\
&= 2 \period_{0}g(y),
\end{align*}
say. On the other hand,
\[\frac{1}{2}a(f,t_{0})=\period_{0}g(y)=u_{0}\int_{\hypmodg}A(f)(p)\d{\mu(p)}.\]
It follows that
\begin{equation}\label{eq:pars1}
\int_{\hypmodg} (A(f)(p))^{2}\d{\mu(p)} \leq K\frac{\period_{0}g(y)}{u_{0}},
\end{equation}
where $u_{0}$ is the constant eigenfunction.
By Parseval we have
\[\int_{\hypmodg} (A(f)(p))^{2}\d{\mu(p)} = \sum_{j=0}^{\infty}\abs{a(f,t_{j})}^{2}\geq \sum_{\lambda_{j}\leq y}\abs{a(f,t_{j})}^{2},\]
so that
\begin{equation}\label{eq:pars2}
\int_{\hypmodg} (A(f)(p))^{2}\d{\mu(p)} \geq \sum_{\lambda_{j}\leq y}\abs{\period_{j}}^{2}\left(\int_{0}^{\theta}\frac{\xi_{\lambda_{j}}(v)}{\cos^{3}v}\d{v}\right)^{2}.
\end{equation}
Since $\lambda_{j}\leq y$, from the bound (<ref>) it follows that:
\[\xi_{\lambda_{j}}\geq 1-\frac{2+\lambda_{j}}{2}\tan^{2}v\geq 1-\frac{2+y}{2}\tan^{2}v.\]
\begin{align*}
\int_{0}^{\theta}\frac{\xi_{\lambda_{j}}(v)}{\cos^{3}v}\d{v} &\geq\int_{0}^{\theta}\left(1-\frac{2+y}{2}\tan^{2}v\right)\frac{dv}{\cos^{3}v}\\
&=\frac{1}{8\tan^{2}\theta}\left(\tan\theta \sec\theta (2\sec^{2}\theta - 3)+(1+4\tan^{2}\theta)\log\left(\frac{1+\tan\frac{\theta}{2}}{1-\tan\frac{\theta}{2}}\right)\right)\\
\end{align*}
We are interested in the behaviour of $h$ as $y\tendsto\infty$, that is, $\theta\tendsto0^{+}$.
After a tedious but elementary computation we find that
\[\lim_{\theta\tendsto0^{+}}\frac{h(\theta)}{\tan\theta}=\frac{2}{3}.\]
This means that $h(\theta)\geq(\frac{2}{3}-\epsilon)\tan\theta$ for small enough $\theta$ and for some $\epsilon>0$. In other words, we have proved that
\begin{equation}\label{eq:pars3}
\int_{0}^{\theta}\frac{\xi_{\lambda_{j}}(v)}{\cos^{3}v}\d{v}\geq cy^{-1/2},
\end{equation}
for some constant $c>0$, as $y\tendsto\infty$. Now, combining (<ref>), (<ref>) and (<ref>) we get
\[\sum_{\lambda_{j}\leq y}\abs{\period_{j}}^{2}\leq \frac{K\period_{0}}{c^{2}u_{0}}g(y)y\ll y^{1/2}.\]
The result follows from observing that $\lambda_{j}=1+t_{j}^{2}$.
As pointed out earlier, we can actually express $\xi$ in an elementary form. We suspect that this is always possible in
odd dimensional hyperbolic space. In even dimensions the special functions are more complicated Legendre or hypergeometric functions.
Let $r=\tan v$, then
\begin{equation}\label{fourierc}
a(f,t_{j})=2\period_{j}\int_{0}^{\infty}f(1+r^{2})\xi(\arctan r)\sqrt{1+r^{2}}\d{r}.
\end{equation}
Apply the transformation $\tan v = \sinh w$ in (<ref>). It becomes
\[\xi''(w)+2\tanh w\, \xi'(w)+\lambda\xi(w)=0.\]
Since $\lambda =s_{j}(2-s_{j})$, we have $1-\lambda=(s_{j}-1)^{2}$. It is then easy to see that the solution with our initial conditions is
\begin{equation}\label{eq:xiw}
\xi(w)=\frac{\cosh w(s_{j}-1)}{\cosh w},
\end{equation}
or in terms of $r$,
\begin{equation}\label{eq:xir}
\xi(r)=\frac{\cosh((s_{j}-1)\arcsinh r)}{\sqrt{1+r^{2}}}.
\end{equation}
Thus we can finally write the explicit form for $\xi$ in terms of $v$ as
\begin{equation}\label{eq:xiv}
\xi(v)=\frac{\cosh((s_{j}-1)\arcsinh\tan v)}{\sec v}.
\end{equation}
We will now show how to estimate the spectral coefficients $a(f,t_{j})$.
With the explicit form (<ref>) for $\xi$, we can write
\begin{equation}
a(f,t_{j})=2\period_{j}\int_{0}^{\infty}f(1+r^{2})\cosh((s_{j}-1)\arcsinh r)\d{r}.
\end{equation}
We are thus led to consider the integral transform
\[c(f,t)=\int_{0}^{\infty}f(1+r^{2})\cosh((s-1)\arcsinh r)\d{r},\]
where $s=1+it$.
Now, define
\[f\left(\frac{1}{\cos^{2}v(p)}\right)=\begin{cases} 1,&\text{if $0\leq \abs{v}\leq \Theta$,}\\ 0,&\text{if $\Theta<\abs{v}<\frac{\pi}{2}$,}\end{cases}\]
or equivalently
\[f\left(\frac{1}{\cos^{2}v(p)}\right)=\begin{cases} 1,&\text{if $1\leq \sec v\leq X $,}\\ 0,&\text{if $X<\sec v$.}\end{cases}\]
If we let $r=\tan v$, then we get that
\[f\left(1+r^{2}\right)=\begin{cases} 1,&\text{if $0\leq r\leq U $,}\\ 0,&\text{if $U<r$,}\end{cases}\]
where $U=\tan\Theta=\sqrt{X^{2}-1}$. Notice that
\[U = X\sqrt{1-X^{-2}}=X\left(1+O(X^{-2})\right)=X+O(X^{-1}).\]
In particular,
\[A(f)(p)=\widetilde{N}(p,\Theta)=N(p,X).\]
Now, letting $r=\sinh u$, we can rewrite $c(f,t)$ as
\[c(f,t)=\int_{0}^{\infty}f(\cosh^{2}u)\cosh((s-1)u)\cosh u\d{u}.\]
Notice that $2\cosh((s-1)u)\cosh u=\cosh su+\cosh(2-s)u$, so that
\[c(f,t)=\frac{1}{4}\int_{\reals}f(\cosh^{2}u)\cosh su\d{u}+\frac{1}{4}\int_{\reals}f(\cosh^{2}u)\cosh (2-s)u\d{u},\]
\[f(\cosh^{2}u) =\begin{cases} 1, &\text{ if $\abs{u}\leq\arcsinh U$,}\\ 0, &\text{otherwise.}\end{cases}\]
Since both of the integrals in $c(f,t)$ are of the same type, we define the integral transform $d(f,s)$ given by
\begin{equation}\label{eq:dtransform}
d(f,s)=\int_{\reals}f(\cosh^{2}u)\cosh su\d{u}.
\end{equation}
We list some simple properties of the $d(f,s)$-transform without proof.
Suppose $f$ and $g$ are compactly supported even functions with finitely many discontinuities, let $\alpha\in\reals$, then
\begin{align*}
4c(f,t) &= d(f,s) + d(f,2-s),\\
d(\alpha f, s)&=\alpha d(f,s),\\
d(f\ast g,s)&=d(f,s)d(g,s),\\
\intertext{where $\ast$ is the usual convolution. Also}
d(\ind_{[-T,T]},s)&=\frac{2\sinh sT}{s},\\
\intertext{where $\ind_{[-T,T]}(\cosh^{2}u)$ is the indicator function on $[-T,T]$, and}
\end{align*}
Let $1>\delta>0$, and define $\chi(\cosh^{2}u)=(2\delta)^{-1}\ind_{[-\delta,\delta]}(\cosh^{2}u)$ to be a characteristic function with unit mass with respect to the
Now define,
\[\widetilde{f}^{+}(\cosh^{2}u)=\begin{cases}
1, & \text{if $\abs{u}\leq \arcsinh U+2\delta$,}\\
0, & \text{otherwise.}
\end{cases}\]
\[\widetilde{f}^{-}(\cosh^{2}u)=\begin{cases}
1, & \text{if $\abs{u}\leq \arcsinh U-2\delta$,}\\
0, & \text{otherwise.}
\end{cases}\]
Let $f^{+}=\widetilde{f}^{+}\ast\chi\ast\chi$ and $f^{-}=\widetilde{f}^{-}\ast\chi\ast\chi$.
Then $f^{+}(x)=1$ for $\abs{x}\leq \arcsinh U$ and vanishes for $\abs{x}\geq \arcsinh U+4\delta$, and similarly
$f^{-}(x)$ vanishes for $\abs{x}\geq \arcsinh U$. It follows that
\[A(f^{-})(p) \leq N(p,X) \leq A(f^{+})(p).\]
Hence, in order to estimate $N(p,X)$ we need bounds for $A(f^{+})$ and $A(f^{-})$, which in turn leads us to
investigate $c(f^{\pm},t)$. The case for $f^{-}$ is analogous, so we restrict the treatment below to $f^{+}$.
Without any smoothing, the spectral expansion for $A(f)$ would of course not converge. In two dimensions
it suffices to use a single convolution (linear decay). In our case we need at least two convolutions to
ensure convergence.
On the other hand, any more smoothing in this manner does not yield improvements for the pointwise bound
nor for the application of the large sieve.
The integral transform $c(f,t)$ satisfies the following properties:
* For $s=1+it$ we can write
\begin{equation}\label{eq:oscillatory}
c(f^{+},t) = a(t, \delta)X^{1+it} + b(t,\delta)X^{1-it},
\end{equation}
where $a$ and $b$ satisfy
\[a(t,\delta),\, b(t,\delta) \ll \min(\abs{t}^{-1},\abs{t}^{-3}\delta^{-2}).\]
* For $s\in[1,2]$ we have
\[c(f^{+},t) = \frac{2^{s-2}}{s}X^{s}+\frac{2^{-s}}{2-s}X^{2-s}+O(\delta X^{s}),\]
where the case of $s=2$ is understood as
\[c(f^{+},i) = \frac{X^{2}}{2}+O(\delta X^{2}).\]
We have
\begin{align*}
d(f^{+},s) &=\frac{8\sinh s(\arcsinh U+2\delta)\sinh^{2}s\delta}{(2\delta)^{2}s^{3}}\\
&= \frac{8\sinh^{2}s\delta}{(2\delta)^{2}s^{3}}\sinh (s\log(U+\sqrt{U^{2}+1}) +2s\delta).
\end{align*}
By Taylor expansion $U+\sqrt{U^{2}+1}=2U+O(U^{-1})$, so that
\[(U+\sqrt{U^{2}+1})^{s}=(2U)^{s}+O(sU^{s-2})=(2X)^{s}+O(sX^{s-2}).\]
Now suppose $s\in[1,2)$, then we may assume that $\abs{s}\delta<1$. So,
\begin{align*}
\sinh s(\arcsinh U +2\delta) %\sinh (s\arcsinh U)\cosh 2s\delta+\cosh (s\arcsinh U) \sinh 2s\delta\\
&= \frac{1}{2}\left((2X)^{s}+O(X^{s-2})\right)(1+O(\delta))+O(\delta X^{s}+\delta X^{s-2})\\
&= \frac{1}{2}(2X)^{s}+O(\delta X^{s}).
\end{align*}
It follows that
\[d(f^{+},s) =\frac{4\sinh^{2}s\delta}{(2\delta)^{2}s^{3}}\left((2X)^{s}+O(\delta X^{s})\right).\]
Since $\abs{s}\delta<1$, we also have that $(\sinh s\delta)/s\delta=1+O(\delta)$, and
\[d(f^{+},s) = \frac{1}{s}(1+O(\delta))((2X)^{s}+O(\delta X^{s}))=\frac{2^{s}}{s}X^{s}+O(\delta X^{s}).\]
\begin{equation*}
c(f^{+},t) =\frac{2^{s-2}}{s}X^{s}+\frac{2^{-s}}{2-s}X^{2-s}+O(\delta X^{s}).
\end{equation*}
Now, for the smallest eigenvalue, $s=2$, we get
\[c(f^{+},i) = \frac{1}{4}\left( 2X^{2}+O(\delta X^{2}) + O(\log X)\right),\]
as $d(\chi,0) = 1$.
This proves (<ref>) in the proposition.
We now consider the case when $s$ is complex, that is, $s=1+it$. Assume $t>0$ and $X>1$, to get
\begin{multline*}
\sinh((1+it)(\arcsinh U + 2\delta)) = \frac{1}{2}\biggl((2X+O(X^{-1}))^{1+it}e^{2\delta(1+it)}
\end{multline*}
Thus we can write
\begin{equation}\label{eq:sinhestimate}
\sinh((1+it)(\arcsinh U + 2\delta))=X^{1+it}\upsilon(t,\delta),
\end{equation}
where $\upsilon(t,\delta)$ is bounded for $0<\delta<1$ and any $t$. Hence,
\[d(f^{+},1+it)=X^{1+it}\upsilon(t,\delta)\frac{2}{s}\left(\frac{\sinh s\delta}{s\delta}\right)^{2}.\]
Now, suppose $\abs{s}\delta<1$, then $\sinh(s\delta)/(s\delta)\ll 1$.
So in this case
\[d(f^{+},1+it)=X^{1+it}\abs{t}^{-1}.\]
On the other hand, if $\abs{s}\delta\geq 1$, then $\sinh s\delta=O(1)$ so that
\[d(f^{+},1+it)=X^{1+it}\abs{t}^{-3}\delta^{-2}.\]
Working similarly with $d(f^{+},1-it)$ proves (<ref>).
Before we can prove the theorem, we need to know the local Weyl's law in our setting.
elstrodt1998 prove this for Eisenstein series in Chapter 6 Theorem 4.10.
It is clear that their proof can be extended to include the cuspidal part. This yields the following lemma.
For $T>1$, we have for all $p\in\hyps$ that
\[\sum_{t_{j}\leq T}\abs{u_{j}(p)}^{2}\ll y(p)^{2}T+T^{3}.\]
We now have all the ingredients to prove our main theorem.
First, write the spectral expansion of $A(f^{+})(p)$:
\begin{align*}
A(f^{+})(p) &= \sum_{j}2c(f^{+},t_{j})\period_{j}u_{j}(p)\\
&= X^{2}\period_{0}u_{0}+\sum_{s_{j}\in[1,2)}2\period_{j}u_{j}(p)\left(\frac{2^{s_{j}-2}X^{s_{j}}}{s_{j}}+\frac{2^{-s_{j}}X^{2-s_{j}}}{2-s_{j}}+O(\delta X^{s_{j}})\right)\\
&\phantom{=} +\sum_{t_{j}\in\reals}2c(f^{+},t_{j})\period_{j}u_{j}(p).
\end{align*}
Now the summation over $s_{j}\in[1,2)$ is finite, so
\[A(f^{+})(p)=\sum_{s_{j}\in(1,2]}\frac{2^{s_{j}-1}X^{s_{j}}}{s_{j}}\period_{j}u_{j}(p)+G(f^{+},p)+O(X+\delta X^{2}),\]
\[G(f^{+},p) = \sum_{0\neq t_{j}}2c(f^{+},t_{j})\period_{j}u_{j}(p).\]
Again, by the discreteness of the spectrum we can estimate the contribution of small $t_{j}$'s
\[G(f^{+},p) = \sum_{\abs{t_{j}}\geq 1}2c(f^{+},t_{j})\period_{j}u_{j}(p)+O(X).\]
Now, since $c(f^{+},t)$ is even in $t$, we get by a dyadic decomposition
\begin{align*}\label{eq:dyaestimate}
\sum_{\abs{t_{j}}\geq 1}2c(f^{+},t_{j})\period_{j}u_{j}(p) &\ll \sum_{t_{j}\geq 1}c(f^{+},t_{j})\period_{j}u_{j}(p)\notag\\
&= \sum_{n=0}^{\infty}\left(\sum_{2^{n}\leq t_{j}< 2^{n+1}}c(f^{+},t_{j})\period_{j}u_{j}(p)\right)\notag\\
&\ll \sum_{n=0}^{\infty}\sup_{2^{n}\leq t_{j}<2^{n+1}}c(f^{+},t_{j})\left(\sum_{2^{n}\leq t_{j}<2^{n+1}}\period_{j}u_{j}(p)\right).
\end{align*}
By the Cauchy–Schwarz inequality and Lemmas <ref> and <ref> we have
\begin{align*}
G(f^{+},p) &\ll \sum_{n=0}^{\infty}\sup_{2^{n}\leq t_{j}<2^{n+1}}c(f^{+},t_{j})\left(\sum_{t_{j}<2^{n+1}}\abs{\period_{j}}^{2}\right)^{1/2}\left(\sum_{t_{j}<2^{n+1}}\abs{u_{j}(p)}^{2}\right)^{1/2}+X\\
&\ll \sum_{n=0}^{\infty}\sup_{2^{n}\leq t_{j}<2^{n+1}}c(f^{+},t_{j})2^{2n+2}+X.\\
\intertext{We separate the sum over $n$ depending on whether $t_{j}\delta\leq 1$ or $t_{j}\delta\geq 1$,}
G(f^{+},p)&\ll \sum_{n<\log_{2}\delta^{-1}}2^{2n+2}\sup_{2^{n}\leq t_{j}<2^{n+1}}c(f^{+},t_{j})+\sum_{n>\log_{2}\delta^{-1}}2^{2n+2}\sup_{2^{n}\leq t_{j}<2^{n+1}}c(f^{+},t_{j})+X.\\
\intertext{Hence, by Proposition~\ref{prop:estimates},}
G(f^{+},p)&\ll \sum_{n<\log_{2}\delta^{-1}}2^{2n+2}X2^{-n}+\sum_{n>\log_{2}\delta^{-1}}2^{2n+2}X\delta^{-2}2^{-3n}+X\ll X\delta^{-1}+X.
\end{align*}
Putting all this together we find that
\begin{equation}\label{eq:finalestimate}
A(f^{+})(p) = \sum_{s_{j}\in(1,2]}\frac{2^{s_{j}-1}X^{s_{j}}}{s_{j}}\period_{j}u_{j}(p)+O(X+\delta X^{2}+\delta^{-1} X).
\end{equation}
The optimal choice for $\delta$ comes from equating $\delta X^{2}=\delta^{-1} X$, which gives $\delta=X^{-1/2}$. The result follows from noting
that $u_{0}=\vol(\hypmodg)^{-1/2}$ and $\period_{0}=\vol(S)\vol(\hypmodg)^{-1/2}$.
§ APPLICATIONS OF THE LARGE SIEVE
We will now apply Chamizo's large sieve inequalities to show that the mean square of the error term $\error[X][p]$ satifies the
conjectured bound $O(X^{1+\epsilon})$ over a spatial average.
In the radial aspect Chamizo proves large sieve inequalities with exponential weights for all moments in two dimensions.
We extend his result to three dimensions for the second moment. We can only prove a mean square estimate of
$O(X^{2+2/3})$ in the radial average. This translates to an improvement of $1/6$ compared to the pointwise
bound we obtained in Section <ref>.
More specifically, our aim is to prove the following two theorems.
Let $X>2$ and $X_{1},\ldots,X_{R}\in[X,2X]$ such that $\abs{X_{k}-X_{l}}>\epsilon>0$ for all $k\neq l$.
Suppose $R\epsilon\gg X$ and $R> X^{2/3}$, then
\begin{equation}\label{eq:disclimit1}
\frac{1}{R}\sum_{k=1}^{R}\abs{\error[X_{k}]}^{2}\ll X^{2+2/3}\log X.
\end{equation}
Taking the limit $R\rightarrow\infty$ gives
\begin{equation}\label{eq:intlimit1}
\frac{1}{X}\int_{X}^{2X}\abs{\error[x]}^{2}\d{x}\ll X^{2+2/3}\log X.
\end{equation}
For $p,q\in\hypmodg$, let
\[\tilde{d}(p,q)=\inf_{\gamma\in\Gamma}d(p,\gamma q),\]
be the induced distance on $\hypmodg$.
Let $X>2$ and $p_{1},\ldots,p_{R}\in\hypmodg$ with $\tilde{d}(p_{k},p_{l})>\epsilon>0$ for all $k\neq l$.
Suppose $R\epsilon^{3}\gg 1$ and $R>X$, then
\begin{equation}\label{eq:disclimit2}
\frac{1}{R}\sum_{k=1}^{R}\abs{\error[X][p_{k}]}^{2}\ll X^{2}\log^{2}X.
\end{equation}
Taking the limit as $R\tendsto\infty$ gives
\begin{equation}\label{eq:intlimit2}
\int_{\hypmodg}\abs{\error[X][p]}^{2}\d{\mu(p)}\ll X^{2}\log^{2}X.
\end{equation}
We split the rest of this section into three parts: one for each of the averages after which we prove the generalisation of
the radial large sieve inequality that is used for Theorem <ref>.
§.§ Radial Average
We will prove the following proposition.
Let $X>2$ and $X_{1},\ldots,X_{R}\in[X,2X]$ such that $\abs{X_{k}-X_{l}}\geq \epsilon>0$ for all $k\neq l$. Then, we have
\begin{equation}\label{eq:finalerror}
\sum_{k=1}^{R}\abs{\error[X_{k}]}^{2} \ll X^{3+2/3}\epsilon^{-1}\log X+X^{2+2/3}R+X^{3+1/3}\log X.
\end{equation}
Theorem <ref> follows immediately from the above proposition.
We take $\epsilon\asymp R^{-1}X$.
Hence the bound (<ref>) becomes
\[\sum_{k=1}^{R}\abs{\error[X_{k}]}^{2}\ll X^{2+2/3}R\log X+X^{2+2/3}R+X^{3+1/3}\log X.\]
So if we choose $R>X^{2/3}$ then
\[\frac{1}{R}\sum_{k=1}^{R}\abs{\error[X_{k}]}^{2}\ll X^{2+2/3}\log X.\]
This proves (<ref>). For the integral limit (<ref>) it suffices to consider a limiting partition of $[X,2X]$ with equally spaced points.
The large sieve inequality for radial averaging is given by the following theorem.
Given $p\in\hypmodg$, suppose that $X>1$ and $T>1$. Let $x_{1},\ldots,x_{R}\in [X,2X]$. If $\abs{x_{k}-x_{l}}>\epsilon>0$ for all $k\neq l$, then
\begin{equation}\label{eq:chamizo2}
\sum_{k=1}^{R}\abs[\bigg]{\sum_{\abs{t_{j}}\leq T}a_{j}x^{it_{j}}_{k}u_{j}(p)}^{2}\ll (T^{3}+XT^{2}\epsilon^{-1})\norm{a}^{2}_{\ast},
\end{equation}
\begin{equation}\label{eq:aeq}
\norm{a}^{2}_{\ast}=\sum_{\abs{t_{j}}\leq T}\abs{a_{j}}^{2}.
\end{equation}
In two dimensions, [Theorem 2.2]chamizo11996 proves a corresponding result to the above theorem.
The proof in three dimensions is similar, but we write it down in Section <ref>.
Let $f$ be a compactly supported function with finitely many discontinuities on $[1,\infty)$, and denote
\[\eerror[f] = A(f)(p) - \sum_{1\leq s_{j}\leq 2}c(f,t_{j})\period_{j}u_{j}(p).\]
Then, recall we have shown that (see (<ref>))
\begin{align*}
\eerror[f^{+}] &= O(X\delta^{-1}+X),\\
\eerror[f^{-}] &= O(X\delta^{-1}+X),
\end{align*}
\begin{equation}\label{eq:newerror}
\eerror[f^{-}] < \error+O(X^{2}\delta+X) < \eerror[f^{+}].
\end{equation}
We can now prove the proposition.
For simplicity, we combine the error terms in (<ref>). Suppose that
$1>\delta\gg X^{-1}$, then
\[\eerror[f^{-}] < \error+O(X^{2}\delta) < \eerror[f^{+}].\]
\[\sum_{k=1}^{R}\abs{\error[X_{k}]}^{2}\ll \sum_{k=1}^{R}\abs{\eerror[f][X_{k}]}^{2}+RX^{4}\delta^{2},\]
where $f$ is appropriately chosen as $f^{+}$ or $f^{-}$ depending on $k$. The main strategy is again
to apply dyadic decomposition in the spectral expansion. We use the following notation
for the truncated spectral expansion:
\begin{equation}\label{eq:truncspec}
\ssum = \sum_{T<\abs{t_{j}}\leq 2T}2c(f,t_{j})\period_{j}u_{j}(p).
\end{equation}
We consider three different ranges, which we choose so that the tails of the spectral expansion get
absorbed into the error term. The correct ranges are given by
\begin{align*}
A_{1} &= \Set{t_{j}\given0<\abs{t_{j}}\leq 1},\\
A_{2} &= \Set{t_{j}\given 1<\abs{t_{j}}\leq \delta^{-3}},\\
A_{3} &= \Set{t_{j}\given \abs{t_{j}}>\delta^{-3}}.
\end{align*}
Also, define
\[S_{i}=\sum_{t_{j}\in A_{i}}2c(f,t_{j})\period_{j}u_{j}(p).\]
We can now write
\[\eerror[f] = S_{1}+S_{2}+S_{3}.\]
For the tail we have
\begin{align*}
\sum_{t_{j}\in A_{3}} 2c(f,t_{j})\period_{j}u_{j}(p) &\ll \sum_{\abs{t_{j}}\geq \delta^{-3}}c(f,t_{j})\period_{j}u_{j}(p)\\
&\ll \sum_{t_{j}> \delta^{-3}}\min(\abs{t}^{-1},\abs{t}^{-3}\delta^{-2})X\period_{j}u_{j}(p).
\end{align*}
With a dyadic decomposition we get
\begin{align*}
\sum_{t_{j}\in A_{3}}2c(f,t_{j})\period_{j}u_{j}(p) &\ll X\delta^{-2}\sum_{n=0}^{\infty}\left(\sum_{2^{n}\delta^{-3}<t_{j}\leq 2^{n+1}\delta^{-3}}t^{-3}\period_{j}u_{j}(p)\right)\\
&\ll X\delta^{-2}\sum_{n=0}^{\infty}\delta^{9}2^{-3n}\left(\sum_{2^{n}\delta^{-3}<t_{j}\leq 2^{n+1}\delta^{-3}}\period_{j}u_{j}(p)\right),\\
\intertext{and then from the Cauchy--Schwarz inequality it follows that}
&\ll X\delta^{7}\sum_{n=0}^{\infty}2^{-3n}\left(\sum_{t_{j}\leq
\delta^{-3}2^{n+1}}\abs{\period_{j}}^{2}\right)^{1/2}\left(\sum_{t_{j}\leq\delta^{-3}2^{n+1}}\abs{u_{j}(p)}^{2}\right)^{1/2}\\
&\ll X\delta^{7}\sum_{n=0}^{\infty}2^{-3n}(2^{n/2}\delta^{-3/2})(2^{3n/2}\delta^{-9/2}) \ll X\delta,
\end{align*}
as required.
Next, for the first interval we have
\[ S_{1} = \sum_{t_{j}\in A_{1}}2c(f,t)\period_{j}u_{j}(p)\ll X\sum_{\abs{t_{j}}<1}\abs{t}^{-1}\period_{j}u_{j}(p)\ll X.\]
\[S_{1}+S_{3}=O(X^{2}\delta).\]
Finally, we split the summation in $S_{2}$ into dyadic intervals by letting $T=2^{n}$ for $n=0, 1, \ldots, [\log_{2}\delta^{-3}]$.
We then have
\begin{equation}\label{eq:errorrad}
\sum_{k=1}^{R}\abs{\eerror[f][X_{k}]}^{2}\ll\sum_{k=1}^{R}\abs*{\sum_{1\leq 2^{n}\leq \delta^{-3}}\ssum[2^{n}][X_{k}]}^{2}+O(RX^{4}\delta^{2}).
\end{equation}
Applying the Cauchy–Schwarz inequality to the sum over the dyadic intervals yields
\[\abs[\bigg]{\sum_{1\leq 2^{n}<\delta^{-3}}\ssum[2^{n}][X_{k}]}^{2}\ll \log X \sum_{1\leq 2^{n}<\delta^{-3}}\abs{\ssum[2^{n}][X_{k}]}^{2}.\]
Substituting this back into (<ref>) we get
\begin{equation}\label{eq:errorest}
\sum_{k=1}^{R}\abs{\eerror[f][X_{k}]}^{2}\ll\log X \sum_{1\leq 2^{n}\leq X^{-1}\delta^{-3}}\sum_{k=1}^{R}\abs{\ssum[2^{n}][X_{k}]}^{2}+O(RX^{4}\delta^{2}).
\end{equation}
Recall from Proposition <ref> that we can write
\[c(f,t) = X(a(t,\delta)X^{it}+b(t,\delta)X^{-it}),\]
where $a$ and $b$ satisfy
\[a(t,\delta),\,b(t,\delta) = \min(\abs{t}^{-1},\abs{t}^{-3}\delta^{-2}).\]
Keeping in mind our notation with $T=2^{n}$ and (<ref>), we apply
Theorem <ref> to get
\begin{equation}\label{eq:sestimate2}
\sum_{k=1}^{R}\abs{\ssum[T][X_{k}]}^{2}\ll (T^{3}+XT^{2}\epsilon^{-1})\norm{a}^{2}_{\ast},
\end{equation}
\begin{align*}
\norm{a}^{2}_{\ast} &\ll \sum_{T<\abs{t_{j}}\leq 2T}\abs{\min(\abs{t_{j}}^{-1},\abs{t_{j}}^{-3}\delta^{-2})X\period_{j}}^{2}\\
&\ll X^{2}\min(T^{-2},T^{-6}\delta^{-4})\left(\sum_{T\leq \abs{t_{j}}<2T}\abs{\period_{j}}^{2}\right)\\
&\ll X^{2}\min(T^{-1},T^{-5}\delta^{-4}).
\end{align*}
This simplifies (<ref>) to
\[\sum_{k=1}^{R}\abs{\ssum[T][X_{k}]}^{2}\ll (T^{3}+XT^{2}\epsilon^{-1})X^{2}\min(T^{-1},T^{-5}\delta^{-4}).\]
Therefore (<ref>) becomes
\begin{align*}
\sum_{k=1}^{R}\abs{\error[X_{k}]}^{2} %&\ll \log X\sum_{1\leq T< \delta^{-3}}\sum_{k=1}^{R}\abs{\ssum[T][X_{k}]}^{2}+RX^{4}\delta^{2}\\
&\ll \log X\sum_{1\leq T<\delta^{-3}}(T^{3}+XT^{2}\epsilon^{-1})X^{2}\min(T^{-1},T^{-5}\delta^{-4})+RX^{4}\delta^{2}.
\end{align*}
We split the summation depending on whether $T<\delta^{-1}$ and get
\begin{align*}
\sum_{k=1}^{R}\abs{\error[X_{i}]}^{2} &\ll X^{2}\log X\left(\sum_{1\leq T\leq\delta^{-1}}T^{2}\right)+X^{3}\epsilon^{-1}\log X\left(\sum_{1\leq T\leq\delta^{-1}}T\right)\\
&\phantom{\ll}+X^{2}\delta^{-4}\log X\left(\sum_{\delta^{-1}\leq T\leq \delta^{-3}}T^{-2}\right)\\
&\phantom{\ll}+X^{3}\delta^{-4}\epsilon^{-1}\log X\left(\sum_{\delta^{-1}\leq T\leq \delta^{-3}}T^{-3}\right)+RX^{4}\delta^{2}.
\end{align*}
With trivial estimates we have
\[\sum_{k=1}^{R}\abs{\error[X_{k}]}^{2} \ll X^{2}\delta^{-2}\log X+X^{3}\epsilon^{-1}\delta^{-1}\log X+RX^{4}\delta^{2}.\]
The optimal choice for $\delta$ comes from $X^{4}\delta^{2}=X^{2}\delta^{-1}$, that is, $\delta=X^{-2/3}$, since $\epsilon R\asymp X$. This gives
\begin{equation*}
\sum_{k=1}^{R}\abs{\error[X_{k}]}^{2} \ll X^{3+1/3}\log X+X^{3+2/3}\epsilon^{-1}\log X+RX^{2+2/3}.
\end{equation*}
§.§ Spatial Average
We now consider the spatial average. In this case the corresponding large sieve inequality was
already proved by Chamizo in <cit.> for $n$ dimensions for any cocompact group.
It would not be difficult to extend it to cofinite groups in three dimensions.
We state it as Theorem <ref> simplified to our setting. With similar strategy as in Section <ref>, we prove the following
proposition which readily yields Theorem <ref>.
Suppose $X>1$ and let $p_{1},p_{2},\ldots,p_{R}\in\hypmodg$ with $\tilde{d}(p_{k},p_{l})>\epsilon$ for
some $\epsilon>0$. Then we have
\[\sum_{k=1}^{R}\abs{\error[X][p_{k}]}^{2} \ll X^{4}R^{-1}\log^{2}X+X^{2}\epsilon^{-3}.\]
As before, Theorem <ref> follows easily.
We pick $\epsilon^{-3}\ll R$ and $R>X$. Then
\[\frac{1}{R}\sum_{k=1}^{R}\abs{\error[X][p_{k}]}^{2} \ll X^{4}R^{-2}\log^{2}X+X^{2}\ll X^{2}\log^{2}X.\]
For the integral limit we take hyperbolic balls of radius $\epsilon/2$ uniformly spaced in $M$.
For small radii the
volume of such a ball is $(4/3)\pi (\epsilon/2)^{3}$, <cit.>. This is compatible
with our assumption that $R\epsilon^{3}$ is bounded from below since $M$ is of finite volume.
For the proof of Proposition <ref> we use the large sieve inequality in the following form.
Given $T>1$, $p_{1},\ldots,p_{R}\in M=\hypmodg$, if
$\tilde{d}(p_{k},p_{l})>\epsilon>0$ for all $k\neq l$, then
\[\sum_{k=1}^{R}\abs*{\sum_{\abs{t_{j}}\leq T} a_{j}u_{j}(p_{k})}^{2}\ll (T^{3}+\epsilon^{-3})\norm{a}_{\ast}^{2},\]
where $\norm{a}_{\ast}$ is as in (<ref>).
We can then prove Proposition <ref>.
By a direct application of the Cauchy–Schwarz inequality we have
\[\left(\sum_{k=1}^{n}a_{k}\right)^{2} \ll \sum_{k=1}^{n}k^{2}a_{k}^{2}.\]
\begin{equation}\label{eq:logbound}
\abs*{\sum_{1\leq T\leq\delta^{-3}}\ssum[T][X][p_{k}]}^{2} \ll\sum_{1\leq T\leq\delta^{-3}}\abs{\log T}^{2}\abs{\ssum[T][X][p_{k}]}^{2}.
\end{equation}
Repeating this with the identity
\[\left(\sum_{k=1}^{n}a_{k}\right)^{2} \ll \sum_{k=1}^{n}(n+1-k)^{2}a_{k}^{2},\]
and combining with (<ref>) yields
\[\abs*{\sum_{1\leq T\leq\delta^{-3}}\ssum[T][X][p_{k}]}^{2} \ll\sum_{1\leq T\leq\delta^{-3}}\abs{c_{T}}^{2}\abs{\ssum[T][X][p_{k}]}^{2},\]
where $c_{T}=\min\left(\abs{\log T\delta^{3}+1},\log T\right)$.
Repeating the analysis from the proof of Proposition <ref> and applying Theorem <ref> gives
\begin{align*}
\sum_{k=1}^{R}\abs{\error[X][p_{k}]}^{2} &\ll \sum_{1\leq T\leq \delta^{-3}}\abs{c_{T}}^{2}(T^{3}+\epsilon^{-3})X^{2}T^{-1}\min(1,T^{-4}\delta^{-4})+RX^{4}\delta^{2}\\
&\ll X^{2}\delta^{-2}\log^{2}X + X^{2}\epsilon^{-3} + RX^{4}\delta^{2},
\end{align*}
as before. We optimise by setting $X^{2}\delta^{-2}=RX^{4}\delta^{2}$, which gives $\delta = R^{-1/4}X^{-1/2}$. This yields
\[\sum_{k=1}^{R}\abs{\error[X][p_{k}]}^{2} \ll X^{3}R^{1/2}\log^{2}X+X^{2}\epsilon^{-3}.\]
§.§ A Large Sieve Inequality
We need two technical lemmas to prove Theorem <ref>. The first one is Lemma 3.2 in <cit.>.
Let $b=(b_{1},\ldots,b_{R})\in\complex^{R}$ be a unit vector and let $A=(a_{ij})$ be an $R\times R$ matrix over $\complex$ with
$\abs{a_{ij}}=\abs{a_{ji}}$. Then
\[\abs{b\cdot Ab}=\abs{\sum_{i,j=1}^{R}b_{i}\overline{b}_{j}a_{ij}}\leq \max_{i}\sum_{j=1}^{R}\abs{a_{ij}}.\]
We also need to compute the inverse Selberg transform for the Gaussian at different frequencies.
Let $h(1+t^{2})=e^{-t^{2}/(2T)^{2}}\cos(rt)$. The inverse Selberg transform $k$ of $h$ satisfies for all $x>0$
\begin{align*}
k(\cosh x) &\ll T^{3}\frac{(x+r)e^{-T^{2}(x+r)^{2}}+(x-r)e^{-T^{2}(x-r)^{2}}}{\sinh x},
\shortintertext{and}
k(1) &\ll \min(T^{3},r^{-3}).
\end{align*}
According to <cit.>, the inverse Selberg transform $k$ of $h$ for $x\geq1$
is given by
\[-2\pi k(x)=\frac{d}{dx}\frac{1}{2\pi}\int_{-\infty}^{\infty}h(1+t^{2})e^{-it\arccosh x}\d{t}.\]
Hence, by a direct computation
\begin{equation*}
-2\pi k(\cosh x) = \frac{1}{2\pi i}\frac{1}{\sinh x}\int_{-\infty}^{\infty}e^{-t^{2}/(2T)^{2}}\cos(rt)te^{-itx}\d{t},
\end{equation*}
which is just the Fourier transform of a Gaussian times $t\cos(rt)$. Hence, by standard results
<cit.> and <cit.>
we have
\[k(\cosh x) = \frac{2\sqrt{\pi}}{4\pi^{2}}T^{3}g(x),\]
\[g(x)=\frac{(x+r)e^{-T^{2}(x+r)^{2}}+(x-r)e^{-T^{2}(x-r)^{2}}}{\sinh x}.\]
Taking the limit gives
\begin{align*}
\lim_{x\tendsto 0} g(x) %&= \lim_{x\tendsto 0}\frac{e^{-T^{2}(x+r)^{2}}+e^{-T^{2}(x-r)^{2}}-2T^{2}((x+r)^{2}e^{-T^{2}(x+r)^{2}}+(x-r)^{2}e^{-T^{2}(x-r)^{2}}}{\cosh x}\\
\end{align*}
Let $u(x)=2e^{-x^{2}}(1-2x^{2})$. Hence,
\[k(1) = \frac{1}{2\pi^{3/2}}T^{3}u(rT).\]
Since $u(x)$ is bounded, we get trivially that $k(1)\ll T^{3}$. On the other hand,
\[T^{3}u(rT)=\frac{1}{r^{3}}(Tr)^{3}u(rT)\ll r^{-3}.\]
It follows that $k(1)\ll\min(T^{3},r^{-3})$.
Let $S$ be the left-hand side of (<ref>).
Since $\complex^{R}$ is self-dual, it follows from Riesz representation theorem that
there exists a unit vector $\vec{b}=(b_{1},\ldots,b_{R})$ in $\complex^{R}$ such that
\[ S = \left(\sum_{k=1}^{R}b_{k}\left(\sum_{\abs{t_{j}}\leq T}a_{j}x^{it_{j}}_{k}u_{j}(p)\right)\right)^{2}.\]
Then, by the Cauchy–Schwarz inequality
\[ S\leq \norm{a}^{2}_{\ast}\widetilde{S},\]
\[\widetilde{S} = \sum_{\abs{t_{j}}\leq T}\abs*{\sum_{k=1}^{R}b_{k}x^{it_{j}}_{k}u_{j}(p)}^{2}.\]
In order to understand the sum $\widetilde{S}$, we smooth it out by a Gaussian centered around zero.
This allows us to apply Lemma <ref>, which shows that the
Selberg transform for a Gaussian is easy to compute. Thus,
\[\widetilde{S} \ll \sum_{j}e^{-t_{j}^{2}/(4T^{2})}\abs*{\sum_{k=1}^{R}b_{k}x^{it_{j}}_{k}u_{j}(p)}^{2}.\]
After we open up the squares and interchange the order of summation, we apply Lemma <ref> to get
\[S\ll \norm{a}^{2}_{\ast}\max_{k}\sum_{l=1}^{R}\abs{S_{kl}},\]
\[S_{kl} = \sum_{j}e^{-t_{j}^{2}/(4T^{2})}\cos(r_{kl} t_{j})\abs{u_{j}(p)}^{2},\]
\[r_{kl} = \abs[\bigg]{\log\frac{x_{k}}{x_{l}}}.\]
We can identify $S_{kl}$ as the diagonal contribution in the spectral expansion of an automorphic kernel with $h(1+t^{2})=e^{-t^{2}/(4T^{2})}\cos(r_{kl}t)$.
It follows from Lemma <ref> that
\begin{equation}\label{eq:specexp}
S_{kl}\ll\min(T^{3},r_{kl}^{-3})+\sum_{\gamma\neq\id}T^{3}e^{-T^{2}(d(\gamma p, p)-r_{kl})^{2}}.
\end{equation}
The standard hyperbolic lattice point problem (e.g. <cit.>) gives
\[\#\Set{\gamma\in\Gamma\given\delta(p,\gamma q)\leq x}\ll x^{2},\]
where the implied constant depends on $\Gamma$ and $p$. We can rewrite this as
\[\log(1+\#\Set{\gamma\in\Gamma\given r<d(p,\gamma q)\leq r+1})\ll r^{2}+1.\]
This shows that the series in (<ref>) converges as $T\tendsto\infty$, so that
\[S_{kl}\ll\min(T^{3},r_{kl}^{-3}).\]
Hence, by the mean value theorem
\[\sum_{l=1}^{R}\abs{S_{kl}}\ll\sum_{l=1}^{R}\min(T^{3},X^{3}\abs{x_{k}-x_{l}}^{-3}).\]
The case $l=k$ yields $T^{3}$. So suppose $l\neq k$, then
separate the $x_{l}$ for which $T\leq X\abs{x_{k}-x_{l}}^{-1}$. By the spacing condition, there
are at most $2XT^{-1}\epsilon^{-1}$ such points. Hence,
\begin{align}
\sum_{l=1}^{R}\abs{S_{kl}}&\ll T^{3}XT^{-1}\epsilon^{-1}+\int_{1}^{\infty}\frac{X^{3}}{\abs{XT^{-1}+\epsilon u}^{3}}\d{u} + T^{3}\label{eq:improvement}\\
&\ll T^{2}X\epsilon^{-1}+T^{3}.\notag
\end{align}
|
1511.00407
|
Hefei National Laboratory for Physical Sciences at Microscale and Department
of Modern Physics, University of Science and Technology of China, Hefei,
Anhui 230026, China
CAS Center for Excellence and Synergetic Innovation Center in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China
Hefei National Laboratory for Physical Sciences at Microscale and Department
of Modern Physics, University of Science and Technology of China, Hefei,
Anhui 230026, China
CAS Center for Excellence and Synergetic Innovation Center in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China
Hefei National Laboratory for Physical Sciences at Microscale and Department
of Modern Physics, University of Science and Technology of China, Hefei,
Anhui 230026, China
CAS Center for Excellence and Synergetic Innovation Center in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China
Hefei National Laboratory for Physical Sciences at Microscale and Department
of Modern Physics, University of Science and Technology of China, Hefei,
Anhui 230026, China
CAS Center for Excellence and Synergetic Innovation Center in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China
Quantum repeater holds the promise for scalable long-distance quantum communication. Towards a first quantum repeater based on memory-photon entanglement, significant progresses have made in improving performances of the building blocks. Further development is hindered by the difficulty of integrating key capabilities such as long storage time and high memory efficiency into a single system. Here we report an efficient light-matter interface with sub-second lifetime by confining laser-cooled atoms with 3D optical lattice and enhancing the atom-photon coupling with a ring cavity. An initial retrieval efficiency of 76(5)% together with an 1/e lifetime of 0.22(1) s have been achieved simultaneously, which already support sub-Hz entanglement distribution up to 1000 km through quantum repeater. Together with an efficient telecom interface and moderate multiplexing, our result may enable a first quantum repeater system that beats direct transmission in the near future.
Quantum communication <cit.> relies on photon transimission over long distance. Direct transmission is limited to moderate distances (less than 500 km <cit.>) due to exponential decay of photons. Quantum repeater <cit.> is an ultimate solution to go significantly beyond this limitation. There are many quantum repeater schemes proposed so far <cit.>. Considering the experimental capabilities, the memory-photon entanglement based schemes <cit.> are much more feasible than the error correction based schemes <cit.>. Towards a first quantum repeater based on memory-photon entanglement, significant progresses have made in improving performances of the building blocks <cit.>. Further development is hindered by the difficulty of integrating key capabilities such as long storage time <cit.> and high memory efficiency <cit.> into a single system. So far, storage lifetime has been improved to the sub-second regime for single excitations in an atomic ensemble <cit.> and to the sub-minute regime for classical light storage <cit.>, nevertheless, storage efficiency in these experiments is typically very low ($\sim$16%). If we define a threshold of 50% for the memory efficiency, the longest storage time is limited to 1.2 ms <cit.>, which is far away from the second regime requirement <cit.> of a quantum repeater. Memory efficiency is essentially important as it intervenes in every entanglement swapping operation between adjacent quantum repeater nodes. According to theoretical estimations <cit.>, 1% increase of retrieval efficiency can improve long-distance entanglement distribution rate by 10%$\sim$14%.
In this paper, we report an efficient quantum memory with sub-second regime lifetime by making use of a 3D optical lattice confined atomic ensemble inside a ring cavity. The quantum memory is nonclassically correlated with a single photon, thus forms a light-matter interface for quantum repeaters <cit.>. Optical lattice limits atomic motion in all direction thus suppresses various motion-induced decoherence, and ring cavity enhances atom-photon coupling thus improves the retrieval efficiency. We further use the magic trap technique <cit.> to compensate the lattice induced differential light shift. To be compatible with sub-second regime storage, the ring cavity is stabilized with a large-detuned reference beam. By taking all these measures, we finally realize a light-matter interface with an initial efficiency of 76(5)% and a $1/\rm{e}$ lifetime of 0.22(1) s. This is a significant step forward in realizing a high-performance light-matter interface filling the requirement of quantum repeaters.
Experimental setup and atomic levels. The $^{87}\rm{Rb}$ atoms are trapped with a 3D optical lattice by interfering four circularly polarized 1064 nm laser beams ($\mathrm{L}_i$, $i\!=\!\{1,2,3,4\}$). The atoms are initially prepared in the ground state $\left|F\!=\!2,m_F\!=\!0\right>$. Another ground state of $\left|F\!=\!1,m_F\!=\!0\right>$ is employed for storage. The ring cavity, in the horizontal plane (the yz-plane), consists of one partially reflecting mirror PR of 92.0(3)% reflection rate, two highly reflecting mirrors $\rm{HR}_1$ and $\rm{HR}_2$, and two plano-convex lenses $f_1$ and $f_2$ with a same focus length of 250 mm. The orthogonal linear polarized write and read pulses of the DLCZ memory are counter-propagating through the atoms, having a separation angle of $\theta=3^{\circ}$ with respect to the cavity mode. The write-out and read-out photons are cavity-enhanced and escape from the two coupling ports of the mirror PR. The cavity-locking beam is combined into the read-out channel with a polarized beam splitter. Leaks of the locking pulses from the mirror $\rm{HR}_{2}$ are detected by a fast photodiode (PD), and fed forward to stabilize the cavity length by displacing the mirror $\rm{HR}_{1}$ with a piezoelectric transducer (PZT).
The experimental setup is shown in Fig. <ref>. An ensemble of $^{87}\!\rm{Rb}$ atoms is first prepared through magneto-optical trapping (MOT) and cooled down to about $12\,\rm{\mu K}$ via optical molasses. The atoms are finally loaded into a 3D optical lattice via interfering four 1064-nm laser beams at the setup centre, and optically pumped to the ground state $\left|F\!=\!2, m_{F}\!=\!0\right\rangle$. The optical lattice is formed by four circularly polarized laser beams $\rm{L}_{\emph{i}}$ ($i\!=\!\{1,2,3,4\}$), with the same beam waist diameter of $500\,\rm{\mu m}$ overlapping through the atoms, and $12^\circ$ axisymmetric angled to the bias magnetic field $B$. The 1064 nm laser output is $50\%\!:\!50\%$ split into two beams $\rm{L}_{1}$ and $\rm{L}_{2}$, then reflected backward and used as the beams $\rm{L}_{3}$ and $\rm{L}_{4}$ respectively. As the vacuum glass cell is not anti-reflecting coated for 1064 nm, power of $\rm{L}_3$ ($\rm{L}_4$) is $19\%$ lower than $\rm{L}_1$ ($\rm{L}_2$). Lattice periods are 5.9 $\rm{\mu m}$, 2.8 $\rm{\mu m}$ and 0.54 $\rm{\mu m}$ for the x, y and z directions respectively. The lattice-trapped ensemble is 0.2 mm wide in the xy plane and 0.8 mm long in the z direction. The optical depth is about 1.6 in the z direction. A temporal gap of 90 ms is left in the lattice phase to wait for all the unbounded atoms flying away. The write pulse is vertically polarized, of pulse width 120 ns and beam power $3.8\,\rm{\mu W}$; while the read pulse is horizontally polarized, of pulse width 240 ns and beam power $330\,\rm{\mu W}$. The write-out photons are collected by a single-mode fibre, and measured by single-photon detector $D_1$ after a Fabry-Perot cavity filter; the read-out photons are collected and sent through a vapour cell atomic filter, and then measured by single-photon detectors $D_2$ and $D_3$ after $50\%\!:\!50\%$ splitting. In order to suppress the beam leakages and their influence on the stored single excitations in the ensemble, most of the laser beams are controlled by double-passed acousto-optic modulators and the laser beams for the MOT are blocked with a mechanical chopper during memory operations.
Confinement of atom motion with a 3D optical lattice suppresses all motion-induced decoherence<cit.>, but gives rise to a new decoherence of differential light shift. By adding an appropriate bias magnetic field $B$, differential light shift from the vector polarizability compensates that from the scalar part <cit.>. Without the ring cavity, we first calibrate the magic condition via the electromagnetically induced transparency (EIT) memory <cit.>. The single-photon-level probe pulse to be stored is resonant with the D1 line $\left|F\!=\!2\right>\!\to\!\left|F'\!=\!2\right>$ and transmits in the write-out direction in Fig. <ref>. The control pulse is resonant with the transition $\left|F\!=\!1\right>\!\to\!\left|F'\!=\!2\right>$ and transmits in the write direction. The probe and control pulses are controlled to enable a stopped-light configuration <cit.>. By optimizing the retrieval amplitude at $t\!=\!0.5$ s, we find the best magic compensating field to be $4.56\,\rm{Gauss}$. With a lattice potential of $U_{0}\!=\!146\,\rm{\mu K}$, we measure the retrieval signal decay as a function of storage time in Fig. <ref> (a). The $1/e$ time constant is fitted to be 0.51(3) s. Changing the lattice laser power $U_{0}$, the storage lifetime are more or less the same, with an average value of 0.53 s (Fig. <ref> (b)). This indicates that the differential light shift has been successfully eliminated.
Lifetime measurement of the optical lattice-trapped EIT memory without a cavity. (a) Retrieval amplitude normalized to the point of $t=10\,{\rm{ms}}$. Fitting with an exponential function (red line), we get an $1/e$ lifetime of $\tau=0.51(3)\,\rm{s}$. Lattice potential is set to $U_0=146\,{\rm \mu K}$ for this measurement. (b) Lifetime measured for different trap potentials. The average value is $\tau=0.53\,\rm{s}$ (red line).
Afterwards, we set up a ring cavity around the lattice-trapped atoms to enhance the atom-photon coupling strength and carry out the DLCZ <cit.> memory with single photons. The orthogonal linear-polarized write and read pulses are counter-propagating through the atoms (Fig. <ref>). The write and read beams are red detuned by 40 MHz to the D1 transitions. The write-out and read-out photons are configured to be resonant with the ring cavity. The cavity has a finesse of $\mathcal{F}\!=\!52$, and a linewidth of 9.2 MHz. In our previous experiments <cit.>, the cavity was intermittently stabilized to avoid influence from the locking beam on to the memory. In order to obtain a high retrieval efficiency for long storage durations, we have developed the cavity stabilization technique with large detuning of atomic ransitions. The cavity locking beam is from a reference laser of $\lambda\!=\!800\,\rm{nm}$ with 5 nm detuned to the D1 line transitions, and is stabilized with an ultra-stable cavity due to the unavailability of atomic lines. The cavity locking beam has a power of $1\,\rm{\mu W}$, which introduces a differential light shift of 0.3 Hz in side the cavity. In our experiment, we turn on the locking beam for 10 ms to pull back the cavity resonance before applying the read pulse.
Millisecond regime decay of retrieval efficiency with DLCZ storage. Retrieval efficiency $R$ is normalized to the initial value at $t=0$. Data points in black square refer to the angled case $\theta=3^{\circ}$, while points in red dot refer to the collinear case $\theta=0^{\circ}$. For a reference of $R_{\rm{nor}}=0.5e^{-1}+0.5$ (gray dashed line), the decay time is 0.23 ms for the angled case, while 2.52 ms for the collinear case. The lattice potential $U_0$ has a fixed value of $70\,\rm{\mu K}$.
By applying a write pulse, a write-out photon is detected with an overall probability of $p_{\rm{w}}\!\sim\!10^{-3}$ and heralds the creation of a collective excitation in the atomic ensemble. Since $p_{\rm{w}}$ is low, we repeat the write trials till a write-out signal photon is detected. The retrieval process with an adjustable delay only starts after a successful write-out event. Retrieval efficiency is measured as $R=p_{\rm{r|w}}-p_{\rm{r}}$, where $p_{\rm{r}|\rm{w}}$ ($p_{\rm{r}}$) refers to the conditional (unconditional) detection probability of a read-out photon.
Unexpectedly, we observe significant efficiency decay within the initial sub-millisecond regime as shown in Fig. <ref> in black. We suspect that this is due to imperfect lattice potentials which result in some free atoms not confined within single lattice sites. Thus we make the same measurement for the collinear case <cit.> ($\theta=0$), with the result shown in Fig. <ref> in red. The retrieval efficiency drops by about 50% in both measurements but with different time scales. The angled case drops faster since the phase grating gets distorted for transversal movement of a half spinwave wavelength ($\lambda_{\rm{sw}}/2=7.6\,\rm{\mu m}$). While in the collinear case, since the spinwave wavelength is largely increased to 4 cm, transversal motion influences the retrieval efficiency by distorting the transversal mode of the read-out photon, which is sensitive to the movement scale of the cavity mode waist ($w_{\rm{cav}}=60\,\rm{\mu m}$). Considering an atom temperature of 12 $\rm{\mu K}$, the time required for movement over $\lambda_{\rm{sw}}/2$ or $w_{\rm{cav}}$ coincide with the decay time observed in Fig. <ref>. In order to minimize these free atoms, we dynamically increase $U_0$ from $70\,\rm{\mu K}$ to $180\,\rm{\mu K}$ after all the unloaded atoms flying out the lattice region. By doing this, the dropping rate of retrieval efficiency gets reduced remarkably to $1\!-\!R(10\,\mathrm{ms})/R(0\,\mathrm{\mu s})\!\approx\!20\%$, which is a significant improvement compared to previous experiments with 1D lattice <cit.> and hollow beam dipole trap <cit.>.
Intrinsic retrieval efficiency $\chi$ versus storage time for DLCZ storage. Data points are fitted with a double-exponential decay function (red line). The initial intrinsic retrieval efficiency is $\chi(0)=76(5)\%$.
For a threshold of $\chi=50\%$ (blue dashed line), our memory persists until 51 ms. While for a threshold of $\chi=e^{-1}\chi(0)$ (orange dotted line), our memory persists until 0.22 s.
In Fig. <ref> we show the final result of retrieval efficiency measurement in the range of 0$\sim$500 ms. The intrinsic retrieval efficiency $\chi$ is obtained through $\chi=R/(\eta _{\mathrm{cav}}\,\eta _{\mathrm{t}}\,\eta _{\mathrm{spd}})$, where $\eta _{\mathrm{cav}}=68(3)\%$ refers to the photon emanation rate from the ring cavity, $\eta _{\mathrm{t}}=55(2)\%$ refers to the propagation efficiency from the cavity to the detectors, and $\eta _{\mathrm{spd}}=63(2)\%$ refers to the detection efficiency for the single-photon detectors. We fit the data with a double-exponential decay function $\chi\left(t\right)\!=\!\chi_1\exp\left(-t/\tau_1\right)\!+\!\chi_2\exp\left(-t/\tau_2\right)$, where $\tau_1$ and $\tau_2$ are the two characteristic decay parameters. The fitted parameters are $\tau_1=$0.13(4) ms, $\tau_2$=285(12) ms, $\chi_1$=$15.8\pm1.8$% and $\chi_2$=$59.8\pm4.0$% respectively.
The initial intrinsic retrieval efficiency is $\chi(0)=76(5)\%$, which is comparable with the best previous results on efficiency in the single-quanta regime <cit.>. The $1/e$ decay time for the retrieval efficiency is $0.22(1)\,\rm{s}$, which is also comparable with the best previous results on lifetime in the single-quanta regime <cit.>. For an threshold of $\chi\geq50$%, our memory can persist until $51\,\rm{ms}$, while the best previous result is merely 1.2 ms <cit.>. We would like to stress that efficient storage far beyond the millisecond regime is essentially important for quantum repeaters <cit.>. To further identify that our experiment genuinely operates in the single-quanta regime, we measure the anticorrelation parameter <cit.> $\alpha$ for different storage times, with the result shown in Table <ref>. The write-out photon is detected with $D_{1}$, while the read-out photon is 50:50 split and detected with two separate detectors $D_{2}$ and $D_{3}$. The $\alpha$ parameter is defined as
\begin{equation}
\alpha\!=\!p_{23|1}/(p_{2|1} \cdot p_{3|1}),
\end{equation}
where $p_{j|i}$ ($p_{jk|i}$) is the conditional single (two-fold coincidence) detection probability. The measured $\alpha$ parameter at $t=0\,\rm{s}$ and $t=0.5\,\rm{s}$ are $0.11(5)$, and $0.30(21)$ respectively, both of which are well below the classical threshold of $\alpha\geq1$, implying the non-classical behaviour of our memory.
Measurement of the $\alpha$ parameter.
$t$(ms) $\alpha$ $n_{123}$ $R$(%) $\chi$(%) $T_{\rm{m}}$(hour)
0 0.11(5) 6$<$55 17.1(5) 73(5) 1.4
10 0.16(7) 6$<$37 12.4(4) 53(4) 5.3
100 0.28(11) 7$<$25 9.7(3) 41(3) 6.1
300 0.09(9) 1$<$11 5.4(2) 23(2) 8.6
500 0.30(21) 2$<$7 2.6(1) 10(1) 24.7
Notes: $t$, storage time; $n_{123}$, number of triple coincidence events of the detectors $D_1$, $D_2$ and $D_3$; $R$, measured retrieval efficiency; $\chi$, intrinsic retrieval efficiency; $T_{\rm{m}}$, measurement duration. For the column of $n_{123}$, the right part of each inequality refers to the expected triple coincidence when $\alpha=1$.
Realization of a light-matter interface with high efficiency and long lifetime has significant applications in long-distance quantum communication. If we consider a DLCZ quantum repeater with multiplexing <cit.>, our current result on efficiency and lifetime already supports sub-Hz entanglement distribution up to 1000 km <cit.>, assuming channel loss of 0.16 db/km <cit.>, perfect telecom interface, single-photon detectors with 100% efficiency and moderate multiplexing. Our work thus demonstrates for the first time that efficiency and lifetime meet the requirement of quantum repeater simultaneously. Higher intrinsic retrieval efficiency will be possible by using a cavity with higher finesse. Longer storage time will be possible by using dynamical decoupling <cit.>. Besides, as the optical lattice tightly confines atoms in all directions, our system has a feasible spatial multiplexing capacity <cit.> with long lifetime. In order to realize a practical quantum repeater, coupling losses have to be minimized and telecom interface with high efficiency should be integrated. Our result will also be very useful for creating large-scale cluster states for one-way quantum computing <cit.>.
This work was supported by the National Natural Science Foundation of China, National Fundamental Research Program of China (under Grant No. 2011CB921300), and the Chinese Academy of Sciences. X.-H.B. acknowledge support from the Youth Qianren Program.
§ SUPPLEMENTARY INFORMATION
Below we present calculations of entanglement distribution rate through quantum repeater and make comparisons with direct transmission <cit.>. For direction transmission, we consider ideal single photons with a repetition rate of 10 GHz and using low-loss fibers with 0.16 dB/km and detectors of 100% efficiency. The calculation result is shown in Fig. <ref> as dashed lines. If dark count is not considered, the distribution rate simply drops exponentially. By choosing a threshold distribution rate of $R=0.01~\rm{Hz}$, the maximal distance is 750 km. When a reasonable dark count probability of $10^{-9}$ is considered, the distribution rate gets cutoff for a distance of 530 km. We note that the cutoff distance for direct transmission may vary slightly by assuming different parameters <cit.>.
Entanglement distribution rate for quantum repeater versus single-photon distribution rate through direct transmission. Dashed lines correspond direct transmission (Gray: no dark count, Black: dark count probability of $10^{-9}$ considered). Solid lines correspond to quantum repeater with different memory parameters (lifetime + efficiency). Red: 0.2 s + 16% (Radnaev el al. in 2010 <cit.>), Green: 3.2 ms + 73% (Bao et al. in 2012 <cit.>), Blue: 0.22 s + 76% (result of this paper).
For quantum repeaters, we consider the DLCZ protocol <cit.> with multiplexing <cit.>. We optimize the nesting level $n$ and set the limit of $n\leq3$. We assume using the same fiber and detectors as direct transmission. We also assume perfect telecom interface. Pair emission probability $p$ and multiplexing number $N_m$ are selected such that $N_m p=1$ and the fidelity due to multi-photon errors is 95%. We note that low dark count probability of $10^{-9}$ has negligible influence in quantum repeater with small nesting levels <cit.> since entanglement connection and detection are conducted heraldedly. For the quantum memory parameters (lifetime + efficiency), we consider three representative results in the single quantum regime with cold atomic ensembles, which includes the result of 0.2 s + 16% by Radnaev el al. in 2010 <cit.>, the result of 3.2 ms + 73% by Bao et al. in 2012 <cit.> and the current result of 0.22 s + 76% in this paper. We calculate the generation rate of a pair of two-photon entanglement over distance $L$ for the three groups of memory parameters. In our calculation, a memory decay model of $e^{-t/\tau}$ is assumed for simplicity. The results are shown as solid lines in Fig. <ref>. The multiplexing number $N_m$ varies from 40 up to 1000 for different distances and different memory parameters. It is clear that the previous memories merely allow entanglement distribution with similar or even worse scaling than direct transmission. For a threshold $R=0.01~\rm{Hz}$, the maximal distance is 280 km, which is not comparable with direction transmission. In contrast, our new result shows a much better scaling than direction transmission and enables entanglement distribution much longer than the cutoff distance of direct transmission. For a threshold $R=0.01~\rm{Hz}$, our new result supports a quantum repeater over 860 km. If the memories at the two terminals are retrieved and detected immediately after storage (valid for quantum key distribution), the effective entanglement distribution rate goes much higher (0.04 Hz @1000 km).
We conclude that our new result on memory efficiency and lifetime for the first time enables to build a quantum repeater system that beats direct transmission.
|
1511.00113
|
Let $\DirIntro$ be the set of all $d$-regular directed graphs on $n$ vertices.
Let $\rangr$ be a graph chosen uniformly at random from $\DirIntro$ and $\Mrand$
be its adjacency matrix. We show that $\Mrand$ is invertible with probability at
least $1-C\ln^{3} d/\sqrt{d}$ for $C\leq d\leq cn/\ln^2 n$, where $c, C$ are
positive absolute constants. To this end, we establish a few properties
of $d$-regular directed graphs. One of them, a Littlewood–Offord type anti-concentration property, is of independent interest.
Let $J$ be a subset of vertices of $\rangr$ with $|J|\approx n/d$.
Let $\delta _i$ be the indicator of the event that the vertex $i$ is connected to
$J$ and define $\delta = (\delta_1, \delta_2, ..., \delta_n)\in \{0, 1\}^n$. Then for every $v\in\{0,1\}^n$ the probability that $\delta=v$
is exponentially small. This property holds even if a part of the graph is “frozen."
AMS 2010 Classification:
60C05, 60B20, 05C80, 15B52,
Adjacency matrices, anti-concentration, invertibility, Littlewood–Offord theory, random digraphs, random graphs, random matrices, regular graphs,
singular probability, singularity, sparse matrices
§ INTRODUCTION
For $1\leq d \leq n$ an undirected (resp., directed) graph $G$ is called
$d$-regular if every vertex has exactly $d$ neighbors (resp., $d$
in-neighbors and $d$ out-neighbors). In this definition we allow graphs to
have loops and, for directed graphs, opposite (anti-parallel) edges, but
no multiple edges. Thus directed graphs (digraphs) can be viewed as
bipartite graphs with both parts of size $n$. For a digraph $G$ with $n$ vertices
its adjacency matrix $(\mu_{ij})_{i,j\leq n}$ is defined by
\mu_{ij}=
\begin{cases}1,&\mbox{if there is an edge from $i$ to $j$;}\\0,&\mbox{otherwise.}\end{cases}
For an undirected graph $G$ its adjacency matrix is defined
in a similar way (in the latter case the matrix is symmetric).
We denote the sets of all undirected (resp., directed) $d$-regular graphs
by $\UndirIntro$ and $\DirIntro$, respectively, and the
corresponding sets of adjacency matrices by $\McUndir$ and $\Mc$.
Clearly $\McUndir \subset \Mc$ and $\Mc$ coincides with the set of $n\times n$ matrices with $0/1$-entries and such that every row and every column has exactly $d$ ones. By the probability on $\UndirIntro$, $\DirIntro$, $\McUndir$, and $\Mc$ we always mean the normalized counting measure.
Spectral properties of adjacency matrices of random $d$-regular graphs
attracted considerable attention of researchers
in the recent years.
Among others, we refer the reader to <cit.>, <cit.>, <cit.>, <cit.>, <cit.>,
and <cit.> for results
dealing with the eigenvalue distribution. At the same time, much less
is known about the singular values of the matrices.
The present work is motivated by related general questions on
singular probability.
One problem was mentioned by Vu in his survey <cit.>
(see also 2014 ICM talks by Frieze and Vu <cit.>, <cit.>). It asks if for $3\leq d\leq n-3$
the probability that
a random matrix uniformly distributed on $\McUndir$ is singular goes to zero as
$n$ grows to infinity.
Note that in the case $d=1$ the matrix is a permutation matrix, hence non-singular;
while in the case $d=2$ the conjecture fails (see <cit.> and, for the directed case, <cit.>). Note also that
$M\in \Mc$ is singular if and only if the “complementary" matrix $M'
\in {\cal{M}}_{n, n-d}$ obtained by interchanging zeros and ones is singular, thus the cases $d=d_0$ and $d=n-d_0$ are essentially the same.
The corresponding question for non-symmetric
adjacency matrices is the following (cf., <cit.>):
Is it true that for every $3\leq d\leq n-3$
\begin{equation}\tag{$*$}\label{the question}
p_{n,d}:= \P _{\Mc} \left(\{M\in \Mc:\,M\mbox{ is singular}\} \right)
% =\frac{|\{M\in \Mc:\,M\mbox{ is singular}\}|}{|\Mc|}
\longrightarrow
0\quad \mbox{ as } \quad n\to\infty?
\end{equation}
The main difficulty in such singularity questions
stems from the restrictions on row- and column-sums, and from possible
symmetry constraints for the entries.
The question (<ref>) has been recently studied in
<cit.> by Cook who obtained the bound $p_{n,d} \leq d^{-c}$ for
a small universal constant $c>0$ and $d$ satisfying
$\omega(\ln^2 n)\leq d\leq n-\omega(\ln^2 n)$, where $f\geq \omega(a_n)$
means $f/a_n \to \infty$ as $n\to \infty$.
The main result of our paper is the following theorem.
There are absolute positive constants $c, C$ such that for
$C\le d\le cn/\ln^2 n$ one has
p_{n, d} \leq
% \frac{|\{M\in \Mc:\,M\mbox{ is singular}\}|}{|\Mc|}\leq
\frac{C\ln ^{3}d}{\sqrt{d}}.\;\;
Thus we proved that $p_{n,d} \to 0$ as $d\to \infty$, which in
particular verifies (<ref>) whenever $d$ grows to
infinity with $n$, without any restrictions on the rate of convergence.
(Recall that the proof in <cit.> requires $d\geq \omega(\ln^2 n)$.)
We would also like to notice that even for the range
$\omega(\ln^2 n)\leq d\le c n/\ln^2 n$, our bound on probability in Theorem A is better than in <cit.>.
Of course, it would be nice to obtain a bound going to zero with $n$ and not
with $d$ for the range $d\ge 3$ as well.
In the remaining part of the introduction we describe methods
and techniques used in this paper. We also explain several novel
ideas that allow us to drop the restriction
$d\geq \omega(\ln^2 n)$ and to treat very sparse matrices. In particular,
we introduce the notion of almost constant vectors and show how
to eliminate matrices having almost constant null vectors;
we show a new approximation argument dealing with tails of
properly rescaled vectors; we prove an anti-concentration property
for graphs, which is of independent interest; and we provide
a more delicate version of the so-called “shuffling" technique.
This paper can be naturally split into two distinct parts. In the first
one we establish certain properties of random $d$-regular digraphs. In the
second part we use them (or to be more
precise, their “matrix” equivalents) to deal with the singularity of
adjacency matrices. However in the introduction we reverse this
order and discuss first the “matrix" part as it provides a general
perspective and motivations for graph results.
Singularity of random square matrices is a subject with a long history and many
results. In <cit.> (see also <cit.>) Komlós proved
that a random $n\times n$ matrix with independent $\pm 1$ entries
(Bernoulli matrix) is singular
with probability tending to zero as $n\to\infty$.
Upper bounds for the singular probability of random Bernoulli
matrices were successively improved to $c^n$ (for some $c\in(0,1)$) in
<cit.>; to $\bigl(3/4+o(1)\bigr)^n$ in <cit.>; and to
$\bigl(1/\sqrt{2}+o(1)\bigr)^n$ in <cit.>. Recall that the
conjectured bound is $\bigl(1/2+o(1)\bigr)^n$. The corresponding problem for symmetric Bernoulli matrices was considered
in <cit.>, <cit.>, <cit.>.
Recently, matrices with independent rows and with row-sums constrained
to be equal to zero were studied in <cit.>.
In all these works, a fundamental role is played by what is nowadays
called the Littlewood-Offord theory. In its classical form,
established by Erdős <cit.>, the Littlewood-Offord inequality states that for every fixed $z\in\R$, a fixed vector $a=(a_1,a_2,\dots,a_n)\in\R^n$
with non-zero coordinates, and for independent random signs $r_k$, $k\le n$, the probability $\P\left\{\sum_{k=1}^n r_ka_k=z\right\}$
is bounded from above by $n^{-1/2}$.
This combinatorial result has been substantially strengthened and
generalized in subsequent years, leading to a much better
understanding of interrelationship between
the law of the sum $\sum_{k=1}^n r_ka_k$ and the arithmetic
structure of the vector $a$.
For more information and further references, we refer the
reader to <cit.>, <cit.>, and <cit.>.
The use of the Littlewood-Offord theory in context of random matrices
can be illustrated as follows. Given an $n\times n$ matrix $A$ with
i.i.d. elements, $A$ is non-singular if and only if the inner product
of a normal vector to the span of any subset of $n-1$ columns of $A$
with the remaining column is non-zero. Thus, knowing the “typical”
arithmetic structure of the random normal vectors and conditioning on
their realization, one can estimate the probability that $A$ is
singular. Moreover, a variant of this approach allows us to obtain sharp
quantitative estimates for the smallest singular value of the matrix
with independent subgaussian entries <cit.>.
Similarly to the aforementioned works, the Littlewood-Offord
inequality plays a crucial role in the proof of Theorem A.
Note that if $\Mrand$ is a random matrix uniformly distributed on
$\Mc$ then every
two entries/rows/columns of $\Mrand$ are probabilistically dependent;
moreover, a realization of the first $n-1$ columns uniquely defines
the last column of $\Mrand$. This makes a straightforward application
of the Littlewood-Offord
theory (as illustrated in the previous paragraph) impossible.
In <cit.>, a sophisticated approach based on the “shuffling” of two
rows was developed to deal with that problem.
The shuffling consists in a random perturbation of two rows of a
fixed matrix $M\in\Mc$ in such a way that the sum of the rows remains
unchanged. We discuss this procedure in more details in
Section <ref>. It can be also defined in terms of “switching"
discussed below.
The proof in <cit.> can be divided into two steps: at the first
step, one proves that the event that a random matrix $\Mrand$
does not have any (left or right)
null vectors with many ($\geq C n d^{-c}$) equal coordinates has
probability close to one, provided that $d\ge \omega (\ln^2 n)$.
Then one shows that conditioned on this event, a random matrix $\Mrand$ is
non-singular with large probability.
In our paper, we expand on some of the techniques developed in
<cit.> by adding new crucial ingredients.
On the first step, in Section <ref>, we show
that for $C\leq d\leq cn$, with probability going to one with $n$, a random
matrix $\Mrand$ does not have any null vectors having at least
$n(1-1/\ln d)$ equal coordinates, (we call such vectors almost constant). Note that we rule out a much smaller set of null vectors. This allows us to drop the lower bound on $d$, but requires a delicate
adjustment of the second step.
Key elements of the first step consists of a new anti-concentration
property of random graphs and their adjacency matrices as well as of
using a special form of an $\varepsilon$-net
build from the “tails” of appropriately rescaled
vectors $x\in\R^n$.
Then, conditioning on the event that $\Mrand$ does not have
almost constant null vectors, we show in Section <ref>
that a random matrix $\Mrand$ is non-singular with high probability.
This relies on a somewhat modified and simplified version of the
shuffling procedure for the matrix rows.
As the shuffling involves supports of only two rows we get at this step that
probability converges with $d$ and not with $n$. We would like
to emphasize that this is the only step which does not allow to have the convergence to zero with $n$.
We now turn our attention to Section <ref>, which deals with the set $\DirIntro$ of $d$-regular digraphs.
Our analysis
is based on an operation called “the simple switching,”
which is a standard tool to work with regular graphs. As an
illustration, let $G\in\DirIntro$ and let $i_1\neq i_2$ and
$j_1\neq j_2$ be vertices of $G$ such that $(i_1,j_1)$ and $(i_2,j_2)$
are edges of $G$ and $(i_1,j_2)$, $(i_2,j_1)$ are not. Then the
simple switching consists in replacing the edges $(i_1,j_1)$,
$(i_2,j_2)$ with $(i_1,j_2)$ and $(i_2,j_1)$, while leaving all other
edges unchanged. Note that the operation does not destroy
$d$-regularity of the graph. The simple switching was introduced (for
general graphs) by Senior <cit.> (in that paper, it was called
“transfusion”); in the context of $d$-regular graphs it was first
applied by McKay <cit.>. As in <cit.>,
we use this operation to compare cardinalities of certain subsets of $\DirIntro$.
We note that one could use the configuration model, introduced by Bollobàs <cit.> in the context of random regular graphs,
to prove our results for sparse graphs. We prefer to use the switching method in order to have a unified proof
for all ranges of $d$.
As in the matrix counterpart we work with a random graph $\rangr$
uniformly distributed on $\DirIntro$.
For a finite set $S$, we denote by $|S|$ its cardinality.
For a positive integer $n$ we denote by $[n]$ the set $\{1,2,\dots,n\}$.
For every subset $S\subset [n]$, let $\innbrs_\rangr(S)$ be the set of all vertices
of $\rangr$ which are in-neighbors to some vertex in $S$. Further, for every two
subsets $I,J$ of $[n]$, we denote by $\edgrd(I,J)$ the collection of edges of $\rangr$ starting from a vertex in $I$ and ending at a vertex from $J$.
In a simplified form, our first statement about graphs
(Theorem <ref> in
Section <ref>)
can be formulated as follows:
Let $8\leq d\leq n$, $\varepsilon\in (0,1)$, and $k\geq 2$. Assume that
$\varepsilon^2 \geq d^{-1} \max\{8, \ln d\}$ and $k\leq c\varepsilon n/d$
for a sufficiently small absolute positive constant $c$.
\P\bigl\{\exists S\subseteq [n], \,
\vert S\vert =k \, \text{ such that }\, \,
\lvert\innbrs_\rangr(S)\rvert \leq (1-\varepsilon)d\lvert S\rvert \bigr\} \leq
\exp\left(-\frac{\eps^2 d k}{8}\, \ln\left(\frac{3ec\eps n}{k d}\right)\right).
Note that $|S|\leq \lvert\innbrs_\rangr(S)\rvert\leq d|S|$.
Thus, roughly speaking, our result says
that “typically,” whenever a set $S$ is not too large,
the set of all in-neighbors of $S$ has cardinality close to the maximal possible one.
In the
case of undirected graphs such results are known (see e.g. <cit.> and references therein).
We note that in fact we prove a more general statement, in which we estimate the probability
conditioning on a “partial” realization of a random graph $\rangr$, when a certain subset of its edges is fixed
(see Theorem <ref>).
In our second result, we estimate the probability that $\edgrd(I,J)$ is empty for
large sets $I$ and $J$ (see Theorem <ref> in
Section <ref>):
There exist absolute positive constants $c, C$
such that the following holds. Let $2\leq d\leq n/24$ and $C n \ln d/d \leq \ell \leq r\leq n/4$. Then
\P\bigl\{\edgrd(I,J)=\emptyset\;\;\mbox{for some }I,J\subset[n]\mbox{ with }|I|\geq \ell,\;|J|\geq r\bigr\}
\leq \exp\left(-c r\ell d/n\right).
Note that the first statement can be reformulated in terms of sets
$\edgrd(I,J)$ (however, the range of cardinalities for $I$ and $J$
will be different compared to the second result). These statements
can be seen as manifestations of a general phenomenon that a random
graph $\rangr$ with a large probability has good regularity properties.
Let us also note that analogous statements for the Erdős–Rényi graphs
(in this random model an edge between every two vertices is included/excluded
in a graph independently of other edges) follow from standard Bernstein-type inequalities. For related results on $d$-regular random graphs, we refer
the reader to <cit.> where concentration properties of co-degrees
were established in the undirected setting, and to <cit.> for concentration of co-degrees and of
the “edge counts” $|\edgrd(I,J)|$ for digraphs. In paper <cit.> which
serves as a basis for the main theorem of <cit.> mentioned above, rather strong concentration properties of $|\edgrd(I,J)|$ are established;
however, the results provided in that paper are valid only for
$d\geq \omega(\ln n)$.
The proof in <cit.> is based on the method of exchangeable pairs introduced by Stein and developed
for concentration inequalities by Chatterjee (see survey <cit.> for more information and references).
On the contrary, our proof of the afore-mentioned statements is simpler, completely self-contained and works for $d\geq C$.
As we mentioned above, we use the following
Littlewood-Offord type anti-concentration result
matching anti-concentration properties of a
weighted sum of independent random variables or vectors
studied in the Littlewood-Offord theory.
This result is of independent interest, and we
formulate it here as a theorem (see also Theorem <ref> in
Section <ref>).
For every $J\subset [n]$ and $i\in[n]$ we define
$\delta_i^{J}(\rangr)\in \{0,1\}$ as the indicator
of the event $\{i\in \innbrs_\rangr(J)\}$ and denote
$\delta^J(\rangr):=(\delta_1^J(\rangr),\ldots, \delta_n^J(\rangr))\in
\{0,1\}^n$.
There are two positive absolute constants $c$ and $c_1$ such that the
following holds.
Let $32\leq d\leq cn$ and $I, J$ be disjoint subsets of $[n]$ such that
$\vert I\vert \leq d\vert J\vert/32$ and $8\leq \vert J \vert \leq 8 c n/d$.
Let vectors $a^i\in\{0,1\}^n$, $i\in I$, be
such that the event
\e:=\{\innbrs_\rangr(i)=\supp a^i\mbox{ for all }i\in I\}
has non-zero probability (if $I=\emptyset$ we set $\e=\DirIntro$).
Then for every $v\in \{0,1\}^n$ one has
\P\{ \delta^J(\rangr) =v \mid \e\}\leq 2\exp\left(- c_1 d
\vert J\vert \ln\big(\frac{n}{ d\vert J\vert }\big)\right).
We note that the probability estimate in the previous statement
matches the one for the corresponding quantity $\delta^J$ in the Erdős–Rényi model.
The paper is organized as follows.
Sections <ref> deals with all results related to graphs.
Section <ref> provides links between the graph
results of Section <ref> and the matrix results used in
Section <ref>.
Finally, Section <ref> presents the proof of the main theorem,
including a number of auxiliary combinatorial lemmas.
In this paper letters $c, C$, $c_0, C_0$, $c_1, C_1$, ... always denote
absolute positive constants (i.e. independent of any parameters), whose
precise value may be different from line to line.
Main results of this paper were announced in <cit.>.
Aknowledgment. This work was conducted while the second named author was a
Research Associate at the University of Alberta,
the third named author was a graduate student and held
the PIMS Graduate Scholarship,
and the last named author was a CNRS/PIMS PDF at
the same university.
They all would like to thank the Pacific Institute
and the University of Alberta for the support.
A part of this work was also done when the first four authors
took part in activities of the annual program “On Discrete Structures: Analysis and Applications" at the Institute for Mathematics and its Applications (IMA), Minneapolis, MN, USA.
These authors would like to thank IMA
for the support and excellent working conditions.
All authors would like to thank Michael Krivelevich for many
helpful comments on the “graph" part of this paper. We would also like to thank Justin Salez for helpful comments.
§ EXPANSION AND ANTI-CONCENTRATION FOR RANDOM DIGRAPHS
§.§ Notation and preliminaries
For a real number $x$, we denote by $\lfloor x\rfloor$ the largest integer smaller than or equal to $x$ and
by $\lceil x\rceil$ the smallest integer larger than or equal to $x$.
Further, for every $a\geq 1$, we denote by $[a]$ the set
$\{i\in\N:\, 1\leq i\leq \lfloor a\rfloor\}$.
Let $d\leq n$ be positive integers. A $d$-regular directed graph (or
$d$-regular digraph) on $n$ (labeled) vertices is a graph in which every
vertex has exactly $d$ in-neighbors and $d$ out-neighbors. We allow the graphs
to have loops and opposite/anti-parallel edges but do not allow multiple edges.
Thus this set coincides with the set of $d$-regular bipartite graphs with both
parts of size $n$. The set of vertices of such graphs is always identified with $[n]$.
The set of all these graphs is denoted by $\DirIntro$. When $n$ and $d$ are clear
from the context, we will use a one-letter notation $\D$. Note that the set of adjacency matrices for graphs in $\D$ coincides with the set of $n\times n$
matrices with $0/1$-entries such that every row and every column has exactly $d$ ones.
By a random graph on $\D$ we always
mean a graph uniformly distributed on $\D$ (that is, with respect to the normalized counting measure).
Let $\nrangr=([n],E)$ be an element of $\D$, where $E$ is the set of its
directed edges.
Thus $(i,j)\in E$, $i,j\leq n$, means that there is an edge going from vertex $i$ to vertex $j$.
We will denote the adjacency
matrix of $\nrangr$ by $M=M(\nrangr)$; its rows and columns by $R_i=R_i(M)=R_i(\nrangr)$ and
$X_i=X_i(M)=X_i(\nrangr)$, $i\leq n$, respectively.
Given a graph $\nrangr\in\D$ and a subset $S\subset [n]$ of its vertices, let
\begin{align*}
\outnbrs(S)&=\outnbr(S):=\bigl\{v\leq n:\, \exists i\in S \, \, (i,v) \in E \bigr\} = \bigcup _{i\in S} {\rm supp} R_i,\\
\innbrs(S)&=\innbr(S):=\bigl\{v\leq n:\, \exists i\in S \, \, (v,i) \in E
%v\to i\mbox{ for some }i\in S
\bigr\}=\bigcup _{i\in S} {\rm supp} X_i.
\end{align*}
Similarly, we define the out-edges and the in-edges as follows
\begin{align*}
\outedg(S)&:=\bigl\{e\in E:\, e=(i,j) \text{ for some } i\in S \text{ and } j\leq n \bigr\},\\
\inedg(S)&:=\bigl\{e\in E:\, e=(i,j) \text{ for some } i\leq n \text{ and } j\in S\bigr\}.
\end{align*}
For one-element subsets of $[n]$ we will use lighter notations
$\innbr(i)$, $\outedg(i)$, $\inedg(i)$ instead of
$\innbr(\{i\})$, $\outedg(\{i\})$, $\inedg(\{i\})$.
Given a graph $\nrangr=([n],E)$, for every $I,J\subset [n]$ the set of all edges departing from $I$ and landing in $J$ is denoted by
\edg(I\times J)= \edg(I,J)=\bigl\{ e\in E:\, e=(i,j) \text{ for some } i\in I \text{ and } j\in J\bigr\}.
Further, we let
\DO(I,J) = \bigl\{ \nrangr\in \D:\, \edg(I, J)=\emptyset\bigr\}.
Note that $\DO(I,J)$ is the set of all graphs whose adjacency matrices have zero $I\times J$-minor,
hence the superscript “$0$”.
Given $\nrangr\in\D$, for $u, v\leq n$ the sets of common out-neighbors and common in-neighbors will be denoted as
\begin{align*}
\outco(u, v)&=\{j\leq n\, :\, (u, j), (v, j)\in E\} = \supp R_u \cap \supp R_v,\\
\inco(u, v)&=\{i\leq n \, :\, (i, u), (i, v)\in E\} = \supp X_u \cap \supp X_v.
\end{align*}
For every $S\subset [n]$ and $F\subset [n]\times [n]$, we define
\D(S,F)=\bigl\{ \nrangr\in\D:\, \inedg( S)=F\bigr\}.
Informally speaking, $\D(S,F)$ is the subset of $d$-regular graphs for which
the in-edges of $S$ are “frozen” and, as a set, coincide with $F$.
Note that a necessary (but not sufficient) condition for $\D(S,F)$ to be non-empty is
\forall i\leq n\, \, \, \, \vert \{\ell\in [n]:\, (i, \ell)\in F\}\vert \leq d \quad \mbox{ and } \quad \forall j\in S\, \, \, \,
\vert \{\ell\in [n]:\, (\ell, j)\in F\}\vert = d.
For every $\varepsilon \in (0,1)$, denote
\Dco(\varepsilon)=\bigl\{ \nrangr\in\D:\, \forall i\neq j\leq n\,\,\, \, \, \,
\vert \outco(i,j)\vert\leq \varepsilon d\bigr\} = \bigcap_{i<j}
\D_{i,j}^{co}(\varepsilon),
\D_{i,j}^{co}(\varepsilon):=\bigl\{ \nrangr\in\D:\, \vert \outco(i,j)\vert\leq
\varepsilon d\bigr\}.
Let $A$, $B$ be sets, and $R\subset A\times B$ be a relation.
Given $a\in A$ and $b\in B$, the image of $a$ and preimage of $b$ are defined by
R(a) = \{ y \in B \, : \, (a,y) \in R\} \quad \mbox{ and } \quad
R^{-1}(b) = \{ x \in A \, : \, (x,b) \in R\}.
We also set $R(A)=\cup _{a\in A} R(a)$.
Further in this section, we often define relations between sets in order to estimate their cardinality,
using the following simple claim.
Let $s, t >0$.
Let $R$ be a relation between two finite sets $A$ and $B$ such that for
every $a\in A$ and every $b\in B$ one has $|R(a)|\geq s$ and $|R^{-1}(b)|\leq t$.
s |A|\leq t |B|.
Without loss of generality we assume that $A=[k]$ and $B=[m]$ for some positive integers $k$ and $m$.
For $i\leq k$ and
$j\leq m$, we set $r_{ij} =1$ if $(i,j)\in R$ and $r_{ij} =0$ otherwise. Counting the number
of ones in every row and every column of the matrix $\{r_{ij}\}_{ij}$ we obtain
\sum _{i=1}^k \sum _{j=1}^m r_{ij} = \sum _{i=1}^k |R(i)| \geq s k = s |A|
\quad \mbox{ and } \quad
\sum _{j=1}^m \sum _{i=1}^k r_{ij} = \sum _{j=1}^m |R^{-1}(j)| \leq t m = t |B|,
which implies the desired estimate.
§.§ An expansion property of random digraphs
In this section, we
establish certain expansion properties of random graphs uniformly distributed on $\D$, which
can roughly be described as follows: given a subset $S\subset [n]$ of cardinality $|S|\leq c n/d$,
with high probability the number of in-neighbors of $S$ is of order $d|S|$.
Beside its own interest, this result is used in the proof of the anti-concentration property for graphs which will be given in
Section <ref>. In fact we will need a
statement where we control the number of in-neighbors of
a subset of vertices while “freezing” (i.e. conditioning on a realization of) a set of edges inside the graph.
Let $8\leq d\leq n$, $\varepsilon\in (0,1)$, and $k\geq 2$. Assume that
\varepsilon^2 \geq \frac{\max\{8, \ln d\}}{d}\quad\text{and}\quad k\leq \frac{c\varepsilon n}{d}
for a sufficiently small absolute positive constant $c$.
Let $I\subset [n]$ be of cardinality at most $n/8$.
\Gamma _k=\bigl\{ \nrangr\in\D \, :\, \exists S\subseteq I^c, \,
\vert S\vert =k, \, \, \text{ such that }\, \,
\lvert\innbr(S)\rvert \leq (1-\varepsilon)d\lvert S\rvert \bigr\}
\Gamma=\bigl\{ \nrangr\in\D\, :\, \exists S\subseteq I^c, \,
\vert S\vert \leq c\eps n/d,\, \, \text{ such that }\, \,
\lvert\innbr(S)\rvert \leq (1-\varepsilon)d\lvert S\rvert \bigr\}
= \bigcup _{\ell=2}^{c\eps n/d} \Gamma _\ell .
Then for every $F\subset [n]\times [n]$ with $\D(I,F)\neq \emptyset$ we have
\P\left(\Gamma _k\mid \D(I,F) \right) \leq
\exp\left(-\frac{\eps^2 d k}{8}\, \ln\left(\frac{3ec\eps n}{k d}\right)\right) .
In particular,
\P\left(\Gamma \mid \D(I,F) \right) \leq
\exp\left(-\frac{\eps^2 d}{8} \ln\left(\frac{ec\eps n}{d}\right)\right) .
Let us describe the idea of the proof of Theorem <ref>. Suppose we are given
a set of vertices $S$ of an appropriate size. Since $\vert \inedg(S)\vert =d\vert S\vert$, then
we always have
\vert S\vert \leq \vert \innbr(S)\vert\leq d\vert S\vert.
We want to prove that the number of graphs satisfying $\vert \innbr(S)\vert\leq (1-\varepsilon)d\vert S\vert$
is rather small.
In order to estimate the number of in-neighbors of $S$, our strategy is to build
$S$ by adding one vertex at a time and trace how the number of
in-neighbors is changing. Namely, if $S=\{v_i\}_{i\leq s}$ then to build $S$ we start by setting $S_1:=\{v_1\}$ – a set for which we know that it has exactly $d$ in-neighbors. Now we add the vertex
$v_2$ to $S_1$ to get $S_2:=\{v_1,v_2\}$. We need to trace how
the number of in-neighbors to $S_2$ changed compared to that of $S_1$. More precisely,
we need to count the number of graphs for which the number of in-neighbors
has increased by at most $(1-\varepsilon/2)d$. To this end,
we count the number of graphs having the property that the number of common in-neighbors to $v_1$ and $v_2$ is at least
$\varepsilon d/2$. We count such graphs by applying the simple switching.
One should be careful here to switch the edges without interfering with the frozen area of the graph.
We continue in a similar manner by adding one vertex at a time and controlling the number of
common in-neighbors between the added vertex and the existing ones.
Now, note that the condition $\vert \innbr(S)\vert \leq (1-\varepsilon)d\vert S\vert$
implies that for a large proportion of the vertices added, the number of common in-neighbors with the existing vertices is
at least $\varepsilon d/2$. We use this together with the cardinality estimates obtained via the simple switching at
each step to get the required result.
We use the following notation.
Given $S\subset [n]$ and $\delta\in (0,1)$,
we set $\Gamma(S, \emptyset)=\D$ and for a non-empty $J\subset [n]$, let
\Gamma(S, J)=\Gamma(S, J, \delta)=\Bigl\{G\in\D \, : \, \forall j\in J \, \mbox{ one has } \Big| \bigcup_{i\in S, i<j} \inco(i, j) \Big| \geq \delta d\Bigr\}
(the number $\delta$ will always be clear from the context). We also use a simplified notation $\Gamma(S, j):= \Gamma(S, \{j\})$.
Note that $\Gamma(S, J)$ contains all graphs in which every vertex $j\in J$ has many common in-neighbors
with the set $\{i\in S\, \, :\, \, i<j\}$. In the next lemma, we estimate cardinalities of $\Gamma(S,j)$, conditioning
on a “partial” realization of a graph.
Let $\delta \in (0,1)$, $2 \leq d\leq n/12$,
$1\leq k\leq \delta n/{(4e d)}$ and
$F, H\subset [n]\times[n]$. For every $I\subset [k+d]^c$ satisfying
\vert I\vert \leq \frac{n}{8},
one has
\lvert \Gamma([k], k+1)\cap \D([k],F)\cap \D(I,H) \rvert
\leq \gamma_k \lvert \D([k],F)\cap \D(I,H)\rvert ,
\gamma _k= \left(\frac{2 ekd}{\delta n}\right)^{\delta d}.
Less formally, the above statement asserts that, considering a subset of $\D$
with prescribed (frozen) sets of in-edges for $[k]$ and $I$, for a vast majority of such graphs
the $(k+1)$-th vertex will have a small number of common in-neighbors with the first $k$ vertices.
We assume that the intersection $\D([k],F)\cap \D(I,H)$ is non-empty.
Then we have $F([n])=[k]$ and $F^{-1}([k])=\innbr ([k])$
(recall notation for images and preimages of a relation).
Without loss of generality, $\innbr ([k]) = [n_1]^c$ for some $n_1\leq n$.
Note that
k\leq \big\vert \innbr ([k])\big\vert \leq kd,
hence $n-kd \leq n_1\leq n-k$.
For $0\leq q\leq d$ denote
Q(q):=\bigl\{ \nrangr \in \D([k],F)\cap \D(I,H):\, \big\vert
\innbr([k])\cap \innbr(k+1) \big\vert =q\bigr\}.
Q:= \Gamma([k], k+1) \cap \D([k],F)\cap \D(I,H)=\bigcup _{q=\lceil\delta d\rceil}^d Q(q).
We proceed by comparing the cardinalities $Q(q)$ and $Q(q-1)$ for every $1\leq q\leq d$.
To this end, we will define a relation $R_q\subset Q(q)\times Q(q-1)$.
Let $\nrangr \in Q(q)$. Then there exist
$n_1 <i_1<...<i_{q}$ such that for every $\ell\leq q$ we have
i_{\ell}\in \innbr([k])\cap \innbr(k+1).
For every $\ell \leq q$, there are at most $d^2$ edges inside
\edg\big([n_1], \outnbr(i_\ell)\big).
Further, there are $(n_1-(d-q))d$ edges in $\outedg\big([n_1]\setminus \innbr(k+1)\big)$ and
at most $d\vert I\vert$ edges in $\edg\big([n_1]\setminus \innbr(k+1), I\big)$.
Therefore, for every $\ell \leq q$, the cardinality of the set
E_\ell:=\edg\big([n_1]\setminus \innbr(k+1), I^c\setminus \outnbr(i_\ell)\big)
can be estimated as
|E_\ell|\geq \big(n_1-(d-q)-\vert I\vert \big)d-d^2\geq (7n/8 - kd - 2d) d\geq nd/2
(here, we used the conditions $|I|\leq n/8$ and $n_1\geq n-kd$ together with the restrictions on $k$).
Now, we turn to constructing the relation $R_q$.
We let a pair $(\nrangr, \nrangr')$ belong to $R_q$ for some $\nrangr' \in Q(q-1)$ if
$\nrangr'$ can be obtained
from $\nrangr$ in the following way. First we choose $\ell \leq q$ and an edge $(i,j)\in E_\ell$. We destroy the edge
$(i_\ell, k+1)$ to form the edge $(i,k+1)$, then destroy the edge $(i,j)$ to form the edge $(i_\ell,j)$
(in other words, we perform the simple switching on the vertices $i,i_\ell,j,k+1$). Note that
the conditions $i\notin \innbr(k+1)$ and $j\notin \outnbr(i_\ell)$, which are implied by the definition of $E_\ell$,
guarantee that the simple switching does not create multiple edges, and we obtain a valid graph in $Q(q-1)$.
The definition of $R_q$ implies that for every $G\in Q(q)$ one has
\begin{equation}\label{eq-image-Rq-lem51}
\vert R_q(\nrangr)\vert\geq\sum\limits_{\ell=1}^q |E_\ell|\geq \frac{qnd}{2}.
\end{equation}
Now we estimate the cardinalities of preimages.
Let $\nrangr' \in R_q\bigl( Q(q)\bigr)$.
In order to reconstruct a graph $\nrangr$ for which
$(\nrangr, \nrangr')\in R_q$, we need to perform a simple switching which
\mbox{destroys an edge from} \quad \edgpr \bigl([n_1], k+1\bigr) \quad \quad
\mbox{and}\quad \quad \mbox{adds an edge to} \quad \edgpr \bigl([n_1]^c, k+1\bigr) .
There are at most $d-q+1$ choices to destroy an edge in $\edgpr \bigl([n_1], k+1\bigr)$,
and at most $n-n_1\leq kd$ possibilities to create an edge connecting $[n_1]^c$ with $(k+1)$-st vertex.
Assume that we destroyed an edge $(v, k+1)$ and added an edge $(u, k+1)$.
The second part of the simple switching is to destroy an excessive out-edge
of $u$ and create a corresponding edge (with the same end-point) for $v$. It
is easy to see that we have at most $d$ possibilities for the second part
of the switching. Therefore,
\begin{equation*}
\lvert R_q^{-1}(\nrangr')\rvert \leq kd^3.
\end{equation*}
Using this bound, Claim <ref>, and (<ref>),
we obtain that
\lvert Q(q)\rvert \leq \left(\frac{2 kd^2}{qn}\right)\cdot \lvert Q(q-1)\rvert
and, applying the estimate successively,
\lvert Q(q)\rvert \leq \left(\frac{2 kd^2}{n}\right)^{q} \, \,
\frac{1}{q!}\, \, \lvert Q(0)\rvert.
Since $q!\geq 2(q/e)^q$ and $2e kd/(\delta n)\leq 1/2$, this implies
\vert Q\vert= \sum_{q=\lceil\delta d\rceil}^d\lvert Q(q)\rvert
\leq \frac{1}{2}\,\, \sum_{q=\lceil\delta d\rceil}^d \left(\frac{2e kd}{\delta n}\right)^{q} \,
\lvert Q(0)\rvert
\leq
\left(\frac{2e kd}{\delta n}\right)^{\delta d}\,
\lvert Q(0)\rvert .
Using that $Q(0)\subset \D([k],F)\cap \D(I,H)$, we obtain the desired result.
Now, we iterate the last lemma to obtain the following statement.
Let $\delta$, $n$, $d$, $k$ and $\gamma _k$ be as in Lemma <ref> and let $\ell\leq k$.
Further, let $I\subset [n]$ satisfy
\vert I\vert \leq n/8
and let $H\subset [n]\times[n]$.
Then for every subsets $J\subset S\subset I^c$ such that $\vert S\vert =k$ and $\vert J\vert =\ell$, one has
\lvert \Gamma(S, J)\cap \D(I,H) \rvert \leq {\gamma_k}^{\ell} \, \lvert \D(I,H) \rvert .
Without loss of generality we assume that the intersection
$\Gamma(S, J)\cap \D(I,H)$ is non-empty, that $S=[k]$ and $I\subset [k+d]^c$.
Write $J=\{j_1, ..., j_{\ell}\}$ for some $j_1<...<j_{\ell}$. For $1\leq s\leq \ell$ denote
$J_s=\{j_1, ..., j_{s}\}$, $J_0=\emptyset$ and let $k_s=j_{s}-1$.
Note that for every $1\leq s\leq \ell$, we have
\Gamma (S, J_s)= \Gamma ([k_s], J_s).
Note also that
\begin{equation}\label{eq-coro-split}
\Gamma (S, J_s)= \Gamma ([k_s], k_s+1)\cap \Gamma (S, J_{s-1}).
\end{equation}
\begin{equation}\label{product-al}
\big\vert \Gamma (S, J) \cap \D(I,H) \big\vert =
|\D(I,H)|\, \prod _{s=1}^{\ell}
\frac{|\Gamma (S, J_s)\cap \D(I,H)|}{|\Gamma (S, J_{s-1})\cap \D(I,H)|}.
\end{equation}
For $1\leq s\leq \ell$ define
{\cal{F}}_s = \left\{F \subset [n]\times[n] \, : \, \D([k_{s}], F)\cap \D(I,H)
\subset \Gamma (S, J_{s-1}) \right\}.
Then by (<ref>) we have
\Gamma (S, J_{s})\cap \D(I,H)= \bigsqcup_{F\in\mathcal{F}_s}
\Gamma ([k_s], k_s+1)\cap \D([k_{s}], F)\cap \D(I,H).
Applying Lemma <ref> we obtain
\begin{align*}
\big\vert \Gamma (S, J_s)\cap \D(I,H) \big\vert &= \sum _{F \in {\cal{F}}_s} | \Gamma ([k_s], k_s+1)\cap \D([k_{s}], F)\cap \D(I,H)| \\
&\leq \gamma_{k_s} \sum _{F \in {\cal{F}}_s} | \D([k_{s}], F)\cap \D(I,H)|\\
&\leq \gamma_k | \Gamma (S, J_{s-1}) \cap \D(I,H)| ,
\end{align*}
where the last inequality follows from the definition of ${\cal{F}}_s$ and $k_s\leq k$.
This and (<ref>) imply the result.
We are now ready to prove Theorem <ref>.
In the proof, we will use Corollary <ref>, together with an easy observation
that the condition $|\innbr(S)| \leq (1-\varepsilon ) d|S|$ for an (ordered) subset $S$ of vertices implies
that proportionally many vertices in $S$ have at least $\varepsilon d/2$ common in-neighbors
with the union of the preceeding vertices.
Proof of Theorem <ref>.
Let $\nrangr\in \Gamma _k$ and $S$ be as in the definition of $\Gamma _k$. For $j\in S$ consider
A_j = \bigcup_{i\in S, i<j} \inco(i, j)
and denote by $m_j$ its cardinality. Note that for
$j_0=\min \{j \, : \, j\in S\}$ one has $A_{j_0}=\emptyset$ and $m_{j_0}=0$. Note also
\lvert\innbr(S)\rvert = \sum_{j\in S} (d- m_j).
Let $\delta =\eps/2$ and consider $J' :=\{ j\in S \, : \, m_j \geq \delta d\}.$ Since $m_{j_0}=0$,
(1-\eps )\, d\, |S| \geq \lvert\innbr(S)\rvert \geq \sum_{j\in S\setminus J'} (d- m_j) > (1-\eps/2)\, d \, (|S|-|J'|),
which implies
|J'| > \frac{\eps}{2-\eps}\, |S| > \frac{\eps}{2} \, |S|.
Hence, for every $\nrangr\in \Gamma _k$ there exists $S\subset I^c$ with $|S|=k$,
and $J \subset S$ such that
\begin{equation*}
|J |=\lceil \eps k /2 \rceil:=\ell \quad \mbox{ and } \quad m_j\geq \delta d \, \,
\mbox{ for all } \, \, j\in J.
\end{equation*}
\Gamma _k \subset \bigcup _{|S|=k} \, \, \bigcup_{J\subset S, |J|=\ell}\,\, \Gamma(S, J).
By Corollary <ref> we have
\lvert \Gamma _k \cap \D(I,F) \rvert \leq {n\choose k} \, {k \choose \ell} \,
{\gamma_k}^{\ell} \, |\D(I,F) | \leq \left(\frac{en}{k}\right)^k\,
\left(\frac{e k}{\ell}\right)^{\ell} \, {\gamma_k}^{\ell} \, |\D(I,F)|.
We assume that $\eps k \geq 2$ (the case $\eps k <2$, in which $\ell =1$, is treated similarly). Using $\eps \geq \max\{\sqrt{\ln d /d}, \sqrt{8/d}\}$, by direct calculations we observe
\left(\frac{en}{k}\right)^k\, \left(\frac{e k}{\ell}\right)^{\ell} \,
\left(\frac{4 e kd}{\eps n}\right)^{\eps d\ell/2} \leq
\left(\frac{en}{k}\right)^k\, \left(\frac{2 e}{\varepsilon}\right)^{\eps k/2} \,
\left(\frac{4ekd}{\eps n}\right)^{\eps^2 dk /4}
\leq \left(\frac{C_1 k d}{\eps n}\right)^{\eps^2 dk /8}
for a sufficiently large absolute constant $C_1>0$.
Taking $c\leq 1/(3 e C_1)$, we obtain the desired estimate for $\Gamma_k$.
The second assertion of the theorem regarding $\Gamma$ follows immediately.
As we have already noted, Theorem <ref> essentially postulates that a
random $d$-regular digraph typically has good expansion properties in the sense that
every sufficiently small subset $S$ of its vertices has almost $d|S|$ in-neighbors and $d|S|$ out-neighbors.
In the undirected setting, expansion properties of graphs are a subject of very intense research
(see, in particular, <cit.> and references therein).
As the conclusion for this subsection, we would like to recall some of the known expansion properties
of undirected random graphs and compare them with the main result of this part of our paper.
Let $G=(V,E)$ be an undirected graph on $n$ vertices. Given a subset $U \subset V$, by $\partial_V U$
we denote a set of all vertices adjacent to the set $U$ but not in $U$, i.e.
\partial_V U:=\{i\not\in U:\ \exists j\in U\, (i,j)\in E\}
= \innbr(U)\setminus U.
Similarly, let $\partial_E U$ be the set of all edges of $G$ with
exactly one endpoint in $U$.
For every $\lam\in(0,1]$, we define the $\lam$-vertex isoperimetric number
i_{\lam,V}(G):=\min_{\vert U\vert \leq \lam n} \frac{ \vert \partial_V U\vert}{\vert U\vert},
and, for every $\lam\in(0,1/2]$, the $\lam$-edge isoperimetric number
i_{\lam,E}(G):=\min_{\vert U\vert \leq \lam n} \frac{ \vert \partial_E U\vert}{\vert U\vert}.
For $\lam=1/2$, the above quantities are simply called the vertex and the edge
isoperimetric numbers, denoted by $i_V(G)$ and $i_E(G)$.
Since $\vert\partial_V U\vert \leq \vert \partial_E U\vert\leq d\vert\partial_V U\vert$, for every $\lam\in(0,1/2]$ we have
\begin{equation}\label{relation-isop-edge-vertex}
i_{\lam,V}(G)\leq i_{\lam,E}(G)\leq d i_{\lam,V}(G).
\end{equation}
Now, let $\rangr$ be a $d$-regular graph uniformly distributed on the set $\UndirIntro$.
In <cit.> it was shown that for large enough fixed $d$
\begin{equation}\label{eq-bollobas}
i_E(\rangr)\geq d/2 -\sqrt{d\ln 2},
\end{equation}
with probability going to one with $n$.
This result was generalized in
<cit.>, where is was shown that
i_{\lam,E}(\rangr)\geq d(1-\lam +o(1))
with probability going to one with $n$, where $o(1)$ depends on $d$ and
can be made arbitrarily small by increasing $d$.
Note that the relation (<ref>) together with
results from <cit.> immediately implies
$$i_{\lam,V}(\rangr)\geq 1-\lam +o(1)$$
(where the bound should be interpreted in the same way as before), however
the bound is far from being optimal.
An estimate for the second eigenvalue of $\rangr$ proved in <cit.> implies that for a fixed $d$
with large probability (going to one with $n$)
i_V(\rangr)\geq 1-8/d + O(1/d^2).
Moreover, for every $d$ and $\delta>0$ for small enough $\lam=\lam(d, \delta)>0$ the parameter $i_{\lam,V}$
(corresponding to expansions of small subsets of $V$) can be estimated as
$$i_{\lam,V}(\rangr)\geq d-2-\delta$$
(see <cit.>).
Our main result of this subsection can be interpreted as an expansion
property of regular digraphs for small vertex subsets. We define
the vertex isoperimetric number $i_{\lam,V}$ for digraphs by the same
formula as for undirected graphs. Theorem <ref> has the
following consequence, which, in particular, provides quantitative
estimates of $i_{\lam,V}$ for $d$ growing together with $n$ to infinity.
Let $8\leq d\leq n$ and $\varepsilon\in (0,1)$. Assume that
\varepsilon^2 \geq \frac{\max\{8, \ln d\}}{d}, \quad d\leq \frac{c\eps n}{2}
\quad\text{and}\quad \lam(\varepsilon) := \frac{c\varepsilon}{d}.
Further, let $\rangr$ be uniformly distributed on $\D$.
i_{\lam(\varepsilon), V}(\rangr) \geq (1-\varepsilon)d-1
with probability at least
1- \exp\left(-\frac{\eps^2 d}{8} \ln\left(\frac{ec\eps n}{d}\right)\right).
§.§ On existence of edges connecting large vertex subsets
In this part, we consider the following problem.
Let $\rangr$ be uniformly distributed on $\D$ and let $I$ and $J$ be two (large enough) subsets of $[n]$. We want to estimate the probability that $\rangr$ has no edges connecting a vertex from $I$ to a vertex from $J$. The main result of the subsection is the following theorem.
There exist absolute constants $c>0$ and
$C, C_1\geq 1$ such that the following holds.
Let $C_1\leq d\leq n/24$ and let natural
numbers $\ell$ and $r$ satisfy
\begin{equation*}
\frac{n}{4}\geq r\geq\ell \geq \frac{C n \ln (en/r)}{d}.
\end{equation*}
\P\left\{
\bigcup
\DO(I,J) \right\}
\leq \exp\left(-\frac{c r\ell d}{n}\right),
where the union is taken over all $I, J\subset [n]$ with
$\vert I\vert \geq \ell$ and $\vert J\vert \geq r$.
Obviously, the roles of $\ell$ and $r$ in this theorem are
interchangeable and the assumptions on $\ell$ and $r$ imply
that $\ell \geq Cn/d$ and
$r\geq C_1 n \ln d/d$.
We would like to notice that adding an assumption
$\ell \geq 4d^2$ in this theorem,
we could simplify its proof (we would not need quite technical Lemma <ref> below)
The statement of the theorem can be related to known results on the independence number of
random undirected graphs. Recall that the independence number $\alpha(\nrangr)$ of a
graph $\nrangr$ is the cardinality of the largest subset of its vertices such that no two vertices of the subset are adjacent.
Suppose now that $\rangr$ is uniformly distributed on $\UndirIntro$.
For $d\to\infty$ with $d\leq n^{\theta}$ for some fixed $\theta<1$, it was shown in <cit.> and <cit.>
that, as $n$ goes to infinity, the ratio $\alpha(\rangr)/\bigl(2n d^{-1}\ln d\bigr)$ converges to $1$ in probability.
Moreover, in <cit.> it was verified that in the range $n^{\theta}\leq d\leq 0.9n$ (for a sufficiently
large $\theta<1$), the asymptotic value of $\alpha(\rangr)$ is $2\ln d/\ln (n/(n-d))$, which is equivalent to $2n\ln d/d$ when
$d/n$ is small. Taking $I=J$ in Theorem <ref>, we observe a bound of the same order for random digraphs,
which can be interpreted as a large deviation estimate for the independence
number as follows.
There exist absolute positive constants $c,C$ such that for every
$2\leq d\leq n/24$ and a random digraph $\rangr$ uniformly distributed
on $\D$ one has
\begin{equation*}
\P\left\{\alpha(\rangr)> C\, \frac{n\ln d}{d}\right\} \leq
\exp\left(-\frac{c n\ln^2d}{d}\right).
\end{equation*}
We first give an idea of the proof of Theorem <ref>. Fix
two sets of vertices $I$ and $J$ of sizes $\ell$ and $r$. Our strategy is
to start with two small subsets of $I$, $J$ and to arrive to
$I$, $J$ by adding one vertex at a time.
Suppose that $I_1\subset I$ and $J_1\subset J$
and $S$ is a subset of graphs from $\D$ with no edges departing from $I_1$ and landing in $J_1$. We
add a vertex from $J\setminus J_1$ to $J_1$ to form a set $J_2$ and check whether
the property of having no edges connecting $I_1$ to $J_2$ is preserved, using the simple switching.
More precisely, when conditioning on the set of graphs $S$,
we estimate the proportion of graphs in $S$ such that there are no edges departing from $I_1$
to the vertex added. We perform an analogous procedure by adding a vertex to $I_1$ and
continue until the whole sets $I$ and $J$ are reconstructed.
Note that a similar argument can be applied in the undirected setting to estimate probability of large
deviation for the independence number (the sets $I$ and $J$ shall be equal in this situation).
We omit the proof of the undirected case as it is a simple adaptation of the argument
of Theorem <ref> and is not of interest in the present paper.
We start with a lemma which can be described as follows:
given two sets of vertices $[p]$ and $[k]$, among
graphs having no edges departing from $[p]$ to $[k]$, we count
how many have no departing edges from $[p]$ to the vertex $k+1$.
The proof of Theorem <ref> will then follow by iterating this lemma.
Let $20\leq d\leq n/24$ and $4e^2n/d \leq p, k \leq n/4$. Then
\max\left\{ \vert \DO([p],[k+1])\vert,\, \, \vert \DO([p +1],[k])\vert \right\} \leq \exp\left(-\frac{pd}{4e^2n}\right)\, \vert \DO([p],[k])\vert.
To prove this lemma we need the following rather technical statement,
which shows that for most graphs under consideration every
two vertices have a relatively small number of common out-neighbors.
For reader's convenience we postpone its proof to the end of this section.
Let $\varepsilon\in (0,1)$, $0\leq k\leq n/4$, $0\leq p\leq n$, and $d\leq \varepsilon n/12$.
\lvert \DO([p], [k])\setminus \Dco(\varepsilon) \rvert
\leq \frac{n^2}{2}\, \left(\frac{2e d}{\varepsilon n}\right)^{\varepsilon d}\,
\lvert \DO ([p], [k])\rvert ,
where $\DO([p], [k])=\D$ if $p=0$ or $k=0$.
Proof of Lemma <ref>.
We prove the bound for $\DO([p],[k+1])$, the other bound is obtained by passing to the transpose graph.
Fix $q:= \lceil pd/(2e^2n)\rceil $. Denote
Q := \DO([p],[k+1])\cap \Dco(1/2)
Q(q):= \{\nrangr\in \DO([p],[k]):\ \vert \edg([p], k+1)\vert =q\}.
To estimate cardinalities we construct a relation $R$ between $Q$ and $Q(q)$. We say that $(\nrangr,\nrangr')\in R$ for some
$\nrangr\in Q$ and $\nrangr' \in Q(q)$ if
$\nrangr'$ can be obtained
from $\nrangr$ using the simple switchings as follows. First
choose $q$ vertices $1\leq v_1<v_2<\ldots<v_q\leq p$. There are
$p \choose q$ such choices. Then choose $q$ edges in
$\edg([p]^c, k+1)$, say $(w_i, k+1)$, $i\leq q$, with $p<w_1<w_2<\ldots<w_q\leq n$. There are
$d \choose q$ such choices. Finally for every $i\leq q$ choose
j(i)\in\outnbr(v_i)\setminus \outnbr(w_i).
Since $\nrangr\in \Dco(1/2)$, for every $i\leq q$ there are at least $d/2$ choices
of $j(i)$. For every $i\leq q$ we destroy edges $(w_i, k+1)$,
$(v_i, j(i))$ and create edges $(v_i, k+1)$,
$(w_i, j(i))$. We have
\begin{equation}\label{eq-image-lem-th3}
\lvert R (\nrangr)\rvert \geq {p\choose q}\, {d\choose q}
\, \left(\frac{d}{2}\right)^q \geq
\left(\frac{p d^2}{2q^2}\right)^q.
\end{equation}
Now we estimate the cardinalities of preimages.
Let $\nrangr' \in R(Q)$. We bound $|R^{-1}(\nrangr')|$ from above.
To reconstruct a possible $\nrangr\in Q$ with $(\nrangr, \nrangr')\in R$, we perform simple switchings as follows.
Write $\edgpr([p],k+1)$ as $(v_1, k+1), \ldots, (v_q, k+1)$ with
$1\leq v_1<\ldots < v_q\leq p$. Choose $q$ vertices
$p<w_1<\ldots w_q\leq n$ such that
w_i\in [p]^c\setminus \innbrpr(k+1)
for all $i\leq q$. There are
{n-p-(d-q) \choose q} \leq \left(\frac{e n}{q}\right)^q
such choices. For every $i\leq q$ find
j\in \left(\outnbrpr(w_i) \cap [k+1]^c \right)\setminus
\outnbrpr(v_i)
(there are at most $d$ such choices).
For every $i\leq q$ we destroy edges $(v_i, k+1)$,
$(w_i, j(i))$ and create edges $(w_i, k+1)$,
$(v_i, j(i))$. We obtain
\begin{equation*}
\lvert R^{-1}(\nrangr')\rvert \leq
\left(\frac{e n}{q}\right)^q \, d^q.
\end{equation*}
Claim <ref> together with the last bound and
(<ref>) yields
\lvert Q\rvert
\leq \left(\frac{2 e n q}{pd}\right)^q
\lvert Q(q)\rvert.
By the choice of $q$, we have
$\lvert Q\rvert \leq \exp(-pd/(2e^2n)) \lvert Q(q)\rvert.$
This, together with Lemma <ref>, implies
\begin{align*}
\vert\DO([p],[k])
%\cap \Dco(1/2)
\vert
&\geq \vert Q(q)\vert \geq \exp\left(\frac{pd}{2 e^2 n}\right)\,
\vert \DO([p],[k+1])\cap \Dco(1/2) \vert \\
&= \exp\left(\frac{pd}{2 e^2 n}\right)\,
\left( \vert \DO([p],[k+1]) \vert -
\vert \DO([p],[k+1])\setminus \Dco(1/2) \vert
\right)\\
\geq \exp\left(\frac{pd}{2 e^2 n}\right)\,
\left(1-\frac{n^2}{2}\, \left(\frac{4 e d}{n}
\right)^{d/2}\right) \,
\vert \DO([p],[k+1]) \vert ,
%\geq \frac{1}{2}\,
%\exp\left(\frac{pd}{2 e^2 n}\right)\,
%\vert \DO([p],[k+1]) \vert
\end{align*}
which implies the desired result.
Proof of Theorem <ref>.
It is enough to prove the theorem for the union over all $|I|=\ell$ and $|J|=r$.
By the union bound, we have
\begin{align}\label{eq-start-graph-spread}
\P\left\{ {\bigcup}\, \DO(I,J) \right\}
\leq {n\choose \ell} \, {n\choose r} \,
\frac{\lvert\DO([\ell], [r])\rvert}{\lvert\D\rvert}
%\nonumber\\ &
\leq \left(\frac{en}{r}\right)^{2r}
\, \frac{\lvert\DO([\ell], [r])\rvert}{\lvert\D\rvert}.
\end{align}
Setting $\DO([0], [0])=\D$ and using $\DO([k], [k])\supset \DO([k+1], [k+1])$, we get
\begin{align}
\frac{\lvert\DO([\ell], [r])\rvert}{\lvert\D\rvert}
&= \prod_{k=0}^{\ell-1}\, \frac{ \lvert\DO([k+1],
[k+1])\rvert}{\lvert\DO([k], [k])\rvert}
\,\, \prod_{k=\ell}^{r-1}\, \frac{ \lvert\DO([\ell],
[k+1])\rvert}{\lvert\DO([\ell], [k])\rvert}\nonumber\\
&\leq \prod_{k=\lceil\ell/2\rceil}^{\ell-1}\, \frac{ \lvert\DO([k+1],
[k+1])\rvert}{\lvert\DO([k], [k])\rvert}
\,\, \prod_{k=\ell}^{r-1}\, \frac{ \lvert\DO([\ell],
[k+1])\rvert}{\lvert\DO([\ell], [k])\rvert}.\label{eq-th3-start-product}
\end{align}
Further, we write
\frac{ \lvert\DO([k+1],
[k+1])\rvert}{\lvert\DO([k], [k])\rvert} =\frac{ \lvert\DO([k+1],
[k+1])\rvert}{\lvert\DO([k], [k+1])\rvert} \cdot \frac{ \lvert\DO([k],
[k+1])\rvert}{\lvert\DO([k], [k])\rvert},
and applying Lemma <ref>,
for every $\lceil \ell/2\rceil \leq k\leq \ell-1$ we observe
\begin{equation*}\label{eq-th3-product-part1}
\frac{ \lvert\DO([k+1],
[k+1])\rvert}{\lvert\DO([k], [k])\rvert}
\leq \exp\left(-\frac{kd}{2 e^2 n}\right),
\end{equation*}
and for every $\ell\leq k\leq r-1$,
\begin{equation*}\label{eq-th3-product-part2}
\frac{ \lvert\DO([\ell],
[k+1])\rvert}{\lvert\DO([\ell], [k])\rvert}
\leq \exp\left(-\frac{\ell d}{4 e^2 n}\right).
\end{equation*}
Thus (<ref>) implies
\begin{equation*}
\frac{\lvert\DO([\ell], [r])\rvert}{\lvert\D\rvert}
\leq \exp\left(-\frac{\ell rd}{8 e^2 n}\right).
\end{equation*}
Combining this bound and (<ref>) and using
that $\ell \geq C n \ln (en/r) /d$ we complete the proof.
Proof of Lemma <ref>.
\begin{equation*}
\lvert \DO([p], [k])\setminus \Dco(\varepsilon)\rvert
\leq \sum_{i< j} \lvert \DO([p], [k])\setminus \D_{i,j}^{co}(\varepsilon)\rvert .
\end{equation*}
Fix $1\leq i<j\leq n$. For $0\leq q\leq d$, denote
Q(q):=\bigl\{ \nrangr \in \DO([p], [k]):\, \vert \outco(i,j)\vert =q\bigr\}
Q:=\DO([p], [k])\setminus \D_{i,j}^{co}(\varepsilon)=
\bigsqcup _{q=\lfloor\varepsilon d\rfloor + 1}^d Q(q).
First, for every $1\leq q\leq d$ we will compare the cardinalities of $Q(q)$ and $Q(q-1)$.
To this end, we will define a relation $R_q$ between the sets $Q(q)$ and $Q(q-1)$ in the following way.
Let $\nrangr \in Q(q)$. Then there exist
$j_1<...<j_{q}$ such that for every $\ell\leq q$
j_{\ell}\in \outnbr(i)\cap \outnbr(j).
Note that for every $\ell \leq q$, there are $d^2$ edges inside
\outedg\big(\innbr(j_\ell)\big).
Also, there are at least $(n-k-(2d-q))d$ edges in $\inedg\big([k]^c\setminus \outnbr(\{i,j\})\big)$.
Therefore, for $\ell \leq q$, the set
E_\ell:=\edg\big([n]\setminus \innbr(j_\ell), [k]^c\setminus \outnbr(\{i,j\})\big)
is of cardinality at least
|E_\ell|\geq (n-k-(2d-q))d-d^2\geq nd/2.
We say that $(\nrangr, \nrangr') \in R_q$ for some $\nrangr' \in Q(q-1)$
if $\nrangr'$ can be obtained from $\nrangr$ in the following way. First
we choose $\ell \leq q$ and an edge $(u,v)\in E_\ell$. Note $v\in [k]^c$
and $u\ne i$. Since $v\not\in \outnbr(j)$ then we can destroy the edge
$(j,j_\ell)$ and create the edge $(j,v)$. Since $u\not\in\innbr(j_\ell)$
then we can destroy the edge $(u,v)$ and create the edge $(u,j_\ell)$.
Thus, we obtain $\nrangr'$ by the simple switching on vertices $u,v,j,j_\ell$.
It is not difficult to see that we have not created any edges between $[p]$ and $[k]$,
hence $\nrangr'$ indeed belongs to $Q(q-1)$.
Counting the admissible simple switchings, we get for every $\nrangr\in Q(q)$,
\begin{equation}\label{eq-image-Rq-lem5}
\vert R_q(\nrangr)\vert\geq \frac{qnd}{2}.
\end{equation}
Now we estimate the cardinalities of preimages.
Let $\nrangr' \in R_q\bigl( Q(q)\bigr)$.
In order to reconstruct a possible $\nrangr$ for which
$(\nrangr, \nrangr')\in R_q$, we need to perform the simple switching which removes an edge $(j, v)$
with $v\not\in \outnbrr(i)$ and recreates an edge $(j, w)$ for some
w\in \outnbrr(i)\setminus \outnbrr(j).
There are at most $d-q+1$ choices for such $v$ and
at most $d-q+1$ choices for such $w$.
For the second part of the switching, we have
at most $d$ possible choices. Therefore,
\begin{equation*}
\lvert R_q^{-1}(\nrangr')\rvert \leq d(d-q+1)^2\leq d^3.
\end{equation*}
Using this bound, (<ref>), and Claim <ref>,
we obtain that
\lvert Q(q)\rvert \leq \left(\frac{2 d^2}{qn}\right)\cdot \lvert Q(q-1)\rvert
and, applying this successively,
\begin{equation*}
\lvert Q(q)\rvert \leq \left(\frac{2 d^2}{n}\right)^{q} \, \,
\frac{1}{q!}\, \, \lvert Q(0)\rvert.
\end{equation*}
$q!\geq 2(q/e)^q$ and $2e d/(\varepsilon n)\leq 1/2$, this implies
\vert Q\vert= \sum_{q=\lfloor\varepsilon d\rfloor + 1}^d\lvert Q(q)\rvert
\leq \frac{1}{2}\,\, \sum_{q=\lfloor\varepsilon d\rfloor + 1}^d
\left(\frac{2e d}{\varepsilon n}\right)^{q} \, \lvert Q(0)\rvert
\leq \left(\frac{2e d}{\varepsilon n}\right)^{\varepsilon d}\,
\lvert Q(0)\rvert .
Using that $Q(0)\subset \DO([p],[k])$ and that there are $n(n-1)/2$ pairs $i<j$, we obtain the desired result.
§.§ An anti-concentration property for random digraphs
For every $\nrangr\in \D$, $J\subset [n]$ and $i\in[n]$,
we define $\delta_i^{J}\in \{0,1\}$ by
\delta_i^J=\delta_i^J(\nrangr) :=\left\{
\begin{array}{ll}
1 & \mbox{if } i\in \innbr(J), \\
0 & \mbox{otherwise}.
\end{array}
\right.
Denote $\delta^J:=(\delta_1^J,\ldots, \delta_n^J)\in \{0,1\}^n$.
The vector $\delta^J$ can be regarded as an indicator of the vertices that are connected to $J$,
without specifying how many edges connect a vertex to $J$.
Taking $v\in\{0,1\}^n$ and conditioning on the realization $\delta^J=v$,
we obtain a class of graphs with a particular arrangement of the edges. Namely,
if a vertex $i$ of a graph in the class is not connected to $J$ then all graphs in the class have the same property. In this section we estimate the cardinalities of such classes generated by vertices of the cube, under additional assumption that a part of the graph is “frozen." We show that if the size of the set $J$ is at most $cn/d$ then a large proportion of such classes are “approximately” of the same size. In other words, we prove that the distribution of $\delta^J$ is similar to that of a random vector uniformly distributed on the discrete cube $\{0,1\}^n$ in the sense that for each fixed $v\in\{0,1\}^n$ the probability that $\delta^J=v$ is exponentially small. This makes a link to the anti-concentration results in the Littlewood-Offord theory. We start with a simplified version of this result, when there is no “frozen" part. In this case it is a rather straightforward consequence of Theorem <ref>.
Let $8\leq d\leq n$ and $J\subset[n]$. Let $v\in\{0,1\}^n$ and $m:=|\supp v|$.
\begin{equation*}\label{P-ev-I1}
\P\{ \delta^J=v\}\leq \binom{n}{m}^{-1}\le
\exp\left(-m\ln\frac{ n}{m}\right).
%\bf{1}_{|J|\leq m\le d|J|},
\end{equation*}
Moreover, if $|J|\leq c n/d$, then
\begin{equation*}
\label{P-ev-I0}
\P\{ \delta^J=v\}\leq
\exp\left(-cd|J|\ln\frac{ cn}{d|J|}\right),
\end{equation*}
where $c$ is an absolute positive constant.
Note that $\max \{d, |J|\}\leq|\supp \delta^J|\le d|J|$, therefore $\P\{ \delta^J=v\}=0$ unless $\max \{d,|J|\}\leq m \leq d|J|$.
Without loss of generality we assume that $\max\{d,|J|\}\leq m \leq d|J|$.
Consider the following subset of the discrete cube
T=\{w\in\{0,1\}^n \, :\, |\supp w|=m\}.
Clearly, every $w\in T$ can be obtained by a permutation of
the coordinates of $v$.
Since the distribution of a random graph is invariant under permutations,
$\P\{ \delta^J=v\}=\P\{ \delta^J=w\}$ for every $w\in T$. Therefore,
\p\{\delta^J=v\}\le |T|^{-1}=\binom{n}{m}^{-1}\le \exp\left(-m\ln\frac{n}{m}\right),
which proves the first bound and the “moreover" part in the case
$m\geq d|J|/2$.
Suppose now that $\vert J\vert \leq c n/d$ and $m\leq d|J|/2$.
Applying Theorem <ref> with $S=J$, $I=\emptyset$, and $\varepsilon=1/2$,
we observe
\P\{ |\supp \delta^J|\le d|J|/2 \}\leq
\exp\left(-cd|J|\ln\frac{ cn}{d|J|}\right),
which completes the proof of the “moreover" part.
Now we turn to the main theorem of this section, which will play a key role in the “matrix" part of our paper. We obtain an anti-concentration property for the vector $\delta^J$ even under an assumption that a part of edges is “frozen.” It requires a more delicate argument.
There exist two absolute positive constants $c, \tilde c$ such that
the following holds.
Let $32\leq d\leq cn$ and let $I, J$ be disjoint subsets of $[n]$ such that
\begin{equation*}\label{eq-choiceI-th-anticoncentration}
\vert I\vert \leq \frac{d\vert J\vert}{32} \quad \text{ \ and \ }\quad
8\leq \vert J \vert \leq \frac{8 c n}{d} .
\end{equation*}
Let $F\subset [n]\times[n]$ be such that $\D(I,F)\neq\emptyset$ and let $v\in \{0,1\}^n$. Then
\P\{ \delta^J =v \mid \D(I,F)\}\leq
2\exp\left(- \tilde c d\vert J\vert \ln\left(\frac{n}{d\vert J\vert}\right)\right).
To prove this theorem we first estimate the size
of the class of graphs given by a realization of a subset of coordinates of $\delta^J$.
More precisely, restricted to a subset of graphs with predefined out-edges for
the first $i-1$ vertices of $\delta^J$, we count
the number of graphs for which the vertex $i$ is connected to $J$.
In other words, conditioning on the realization of the first
$i-1$ coordinates of $\delta^J$, we estimate the probability that $\delta_i^J=1$.
In Lemma <ref> below we show that this probability is of the order $d\vert J\vert /n$ for a wide range of $i$-s. In a sense, this shows that
the sets of out-edges restricted on $J$ for different vertices behave like independent. Indeed, in the Erdős–Rényi model, when the edges are distributed independently
with probability of having an edge equals $d/n$, the probability that a vertex $i$ is connected to $J$ is of order $d\vert J\vert /n$.
We need the following notation. For $\eps \in (0,1)$ and $J\subset [n]$ let
\Lambda(\varepsilon ,J)= \big\{\nrangr\in \D:\ \vert \innbr(J)\vert
\geq (1-\varepsilon) d\vert J\vert \big\} .
Let $2\leq d\leq n/32$.
Let $F\subset[n]\times[n]$ and $I, J$ be disjoint subsets of $[n]$ satisfying
\begin{equation}\label{eq-choiceI-lem-anticoncentration}
\vert I\vert \leq \frac{d\vert J\vert}{32} \quad \text{ \ and \ }
\quad 8 \leq \vert J \vert \leq \frac{n}{4d}.
\end{equation}
Then there exists a permutation $\sigma\in \Pi_n$ such that for every
2\vert I\vert\leq i_1< i_2< \ldots < i_{d\vert J\vert/16},
every $s \leq d\vert J\vert/16$ and $\mathcal{H}\subset 2^{[n]\times[n]}$
{\widetilde{\Gamma}}:= \left\{ \nrangr\in \D(I,F)\cap \Lambda \left(\tfrac{1}{2}, J\right)\, : \,
\edg\big(\sigma([2\vert I\vert]\cup\{i_1,\ldots,i_{s-1}\}), I^c\big)\in \mathcal{H} \right\} \ne \emptyset
one has
\begin{equation*}\label{eq-lem-anticoncentration}
\frac{d\vert J\vert}{9n}\leq \P \left\{ \delta_{\sigma(i_s)}^J =1 \mid {\widetilde{\Gamma}} \right\}
\leq \frac{2d\vert J\vert}{n}.
\end{equation*}
As the proof of lemma is rather technical, we postpone it to the end
of this section.
Proof of Theorem <ref>.
Fix $F\subset [n]\times[n]$ with $\D(I,F)\neq\emptyset$ and $v\in \{0,1\}^n$.
Let $\sigma$ be a permutation given by Lemma <ref>.
Denote $B:= \D(I,F)\cap \Lambda \left(\tfrac{1}{2}, J\right)$.
Since $J\subset I^c$, applying Theorem <ref> with $\eps =1/2$ and $k=|J|$,
we get that for some appropriate constant $\tilde c$
\begin{equation*}\label{new-eq-13}
\P\left\{ \Lambda \left(\tfrac{1}{2}, J\right) \mid \D(I,F) \right\}
\geq 1-\exp\left(- \tilde c d\vert J\vert \ln\Big(\frac{n}{d\vert J\vert}\Big)\right),
\end{equation*}
which in particular implies that $B$ is non-empty.
Using this we have
\begin{align*}
\P\{ \delta^J =v \mid \D(I,F) \} &\leq \P\left\{ \delta^J =v \mid
\D(I,F)\cap \Lambda \left(\tfrac{1}{2}, J\right) \right\} + \P\left\{ \Lambda^c
\left(\tfrac{1}{2}, J\right)\mid \D(I,F) \right\}\nonumber\\
& \leq \P\left\{ \delta^J =v \mid
B\right\}+ \exp\left(- \tilde c d\vert J\vert \ln\left(\frac{n}{d\vert J\vert}\right)\right).\label{eq-anti-new-th}
\end{align*}
Therefore, it is enough to estimate the first term in the previous inequality.
Note that if $\vert\supp v\vert < d\vert J\vert/2$ then
\P\left\{ \delta^J =v \mid
\D(I,F)\cap \Lambda \left(\tfrac{1}{2}, J\right) \right\}=0.
Assume that $\vert\supp v\vert\geq d\vert J\vert/2$ and denote $m=d|J|/16$. Since
$2\vert I\vert\leq m$, there exist
2\vert I\vert \leq i_1< i_2< \ldots< i_{m}
such that for every $s\leq m$ one has $v_{\sigma(i_s)}=1.$
Let $Q_1=[2\vert I\vert]$. For every $2\leq s\leq m+1$, define
$Q_s:=Q_1\cup\{i_1,\ldots,i_{s-1}\}$ and
\mathcal{H}_s:=\bigl\{H\subset \sigma(Q_s)\times I^c
\, :\,
\forall \ell \in Q_s\, \, \, \, \,
\Leftrightarrow ``\forall j\in J \, \, \, \, (\sigma(\ell),j)\not\in H"\bigr\}.
In words, $\mathcal{H}_s$ is the collection of all possible realizations of configurations of edges connecting $\sigma(Q_s)$
to $I^c$, such that $\sigma(\ell)$ is not connected to $J\subset I^c$ if and only if $v_{\sigma(\ell)}=0$ ($\ell\in Q_s$).
Note that
\begin{equation*}
A_s:= \big\{\nrangr \in \D\, : \, \forall \ell \in Q_s\quad \delta^J_{\sigma(\ell)}=v_{\sigma(\ell)} \big\} =
\big\{\nrangr \in \D \, : \, \edg\big(\sigma(Q_s), I^c\big)\in \mathcal{H}_s\big\}.
\end{equation*}
% m:=d\vert J\vert /16, \,\, \mbox{ and }\,\,
B_s:= \big\{\nrangr \in \D\, : \,
\delta^J_{\sigma(i_s)}=1 \big\}.
Since $v_{\sigma(i_s)}=1$ for every $s\leq m$ then $A_{s+1}= B_s \cap A_s$ and
\P\{ A_{s+1} \mid B\} = \P\{ B_s \cap A_s \mid B\}
= \P\{ B_s \mid B\cap A_s \}\, \P\{ A_{s} \mid B\}.
\P\{ \delta^J =v \mid B\}\leq \P\{ A_{m+1} \mid B\}
\leq \prod_{s = 1}^{m} \P\{ B_s \mid B\cap A_s \} .
By the assumptions of the theorem and
Lemma <ref>,
for every $s\leq m$ we have
\P\{ B_s \mid B\cap A_s \}\leq \frac{2d|J|}{n},
which implies
\begin{equation*}
\label{eq-anticoncentration-finalestimate}
\P\{ \delta^J =v \mid B\}\leq \left( \frac{2d|J|}{n}\right)^m\leq
\exp\left(-\frac{d |J|}{16}\ln\left(\frac{n}{2d\vert J\vert}\right)\right) .
\end{equation*}
This completes the proof.
It remains to prove Lemma <ref>. To get the
lower bound we employ the simple switching to graphs whose $i$-th
vertex is not connected to $J$ and transform them into graphs with
the $i$-th vertex connected to $J$. To get the upper bound, we do
the opposite trick to transform graphs with only one edge relating
vertex $i$ to $J$ to a graph with no connections from $i$ to $J$. Then
we show that if $i$ is connected to $J$, it is more likely that the
number of corresponding out-edges is small. This is very natural if
we have in mind the result proven in Theorem <ref>.
Indeed, if vertices of a graph had a large number of out-edges
connecting them to $J$, then the number of in-neighbors to $J$
would be small, while Theorem <ref> states that
$\innbr(J)$ is rather large.
Proof of Lemma <ref>.
Let $\sigma$ be a permutation such that the sequence
\big\{\vert \outnbr\big(\sigma(\ell)\big)\cap I\vert \big\}_{\ell=1}^{n}
is non-increasing. Note that $\sigma$ depends only on $F$ when $\nrangr\in \D(I,F)$.
First we note that for every
$\nrangr\in \D(I,F)$
\begin{equation}\label{eq-many-1-out-of-I}
\forall i\geq 2\vert I\vert\, \, \, \quad \big\vert \outnbr\big(\sigma(i)\big)\cap I^c \big\vert \geq d/2 .
%%\text{ \ for any \ }\ell\geq i.
\end{equation}
Indeed, otherwise there would exist $\nrangr\in \D(I,F)$ and $i_0\geq 2\vert I\vert$ such that
\big\vert \outnbr\big(\sigma(i_0)\big)\cap I\big\vert > d/2.
Since $\big\{\vert \outnbr\big(\sigma(\ell)\big)\cap I\vert\big\}_{\ell\leq n}$ is non-increasing,
then for every $\ell \leq i_0$
we would have
\big\vert \outnbr\big(\sigma(\ell)\big)\cap I \big\vert > d/2.
This would imply
\big\vert \edg(\sigma([i_0]), I)\big\vert > i_0 d/2
\geq d\vert I\vert,
which is impossible.
Fix $s\leq d\vert J\vert/16$.
For $0\leq k\leq p:=\min\{d,\vert J\vert\}$ denote
\widetilde{\Gamma}_k:= \big\{\nrangr\in \widetilde{\Gamma}:\ \big\vert \edg(\sigma(i_s),J)\big\vert =k\big\}.
\widetilde{\Gamma}=\bigsqcup_{k\leq p} \widetilde{\Gamma}_k.
The statement of the lemma is equivalent to the following estimate
\begin{equation}\label{eq-lem-anticoncentration-goal}
\frac{d\vert J\vert}{9n}\, \big\vert \widetilde{\Gamma}\big\vert \leq
\big\vert \widetilde{\Gamma}\setminus \widetilde{\Gamma}_0\big\vert
\leq \frac{2 d\vert J\vert}{n} \, \big\vert \widetilde{\Gamma}\big\vert .
\end{equation}
We first show that
\begin{equation}\label{eq-lem-anticoncentration-claim1}
\big\vert \widetilde{\Gamma}_0\big\vert \leq \frac{8n}{d\vert J\vert}\, \big\vert \widetilde{\Gamma}_1\big\vert .
\end{equation}
Note that (<ref>) implies the left hand side of
Indeed, since $\widetilde{\Gamma}_1\subseteq \widetilde{\Gamma}\setminus \widetilde{\Gamma}_0$, then
(<ref>) yields that
\big\vert \widetilde{\Gamma}_0\big\vert \leq \frac{8n}{d\vert J\vert}\,
\big\vert \widetilde{\Gamma}\setminus \widetilde{\Gamma}_0\big\vert.
Adding $\big\vert \widetilde{\Gamma}\setminus \widetilde{\Gamma}_0\big\vert$ to both sides we obtain
the left hand side of (<ref>).
In order to prove (<ref>), we define a relation $R$
between the sets $\widetilde{\Gamma}_0$ and $\widetilde{\Gamma}_1$.
Let $\nrangr\in \widetilde{\Gamma}_0$. Since $G\in \Lambda \big(\frac{1}{2}, J\big)$
and $2\vert I\vert +s \leq d\vert J\vert/8$, then
\begin{equation}\label{eq-lem-anticoncentration-innbrJ}
\big\vert \innbr(J)\setminus \sigma([2\vert I\vert]\cup \{i_1,\ldots,i_{s-1}\})\big\vert \geq \frac{3d\vert J\vert }{8}.
\end{equation}
T:=\big(\innbr(J)\setminus \sigma([2\vert I\vert]\cup \{i_1,\ldots,i_{s-1}\})\big)\times (\outnbr\big(\sigma(i_s)\big)\cap I^c).
Since $\nrangr\in \widetilde{\Gamma}_0$, that is $\big\vert \edg(\sigma(i_s),J)\big\vert =0$, we have
\begin{equation*}\label{eq-lem-anticoncentration-edgT}
\big\vert\edg\big(T\big)\big\vert \leq (d-1)\big\vert \outnbr\big(\sigma(i_s)\big)\cap I^c\big\vert .
\end{equation*}
This together with (<ref>), (<ref>), and
$\vert J\vert \geq 8$ implies that
the set $S:=T\setminus \edg(T)$ satisfies
\begin{equation}\label{eq-lem-anticoncentration-sizeS}
\vert S\vert \geq \left(\frac{3d\vert J\vert }{8}-d+1\right) \cdot \big\vert
\outnbr\big(\sigma(i_s)\big)\cap I^c \big\vert
\geq \frac{d^2\vert J\vert}{8}.
\end{equation}
We say that $(\nrangr,\nrangr')\in R$ for some $\nrangr'\in \widetilde{\Gamma}_1$ if $\nrangr'$ can be obtained from
$G$ in the following way.
First choose $(v,j)\in S$. Since $j\in \outnbr\big(\sigma(i_s)\big)\cap I^c$ and
$(v,j)\not\in \edg(T)$ then we can destroy the edge $(\sigma(i_s),j)$ and create the edge $(v,j)$.
Since $v\in \innbr(J)$, there is $j'\in J$ such that $(v, j')$ is an edge in $\nrangr$. Since
$\nrangr\in \widetilde{\Gamma}_0$, $(\sigma(i_s), j')\not\in \nrangr$. Thus we can destroy the edge
$(v,j')$ and create the edge $(\sigma(i_s),j')$, completing the simple switching.
By (<ref>) we get
\begin{equation*}\label{eq-imageR1-anticoncentration}
\vert R(\nrangr)\vert\geq \frac{d^2\vert J\vert}{8}.
\end{equation*}
Note that the above transformation of $\nrangr$ does not decrease
$\vert \innbr(J)\vert$ which guarantees that $\nrangr'\in \Lambda \big(\frac{1}{2}, J\big)$.
Now we estimate the cardinalities of preimages. Let $\nrangr'\in R(\widetilde{\Gamma}_0)$.
In order to reconstruct a possible $\nrangr$ for which $(\nrangr,\nrangr')\in R$,
destroy the only edge $(\sigma(i_s), j')$ in $\edgpr(\sigma(i_s), J)$
and create an edge $(\ell, j')$ for $\ell \not\in \sigma([2\vert I\vert]\cup\{i_1,\ldots,i_{s-1}\})$.
There are at most $n-2\vert I\vert -(s-1)\leq n$ possible choices at this step.
To complete the simple switching, we destroy one of the edges in $\edgpr(\ell, J^c\cap I^c)$
and create an edge connecting $\sigma(i_s)$ to $J^c\cap I^c$.
There are at most $d$ possible choices here. Therefore,
\begin{equation*}\label{eq-preimageR1-anticoncentration}
\vert R^{-1}(\nrangr')\vert\leq nd.
\end{equation*}
By Claim <ref>, this implies the inequality (<ref>).
We now show that for every $k\in\{1,\ldots, p\}$, one has
\begin{equation}\label{eq-lem-anticoncentration-claim2}
\vert \widetilde{\Gamma}_k\vert\leq \frac{2d\vert J\vert}{k n}\, \vert \widetilde{\Gamma}_{k-1}\vert .
\end{equation}
Note that (<ref>) implies the right hand side of
Indeed, by (<ref>),
\vert \widetilde{\Gamma}_k\vert\leq \left(\frac{2d\vert J\vert}{n}\right)^k \, \frac{1}{k!}\, \, \vert
\widetilde{\Gamma}_{0}\vert,
\vert \widetilde{\Gamma}\vert = \vert \widetilde{\Gamma}_0\vert +\sum_{k=1}^p \vert \widetilde{\Gamma}_k\vert
\leq \exp\left(\frac{2d\vert J\vert}{n}\right) \vert \widetilde{\Gamma}_0\vert ,
which implies
\big\vert \widetilde{\Gamma}\setminus \widetilde{\Gamma}_0\big\vert
\leq
\big\vert \widetilde{\Gamma}\big\vert-\exp\left(-\frac{2 d\vert J\vert}{n}\right) \, \big\vert \widetilde{\Gamma}\big\vert
\leq \frac{2 d\vert J\vert}{n} \, \big\vert \widetilde{\Gamma}\big\vert .
In order to prove (<ref>) for every $k\in\{1,\ldots, p\}$,
we construct a relation $R_k$ between the sets $\widetilde{\Gamma}_k$
and $\widetilde{\Gamma}_{k-1}$.
Let $\nrangr\in \widetilde{\Gamma}_k$. Note that
\begin{equation}\label{eq-lem-anticoncentration-Sk-1}
\big\vert \sigma([2\vert I\vert]\cup\{i_1,\ldots,i_s\})\cup \innbr\big( J\big)\big\vert \leq 2\vert I\vert +s+d\vert J\vert \leq \frac{9d\vert J\vert}{8}.
\end{equation}
By (<ref>), we get
\begin{equation}\label{eq-lem-anticoncentration-Sk-2}
\big\vert I^c\cap J^c \setminus \outnbr(\sigma(i))\big\vert \geq
n- \frac{d|J|}{32} - \frac{n}{4d} - d \geq \frac{27n}{32} - \frac{d|J|}{32} .
\end{equation}
S_k:= \edg\big( \sigma([2\vert I\vert]^c\setminus \{i_1,\ldots,i_s\})\setminus \innbr\big(J\big), I^c\cap J^c \setminus \outnbr(\sigma(i_s))\big).
Using (<ref>), (<ref>) we observe that
\begin{equation}\label{eq-lem-anticoncentration-size-Sk}
\vert S_k\vert \geq d \left(\frac{27n}{32} - \frac{d|J|}{32}-\frac{9d\vert J\vert}{8}\right)
\geq \frac{nd}{2}.
\end{equation}
We say that $(\nrangr,\nrangr')\in R_k$ for some $\nrangr'\in \widetilde{\Gamma}_{k-1}$ if $\nrangr'$
can be obtained from $\nrangr$ in the following way.
Let $(\sigma(i_s), j_1)$ be one of the $k$ edges in $\edg(\sigma(i_s),J)$.
Destroy an edge $(v,j)\in S_k$. Since $j\not\in \outnbr\big(\sigma(i_s)\big)$,
then we can create the edge $(\sigma(i_s), j)$.
Since $v\not\in \innbr(j_1)$, then we can destroy the edge $(\sigma(i_s), j_1)$ and
create the edge $(v,j_1)$, thus completing the simple switching.
Therefore by (<ref>) we get
\begin{equation*}\label{eq-imageRk-anticoncentration}
\vert R_k(\nrangr)\vert\geq \frac{knd}{2}.
\end{equation*}
Note that the above transformation of $\nrangr$ does not decrease
$\vert \innbr(J)\vert$ which guarantees that $\nrangr'\in \Lambda \big(\frac{1}{2}, J\big)$.
Now we estimate the cardinalities of preimages. Let $\nrangr'\in R_k(\widetilde{\Gamma}_k)$.
In order to reconstruct a possible $\nrangr$ for which $(\nrangr,\nrangr')\in R_k$,
destroy an edge $(v,j_1)$ from $\edgpr(\sigma([2\vert I\vert]^c\setminus\{i_1,\ldots,i_s\}), J)$
to create the edge $(\sigma(i_s), j_1)$ for $j_1\in J$. There are at most
$d\vert J\vert $ such choices. To complete the simple switching,
we destroy an edge $(\sigma(i_s), j_2)$ in $\edgpr(\sigma(i_s), I^c\cap J^c)$ and create the
edge $(v, j_2)$. There are at most $d$ possible choices here.
\begin{equation*}\label{eq-preimageRk-anticoncentration}
\vert R_k^{-1}(\nrangr')\vert\leq d^2\vert J\vert .
\end{equation*}
Claim <ref> implies the inequality (<ref>), and completes the proof.
§ ADJACENCY MATRICES OF RANDOM DIGRAPHS
In this section we continue to study density properties of random $d$-regular directed (rrd) graphs. We interpret results obtained in the previous section in terms of adjacency matrices and provide consequences of the anti-concentration property, Theorem <ref>, needed to investigate the invertibility of adjacency matrices.
§.§ Notation
For $1\leq d\leq n$ we denote by $\mathcal{M}_{n,d}$ the set of
$n\times n$ matrices with $0/1$-entries and such that every row
and every column has exactly $d$ ones. By a random matrix on
$\mathcal{M}_{n,d}$ we understand a matrix uniformly distributed on
$\mathcal{M}_{n,d}$, in other words the probability on $\mathcal{M}_{n,d}$
is given by the normalized counting measure.
Whenever it is clear from the context, we
usually use the same letter $M$ for an element of $\Mc$ and for a random matrix.
For $I\subset [n]$ by $P_I$ we
denote the orthogonal projection on the coordinate subspace $\R^I$ and
$I^c:=[n]\setminus I$. For a matrix $M\in \Mc$ we say that a non-zero
vector $x$ is a null-vector of $M$ if either $Mx=0$ (a right null-vector) or
$x^T M=0$ (a left null vector).
Let $M=\{\mu_{ij}\}\in \Mc$. The $i$'th row of $M$ is denoted by $R_i=R_i(M)$
and the $i$'th column by $X_i=X_i(M)$, respectively.
For $j\leq n$, we denote $\supp\, X_j=\{i\leq n\,:\, \mu_{ij}=1\}$ and
for every subset $J\subset [n]$ we let
\begin{equation*}
S_J:=\bigcup_{j\in J}\, \supp\, X_j,
\end{equation*}
Clearly, $|J|\leq |S_J|\leq d|J|$ and $ n-d|J|\leq |(S_J)^c|\leq n-|J|$.
For $x\in \R^n$ we denote its coordinates by $x_i$, $i\leq n$, its $\ell_{\infty}$-norm by $\|x\|_{\infty}=\max_i |x_i|$
and for a linear operator $U$ from $X$ to $Y$ by $\|U\, :\, X\to Y\|$ we denote
its operator norm.
§.§ Maximizing columns support
In this section we reformulate Theorem <ref>
in terms of adjacency matrices. It corresponds to bounding from below
the number of rows which are non-zero on a given set of columns. More precisely,
for every subset $J\subset [n]$ we have $|S_J|\leq d|J|$.
We prove that for almost all matrices in $\Mc$, this inequality is close to being sharp
whenever $J$ is of the appropriate size (less than some proportion of $n/d$).
This means that $S_J$ is of almost maximal size.
Let $8\leq d\leq n$ and $\varepsilon\in (0,1)$ satisfy
\varepsilon^2 \geq \frac{\max\{8, \ln d\}}{d}.
\begin{equation*}
\Omega_\varepsilon=\Big\{M\in \Mc\,:\,\forall J\subset[n],\,
|J|\leq \frac{c_0 \eps n}{d}, \, \, \text{ one has }\, \,|S_J|\geq
(1-\varepsilon) d|J|\Big\},
\label{Oo}
\end{equation*}
where $c_0$ is a sufficiently small absolute positive constant.
\begin{equation*}
\label{POo}
\mathbb{P}(\Omega_\varepsilon)\geq 1-\exp\left(-\frac{\eps^2 d}{8}
\ln\left(\frac{ec_0\eps n}{d}\right)\right).
\end{equation*}
In fact Theorem <ref> gives slightly more, namely
the corresponding estimates when $|J|=k$ for a fixed
$k\leq c_0 \varepsilon n/{d}$.
However we don't use it below.
The following proposition is a direct consequence of Lemma <ref>
(applied with $2\eps$ instead of $\eps$ and with $p=k=0$). It shows
that for a big proportion of matrices in $\Mc$, every two rows have almost disjoint supports.
Let $\varepsilon\in (0,1)$ and $8\leq d\leq \eps n/6$.
\begin{equation*}
\label{O2e}
\Omega^2_{\varepsilon}=\Big\{M\in \Mc:\,\forall i,j \in [n]\,
\quad |\supp(R_i+R_j)|\geq 2(1-\varepsilon)d\Big\}.
\end{equation*}
\begin{equation*}
\mathbb{P}(\Omega^2_{\varepsilon})\geq 1- \frac{n^2}{2}\, \left(\frac{e d}{\varepsilon n}\right)^{\varepsilon d}.
\label{PO2e}
\end{equation*}
§.§ Large zero minors
In this section we reformulate Theorem <ref>
in terms of adjacency matrices.
It states that almost all matrices in $\Mc$ do not contain
large zero minors.
Given $0\leq \alpha,\beta\leq 1$ we define
\begin{align}
\eoo(\alpha,\beta)=\{M\in\Mc\,\, :\, \, &\exists I,J\subset[n]\quad\text{such that}\quad
|I|\geq \alpha n,\,|J|\geq \beta n, \notag
\\
&\text{ and }\quad \forall i\in I\, \forall j\in J\quad \mu_{ij}=0\}.
\label{eo-def}
\end{align}
In other terms, the elements of $\eoo(\alpha, \beta)$ are the matrices in $\Mc$ having a
zero submatrix of size at least $\alpha n\times \beta n$. Theorem <ref>, reformulated below, shows that this set is small
whenever $\alpha$ and $\beta$ are not very small.
There exist absolute positive constants $c, C$
such that the following holds. Let $2\leq d\leq n/24$ and
$0<\alpha\leq \beta\leq 1/4$. Assume that
\alpha \geq \frac{C \ln (e/\beta)}{d}.
\mathbb{P}\left(\eoo(\alpha,\beta)\right) \leq
\exp\left(-c \alpha \beta dn\right).
We usually apply this theorem with the following choice of parameters:
$\alpha = p/(2q)$, $\beta = p/2$, where $q= c_1 p^2 d$ for a sufficiently small absolute
positive constant $c_1$.
Then we have
\begin{equation}
\label{remzero}
\mathbb{P}\left(\eoo\left(\frac{p}{2q},\frac{p}{2}\right)\right) \leq \exp(-c_2 n).
\end{equation}
We will also need the following simple lemma.
Let $1\leq d\leq n$ and $0<\alpha, \beta<1$. Let
\begin{align*}
\Omega_{\alpha,\beta}=\Big\{M\in \Mc\,:\,\,\,
% &
\forall J,
% \, \, \mbox{ with }
\, \, |J|\geq\beta n, \, \,
\mbox{ one has } \, \,
% \\ &
R_i\cap J|\geq\beta/2\alpha\}|\geq(1-\alpha)n\Big\}.
\end{align*}
Then provided that $\alpha n$ is an integer, we have
\left(\eoo(\alpha,\beta/2)\right)^c \subset \Omega_{\alpha,\beta}.
Let $M\in \Omega_{\alpha,\beta}^c$. Then there exist $J\subset [n]$ with $|J|\geq \beta n$
and $I\subset [n]$ with $|I|= \alpha n$ such that
\forall i\in I\, \, \quad |\supp\, R_i\cap J|<\beta/2\alpha .
This shows that the minor $\{\mu_{ij}\,:\, i\in I,\, j\in J\}$ has strictly less than
$\beta n/2$ ones, which means that at least $\beta n/2$ columns of this minor
are zero-columns. Thus
\exists I\subset [n], \,|I|= \alpha n, \,\,\, \, \exists J_0 \subset [n], \,|J_0|\geq \beta n/2,
\,\,\, \, \forall i\in I,\, \, \forall j\in J_0\, :\, \mu_{ij}=0.
In other words, there is a zero minor of size $\alpha n\times \beta n/2$.
This proves the lemma.
§.§ An anti-concentration property for adjacency matrices
For every $F\subset [n]\times[n]$ and $I\subset[n]$, let
\Mc(I,F)=\left\{\M=\{\mu_{ij}\}\in\Mc \, :\, \mu_{ij}=1 \, \, \mbox{ if and only if }
\, \, j\in I,\, (i, j) \in F\right\}.
Thus matrices in $\Mc(I,F)$ have the same columns indexed by $I$ and the places of ones in
these columns are given by $F\cap ([n]\times I)$. Of course this set
can be empty.
For every $\M\in \Mc$, $J\subset [n]$ and $i\leq n$, we define $\delta_i^{J}\in \{0,1\}$ as follows
\delta_i^J=\delta_i^J(\M) :=\left\{
\begin{array}{ll}
1 & \mbox{if } \supp R_i\cap J\ne \emptyset ,\\
0 & \mbox{otherwise.}
\end{array}
\right.
We also denote $\delta^J:=(\delta_1^J,\ldots, \delta_n^J)\in \{0,1\}^n$.
The quantity $\delta^J$ indicates the rows whose supports intersect with $J$, i.e. the rows that have at least one $1$ in columns indexed by $J$.
The following is a reformulation of Theorem <ref>, concerning
the anti-concentration property of graphs, in terms of adjacency matrices.
There are absolute positive constants $c, \tilde c$ such that the following holds.
Let $32\leq d\leq cn$ and $I, J$ be disjoint subsets of $[n]$ such that
\begin{equation}\label{eq-choiceI-th-anticoncentration-reformulation}
\vert I\vert \leq \frac{d\vert J\vert}{32} \quad \text{ \ and \ }\quad
8\leq \vert J \vert \leq \frac{8 c n}{d}.
\end{equation}
Let $F\subset [n]\times[n]$ be such that $\Mc(I,F)\neq\emptyset$ and $v\in \{0,1\}^n$. Then
\P\{ \delta^J =v \mid \Mc(I,F)\}\leq 2\exp\left(-\tilde c d\vert J\vert \ln\left(\frac{n}{ d\vert
J\vert }\right)\right).
This theorem has the following consequence.
There are absolute positive constants $c, c'$ such that the following holds. Let $32\leq d\leq cn$,
$\lambda\in \R$, $a>0$, and $I, J,J_\lambda$ be a partition of $[n]$ satisfying
Let $q\leq n/2$ be such that
\begin{equation}\label{eq-choice-q-anticoncentration-matrix}
2^{q+1} \leq \exp\left( c' d\vert J\vert\ln\left(\frac{n}{d\vert J\vert}\right)\right)
%%\quad c_{\ref{th-anticoncentration-matrix part}}=1/288,
\end{equation}
and $y$ be a vector in $\R^n$ satisfying
\begin{equation}
\label{eq-condition-y-th-matrix part}
\forall \ell\in J_\lambda\ \ y_\ell =\lambda \text{\quad and \quad } \forall j\in J\ \ y_j-\lambda \geq 2a.
\end{equation}
Then for every $S\subset [n]$ with $\vert S\vert \geq n-q$, one has
\begin{equation}\label{PS}
\P\{ \Vert P_S\M y\Vert_\infty < a\} \leq \exp\left( -c'd\vert J\vert\ln\left(\frac{n}{d\vert J\vert}\right)\right).
\end{equation}
The above statement with essentially the same proof holds when
(<ref>) is replaced by
\begin{equation*}
\forall \ell\in J_\lambda\ \ y_\ell =\lambda \text{\quad and \quad } \forall j\in J\ \ \lambda - y_j \geq 2a.\end{equation*}
To prove Proposition <ref> we need the following lemma.
Let $\lambda\in \R$, $a>0$, and $I, J,J_\lambda$ be a partition of $[n]$ satisfying (<ref>).
Let $y$ be a vector in $\R^n$ satisfying (<ref>).
Then for every $i\leq n$ and every $F\subset [n]\times[n]$ there exists $v_i\in\{0,1\}$ such that
\begin{equation}\label{eq-lem-from matrix to anticoncentration}\nonumber
\{\M\in \Mc(I,F)\, \mid\, \delta_i^J(M)=v_i\}\subseteq
\{\M\in\Mc(I,F)\, \mid \, \vert (\M y)_i\vert\geq a\}.
\end{equation}
Fix $i\in [n]$ and $F\subset [n]\times[n]$. We argue by contradiction.
Assume that the above inclusion
is violated in both cases, $v_i=0$ and $v_i=1$. Then there exist two matrices
$\M_1, \M_2\in \Mc(I,F)$ such that
A_1:=\supp R_i^1\cap J\neq \emptyset, \quad \supp R_i^2\cap J= \emptyset, \quad
\vert (\M_1 y)_i\vert <a \quad \text{ and } \quad \vert (\M_2 y)_i\vert <a,
where $R_i^j=R_i(M^j)$ denotes the $i$-th row of $M_j$, $j=1,2$. Note that since $M_1, M_2\in \Mc(I,F)$ then
\supp R_i^1\cap I=\supp R_i^2\cap I := A_2.
Let $s_1:= |A_1|$ and $s_2:= |A_2|$. Using (<ref>), we observe
(\M_1y)_i = \sum_{j\in A_1} y_j + \sum_{j\in A_2} y_j + \lambda (d-s_1-s_2) \quad \text{ and }\quad (\M_2 y)_i =\sum_{j\in A_2} y_j + \lambda (d-s_2).
(\M_1y)_i -(\M_2y)_i= \sum_{j\in A_1} (y_j-\lambda) \geq 2s_1 a\geq 2a,
which is impossible as $ \vert (\M_1 y)_i\vert <a \text{ and } \vert (\M_2 y)_i\vert <a$.
Proof of Proposition <ref>.$\quad$
Since (<ref>) and (<ref>) are homogeneous in $y$, without loss of generality we assume that $a=1$.
Fix $S\subset [n]$ with $|S|\geq n-q$. Let $\cal{F}$ be the set of all
$F\subset [n]\times[n]$ such that $\Mc(I,F)$ is not empty.
Note that
$\{\Mc(I,F)\}_{F\in \mathcal{F}}$ form a partition of $\Mc$. Therefore
it is enough to prove that for every $F\in \mathcal{F}$,
p_0 := \P\{ \Vert P_S\M y\Vert_\infty < 1\mid \Mc(I,F)\} \leq \exp\left( -c'd\vert J\vert\ln\left(\frac{n}{d\vert J\vert}\right)\right).
Fix $F\in \mathcal{F}$. Let
$v_1,\ldots, v_n\in \{0,1\}$ be given by
Lemma <ref>.
Note that
\begin{equation*}\label{eq1-proof-th-anticoncentration-matrix part}
\Vert P_S\M y\Vert_\infty <1\quad\quad \text{ iff }\quad \quad
\forall i\in S \quad \vert (\M y)_i\vert < 1,
\end{equation*}
therefore if $\|P_S\M y\|_\infty <1$ then
$\{i\, :\,\, \delta_i^J(M) =v_i\}\subset S^c$.
p_0 \leq \P\{ \{i:\ \delta_i^J(M)= v_i\}\subseteq S^c \mid \Mc(I,F)\}.
Now for every $K\subset [n]$, define $v^K\in \{0,1\}^n$ by
\begin{array}{ll}
v_i & \mbox{if } i\in K, \\
1-v_i & \mbox{otherwise.}
\end{array}
\right.
Since $m:=\vert S^c\vert \leq q$, by
Theorem <ref> we obtain
\begin{align*}
%\P\{ &\Vert P_S\M y\Vert_\infty <1\mid \Mc(I,F)\}
%&\leq \P\{ \vert\{i\in S:\ \delta_i^J= v_i\}\vert \leq q\mid \Mc(I,F)\}\nonumber\\
%\leq \P\{ \{i:\ \delta_i^J= v_i\}\subseteq S^c \mid \Mc(I,F)\}\nonumber\\
&\leq \sum_{\ell =0}^{m} \P\{ \exists K\subset S^c:\ \vert K\vert =\ell \text{ and }\delta^J(M)= v^K \mid \Mc(I,F)\}\nonumber \\
& \leq \sum_{\ell =0}^{m} {m \choose \ell} \max_{\vert K\vert =\ell} \P\{\delta^J(M)= v^K \mid \Mc(I,F)\}
%\nonumber \\&
\leq 2^{q+1}\exp\left( -\tilde c d\vert J\vert\ln\left(\frac{n}{d\vert J\vert}\right)\right).
\end{align*}
Taking $c'=\tilde c/2$ and using (<ref>) we complete the proof.
§ INVERTIBILITY OF ADJACENCY MATRICES
In this section we investigate the invertibility of adjacency
matrices $M\in\Mc$ of random $d$-regular directed graphs and prove Theorem A.
§.§ Almost constant null-vectors
We say that a non-zero vector is “almost constant" if for some
at least $(1-p)n$ of its coordinates
are equal to each other.
for $0 < p <1/2$ consider the following set of vectors
\begin{equation}
\label{AC}
\lam_x\in\R\quad |\{i\ :\;x_i=\lam_x\}|\geq
\end{equation}
In this section we estimate the probability of the event
\begin{equation}
\label{eAC}
\eac:=\{M\in\Mc\,:\, \forall x\in AC(p)\quad\M x\neq 0
\quad\text{and}\quad x^T\M\neq0\},
\end{equation}
which relates almost constant vectors to null vectors of $M$.
We show that that this probability is close to one, in other
words we show that
with high probability
a matrix $M\in\Mc$ cannot have almost constant null vectors.
This will be used in the proof of the main theorem allowing one
to restrict the proof to the event $\eac$. More precisely,
we prove the following theorem.
There are absolute positive constants $C, c$ such that for $C\le d\le cn$ and $p\le c/\ln d$ one has
\begin{equation}
\p\big(\eac\big)\geq 1-\left(\frac{Cd}{n}\right)^{cd}.
\label{pAC}
\end{equation}
We start with some comments on the structure of
almost constant vectors. Since $p<1/2$, if
$x\in AC(p)$ then only one real number $\lam_x$ satisfies (<ref>).
For every $x\in AC(p)$ we fix such $\lam_x \in \R$. We set
\begin{equation*}
AC^+(p)=\{x\in AC(p)\,:\,\lam_x\geq 0\}.
\end{equation*}
Note that $\lam_{-x} = - \lam_{x} $, therefore
\eac=\{M\in\Mc\,:\, \forall x\in AC^+(p)\quad\M x\neq 0
\quad\text{and}\quad x^T\M\neq0\}.
since $(x^TM)^T=M^Tx$ and $M^T$ has the same distribution as $M$ then
\begin{align*}
\P(\{M\in\Mc\,:\, \forall x\in AC^+(p)\quad\M x\neq 0
\})=\P(\{M\in\Mc\,:\, \forall x\in AC^+(p)\quad x^T\M \neq 0
\}).
\end{align*}
Therefore it is enough to consider the event
\begin{align*}
\e^{AC+}(p)=\{M\in\Mc\,:\, \forall x\in AC^+(p)\quad\M x\neq 0
\}\notag\\
\end{align*}
and to prove that
\begin{equation*}
\p\big(\e^{AC+}(p)\big)\geq 1-\frac{1}{2}\left(\frac{Cd}{n}\right)^{cd}.
\label{pAC+}
\end{equation*}
To this end we split $AC^{+}(p)$ into two complementary sets and
treat them separately in two lemmas.
For a vector $x = (x_i) \in\R^{n}$ we define the rearrangement
$x^\star=(x_i^\star)_i $ as follows: $x_i^\star = x_{\pi(i)}$, where
$\pi:[n]\to[n]$ is a permutation of
$[n]$ such that $(|x_i^\star|)_i$ is a decreasing sequence,
that is, $\vert x_{\pi(1)}\vert \ge \vert x_{\pi(2)}\vert \ge \ldots \geq \vert x_{\pi(n)}\vert$.
Contrary to the usual decreasing rearrangement of absolute values
of a sequence, here values $x_i^\star$ can be negative.
In the proof, we choose appropriately a positive integer $m_0$ and consider
a certain subset of $AC^{+}(p)$. For a vector $x$ in this subset
we “ignore" its first $m_0$ coordinates $x^\star _i$, i.e. we consider $P_I x^\star$ with $I=[m_0]^c$.
Then we show that this vector can be split into a sum of two vectors with disjoint supports and such that the second vector has equal coordinates
on its support. To approximate such vectors in $\ell _\infty$-metric we construct a net in the following way.
Let $\eta >0$ be a reciprocal of an integer.
For every $H\subset [n]$ of cardinality $k_1:=pn - m_0$ (we choose $p$ so that $pn$ is an integer) fix an $\eta$-net
$\Delta_H$ in the cube $P_H ([-1, 1]^n)$.
Such $\Delta_H$ can be chosen with $|\Delta_H|\le(1/\eta)^{ k_1}$.
Given $L\subset [n]$ of cardinality $k_2:=(1-p)n$, consider the one-dimensional space generated by the vector $v_L$ with $\supp v_L=L$
and all coordinates on $L$ equal to one. Fix an $\eta$-net $\Lambda_L$ in the segment $[-v_L,\,v_L]$. Clearly, $\Lambda_L$ can be chosen with $|\Lambda_L|= 1/\eta $. Note also that for every $z\in \Lambda_L$ one has $\supp z=L$ and $z_i=z_j$ whenever $i, j\in L$, that is $z\in AC(p)$ and $z_i=\lam_z$ for $i\in \Lambda$. Given disjoint subsets $H$, $L$ of $[n]$ of cardinalities $k_1$ and $k_2$
respectively, consider $\Delta_H\oplus\Lambda_L=\{w+z \, :\, w\in \Delta_H, \, z\in \Lambda_L \}$. Then
\Delta_H\oplus\Lambda_L \subset AC(p) \quad \quad \mbox{ and }\quad \quad \left| \Delta_H\oplus\Lambda_L \right|\leq (1/\eta)^{ k_1+1}\leq(1/\eta)^{pn}.
Finally we observe that the vector $P_Ix^\star$ can be approximated by the vectors in the union of $\Delta_H\oplus\Lambda_L$ over all such choices of $H$ and $L$.
In fact we will use only a subset of this union.
Fix a parameter $a>0$ and a positive integer $r$. For
$H$, $L$ as above
\Gamma(H, L)=\Gamma_{a,r}(H, L):=\{y\in \Delta_H \oplus \Lambda_L\,:\, \exists
J\subset H, \, |J|=r,\,\,
\text{ such that }\, \, \forall i\in J \,\, \, y_i-\lambda_y\geq 2a\}.
Clearly, $\vert \Gamma(H,L)\vert \leq (1/\eta)^{pn}$.
Finally, set
\begin{align*}
\mathcal{N} = \mathcal{N}_{a,r}:= \bigcup_{|L|=k_2,|H|=k_1}
\Gamma(H, L),
\end{align*}
where the union is taken over all disjoints subsets $H$ and $L$ of $[n]$ of cardinalities $k_1$ and $k_2$ correspondingly.
\begin{equation}\label{eq-size-subnet}
\vert {\mathcal{N} }\vert
\leq {n\choose k_2}\, {n-k_2 \choose k_1}\, \left(\frac{1}{\eta}\right)^{pn}
\leq {n \choose pn}\, {pn \choose m_0}\, \left(\frac{1}{\eta}\right)^{pn}
\leq \left(\frac{2 e}{\eta p}\right)^{pn}.
\end{equation}
We are ready now to prove two lemmas needed for Theorem <ref>.
In both of them we use the following set associated with $x\in AC^+(p)$
and a given $m_0$,
J_x =J_x(m_0) :=\{i> m_0\;:\;|x_i^\star-\lambda_x| \geq 1/(2d)\}.
There are absolute positive constants $c$ and $c_1$ such that the
following holds.
Let $32\leq d\leq cn$, $m_0\geq 1$, and $r\geq 8$ be integers such that
$1\leq 2 c_1 r \ln(n/(dr))$. Let $p\in (0, 1/2)$ be such that $pn$ is an integer.
Assume that
\begin{equation}
\label{eq-ac-condition-p-lem1}
m_0\leq 2 c_1 r \ln\left(\frac{n}{dr}\right),\quad
r\leq \frac{8 c n}{d}, \quad
p\leq \frac{d r}{32 n}, \ \mbox{ and } \
p\left(\ln(e/p)+ \ln(18 d^2)\right)\leq \frac{c_1 d r}{n}\ln \left(\frac{n}{dr}\right).
\end{equation}
Consider the following subset of almost constant vectors
\begin{align*}
T_1=\{x\in AC^+(p)\,:\,\,&|x^\star_{m_0}|=1\quad\text{and}\quad
|J_x|\geq 2r\}\label{2b}
\end{align*}
and the corresponding event
\begin{align*}
\e_{T_1}=\{M\in\Mc\,:\, \forall x\in T_1\quad\M x\neq 0\}.
\end{align*}
\begin{equation*}
\mathbb{P}(\e_{T_1})\geq 1-2\exp\left(- c_1dr \ln\left(\frac{n}{d r}\right)\right).
\end{equation*}
We apply this lemma with $r=c_2 n/d$,
$m_0 = c_3 n/d$, so that the probability is exponentially (in $n$)
close to one.
In fact we show a stronger estimate which
could be of independent interest, namely
\begin{align*}
\p \big(\{M\in\Mc\,:\, \exists x\in T_1
\text{ \ such that \ }
& \| Mx \|_{\infty} < 1/(8d) \}\big)
\le
2\exp\left(- c_1dr \ln\left(\frac{n}{d r}\right)\right). %\label{more}
\end{align*}
Proof of Lemma <ref>.
We prove a stronger bound from Remark <ref>.
We start by few general comments on the
strategy behind the proof.
By the construction of $T_1$, for $x=(x_i)_i\in T_1$ we have
\begin{align*}
\max\Big\{|\{i> m_0\,:\,x_i^\star-\lambda_x\geq 1/(2d)\}|,\;|\{i>
m_0\,:\,\lambda_x-x_i^\star\geq 1/(2d)\}|\Big\}\geq r.
\end{align*}
Therefore denoting
T_1^+:=\{x\in T_1\, :\, |\{i> m_0\,:\,x_i^\star-\lambda_x\geq 1/(2d)\}| \geq r\}
T_1^-:=\{x\in T_1\, :\, |\{i> m_0\,:\,\lambda_x-x_i^\star \geq 1/(2d)\}| \geq r\},
we have $T_1\subseteq T_1^+\cup T_1^-$.
Thus it is sufficient to show that
\begin{align}
p_0:= \p \big(\{M\in\Mc\,:\, \exists x\in T_1^+ \,\,\,\,
%\text{ \ such that \ }
& \| Mx \|_{\infty} < 1/(8d) \}\big)
\le
\exp\left(- c_1dr \ln\left(\frac{n}{d r}\right)\right)
\label{more+}
\end{align}
and similarly for $T_1^-$. Below we prove (<ref>) only.
Its counterpart for $T_1^-$ follows the same lines, one just needs to apply Proposition <ref>
with Remark <ref> below
(with a slight modification of the net constructed above).
To prove (<ref>) we first approximate vectors in $T_1^+$ by elements of the net ${\cal N}$ constructed above. By the union bound, this will reduce (<ref>) to estimates on the net.
Then, applying Proposition <ref>,
we obtain a probability bound for a fixed vector from the net.
As usual, the balance between the probability bound and the size of the net plays the crucial role.
Fix two parameters $\eta:=1/(9 d^2)$ and $a= 1/(4d)-\eta$, and take $k_1=pn-m_0$, $k_2=(1-p)n$ as in the construction
of the net $\cal N$ above.
We start by showing how an element of $T_1^+$ is approximated by
an element from $\cal N$.
Let $x\in T_1^+$ and assume for simplicity that
$\vert x_1\vert\ge \vert x_2\vert\ge \ldots \ge \vert x_n\vert$ (that is, $x=x^\star$).
Recall that $\lambda_x$ is the unique real number satisfying
(<ref>). By the definition of $T_1^+$ it is easy to see
that there exists a partition $J, J_0, I$ of $[n]$ such that
\begin{align}
&|J|= r,
\quad |J_0| = k_2,
\quad |I| =n-r-k_2,
\notag \\
&\forall i\in J_0\quad x_i=\lam_x \notag \
\quad{\text{with} \ }\quad \lam_x\ge 0, \\
&\forall j\in J\quad \,
j>m_0 \quad{\text{and} \ }\quad x_j\geq \lam_x+1/(2d).\notag
\end{align}
Since $|x_{m_0}|=1$ and there is $i> m_0$ such that $x_i\geq \lam_x+1/(2d)$, we observe that $\lam_x< 1$. Since for $i\leq m_0$ we have either $x_i\ge 1$ or $x_i\le -1$, then $J_0\cap I_0=\emptyset$, where $I_0=[m_0]$. Note also that $J\cap I_0=\emptyset$, hence $I_0\subset I$. Set $H=J\cup (I\setminus I_0)$ and $L= J_0$. Then $\vert H\vert =k_1$, $\vert L\vert =k_2$, and $A:=I_0^c= H\cup L$.
By the definition of $\Delta_H$ and $\Lambda_L$ there exist
$y'\in\Delta_H$ and $y''\in\Lambda_L$ such that
\| P_{H}x - P_{H}y'\|_{\infty} \le \eta \quad \quad \mbox{ and } \quad \quad\| P_{L} x - P_{L} y''\|_{\infty} \le \eta.
Therefore $y:=y'+y''\in \Delta_{H}\oplus \Lambda_{L}$ satisfies
$\| P_{A}x - P_{A}y\|_{\infty} \le \eta$.
Moreover, by the construction of the net $y\in AC(p)$,
\forall i\in L\quad \ y_i=y''_i=\lambda_y\quad \text{\quad \ and \quad } \quad \forall i\in J\,\,\, y_i-\lambda_y \geq x_i -\lambda_x -2\eta \geq
Thus we showed that for every $x\in T_1^+$ there exist $H, L \subset [n]$ with $|H|=pn-m_0$, $|L|=(1-p)n$, and $y\in \Gamma (H, L) = \Gamma_{a,r} (H, L)$ such that $\|P_A x - P_A y\|\leq \eta$. Note also, that given
$H$ and $L$ one can “reconstruct" $I_0$ as $I_0=[n]\setminus (H\cup L)$.
Moreover, denoting
\begin{equation*}
S:=S_{I_0}^c=[n]\setminus \supp\sum_{i\in I_0}X_i.
\end{equation*}
and observing that $P_{S}MP_{I_0}=0$ (indeed, for every $i\in S$ and $j\in I_0$ one has $\mu_{ij}=0$), we get
\begin{align*}
\Vert P_{S} M y\Vert_{\infty}
&= \Vert P_{S} M x + P_{S} M (y-x)\Vert_{\infty}
= \Vert P_{S} M x + P_{S} M P_{A}(y-x)\Vert_{\infty}\nonumber\\
&\leq \Vert P_{S} M x\Vert_{\infty}+ \Vert P_{S} M P_{A}(y-x)\Vert_{\infty}
<\|Mx\|_{\infty} + \Vert M\, : \, \ell_{\infty} \rightarrow \ell_{\infty} \Vert \, \eta
\nonumber\\&
\leq 1/(8d) +\eta d < a,
\end{align*}
provided that $\|Mx\|_{\infty}\leq 1/(8d)$.
Thus, by the union bound, we obtain
p_0 \leq \sum _{y\in {\mathcal{N}}} \p\big(\{M\in\Mc\,\, :\, \, \Vert P_{S} M y \Vert_{\infty} <a\}\big),
where $S=S(y)=S^c_{I_0}$, $I_0=I_0 (y)=[n]\setminus (H\cup L)$ whenever
$y\in \Gamma (H, L)$.
Finally we estimate the probabilities in the sum. Let $H, L \subset [n]$ be such that $|H|=pn-m_0$, $|L|=(1-p)n$, and $y\in \Gamma (H, L)$, $J$ be from the definition $\Gamma (H, L)$ and $S$ be as above. Let $I=
[n]\setminus (J\cup L)$. Then $I$, $J$, $L$ form a partition of $[n]$ with $|J|=r$ and $|I|= pn-r$. By assumptions of the lemma, this partition
satisfies (<ref>).
Note also that assumptions on $m_0$ and $r$ imply $m_0 d< n$, hence $\vert S\vert \geq n-m_0 d>0$, and
(<ref>) is satisfied with $q:=m_0 d$ (with $c_1=c'/2$).
By the definition of $\Gamma (H, L)$ the vector $y$ satisfies
\forall i\in L \quad y_i=\lambda_y \quad \ \text{ \ and \ } \quad\forall i\in J\, \, \, y_i-\lambda_y\geq 2a.
Applying Proposition <ref>
with the partition $\{I, J, L\}$, the vector $y$, and the
set $S$, we obtain
\begin{equation*}
\p\big(\{M\in\Mc\,:\, \Vert P_{S} M y\Vert_{\infty} <a\}\big)\leq
\exp\left(- 2c_1dr \ln\left(\frac{n}{d r}\right)\right).
\end{equation*}
Since $\eta=1/(9d^2)$, by (<ref>) and (<ref>), this implies
\begin{align*}
p_0\leq |{\mathcal{N}}| \, \exp\left(- 2c_1dr \ln\left(\frac{n}{d r}\right)\right) \leq
\left(\frac{18 d^2 e}{p}\right)^{pn}\, \exp\left(- 2c_1dr \ln\left(\frac{n}{d r}\right)\right)
%&\leq {n\choose pn}\, {pn \choose pn-r}\, {pn-r \choose m_0}\, %\left(\frac{1}{\eta}\right)^{pn} \, \exp\Big(- \frac{2 c_1 d^2 r^2}{n}\Big)\\
%&\leq \left(\frac{e}{p}\right)^{pn} \, 4^{pn} \, \left(9 d^3\right)^{pn} \, %\exp\Big(- \frac{2 c_1 d^2 r^2}{n}\Big)
\leq \exp\left(- c_1dr \ln\left(\frac{n}{d r}\right)\right),
\end{align*}
which completes the proof.
In the next lemma we prove an analogous statement for the set which is complementary to $T_1$. Recall that $\Omega_{\varepsilon}$ was introduced
in Theorem <ref> and let $c_0$ be the same constant as in that theorem.
Let $\varepsilon\in (0,1/4)$. Let $m_0, m_1, r$ be positive integers such that
m_1=m_0+2r < \min\{ m_0/(2\eps), \, c_0 \eps n/d\}.
%\quad \mbox{ and } \quad
%\frac{m_0}{4\eps}\leq r \leq \frac{m_0}{16\eps^2} \quad \mbox{ and } \quad m_0\leq %\frac{4 c_0 \eps^3 n}{d}.
Consider the following subset of almost constant vectors
\begin{align*}
T_2=\{x\in AC^+(p)\,:\,\, {\text{either} \ }\quad|x^\star_{m_0}|=0\, \text{ \ or \ } \, (|x^\star_{m_0}|=1\quad\text{and}\quad |J_x|< 2r\, )\}.
\end{align*}
\begin{align*}
\Omega_{\varepsilon}\subset \e_{T_2}:=\{M\in\Mc\,:\, \forall x\in T_2\quad\M x\neq 0\}.
\end{align*}
To prove the lemma we need the following simple observation.
Let $\eps\in (0, 1/2)$, $1\leq d\leq n$, and
$1\leq m\leq c_0\eps n/d$, where $c_0$ is the constant from the definition
of $\Omega_{\varepsilon}$.
Let $M\in \Omega_{\varepsilon}$ and $I$ be the set of indices corresponding to rows having exactly one $1$ in columns indexed by $[m]$, i.e.
\begin{equation*}
I=\{i\in S_{[m]}\, :\, |\supp R_i\cap [m]|=1\}.
\end{equation*}
Then $\vert I\vert\geq (1-2\eps) d m >0.$
Since $M\in \Omega_{\varepsilon}$,
\begin{equation*}
|S_{[m]}|\geq (1-\varepsilon)dm.
\label{Smx}
\end{equation*}
Since rows $R_i$, $i\in I$, have exactly one $1$ on $[m]$, while
other rows indexed by $S_{[m]}$ have at least two ones on $[m]$,
we observe
\vert I\vert +2(|S_{[m]}|-\vert I\vert )\leq dm .
This implies the desired result.
Proof of Lemma <ref>.
Let $M\in\Omega_{\varepsilon}$ and $x\in T_2$.
For simplicity assume that $x=x^\star$, so that
Our proof consists of the following three cases.
Case 1: $\vert x_{m_0}\vert =0$.
Let $m_x\geq 1$ be the largest integer such that $\vert x_{m_x}\vert \neq 0$. Clearly $m_x< m_0$. Let $I_x$ be the set of indices corresponding to rows having exactly one $1$ in columns indexed by $[m_x]$. By Claim <ref>, $I_x\ne \emptyset$.
Thus there exists a row $R_i$, $i\in I_x$, and a unique $j\in [m_x]$ such that
$\mu_{ij}=1$. This implies
(Mx)_{i} = \langle R_i,x\rangle = x_j\ne 0.
Case 2: $\vert x_{m_0}\vert=1$, $|J_x|<2 r$, and $\lambda_x < 1/({2d})$. In this case the cardinality of the set
\{i\geq m_0\, :\, \vert x_i\vert \geq \lambda_x +1/(2d)\}
is less than $2r$. Since $m_1-m_0 = 2r$, we have
$ \vert x_{m_1}\vert <\lambda_x + 1/(2d) < 1/d.$
Let $J_j=[m_j]$, $j=0,1$. We first show that there is a row $R_{i}$ such that
\begin{equation}
|\supp R_{i}\cap J_{0}|=1\quad \mbox{ and } \quad |\supp R_{i}\cap(J_{1}\setminus J_{0})|=0.
\label{Rik}
\end{equation}
Let $I$ be the set of indices corresponding to rows having exactly one $1$ in $J_1$. By Claim <ref>, $|I|\ge (1-2\eps) d m_1$.
Since the number of nonzero rows on $J_1\setminus J_{0}$ is at most $d\vert J_1\setminus J_{0}\vert$, the number of rows satisfying (<ref>) is at least
(1-2\eps) d m_1- d(m_1-m_{0})= d(m_{0}- 2\eps m_{1}) >0,
that is there exists a row $R_{i}$ satisfying (<ref>). Denote
$j_0\in J_{0}$ the only coordinate of $P_{J_1 }R_{i}$ which is equal to one,
i.e. $\mu_{i j_0}=1$ and for every
$j\in J_1\setminus\{j_0\}$, $\, \mu_{i j}=0$. Therefore if $j\in \supp R_{i}$ and $j\neq j_0$
then $j>m_1$ and $\vert x_j\vert \leq \vert x_{m_1}\vert$.
Since $\vert x_{m_1}\vert < 1/d$, we observe
|(Mx)_{i}| = \vert \langle R_{i},x\rangle\vert \geq \vert x_{j_0}\vert -\sum_{\underset{j\neq j_1}{j\in \supp R_{i_1}}} \vert x_j\vert
\geq |x_{m_0}|-(d-1)|x_{m_1}|\geq 1-\frac{d-1}{d}>0.
Case 3: $\vert x_{m_0}\vert =1$, $|J_x|< 2r$, and $\lambda_x \geq 1/(2d)$.
Consider the set
J:= \{ i\leq n\,: 0<x_i<\lambda_x+ 1/(2d)\}.
Then $A:=J^c\subset [m_0]\cup J_x$. Thus
$\vert S_{A}\vert \leq (m_0+2r)d <n $.
Therefore, there exists a row $R_j$, $j\notin S_{A}$, such that $\supp R_j\subset J$. This implies
(M x)_j = \langle R_j, x\rangle =\sum_{j\in J} x_j >0.
Thus in all cases $Mx\ne 0$, which completes the proof.
Proof of Theorem <ref>.
Recall that as we mentioned after the theorem it is enough to bound the probability of the event $\e^{AC+}$.
Let $c$, $c_0$, $c_1$ be constants from Lemmas <ref> and <ref>.
We choose $\eps>0$ to be small enough constant ($\eps =\min \{1/8, c_1 c_0/4, 32c/c_0\}$ would work), $m_0=\lfloor 2 c_0 \eps ^2 n/d\rfloor$ and
$r=\lfloor m_0/(8\eps )\rfloor\approx c_0 \eps n/(4d)$, so that assumptions
of Lemmas <ref> and <ref> on $m_0$ and $r$ are satisfied. Finally,
for a sufficiently small absolute positive constant $c_2$ we choose the biggest $p\leq c_2 /\ln d$ such that $pn$ is an integer. Then assumptions of Lemma <ref> on $p$ are also satisfied (note that it is enough to prove the theorem with the biggest possible $p$). Therefore, applying these lemmas together with Theorem <ref>, we have
\mathbb{P}(\e_{T_1})\geq 1- 2\exp(-c_3 n) \text{ \ and \ }
\mathbb{P}(\e_{T_2})\geq \mathbb{P}(\Omega_{\eps})\geq
1-\exp\left(-c_4 d \ln\left(\frac{c_5 n}{d}\right)\right),
where $T_1$, $T_2$ are events from the lemmas and $c_i$'s are absolute positive constants. Since $\e^{AC+}\supseteq \e_{T_1}\cap \e_{T_2}$ we obtain the desired result
by adjusting absolute constants.
§.§ Auxiliary results
§.§.§ Simple facts
We will need the two following simple facts.
Let $p\in (0, 1/3]$ and $x\in\R^n$. Assume that
\forall \lam\in\R\quad |\{i\,:\,x_i=\lam\}|\leq(1-p)n.
Then there exists $J\subset[n]$ such that
pn\leq|J|\leq (1-p)n \quad\text{ and }\quad \forall i\in J\,\, \forall j\notin J \quad
x_i\neq x_j.
We apply this claim twice, once in the following form. Let $m\le n$ and $\ell \leq m/3$. Let $S\subset[n]$, $|S|=m$, and let
$v\in \R^n$ satisfy $\, \forall \lam\in\R\quad |\{i\in S\,:\,x_i\ne\lam\}|\geq \ell$.
Then there exists $J\subset S$ such that
\ell \leq|J|\leq m-\ell \quad\text{ and }\quad \forall i\in J\,\, \forall j\in S\setminus J \quad
v_i\neq v_j.
Proof of Claim <ref>.
Let $\{\lam_1,...,\lam_k\}$ be the set of all distinct values of
coordinates of $x$.
For $j\leq k$,
let $I_j=\{i\,:\, x_i=\lam_j\}$ and $m_j=|I_j|$.
Clearly, $m_j\leq(1-p)n$ for every $j$.
By relabeling
assume that $m_1\geq m_2\geq...\geq m_k$.
If $m_1\geq pn$, choose $J=I_1$. Otherwise set
$J=I_1\cup...\cup I_{\ell}$,
where $\ell$ is the smallest number satisfying
$m:=|J|=m_1+...+m_{\ell}\geq pn$.
Since $m_j\leq m_1<pn$ for $j\leq k$, then
$m< 2pn$, and this implies
pn\leq|J|<2pn \leq (1-p)n.
Let $A$, $A_1$, ..., $A_m$ be sets such that every $x\in A$ belong
to at least $k$ of sets $A_i$'s. Then we say that $\{A_i\}_i$ forms
a $k$-fold covering of $A$.
The proof of the following fact uses a standard argument in measure theory,
so we omit it.
Let $(X, \mu)$ be a measure space. Let $A$, $A_1$, ..., $A_m$ be subsets
of $X$ such that $\{A_i\}_i$ forms a $k$-fold covering of $A$. Then
k \mu (A) \leq \sum _{i=1}^{m} \mu (A_i).
§.§.§ Combinatorial results
In this section we prove a Littlewood-Offord type result,
which will be one of key steps in the shuffling procedure.
Consider a probability space
\Omega_0 = \{B\subset [2d]\,:\,|B|=d\}
with the probability given by the normalized counting measure.
For a vector $v\in \R^{2d}$ and $B\in \Omega_0$ denote
v_B = \sum _{i\in B} \, v_i .
Let $1\leq k \leq d$. Let $v\in\R^{2d}$ and $a\in \R$. Assume
there exists $J\subset[2d]$ such that $|J|=k$ and for every
$i\in J$ and every $j\not\in J$ one has $v_i\neq v_j$. Then
\p (v_B = a) \leq \frac{10}{\sqrt{ k}}.
To prove Proposition <ref> we need two combinatorial lemmas.
We start with so-called anti-concentration Littlewood-Offord type lemma
(<cit.>, see also <cit.>). Usually it is stated for $\pm 1$ Bernoulli random variables, but by shifting and rescaling it holds for any
two-valued random variables, where by a two-valued random variable we mean
a variable that takes two different values, each with probability half.
$\xi_1$, $\xi_2$, ..., $\xi _m$ be independent two-valued random variables.
Let $x\in \R^m$.
\sup_{a\in \R}\p\Big(\sum_{i=1}^m\xi_i x_i=a\Big)\leq |\supp x|^{-1/2}.
Let $\Pi_{2d}$ be the permutation group with a probability given by the
normalized counting measure and denoted by $\p_{\Pi_{2d}}$. By $\pi$ we
denote a random permutation. The proof of the next lemma is rather
straightforward, we postpone it to the end of the section.
Let $1\leq k\leq d$.
Let $x\in\R^{2d}$ and $J \subset[2d]$ be such that
$|J|=k$ and for every $i\in J$ and every $j\not\in J$ one has
$x_i\neq x_j$. For $\pi\in\Pi_{2d}$ let
E=E(\pi)=\{(x_{\pi(i)},x_{\pi(i+d)})\,: \, i\leq d,\, \, x_{\pi(i)}\neq x_{\pi(i+d)}\}.
\p_{\Pi_{2d}}\left( |E|\leq \frac{k}{50} \right)\leq \left(\frac{k}{1.1 \, d}\right)^{k/3}.
Proof of Proposition <ref>. Let $B$ be a (set-valued) random
variable on $\Omega _0$. Let $\delta=(\delta_1,...,\delta_d)$ be a vector of independent
Bernoulli random $0/1$ variables ($\p(\delta_i=1)=1/2$ for $i\leq d$), and $\Omega$
denote the corresponding probability space. Consider a random
(on $\Pi_{2d}\times\Omega$) set of indexes
A(\delta,\pi)=\{\pi(i)\,:\, \delta_i=1\}\cup\{\pi(i+d)\,:\, \delta_i=0\}\subset [2d].
Note that $|A(\delta,\pi)|=d$.
It is not difficult to see that
for every fixed $\delta$,
$A(\delta,\pi)$ has the same distribution as $B$.
Therefore, $A(\delta,\pi)$ on $\Pi_{2d}\times\Omega$
has the same distribution as $B$ on $\Omega_0$. Thus,
given $v\in\R^{2d}$, the random variable $v_B$
has the same distribution as $v_{A(\delta,\pi)}$.
Now we introduce the following random variables on $\Pi_{2d}\times\Omega$:
\xi_i=\xi_i^v=\delta_iv_{\pi(i)}+(1-\delta_i)v_{\pi(i+d)}.
Note that $\p(\xi_i=v_{\pi(i)})=\p(\xi_i=v_{\pi(i+d)}) =1/2$ and that
v_{A(\delta,\pi)}=\sum_{i\in {A(\delta,\pi)}}v_{i}=\sum_{i=1}^d\xi_i.
Moreover, if we condition on $\pi$, the random variables $\bar{\xi}_i=\xi_i|_\pi$ are independent,
hence we can apply Lemma <ref>. Denote by $m(\pi)$ the number of two-valued $\bar{\xi}_i$'s.
\p_{\Omega}\Big(\sum_{i=1}^d\bar \xi_i=a\Big)\leq \frac{1}{\sqrt{m(\pi)}}.
Finally, we note that by Lemma <ref> we have many permutations with large $m(\pi)$, namely
\p_{\Pi_{2d}}\big(m(\pi)\leq k/50\big)\leq \left( \frac{k}{1.1 d}\right)^{k/3}.
\p\Big(\sum_{i=1}^d \xi_i=a\Big)\leq \sqrt\frac{50}{k} +
\left( \frac{k}{1.1 d}\right)^{k/3}.
This implies the desired result.
Proof of Lemma <ref>.
Without loss of generality we can assume that $x_i=1$ for $i\leq k$ and
$x_i=0$ for $i>k$. Let $\pi$ be a random permutation uniformly distributed
over $\Pi_{2d}$. The basic idea of the proof is to condition on a realization
of a set $\{i\leq d:\,x_{\pi(i)}=1\}$ and show that the conditional probability
of the event $|E|<k/50$ is small regardless of that realization.
Let $A=\{i\leq d:\,x_{\pi(i)}=1\}$ be a random subset of $[d]$.
Fix a subset $A_0\subset [d]$ with $|A_0|\leq k$
(then the event $A=A_0$ has a non-zero probability). Denote $m:=|A_0|$.
Further, define a random subset
$E_0=\{i\in A_0:\,x_{\pi(i+d)}=1\}$.
Clearly, we have $|E|=m-|E_0|+(k-m-|E_0|)=k-2|E_0|$ everywhere on the event $\{A=A_0\}$.
Let a parameter $\beta_1\in(0, 0.1)$ be chosen later.
Consider three cases.
Case 1: $m\leq(1-\beta_1)k/2$. Then clearly we have $|E|\geq k- 2m\geq \beta_1 k$
everywhere on the event $\{A=A_0\}$.
Case 2: $m\geq(1+\beta_1)k/2$. Since $|E_0|+m\leq k$ (deterministically),
we have $|E|\geq 2m-k\geq \beta_1 k$ everywhere on $\{A=A_0\}$.
Case 3: $(1-\beta_1)k/2\leq m\leq(1+\beta_1)k/2$. Note that
the set $\{\pi(d+1),\pi(d+2),\dots,\pi(2d)\}$ contains $k-m$ ones and $d-k+m$ zeros. Thus, for every $\ell\leq k-m$ we have
p_\ell:=\P(|E_0|=\ell\,|\,A=A_0)=\frac{1}{d!}{m\choose \ell}{d-m\choose k-m-\ell}(k-m)!(d-k+m)!,
where the second factor is the number of choices of subsets $E_0$ of $A_0$ of cardinality $\ell$, the third factor is the number of
possible choices of the set $\{d+1\leq i\leq 2d:\,x_{\pi(i)}=1\mbox{ and }x_{\pi(i-d)}=0\}$ provided that $|E_0|=\ell$,
and the factors $(k-m)!$ and $(d-k+m)!$ are the numbers of all permutations of ones and zeros in the last $d$ positions.
p_\ell={d\choose m}^{-1}{k-m\choose \ell}{d-k+m\choose m-\ell}
%\leq \left(\frac{m}{d}\right)^m
We choose $\beta >3/4$ from $(1-\beta)(1-\beta_1)/2=\beta _1$ and set $\beta_2=1-\beta$.
Using Chernoff bounds, we observe
\begin{align*}
\sum_{\ell\geq \beta m} {d-k+m\choose m-\ell} =
\sum_{\ell\leq (1-\beta) m} {d-k+m\choose \ell} \leq
\left(\frac{e(d-k+m)}{\beta_2 m}\right)^{\beta_2 m}
\leq \left(\frac{e d}{\beta_2 m}\right)^{\beta_2 m}.
\end{align*}
Since $k\leq 2m/(1-\beta_1)$ and $2/(1-\beta_1)-1-\beta=2\beta_2$, then
for $\ell\geq \beta m$,
{k-m\choose \ell} = {k-m\choose k- m-\ell} \leq
\left(\frac{e(k-m)}{k- m-\beta m}\right)^{k- m-\beta m}
\leq \left(\frac{e(1+\beta_1)}{2(1-\beta_1)\beta_2}\right)^{2\beta_2 m}
\leq \left(\frac{3}{2\beta_2}\right)^{2\beta_2 m}.
Therefore we have
\begin{align*}
\sum_{\ell\geq \beta m}p_\ell\leq \left(\frac{m}{d}\right)^m \, \left(\frac{3}{2\beta_2}\right)^{2\beta_2 m} \, \left(\frac{e d}{\beta_2 m}\right)^{\beta_2 m} \leq \left(\frac{m}{d}\right)^{\beta m} \, \left(\frac{2}{\beta_2 }\right)^{3 \beta_2 m}\leq
\left(\frac{(1+\beta _1) k}{2d} \left(\frac{2}{\beta_2 }\right)^{3\beta_2/\beta}\right)^{\beta m}.
\end{align*}
Choosing $\beta _1=1/50$ we obtain
\begin{align*}
\sum_{\ell\geq \beta m}p_\ell\leq \left(\frac{k}{1.1 \, d}\right)^{k/3}.
\end{align*}
On the event $\{ A=A_0 \}$ we have $|E|\geq m-|E_0|$, hence
\P(|E|\leq (1-\beta)m\,|\,A=A_0)\leq\sum_{\ell\geq \beta m}p_\ell.
Using that $m\geq(1-\beta_1)k/2$ and that $(1-\beta)(1-\beta_1)/2=\beta_1$ we obtain
\P(|E|\leq \beta_1 k\,\, |\, \,A=A_0)\leq
\left(\frac{k}{1.1 \, d}\right)^{k/3},
which completes the proof.
§.§ Proof of the main theorem
In this section, we complete the proof of the main result for adjacency matrices.
The general scheme is similar to the one in <cit.>.
The main idea of the proof of Theorem A can be roughly described as follows:
after throwing away all small “bad” events (namely, the existence of almost
constant null vectors, big zero minors, and rows with largely overlapping supports)
we split the remaining singular matrices from $\Mc$ into two sets
E_1=\{M\in\Mc:\,\rk M= n-1\} \quad \mbox{ and } \quad E_2=\{M\in\Mc:\,\rk M\le n-2\}.
Then, combining linear-algebraic arguments (Lemmas <ref> and <ref>) with
the shuffling procedure (Lemma <ref>), we show that $E_1$ and $E_2$ have a small proportion inside the sets $\Mc$ and $\{M\in\Mc:\,\rk M\le n-1\}$, respectively. This implies that $E_1\cup E_2$ has small probability.
The argument is rather technical and involves various events and “linear-algebraic” objects. To make the reading more convenient, we first group the notation used in this section.
§.§.§ Notation
For every $k\leq n$, let
\ek=\{M\in\Mc\,:\, \rk \M\leq k\}.
Our purpose is to estimate the probability of the event $\en$.
Let $M$ be a matrix from $\Mc$ with rows $R_i$, $i\leq n$.
For every $i\in[n]$, we denote by $M^i$ the $(n-1)\times n$ minor of $M$
obtained by removing the row $R_{i}$.
Further, take a pair of distinct indices $(i, j)$, $i\ne j\le n$.
By $\MMij$ we denote the $(n-2)\times n$ minor of $M$ obtained by
removing the rows $R_i,R_j$. Additionally, define
\begin{align*}
&\Vij=\Vij(M)=\spn\{R_k,\,k\neq i,j \}%\label{V12}
\quad \mbox{ and } \quad
&\Fij=\Fij(M)=\spn\{\Vij,\,R_i+R_j \}.%\label{F12}
\end{align*}
In what follows, we write simply $\Vij$ and $\Fij$ instead of $\Vij(M)$ and $\Fij(M)$ as
the matrix $M$ will always be clear from the context.
Note that the random vector $R_i+R_j$ and the random subspaces $\Vij$ and $\Fij$
are fully determined by the $(n-2)\times n$ matrix $\MMij$.
As we see later, to be able to successfully apply the aforementioned shuffling to a pair of rows $R_i,R_j$, we will need at our disposal a vector orthogonal to the subspace $\Fij$ such that its restriction to the union of the supports of $R_i$ and $R_j$ has many pairs of distinct coordinates. Of course, such a vector may not exist for some matrices $M\in\Mc$. We start by defining for every $q\in [n]$ and $i\ne j\le n$ a “good” subset of $\Mc$ as follows:
\begin{align}
\label{eijq}
\e^{i,j}(q)=\{M\in\Mc:\,\,&\exists v\perp \Fij
\quad\text{such that}
\\
&\forall\lam\in\R\,\,\, \, |\{k\in \supp (R_i+R_j):\;v_k\neq\lam\}|\geq q \}.\notag
\end{align}
For a matrix in this set
we fix one vector satisfying (<ref>), in fact we
define it as a function of the matrix. The crucial fact for our proof is
that since $\Fij$ and $R_i+R_j$ are uniquely
determined by $\MMij$, we can fix such a vector for the
class of matrices “sharing” the same minor $\MMij$.
Given $M\in \e^{i,j}(q)$, consider the equivalence class
{\cal{H}}_M^{i,j}(q) = \{\widetilde M\in \e^{i,j}(q) \, \, :\, \, \widetilde M^{i,j} =M^{i,j} \}.
For every equivalence class ${\cal{H}}_M^{i,j}(q)$ fix one vector
$v=v(M,q, i, j)$ satisfying
\begin{equation}
v \perp F_{i,j} \;\; \mbox{and}
\;\; \forall\lam\in\R\;\;|\{k\in \supp (R_i+R_j):\;v_k\neq\lam\}|\geq q.
\label{v}
\end{equation}
Whenever $q$ and the indices $i,j$ are clear from context, we write $v(M)$ instead of $v(M,q,i,j)$.
One of the key ideas of the proof of Theorem A is to show
that for most matrices $M$ in ${\cal{H}}_M^{1,2}(q)$, the vector
$v(M)$ does not belong to the kernel of $M$. To this end we introduce a subset of $\ea$
\begin{equation*}
\K=\{M\in\ea:\, v(M)\in \ker M\}.
\label{K}
\end{equation*}
In Lemma <ref> below we will show that the ratio $|\K|/|\Mc|$ goes to zero as $d\to \infty$.
As we mentioned above, in the proof we essentially restrict ourselves to the set of matrices,
which have no almost constant null-vectors, no
big zero minors, and no rows with largely overlapping supports. To define this set formally, let
$0< p\leq 1/3$, $2\leq q\leq d/2$, and
$\varepsilon\in (0, 1)$. Denote
\begin{align*}
\eg=\eg(p,q,\varepsilon):=\eac\cap{\Omega}^2_{\varepsilon}\setminus\eoo\big({p}/{2q}, {p}/{2}\big),
\label{eg}
\end{align*}
where ${\Omega}^2_{\varepsilon}$, $\eoo({p}/{2q}, {p}/{2})$, and $\eac$ were
introduced in Proposition <ref>,
(<ref>), and (<ref>), respectively.
By Proposition <ref>, Theorem <ref> , and (<ref>)
we have
\begin{equation}
\p\big(\eg^c\big)\leq \frac{n^2}{2}\left(\frac{e d}{\eps n}\right)^{\eps d} +
\left(\frac{C d}{n}\right)^{c d} + e^{-c n} \leq
n^2\, \left(\frac{e d}{\eps n}\right)^{\eps d}
\leq \left(\frac{e d}{\eps n}\right)^{\eps d/2}
\label{nu}
\end{equation}
provided that $p\leq c_1/\ln d$, $q=c_2 p^2 d$, $c_3/\eps^2 \leq d\leq \eps n/6$ and $\eps$ is small enough.
Further we will need two more auxiliary events dealing with the
$(n-2)\times n$ minors $\MMij$ of $M$. For $i\ne j,$ introduce
\begin{equation*}
\label{eijY}
\e^{i,j}_{n-2}=\{M\in\Mc\,:\, \rk \MMij=n-2\quad\text{and}\quad R_i+R_j\notin V_{i,j}\},
\end{equation*}
\begin{equation*}
\label{rkij}
\erkij=\{M\in\Mc\,:\, R_i, R_j\in V_{i,j}\}.
\end{equation*}
Note that for every $M\in\erkij$ we have $\rk \M=\rk \MMij$. In the next section we prove several
statements involving events $\e^{i,j}_{n-2}$, $\erkij$, and $\K$.
§.§.§ Proof of Theorem A
The next lemma encapsulates the shuffling procedure.
Recall that ${\Omega}^2_{\varepsilon}$ was defined in
Proposition <ref>.
Let $\varepsilon \in (0,1)$ and $2\varepsilon d< q\leq 2d/3$. Then
\p\left(\K\, \big| \, \ea\cap{\Omega}^2_{\varepsilon}\right)\leq
\frac{10}{\sqrt{(q-2\varepsilon d)}}.
Note that
\begin{equation*}
\K= \{M\in \ea:\, \langle v(M), R_1 \rangle =0 \}.
\end{equation*}
Let $M\in\ea\cap{\Omega}^2_{\varepsilon}$. Denote
\begin{align*}
S_{1,2}=S_{1,2}(M)=\supp R_1\cup \supp R_2,\quad
s_{1,2}=s_{1,2}(M)=\supp R_1\cap \supp R_2
%%, \quad \mbox{ and } \quad
%% S=S(M) = S_{1,2}\setminus s_{1,2}
\end{align*}
and set
S=S(M) = S_{1,2}\setminus s_{1,2},\quad m_1= |S_{1,2}|, \quad m_2=|s_{1,2}|,
\quad \mbox{ and } \quad m = |S|.
Note that $m_1=2d-m_2$ and $m=m_1-m_2 = 2(d-m_2)$.
By the definition of ${\Omega}^2_{\varepsilon}$, we have
m_1\geq 2(1-\varepsilon)d\quad \mbox{ and } \quad m_2\leq 2\varepsilon d.
By (<ref>), the vector $v:=v(M)$ satisfies $\, \forall \lam\in\R\quad |\{i\in S\,:\,x_i\ne\lam\}|\geq q-m_2$. Since $q-m_2\leq m/3$, by
Claim <ref> (see Remark <ref>)
there exists $J\subset S$ such that
\begin{equation}\label{vec-v}
q-m_2\leq|J|\leq m-(q-m_2) \;\text{ and }\; \forall i\in J\, \, \, \,
\forall j\in S\setminus J \, \, \;\; v_i\neq v_j.
\end{equation}
We compute the desired probability as follows.
For every (fixed) $M\in \ea\cap{\Omega}^2_{\varepsilon}$ consider the equivalence class
{\cal{F}}_M := {\cal{H}}_M^{1,2}(q) \cap{\Omega}^2_{\varepsilon}
= \left\{\widetilde M\in \ea\cap{\Omega}^2_{\varepsilon} \, \, :\, \, \widetilde M^{1,2} =M^{1,2} \right\}.
Note that by construction $S_{1,2}(\widetilde M)=S_{1,2}(M)=S_{1,2}$, $s_{1,2}(\widetilde M)=s_{1,2}(M)=s_{1,2}$ and $v(\widetilde M)=v(M)=v$
for every matrix $\widetilde M$ in ${\cal{F}}_M$. Therefore it is enough to show that the proportion of
matrices $\widetilde M \in {\cal{F}}_M$ satisfying
$\langle v, R_1(\widetilde M)\rangle =0$ is small.
Every matrix $\widetilde M \in {\cal{F}}_M$ is determined by its minor $M^{1,2}$ (which is
fixed on ${\cal{F}}_M$) and its first row $R_1(\widetilde M)$. Thus to determine a matrix in
${\cal{F}}_M$ it is enough to choose a support of the first row, which is a subset of $S_{1,2}$.
Since $m_2$ elements in $\supp R_1$ are fixed (as $s_{1,2}$ is fixed) then we have to calculate how many $(d-m_2)$-element subsets $B$ of $m$-element set $S$ exist so that
\langle v, R_1\rangle = \sum _{i\in B\cup s_{1,2}} v_i =0,
that is
\sum _{i\in B} v_i =a := - \sum _{i\in s_{1,2}} v_i .
For vectors $v=v(M)$ satisfying (<ref>) this was calculated in
Proposition <ref> (note that $m=2(d-m_2)$, $a$ is independent of
$B\subset S$ and apply the proposition with $q-m_2$ and $m/2$ instead of
$k$ and $d$). Applying this for all classes ${\cal{F}}_M$, we obtain the
desired bound.
In what follows, we will show that, up to intersection with $\eg \cap \ea\cap \e^{1,2}_{n-2}$ (resp., $\eg \cap \ea\cap \erk$), the event
$E_1=\en\setminus \e_{n-2}$ (resp., $E_2= \e_{n-2}$) is a subset of $\K$, hence has a small probability. Our treatment of singular matrices $M$ with $\rk M=n-1$ and $\rk M\leq n-2$ is slightly different, although the general idea is the same – at the first step, given a singular matrix $M$, we fix a left null vector $y=y(M)$ and a right null-vector $x=x(M)$ and choose a row $R_i$ such that $\rk M^i=\rk M$ and $x$ has many distinct coordinates on $R_i$. On the next step, we choose a second row $R_j$ so that the minor $M^{i, j}$ is of maximal rank, that is $\rk M^{i, j} = n-2$ in the case $\rk M=n-1$ and $\rk M^{i, j} = \rk M$ in the case $\rk M\leq n-2$. We also show that there are many choices for such $i$ and $j$. Finally, using the shuffling, we show, in a sense, that we can increase the rank of a matrix by “playing" with rows $i$ and $j$ only, i.e. that the events $\e _{n-2}$ and $\en$ are small inside $\en$ and $\Mc$ respectively. The next lemma describes the set of “good" $i$'s for the first step.
Let $0< p\leq 1/3$, and $ q\geq 2$ be such that $ pn/2q$ is an integer.
Further, let
M\in\en\cap\eac\setminus\eoo\big({p}/{2q}, {p}/{2}\big)
and $x\in\ker M\setminus\{0\}$, $y\in\ker M^T\setminus\{0\}$.
Consider the set of indices
\begin{equation*}
I_M(x,y)=\{i\in\supp y:\, \forall\lam\in\R\, \,
|\{j\in\supp R_{i}\;:\;x_j\neq\lam\}|\geq q\}.
\end{equation*}
\begin{equation*}
|I_M(x,y)|\geq pn/2.
\label{|Ixy|}
\end{equation*}
Note that for $y\in\ker M^T$ we have $\sum _i y_i R_i=0$ and
$I_M(x,y)\subset \supp y$. Therefore removing the row $R_i$,
$i\in I_M(x,y)$, we do not decrease the rank of $M$, that is
$\rk M^{i}=\rk M$.
Proof of Lemma <ref>.
Since $x\notin AC(p)$ and $p\leq 1/3$, by Claim <ref> there
exists a subset $J_x\subset[n]$ such that
pn\leq |J_x|\leq (1-p)n \quad \text{and} \quad
\forall i\in J_x \,\, \forall j\not\in J_x \quad x_i\neq x_j.
Now we compute how many rows have more than $q$ ones in $J_x$ and
more than $q$ ones in $J_x^c$. Since $M\not\in \eoo({p}/{2q}, p/2)$
then applying Lemma <ref> with $\alpha=p/(2q)$ and $\beta=p$,
we get
|\{i\;:\;|\supp R_i\cap J_x|\geq q\}|\geq(1- p/2q)n \quad \mbox{ and } \quad
|\{i\;:\;|\supp R_i\cap J^c_x|\geq q\}|\geq(1- p/2q)n.
\begin{align*}
|\{i\;:\;|\supp R_i\cap J_x|\geq q \quad \,\,\text{and}\quad \,\,|\supp R_i\cap
J^c_x|\geq q\}|\geq(1- p/q)n.
\end{align*}
By the construction of the set $J_x$ this implies that the set
\begin{align*}
I:=\{i\;:\;\forall\lam\in\R\quad|\{j\in \supp R_i\;:\;x_j\neq\lam\}|
\geq q\}
\end{align*}
has cardinality at least $(1-p/q)n$.
Finally, since $y\notin AC(p)$, we have that $|\supp y|\geq pn$, which implies
|I_M(x,y)| = |I\cap\supp y| \geq pn -pn/q \geq pn/2,
and completes the proof.
Now we consider the set of matrices $M\in \eg$ with $\rk M\leq n-2$.
Let $p,q$ satisfy the assumptions of Lemma <ref>, $\eps\in (0, 1)$,
and let $\e=\enn\cap\eg$ with $\eg=\eg(p,q,\varepsilon)$. Then
\begin{equation*}
\p\big(\e\big)\leq2 p^{-2}\, \, \p\big(\erk\cap\e^{1,2}(q)\cap\e\big).
\end{equation*}
Fix $M\in\e$. There exist $x\in\ker M\setminus\{0\}$ and
$y\in\ker M^T\setminus\{0\}$, that is
\forall i\leq n \,\,\, x\perp R_i\quad \text{ and }\quad \sum_{i\in \supp y} y_i
Note that by the definition of $\eg$ we have $x,y\notin AC(p)$. We compute
how many ordered pairs $(i,j)$, $i\ne j$, satisfy
\begin{equation*}
R_i, R_j\in V_{i,j}\quad \text{ and }\quad \forall\lam\in\R\,\,|\{k\in \supp(R_i+R_j):\;x_k\neq\lam\}|\geq q.
\end{equation*}
By Lemma <ref>, the set $I_M(x,y)$ satisfies $|I_M(x,y)|\geq pn/2$,
and for every $i\in I_M(x,y)$ we have $\rk M^{i}=\rk M$.
Next, since $\rk M^{i}<n-1$, the set $\ker (M^{i})^T\setminus\{0\}$ is non-empty, i.e.
\exists y^{(i)}\in\R^n\setminus\{0\}\, \, \mbox{ such that }\, \, y^{(i)}_{i}=0,\quad \sum_{j=1}^n y^{(i)}_j R_j =0.
Clearly $y^{(i)} \in\ker M^T\setminus\{0\}$, and, since $M\in \eac$, $y^{(i)}$ has at least
$pn$ non-zero coordinates; moreover,
\forall j\leq n \quad\text{ such that }\quad
y^{(i)}_{j}\neq0\quad\text{ one has }\quad R_j\in V_{i,j}.
This shows that for every $M\in \e$ there are at least $(pn)^2/4$ pairs
$(i, j)$ with $i<j$ satisfying $R_i, R_j\in V_{i,j}$. Obviously $x\perp\Fij$ for every $i,j\leq n$. Hence for each pair $(i, j)$ we have
\begin{equation*}
R_i, R_j\in V_{i,j}, \;\; x\perp\Fij\,\, \, \mbox{ and } \, \, \, \forall\lam\in\R\,\,|\{k\in \supp(R_i+R_j):\;x_k\neq\lam\}|\geq q,
\end{equation*}
implying that $M$ belongs to at least $(pn)^2/4$ distinct subsets among $\{\erkij\cap\e^{i,j}(q)\cap\e\}_{i<j}$.
Thus, $\{\erkij\cap\e^{i,j}(q)\cap\e\}_{i<j}$
forms a ($(pn)^2/4$)-fold covering for $\e$. Since
\p\left( \erkij \cap\e^{i,j}(q)\cap\e \right)
\p\left( \erk \cap\e^{1,2}(q)\cap\e \right)
then applying Claim <ref>, we obtain
\begin{align*}
\frac{(pn)^2}{4} \, \p(\e)
\sum _{i<j} \p\left( \erkij \cap\e^{i,j}(q)\cap\e \right)
\frac{n(n-1)}{2} \, \p\left( \erk \cap\e^{1,2}(q)\cap\e \right).
\end{align*}
This completes the proof of the lemma.
Now we treat the case when a matrix has rank $n-1$.
Let $p,q$ satisfy the conditions of Lemma <ref>, $\eps\in (0, 1)$, and let
$\e=(\en\setminus\enn)\cap\eg$ with $\eg=\eg(p,q,\varepsilon)$. Then
\begin{equation*}
\p\big(\e\big)\leq2 p^{-2}\, \, \p\big(\e^{1,2}_{n-2}\cap\e^{1,2}(q)\cap\e).
\end{equation*}
Repeating the first step of the proof of Lemma <ref>, we fix $M\in\e$,
$x\in\ker M\setminus\{0\}$, $ y\in\ker M^T\setminus\{0\}$. Then by
Lemma <ref> the set of indices $I_M(x,y)$
has cardinality $|I_M(x,y)|\geq pn/2$ and for every $i\in I_M(x,y)$
the $(n-1)\times n$ minor $M^{i}$ satisfies
$\rk M^{i}=\rk M$.
Now, we calculate how many ordered pairs $(i,j)$, $i\ne j$,
exist such that
\begin{equation*}
\rk M^{i,j}=n-2\quad \quad \mbox{ and } \quad \quad R_{i}+R_{j}\notin V_{i,j}.
\end{equation*}
Let $i\in I_M(x,y)$. Since $y\notin AC(p)$, there are at least
$pn$ choices of $j$ such that $y_{j}\neq y_{i}$. Fix such a $j$. We claim that then
$R_{i}+R_{j}\notin V_{i,j}$.
Indeed, otherwise
R_{i}+R_{j}=\sum_{\ell\neq i,j}z_{\ell} R_{\ell}
for some $z_\ell\in\R$,
hence there exists $w\in \ker M^T \setminus \{0\}$ such that $w_i=w_j$.
Since the dimension of $\ker M^T$ is one, we have $y=\lambda w$ for some $\lambda \in \R$,
which contradicts the condition $y_{i}\neq y_{j}$.
Therefore, there are at least $(pn)^2/2$ pairs $(i,j)$ with $i\ne j$ satisfying
\rk \MMij=n-2, \quad \quad R_{i}+R_{j}\notin V_{i,j},
x\perp\Fij, \quad \quad \forall\lam\in\R\, \, \, \, |\{k\in \supp(R_{i}+R_{j})\;:\;x_k\neq\lam\}|\geq q.
In other words, the matrix $M$ belongs to at least $(pn)^2/2$ events $\e^{i,j}_{n-2}\cap\e^{i,j}(q)\cap \e$.
Thus, we proved that
$\{\e^{i,j}_{n-2}\cap\e^{i,j}(q)\cap \e\}_{i<j}$
forms a $((pn)^2/4)$-fold covering of $\e$. Since for every $i<j$,
\p\left(\e^{i,j}_{n-2}\cap\e^{i,j}(q)\cap \e\right)=\p\left(\e^{1,2}_{n-2}\cap\e^{1,2}(q)\cap \e\right),
then applying Claim <ref>, we obtain
\begin{align*}
\frac{(pn)^2}{4} \, \p\left( \e\right) \leq \sum_{i<j} \p\left(\e^{i,j}_{n-2}\cap\e^{i,j}(q)\cap \e\right)
= \frac{n(n-1)}{2} \, \p\left(\e^{1,2}_{n-2}\cap\e^{1,2}(q)\cap \e\right),
\end{align*}
and the proof is complete.
We now can finish Theorem A.
Proof of Theorem A.
Let $p$, $q$, $\eps$ be chosen later to satisfy assumptions in the corresponding statements.
By Lemmas <ref>, <ref> and (<ref>) we obtain
\begin{align*}
\p(\en)&\leq \p\big(\enn\cap\eg\big)+ \p\big((\en\setminus \enn)\cap\eg\big)+ \left(\frac{e d}{\eps n}\right)^{\eps d/2}\\
&\leq2p^{-2}\big(\p(A)+\p(B)\big)+ \left(\frac{e d}{\eps n}\right)^{\eps d/2},
\end{align*}
\begin{align*}
A=\erk\cap\e^{1,2}(q)\cap\enn\cap\eg \quad \mbox{ and } \quad
B=\e^{1,2}_{n-2}\cap\e^{1,2}(q)\cap(\en\setminus \enn)\cap\eg.
\end{align*}
We show now that
A\cup B\subset \K\cap\e^{1,2}(q)\cap{\Omega}^2_{\varepsilon}.
In other words, we verify that
for a matrix $M\in A\cup B$ the vector
$v(M)\in F^\perp_{1,2}$ (see Definition <ref>) belongs to $\ker M$.
Indeed, if $M\in A$, then $R_1, R_2\in\V$.
Using the condition $v(M)\in F^\perp_{1,2}$ we immediately get $v(M)\in\ker M$.
If $M\in B$ then $\rk M=n-1$, $\rk M^{1,2}=n-2$, and $R_1+R_2\notin \V$.
Therefore $\dim F_{1,2}=n-1$. Since $\ker M\subseteq F^\perp_{1,2}$
and $\dim\ker M=\dim F^\perp_{1,2}=1$ we infer $\ker M= F^\perp_{1,2}$
and thus $v(M)\in\ker M$.
Finally note that $A$ and $B$ are disjoint, so $\p(A)+\p(B)=\p(A\cup B)$.
Applying Lemma <ref> we obtain
\begin{align*}
\p(\en)\leq 2 p^{-2} \p\left(\K\cap\e^{1,2}(q)\cap{\Omega}^2_{\varepsilon}
\right) + \left(\frac{e d}{\eps n}\right)^{\eps d/2}
%\\ &
=\frac{20}{p^2\sqrt{(q-2\varepsilon d)}}+ \left(\frac{e d}{\eps n}\right)^{\eps d/2}.
\end{align*}
Finally we choose the parameters. Let $c_1$, $c_2$ be sufficiently small
positive absolute constants. Choose $p=c_1/\ln d$ and $q$ to be the largest
integer not exceeding $c_2 p^2 d = c_1 c_2 d/\ln^2 d$ (we slightly adjust
$c_1, c_2$ so that $pn/2q$ is also an integer). Let $\eps = q/(4d)\approx 1/\ln ^2 d$
(note that then the condition $c_3/\eps^2 \leq d\leq \eps n/6$ needed in (<ref>) is also satisfied).
Then we obtain the desired bound.
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$\eps = q/(4d)\approx 1/((\ln \ln d) \ln d)$. This would lead to the restriction
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Core decomposition is a fundamental graph problem with a large number of applications. Most existing approaches for core decomposition assume that the graph is kept in memory of a machine. Nevertheless, many real-world graphs are big and may not reside in memory. In the literature, there is only one work for I/O efficient core decomposition that avoids loading the whole graph in memory. However, this approach is not scalable to handle big graphs because it cannot bound the memory size and may load most parts of the graph in memory. In addition, this approach can hardly handle graph updates. In this paper, we study I/O efficient core decomposition following a semi-external model, which only allows node information to be loaded in memory. This model works well in many web-scale graphs. We propose a semi-external algorithm and two optimized algorithms for I/O efficient core decomposition using very simple structures and data access model. To handle dynamic graph updates, we show that our algorithm can be naturally extended to handle edge deletion. We also propose an I/O efficient core maintenance algorithm to handle edge insertion, and an improved algorithm to further reduce I/O and CPU cost by investigating some new graph properties. We conduct extensive experiments on $12$ real large graphs. Our optimal algorithm significantly outperform the existing I/O efficient algorithm in terms of both processing time and memory consumption. In many memory-resident graphs, our algorithms for both core decomposition and maintenance can even outperform the in-memory algorithm due to the simple structures and data access model used. Our algorithms are very scalable to handle web-scale graphs. As an example, we are the first to handle a web graph with $978.5$ million nodes and $42.6$ billion edges using less than $4.2$ GB memory.
§ INTRODUCTION
Graphs have been widely used to represent the relationships of entities in a large spectrum of applications such as social networks, web search, collaboration networks, and biology. With the proliferation of graph applications, research efforts have been devoted to many fundamental problems in managing and analyzing graph data. Among them, the problem of computing the $k$-core of a graph has been recently studied <cit.>. Here, given a graph $G$, the $k$-core of $G$ is the largest subgraph of $G$ such that all the nodes in the subgraph have a degree of at least $k$ <cit.>. For each node $v$ in $G$, the core number of $v$ denotes the largest $k$ such that $v$ is contained in a $k$-core. The core decomposition problem computes the core numbers for all nodes in $G$. Given the core decomposition of a graph $G$, the $k$-core of $G$ for all possible $k$ values can be easily obtained. There is a linear time in-memory algorithm, devised by Batagelj and Zaversnik <cit.>, to compute core numbers of all nodes.
Applications. Core decomposition is widely adopted in many real-world applications, such as community detection <cit.>, network clustering <cit.>, network topology analysis <cit.>, network visualization <cit.>, protein-protein network analysis <cit.>, and system structure analysis <cit.>. In addition, many researches are devoted on the core decomposition for specific kinds of networks <cit.>. Moreover, due to the elegant structural property of a $k$-core and the linear solution for core decomposition, a large number of graph problems use core decomposition as a subroutine or a preprocessing step, such as clique finding <cit.>, dense subgraph discovery <cit.>, approximation of betweeness scores <cit.>, and some variants of community search problems <cit.>.
Motivation. Despite the large amount of applications for core decomposition in various networks, most of the solutions for core decomposition assume that the graph is resident in the main memory of a machine. Nevertheless, many real-world graphs are big and may not reside entirely in the main memory. For example, the social network contains $1.32$ billion nodes and $140$ billion edges[<http://newsroom.fb.com/company-info>]; and a sub-domain of the web graph contains $978.5$ million nodes and $42.6$ billion edges[<http://law.di.unimi.it/datasets.php>]. In the literature, the only solution to study I/O efficient core decomposition is proposed by Cheng et al. <cit.>, which allows the graph to be partially loaded in the main memory. adopts a graph partition based approach and partitions are loaded into main memory whenever necessary. However, cannot bound the size of the memory and to process many real-world graphs, still loads most edges of the graph in the main memory. This makes unscalable to handle web-scale graphs. In addition, many real-world graphs are usually dynamically updating. The complex structure used in makes it very difficult to handle graph updates incrementally.
Our Solution. In this paper, we address the drawbacks of the existing solutions for core decomposition and propose new algorithms for core decomposition with guaranteed memory bound. Specifically, we adopt a semi-external model. It assumes that the nodes of the graph, each of which is associated with a small constant amount of information, can be loaded in main memory while the edges are stored on disk. We find that this assumption is practical in a large number of real-world web-scale graphs, and widely adopted to handle other graph problems <cit.>. Based on such an assumption, we are able to handle core decomposition I/O efficiently using very simple structures and data access mechanism. These enable our algorithm to efficiently handle graph updates incrementally under the semi-external model.
Contributions. We make the following contributions:
(1) The first I/O efficient core decomposition algorithm with memory guarantee. We propose an I/O efficient core decomposition algorithm following the semi-external model. Our algorithm only keeps the core numbers of nodes in memory and updates the core numbers iteratively until convergency. In each iteration, we only require sequential scans of edges on disk. To the best of our knowledge, this is the first work for I/O efficient core decomposition with memory guarantee.
(2) Several optimization strategies to largely reduce the I/O and CPU cost. Through further analysis, we observe that when the number of iterations increases, only a very small proportion of nodes have their core numbers updated in each iteration, and thus a total scan of all edges on disk in each iteration will result in a large number of waste I/O and CPU cost. Therefore, we propose optimization strategies to reduce these cost. Our first strategy is based on the observation that the update of core number for a node should be triggered by the update of core number for at least one of its neighbors in the graph. Our second strategy further maintains more node information. As a result, we can completely avoid waste I/Os and core number computations, in the sense that each I/O is used in a core number computation that is guaranteed to update the core number of the corresponding node. Both optimization strategies can be easily adapted in our algorithm framework.
(3) The first I/O efficient core decomposition algorithm to handle graph updates. We consider dynamical graphs with edge deletion and insertion. Our semi-external algorithm can naturally support edge deletion with a simple algorithm modification. For edge insertion, we first utilize some graph properties adopted in existing in-memory algorithms <cit.> to handle graph updates for core decomposition. We propose a two-phase semi-external algorithm to handle edge insertion using these graph properties. We further explore some new graph properties, and propose a new one-phase semi-external algorithm to largely reduce the I/O and CPU cost for edge insertion. To the best of our knowledge, this is the first work for I/O efficient core maintenance on dynamic graphs.
(4) Extensive performance studies. We conduct extensive performance studies using $12$ real graphs with various graph properties to demonstrate the efficiency of our algorithms. We compare our algorithm, for memory-resident graphs, with <cit.> and the in-memory algorithm <cit.>. Both our core decomposition and core maintenance algorithms are much faster and use much less memory than . In many datasets, our algorithms for core decomposition and maintenance are even faster than the in-memory algorithm due to the simple structure and data access model used. Our algorithms are very scalable to handle web-scale graphs. For instance, we consume less than $4.2$ GB memory to handle the web-graph with $978.5$ million nodes and $42.6$ billion edges.
Outline. problem provides the preliminaries and problem statement. existing introduces some existing solutions for core decomposition under different settings. decomposition presents our semi-external core decomposition algorithm and explores some optimization strategies to reduce I/O and CPU cost. maintenance discusses how to design semi-external algorithms to maintain core numbers incrementally when the graph is dynamically updated, and investigates some new graph properties to improve the algorithm when handling edge insertion. experiment evaluates all the introduced algorithms using extensive experiments. relatedwork reviews the related work and conclusion concludes the paper.
§ PROBLEM STATEMENT
Consider an undirected and unweighted graph $G=(V,E)$, where $V(G)$ represents the set of nodes and $E(G)$ represents the set of edges in $G$. We denote the number of nodes and the number of edges of $G$ by $n$ and $m$ respectively. We use $\nb(u,G)$ to denote the set of neighbors of $u$ for each node $u\in V(G)$, i.e., $\nb(u,G)=\{v|(u,v)\in E(G)\}$. The degree of a node $u\in V(G)$, denoted by $\dg(u,G)$, is the number of neighbors of $u$ in $G$, i.e., $\dg(u,G)=|\nb(u,G)|$. For simplicity, we use $\nb(u)$ and $\dg(u)$ to denote $\nb(u,G)$ and $\dg(u,G)$ respectively if the context is self-evident. A graph $G'$ is a subgraph of $G$, denoted by $G'\subseteq G$, if $V(G')\subseteq V(G)$ and $E(G')\subseteq E(G)$. Given a set of nodes $V_c\subseteq V$, the induced subgraph of $V_c$, denoted by $G(V_c)$, is a subgraph of $G$ such that $G(V_c)=(V_c,\{(u,v)\in E(G)| u, v \in V_c\})$.
($k$-Core) Given a graph $G$ and an integer $k$, the $k$-core of graph $G$, denoted by $G_k$, is a maximal subgraph of $G$ in which every node has a degree of at least $k$, i.e., $\forall v\in V(G_k), d(v,G_k)\geq k$ <cit.>.
Let $\kmax$ be the maximum possible $k$ value such that a $k$-core of $G$ exists. According to <cit.>, the $k$-cores of graph $G$ for all $1\leq k\leq \kmax$ have the following property:
$\forall 1\leq k < \kmax: G_{k+1}\subseteq G_k$.
Next, we define the core number for each $v\in V(G)$.
(Core Number) Given a graph $G$, for each node $v\in V(G)$, the core number of $v$, denoted by $\core(v,G)$, is the largest $k$, such that $v$ is contained in a $k$-core, i.e., $\core(v,G)=\max\{k|v\in V(G_k)\}$. For simplicity, we use $\core(v)$ to denote $\core(v,G)$ if the context is self-evident.
Based on kcore and corenumber, we can easily derive the following lemma:
Given a graph $G$ and an integer $k$, let $V_k=\{v\in V(G)|\core(v)\geq k\}$, we have $G_k = G(V_k)$.
Problem Statement. In this paper, we study the problem of Core Graph Decomposition (or Core Decomposition for short), which is defined as follows: Given a graph $G$, core decomposition computes the the $k$-cores of $G$ for all $1\leq k \leq \kmax$. We also consider how to update the $k$-cores for of $G$ for all $1\leq k\leq \kmax$ incrementally when $G$ is dynamically updated by insertion and deletion of edges.
According to kcore, core decomposition is equivalent to computing $\core(v)$ for all $v\in V(G)$. Therefore, in this paper, we study how to compute $\core(v)$ for all $v\in V(G)$ and how to maintain them incrementally when graph is dynamically updating .
Considering that many real-world graphs are huge and cannot entirely reside in main memory, we aim to design I/O efficient algorithms to compute and maintain the core numbers of all nodes in the graph $G$. To analyze the algorithm, we use the external memory model introduced in <cit.>. Let $M$ be the size of main memory and let $B$ be the disk block size. A read I/O will load one block of size $B$ from disk into main memory, and a write I/O will write one block of size $B$ from the main memory into disk.
Assumption. In this paper, we follow a semi-external model by assuming that the nodes can be loaded in main memory while the edges are stored on disk, i.e., we assume that $M\geq c\times |V(G)|$ where $c$ is a small constant. This assumption is practical because in most social networks and web graphs, the number of edges is much larger than the number of nodes. For example, in SNAP[<http://snap.stanford.edu/data/>], among $79$ real-world graphs, the largest graph contains $65$ M nodes and $1.8$ G edges. In KONET[<http://konect.uni-koblenz.de/networks/>], among $239$ real-world graphs, the largest graph contains $68$ M nodes and $2.6$ G edges. In WebGraph[<http://law.di.unimi.it/>], among $75$ real-world graphs, the largest graph contains $721$ M nodes and $137.3$ G edges, and the second largest graph contains $978$ M nodes and $42.6$ G edges. In our proposed algorithm of this paper, when handling the two largest graphs in WebGraph, we only require $3.1$ GB and $4.2$ GB memory respectively, which is affordable even by a normal PC.
Graph Storage. In this paper, we use an edge table on disk to store the edges of $G$. e.g., we store $\nb(v_1)$, $\nb(v_2)$, $\ldots$, $\nb(v_n)$ consecutively as adjacency lists in the edge table. We also use a node table on disk to store the offsets and degrees for $v_1$, $v_2$, $\ldots$, $v_n$ consecutively. To load the neighbors of a certain node $v_i\in V(G)$, we can access the node table to get the offset and $\dg(v_i)$ for $v_i$, and then access the edge table to load $\nb(v_i)$.
A Sample Graph $G$ and its Core Decomposition
Consider a graph $G$ in core, the induced subgraph of $\{v_0,$ $v_1,$ $v_2,$ $v_3\}$ is a $3\text{-}core$ in which every node has a degree at least $3$. Since no $4$-core exists in $G$, we have $\core(v_0)=\core(v_1)=\core(v_2)=\core(v_3)=3$. Similarly, we can derive that $\core(v_4)=\core(v_5)=\core(v_6)=\core(v_7)=2$ and $\core(v_8)=1$. When an edge $(v_7,v_8)$ is inserted in $G$, $\core(v_8)$ increases from $1$ to $2$, and the core numbers of other nodes keep unchanged.
§ EXISTING SOLUTIONS
In this section, we introduce three state-of-the-art existing solutions for core decomposition in different settings, namely, in-memory core decomposition, I/O efficient core decomposition, and in-memory core maintenance.
In-memory Core Decomposition. The state-of-the-art in-memory core decomposition algorithm, denote by $\imcore$, is proposed in <cit.>. The pseudocode of $\imcore$ is shown in imcore. The algorithm processes the node with core number $k$ in increasing order of $k$. Each time, $k$ is selected as the minimum degree of current nodes in the graph (line 3). Whenever there exists a node $v$ with degree no larger than $k$ in the graph (line 4), we can guarantee that the core number of $v$ is $k$ (line 5) and we remove $v$ with all its incident edges from the graph (line 6). Finally, the core number of all nodes are returned (line 7). With the help of bin sort to maintain the minimum degree of the graph, $\imcore$ can achieve a time complexity of $O(m+n)$, which is optimal.
$\imcore($Graph $G)$
$G'\leftarrow G$;
$G' \neq \emptyset$
$k\leftarrow \min_{v\in V(G')}\dg(v, G')$;
$\exists v\in V(G'): \dg(v, G')\leq k$
$\core(v)\leftarrow k$;
remove $v$ and its incident edges from $G'$;
return $\core(v)$ for all $v\in V(G)$;
I/O Efficient Core Decomposition. The state-of-the-art efficient core decomposition algorithm is proposed in <cit.>. The algorithm, denoted as $\emcore$, is shown in emcore. It first divides the whole graph $G$ into partitions on disk (line 1). Each partition contains a disjoint set of nodes along with their incident edges. An upper bound of $\core(v)$, denoted by $\cub(v)$, is computed for each node $v$ in each partition $P_i$. Then the algorithm iteratively computes the core numbers for nodes in a top-down manner.
In iteration, the nodes with core values falling in a certain range $[k_l, k_u]$ is computed (line 6-14). Here, $k_l$ is estimated based on the number of partitions that can be loaded in main memory (line 6). In line 7, the algorithm computes the set of partitions each of which contains at leat one node $v$ with $\cub(v)$ falling in $[k_l,k_u]$, and in line 8, all such partitions are loaded in main memory to form an in-memory graph $G_{mem}$. In line 9, an in-memory core decomposition algorithm is applied on $G_{mem}$, and those nodes in $G_{mem}$ with core numbers falling in $[k_l,k_u]$ get their exact core numbers in $G$. After that, for all partitions loaded in memory (line 10), those nodes with exact core numbers computed are removed from the partition (line 11), and the their core number upper bounds and degrees are updated accordingly (line 12). Here the new node degrees have to consider the deposited degrees from the removed nodes. Finally, the in-memory partitions are merged and written back to disk (line 13), and $k_u$ is set to be $k_l-1$ to process the next range of $k$ values in the next iteration.
The I/O complexity of $\emcore$ is $O(\frac{\kmax\cdot (m+n)}{B})$. The CPU complexity of $\emcore$ is $O(\kmax\cdot(m+n))$. However, the space complexity of $\emcore$ cannot be well bounded. In the worst case, it still requires $O(m+n)$ memory space to load the whole graph into main memory. Therefore, $\emcore$ is not scalable to handle large-sized graphs.
$\emcore($Graph $G$ on Disk$)$
divide $G$ into partitions ${\cal P}=\{P_1, P_2, \ldots, P_t\}$ on disk;
partition $P_i\in {\cal P}$
compute $\cub(v)$ for all $v\in V(P_i)$;
$k_u\leftarrow +\infty$;
estimate $k_l$;
${\cal P}_{mem}\leftarrow \{P_i\in {\cal P}|\exists v\in V(P_i):\cub(v)\in [k_l, k_u]\}$;
$G_{mem}\leftarrow$ load partitions in ${\cal P}_{mem}$ in main memory;
$\core(v)\leftarrow \core(v,G_{mem})$ for all $\core(v,G_{mem})\in [k_l, k_u]$;
partition $P_i\in {\cal P}_{mem}$
remove nodes $v$ with $\core(v,G_{mem})\in [k_l, k_u]$ from $P_i$;
update $\cub(v)$ and $\dg(v)$ for all $v\in V(P_i)$;
write $P_i$ back to disk (merge small partitions if necessary);
$k_u\leftarrow k_l-1$;
return $\core(v)$ for all $v\in V(G)$;
In-memory Core Maintenance. To handle the case when the graph is dynamically updated by insertion and deletion of edges, the state-of-the-art core maintenance algorithms are proposed in <cit.> and <cit.>, which are based on the same findings shown in the following theorems:
If an edge is inserted into (deleted from) graph $G$, the core number $\core(v)$ for any $v\in V(G)$ may increase (decrease) by at most $1$.
If an edge $(u,v)$ is inserted into (deleted from) graph $G$, suppose $\core(v)\leq \core(u)$ and let $V'$ be the set of nodes whose core numbers have changed, if $V'\neq \emptyset$, we have:
$G(V')$ is a connected subgraph of $G$;
$v\in V'$; and
$\forall v'\in V': \core(v')=\core(v)$;
Based on change1 and change2, after an edge $(u,v)$ is inserted into (deleted from) graph $G$, suppose $\core(v)\leq \core(u)$, instead of computing the core numbers for all nodes in $G$ from scratch, we can restraint the core computation within a small range of nodes $V'$ in $G$. Specifically, we can follow a two-step approach: In the first step, we can perform a depth-first-search from node $v$ in $G$ to compute all nodes $v'$ with $\core(v')=\core(v)$ and are reachable from $v$ via a path that consists of nodes with core numbers equal to $\core(v)$. Such nodes form a set $V'$ which is usually much smaller than $V(G)$. In the second step, we only restraint the core number updates within the subgraph $G(V')$ in memory, and each update increases (decreases) the core number of a node by at most $1$. The algorithm details and other optimization techniques can be found in <cit.> and <cit.>.
§ I/O EFFICIENT CORE DECOMPOSITION
§ I/O EFFICIENT CORE MAINTENANCE
In this section, we discuss how to incrementally maintain the core numbers when edges are inserted into or deleted from the graph under the semi-external setting.
§.§ Edge Deletion
Algorithm Design. In change1, we know that after an edge deletion, the core number for any $v\in V(G)$ will decrease by at most $1$. Therefore, after an edge deletion, the old core numbers of nodes in the graph are upper bounds of their new core numbers. Recall that in semicores, as long as $\coreub(v)$ is initialized to be an arbitrary upper bound of $\core(v)$ for all $v\in V(G)$, $\coreub(v)$ can be finally converged to $\core(v)$ after the algorithm terminates. Therefore, semicores can be easily modified to handle edge deletion.
$\semidels($Graph $G$ on Disk, Edge $(u,v))$
delete $(u,v)$ from $G$;
$\cnt(u)\leftarrow \cnt(u)-1$;
$v_{min}\leftarrow u$; $v_{max}\leftarrow u$;
$\cnt(v)\leftarrow \cnt(v)-1$;
$v_{min}\leftarrow v$; $v_{max}\leftarrow v$;
$\cnt(u)\leftarrow \cnt(u)-1$; $\cnt(v)\leftarrow \cnt(v)-1$;
$v_{min}\leftarrow \min\{u,v\}$; $v_{max}\leftarrow \max\{u,v\}$;
line 4-14 of semicores;
Specifically, we show our algorithm $\semidels$ for edge deletion in semidels. Given an edge $(u,v)\in E(G)$ to be removed, we first delete $(u,v)$ from $G$ (line 1). We will discuss how to update $G$ on disk after edge deletion / insertion later. In line 2-8, we update $\cnt(u)$ and $\cnt(v)$ due to the deletion of edge $(u,v)$, and we also compute the initial range $v_{min}$ and $v_{max}$ for node checking. Here, we consider three cases. First, if $\coreub(u)<\coreub(v)$, we only need to decrease $\cnt(u)$ by $1$, and set $v_{min}$ and $v_{max}$ to be $u$. Second, if $\coreub(v)<\coreub(u)$, we decrease $\cnt(v)$ by $1$, and set $v_{min}$ and $v_{max}$ to be $v$. Third, if $\coreub(v)=\coreub(u)$, we decrease both $\cnt(v)$ and $\cnt(u)$ by $1$, and set $v_{min}$ and $v_{max}$ to be $\min\{u,v\}$ and $\max\{u,v\}$ respectively. Now we can use semicores to update the core numbers of other nodes (line 11).
Graph Maintenance. We introduce how to maintain the graph on disk when edges are inserted into / deleted from the graph. Recall that our graph is stored in terms of adjacency lists on disk. If we simply update the lists after each edge insertion / deletion, the cost will be too high. To handle this, we allow a memory buffer to maintain the latest inserted / deleted edges. We also index the edges in the memory buffer. When the buffer is full, we update the graph on disk and clear the buffer. Noticed that each time when we load $\nb(v)$ for a certain node $v$ from disk, we also need to obtain the inserted / deleted edges for $v$ from the memory buffer, and use them to compute the updated $\nb(v)$.
Iteration$\setminus$$v$ $v_0$ $v_1$ $v_2$ $v_3$ $v_4$ $v_5$ $v_6$ $v_7$ $v_8$
Old Value $3$ $3$ $3$ $3$ $2$ $2$ $2$ $2$ $1$
Iteration $1$ $2$ $2$ $2$ $2$ $2$ $2$ $2$ $2$ $1$
Illustration of $\semidels$ (Delete $(v_0,v_1)$)
Suppose after semicores, we delete edge $(v_0,v_1)$ from $G$ (core). Using semidels, we first update both $\cnt(v_0)$ and $\cnt(v_1)$ from $3$ to $2$ and then invoke line 4-14 of semicores with $v_{min}=0$ and $v_{max}=1$. Only $1$ iteration is needed with $4$ node computations as shown in ex:semidels.
§.§ Edge Insertion
The Rationality. After a new edge $(u,v)$ is inserted into graph $G$, according to change1, we know that the core number for any $v\in V(G)$ will increase by at most $1$. As a result, the old core number of a node in the graph may not be an upper bound of its new core number. Therefore, semicores cannot be applied directly to handle edge insertion. However, according to change2, after inserting an edge $(u,v)$ (suppose $\coreub(v)\leq \coreub(u)$), we can find a candidate set $V_c$ consisting of all nodes $w$ that are reachable from node $v$ via a path that consists of nodes with $\coreub$ equals $\coreub(v)$, and we can guarantee that those nodes with core numbers increased by $1$ is a subset of $V_c$. Consequently, if we increase $\coreub(v)$ by $1$ for all $v\in V_c$, we can guarantee that for all $u\in V(G)$, $\coreub(u)$ is an upper bound of the new core number of $u$. Thus we can apply semicores to compute the new core numbers.
Algorithm Design. Our algorithm $\semiadd$ for edge insertion is shown in semiadd. In line 1, we insert $(u,v)$ into $G$. In line 1-4, we update $\cnt(u)$ and $\cnt(v)$ caused by the insertion of edge $(u,v)$. We use $\act(w)$ to denote whether $w$ is a candidate node with core number increased which is initialized to be false except for node $u$. In line 8-21, we iteratively update $\act(w)$ for $w\in V(G)$ until convergency. In each iteration (line 9-20), we find nodes $v'$ with $\act(v')=$ true and $\coreub(v')$ not being increased (line 11). For each such node $v'$, we increase $\coreub(v')$ by $1$ (line 12), and load $\nb(v')$ from disk. Since $\coreub(v')$ is changed, we need to compute $\cnt(v')$ (line 14) and update the $\cnt$ values for the neighbors of $v'$ (line 15-16). In line 17-20, we set $\act(u')$ to be true for all the neighbors $u'$ of $v'$ (line 17-18) if $u'$ is a possible candidate (line 18), and we update the range of nodes to be checked in the next iteration (line 20). After all iterations, we compute the range of the candidate nodes (line 22-24). Now we can guarantee that $\coreub(v')$ is an upper bound of the new core number of $v'$. Therefore, we invoke line 4-14 of semicores to compute the core numbers of all nodes in the graph (line 25).
Iteration$\setminus$$v$ $v_0$ $v_1$ $v_2$ $v_3$ $v_4$ $v_5$ $v_6$ $v_7$ $v_8$
Old Value $2$ $2$ $2$ $2$ $2$ $2$ $2$ $2$ $1$
Iteration $1.1$ $2$ $2$ $2$ $2$ $3$ $3$ $3$ $3$ $1$
Iteration $1.2$ $2$ $2$ $3$ $3$ $3$ $3$ $3$ $3$ $1$
Iteration $1.3$ $3$ $3$ $3$ $3$ $3$ $3$ $3$ $3$ $1$
Iteration $2.1$ $2$ $2$ $2$ $3$ $3$ $3$ $3$ $2$ $1$
Illustration of $\semiadd$ (Insert $(v_4,v_6)$)
Suppose after deleting edge $(v_0, v_1)$ from the graph $G$ (core) in semidels, we insert a new edge $(v_4, v_6)$ into $G$. The process to compute the new core numbers of nodes in $G$ is shown in ex:semiadd. Here, we use $3$ iterations $1.1$, $1.2$, and $1.3$ to compute the candidate nodes, and use $1$ iteration $2.1$ to compute the new core numbers. In iteration $1.1$, when $v_4$ is computed, it triggers its smaller neighbors $v_2$ and $v_3$ to be computed in the next iteration and triggers its larger neighbor $v_5$ to be computed in the current iteration. The total number of node computations is $12$.
$\semiadd($Graph $G$ on Disk, Edge $(u,v))$
insert $(u,v)$ into $G$;
swap $u$ and $v$ if $\coreub(u)>\coreub(v)$;
$\cnt(u)\leftarrow \cnt(u)+1$;
if $\coreub(v)=\coreub(u)$ then $\cnt(v)\leftarrow \cnt(v)+1$;
$\corg\leftarrow \coreub(u)$;
$\act(w)\leftarrow \textbf{false}$ for all $w\in V(G)$; $\act(u)\leftarrow$ true;
$v_{min}\leftarrow u$; $v_{max}\leftarrow u$; $\update\leftarrow \textbf{true}$;
$\update \leftarrow \textbf{false}$; $v'_{min}\leftarrow v_n$; $v'_{max}\leftarrow v_1$;
$v'\leftarrow v_{min}$ to $v_{max}$
$\act(v')=\textbf{true}$ and $\coreub(v')=\corg$
$\coreub(v')\leftarrow \coreub(v')+1$;
load $\nb(v')$ from disk;
$\cnt(v')\leftarrow \computecnt(\nb(v'), \coreub(v'))$;
$u'\in \nb(v')$ s.t. $\coreub(u')=\coreub(v')$
$\cnt(u')\leftarrow \cnt(u')+1$;
$u'\in \nb(v')$
$\coreub(u')=\corg$ and $\act(u')=\textbf{false}$
$\act(u')\leftarrow \textbf{true}$;
$\updaterange(v'_{min}, v'_{max}, v_{max}, \update, u', v')$;
$v_{min}\leftarrow v'_{min}$; $v_{max}\leftarrow v'_{max}$;
$v_{min}\leftarrow u$; $v_{max}\leftarrow u$;
$v\in V(G)$ s.t. $\act(v)=\textbf{true}$
$v_{min}\leftarrow \min\{v_{min}, v\}$; $v_{max}\leftarrow \max\{v_{max}, v\}$;
line 4-14 of semicores;
§.§ Optimization for Edge Insertion
The Rationality. semiadd handles an edge insertion using two phases. In phase 1, we compute a superset $V_c$ of nodes whose core numbers will be updated, and we increase the core numbers for all nodes in $V_c$ by $1$. In phase 2, we compute the core numbers of all nodes using semicores. One problem of semiadd is that the size of $V_c$ can be very large, which may result in a large number of node computations and I/Os in both phase 1 and phase 2 of semiadd. Therefore, it is crucial to reduce the size of $V_c$.
Now, suppose and edge $(u,v)$ is inserted into the graph $G$; $\cnt(u)$ and $\cnt(v)$ are updated accordingly; and $\coreub(w)$ values for all $w\in V(G)$ have not been updated . Without loss of generality, we assume that $\coreub(u)<\coreub(v)$ and let $\corg=\coreub(u)$. Let $V_c$ be the set of candidate nodes computed in semiadd, i.e., $V_c$ consists of all nodes that are reachable from $u$ via a path that consists of nodes with $\coreub$ equals $\corg$. Let $V_c^*\subseteq V_c$ be the set of nodes with $\coreub$ updated to be $\corg+1$ after inserting $(u,v)$. We have the following lemmas:
(a) For $v'\in V_c\setminus V_c^*$, $\cnt(v')$ keeps unchanged; (b) For $v'\in V_c^*$, $\cnt(v')$ will not increase.
This lemma can be easily verified according to semicores and change1.
If $\cnt(v')\geq \corg+1$ for all $v'\in V_c$, then we have $V_c^*=V_c$.
If we increase $\coreub(v')$ by $1$ for all $v'\in V_c$, it is easy to verify that $\cnt(v')$ for all $v'\in V_c$ keep unchanged. Now suppose $\cnt(v')\geq \corg+1$ for all $v'\in V_c$, we can derive that the locality property in locality holds for every $v'\in V(G)$. Therefore, the new $\coreub(v')$ is the core number of $v'$ for every $v'\in V(G)$. This indicates that $V_c^*=V_c$.
For any $v'\in V_c$, if $v'\in V_c^*$, then we have $\cnt(v')\geq \corg+1$.
Since $v'\in V_c^*$, we know that the new $\cnt(v')$ is no smaller than $\corg+1$. According to o1 (b), the original $\cnt(v')$ is also no smaller than $\corg+1$, since $\cnt(v')$ will not increase. Therefore, the lemma holds.
For each $v'\in V_c$, we define $\cnts(v')$ as:
\begin{equation}
\label{eq:o4}
\cnts(v') = |\{u'\in \nb(v')\ |\ \coreub(u')>\corg \ or \ u'\in V_c^*\}|
\end{equation}
We have:
(a) If $v'\in V_c^*$, then the updated $\cnt(v')=\cnts(v')$; and
(b) $v'\in V_c^* \Leftrightarrow \cnts(v')\geq \corg+1$.
For (a): for all $v'\in V_c^*$, since $\coreub(v')$ will become $\corg+1$, all nodes $u'\in V_c\setminus V_c^*$ will not contribute to $\cnt(v')$ according to semicores. Therefore, (a) holds.
For (b): $\Rightarrow$ can be derived according to (a). Now we prove $\Leftarrow$. Suppose $\cnts(v')\geq \corg+1$, to prove $v'\in V_c^*$, we prove that if we increase $\coreub(u')$ to $\corg+1$ for all $u'\in V_c$ and apply semicores, then $\coreub(v')$ will keep to be $\corg+1$ after convergency. Note that for all nodes $u' \in V_c^*$ and $u'\in \nb(v')$, $\coreub(u')$ will keep to be $\corg+1$ and will contribute to $\cnt(v')$, and all nodes $u'\in \nb(v')$ with $\coreub(u')>\corg$ will also contribute to $\cnt(v')$. According to o4, we have $\cnt(v')\geq \cnts(v') \geq \corg+1$. Therefore, $\coreub(v')$ will never decrease according to semicores. This indicates that $v'\in V_c^*$.
According to o4 (b), $\cnts(v')$ can be defined using the following recursive equation:
\begin{multline}
\label{eq:cnts}
\cnts(v')= |\{u'\in \nb(v')\ |\ \coreub(u')>\corg\ \text{or}\ \\(\coreub(u')=\corg\ \text{and}\ \cnts(u')\geq \corg+1)\}|
\end{multline}
To compute $\cnts(v')$ for all $v'\in V_c$, we can initialize $\cnts(v')$ to be $\cnt(v')$, and apply cnts iteratively on all $v'\in V_c$ until convergency. However, this algorithm needs to compute $V_c$ first, which is inefficient. Note that according to cnts and o4 (b), we only care about those nodes $u'$ with $\cnts(u')\geq \corg+1$. Therefore, we do not need to compute the whole $V_c$ by expanding from node $u$. Instead, for each expanded node $u'$, if we guarantee that $\cnts(u')< \corg+1$, we do not need to expand $u'$ further. In this way, the computational and I/O cost can be largely reduced.
$\semiadds($Graph $G$ on Disk, Edge $(u,v))$
line 1-5 of semiadd;
$\status(w)\leftarrow \statuse$ for all $w\in V(G)$; $\status(u)\leftarrow \statusq$;
$v_{min}\leftarrow u$; $v_{max}\leftarrow u$; $\update\leftarrow \textbf{true}$;
$\update \leftarrow \textbf{false}$; $v'_{min}\leftarrow v_n$; $v'_{max}\leftarrow v_1$;
$v'\leftarrow v_{min}$ to $v_{max}$
load $\nb(v')$ from disk;
$\cnt(v')\leftarrow \computecnts(\nb(v'),\corg)$;
$\status(v')\leftarrow \statusy$; $\coreub(v')\leftarrow \corg+1$;
$u'\in \nb(v')$ s.t. $\coreub(u')=\corg+1$
$\cnt(u')\leftarrow \cnt(u')+1$;
$\cnt(v')\geq \corg+1$
$u'\in \nb(v')$ s.t. $\coreub(u')=\corg$
$\cnt(u') \geq \corg +1$ and $\status(u') = \statuse$
$\status(u')\leftarrow \statusq$;
$\updaterange(v'_{min}, v'_{max}, v_{max}, \update, u', v')$;
$\status(v')=\statusy$ and $\cnt(v')< \corg+1$
load $\nb(v')$ from disk if not loaded;
$\cnt(v')\leftarrow \computecnt(\nb(v'),\corg)$;
$\status(v')\leftarrow \statusn$; $\coreub(v')\leftarrow \corg$;
$u'\in \nb(v')$ s.t. $\coreub(u')=\corg+1$
$\cnt(u')\leftarrow \cnt(u')-1$;
$u'\in \nb(v')$ s.t. $\status(u')=\statusy$
$\cnt(u')\leftarrow \cnt(u')-1$;
$\updaterange(v'_{min}, v'_{max}, v_{max}, \update, u', v')$;
$v_{min}\leftarrow v'_{min}$; $v_{max}\leftarrow v'_{max}$;
Procedure $\computecnts(\nb(v'),\corg)$
$s\leftarrow 0$;
$u'\in \nb(v')$
if $\coreub(u')>\corg$ or ($\coreub(u')=\corg$ and $\cnt(u')\geq \corg+1$ and $\status(u')\neq \statusn$) then $s\leftarrow s+1$;
return $s$;
Algorithm Design. Based on the above discussion, for each node $w\in V(G)$, we use $\status(w)$ to denote the status of node $w$ during the processing of node expansion. Each node $w\in V(G)$ has the following four status ($\status(w)$):
$\statuse$: $w$ has not been expanded by other nodes.
$\statusq$: $w$ is expanded but $\cnts(w)$ is not calculated.
$\statusy$: $\cnts(w)$ is calculated with $\cnts(w)\geq \corg+1$.
$\statusn$: $\cnts(w)$ is calculated with $\cnts(w)< \corg+1$.
With $\status(w)$ and according to o4 (a) and o1 (a), we can reuse $\cnt(w)$ to represent $\cnts(w)$ for each $w\in V(G)$. That is, if $\status(w)=\statusy$, $\cnt(w)$ can represent $\cnts(w)$ which is calculated using cnts, otherwise, if $\status(w)=\status$ $\cnt(w)$ is calculated using semicores.
Our new algorithm $\semiadds$ for edge insertion is shown in semiadds. The initialization phase is similar to that in semiadd (line 1). In line 6, we initialize $\status(w)$ to be $\statuse$ except $\status(u)$ which is initialized to be $\statusq$. The algorithm iteratively update $\status(v')$, $\coreub(v')$, and $\cnt(v')$ for all $v'\in V(G)$. In each iteration (line 5-28), we check $v'$ from $v_{min}$ to $v_{max}$ (line 6), and for each such $v'$ to be checked, we consider the following status transitions:
From $\statusq$ to $\statusy$ (line 7-12): If $\status(v')=\statusq$ (line 7), we load $\nb(v')$ from disk (line 8) and compute $\cnt(v')$ using cnts by invoking $\computecnts(\nb(v'),\corg)$ which is shown in line 29-33. Compared to cnts, we add a new condition for $u'\in \nb(v')$: $\status(u')\neq \statusn$ (line 32). This is because for node $u'$ with $\status(u')= \statusn$, it is computed using semicores other than cnts, and it cannot contribute to $\cnt(v')$. After computing $\cnt(v')$, in line 10, we set $\status(v')$ to be $\statusy$ and increase $\coreub(v')$ to be $\corg+1$. Since $\coreub(v')$ is increased to be $\corg+1$, we need to increase $\cnt(u')$ for all neighbor $u'$ of $v'$ with $\coreub(u')=\corg+1$ (line 11-12).
From $\statuse$ to $\statusq$ (line 13-17): After setting $v'$ to be $\statusq$, if $\cnt(v')\geq \corg+1$, $v'$ will not set to be $\statusn$ in this iteration. In this case (line 13), we can expand $v'$. That is, for all neighbors $u'$ of $v'$ with $\coreub(u')=\corg$ (line 14), if $\cnt(u')\geq \corg+1$ (refer to o3) and $u'$ has not be expanded ($\status(u')=\statuse$), we set $\status(u')$ to be $\statusq$ so that $u'$ can be expanded, and update the range of nodes to be checked (line 15-17).
From $\statusy$ to $\statusn$ (line 18-27): If $\status(v')$ is $\statusy$ and $\cnt(v')<\corg+1$, we need to change the status of $v'$ (line 18). Here, in line 19, we load $\nb(v')$ from disk if it is not loaded in line 8. In line 20, we compute $\cnt(v')$ using semicores. In line 21, we set $\status(v')$ to be $\statusn$, and update $\coreub(v')$ to be $\corg$ according to o1 (a). Since $\coreub(v')$ is changed from $\corg+1$ to $\corg$, for all neighbors $u'$ of $v'$ with $\coreub(u')=\corg+1$, we need to decrease $\cnt(u')$ (line 22-23). In addition, according to cnts, the status change from $\statusy$ to $\statusn$ for $v'$ will trigger each neighbor $u'$ of $v'$ to decrease its $\cnt(u')$ if $\status(u')=\statusy$ (line 24-25). For each such $u'$, if $\cnt(u')$ is decreased below $\corg$, $\status(u')$ need to be updated in the same of later iterations (line 26-27).
Compared to semiadd that requires two phases to update the core numbers, semiadds requires only one phase without invoking semicores for core number updates.
Iteration$\setminus$$v$ $v_0$ $v_1$ $v_2$ $v_3$ $v_4$ $v_5$ $v_6$ $v_7$ $v_8$
Old Value $2$ $2$ $2$ $2$ $2$ $2$ $2$ $2$ $1$
Iteration $1$
Iteration $2$
New Value $2$ $2$ $2$ $3$ $3$ $3$ $3$ $2$ $1$
Illustration of $\semiadds$ (Insert $(v_4,v_6)$)
Suppose after deleting edge $(v_0, v_1)$ from graph $G$ (core) in semidels, we insert edge $(v_4, v_6)$ into $G$. The process to update the status of nodes in each iteration is shown in ex:semiadds. In iteration $1$, when we check $v_4$, we update $\status(v_4)$ from $\statusq$ to be $\statusy$, and update the status of its neighbors ($v_2$, $v_3$, $v_5$, and $v_6$) to be $\statusq$. In iteration $2$, for $v_2$ with status $\statusq$, we can calculate that $\cnt(v_2)=2<\corg+1=3$. Therefore, we set $\status(v_2)$ to be $\statusn$, and decrease $\cnt(v_4)$ accordingly. The cells involving a node computation are marked grey. Totally 2 iterations are needed. The four nodes $v_3$, $v_4$, $v_5$, and $v_6$ with $\status$ being $\statusy$ have their core numbers updated. Compared to semiadd, we decrease the number of node computations from $12$ to $5$.
§ PERFORMANCE STUDIES
In this section, we experimentally evaluate the performance of our proposed algorithms for both core decomposition and core maintenance. performance compares our solutions with state-of-the-art algorithms; dynamic shows the efficiency of our maintenance algorithm; and we reports the algorithm scalability in scal.
All algorithms are implemented in C++, using gcc complier at -O3 optimization level. All the experiments are performed under a Linux operating system running on a machine with an Intel Xeon 3.4GHz CPU, 16GB RAM and 7200 RPM SATA Hard Drives (2TB). The time cost of algorithms are measured as the amount of wall-clock time elapsed during the program's execution. We adhere to standard external memory model for I/O statistics <cit.>.
Datasets. We use two groups of datasets to demonstrate the efficiency of our semi-external algorithm. Group one consists of six graphs with relatively smaller size: , , , , and . Group two consists of six big graphs: , , , , and . The detailed information for the $12$ datasets is displayed in datasets.
Datasets $|V|$ $|E|$ density $\kmax$
317,080 1,049,866 3.31 113
1,134,890 2,987,624 2.63 51
2,394,385 5,021,410 2.10 131
3,774,768 16,518,948 4.38 64
3,997,962 34,681,189 8.67 360
3,072,441 117,185,083 38.14 253
118,142,155 1,019,903,190 8.63 1506
41,291,594 1,150,725,436 27.86 3224
41,652,230 1,468,365,182 35.25 2488
50,636,154 1,949,412,601 38.49 4510
105,896,555 3,738,733,648 35.30 5704
978,408,098 42,574,107,469 43.51 4244
In group one (small graphs), is a co-authorship network of the computer science bibliography DBLP. is a social network based on the user friendship in Youtube. is a network containing all the users and discussion from the inception of Wikipedia till January 2008. is citation graph includes all citations made by patents granted between 1975 and 1999. (LiveJournal) is a free online blogging community where users declare friendships of each other. is a free online social network.
In group two (big graphs), is a graph obtained from the 2001 crawl performed by the WebBase crawler. is a fairly large crawl of the <.it> domain. is a social network collected from Twitter where nodes are users and edges follow tweet transmission. is a graph obtained from a 2005 crawl of the <.sk> domain. is a graph gathering a snapshot of about $100$ million pages for the DELIS project in May 2007. Finally, is a web graph underlying the ClueWeb12 dataset. All datasets can be downloaded from SNAP[<http://snap.stanford.edu/index.html>] and LAW[<http://law.di.unimi.it/index.php>].
§.§ Core Decomposition
Small Graphs. To explicitly reveal the performance of our core decomposition algorithms, we select the external-memory core decomposition algorithm $\emcore$ <cit.> and the classical in-memory algorithm <cit.>, denoted by $\imcore$ for comparison.
As shown in exp:performance (a), the total running time of $\semicores$ is $10$ times faster than that of the $\emcore$ on average. It is remarkable that $\semicores$ can be even faster than the in-memory algorithm $\imcore$. exp:performance (c) shows that algorithm $\semicores$ requires less memory than $\emcore$ and $\imcore$. Among all algorithms, $\semicore$ uses least memory since it does not rely on the numbers for all nodes comparing to $\semicores$. By contrast, $\emcore$ consumes a large amount of memory. Especially in and , $\emcore$ consumes almost the same memory size as $\imcore$. exp:performance (e) shows the I/O consumption of all algorithms except $\imcore$. $\semicores$ and $\emcore$ usually consume the least I/Os. However, due to the simple read-only data access of $\semicores$, $\semicores$ is much more efficient than $\emcore$ (refer to exp:performance (a)).
[Time Cost (Small Graphs)]
[Time Cost (Big Graphs)]
[Memory Usage (Small Graphs)]
[Memory Usage (Big Graphs)]
[I/Os (Small Graphs)]
[I/Os (Big Graphs)]
Core Decomposition on Different Datasets
[Average Time (Small Graphs)]
[Average Time (Big Graphs)]
[Average I/Os (Small Graphs)]
[Average I/Os (Big Graphs)]
Core Maintenance on Different Datasets
Big Graphs. We report the performance of our algorithms on big graphs in exp:performance (b), (d), and (f). The largest dataset contains nearly $1$ billion nodes and $42.6$ billion edges. We can see from exp:performance (a) that $\semicores$ can process all datasets within $10$ minutes except . In exp:performance, we can see that $\semicores$ totally costs less than $4.2$ GB memory to process the largest dataset . This result demonstrates that our algorithm can be deploy in any commercial machine to process big graph data. exp:performance (f) further reveals the advance of optimization in terms of I/O cost, since $\semicores$ spends much less I/Os than $\semicore$ and $\semicorep$ in all datasets.
§.§ Core Maintenance
We test the performance of our maintenance algorithms ($\semiadd$, $\semiadds$, and $\semidels$). The state-of-the-art streaming in-memory algorithms in <cit.>, denoted by $\imadd$ and $\imdel$ are also compared in small graphs. We randomly select $100$ distinct existing edges in the graph for each test. To test the performance of edge deletion, we remove the $100$ edges from the graph one by one and take the average processing time and I/Os. To test the performance of edge insertion, after the $100$ edges are removed, we insert them into the graph one by one and take the average processing time and I/Os. The experimental results are reported in exp:update.
From exp:update, we can see that $\semidels$ is more efficient than $\semiadds$ in both processing time and I/Os for all datasets. This is because $\semidels$ simply follows $\semicores$ and does not rely on the calculation of other new graph properties. From exp:update (a), we can find that our core maintenance algorithm $\semiadds$ is comparable to the state-of-the-art in-memory algorithm $\imadd$ for edge insertion. $\semidels$ is even faster than $\imdel$ for edge deletion. This is due to the simple structures and data access model used in $\semidels$. $\semiadds$ outperforms $\semiadd$ in both processing time and I/Os for all datasets.
[Vary $|V|$ ()]
[Vary $|V|$ ()]
[Vary $|E|$ ()]
[Vary $|E|$ ()]
Scalability of Core Decomposition
[Vary $|V|$ ()]
[Vary $|V|$ ()]
[Vary $|E|$ ()]
[Vary $|E|$ ()]
Scalability of Core Maintenance
§.§ Scalability Testing
In this experiment, we test the scalability of our core decomposition and core maintenance algorithms. We choose two big graphs and for testing. We vary number of nodes $|V|$ and number of edges $|E|$ of and by randomly sampling nodes and edges respectively from 20% to 100%. When sampling nodes, we keep the induced subgraph of the nodes, and when sampling edges, we keep the incident nodes of the edges. Here, we only report the processing time. The memory usage is linear to the number of nodes, and the curves for I/O cost are similar to that of processing time.
Core Decomposition. exp:scal-static (a) and (b) report the processing time of our proposed algorithms for core decomposition when varying $|V|$ in and respectively. When $|V|$ increases, the processing time for all algorithms increases. $\semicores$ performs best in all cases and is over an order of magnitude faster than $\semicore$ in both and . exp:scal-static (a) and (b) show the processing time of our core decomposition algorithms when varying $|E|$ in and respectively. When $|E|$ increases, the processing time for all algorithms increases, and $\semicores$ performs best among all three algorithms. When $|E|$ increases, the gap between $\semicores$ and $\semicore$ also increases. For example, in , when $|E|$ reaches $100\%$, $\semicores$ is more than two orders of magnitude faster than $\semicore$.
Core Maintenance. The scalability testing results for core maintenance are shown in exp:scal-dynamic. As shown in exp:scal-dynamic (a) and exp:scal-dynamic (b), when increasing $|V|$ from $20\%$ to $100\%$, the processing time for all algorithms increases. $\semidels$ performs best, and $\semiadds$ is faster than $\semiadd$ for all testing cases. The curves of our core maintenance algorithms when varying $|E|$ are shown in exp:scal-dynamic (c) and exp:scal-dynamic (d) for and respectively. $\semidels$ and $\semiadds$ are very stable when increasing $|E|$ in both and , which shows the high scalability of our core maintenance algorithms. $\semiadd$ performs worst among all three algorithms. When $|E|$ increases, the performance of $\semiadd$ is unstable because $\semiadd$ needs to locate a connected component whose size can be very large in some cases.
§ RELATED WORK
Core Decomposition. $k$-core is first introduced in <cit.>. Batagelj and Zaversnik <cit.> give an linear in-memory algorithm for core decomposition, which is presented detailed in existing. This problem is also studied for the weighted graphs <cit.> and directed graphs <cit.>. Cheng et al. <cit.> propose an I/O efficient algorithm for core decomposition. <cit.> gives a distributed algorithm for core decomposition. Core decomposition in random graphs is studied in <cit.>. Core decomposition in an uncertain graph is studied in <cit.>. Locally computing and estimating core numbers are studied in <cit.> and <cit.> respectively. <cit.> and <cit.> propose in-memory algorithms to maintain the core numbers of nodes in dynamic graphs.
Semi-external Algorithms. Semi-external model, which strictly bounds the memory size, becomes very popular in processing big graphs recently. For example, <cit.> proposes a semi-external algorithm to find all strong connected components for a massive directed graph. <cit.> gives semi-external algorithms to compute a DFS tree for a graph in the disk using divide & conquer strategy. <cit.> studies maximum independent set under the semi-external model.
§ CONCLUSIONS
In this paper, considering that many real-world graphs are big and cannot reside in the main memory of a machine, we study I/O efficient core decomposition on web-scale graphs, which has a large number of applications. The existing solution is not scalable to handle big graphs because it cannot bound the memory size and may load most part of the graph in memory. Therefore, we follow a semi-external model, which can well bound the memory size. We propose an I/O efficient semi-external algorithm for core decomposition, and explore two optimization strategies to further reduce the I/O and CPU cost. We further propose semi-external algorithms and optimization techniques to handle graph updates. We conduct extensive experiments on $12$ real graphs, one of which contains $978.5$ million nodes and $42.6$ billion edges, to demonstrate the efficiency of our proposed algorithm.
|
1511.00190
|
[2000]Primary 81R50, 58B32, 20D05
S. Majid]Shahn Majid
Queen Mary University of London
School of Mathematical Sciences, Mile End Rd, London E1 4NS, UK
We study super-braided Hopf algebras $\Lambda$ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules $\Lambda^1$ over a Hopf algebra $A$ which are quotients of the augmentation ideal $A^+$ under right multiplication and the adjoint coaction. Here super-bosonisation $\Omega=A\rbiprod\Lambda$ provides a bicovariant differential graded algebra on $A$. We introduce $\Lambda_{max}$ providing the maximal prolongation, while the canonical braided-exterior algebra $\Lambda_{min}=B_-(\Lambda^1)$ provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator $\sharp$ by super-braided Fourier transform on $B_-(\Lambda^1)$ and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on $k[S_3]$ with its 3D calculus and obeying the $q$-Hecke relation $\sharp^2=1+(q-q^{-1})\sharp$ in middle degree on $k_q[SL_2]$ with its 4D calculus. Our work also provided a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras $A$ whereby any subcoalgebra $\CL\subseteq A$ defines a sub braided-Lie algebra and $\Lambda^1\subseteq \CL^*$ provides the required data $A^+\to \Lambda^1$.
§ INTRODUCTION
Differential exterior algebras $\Omega$ on quantum groups were extensively studied in the 1990s starting with <cit.> and have a critical role as examples of noncommutative geometry more generally. However, one problem which has remained open since that era is the general construction of a Hodge star operator $\sharp$ in noncommutative geometry, even in the quantum group case. Until now the Hodge operator has been treated mainly in an ad-hoc manner in particular examples, motivated typically by $\sharp^2=\pm\id$ as a requirement, e.g. <cit.>. We also note a framework<cit.> based on a pair of differential structures and contraction with a generalised metric, and <cit.> in another q-deformation framework. By contrast our new approach depends canonically on the braided-Hopf algebra structure of the exterior algebra which applies at least for bicovariant calculi on quantum groups and covariant calculi on quantum-braided planes. Moreover, this new approach gives very different, and we think more interesting, answers than the previous approaches. Specifically, Section 3.2 includes the example of $k(S_3)$, the function algebra on the permutation group $S_3$, with its 2-cycle calculus, where $\sharp$ in middle degree obeys
\[ \sharp^3=\id\]
so that $\sharp$ as a whole has order 6. We also cover electromagnetism on this finite group as in <cit.> but using the new Hodge star and again achieving a full diagonalisation of the Laplace-Beltrami operator $\extd\delta+\delta\extd$. Similarly, Section 4.2 compute our Hodge operator $\sharp$ for $k_q[SL_2]$ with its standard 4D calculus<cit.> and in middle degree we find, rather unexpectedly, that it obeys the well-known $q$-Hecke relation
\[ \sharp^2=\id+(q-q^{-1})\sharp\]
when suitably normalised. Both of these examples are very different from requiring $\sharp^2=\pm\id$. While the nicest version of the theory assumes a central bi-invariant metric and volume form, our Fourier approach is more general as illustrated in Section 3.1 on the quantum plane $\A_q^2$.
Conceptually, we adopt a novel point of view<cit.> on what the Hodge star is even classically. Namely, at every point of a manifold $M$ of dimension $n$ the exterior algebra of differential forms has fibre the exterior algebra $\Lambda(\R^n)$ with generators $\theta_i=\extd x^i$ in local coordinates and the usual `Grassmann algebra' relations $\theta_i\theta_j+\theta_j\theta_i=0$. This is a finite-dimensional super-Hopf algebra and we can apply a super version of usual Hopf algebra Fourier transform $\CF:\Lambda\to \Lambda^*$ using the Berezin integral $\int {\rm Vol}=1$ where ${\rm Vol}=\theta_1\cdots\theta_n$. This then extends to the whole manifold and in the presence of a metric gives the classical Hodge operator $\sharp:\Omega^m\to \Omega^{n-m}$. This point of view was also recently used for the Hodge star on supermanifolds<cit.>. The same approach applies to bicovariant differential exterior algebras on Hopf algebras, where we recall that these are parallelizable via an exterior algebra of left-invariant differential forms $\Lambda$ forming a super-braided-Hopf algebra, so we can do super-Fourier transform on this.
In algebraic terms, $\Omega$ in the bicovariant case is a super-Hopf algebra equipped with a split super-Hopf algebra projection $\Omega\twoheadrightarrow H$. Hence by a super-version of the Radford-Majid theorem <cit.> one knows that $\Omega\isom A\rbiprod \Lambda$ is the super-bosonisation of $\Lambda$ as a super-braided Hopf algebra in the braided category of crossed (or Drinfeld-Radford-Yetter) modules <cit.>, where we assume that $A$ has invertible antipode. Moreover, in the standard setting $\Omega$ is generated by degrees 0,1 and hence $\Lambda$ has primitive generators $\Lambda^1$. As a result we can focus on such primitively-generated super-braided Hopf algebras $\Lambda$ and translate elements of noncommutative geometry in these terms. Particularly, the exterior derivative restricts to $\extd:\Lambda\to \Lambda$ to make $\Lambda$ into a differential graded algebra (but not within the category, $\extd$ is not a morphism) and in Section 2 we give an explicit construction of this in the case $\Lambda=\Lambda_{max}$ corresponding to the maximal prolongation of $\Omega^1$. Any other exterior algebra corresponds to a quotient of this and at the other extreme we revisit the more well-known case $\Lambda=\Lambda_{min}=B_-(\Lambda^1)$ given by the canonical (super) braided-linear space associated to an object of an Abelian braided category. This $B_\pm(\Lambda^1)$ construction appeared in the case where $\Lambda^1$ is rigid in <cit.> while quadratic primitively generated braided-Hopf algebras appeared in <cit.>. The $B_+$ construction as an algebra is often called a `Nichols algebra' cf<cit.> or `Nichols-Woronowicz algebra' cf<cit.> but we note that neither of these works considered $B_\pm$ as braided-Hopf algebras, that structure being introduced following the development of braided-Hopf algebras as above. The super-braided Hopf algebra interpretation of the Woronowicz construction of bicovariant differential exterior algebras was in <cit.> among other works. In this context the universal property of $B_-$ corresponds in some sense to the minimal relations needed to ensure Poincaré duality, a remark that will be reflected in our approach to the Hodge star. It was also observed recently <cit.> that the exterior derivative $\extd$ on a bicovariant exterior algebra is not only a super-derivation but also a super-coderivation
\[ \Delta \extd=(\extd\tens \id + (-1)^{|\ |}\tens\extd)\Delta\]
where $|\ |$ is the degree operator and $\Delta$ is the super-coproduct. This turns out to be key to going the other way of building $(\Omega,\extd)$ from data $\extd$ on $\Lambda$. Although these matters are somewhat familiar, the braided approach to the differential structure requires proofs which we provide as part of a necessary systematic treatment. Section 2.4 similarly provides a canonical construction for the differential exterior algebra on $B_+(\Lambda^1)$ as a quantum-braided plane.
As part of this, and critical for Fourier transform on $B_\pm(\Lambda^1)$, is the notion of braided-exponential in our approach to these algebras (being used notably in <cit.> to inductivively build up the quasitriangular structure of quantum groups $U_q(g)$ as a succession of $q$-exponentials). Here <cit.>
\[ B_\pm(\Lambda^1)=T_\pm\Lambda^1/\oplus_m\ker[m;\pm\Psi]!\]
\[ \exp=\sum_m ([m, \pm\Psi]!^{-1}\tens\id)\coev_{\Lambda^1{}^{\tens m}}\]
are defined in terms of braided factorials $[m,\Psi]!$ as in <cit.>. We recall this theory in Section 2.2. In the $B_+$ case we have previously proposed braided Fourier theory on the Fomin-Kirillov algebra and its super-version as Hodge star in<cit.>, but without a systematic treatment.
Our central results appear in Section 3. If we have a unique bi-invariant top degree (say of degree $n$) then super-braided-Hopf algebra Fourier transform gives us a map $\CF:\Lambda^m\to \Lambda^{*(n-m)}$ and in the presence of a quantum metric a Hodge star map $\sharp:\Lambda^m\to \Lambda^{n-m}$. This extends in the context above to the geometric $\sharp:\Omega^m\to \Omega^{n-m}$. Proposition <ref> establishes in some generality that $\sharp$ commutes with the braided antipode $S$ and is involutive in degrees $0,1,n-1,n$. This in turn follows from some general results about super-braided Fourier transform in Section 3.1 which builds on our previous diagrammatic work, particularly<cit.>. Section 3.1 also covers the Hodge operator on the well-known quantum plane $\A_q^2$, where $\Lambda=B_-(\Lambda^1)=\A_q^{0|2}$, the fermionic quantum plane.
In Section 4 we focus on the case of coquastriangular Hopf algebras $(A,\CR)$ <cit.>. In line with the braided-Hopf algebra methods of the present paper, we first present a starting point for the construction of $\Lambda^1$ itself, namely as the dual of a braided-Lie algebra. We show that every subcoalgebra $\CL\subseteq A$ is a braided-Lie algebra in the sense introduced in <cit.>. This gives a significantly cleaner result than previous attempts such as <cit.> and builds on our recent work <cit.>. Everything is worked out in detail for $\C_q[SL_2]$ recovering previously `R-matrix' formulae when we take the standard matrix subcoalgebra, including the 4D braided-Lie algebra<cit.> and the Woronowicz 4D calculus<cit.> from $\Lambda=\CL^*$. We then compute the canonical braided-Fourier transform on the latter with results as described above.
We work over a general ground field $k$ and $q\in k^\times$. In examples, we will assume characteristic zero for our calculations. We use the Sweedler notation $\Delta a=a\o\tens a\t$ for coproducts and $\Delta_R v=v\rz\tens v\co$ for right coactions (summations understood). We denote the kernel of the counit by $A^+$ and $\pi_\eps:A\to A^+$ defined by $\pi_\eps(a)=a-\eps(a)$ is the counit projection. We will make extensive use of the theory of braided-Hopf algebras <cit.> including the diagrammatic notation used for this kind of algebra in <cit.>.
§ BRAIDED CONSTRUCTION OF EXTERIOR ALGEBRAS ON HOPF ALGEBRAS
In this preliminary section we give a self-contained braided-Hopf algebra approach to bicovariant exterior algebras on Hopf algebras building on our recent work <cit.>. We recall first that a differential graded algebra means a graded algebra $\Omega=\oplus_n\Omega^n$ equipped with a super-derivation $\extd$ increasing degree by 1 and squaring to 0. The standard setting is where $\Omega^1$ is spanned by elements of the form $a\extd b$ where $a,b\in A=\Omega^0$, and $\Omega$ is generated by degrees 0,1; in this case we say that we have an exterior algebra over $A$. Any first order differential structure $(\Omega^1,\extd)$ over $A$ can be extended to a maximal prolongation.
§.§ Maximal prolongation on a Hopf algebra
When $A$ is a Hopf algebra we can ask that left and right comultiplication extends to a bicomodule structure with coactions $\Delta_L,\Delta_R$ on $\Omega^1$ and $\extd$ a bicomodule map. In this case it is known that $\Omega$ becomes a super-Hopf algebra with coproduct that of $A$ on degree zero and $\Delta_L+\Delta_R$ on degree 1, see <cit.>. In this case $\pi:\Omega\to A$ which sends all degrees $>0$ to zero and is the identity on degree 0, forms a Hopf algebra projection split by the inclusion of $A$. As a result, assuming that the antipode of $A$ is invertible and by a super version of <cit.>, we have $\Omega\isom A\rbiprod \Lambda$ where $\Lambda$ is a super-braided Hopf algebra in the braided category of right $A$-crossed (or Drinfeld-Yetter) modules. We recall that a right $A$-crossed module means a vector space $\Lambda^1$ which is both a right module and a right comodule such that
\[ \Delta_R(v\ra a)=v\rz\ra a\t\tens (Sa\o)v\co a\th,\quad\forall v\in \Lambda^1,\ a\in A. \]
In this case there is an associated map $\Psi:\Lambda^1\tens \Lambda^1\to \Lambda^1\tens \Lambda^1$ defined by $\Psi(v\tens w)= w\rz\tens v\ra w\co$. A similar map for any pair of crossed modules makes the category of these braided when the antipode $S$ is invertible. Here $A^+=\ker\eps$ is itself a right crossed module by right multiplication and $\Ad_R(a)=a\t\tens (Sa\o)a\th$ and the result in <cit.> that first order calculi $(\Omega^1,\extd)$ are classified by ad-stable right-ideals can be viewed as saying that they are classified by surjective morphisms $\varpi:A^+\to \Lambda^1$.
This point of view was recently used in <cit.> to generalise beyond the surjective case, where we do not assume that $\varpi$ is surjective. The exterior algebra is similarly given as bosonisation of a pair $(\Lambda,\extd)$ consisting of a primitively generated (by degree 1) super-braided Hopf algebra $\Lambda$ in the crossed module category equipped with a super-derivation (the restriction of $\extd$) which is a right $A$-comodule map and obeys $\extd^2=0$. This is required to have the further characteristic properties<cit.>
\begin{equation}\label{mc} \extd \varpi(a)+(\varpi\pi_\eps a\o)(\varpi\pi_\eps a\t)=0\end{equation}
\begin{equation}\label{dmod} (\extd \eta)\ra a-\extd(\eta\ra a)=(\varpi\pi_\eps a\o)\eta\ra a\t - (-1)^{|\eta|} (\eta\ra a\o) \varpi\pi_\eps a\t\end{equation}
\begin{equation}\label{dDelta} \und\Delta\extd\eta-(\extd\tens\id+(-1)^{|\ |}\tens\extd)\und\Delta\eta=(-1)^{|\eta\Bo|}\eta\Bo\rz\tens (\varpi\pi_\eps \eta\Bo\co)\eta\Bt\end{equation}
for all $a\in A^+$ and $\eta\in \Lambda$, where we underlined the braided-coproduct. It is shown in <cit.> that given such $(\Lambda,\extd)$, we obtain a bicovariant calculus $(\Omega,\extd)$ with $\extd a=a\o\varpi\pi_\eps a\t$ on degree 0 and that $\extd$ is also a supercoderivation. Here (<ref>) is called the Maurer-Cartan equation cf<cit.>. These results also clarify the surjective case:
In the case where $\varpi:A^+\to \Lambda^1$ is surjective, if $\extd$ is a super-derivation on $\Lambda$ with $\extd^2=0$ and obeys the Maurer Cartan equation then (<ref>)-(<ref>) hold.
First, it is straightforward (but in the 2nd case somewhat involved) to check that if (<ref>)-(<ref>) hold on $\omega$ then they hold on $\omega\eta$ for all $\eta\in\Lambda^1$. In both cases we use the super-derivation rule to break down $\extd(\omega\eta)$. For the second case we also need the super-braided-homomorphism property of the coproduct, in the form
\[ \und\Delta(\omega\eta)=\omega\Bo\tens \omega\Bt\eta+(-1)^{|\omega\Bt|}\omega\Bo\eta\rz\tens \omega\Bt\ra\eta\co \]
We use (<ref>) on degree 1 in the form
\[ \und\Delta \extd \eta=\extd\eta\tens 1+1\tens\extd\eta-\eta\rz\tens \varpi\pi_\eps \eta\co\]
in the start of the induction and so as to be able to similarly compute $\und\Delta(\omega\extd\eta)$. We omit further details of the induction but we still need to establish both properties on degree 1. Thus
\begin{eqnarray*} (\extd \varpi(a))\ra b-\extd(\varpi(a)\ra b)&=&-((\varpi\pi_\eps a\o)(\varpi\pi_\eps a\t))\ra b+(\varpi\pi_\eps(a\o b\o ))(\varpi\pi_\eps(a\t b\t))\\
&=& (\varpi\pi_\eps b\o) \varpi(a b\t)+ (\varpi(a b\o))(\varpi \pi_\eps b\t)\\
&=& (\varpi\pi_\eps b\o) (\varpi (a)\ra b\t)+ (\varpi(a)\ra b\o)(\varpi \pi_\eps b\t)
\end{eqnarray*}
for all $a\in A^+, b\in A$. We use $\pi_\eps(ab)=\pi_\eps(a)b+\eps(a)\pi_\eps b$ for all $a,b\in A$. For (<ref>) we check the degree 1 version as
\begin{eqnarray*} \und\Delta \extd\varpi(a)&=&-\und\Delta((\varpi\pi_\eps a\o)(\varpi\pi_\eps a\t))\\
&=&\extd \varpi a\tens 1+1\tens\extd \varpi a-\varpi\pi_\eps a\o\tens \varpi\pi_\eps a\t-(\varpi\pi_\eps a\t)\rz\tens (\varpi\pi_\eps a\o)\ra(\varpi\pi_\eps a\t)\co\\
&=&\extd \varpi a\tens 1+1\tens\extd \varpi a-\varpi\pi_\eps a\o\tens \varpi\pi_\eps a\t-\varpi\pi_\eps a\th\tens \varpi((\pi_\eps a\o)Sa\t a\fo)\\
&=&\extd \varpi a\tens 1+1\tens\extd \varpi a- \varpi \pi_\eps a\t\tens \varpi\pi_\eps ((Sa\o)a\th)
\end{eqnarray*}
for all $a\in A$. The latter part of the calculation here amounts to the identity
\begin{equation}\label{PsiDelta} \Psi(\varpi \pi_\eps\tens \varpi\pi_\eps)\Delta=(\varpi \pi_\eps\tens \varpi\pi_\eps)(\Delta-\Ad_R)\end{equation}
for the crossed module braiding.
Hence in order to construct $(\Lambda,\extd)$ in the surjective case it suffices to take (<ref>) as a definition $\extd \varpi(a):=-(\varpi\pi_\eps a\o)(\varpi\pi_\eps a\t)$ and show that this is well-defined and extends as a super-derivation of square zero. This will then make $(\Lambda,\extd)$ itself into a DGA over $k$.
Let $\varpi: A^+\twoheadrightarrow \Lambda^1$ be a surjective morphism in the category of right crossed modules. Then
\[ \Lambda_{max}= T\Lambda^1/\< (\varpi\pi_\eps\tens \varpi\pi_\eps)\Delta\ker\varpi\>\]
together with $\extd$ defined by the Maurer-Cartan equation gives a super-braided Hopf algebra in the category which is also a differential graded algebra obeying (<ref>)-(<ref>). Its bosonisation $\Omega_{max}=A\rbiprod \Lambda_{max}$ is the maximal prolongation differential calculus extending $(\Omega^1,\extd)$.
We quotient by the minimal subspace in degree 2 for which $\extd:\Lambda^1\to \Lambda^2$ is well-defined by the Maurer-Cartan equation. Let
\[ \del=\sum_{j=1}^{m}(-1)^{j+1}\Delta_j,\quad \del:A^{\tens m}\to A^{\tens(m+1)}\]
be the usual cobar coboundary, where $\Delta_j$ denotes the coproduct in the $i$'th position, and define $\extd$ on degree $m$ by $\extd .(\varpi\pi_\eps)^m=-\cdot (\varpi\pi_\eps)^{\tens (m+1)}\del$. This is well-defined for the same reason as before because $\del$ is the sum of terms each acting via the the coproduct. Clearly $\del$ is a super-derivation and squares to 0, so $\extd$ has the same features. We also need to check that we have a super-braided Hopf algebra. Since the algebra is quadratic, the main relation to check is
\[ \und\Delta ((\varpi\pi_\eps a\o)(\varpi\pi_\eps a\t)):=(\varpi\pi_\eps a\o)(\varpi\pi_\eps a\t)\tens 1+1\tens (\varpi\pi_\eps a\o)(\varpi\pi_\eps a\t)\]
\[\quad\quad\quad\quad\qquad\qquad+ \varpi\pi_\eps a\o\tens \varpi\pi_\eps a\t-\Psi(\varpi\pi_\eps a\o\tens \varpi\pi_\eps a\t)\]
vanlshes whenever $a\in \CI=\ker\varpi\pi_\eps$. This is clear for the first two terms and for the remaining two we use (<ref>) to obtain $(\varpi\pi_\eps\tens\varpi\pi_\eps)\Ad_R(a)$ (much as in the proof of Lemma <ref>) which indeed vanishes as $\CI$ is Ad-stable because $\varpi$ was a morphism. Hence by the lemma we have the required data $(\Lambda_{max},\extd)$ and obtain a bicovariant calculus after bosonisation, something one can also check directly from $\CI$ an Ad-stable right ideal and the structure of $A\rbiprod \Lambda_{max}$. It is also clear from the construction, since we imposed the minimal relations compatible with the Maurer-Cartan equation, that our calculus is isomorphic to the maximal prolongation.
§.§ Braided linear spaces
Here we take an aside to recall the theory of braided-linear spaces introduced in <cit.> but in a cleaner form as braided operators rather than braided matrices. Braided linear spaces was our term for primitively generated graded braided Hopf algebras, with particular emphasis in <cit.> on what have later been called `Nichols-Woronowicz algebras'<cit.>. If $V$ is an object in an Abelian braided category then it inherits a morphism $\Psi=\Psi_{V,V}:V\tens V\to V\tens V$ obeying the braid relations. Our setting is categorical but we use only the pair $(V,\Psi)$ and tensor powers of $V$ in the following definition.
<cit.> Let $(V,\Psi)$ be an object in an Abelian monoidal category and a braiding on it. The braided binomials here are defined recursively by
\begin{gather*}
\left[ {n\atop r}; \Psi\right] =\Psi_r\Psi_{r+1}\cdots\Psi_{n-1}(\left[ {n-1\atop r-1};\Psi\right] \tens\id)+\left[ {n-1\atop r}; \Psi\right]\tens\id,\quad \left[ {n\atop 0}; \Psi\right]=\left[ {n\atop n}; \Psi\right]=\id.
\end{gather*}
where $0<r<n$ and $\Psi_i$ denotes $\Psi$ acting in the $i,i+1$ tensor factors. We also define `braided integers'
\[ \left[n; \Psi\right]:=\left[ {n\atop 1}; \Psi\right]=\Psi_1\Psi_2\cdots\Psi_{n-1}+\left[ {n-1\atop 1}; \Psi\right]\tens\id=\id+\Psi_1+\Psi_1\Psi_2+\cdots+\Psi_1\Psi_2\cdots\Psi_{n-1}\]
and `braided factorials' $[n,\Psi]!=(\id\tens [n-1,\Psi]!)[n,\Psi]$ where $[1,\Psi]!=\id_V$. We take the convention $[0,\Psi]!=\id_{\und 1}$.
These are operator versions of binomial coefficients and generalise $q$-binomials when applied to the category of $\Z$-graded vector spaces with braiding given by powers of $q$. Relevant to us, the braided factorials also generalise symmetrizers and antisymmetrizers. We need the following main theorem about them:
\[ ([r;\Psi]!\tens[n-r;\Psi]!)\left[ {n\atop r}; \Psi\right] =[n,\Psi]!,\quad 0\le r\le n. \]
The proof in <cit.> is written in matrix terms but immediately translates as operators in our setting. In fact, these results amount to identities in the group algebra of the braid group and are best done diagrammatically. The key observations are that
\begin{equation}\label{rr-1} ([r,\Psi]\tens\id)\Psi_r\cdots\Psi_{n-1}=\Psi_r\cdots\Psi_{n-1}([r-1,\Psi]\tens\id)+\Psi_1\cdots\Psi_{n-1}\end{equation}
since the first $r-1$ terms in $[r,\Psi]$ commute with $\Psi_r\cdots\Psi_{n-1}$ and
\begin{equation}\label{functbinom} \Psi_1\cdots\Psi_{n-1}(\left[ {n-1\atop r}; \Psi\right] \tens\id)=(\id\tens\left[ {n-1\atop r}; \Psi\right] )\Psi_1\cdots\Psi_{n-1}\end{equation}
by functoriality (since the braided-binomial is a morphism) or directly by induction using Definition <ref> and repeated use of the braid relations. Using these properties, <cit.> then proves by induction on $n$ that
\[ ([r,\Psi]\tens\id)\left[ {n\atop r}; \Psi\right]=(\id\tens\left[ {n-1\atop r-1}; \Psi\right])[n,\Psi]\]
from which the theorem follows by repeated application. Also, by writing the above definitions as diagrams and turning the diagrams up-side down, we have co-binomial maps and co-integers defined by
\[ \left[ {n\atop r}; \Psi\right]' =(\id\tens\left[ {n-1\atop r-1};\Psi\right]')\Psi_1\cdots\Psi_{r-1}+\id\tens\left[ {n-1\atop r}; \Psi\right]',\quad \left[ {n\atop 0}; \Psi\right]'=\left[ {n\atop n}; \Psi\right]'=\id.\]
\[ [n,\Psi]'=1+\Psi_{n-1}+\Psi_{n-2}\Psi_{n-1}+\cdots+\Psi_1\cdots\Psi_{n-1}\]
\[ [n,\Psi]'!:=[n,\Psi]'([n-1,\Psi]'\tens\id)\cdots ([2,\Psi]'\tens\id)=[n,\Psi]!\]
\[ \left[ {n\atop r}; \Psi\right]'([r;\Psi]!\tens[n-r;\Psi]!) =[n,\Psi]!,\quad 0\le r\le n \]
where the factorials coincide by repeated use of the braid relations or because both cases can be written as $\sum_{\sigma\in S_n}\Psi_{i_1}\cdots\Psi_{i_{l(\sigma)}}$ where $\sigma=s_{i_1}\cdots s_{i_{l(\sigma)}}$ is a reduced expression in terms of simple transpositions $s_i=(i,i+1)$.
Next we consider the tensor algebra $TV$ in an Abelian braided tensor category as a direct sum of different degrees $T_nV:=V^{\tens n}$ and product given by concatenation of $\tens$. Here $T_0V=k$ or more precisely the unit object of the category. The unit $\eta$ of the algebra $TV$ is the identity map from $k\to T_0V$. Thus
\[ (V\tens \cdots \tens V)\tens (V\tens \cdots\tens V)\to V\tens\cdots \tens V\]
is the identity map with suitable rebracketing (with $\Phi$ as necessary in the general case). We also consider the identity maps
\[ \eta_n: V^{\tens n}\to T_nV\]
with $\eta_0=\eta$. Although all these maps are the identity, we are viewing them in different ways. We will consider two different braided Hopf algebra structures $T_\pm V$ on $TV$, as a braided-Hopf algebra or as a super-braided Hopf algebra in the category.
<cit.><cit.> The tensor algebra has a braided Hopf algebra/super Hopf algebra structure $T_\pm V$ with coproduct
\[\Delta_{T_nV}=\sum_{r=0}^n(\eta_r\tens\eta_{n-r})\circ\left[ {n\atop r}; \pm\Psi\right]. \]
for the two cases. The counit is $\eps_{T_n V}=0$ for all $n>0$.
This is the content of <cit.> in the free case where we impose no relations, but we rework the proof in the current more formal notations and we state the super case explicitly. We start with the linear coproduct
\[ \Delta_{T_1V}=\id_V\tens \eta+\eta\tens \id_V\]
and for $T_+V$ we extend this as a Hopf algebra in the braided category, while for $T_-V$ we extend as a super-Hopf algebra in the braided category. We do the first case; the other is exactly the same by replacing $\Psi$ by $-\Psi$. The proof is by induction assuming the formula for $\Delta_{T_{n-1}V}$,
\begin{eqnarray*} \Delta_{T_nV}&=&(\Delta_{T_{n-1}V}).(\id_V\tens\eta+\eta\tens \id_V)=\left(\eta\tens\id_{V^{\tens n-1}}+\sum_{r=1}^{n-1}\left[ {n\atop r}; \Psi\right]\right)\und\cdot(\id_V\tens\eta+\eta\tens \id_V)\\
&=&\eta\tens\id_{V^{\tens n}}+(\eta_1\tens\eta_{n-1})\Psi_1\cdots\Psi_{n-1}+\sum_{r=1}^{n-1}(\eta_r\tens\eta_{n-r})(\left[ {n-1\atop r};\Psi\right]\tens\id_V)\\
&&+ \sum_{r=1}^{n-1}(\eta_{r+1}\tens\eta_{n-1-r})\Psi_{r+1}\cdots\Psi_{n-1}\left[ {n-1\atop r}; \Psi\right]+\id_{V^{\tens n}}\tens\eta\\
&=&\eta\tens\id_{V^{\tens n}}+\id_{V^{\tens n}}\tens\eta\\
&&+\sum_{r=1}^{n-1}(\eta_r\tens\eta_{n-r})(\left[ {n-1\atop r}; \Psi\right]\tens\id_V)+ \sum_{r=1}^{n-1}(\eta_{r}\tens\eta_{n-r})\Psi_{r}\cdots\Psi_{n-1}\left[ {n-1\atop r-1}; \Psi\right]\\
&=&\eta\tens\id_{V^{\tens n}}+\id_{V^{\tens n}}\tens\eta+\sum_{r=1}^{n-1}(\eta_r\tens\eta_{n-r})\left[ {n\atop r}; \Psi\right]=\sum_{r=0}^n(\eta_r\tens\eta_{n-r})\circ\left[ {n\atop r}; \Psi\right]\\
\end{eqnarray*}
where in the first line we split off the $r=0$ part of $\Delta_{T_{n-1}V}$ and $\und\cdot$ is the braided tensor product. We then compute out the latter and for the 4th equality we renumber $r+1\mapsto r$ in the second sum and absorb the otherwise missing $r=1$. For the 5th equality we use Definition <ref> and finally combine terms to obtain the desired expression for $\Delta_{T_n}V$. Again, this is really a result at the level of the braid group algebra and can be done with diagrams. In this situation we are now ready to define the (super)Hopf algebra quotients
\begin{equation}\label{BV} B_\pm(V)=T_\pm V/\oplus_m \ker [m,\pm\Psi]!\end{equation}
as the braided-symmetric algebra and braided exterior algebra on $V$ respectively. That the coproduct descends to $B_\pm(V)$ follows immediately from Proposition <ref> and Theorem <ref>. That $\oplus_m[m,\pm\Psi]!$ is an ideal or equivalently that the product in $TV$ descends to the quotient follows from the arrow-reversed version of Theorem <ref> where the factorials are on the right. It is easy to see that when our construction is in a braided category and $\phi:V\to W$ is a morphism then $\phi^{\tens}$ (the relevant power) in each degree is a morphism $B_\pm(V)\to B_\pm(W)$ of (super)braided-Hopf algebras. This is because, by functoriality of the braidings, the braidings and braided factorials are intertwined by $\phi$ on each strand in the diagrammatic picture. The $B_+(V)$ case is also called the Nichols-Woronowicz algebra of $V$ due to the structure of the algebra, but the above description and the fact that it is a (super) braided-Hopf algebra is due to the author. The earliest examples were the braided-line and braided quantum-plane (see <cit.>) while other early examples were $U_q(n_+)$ in the work of Lusztig<cit.>.
Our own motivation to consider (<ref>) to all degrees of relations was in the case when $V$ has a right dual $V^*$. Recall that $V^{*\tens n}$ is right-dual to $V^{\tens n}$ by the nested use of $\ev_V$ and we use the same nesting convention for a duality pairing $\<\ ,\ \>$ on tensor products. In this case <cit.> the tensor algebras $T_\pm V^*$ and $T_\pm V$ are dually paired by
\begin{equation}\label{TVeval} \<\ , \>|_{T_nV^*\tens T_m V}=\delta_{n,m}\ev_{V^{\tens n}}(\id\tens [n,\pm\Psi]!) \end{equation}
and $B_\pm(V^*),B_\pm(V)$ are clearly the quotients by the kernel of the pairing.
That the product on one side is the coproduct on the other follows immediately from Proposition <ref> and Theorem <ref>. This means that $B_\pm(V^*), B_\pm(V)$ are nondegenerately paired (super) Hopf algebras in the braided category and the relations of $B_\pm(V)$ are the minimal relations compatible with this duality. The merit of this approach is that we also have the immediate result, which will need later:
<cit.> If $V$ has a right dual and $B_\pm(V)$ has a finite top degree then it has a right dual via $\ev=\<\ ,\ \>$ and coevaluation map $\coev:\und 1\to B_\pm(V)\tens B_\pm(V^*)$ is given by
\[ \exp_V:=\coev=\sum_m ( [m,\pm\Psi]!^{-1}\tens\id)\coev_{V^{\tens m}}\]
where the construction is independent of the choice of inverse image of $[m,\pm\Psi]!$.
This makes more precise the notion of braided-exponentials in <cit.> without formally assuming that the braided factorials are invertible. It was used explicitly in <cit.> to construct quasitriangular structures. If we take the well-known case $\Lambda^1=kx$ in the braided category of $\Z/(n+1)$-graded vector spaces with braiding $\Psi(v\tens w)=q^{\deg(w)\deg(v)}w\tens v$, where $x$ has degree 1, and $q$ a primitive $n+1$-th root of $1$, we have $B_+(kx)=k[x]/(x^{n+1})$ and $\exp$ here is the truncated $q$-exponential where $m!$ is replaced by $[m,q]!$ and $[m,q]=(1-q^m)/(1-q)$ are $q$-integers.
§.§ Minimal prolongation on a Hopf algebra
We now return to our setting of a Hopf algebra $A$ with invertible antipode and a surjective morphism $\varpi:A^+\to \Lambda^1$ in the braided category of right $A$-crossed modules. The following braided-Hopf algebra version of Woronowicz's construction<cit.> is largely known eg <cit.> but we provide a new direct construction for $\extd$ on $B_-(\Lambda^1)$ going through (<ref>)-(<ref>) from <cit.>. This is a rather different from the approach in <cit.>, which was to formally adjoin an inner element $\theta$ and define $\extd=[\theta, \ \}$.
Let $\varpi:A^+\to \Lambda^1$ be a surjective morphism is the category of right crossed modules and $\Lambda_{min}=B_-(\Lambda^1)$. This is a quotient of $\Lambda_{max}$ and inherits $\extd$ obeying (<ref>)-(<ref>). Its super-bosonisation $\Omega_{min}=A\rbiprod \Lambda_{min}$ recovers the Woronowicz bicovariant calculus<cit.> on $A$ associated to the Ad-stable right ideal $\ker\varpi$.
We start with the identity (<ref>) and Ad-invariance of $\CI$ implies now that $(\varpi\pi_\eps \tens \varpi\pi_\eps)\Delta(\CI)\subset \ker[2,-\Psi]=\id-\Psi$, meaning that the relations $(\varpi\pi_\eps \tens \varpi\pi_\eps)\Delta(\CI)=0$ of the maximal prolongation in Proposition <ref> already hold among the quadratic relations in $\Lambda_{min}=B_-(\Lambda^1)$. The latter is therefore a quotient of the $\Lambda_{max}$. Next we consider the coboundary $\del_m: A^{\tens m}\to A^{\tens(m+1)}$ as the proof of Proposition <ref> and regard $A$ as a right crossed module by $\Ad_R$ and $a\ra b=\pi_\eps(a)b$. Then $\pi_\eps$ becomes a morphism. Lemma <ref> below shows that $\del$ descends to $B_-(A)\to B_-(A)$ in each degree as it respects the kernels of the relevant braided-factorials. Next, $\pi_\eps$ being a crossed module morphism induces by $\pi_\eps^{\tens m}$ in degree $m$ a map $B_-(A)\to B_-(A^+)$, under which $\del$ descends to a map $-\tilde \extd:B_-(A^+)\to B_-(A^+)$ given by $-\tilde \extd\pi_\eps^{\tens m}=\pi_\eps^{\tens m}\del_m$ because the kernel of $\pi_\eps^{\tens m}$ is spanned by elements where at least one of the tensor factors is 1. When we apply $\del_m$ then every term has at least one tensor factor $1$ which is then killed by the final $\pi_\eps^{\tens m}$. This is the first cell of
\[
\xymatrix{
B_-(A) \ar[d]^{\del} \ar[r]^{\pi_\eps^{\tens}} & B_-(A^+) \ar[d]^{-\tilde \extd} \ar[r]^{\varpi^{\tens}} & B_-(\Lambda^1) \ar[d]^{-\extd} \\
B_-(A) \ar[r]^{\pi_\eps^{\tens}} & B_-(A^+) \ar[r]^{\varpi^{\tens}} & B_-(\Lambda^1)
\]
Similarly, $\varpi:A^+\to \Lambda^1$ being a morphism of crossed modules induces $B_-(A^+)\to B_-(\Lambda^1)$ given by $\varpi^{\tens n}$ in degree $n$, and $\tilde\extd$ descends to a map $\extd:B_-(\Lambda^1)\to B_-(\Lambda^1)$ defined by $\extd \varpi^{\tens m}=\varpi^{\tens m}\tilde\extd$. This is because the kernel of $\varpi^{\tens m}$ consists of terms where at least one of the tensor factors is in $\CI$. When we compute the $\tilde\extd$ of such terms using $\del_m$, either a $\Delta_j$ does not act on this tensor factor, in which case this tensor factor is present in the output of $\Delta_j$ and the whole term is killed by the action of $(\varpi\pi_\eps)^{\tens m}$, or $\Delta_j$ does act on this element. But then $\cdots\tens (\varpi\pi_\eps\tens\varpi)\pi_\eps(\Delta \CI)\tens\cdots$ is in the kernel of $\id-\Psi$ in the relevant place as seen above, hence vanishes in $B_-(\Lambda)$. We are using the fact that the kernel in each degree contains the degree 2 relations between adjacent tensor factors. In this way, $\extd$ equips $\Lambda_{min}=B_-(\Lambda^1)$ with a differential as a quotient of the construction for $\Lambda_{max}$ in Proposition <ref>. One can show that if $\Omega^1$ is inner by $\theta\in \Lambda^1$ then the same applies to $\Omega$, but we are not assuming this. The algebra structure of the bosonosation $\Omega=A\rbiprod\Lambda_{min}$ is more well-known to be isomorphic to the one in <cit.>. The following lemma was needed to complete the proof. Here $\Psi$ is the braiding for the crossed-module structure on $A$ whereby $\pi_\eps$ becomes a morphism.
Let $A$ be a Hopf algebra and $\Psi_i=\Psi$ the induced braiding
\[ \Psi(a\tens b)=b\t\tens a (Sb\o)b\th-\eps(a)b\t\tens (Sb\o)b\th\]
acting in the $i,i+1$ position of a tensor power, $\Ad_i=\Ad_R$, $\Delta_i=\Delta$ in the $i$'th position. Then
\[ [m,-\Psi]!\del_{m-1}=\left(\sum_{j=1}^{m-1}(-1)^{j+1}\Ad_1^{\tens j}\right)[m-1,-\Psi]!\]
Here $\Ad^{\tens j}$ denotes the tensor product right coaction on $j$ copies (acting here in the first position).
\[ \Psi_i\Delta_j=\Delta_j\Psi_i,\quad {\rm if}\ i<j-1,\quad \Psi_i\Delta_j=\Delta_j\Psi_{i-1},\quad {\rm if}\ i>j+1\]
since the operators act on different tensor factors, just the numbering changes in the 2nd case. We also find by direct computation in the Hopf algebra that
\[ \Psi_i\Delta_i=\Delta_i-\Ad_i,\quad \Psi_i\Delta_{i-1}= (\Delta\tens\Ad)_{i-1}-{\Ad}_i,\quad \Psi_i(\Delta\tens\Ad)_i=(\Delta_{i+1}-\Ad_i)\Psi_i\]
where $\Ad=\Ad_R$ and $\Delta\tens\Ad$ is the tensor product right coaction. As a warm-up, using these relations, we show
\begin{eqnarray*} [3,-\Psi]\del_2&=&(1-\Psi_1+\Psi_1\Psi_2)(\Delta_1-\Delta_2)\\
\end{eqnarray*}
Starting with this, we next prove by induction that
\begin{equation}\label{PsiDeltarelns}[m,-\Psi]\del_{m-1}=(\Ad_1-\Delta_1+\del_{m-1})[m-1,-\Psi].\end{equation}
Assuming this for $m-1$ in the role of $m$, for the 2nd equality,
\begin{eqnarray*} [m,-\Psi]\del_{m-1}&=&[m-1,-\Psi]\del_{m-2}+[m-1,-\Psi](-1)^m\Delta_{m-1}+(-1)^{m-1}\Psi_1\cdots\Psi_{m-1}\del_{m-1}\\
&&-\Psi_1\cdots\Psi_{m-1}\Delta_{m-1}+ (-1)^{m-1}\Psi_1\cdots\Psi_{m-1}\del_{m-2}\\
&=&(\Ad_1-\Delta_1+\del_{m-1})[m-2,-\Psi]+\Psi_1\cdots\Psi_{m-2}\Ad_{m-1}+ (-1)^{m-1}\Psi_1\cdots\Psi_{m-1}\del_{m-1} \end{eqnarray*}
where we picked out and computed the $\Psi_1\cdots\Psi_{m-1}\Delta_{m-1}$ term from the sum in $\del_{m-1}$. Looking now at the
last expression, we compute
\begin{eqnarray*}\Psi_1\cdots\Psi_{m-1}\del_{m-2}&=&\sum_{j=1}^{m-2}(-1)^{j+1}\Psi_1\cdots\Psi_{j+1}\Delta_j\Psi_{j+1}\cdots\Psi_{m-2}\\
&=&(-1)^m\Psi_1\cdots\Psi_{m-2}\Ad_{m-1}-(\Ad_1+\sum_{j=2}^{m-1}(-1)^{j+1}\Delta_j )\Psi_1\cdots\Psi_{m-2}
\end{eqnarray*}
where the $\Ad$ terms cancel between the sum and the displaced sum except for the top term of one sum and the bottom term of the other. In the $\Delta$ sum
all the indices of $\Psi$ are two or more smaller than the index of $\Delta$ so commute to the right. Combining with our previous calculation, we have
\begin{eqnarray*} [m,-\Psi]\del_{m-1}&=& (\Ad_1-\Delta_1+\del_{m-1})[m-2,-\Psi]\\
which proves (<ref>).
Next we use this result as initial base for induction on $i$ in a formula
\begin{eqnarray}\label{PsiDeltai}
&&\nquad [m-i+1]\cdots [m]\del_{m-1} =(\sum_{j=1}^{i-1}(-1)^{j+1}\Ad_1^{\tens j} \nonumber \\
&&+\left((-1)^{i+1}A_1^{\tens i}+\sum_{j=i+1}^{m-1}(-1)^{j+1}\Delta_j\right)[m-i])[m-i+1]\cdots[m-1]
\end{eqnarray}
where $[m]\equiv[m,\Psi]$ for brevity and the nesting is rightmost as for braided factorials so $[m-1]\equiv \id\tens[m-1,-\Psi]$. The case $i=1$ is (<ref>) which we have already proven while the case $i=m-1$ or $i=m$, suitably interpreted in the sense of absent sums or products when out of range, proves the lemma. We use identities
\[ \Psi_i\Ad_j^{\tens i-j}=\Ad_j^{\tens i}-\Ad_i,\quad \Psi_i\Ad_j^{\tens k}=\Ad_j^{\tens k}\Psi_{i-1},\quad i>j+k\]
where the commutation relation is due to acting in different spaces, with renumbering due to the notation. The first equation is a direct computation. One also has
\[ \Psi_i\Ad_j^{\tens k}=\Ad_j^{\tens k}\Psi_i,\quad i<j-1,\quad \Psi_i\Ad_j^{\tens k}=\Ad_j^{\tens k}\Psi_i,\quad j\le i<j+k-1\]
which we do not need right now, in the first case due to different tensor products and in the second case because $\Psi$ is a morphism in the crossed module category and hence commutes with $\Ad$ applied to tensor powers that include those on which $\Psi$ acts. Assuming (<ref>) for $i-1$ in the role of $i$, what we need to show to prove (<ref>) for $i$ is
\begin{eqnarray*}
&&[m-i+1]\left(\sum_{j=1}^{i-2}(-1)^{j+1}\Ad_1^{\tens j}+\left((-1)^{i}A_1^{\tens i-1}+\sum_{j=i}^{m-1}(-1)^{j+1}\Delta_j\right)[m-i+1]\right)\\
&&= \left(\sum_{j=1}^{i-1}(-1)^{j+1}\Ad_1^{\tens j}+\left((-1)^{i+1}A_1^{\tens i}+\sum_{j=i+1}^{m-1}(-1)^{j+1}\Delta_j\right)[m-i]\right)[m-i+1].
\end{eqnarray*}
Now, the first sum commutes with $[m-i+1]$ since on the left this is $1-\Psi_i+\Psi_i\Psi_{i+1}+\cdots+(-1)^{m-i}\Psi_i\cdots\Psi_{m-1}$ due to the right-most embedding. These commute past the $\Ad_1^{\tens j}$ getting changed to $[m-i+1]$ embedded on the right (where the numbering is reduced by one). Hence the first term on the left is
$ \sum_{j=1}^{i-2}(-1)^{j+1}A_1^{\tens j}[m-i+1]$. Next $\Psi_i\Psi_{i+1}\cdots\Psi_{m-1}\Ad_1^{\tens i-1}=(\Ad_1^{\tens i}-\Ad_i)\Psi_i\Psi_{i+1}\cdots\Psi_{m-2}$ as the $\Psi_{i+1}$ and higher commute, reducing index by 1, while $\Psi_i$ computes as shown. Hence the middle terms gives
\[[m-i+1](-1)^i\Ad_1^{\tens i-1}[m-i+1]=(-1)^i\Ad_1^{\tens i-1}[m-i+1]+(-1)^{i+1}(\Ad_1^{\tens i}-\Ad_i)[m-i][m-i+1],\]
the first term of which completes our previous sum to give the first desired term. Accordingly we need only show for the remaining term that
\[ [m-i+1]\sum_{j=i}^{m-1}(-1)^{j+1}\Delta_j=\left((-1)^{i+1}\Ad_i+ \sum_{j=i+1}^{m-1}(-1)^{j+1}\Delta_j\right)[m-i].\]
But this is just the same identity (<ref>) already proven but for $[r]\del_{r-1}$, i.e. $r=m-i+1$ in the role of $m$, for the $m$ tensor factors numbered $i,\cdots,m$. This completes our proof of (<ref>) for all $i$ and proves the lemma.
This fleshes out the braided-Hopf algebra interpretation of the Woronowicz exterior algebra on a Hopf algebra<cit.> using <cit.> for the direct treatment of $\extd$.
§.§ Differential calculi on braided linear spaces
For completeness, we give another braided construction namely the exterior algebra $\Omega(B)$ on a Hopf algebras $B$ in a braided Abelian category. This includes braided symmetric algebras $B=B_+(\Lambda^1)$ as above generated canonically by an object $\Lambda^1$. The further data we will need is a surjective morphism $\varpi:B\to \Lambda^1$ in the category such that
\begin{equation}\label{ederiv} \varpi\circ\cdot=\eps\tens\varpi+\varpi\tens\eps\end{equation}
This data arises naturally as follows: suppose $B^\flat$ is a (possibly degenerately) dually paired braided-Hopf algebra from the right (so the pairing is $\ev:B\tens B^\flat\to \und 1$) and $\CL$ a rigid primitive sub-object $\CL\subset B^\flat$ (so that the coproduct restricted to $\CL$ is the additive one). We view the duality pairing restricted to a map $B\tens\CL\to \und 1$ as a map
\[ \varpi:B\to \Lambda^1=\CL^*,\quad \varpi=(\ev\tens\id)(\id\tens\coev_{\CL})\]
which then obeys (<ref>). This is surjective if there does not exist $\eta\in \CL$ which pairs to zero with all of $B$. In the case of $B=B_+(\Lambda^1)$ or any other graded braided Hopf algebra of the form $B=\und 1\oplus \Lambda^1\oplus B_{>1}$ generated in degree 1 by an object $\Lambda^1$, we simply take $\varpi:B\to \Lambda^1$ as the projection to degree 1.
Let $B$ be a Hopf algebra in an Abelian braided category and $\varpi:B\to \Lambda^1$ a surjective morphism obeying (<ref>). Then
\[ \Omega= B\und\tens\Lambda,\quad \Lambda= T\Lambda^1/\<{\rm image}(\id+\Psi_{\Lambda^1,\Lambda^1})\>\]
\[ \extd|_B=(\id\tens\varpi)\Delta,\quad \extd|_\Lambda=0\]
is a differential exterior algebra on $B$ in the category (one in which all structure maps are morphisms).
\[ \includegraphics[scale=0.5]{diffplane.pdf}\]
Diagrams in the proof of Proposition <ref> for quantum differentials on braided planes
The proof is done diagrammatically in Figure <ref> and applies generally but for convenience of exposition we also refer to concrete elements. The braided tensor product $B\und\tens\Lambda^1$ in concrete terms means $(b\tens v)(c\tens w)=b\Psi(v\tens c)w$ and featured already in the definitiion of a braided-Hopf algebra. Part (a) computes $\extd(bc)$ using the braided coproduct homomorphism property and (<ref>). Using the counit axioms and $\varpi$ a morphism we obtain $b\extd c$ for the first term and $(\extd b)c $ for the second when we remember the braided tensor product. Part (b) checks that $\extd$ extends as a graded derivation with respect the braided tensor product. We compute
\[ \extd(\omega b)+\omega\extd b=( \id\tens\cdot)(\id+\Psi^{-1})({\rm something})\]
hence vanishes in $B\tens\Lambda^2$ which agrees with $\extd\omega=0$ for all $\omega\in \Lambda^1$. The same applies starting in $\Lambda\tens B$
This gives a differential structure on our braided-symmetric algebras $B_+(V)$ regarded as noncommutative spaces. If the category is the comodules of a coquastriangular Hopf algebra, for examples, our construction is covariant in that all structure maps are comodule maps. Also note that if $\{e_i\}$ is a basis of $V$, we have explicitly
\[ \extd = \del^i (\ )e_i\]
where $\del^i$ are the (right handed) braided partial derivatives defined by
\[ \Delta b= 1\tens b+ \del^i b\tens e_i+\cdots \]
They are given explicitly at the level of the tensor algebra by
\[ \extd(v_1\tens ...\tens v_n)=(\eta_{n-1}\tens\eta_1)\left[{n\atop n-1},\Psi\right](v_1\tens\cdots\tens v_n)\]
\[ \left[{n\atop n-1},\Psi\right]=1+\Psi_{n-1}+\Psi_{n-1}\Psi_{n-2}+\cdots+\Psi_{n-1}\cdots\Psi_1\]
where the last tensor factor of the result is viewed in $\Lambda^1$.
If we take the quadratic version $S(V):=B_+^{quad}(V)=TV/\<\ker(\id+\Psi)\>$ then our above construction gives
\[ \Omega(S(V))=S(V)\und\tens S(V^*)^!,\quad \extd v=1\tens v\]
where $!$ denotes the Koszul dual. If a quadratic algebra on a vector space $W$ has relations $R\subset W\tens W$ as the subspace being set to to zero then its Koszul dual is the quadratic algebra on $W^\flat$ with relations $R^\perp\subset W^\flat\tens W^\flat$. This is normally done in the category of vector spaces but we do it here in a braided category using the right dual so that $W^\flat=V$.
We let $B=\A_q^2=B_+(V)$ be the quantum plane associated to the standard corepresentation $V={\rm span}\{x,y\}$ in the braided category of right $k_q[GL_2]$-comodules with $q^2\ne 1$. Here
\[ \Psi(x\tens x)= q^2 x\tens x,\ \Psi(x\tens y)=q y\tens x,\ \Psi(y\tens x)=q x\tens y+(q^2-1)y\tens x,\ \Psi(y\tens y)=q^2y\tens y\]
is given by a particular non-standard normalisation of the usual $\CR$ on $k_q[GL_2]$ (one that does not descend to $k_q[SL_2]$). The kernel of $\id+\Psi$ gives us the relations $yx=qxy$ of the quantum plane since $(\id+\Psi)(y\tens x-q x\tens y)=0$. The algebra $\Lambda=\A_q^{0|2}=\Lambda(V)$ is the fermionic quantum plane
\[ \extd x\wedge \extd x=0,\quad \extd y\wedge\extd y=0,\quad \extd y\wedge\extd x=-q^{-1}\extd x\wedge\extd y\]
where this time the same basis is denoted $\{\extd x,\extd y\}$ as a basis of $\Lambda^1=V$ and one can check for example that $(\id+\Psi)(\extd x\tens\extd y)=\extd x\tens \extd x+q\extd y\tens\extd x$ from the stated braiding. Indeed, it known that $\A_q^2$ and $\A_q^{0|2}$ as Koszul dual as first pointed out by Manin<cit.>. The differential on $v\in V\subset B$ is $\extd v=1\tens v$ i.e., $v$ viewed in the $\Lambda^1$ copy of $V\subset\Lambda(V)$. The relations between $B$ and $\Lambda(V)$ are the braided tensor product so $(1\tens v)(w\tens 1)=\Psi(v\tens w)$. This again comes from the same braiding as above but viewed now as defining the relations
\[ (\extd x)x=q^2x\extd x,\quad (\extd x)y=q y\extd x,\quad (\extd y)x=qx\extd y+(q^2-1)y\extd x,\quad (\extd y)y=q^2y\extd y.\]
One can check for example that $\extd(yx-qxy)=0$ as it should. By construction, this exterior algebra on the quantum plane is $k_q[GL_2]$-covariant. The associated partial derivatives are
\[ \del^1(x^my^n)=[m,q^2]x^{m-1}y^n q^n,\quad \del^2(x^my^n)=x^m [n,q^2]y^{n-1}\]
using the braided coproduct on general monomials computed in <cit.> from the braided-integers $[n,\Psi]$. The partial derivatives here were first found by Wess and Zumino in another approach. They are naturally `braided right derivations' with an extra $q^n$ in the first expression, in order that $\extd$ is a left derivation, acting as braided $q^2$-derivatives in each variable. One can check that $\del^2\del^1=q\del^1\del^2$ as operators, also as per the general theory in <cit.>.
This reworks the treatment of quantum-braided planes and their differentials in <cit.><cit.> now as an example of our above canonical construction based on $B_+(\Lambda^1)$, as opposed to a compatible pair of R-matrices $R,R'$ as previously.
§ BRAIDED FOURIER TRANSFORM AND APPLICATION TO HODGE THEORY
Fourier transform on Hopf algebras is part of their classical literature. It was extended to braided-Hopf algebras in <cit.> and related works and applied to braided linear spaces in <cit.>, though not the ones we consider here. We used diagram proofs and will do so again, while another work from that era is <cit.>. We first explain the general (super) formulation and then apply it to the Hodge operator, including $k(S_3)$ as an example.
§.§ Super-braided Fourier theory
In any braided category $\CC$ and $B\in \CC$ a braided Hopf algebra dually paired with a braided Hopf algebra $B^\star$, we have three actions which we will consider and which we collect in Figure 1 in a diagrammatic notation<cit.>. Diagrams are read as operations flowing down the page, with tensor products and the unit object $\und 1$ suppressed. Two strands flowing own and merging denotes the product and one strand flowing down and splitting denotes the coproduct. As in Section 2 when discussing duals, we assume a pairing $\ev:B^\star\tens B\to \und 1$ which we can write diagrammatically as $\cup$ and with respect to which the product on one side is adjoint to the coproduct on the other. The counit of $B$ is also adjoint to the unit $\eta:\und 1\to B^*$ in the sense $\ev(\eta\tens ( ))=\eps_B$ and vice versa $\ev(( )\tens \eta)=\eps_{B^\star}$, where we have canonical isomorphisms $\und 1\tens B\isom B$ etc which we use. In a concrete $k$-linear setting we can suppose that $\und 1=k$ and $\eta(1)=1$ to simplify the above. Reg makes $B$ a right $B^\star$ module algebra in the braided category. The principal ingredient of Reg here is actually a left action $\vdash$
making $B$ a left $B^\star$-module algebra in the braided category. Similarly, we have a straightforward right action $\dashv$ under which $B^\star$ is a right $B$-module algebra <cit.>.
\[ \includegraphics[scale=.6]{FTdiags.pdf}\]
Diagrammatic definitions of relevant actions, Fourier transform $\CF$ and adjoint Fourier transform $\CF^*$ on a braided-Hopf algebra $B$ with right dual $B^\star$.
We also need the notion of a left integral and the simplest thing is to require a morphism $\int: B\to \und 1$ in the sense $(\id\tens\int)\Delta=\eta\tens\int$. However, we do not want to be too strict about this. For example, for the finite anyonic braided line $B=k[x]/(x^{n+1})$ in the braided category of $\Z/(n+1)$-graded spaces with braiding given by an $n+1$-th root of 1 and $|x|=1$, the obvious $\int x^m=\delta_{m,n}$ is not a morphism to $\und 1$. Our approach is to live with this and not necessarily assume any morphism properties; we can still use the diagrammatic notation but be careful not to pull the map through any braid crossings. A more formal approach is to view it as a morphism $B\to K$ where $K=k$ taken with degree $n$ in the case of the anyonic braided line. The uniqueness of the integral when it exists is similar to the Hopf algebra case (see <cit.> for a formal proof).
For Fourier transform we need not only that $B^\star$ is dually paired but that $B$ is actually rigid with dual object $B^*$. Again, this is a very strong assumption, analogous to finite-dimensionality of $B$ and amounting to this in the typical $k$-linear case. It means that there is a coevaluation map $\exp={\rm coev}: \und 1\to B\tens B^*$, denoted by $\cap$ in the diagrammatic notion, which obeys the well-known `bend-straightenning axioms' with respect to $\cup$. We similarly require a right integral which is not necessarily a morphism $\int^\star: B^\star\to \und 1$. We can live with this or suppose formally that $\int^\star:B^\star\to K^*$ where $K^*\tens K=\und 1= K\tens K^*$ as objects. In our $k$-linear setting this will be by the identification with $k$. The theory below could be generalised to include some infinite-dimensional cases or else these could be treated formally eg in a graded case with $B^*$ a graded dual, each component rigid and the result a formal power series in a grading parameter.
<cit.> Let $B$ be a Hopf algebra in a braided category with $B$ rigid and $\int$ a left integral as above. Then $\CF:B\to B^\star$ defined in Figure 1 is called the braided Fourier transform. We similarly define a dual Fourier transform $\CF^\star:B^\star\to B$ if $B^\star$ has a right integral.
These maps are no longer morphisms if the integrals are not, or one can say more formally that $\CF:B\to K\tens B^\star$ and $\CF^\star: B^\star\to K^*\tens B^\star$. The following extends and completes <cit.>.
In the setting of the definitions above
\[ \CF\circ {\rm Reg}= \cdot \circ(\CF\tens\id), \quad \dashv (\CF\tens\id)=\CF\circ\cdot\]
Moreover, if $\int^*$ is a right integral on $B^\star$ then
\[ \CF^\star\CF=\mu S,\quad \mu:=(\int\tens\int^*)\exp\]
If the integrals are both unimodular and morphisms then $\CF\CF^\star=\mu S$ and $[\CF,S]=0$ when $\mu$ is invertible (see Figure 2 for the general case).
Here $\CF {\rm Reg}$ and $\CF^\star\CF$ are already covered in <cit.> so we do not repeat all the details here. We recall only the diagram proof for $\CF^*\CF$ using the lemma in <cit.> at the first equality in Figure 2 and note that we did not need to assume that $\int,\int^*$ are morphisms to $\und 1$ as in <cit.> as long as we keep the integrals to the left. The second line now uses the same lemma but this time on $B^\star$ to compute $\CF\CF^*$ as shown provided $\int^*$ is also a left integral so that the lemma applies and $\int$ is also a right integral. If $\int,\int^*$ are morphisms to $\und 1$ so we can take them through braid crossings to obtain $\mu S$ and then $\mu \CF S=\CF\CF^\star\CF=\mu S\CF$. The general result $\dashv (\CF\tens\id)= \CF\circ\cdot$ follows more simply from the duality pairing and associativity of the product of $B$.
\[ \includegraphics[scale=.45]{FTstarproof.pdf}\]
Diagrammatic computations of $\CF^\star\CF$ and $\CF\CF^\star$ in Proposition <ref>
The map $\CF^\star$ here is a right-integral version of the theory which is being used to define the adjoint Fourier transform and converted to a left version via $\Psi$. The braided antipode $S$ plays the role of the minus sign familiar in classical Fourier theory and $\mu$ plays the role of $2\pi$. If $\mu$ and $S$ are invertible then the stated results imply that $\CF$ is invertible at least in the $k$-linear setting (with $\CF^{-1}=S^{-1}\CF^\star$ in the unimodular trivial morphism case). Also, if we compose $\CF$ with $S$ then the first property above becomes
\begin{equation}\label{SFvdash} S\CF\circ\vdash =\cdot(\id\tens S\CF).\end{equation}
For the elementary example of $B=k[x]/(x^{n+1})$ we have ${\rm Vol}=x^n$ as discussed, $B^\star=k[y]/(y^{n+1})$ where $|y|=-1$ and ${\rm Vol}^*=y^n$ for the top form and $\exp=\sum_{m=0}^n x^m\tens y^m /[m,q]!$ as mentioned at the end of Section 2.2. Here $[m,q]=(1-q^m)/(1-q)$ and $q$ is a primitive $n+1$-th root of 1. We have
\[ \CF(x^m)={y^{n-m}\over [n-m,q]!},\quad \CF^\star(y^m)={q^{(n-m)^2} x^{n-m}\over [n-m,q]!},\quad \mu=[n,q]!^{-1}\]
as well as $S x^m=(-x)^m q^{m(m-1)\over 2}$ and ditto with $x$ replaced by $y$. One can see that $\CF^\star\CF=\mu S$ using $q^{n(n+1)\over 2}=(-1)^n$. Due to the nontrivial braidings of the integrals, however, the right hand side in Figure 2 gives
\[ \CF\CF^\star=q^{2 D+1}\mu S\]
where $D$ is the monomial degree operator. The same method as in the proof above now gives us $\CF S=q^{2D+1}S\CF$ or equivalently $S\CF=\CF S q^{2D+1}$, which one may verify from the stated $\CF,S$.
The fermionic quantum plane in Example <ref> is a super-braided Hopf algebra, i.e. we take $B=\A_q^{0|2}=B_-(\Lambda^1)$ in the category of $k_q[GL_2]$-comodules. If we now denote the braiding in Example <ref> as $\Psi_+$ adapted to $B_+$ as a braided-Hopf algebra, we now take a different normalization $\Psi=\Psi_-=q^{-2}\Psi_+$ (i.e. induced by another differently normalised coquasitriangular structure) so that $\ker(\id-\Psi_-)={\rm image}(\id+\Psi_+)$. The $\Psi_\pm$ here are $q^{\pm 1}$ times the braiding in the standard $q$-Hecke normalisation. There are no new relations in higher degree so $B_-(\Lambda^1)=B_-^{quad}(\Lambda^1)$. For brevity we let $e_1=\extd x$, $e_2=\extd y$ then $\Psi(e_2\tens e_1)=q^{-1}e_1\tens e_2+\lambda e_2\tens e_1$ etc. in the new normalisation,
where $\lambda=1-q^{-2}$. We now develop $\A_q^{0|2}$ as a super-braided Hopf algebra with $e_i$ primitive and underlying braiding $\Psi$ (meaning we actually transpose with super-braiding $\Psi_{sup}$ having additional $\pm$ factors according to the monomial degrees). This implies $S(e_1e_2)=q^{-2}e_1e_2$ as well as $S(1)=1$, $S(e_i)=-e_i$. On the dual side we have $B^\star=B_-(\Lambda^1{}^*)$ with a dual basis of generators $f^1,f^2$, underlying braiding
\[ \Psi(f^i\tens f^i)=f^i\tens f^i,\quad \Psi(f^1\tens f^2)=q^{-1}f^2\tens f^1+\lambda f^1\tens f^2,\quad \Psi(f^2\tens f^1)=q^{-1}f^1\tens f^2,\]
relations $f^2 f^1=-q f^1 f^2$ and $S(f^1f^2)=q^{-2}f_1f_2$. There is up to scale a unique top degree in each case, namely ${\rm Vol}=e_1e_2$ and ${\rm Vol}^*=f^1f^2$ and we find $\<{\rm Vol}^*,{\rm Vol}\>=\ev(f^1 \tens f^2,[2,-\Psi](e_1\tens e_2)\>=-q^{-1}$, so that
\[ \exp=1\tens 1+\sum_{i=1}^2 e_i\tens f^i - q{\rm Vol}\tens{\rm Vol}^*\]
We define integrals via $\int {\rm Vol}=1$ and $\int^*{\rm Vol}^*=1$ but note that these are not morphisms. Rather we use braidings
\[ \Psi(f^1\tens e_1)=e_1\tens f^1+(1-q^2)e_2\tens f^2,\quad \Psi(f^2\tens e_2)=e_2\tens f^2,
\quad \Psi(f^i\tens e_j)=qe_j\tens f^i\]
for $i\ne j$ (these are obtained from the 2nd inverse $\tilde R$ as in <cit.> for $R$ normalised to our case) to find $\Psi(f^i\tens {\rm Vol})=q {\rm Vol}\tens f^i$ and hence $\Psi({\rm Vol}^*\tens{\rm Vol})=q^2 {\rm Vol}\tens{\rm Vol}^*$. We similarly have $\Psi(f^i\tens {\rm Vol}^*)=q{\rm Vol}^*\tens f^i$. From these it is clear that $\int$ and $\int^*$ are not morphisms in the underlying comodule category. Again, there can be further signs according to the super degrees for the actual super-braiding $\Psi_{sup}$ when we read diagrams in the super-braided case. In particular, we find
\[ \Psi_{sup}^{-1}\exp=1\tens 1- f^1\tens e_1-q^2 f^2\tens e_2-q^{-1}{\rm Vol}^*\tens{\rm Vol}\]
needed in the computation of $\CF^\star$. We now read off from the diagrammatic definitions in Figure 1,
\[ \CF\begin{cases}1\cr e_1\cr e_2\cr {\rm Vol}\end{cases}=\begin{cases}-q{\rm Vol}^*\cr f^2\cr -q^{-1}f^1\cr 1\end{cases},\quad
\CF^\star\begin{cases}1\cr f^1\cr f^2\cr {\rm Vol}^*\end{cases}=\begin{cases}-q^{-1}{\rm Vol}\cr -q^2 e_2 \cr q e_1\cr 1\end{cases},\quad\mu=-q.\]
One can verify that $\CF^\star\CF=\mu S$ as it must by Proposition <ref>. We also have
\[ \CF\CF^\star=\mu q^{2(D-1)}S\]
where $D$ is the monomial degree and one can check that this agrees with the lower line in Figure 2 where we use the above computations to read off the right hand side. In this case $\CF S=q^{2(D-1)}S\CF$ or $S\CF=\CF S q^{2(D-1)}$ as one can verify from the stated form of $\CF,S$.
A similar approach can be used for other quantum planes to express their differential exterior algebras as super-braided Hopf algebras with possibly a different underlying coquastriangular structure from one used for the coordinate algebra as a braided-Hopf algebra. In the presence of an invariant quantum metric we reproduce the otherwise ad-hoc approach to $q$-epsilon tensors and Hodge theory on braided-quantum planes in <cit.>. The $f^i$ generate antisymmetric vectors and $\dashv, \vdash$ define an interior product connected to the exterior algebra product via $\CF$. This example should be seen as a warm-up to Section 3.2 where we look at bicovariant differentials on Hopf algebras themselves. As illustrated here, the actual theory is read off the diagrams with the appropriate braiding including signs. We could indeed shift all constructions to this new super-braided category and say that the above example is an ordinary braided-Hopf algebra there, but we not do so since there will normally be other (bosonic) objects also of interest in the original category. In our context the nicest case is where $\int,\int^*$ are morphisms to $\und 1$ when viewed in the original category but do not necessarily respect the super-degree, for example they could be odd maps in the super-sense in which case they are not morphisms in the super version with extra signs (so we need the slightly more general picture as above). We assume they have the same parity of support (both odd or both even maps). Then $\CF,\CF^\star$ also have this party.
If $\int,\int^*$ are unimodular, morphisms in the underlying category and of the same parity $p$ then $\CF\CF^\star=(-1)^p\mu S$. When $\mu,S$ are invertible we have $\CF S=(-1)^p S\CF$ and $\CF^\star=\mu S \CF^{-1}$
For $\CF\CF^\star$ we have to compute the right hand side of the lower diagram in Figure 2, which now has extra signs. We can still bring out $\mu'=(\int^*\tens\int)\Psi_{sup}^{-1}\exp$ since any signs from crossing the $\int$ leg cancel with signs from crossing the $\int^\star$ leg by our assumptions. Next, we can lift $\int^\star$ through the crossing at the price of $(-1)^p$ in computing $\mu'$. We already have $\CF^\star\CF=\mu S$ from Proposition <ref> and can then conclude the rest.
The behaviour of $\CF$ with respect to ${\rm Reg},\dashv$ has an unchanged form as these statements do not involve additional transpositions, except that the actions themselves are computed for the super-braided Hopf algebra eg with the super-braided coproduct and hence the super-braided Leibniz rule expressed in super-braided module algebra structures. The property (<ref>) becomes
\begin{equation}\label{superSFvdash} S\CF\circ\vdash=\cdot((-1)^{pD}\tens S\CF)\end{equation}
due to the crossing of the first input on the right hand side with the integral in $\CF$.
§.§ Hodge theory on Hopf algebras
We are now going to compute our super-Fourier theory for $B=\Lambda_{min}=B_-(\Lambda^1)$ where $\Lambda^1$ is a rigid object in the braided category of right $A$-crossed modules and $A$ is a Hopf algebra with invertible antipode. Here $B$ is a super-braided Hopf algebra in the category and we assume it has a top degree component $K$ of dimension 1, i.e. up to scale a unique top form ${\rm Vol}\in B_-(\Lambda^1)$. This gives us a unimodular integral $B\to k$ by $\int {\rm Vol}=1$ and zero for lower degrees. To see this, note that the formula in Proposition <ref> ensures that $\und\Delta {\rm Vol}={\rm Vol}\tens 1+1\tens{\rm Vol}$ plus terms of intermediate degree, and we never reach the top degree on applying $\und\Delta$ to lower degree. One can think of this more formally as a morphism $B\to K$ with some possibly non-trivial generator. We also have an identification $B^\star=B_-(\Lambda^1{}^*)$ by extending the duality pairing $\Lambda^1{}^*\tens\Lambda^1$ as a braided-Hopf algebra pairing, given that this is now non-degenerate after quotienting by the relations of $B_-$ as explained in Section 2.2.
Hence we obtain a unimodular integral on this too. In the nicest case, the top forms ${\rm Vol},{\rm Vol}^*$ of degree $n$ (say) span the trivial object $\und 1$ so that $\int,\int^*$ are morphisms to $\und 1$ but of parity $n$ mod 2, so we are in the setting of Corollary <ref>.
Next, in non-commutative geometry a metric is $g\in \Omega^1\tens_A\Omega^1$ with an inverse $(\ ,\ ):\Omega^1\tens_A\Omega^1\to A$. One can show that in this case $g$ must be central. Normally, one also requires the metric to be `quantum symmetric' in the sense of the product $\wedge(g)=0$ in $\Omega^2$. We are interested in left-invariant metrics where $g\in \Lambda^1\tens\Lambda^1$.
A bi-invariant metric on a Hopf algebra $A$ with bicovariant calculus is equivalent to an $A$-crossed module isomorphism $g:\Lambda^{1*}\isom \Lambda^1$. The metric is quantum symmetric if and only if $\Psi(g)=g$.
The metric being bi-invariant means that it is an element $g\in \Lambda^1\tens\Lambda^1$ which is invariant under the coaction $\Delta_R$ on the tensor product. The existence of a bimodule map $(\ ,\ )$ requires $g$ to be central which in turn requires that $g$ is invariant under the crossed module right action $\ra$ (since this determines the cross product of $A\rbiprod \Lambda$). So a metric is equivalent to a morphism $\und 1\to \Lambda^1\tens\Lambda^1$ in the crossed module category. Evaluation from the left makes this equivalent a morphism as stated, which we also denote $g$. Here $\Lambda^*$ is again a right crossed module in the usual way (via the antipode). Clearly $\wedge(g)=0$ if and only if $g\in \ker[2,-\Psi]=\ker(\id-\Psi)$ according to the relations of $B_-(\Lambda^1)$.
Given a bi-invariant metric we therefore have $B_\pm(\Lambda^1{}^*)\isom B_\pm(\Lambda^1)$ hence combined with the above remarks in the finite-dimensional case, an isomorphism which we also denote $g:B_\pm(\Lambda^1)^*\to B_\pm(\Lambda^1)$. We are now ready to define the Hodge operator, using the $B_-$ version. We do it in the nicest case but the same ideas can be used more generally as we have seen in Section 3.1.
Suppose that $\Lambda^1$ is finite-dimensional in the category of right $A$-crossed modules, $g$ a bi-invariant metric and $B_-(\Lambda^1)$ finite-dimensional with a 1-dimensional top degree $n$ and central bi-invariant top form ${\rm Vol}$ used to define $\int$. We define the Hodge star
\[ \sharp{\ }=g\circ\CF: B_-(\Lambda^1)^m\to B_-(\Lambda^1)^{n-m}\]
which we extend as a bimdodule map to $\Omega^m\to \Omega^{n-m}$.
By construction our $\sharp$ is a morphism in the crossed-module category. In geometric terms this means that it extends as a bimodule map and is bicovariant under the quantum group action on $\Omega$. We also define $\sharp^\star=(-1)^D\circ \CF^\star\circ g^{-1}$ where $D$ is the degree operator.
In the setting of Definition <ref>, $\mu=\<{\rm Vol},{\rm Vol}\>^{-1}\in k^\times$, $\sharp$ is invertible and $\sharp S=(-1)^nS\sharp$. If the metric $g$ is quantum symmetric then $S=(-1)^D$, $\sharp^\star=\sharp$ and $\sharp^2=\mu$ on degrees $D=0,1,n-1,n$.
Here $\<{\rm Vol},{\rm Vol}\>$ is non-zero since otherwise ${\rm Vol}$ would be zero in $B_-(\Lambda^1)$, and its inverse supplies the coefficient of the top component of $\exp$, which is $\mu$. Since $\mu\ne 0$ we can apply Corollary <ref> to see in particular that $\sharp,S$ graded-commute. That $S|_{0,1,n-1,n}=(-1)^D$
i.e. on the outer degrees is clear on degrees 0,1 and then holds on degrees $n,n-1$ due to $\sharp, S$ graded-commuting. Next, in terms of $\sharp^\star$ with the metric identification, the result in Corollary <ref> becomes $\sharp^\star\sharp=\mu (-1)^D S$ and $\sharp(-1)^D\sharp^\star=\mu (-1)^n S$ since the parity of the integral is $n$ mod 2. Taking the $(-1)^D$ to the left in the latter equation makes it $(-1)^{n-D}$ so that
\[ \sharp^\star=\mu (-1)^D S\sharp^{-1}\]
on all degrees, giving $\sharp^\star|_{0,1,n-1,n}=\mu\sharp^{-1}$ on the outer degrees. On the other hand, we have
\[\exp=1\tens 1 + g+ \cdots + g^{(n-1)}+ \mu {\rm Vol}\tens{\rm Vol}\]
(for some element $g^{(n-1)}\in \Lambda^{n-1}\tens\Lambda^{n-1}$), while the definition of $\sharp^\star$ is such that it is given by integration agains $\Psi^{-1}\exp$ without any signs. Since $g$ (by the quantum symmetry assumption) and $1\tens 1$ are invariant under $\Psi$, these terms are the same, and hence $\sharp^\star=\sharp$ on degrees $n-1,n$ and hence $\sharp^2=\mu$ on these degrees. In that case $\sharp^\star(\sharp\omega)=\mu\omega=\sharp^2\omega$ on all $\omega$ of degree $n-1,n$ tells us that $\sharp^\star=\sharp$ on degrees 0,1 also, and hence that $\sharp^2=\mu$ on these degrees also. This means that
\[ \Psi_{sup}^{-1}\exp=1\tens 1-g +\cdots +(-1)^{n-1}g^{(n-1)}+(-1)^n\mu{\rm Vol}\tens{\rm Vol}\]
for the computation of $\CF^\star$ and similarly without the signs for $\sharp^\star$.
We similarly define left and right interior products
\[ \vdash: \Lambda^1\tens B_-(\Lambda^1)^m\to B_-(\Lambda^1)^{m-1},\quad \dashv: B_-(\Lambda^1)^m\tens\Lambda^1\to B_-(\Lambda^1)^{m-1}\]
by restricting the left and right actions in Section 3.1 (these are the left and right braided-partial derivatives in the sense of <cit.>). We then extend these to bicovariant bimodule maps
\[\vdash:\Omega^1\tens_A\Omega^m\to \Omega^{m-1},\quad \dashv:\Omega^m\tens_A\Omega^1\to \Omega^{m-1}\]
given by
\[ (a\eta)\vdash (b\omega)=(a\eta,b\omega\Bo)\omega\Bt,\quad (b\omega)\dashv (a\eta)=b\omega\Bo(\omega\Bt,a\eta),\quad \forall a,b\in A,\ \eta\in \Lambda^1,\ \omega\in \Lambda\]
where we underline the braided-coproduct of $\Lambda$. In other words, we extend the braided coproduct as a bimodule map $\Omega\to \Omega\tens_A\Omega$ (not to be confused with the super-coproduct of $\Omega$ as a super-Hopf algebra) and then use the quantum metric pairing to evaluate, taken as zero when degrees do not match.
We can now interpret our Fourier theory in Section 3.1 as
\begin{equation}\label{inthodge} S\sharp(\eta\vdash\omega)=\eta(S\sharp\omega),\quad \sharp(\omega\eta)=(\sharp\omega)\dashv\eta,\quad\forall\eta\in \Omega^1,\quad \omega\in \Omega\end{equation}
where $S$ is the super-braided antipode of $B_-(\Lambda^1)$ extended as a bimodule map to $\Omega$. It should not be confused with the super-coproduct of $\Omega$. We also define a left Lie derivative by
\[ \CL_\eta(\omega):=\eta\vdash \extd\omega+ \extd (\eta\vdash\omega),\quad\forall \eta\in \Omega^1,\ \omega\in\Omega\]
and associated codifferential and Hodge Laplacian
\begin{equation}\label{deltaHodgeL} \delta:=(S\sharp)^{-1}\extd(S\sharp),\quad \square:=\extd\delta+\delta\extd.\end{equation}
The use of $(S\sharp)^{-1}$ here is adapted to the left handed $\vdash$ and left-handed partial derivatives defined by $\extd f=\sum_a (\del^a f)e_a$ for any choice of basis $\{e_a\}$ of $\Lambda^1$. One could equally well use $\sharp$ but this would be adapted to $\dashv$ and right-handed partial derivatives. We also define the Leibnizator
\[ L_\delta(\omega,\eta)=\delta(\omega\eta)-(\delta\omega)\eta-(-1)^{|\omega|}\omega\delta\eta\]
as in <cit.>.
The codifferential and Hodge Laplacian in (<ref>) obey
\[ \delta(f\omega)=f\delta\omega+(\extd f)\vdash \omega,\quad \square(f\omega)=(\square f)\omega+f\square\omega+L_\delta(\extd f,\omega)+\CL_{\extd f}\omega\]
for all $f\in A$ and $\omega\in \Omega$. Moreover,
\[ \delta \alpha=\alpha^a\delta e_a+g_{ab}\del^a\alpha^b,\quad \square f=(\del^af)\delta e_a+ g_{ab}\del^a\del^b f\]
\[ \square \alpha=\alpha^a \square e_a +(\square \alpha^a)e_a+ \del^a\alpha^b(L_\delta(e_a,e_b)+\CL_{e_a}e_b)+ \del^a\del^b\alpha^c \left((e_a\vdash(e_be_c))+ g_{bc}e_a-g_{ab}e_c\right)\]
where $\alpha=\alpha^ae_a$ in a basis and $g_{ab}=(e_a,e_b)$ (summation understood). If $\delta\alpha=0$ then
\[ \square\alpha=\alpha^a\delta\extd e_a+\del^a\alpha^b(\delta(e_ae_b)+e_a\vdash\extd e_b)+\del^a\del^b\alpha^c(e_a\vdash(e_be_c))\]
The formula for $\delta(f\omega)$ follows immediately from the derivation property of $\extd$ and the first interior product property in (<ref>). The formula for $\square(f\omega)$ then follows from this and the Leibniz rule for $\extd$ as in <cit.>. These results then give the explicit formulae for $\alpha=\alpha^ae_a$.
Note concerning $\square\alpha$ that $(e_a\vdash(e_be_c))+ g_{bc}e_a-g_{ab}e_c=(e_ae_b)\dashv e_c$ in the classical case, which is antisymmetric in $a,b$, while $L_\delta(\extd f,\omega)+\CL_{\extd f}\omega=2\nabla_{\extd f}\omega$ in the classical case as shown in <cit.>. Here $\nabla$ is the classical Levi-Civita connection referred back to a derivative along 1-forms via the metric. The special case shown in Corollary <ref> is relevant to `Maxwell theory' where $F=\extd \alpha$ and Maxwell's equation $\delta F=J$ has a degree of freedom to change $\alpha$ by an exact form, which freedom can be reduced by fixing $\delta\alpha=0$. Maxwell's equation then becomes $\square\alpha=J$ where $J$ is required to be a coexact `source'.
§.§ Finite group case
To give a concrete example we recall that any ad-stable subset of a finite group not containing the group identity defines an ad-stable ideal in $k(G)^+$ and hence a bicovariant calculus. There is a canonical basis of 1-forms $\{e_a\}$ labelled by the subset and relations $e_a f= R_a(f)e_a$ where $R_a(f)=f((\ )a)$ is right translation. This corresponds to crossed module action $e_a\ra f=f(a)e_a$. There is a natural choice of bi-invariant metric namely $g=\sum_a e_a\tens e_{a^{-1}}$ provided our subset is closed under inversion. Here the left coaction is trivial on $\Lambda^1$ and the right coaction is the crossed module one, namely $\Delta_Re_a=\sum_{g\in G}e_{gag^{-1}}\tens \delta_g$. The element $\theta=\sum_a e_a$ is similarly bi-invariant and makes the calculus inner (so $\extd=[\theta,\ \}$ as a graded commutator). The above results therefore give us a Hodge star on any finite group with bicovariant calculus stable under inversion and for which $B_-(\Lambda^1)$ has (up to scale) a unique top form which is bi-invariant. The braiding is $\Psi(e_a\tens e_b)=e_{aba^{-1}}\tens e_a$, from which the braided factorials can be computed. The metric sends $g(f^a)=e_{a^{-1}}$ where $\{f^a\}$ is a dual basis. The interior products given by the braided-derivatives
\[ e_a\vdash(e_be_c)=g_{ab}e_c-g_{a,bcb^{-1}}e_b,\quad (e_ae_b)\dashv e_c=e_a g_{bc}-e_{aba^{-1}}g_{ac}\]
on evaluating against the super-braided coproduct via the metric. One also has $\extd\theta=0$ and in nice cases (when the associated quandle is locally skew and the field is characteristic zero) it was shown in <cit.> that $H^1(G)=k\theta$. If one similarly has $H^{n-1}(G)=k\theta^\sharp$ as an expression of Poincaré duality (which is often the case, including in the following example) then a source $J$ is coexact if and only if
\[ \delta J=0,\quad \int_G (J,\theta)=0\]
by elementary arguments using (<ref>), where $\int_G$ means a sum over the group. Here $\int:\Omega^n\to k$ is defined as the tensor product of $\int_G$ and $\int$ on $B_-(\Lambda^1)$ and $\int\extd \beta=0$ on any $n-1$-form $\beta$ under our assumption of a unique bi-invariant top form ${\rm Vol}$ of degree $n$.
The standard 3D calculus on the permuation group $S_3$ on 3 elements is given by the conjugacy class of 2-cycles. We recall that $\Lambda^1=k-{\rm span}\{e_u,e_v,e_w\}$ is a $k(S_3)$-crossed module as above, where $a=u,v,w$ are the 2-cycles $u=(12), v=(23), w=uvu=vuv=(13)$. The minimal exterior algebra in this case is known to be a super version of the Fomin-Kirillov algebra<cit.> with relations and exterior derivative
\[ e_u e_v + e_v e_w + e_w e_u=0,\quad e_v e_u + e_w e_v+ e_u e_w=0, \quad e_u^2=0,\quad \extd e_u+e_v e_w+ e_w e_v=0\]
and the two cylic rotations of these where $u\to v\to w\to u$. The dimensions in the different degrees are $\dim(\Lambda)=1:3:4:3:1$ so there is a unique top form up to scale, which we take as ${\rm Vol}=e_ue_ve_ue_w$. This is clearly central and one can check that it is also bi-invariant. This can be done noting that ${\rm Vol}=e_ue_ve_u\theta$ and computing
\[ \Delta_R(e_ue_ve_u)=e_ue_ve_u\tens(\delta_e+\delta_w)+e_we_ue_w\tens(\delta_u+\delta_{vu})+ e_ve_we_v\tens(\delta_{uv}+\delta_v).\]
Hence we have a canonical Hodge star.
The coevaluation element $\exp$ is a computation from Proposition <ref> which in a basis reads
\[ \exp=\sum_{m=0}^n \cdot[m,-\Psi]!^{-1}(e_{a_1}\tens \cdots\tens e_{a_m})\tens f^{a_m}\cdots f^{a_1}\]
where $\{e_a\}$ is our basis of $\Lambda^1$ and $\{f^a\}$ is a dual one and we sum repeated indices. This is in general but in our case and using the metric identification comes out as
\[ \exp=1\tens 1+ \sum_a e_a\tens e_a+ e_v e_w\tens e_w e_v+ e_u e_w\tens e_w e_u - e_u e_v\tens e_u e_w-e_v e_u\tens e_v e_w\]
\[ \quad\quad +e_ue_ve_w\tens e_u e_v e_w+e_ve_we_u\tens e_v e_w e_u+ e_we_ue_v\tens e_we_ue_v-{\rm Vol}\tens{\rm Vol}\]
as one may check by verfying that this is a sum of basis and dual basis of each degree of $\Lambda^m\tens\Lambda^m$ paired via the metric. For example,
\[ \<e_ue_v,e_we_v\>=\ev(e_u\tens e_v,e_w\tens e_v-e_u\tens e_w)=0\]
\[ \<e_ue_v,e_ve_w\>=\ev(e_u\tens e_v,e_v\tens e_w-e_u\tens e_v)=0\]
\[ \<e_ue_v,e_we_u\>=\ev(e_u\tens e_v,e_w\tens e_u-e_v\tens e_w)=0\]
\[ \<e_ue_v,e_ue_w\>=\ev(e_u\tens e_v,e_u\tens e_w-e_v\tens e_u)=-1\]
where we should remember that the $\ev$ pairing is nested starting on the inside.
The resulting Hodge operator is then computed as
\[ \sharp 1=-{\rm Vol},\quad \sharp e_u=e_w e_u e_v,\quad \sharp e_v=e_u e_v e_w,\quad \sharp e_w=e_v e_w e_u\]
\[ \sharp (e_w e_u e_v)=-e_u,\quad \sharp(e_u e_v e_w)=-e_v,\quad \sharp(e_v e_w e_u)=-e_w,\quad \sharp {\rm Vol}=1\]
\[ \sharp(e_ue_v)=e_w e_u;\quad \sharp(e_a e_b)=e_{aba}e_a=\cdot\Psi(e_a\tens e_b)\]
We see that on the different degrees,
\[ \sharp^2|_{0,1,3,4}=-\id,\quad \sharp^3|_2=\id\]
so that $\sharp$ has order 6. The first of these illustrates Proposition <ref> while we note that $\sharp$ on degree 2 coincides with minus the braided-antipode $S$ of $B_-(\Lambda^2)$ (because this is braided-multiplicative along with an extra sign for the super case, and $S|_1=-\id$). The cohomology for this calculus in characteristic zero is known to be $H^0=k, H^1=k, H^2=0, H^3=k, H^4=k$ and one can see that $\sharp$ is an isomorphism $H^m\isom H^{4-m}$ as expected, the cohomologies being spanned by $1, \theta=e_u+e_v+e_w$ in degrees 0,1 and their $\sharp$ in degrees 3,4.
Next we compute $S$ on degree 3 as $S(e_ue_ve_w)=-\cdot\Psi(Se_u\tens S(e_ve_w))=\cdot\Psi(e_u\tens e_ue_v)=e_ue_we_u=-e_ue_ve_w$. A similar computation gives $S{\rm Vol}={\rm Vol}$ so that $S=(-1)^{D}$ on degree $D\ne 2$ as we saw in Proposition <ref>, while we have already observed that $S=-\sharp$ on degree 2.
We now compute $\sharp^\star:=\CF^\star\circ g$ as a check of our theory. The inverse braiding is $\Psi^{-1}(e_a\tens e_b)=e_b\tens e_{b^{-1}ab}$ in general and extends similarly to products, with the metric identification, we have
\[ \Psi_{sup}^{-1}\exp=1\tens 1- g+e_w e_v\tens e_w e_u+ e_w e_u\tens e_w e_v- e_u e_w\tens e_v e_w-e_v e_w\tens e_e e_w\]
\[ \quad -e_ue_ve_w\tens e_u e_v e_w- e_v e_w e_u\tens e_v e_w e_u- e_w e_u e_v\tens e_w e_u e_v-{\rm Vol}\tens{\rm Vol}.\]
Note that $e_ue_ve_w=e_we_ve_u$ and so forth using the relations. Integrating against gives $\sharp^\star=\sharp$ on degrees 0,1,3,4 as in Proposition <ref> while $\sharp^\star=\id$ on degree 2. The latter agrees with $\sharp^\star=\mu (-1)^D S\sharp^{-1}$ in the proof of Proposition <ref>.
Turning to applications we compute
\[ S\sharp 1=-{\rm Vol}, \quad S\sharp e_u=-e_we_ue_v,\quad S\sharp(e_ae_b)=-e_be_{bab},\quad S\sharp(e_we_ue_v)=e_u,\quad S\sharp{\rm Vol}=1\]
For the Laplace operator we note that $\delta e_a=0$, for example
\[ S\sharp\delta e_u= -\extd (e_w e_u e_v)=-\theta e_w e_u e_v-e_w e_u e_v \theta=-e_ue_we_ue_v- e_w e_u e_v e_u=0\]
using the relations. Hence by Corollary <ref>
\[ \square|_0=\sum\del^a{}^2=-2\sum_a\del^a\]
which is (-2 times) the standard graph Laplacian for the corresponding Cayley graph on $S_3$. It is fully diagonalised as usual by the matrix elements of irreducible representations (the eigenvalues are 0, 6, 12 with eigenspaces of dimensions 1,4,1 respectively). We also have $\delta\alpha=g_{ab}\del^a\alpha^b$ and if this vanishes then
\[ \square\alpha=(\sharp^{-1}\extd)^2\alpha=\sum (({1\over 2}\square+3)\alpha^a)e_a-(\alpha,\theta)\theta-\sum\del^a(\del^b+2)\alpha^{aba}e_b.\]
We note that the last term here only has contributions from $a\ne b$. The above expression is a short computation from Corollary <ref> using
\[ \delta(e_ae_b)= e_b-e_{aba},\quad e_a\vdash(e_be_c)=\delta_{a,b}e_c-\delta_{aba,c}e_b\]
from which we see that
\[ \delta\extd e_a=3e_a-\theta,\quad \delta(e_ae_b)+e_a\vdash \extd e_b=e_b-2e_{aba}+\delta_{a,b}\theta.\]
Solving Maxwell theory in the form $\square\alpha=J$, the source $J$ has to be coexact. From the remarks above and $H^3(S_3)=k\theta^\sharp$, this is equivalent to
\[ \delta J=0,\quad \int_{S_3} (J,\theta)=0\]
It is a useful check of our formula for $\square$ on $\Omega^1$ to see directly that when restricted to coexact forms its image indeed is again coexact. Moreover, by computer one finds the same eigenspaces (each 4-dimensional) as in <cit.> with eigenvalues 3,6,9, so that up to an overall constant the Laplacian restricted to coexact 1-forms is the same in spite of the Hodge operators being rather different. Explicitly,
\[ e_u-e_v, \quad e_u-\sum_a(\delta_{au}+\delta_{uau})e_a\]
and their cyclic rotations under $u\to v\to w\to u$ have eigenvalue 3 (and along with their cyclic rotations add up to zero). Multiplying these by the sign function on $S_3$ gives eigenvectors of eigenvalue 9 while for the eigenvectors of eigenvalue 6 we can use the `point sources' in <cit.>,
\[ J_x:=(3\delta_x-1)\theta+3\sum_a\delta_{xa}e_a,\quad x\in S_3\]
where three points that share a common node in the graph have a zero sum of their sources. These are related to the matrix elements $\rho_{ij}$ of the 2-dimensional representation. For example, if we work over $\R$ and $\rho(u)={\rm diag}(1,-1)$, $\rho(v)={1\over 2}{\rm diag}(-1,1)+{\sqrt{3}\over 2}\tau$ where $\tau$ is the transposition matrix, then
\[ J_e=2\rho_{11}e_u+{\rm cyclic}, \quad J_{vu}-J_{uv}=2\sqrt{3}\rho_{21}e_u+{\rm cyclic}.\]
This solves the `Maxwell theory' on $S_3$ for this calculus by diagonalising $\square$ on coexact 1-forms.
In <cit.> and all other such models until now it has been assumed that the Hodge operator should be designed to square to $\pm 1$, whereas our canonical Hodge operator in this example is order 6 and looks very different, but nevertheless gives the same reasonable Laplacians in degrees 0,1. Our construction also works for $S_4$ and $S_5$ with their 2-cycles calculus and can be analysed similarly, while higher $S_n$, $n>5$ are conjectured<cit.> to have infinite-dimensional $B_-(\Lambda^1)$.
§ CALCULUS AND HODGE OPERATOR ON COQUASITRIANGULAR HOPF ALGEBRAS
Here we start with a new, braided-Lie algebra, approach to the construction of bicovariant $(\Omega^1,\extd)$ on quantum groups such as $k_q[G]$ for $G$ a complex semisimple Lie group. We recall that these are all coquasitriangular in that they come with a convolution-invertible map $\CR:A\tens A\to k$ obeying
\[ b\o a\o \CR(a\t\tens b\t)=\CR(a\o\tens b\o)a\t b\t \]
\[ \CR(ab\tens c)=\CR(a\tens c\o)\CR(b\tens c\t),\quad \CR(a\tens bc)=\CR(a\o\tens c)\CR(a\t\tens b) \]
for all $a,b,c\in A$. This is just dual to Drinfeld's theory in <cit.>, see <cit.>. We will need the `quantum Killing form'
\[ \CQ(a\tens b)= \CR(b\o\tens a\o)\CR(a\t\tens b\t),\quad \forall a,b\in A\]
which obeys $\CQ(Sa\tens Sb)=\CQ(b\tens a)$ since $\CR$ is invariant under $S\tens S$.
The construction of differential calculi on a coquasitriangular Hopf algebra $A$ has its roots in R-matrix constructions from the 1990s but the following general construction builds on our recent treatment in <cit.>. It is shown there that $A$ is a left $A$-crossed module by
\begin{equation}\label{adLcrossed} \Ad_L(a)=a\o Sa\th\tens a\t,\quad a\la b=b\t \CR(b\o\tens a\o)\CR(a\t \tens b\th).\end{equation}
Then any subcoalgebra $\CL$ becomes a left $A$-crossed module by restriction and its dualisation $\CL^*$ in the finite-dimensional subcoalgebra case becomes a right $A$-crossed module. It is shown that the quantum Killing form regarded by evaluation on its first input as a map $\CQ:A^+\to \CL^*$ is a morphism of crossed modules. This gives:
cf <cit.> Let $A$ be a coquasitriangular Hopf algebra and $\CL\subseteq A$ a nonzero finite-dimensional subcoalgebra. Then $\Lambda^1={\rm image}(\CQ)$ and $\varpi=\CQ$ defines a bicovariant differential calculus $\Omega^1$ on $A$.
In <cit.> we used a version of this to naturally construct possibly non-surjective differential calculi with $\Lambda^1=\CL^*$, but we also see from this result that $\CL$ itself is the more fundamental object as starting point.
§.§ Braided-Lie algebras
Our new approach is to start with a Hopf algebra $B$ in a braided category $\CC$ and find a `Lie algebra' for it. We then take its dual to define a calculus.
<cit.> A left braided-Lie algebra is a coalgebra $\CL$ in a braided category, together with a morphism $[\ ,\ ]:\CL\tens\CL\to \CL$ subject to the axioms shown diagrammatically in Figure <ref>. The associated braiding $\tilde\Psi$ and braided-Killing form are also shown.
\[ \includegraphics[scale=.8]{blie.pdf}\]
\[ \includegraphics[scale=.5]{bliedefs.pdf} \]
Axioms of a braided-Lie algebra. Read down the page. We also recall the associated braiding $\tilde\Psi$ and braided-Killing form, and the adjoint action of a braided-Hopf algebra on itself.
A principal result in the case of an Abelian braided category is the construction of the braided-enveloping algebra $U(\CL)$ as a bialgebra. This is defined by the relations of commutativity with respect to the associated braiding $\tilde\Psi$. In the category of sets a braided-Lie algebra reduces to a quandle and this was used recently to prove the cohomology theorem for finite group bicovariant calculi<cit.>. Nondegeneracy of the Killing form also turns out to be an interesting characteristic related at one extreme to the Roth property of a finite group<cit.>. The axioms themselves, however, were inspired by the properties of the braided adjoint action of a braided-Hopf algebra on itself as also recalled in Figure 3.
<cit.> If $B$ is a braided-Hopf algebra then $[\ ,\ ]=\Ad:B\tens B\to B$ the braided adjoint action obeys axiom (L1). If $B$ is cocommutative with respect to the braided-adjoint action in the sense of <cit.> (we say $B$ is Ad-cocommutative) then (L2), (L3) are also obeyed.
The first part was done in <cit.>. Braided cocommutativity with respect to a $B$-module is just the axiom (L2) when specialised to $[\ ,\ ]=\Ad$ and the proof that the adjoint action then obeys (L3) appeared in <cit.> in dual form (turn the diagrams there up-side-down). Clearly:
Any subcoalgebra $\CL\subseteq B$ of an Ad-cocommutative braided-Hopf algebra closed under $\Ad$ is a braided-Lie algebra by restriction.
We next recall that if $A$ is coquasitriangular then there is a braided Hopf algebra version $B(A)$ of $A$ called its transmutation. This is also denoted $\und A$ and has the same coalgebra as $A$ but a modified product<cit.>
\begin{equation}\label{BAprod} a\bullet b=a\t b\th \CR(a\th \tens S b\o)\CR(a\o\tens b\t)=a\rz b\t \CR(a\co\tens S b\o)\end{equation}
and lives in $\CM^A$ by $\Ad_R$. Its product is braided-commutative,
\begin{equation}\label{BAcom} a\bullet b=b\th\bullet a\th\CR(Sb\t\tens a\o)\CR(b\fo\tens a\t)\CR(a\fo\tens b\fiv)\CR(a\fiv\tens Sb\o)\end{equation}
which can be written equivalently as
\begin{equation}\label{BAcomRE} \CR(b\o\tens a\o)a\t\bullet\CR(a\th\tens b\t) b\th=b\o\bullet \CR(b\t\tens a\o) a\t\CR(a\th\tens b\th)\end{equation}
while its braided-antipode is
\begin{equation}\label{BAS} \und S a=Sa\t\CR((S^2 a\th)Sa\o\tens a\fo)=(Sa\o)\rz\CR((Sa\o)\co\tens a\t).\end{equation}
Because we have a mix of both types of structure on the same vector space, we will be more careful to underline the braided versions where they are different.
Let $A$ be coquasitriangular. Then $B(A)$ is $\und\Ad$-cocommutative and $[a,b]:=\und\Ad_a(b)=b\rz\CQ(a\tens b\co)$ for all $a,b\in A$ makes $A$ a braided-Lie algebra in the braided category $\CM^A$. By restriction, any subcoalgebra $\CL\subseteq A$ is a braided-Lie algebra in this category. In this case there is a surjection
\[ U(\CL)\twoheadrightarrow B(A)\]
of bialgebras in the category.
We start by computing the left braided-adjoint action by applying $\und S$ to $a\t$ and using the braiding to commute this past $b$ before multiplying up with respect to $\bullet$:
\begin{eqnarray*} \und{\Ad}_a(b)&=& a\o\bullet b\rz\bullet(\und S a\t)\rz\CR((\und S a\t)\co\tens b\co)\\
&=& a\o\bullet \left( b\rz (\und S a\t)\rz\t\right)\CR((\und S a\t)\co\tens b\co\t)\CR(b\co\o\tens S(\und S a\t)\rz\o)\\
&=&a\o\rz b\rz\t (\und S a\t)\rz\th \CR((\und S a\t)\co\tens b\co\t)\CR(b\co\o\tens S(\und S a\t)\rz\o)\\
&&\quad\quad\quad\quad\CR(a\o\co\tens S(b\rz\o(\und S a\t)\rz\t)\\
&=&a\o\rz b\rz\t (Sa\t)\rz\th \CR((Sa\t)\co\t\tens a\th)\CR((Sa\t)\co\o\tens b\co\t)\\
&&\quad \CR(b\co\o\tens S(S a\t)\rz\o)\CR(a\o\co\tens S(b\rz\o(Sa\t)\rz\t))\\
&=&a\o\rz b\rz\t(Sa\t)\rz\t \CR((Sa\t)\co\tens a\th b\co\t)\\
&&\CR(a\o\co\t b\co\o\tens S(Sa\t)\rz\o)\CR(a\o\co\o\tens Sb\rz\o)\\
\end{eqnarray*}
where we use the definitions, the coaction properties and the multiplicativity property of $\CR$. We next unpack the adjoint coactions on $a$, and use multiplicativity of the last $\CR$ to give
\begin{eqnarray*} &=&a\th b\rz\th Sa\sev\CR(S^2 a\nine Sa\six\tens a\ten b\co\t)\CR(Sa\o a\fiv b\co\o\tens S^2 a\ei)\\
&&\CR(a\t\tens b\rz\t)\CR(a\fo\tens S b\rz\o)\\
&=& b\rz\t a\t Sa\sev\CR(S^2 a\nine Sa\six\tens a\ten b\co\t)\CR(Sa\o a\fiv b\co\o\tens S^2 a\ei)\\
&&\CR(a\th\tens b\rz\th)\CR(a\fo\tens S b\rz\o)\\
&=& b\rz\rz a\t Sa\six\CR(S^2 a\ei Sa\fiv\tens a\nine b\co\t)\CR(Sa\o a\fo b\co\o\tens S^2 a\sev)\\
&&\CR(a\th\tens b\rz\co)\\
&=& b\rz a\t Sa\six\CR(S^2 a\ei Sa\fiv\tens a\nine b\co\th)\CR(Sa\o a\fo b\co\t\tens S^2 a\sev)\\
&&\CR(a\th\tens b\co\o)\\
&=& b\rz a\t Sa\sev\CR(S^2 a\ten Sa\six\tens b\co\th)\nu^{-1}(a\ele)\CR(Sa\fiv\tens a\twe)\\
&&\CR(Sa\o a\fo\tens S^2 a\ei)\CR(b\co\t\tens S^2 a\nine)\CR(a\th\tens b\co\o)\\
&=& b\rz a\t Sa\sev\CR(S^2 a\ten Sa\six\tens b\co\th)\CR(Sa\fiv\tens S^2 a\ele)\nu^{-1}(a\twe)\\
&&\CR(Sa\o a\fo\tens S^2 a\ei)\CR(b\co\t\tens S^2 a\nine)\CR(a\th\tens b\co\o)\\
&=&\CR(a\o\tens Sa\ei) b\rz a\t Sa\sev\CR(Sa\fiv\tens S^2 a\twe)\nu^{-1}(a\thir)\\
&&\CR(S^2 a\ele Sa\six\tens b\co\th)\CR(b\co\t\tens S^2 a\ten)\CR(a\fo\tens S^2a\nine)\CR(a\th\tens b\co\o)\\
&=&\CR(a\o\tens Sa\ei) b\rz a\t Sa\sev\CR(Sa\six\tens S^2 a\ele)\nu^{-1}(a\thir)\\
&&\CR( Sa\fiv S^2 a\twe\tens b\co\th)\CR(b\co\o\tens S^2 a\nine)\CR(a\th\tens S^2a\ten)\CR(a\fo\tens b\co\t)\\
&=&\CR(a\o\tens Sa\six) b\rz a\t Sa\fiv \CR(a\th\tens S^2a\ei) \CR(Sa\fo\tens S^2 a\nine)\nu^{-1}(a\ele)\\
&&\CR( S^2 a\ten\tens b\co\t)\CR(b\co\o\tens S^2 a\sev)\CR(a\th\tens S^2a\ei)\\
&=&\CR(a\o\tens Sa\fo) b\rz a\t Sa\th\nu^{-1}(a\sev)\CR( S^2 a\six\tens b\co\t)\CR(b\co\o\tens S^2 a\fiv)\\
&=&\CR(a\o\tens Sa\t) b\rz \nu^{-1}(a\fiv)\CR( S^2 a\fo\tens b\co\t)\CR(b\co\o\tens S^2 a\th)\\
&=& b\rz \CR(a\t\tens b\co\t)\CR(b\co\o\tens a\o)=b\rz\CQ(a\tens b\co)=[a,b]
\end{eqnarray*}
where the 2nd equality is by quasicommutativity of $A$, the 3rd uses multiplicativity of $\CR$ to recognise $\Ad_R$ on $b\rz$. We then expand out by multiplicativity to recognise $\nu^{-1}(a)=\CR(S^2 a\o\tens a\t)$. This is known<cit.> to be convolution inverse to $\nu(a)=\CR(a\o\tens Sa\t)$ and to obey $\nu^{-1}(a\o)a\t=S^2 a\o\nu^{-1}(a\t)$, which we use to move to the right. The seventh equality uses multiplicativity of $\CR$ so that we can use quasicommutativity on $S^2a\ele Sa\six$ and the braid or Yang-Baxter equations on the last three factors to give the 8th equality. On this we use multiplicativity to cancel $a\fo Sa\fiv$ and obtain the 9th equality and two mutually inverse copies of $\CR$ for the 10th. We finally cancel $a\t Sa\th$ and move $\nu^{-1}$ to the left to cancel $\nu$. We then recognise the answer in terms of $\CQ$ and take this for our braided-Lie bracket. The Lemma tells us that we have (L1) for free. Next, we verify (L2) for $[\ ,\ ]=\und\Ad$ noting that (L2) can be written in the form
\[ \Psi\tilde\Psi=(\id\tens[\ ,\ ])(\Delta\tens\id)\]
where in our case $\Psi$ is the braiding of $\CM^A$ and
\begin{eqnarray*}&&\nquad \tilde\Psi(a\tens b)= [a\o,b\rz]\tens a\t\rz\CR(a\t\co\tens b\co)\\
&&=b\rz\tens a\t\rz \CQ(a\o\tens b\co\o)\CR(a\t\co\tens b\co\t)\\
&&=b\t\tens a\fo \CR(((Sb\o)b\th)\o\tens a\o)\CR(a\t\tens ((Sb\o)b\th)\t)\\
&&\quad\quad\quad\quad\quad\CR((Sa\th)a\fiv\tens ((Sb\o)b\th)\th)\\
&&=b\t\tens a\t\CR(((Sb\o)b\th)\o\tens a\o)\CR(a\th\tens ((Sb\o)b\th)\t)\\
&&=b\rz\tens a\t \CR(b\co\o\tens a\o)\CR(a\th\tens b\co\t)
\end{eqnarray*}
using our result for $\und\Ad$. We compute
\begin{eqnarray*}
&&\nquad \Psi\tilde\Psi(a\tens b)=a\t\rz\tens b\rz\rz\CR(b\rz\co\tens a\t\co)\CR(b\co\o\tens a\o)\CR(a\th\tens b\co\t)\\
&&=a\t\rz\tens b\rz\CR(b\co\o\tens a\t\co)\CR(b\co\t\tens a\o)\CR(a\th\tens b\co\th)\\
&&=a\t\t\tens b\rz\CR(b\co\o\tens a\o (Sa\t\o)a\t\th)\CR(a\th\tens b\co\t)\\
&&=a\o\tens b\rz\CR(b\co\o\tens a\t)\CR(a\th\tens b\co\t)=a\o\tens b\rz\CQ(a\t\tens b\co)=a\o\tens[a\t,b]
\end{eqnarray*}
as required, where we used the coaction properties of $\Ad_R$ and the multiplicativity property of $\CR$ to make a cancellation. The above Lemma then tells us that we get (L3) for free. These results then apply for an subcoalgebra $\CL\subseteq A$ since, due to the form of $\und\Ad$, we see that $\und\Ad(\CL\tens\CL)\subseteq \CL$, since $\Ad_R(\CL)\subseteq \CL\tens A$ because $\CL$ is a subcoalgebra (in other words a sub-coalgebra of $A$ is also a subobject and hence a braided sub-coalgebra $\CL\subseteq B(A)$. For the last part, we can equivalently write
\[ \tilde\Psi(a\tens b)=b\th\tens a\th \CR(Sb\t\tens a\o)\CR(b\fo\tens a\t)\CR(a\fo\tens b\fiv)\CR(a\fiv\tens Sb\o)\]
by expanding out our previous expression using the multiplicativity properties of $\CR$. Comparing with the braided-commutativity of $B(A)$ in (<ref>) we see that $a\bullet b=\bullet\tilde\Psi(a\tens b)$ or the relations of $U(\CL)$.
When $\CL\subseteq A$ is finite-dimensional (1) $U(\CL)$ is Koszul dual to a right-handed quadratic version of the bicovariant calculus in Proposition <ref>. (2) The braided-Killing form is
\[ K(a,b)= \sum_i u(e_i\co\o)\CQ(a,e_i\co\t)\CQ(b,e_i\co\th) \<f^i,e_i\rz\>\]
where $\{e_i\}$ is a basis of $\CL$ and $\{f^i\}$ a dual basis and $u(a)=\CR(a\t\tens Sa\o)$.
The braided-Killing form is obtained by reading down the diagram, as (summation understood)
\begin{eqnarray*}K(a,b)&=&\CQ(b,e_i\o)\CQ(a,e_i\rz\o)\<f^i\rz,e_i\rz\rz\rz\>\CR(e_i\rz\rz\co\tens f^i\co)\\
&=&\CQ(b,e_i\co)\CQ(a,e_i\rz\co)\CR(e_i\rz\rz\co\tens Se\rz\rz\rz\co)\<f^i,e_i\rz\rz\rz\rz\>\\
&=&\CQ(b,e_i\co\fo)\CQ(a,e_i\co\th)\CR(e_i\co\t\tens Se\co\o)\<f^i,e_i\rz\>
\end{eqnarray*}
as stated. For the remark about the dualisation we note that $A$ has a right crossed-module structure given by $\Ad_R$ and
\[ a\ra b= a\t \CR(b\o\tens a\o)\CR(a\th\tens b\t),\quad \forall a,b\in A\]
and its crossed module braiding turns out, by similarly using the properties of $\CR$ as above, to coincide with the fundamental $\tilde\Psi$ for the braided-Lie algebra (as computed in the proof of Theorem <ref>). On the other hand this crossed module is the right handed version of (<ref>) which dualized to give the crossed module structure defining the calculus in Proposition <ref>. This means that $U(\CL)$ is the Koszul or quadratic algebra dual of $\Lambda_{quad}$ (where we impose only the degree 2 relations of $B_-(\Lambda^1)$). The braided-Lie bracket and exterior derivative can also be related as part of a general theory of `quantum Lie algebras' in <cit.> when $1\notin \CL$. Here every bicovariant calculus gives a quantum Lie algebra in the sense of <cit.> and meanwhile (one can show that) every non-unital braided-Lie algebra $\CL$ gives a quantum Lie algebra by extending by $1$ and then taking the kernel of the counit.
The above theorem is a new result and is needed to complete the picture. In the special case where $\CL$ has a matrix coalgebra form on a basis $\{t^i{}_j\}$ (such data defines a matrix corepresentation of $A$) we recover the R-matrix braided-Lie algebra construction introduced in <cit.> but now as a corollary of the above.
cf<cit.> Let $A$ be a coquasitriangular Hopf algebra and $t\in M_n(A)$ a matrix corepresentation. Then the matrix subcoalgebra $\CL=\{t^i{}_j\}$ has braided-Lie bracket, categorical braiding and braided Killing form
\[ [t^i{}_j,t^k{}_l]=t^{k_2}{}_{k_3}R^{-1}{}^{k_1}{}_{k_2}{}^i{}_{i_1}R^{k_3}{}_{k_4}{}^{i_1}{}_{i_2}R^{i_2}{}_{i_3}{}^{k_4}{}_l\tilde R^{i_3}{}_j{}^k{}_{k_1}\]
\[ \Psi(t^i{}_j\tens t^k{}_l)=t^{k_2}{}_{k_3}\tens t^{i_2}{}_{i_3} R{}^{i}{}_{i_1}{}^{k_1}{}_{k_2}R^{-1}{}^{i_1}{}_{i_2}{}^{k_3}{}_{k_4}R^{i_3}{}_{i_4}{}^{k_4}{}_l\tilde R^{i_4}{}_j{}^k{}_{k_1}\]
The braided enveloping algebra $U(\CL)$ is generated by the $\{t^i{}_j\}$ with new relations
\[ t^i{}_j\bullet t^k{}_l=t^{k_2}{}_{k_3}\bullet t^{i_2}{}_{i_3} R^{-1}{}^{k_1}{}_{k_2}{}^{i}{}_{i_1}R^{k_3}{}_{k_4}{}^{i_1}{}_{i_2}R^{i_3}{}_{i_4}{}^{k_4}{}_l\tilde R^{i_4}{}_j{}^k{}_{k_1}.\]
We sum over repeated indices in these expressions.
We expand out $\CQ$ using the properties of $\CR$, then the above bracket can also be written explicitly as
\begin{equation}\label{bLiebracket} [a,b]=b\th \CR(Sb\t\tens a\o)\CR(b\fo\tens a\t)\CR(a\th\tens b\fiv)\CR(a\fo\tens Sb\o)\end{equation}
and the categorical braiding in $\CM^A$ is
\begin{equation}\label{bLiebra} \Psi(a\tens b)= b\th\tens a\th\CR(a\o\tens b\t)\CR(Sa\t\tens b\fo)\CR(a\fo\tens b\fiv)\CR(a\fiv\tens Sb\o)\end{equation}
From these we immediately read off the expressions stated, where $R^i{}_j{}^k{}_l=\CR(t^i{}_j\tens t^k{}_l)$ and $\tilde R^i{}_j{}^k{}_l=\CR(t^i{}_j\tens St^k{}_l)$ is the `second inverse'. We likewise read off the relations of $U(\CL)$ from $\tilde\Psi$ or from (<ref>) to give the result stated. In all cases we can move $\tilde R$ and another $R$ to the left hand side, for example the relations can be written compactly as $R_{21}t_1\bullet R t_2=t_2\bullet R_{21}t_1R$
where the suffices refer to the position in $M_n\tens M_n$ with values in $U(\CL)$, also clear from (<ref>). These are the relations of $B(R)$ <cit.>, the braided analog of the more familiar FRT bialgebra $A(R)$.
This derives the explicit R-matrix formulae needed to compute examples. This in turn recovers the 4D braided-Lie algebra of $k_q[SL_2]$ found in <cit.>:
<cit.> For $A=k_q[SL_2]$ with $q^2\ne\pm 1$, its standard matrix coalgebra and rescaled generators
\[ t=\begin{pmatrix}\alpha &\beta\\ \gamma &\delta\end{pmatrix};\quad t=q^{-1}\alpha+q\delta,\quad z=\lambda^{-1}(\delta-\alpha),\quad x_+=\lambda^{-1}\beta,\quad x_-=\lambda^{-1}\gamma\]
where $\lambda=1-q^{-2}$ (we use different symbols for the entries of $t^i{}_j$ to avoid confusion with the quantum group), the nonzero braided-Lie brackets are
\[ [z,z]=q (2)_q\lambda z,\quad [t,t]=(2)_qt,\quad [t,\ ]=(q^3+q^{-3})\id,\quad [x_+,x_-]= z=-[x_-,x_+]\]
\[ [z,x_\pm]=\pm q^{\pm 1}(2)_q\, x_\pm=-q^{\pm2}[x_\pm,z]. \]
Here $(2)_q=q+q^{-1}$ and we used Corollary <ref> and the standard R-matrix for $SL_2$ with nonzero entries $R^1{}_2{}^2{}_1=q-q^{-1}, R^1{}_1{}^2{}_2=R^2{}_2{}^1{}_1=1, R^1{}_1{}^1{}_1=R^2{}_2{}^2{}_2=q$. The braided Killing form is $[4,q^{-2}]/q^{10}$ times the nonzero values
\[ K(z,z)=(1+q^2),\quad K(t,t)=(1+q^6)(1+q^{-4}+\lambda),\quad K(x_+,x_-)=1=q^{-2}K(x_-,x_+). \]
The enveloping algebra $U(\CL)=B_q[M_2]$ is generated by $\alpha,\beta,\gamma,\delta$ with relations
\[ \beta\alpha=q^2\alpha\beta,\quad \gamma\alpha=q^{-2}\alpha\gamma,\quad \delta\alpha=\alpha\delta\]
\[ [\beta,\gamma]=\lambda\alpha(\delta-\alpha),\quad [\gamma,\delta]=\lambda\gamma\alpha,\quad [\delta,\beta]=
\lambda\alpha\beta\]
where $[\ ,\ ]$ at this point denotes commutator not Lie bracket. This is the algebra of $q$-deformed $2\times 2$ braided hermitian matrices which means that geometrically it should be thought of as $q$-Minkowski space<cit.>. There are two natural central elements, the braided determinant $\det_q=\alpha\delta-q^2\gamma\beta$ which should be thought of as the $q$-Lorentzian distance from the origin and $q$-trace ${\rm tr}_q=q^{-1}\alpha+q\delta=t$ which should be thought of as the `time' direction. In these variables (as opposed to the rescaled `Lie algebra' variables) the classical limit is commutative allowing us to think of this as a noncommutative geometry. Over $\C$ our braided-Lie algebra has a natural real form or $*$-involution $\alpha^*=\alpha, \beta^*=\gamma,\delta^*=\delta$ for real $q$, which fits with the mentioned geometric picture.
§.§ Calculus and Hodge operator on $k_q[SL_2]$
In the case of a coquasitriangular Hopf algebra $A$ with a generating matrix subscoalgebra $\{t^i{}_j\}$, Proposition <ref> or dualization of Corollary <ref> recovers a version of a known R-matrix construction<cit.> of quantum group covariant calculi. We let $\{E_\alpha{}^\beta\}$ be the standard basis of $M_n(k)$ and dual to the $\{t^i{}_j\}$ basis of $\CL$. This then becomes a right $A$-crossed module with
\begin{equation}\label{crossmat} \Delta_R E_\alpha{}^\beta=E_m{}^n\tens t^m{}_\alpha S t^\beta{}_n,\quad E_\alpha{}^\beta\ra t^a{}_b=E_m{}^n R^m{}_\alpha{}^a{}_c R^c{}_b{}^\beta{}_n\end{equation}
\[ \Lambda^1=M_n(k),\quad \varpi(a)=\CQ(a\tens t^\alpha{}_\beta)E_\alpha{}^\beta\]
defines the possibly non-surjective bicovariant calculus, which is, however typically surjective for the standard quantum groups $k_q[G]$ with $q$ generic. The $E_\alpha{}^\beta$ have bimodule relations
\[ E_\alpha{}^\beta t^a{}_b=t^a{}_c E_m{}^n R^m{}_\alpha{}^c{}_d R^d{}_b{}^\beta{}_n\]
and the above right covariance. The calculus has an inner form with
\[ \extd t^a{}_b=t^a{}_c (R_{21}R)^c{}_b{}^\alpha{}_\beta E_\alpha{}^\beta-t^a{}_b\theta=[\theta,t^a{}_b],\quad \theta=E_\alpha{}^\alpha. \]
The associated right crossed-module braiding on $\Lambda^1$ will be denoted $\tilde\Psi$ also (it is adjoint to the one for the braided-Lie algebra) and is computed from the right crossed module structure as
\begin{eqnarray*}&&\nquad \tilde\Psi(E_\alpha{}^\beta\tens E_\gamma{}^\delta)=E{}_m{}^n\<E_\gamma{}^\delta, t^{j_1}{}_{j_2} \> \tens E_\alpha{}^\beta\ra t^m{}_{j_1} S t^{j_2}{}_n\\
&&= E{}_m{}^n\tens E{}_p{}^q \<E_\gamma{}^\delta, t^{j_2}{}_{j_3} \> \<E_\alpha{}^\beta,t^{k_2}{}_{k_3}\>\CR(t^p{}_{k_1}\tens t^m{}_{j_1}S t^{j_4}{}_n)\CR(t^{j_1}{}_{j_2}S t^{j_3}{}_{j_4}\tens t^{k_3}{}_q)
\end{eqnarray*}
and expands out as
\[ \tilde\Psi(E_\alpha{}^\beta\tens E_\gamma{}^\delta)=E_{j_2}{}^{j_3}\tens E_{k_2}{}^{k_3} \tilde R^{k_2}{}_{k_1}{}^{j_4}{}_{j_3}\, R^{k_1}{}_\alpha{}^{j_2}{}_{j_1}
R^{j_1}{}_\gamma{}^\beta{}_{k_4}R^{-1}{}^\delta{}_{j_4}{}^{k_4}{}_{k_3} \]
from which we see that $\tilde\Psi(E_\alpha{}^\beta\tens\theta)=\theta\tens E_\alpha{}^\beta$ so that, in particular, $\theta^2=0$ in $\Lambda_{min}=B_-(\Lambda^1)$.
We now focus on $A=k_q[SL_2]$ where the smallest nontrivial irreducible is 2-dimensional, giving us $\Lambda^1=M_2(\C)$. We write basis $E_1{}^1=e_a, E_1{}^2=e_b, E_2{}^1=e_c, E_2{}^2=e_d$ and use the standard $SL_2$ R-matrix as in Example <ref> to give the bimodule relations of the well-known 4D calculus first found in <cit.>,
\[ e_a
\begin{pmatrix}a&b\\ c&d\end{pmatrix}=\begin{pmatrix}qa&q^{-1} b\\
\[ [e_b, \begin{pmatrix}a&b\cr c&d\end{pmatrix}]=q\lambda\begin{pmatrix}0&a\cr 0&c\end{pmatrix}e_a,
\quad [e_c, \begin{pmatrix}a&b\cr c&d\end{pmatrix}]=q\lambda\begin{pmatrix}b&0\cr d&0\end{pmatrix}e_a\]
\[ [e_d,\begin{pmatrix}a\cr c\end{pmatrix}]_{q^{-1}}=\lambda \begin{pmatrix}b\cr d\end{pmatrix}e_b,\quad
[e_d,\begin{pmatrix}b\cr d\end{pmatrix}]_q=\lambda \begin{pmatrix}a\cr c\end{pmatrix}e_c+q\lambda ^2 \begin{pmatrix}b\cr d\end{pmatrix}e_a,\]
where $[x,y]_q\equiv
xy-qyx$ and $\lambda=1-q^{-2}\ne 0$. The exterior differential is necessarily inner with $\theta=e_a+e_d$ which implies that
\[ \extd \begin{pmatrix}a\\ c\end{pmatrix}=\begin{pmatrix}a\\ c\end{pmatrix}((q-1)e_a+(q^{-1}-1)e_d)+\lambda \begin{pmatrix}b\\ d\end{pmatrix} e_b\]
\[ \extd \begin{pmatrix}b\\ d\end{pmatrix}=\begin{pmatrix}b\\ d\end{pmatrix}((q^{-1}-1+q\lambda^2)e_a+(q-1)e_d)+\lambda \begin{pmatrix}a\\ c\end{pmatrix} e_c. \]
Note that we should scale $\extd$ or $\theta$ by $\lambda^{-1}$ in order to have the right classical limit but we have not done this in order to follow the general construction. The right coaction on left-invariant 1-forms is
\[ \Delta_R\theta=\theta\tens 1,\quad\Delta_R(-e_b,e_z,q^{-1}e_c)=(-e_b,e_z,q^{-1}e_c)\tens\begin{pmatrix} a^2 & (2)_qab & b^2\\
ca &1 + (2)_qbc & db \\ c^2 & (2)_qcd & d^2\end{pmatrix}\]
where $e_z:=q^{-2}e_a-e_d$ and the calculation is from $\Delta_R e_\alpha{}^\beta$ with the relevant R-matrix. We use the symmetric $q$-integers so that $(2)_q=q+q^{-1}$. The crossed module braiding comes out as
\[ \tilde\Psi(e_a\tens \begin{pmatrix}e_a& e_b\cr e_c & e_d\end{pmatrix})= \begin{pmatrix}e_a& q^2 e_b\cr q^{-2}e_c & e_d\end{pmatrix}\tens e_a,\]
\[ \tilde\Psi(e_b\tens \begin{pmatrix}e_a& e_b\cr e_c & e_d\end{pmatrix})=\begin{pmatrix}e_a& e_b\cr e_c & e_d\end{pmatrix}\tens e_b+ \lambda q^2 \begin{pmatrix}-e_b& 0 \cr e_z & e_b\end{pmatrix}\tens e_a\]
\[ \tilde\Psi(e_c\tens \begin{pmatrix}e_a& e_b\cr e_c & e_d\end{pmatrix})=\begin{pmatrix}e_a& e_b\cr e_c & e_d\end{pmatrix}\tens e_c+ \lambda \begin{pmatrix}e_c& - q^2 e_z \cr 0 & - e_c\end{pmatrix}\tens e_a\]
\[ \tilde\Psi(e_d\tens e_a)=e_a\tens e_d+ \lambda^2 q^2 e_z\tens e_a-\lambda(e_b\tens e_c-e_c\tens e_b)\]
\[ \tilde\Psi(e_d\tens e_b)=q^{-2} e_b\tens e_d- \lambda e_z\tens e_b \]
\[ \tilde\Psi(e_d\tens e_c)=q^2 \lambda e_z\tens e_c+(q^4-1+q^{-2}) e_c \tens e_d+\lambda (q^4-1)e_c\tens e_z\]
\[ \tilde\Psi(e_d\tens e_d)=e_d\tens e_d +\lambda(e_b\tens e_c-e_c\tens e_b)-\lambda^2 q^2 e_z\tens e_a\]
from which one can see for example that $\tilde\Psi(e_i\tens\theta)=\theta\tens e_i$. This then gives the relations of $\Lambda=B_-(\Lambda^1)$ as usual Grassmann variables $e_a,e_b,e_c$ and<cit.>
\[ e_a e_d+e_d e_a+\lambda e_c e_b=0,\quad e_d e_c+q^2e_c e_d+\lambda e_a e_c=0\]
\[ e_b e_d+q^2 e_d e_b+\lambda e_b e_a=0,\quad e_d^2=\lambda e_c e_b\]
or equivalently
\[ e_b e_z+q^2 e_z e_b=0,\quad e_z e_c+q^2
e_c e_z=0\]
\[
e_z e_a+e_a e_z=\lambda e_c e_b,\quad e_z^2=(1-q^{-4})e_c e_b.\]
The exterior derivative is
\[ \extd e_a=\lambda e_b e_c,\quad \extd e_c=\lambda q^2e_c
e_z,\quad \extd e_b=\lambda q^2e_z e_b,\quad \extd e_d= \lambda e_c e_b;\quad \extd
e_z=(1-q^{-4})e_b e_c.\]
As in degree 0, we note that $\lambda^{-1}\extd$ has the right classical limit not $\extd$ itself. The dimensions here in each degree are $\dim(\Lambda)=1:4:6:4:1$. The following is mostly known e.g.<cit.> but we give a short proof as it is critical for us.
For the above 4D calculus on $k_q[SL_2]$ with $q^2\ne\pm1$ there is a unique bi-invariant central metric
\[ g=e_c\tens e_b+ q^2 e_b\tens e_c + {q^3\over (2)_q}(e_z\tens e_z - \theta\tens\theta).\]
The inverse metric is
\[ (e_b,e_c)=1,\quad (e_c,e_b)=q^{-2},\quad (e_z,e_z)=q^{-3}(2)_q=-(\theta,\theta)\]
and the rest zero in this basis. In the exterior algebra there is also up to scale a unique top form ${\rm Vol}=e_ae_be_ce_d=e_be_ce_ze_a$ and this is bi-invariant and central. Hence the super-braided Fourier transform applies and we have a Hodge star and interior products.
Using the quantum group relations one has
\[ \begin{pmatrix} a^2 & (2)_qab & b^2\\
ca &1 + (2)_qbc & db \\ c^2 & (2)_qcd & d^2\end{pmatrix}\begin{pmatrix} 0 & 0& -q^2\\ 0 & {q^3\over (2)_{q}}& 0\\
-1 & 0& 0\end{pmatrix} \begin{pmatrix} a^2 & (2)_qab & b^2\\
ca &1 + (2)_qbc & db \\ c^2 & (2)_qcd & d^2\end{pmatrix}^t=\begin{pmatrix} 0 & 0& -q^2\\ 0 & {q^3\over (2)_{q}}& 0\\
-1 & 0& 0\end{pmatrix}\]
which gives us the unique generically $q$-invariant element of the tensor square of the space spanned by $\{-e_b,e_z,q^{-1}e_c\}$ (the quadratic elements here generate $k_q[SO_3]$). We can add to this a multiple of $\theta\tens \theta$ since this is also invariant<cit.>, which we have now fixed so that $g$ is central. Thus for example,
\begin{eqnarray*} (e_c\tens e_b+ q^2 e_b\tens e_c)a & =& e_c\tens a e_b+ q^3\lambda e_b\tens be_a+q^2 e_b\tens a e_c\\
&=& a(e_c\tens e_b+ q^2 e_b\tens e_c)+q\lambda b e_a\tens e_b+ q^3\lambda be_b\tens e_a+q^4\lambda^2 a e_a\tens e_a\\
(e_z\tens e_z-\theta\tens\theta)a&=&-(1+q^{-2})(\lambda e_a\tens e_a+e_a\tens e_d+e_d\tens e_a)a\\
&=& a(e_z\tens e_z-\theta\tens\theta)-(1-q^{-4})(q^2\lambda a e_a\tens e_a+q b e_b\tens e_a+ q^{-1} b e_b\tens e_a)
\end{eqnarray*}
using the above commutation relations. Comparing these we see that $[g,a]=0$. Similarly for the other generators of $k_q[SL_2]$. It is also clear that $\wedge(g)=0$. The inverse is immediate. For Vol the element $e_be_ce_z$ is invariant again for reasons coming from the representation theory of $k_q[SO_3]$. As $\theta$ is also invariant, we know that $e_be_ce_z\theta$ is invariant and hence so is Vol being a multiple of this. For centrality, we check for example
\[ ae_ae_be_ce_d=q^{-1}e_a a e_be_ce_d=e_ae_be_cq^{-1} ae_d=e_ae_be_ce_d a\]
discarding unwanted terms using the wedge product relations.
We now use $g$ to identify $\Lambda^1{}^*\isom \Lambda^1$ and compute $\sharp$ using $\int {\rm Vol}=1$.
For $k_q[SL_2]$ with its 4D calculus, $\mu=q^6$, $\sharp 1=q^6 {\rm Vol}$,
\[ \sharp e_a=-q^4 e_ae_be_c,\quad \sharp e_b=-q^4 e_ae_be_d,\quad \sharp e_c=q^6 e_ae_ce_d,\quad \sharp e_d=q^4 e_be_ce_d+\lambda q^4 e_ae_be_c\]
\[ \sharp (e_ae_b)=-q^2 e_ae_b,\quad \sharp(e_ae_c)=q^4e_ae_c,\quad \sharp(e_ae_d)=q^2 e_be_c+\lambda q^4 e_a e_d\]
\[\sharp(e_be_c)=q^4 e_ae_d,\quad \sharp(e_be_d)=q^4 e_be_d+(1-q^4) e_ae_b,\quad \sharp(e_ce_d)=-q^2e_ce_d\]
\[ \sharp(e_ae_be_c)=-q^2 e_a,\quad\sharp(e_ae_be_d)=-q^2 e_b\quad \sharp(e_ae_ce_d)= e_c\quad \sharp(e_be_ce_d)=q^2e_d+\lambda q^2 e_a\]
and $\sharp {\rm Vol}=1$, where $\lambda=1-q^{-2}$ as above. Acting on degree $D$, this obeys
\[ \sharp^2=q^6,\quad (D\ne 2);\quad (\sharp-q^4)(\sharp+q^2)=0,\quad (D=2).\]
We first explicitly compute the exp element in the form
\[ \exp=1\tens 1+ g+ e_{i_1}e_{i_2}({}_2B){}^{-1}_{IJ}\tens e_{j_1} e_{j_2}+ e_{i_1}e_{i_2}e_{i_3} ({}_3B)^{-1}_{I J} \tens e_{j_1}e_{j_2}e_{j_3}+ e_1 e_2 e_3 e_4 ({}_4B)^{-1} \tens e_1 e_2 e_3 e_4\]
where $e_i, 1\le i\le 4$ refer in order to $e_a,e_b,e_c,e_d$ and $I=(i_1,j_2,\cdots,i_m)$ with $i_1<i_2\cdots<i_m$ labels of a basis of $\Lambda^m$ and
\[ {}_mB{}_{IJ}=\<e_{i_1}\cdots e_{i_m},e_{j_1}\cdots e_{j_m}\>= \ev(e_{i_1}\tens e_{i_2}\cdots \tens e_{i_m}, [m, -\tilde\Psi]!(e_{j_1}\tens e_{j_2}\cdots e_{j_m}))\]
\[=g_{i_1 p_1}\cdots g_{i_mp_m}[m,-\tilde\Psi]!^{p_m\cdots p_2 p_1}_{j_1j_2\cdots j_m}\]
In the last line refer operators to matrices, for example $[2,-\tilde\Psi](e_m\tens e_n)=e_p\tens e_q[2,-\tilde\Psi]^{pq}_{mn}$ and we remember the metric identification where $g_{ij}=(e_i,e_j)$. This is the general picture but with bases labelled in the classical way in the present example for generic $q$. We obtain
\[{}_1B^{-1}=\begin{pmatrix} -\lambda q^2 & 0 & 0 & -q^2 \\ 0 & 0 & q^2 & 0 \\ 0 & 1 & 0 & 0 \\ -q^2 & 0 & 0 & 0 \end{pmatrix}
,\quad {}_2B^{-1}=q^2\begin{pmatrix} 0 & \lambda q^2 & 0 & 0 & 0 & -q^2 \\ \lambda & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & q^2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & -q^2 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\]
\[ {}_3B^{-1}=
-\lambda & 0 & 0 & -1 \\
0 & 0 & q^2 & 0 \\
0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0
\end{pmatrix}
,\quad {}_4B^{-1}=q^6\]
in the basis enumerations $12,13,14,23,24,34$ and $123,124,134,234$ respectively for the middle cases here. In particular, we see that $\mu=q^6$. The matrix ${}_1B^{-1}$ here is the inverse of the matrix $g_{ij}$ in our basis and necessarily gives the coefficients of the metric $g\in \Lambda^1\tens\Lambda^1$, and we note also that $\mu=1/\det(g)$ in this basis. We then carefully integrate against this $\exp$, for example
\[ \sharp(e_be_d)=\int e_be_d e_ae_c \tens (\lambda q^2e_ae_b-q^2e_be_d)+\int e_be_d e_c e_d\tens (-q^2 e_ae_b)\]
where we read from the 2nd row of ${}_2B^{-1}$ for the terms in $\exp$ of the form $e_ae_c\tens\cdots$ and from the last row for terms of the form $e_ce_d\tens\cdots$. The other possibilities in our basis for the first tensor factor of $\exp$ have zero integral. We then evaluate the first displayed integral as $-q^2$ and the second integral as $\lambda$ on using the relations of the exterior algebra, to give $q^4e_be_d+(1-q^4)e_ae_b$ as stated. Integrating against $g$ is easier and gives $\sharp$ on degree 3. We could now deduce $\sharp$ on degree 1 using Proposition <ref> that $\sharp^2=\mu$ on degrees $D\ne 2$ but one can also compute it similarly by integrating against ${}_3B^{-1}$ for a direct calculation and then verify $\sharp^2$. The polynomial identify for $\sharp$ on degree 2 is a direct calculation.
We see that $\sharp$ on degree 2 is not of finite order for generic $q$ but is a deformation of order 2. Indeed, $\mu^{-1}=\det(g)$ (see the proof above) suggests a geometric normalisation to $\sharp'=\mu^{-{1\over 2}}\sharp=q^{-3}\sharp$ in our case, then $\sharp'$ is involutive on degree $D\ne 2$ and obeys the standard $q$-Hecke relation
\[ \sharp'{}^2=\id + (q-q^{-1})\sharp'\]
on degree 2. This is the same relation as obeyed by the braiding in the defining representation of the quantum group, which is also the braiding on the generators of the associated quantum plane. One can also compute $\sharp^\star$ directly and verify that it is given on degree 2 by $q^6S\sharp^{-1}$ as in Corollary <ref> (and otherwise equals $\sharp$). Here the braided antipode on degree 2 is obtained from $S(e_ie_j)=-\cdot \tilde\Psi(e_i\tens e_j)$ as
\[ S(e_ae_b)=q^2e_ae_b,\quad S(e_ae_c)=q^{-2}e_ae_c,\quad S(e_ae_d)=e_ae_d-\lambda e_be_c\]
\[ S(e_be_c)=e_be_c +\lambda q^2 e_de_a,\quad S(e_be_d)=\lambda q^2 e_ae_b-e_de_b,\quad S(e_ce_d)=q^2e_ce_d.\]
One may similarly compute $S$ on degrees 3,4
to find $S=(-1)^D\id$ on all degrees $D\ne 2$ as must be the case by Proposition <ref>. Note that this feature of the antipode is not true for the outer degrees of all braided exterior algebras, see Example <ref>. Finally, one can check that $[\sharp,S]=0$ on all degrees as it must by Corollary <ref>, which in degree 2 provides a very good cross-check of both the displayed $S$ and $\sharp$ computations.
The cohomology for this calculus in characteristic zero and for generic $q$ is known to be $H^0=k, H^1=k, H^2=0, H^3=k, H^4=k$, which is the same as in Example <ref>. One can see that $\sharp$ is again an isomorphism $H^m\isom H^{4-m}$, the cohomologies being spanned by $1, \theta=e_a+e_d$ in degrees 0,1 and their $\sharp$ in degrees 3,4, where $\sharp\theta=-q^4e_be_c e_z$ deforms the classical 3-volume. Applications of this theory to $q$-electromagnetism will be considered elsewhere. Here we consider only the general result in Section 3.2 for the Laplacian on functions. We first recall from <cit.> that
\begin{eqnarray}\label{extdSL2mon}
\extd(c^k b^n d^m)&=& \left(q^{m+n-k}-1\right)\, c^k b^n d^m
\lambda q^{n}(k)_{q}\, c^{k-1} b^n\, d^{m+1}\, e_b \nonumber
\\
&&\kern-20pt + \lambda q^{-k}\,
\left(q^{m-1}(m+n)_{q}\, c^{k+1} b^nd^{m-1}+(n)_{q}\,
c^kb^{n-1}d^{m-1}\right)e_c \nonumber
\\
&&\kern-20pt +\lambda^2\, q \left((k+1)_{q}\, (m+n)_{q}\, c^k b^n d^m
q^{-m}(n)_{q}\, (k)_{q}\, c^{k-1}b^{n-1}d^m\right) e_a
\nonumber
\\
&&\kern-20pt +\left(q^{-m-n+k}-1\right)\, c^k b^n d^m e_a
\end{eqnarray}
where $(n)_q=(q^n-q^{-n})/(q-q^{-1})$. Terms with negative powers of $c,b$ are treated as zero.
This is only part of the algebra but there is a similar formula for the other part of the basis with $a$ in place of $d$ (in the classical limit this approach corresponds to patches where $d^{-1}$ and $a^{-1}$ respectively are adjoined). Writing
\[ \extd f= (\del^bf)e_b+(\del^cf) e_c+(\del^zf) e_z+(\del^0 f)\theta\]
as the definition of our partial derivatives in this basis, it is shown in <cit.> that
\[\del^0={q^2\lambda^2\over (2)_q}\Delta_q\ ,\]
\[ \Delta_q(c^k b^n d^m)=q^{-m}(k)_q(n)_q c^{k-1} b^{n-1} d^m+\left({k+n+m\over 2}\right)_q\left({k+n+m\over 2}+1\right)_q c^k b^n d^m \]
is the naturally arising $q$-deformed Laplace-Beltrami operator on $SL_2$.
For $q^2\ne\pm1$ and the quantum metric in Proposition <ref> on the 4D calculus on $k_q[SL_2]$, the Hodge Laplacian on degree 0 is
\[ \square|_0=g_{ij}\del^i\del^j=2 q^{-1}\lambda^2\Delta_q\]
where $i,j$ are summed over our basis, for example $\{e_b,e_c,e_z,e_0\}$, and $g_{ij}=(e_i,e_j)$ are the metric coefficients.
Writing $e_a=(\theta+e_z)/(1+q^{-2})$ and $e_d=(\theta-q^2 e_z)/(1+q^2)$ we have
\[ \del^z={1\over 1+q^{-2}}(\del^a-\del^d),\quad \del^0={1\over 1+q^{-2}}(\del^a+ q^{-2} \del^d),\]
in terms of the partial derivatives in our original basis read off from (<ref>). The former comes out as
out on $c^kb^nd^m$ as
\[ \del^z={\lambda\over (2)_q}(q^{m+n+2}(k)_q-q^{-k}(m+n)_q)+ {\lambda^2\over (2)_q}q^{2-m}(k)_q(n)_q S_c^- S_b^-\]
where $S_c^-$ lowers the degree of $c$ by 1, etc. A similar computation of $\del^a+q^{-2}\del^d$ gives $\del^0$ as stated. We next compute on $c^k b^n d^m$ that
\begin{eqnarray*}&&\kern-20pt \del^b\del^c+q^{-2}\del^c\del^b = \del^b\lambda q^{-k+m-1}(m+n)_q S_c^+ S_d^-+ \del^b\lambda q^{-k}(n)_q S_b^- S_d^-+ q^{-2}\del^c\lambda q^n(k)_q S_c^- S_d^+\\
&&=2 \lambda^2 q^{n-k-1}(k)_q(n)_qS_c^-S_b^-+\lambda^2 q^{n-k-1+m}\left((k)_q(m+n+1)_q+(m+n)_q(k+1)_q\right)
\end{eqnarray*}
Next, we write for brevity
\[ A=q^{m+n+2}(k)_q-q^{-k}(m+n)_q,\quad B=\lambda q^2({k+n+m\over 2})_q({k+n+m\over 2}+1)_q\]
\[ T=\lambda S_c^- S_b^- q^{2-m}(k)_q (n)_q\]
where $k,n,m$ are now the degree operators for the powers of $c,b,d$ respectively when acting on a monomial. Then $\del^z={\lambda\over(2)_q}(A+T)$ and $\del^0={\lambda\over(2)_q}(B+T)$ and
\[ \del^z{}^2-\del^0{}^2={\lambda^2\over (2)_q^2}\left(A^2-B^2+2 T(A-B)\right)={\lambda\over (2)_q}(A+B+ 2T)\left(1-q^{m+n-k}\right)\]
on noting that $A-B={(2)_q\over\lambda}(1-q^{m+n-k})$ commutes with $T$ (since the latter changes both $k,n$ equally and does not change $m$). Putting in these results and the value of $A+B$ we obtain $\del^b\del^c+q^{-2}\del^c\del^b+q^{-3}(2)_q(\del^z{}^2-\del^0{}^2)= 2q^{-1}\lambda^2\Delta_q$. This can also be used to expresses $\del^0$ or $\Delta_q$ in terms of $\del^b,\del^c,\del^z$.
It remains to check that $\delta e_i=0$ so that Corollary <ref> applies and this is the Hodge Laplacian. Indeed $\extd (S\sharp)^{-1}e_i=-q^{-6}\extd\sharp e_i$ and $\extd\sharp e_a=0$ since
\[ \theta\sharp e_a=-q^4\theta e_ae_be_c=-q^4e_de_ae_be_c=q^4e_a e_be_ce_d=q^4e_ae_be_c\theta=-(\sharp e_a)\theta\]
using the relations of the exterior algebra. Meanwhile one finds more strongly that
\[ \theta \sharp e_i=(\sharp e_i)\theta=0;\quad i=b,c,z\]
where $\sharp e_z=q^4 e_be_c\theta$, again using the relations.
Note that we have been working algebraically but, over $\C$, our constructions are compatible with the $*$-involution corresponding to the compact real form $SU_2$.
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1511.00507
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The cross-classified sampling design consists in drawing samples from a two-dimension population, independently in each dimension. Such design is commonly used in consumer price index surveys and has been recently applied to draw a sample of babies in the French ELFE survey, by crossing a sample of maternity units and a sample of days. We propose to derive a general theory of estimation for this sampling design. We consider the Horvitz-Thompson estimator for a total, and show that the cross-classified design will usually result in a loss of efficiency as compared to the widespread two-stage design. We obtain the asymptotic distribution of the Horvitz-Thompson estimator, and several unbiased variance estimators. Facing the problem of possibly negative values, we propose simplified non-negative variance estimators and study their bias under a super-population model. The proposed estimators are compared for totals and ratios on simulated data. An application on real data from the ELFE survey is also presented, and we make some recommendations. Supplementary materials are available online.
Some key words: analysis of variance, Horvitz-Thompson estimator, independence, invariance, Sen-Yates-Grundy estimator, two-phase sampling, two-stage sampling.
Short title: Estimation under cross-classified sampling
§ =20PT INTRODUCTION
The 2011 French Longitudinal Survey on Childhood (ELFE) comprises more than 18,000 children selected on the basis of their place and date of birth. On the one hand, a sample of 320 maternity units has been drawn on the metropolitan territory. On the other hand, a sample of 25 days divided in four time periods and spread almost equally across the four seasons of 2011 has been selected. The babies born at the sampled locations and on the sampled days have been approached through midwives. Data were collected on babies whose parents consented to their inclusion during their stay at the maternity. ELFE is conducted by the National Institute for Demographic Studies (INED), the National Institute for Health and Medical Research (INSERM) and the French Blood Agency (EFS). The objective of observing children born within the same year is to analyze their physical and psychological health together with their living and environmental conditions. This large-scale study of children's development and socialization is the first of its kind in France. The collected data are now available to public and private research teams and many projects are underway in areas such as health, health environment and social sciences. In order to derive reliable confidence intervals for finite population parameters such as totals or ratios, the ELFE sampling design has to be taken into account.
The ELFE sample is drawn according to a non-standard sampling design, called Cross-Classified Sampling (CCS), following Ohlsson (1996). It consists in drawing independently two samples from each component of a two-dimensional population. In the ELFE survey, a sample of maternity units and a sample of days are independently selected. CCS is peculiar in that the same sample of days is used for each of the selected maternity units, unlike two-stage sampling where independent sub-samples are selected inside the primary sampling units (see Särndal et al., 1992). Considering CCS as a particular two-phase sampling design is possible, and may prove to be useful as in section <ref> below, but is fairly artificial. The two populations involved can be made of units with different natures (like days and maternities for the ELFE survey), leading to samples that are not part of one another but play a symmetric role. This sampling design appears in other contexts than the ELFE survey. Some examples include consumer price index surveys, as detailed in Dalén & Olhsson (1995) for the Swedish survey, where outlets and items are sampled, and business surveys (Skinner, 2015), where businesses and products are sampled. Due to its particular properties, CCS deserves a specific attention. However, as noted by Skinner (2015), "the literature on the theory of cross-classified sampling is very limited". In particular, no general theory is derived under the finite population framework. While the papers by Vos (1964) and Ohlsson (1996) focus on simple random sampling without replacement, Skinner (2015) give some results under stratified without replacement simple random sampling and under with replacement unequal probability sampling.
In the present paper, we develop a general theory for estimation and variance estimation under CCS. The asymptotic normality of the Horvitz-Thompson estimator is derived under some mild conditions. A comparison with a two-stage sampling design is carried out in a general framework. We also raise an issue, not reported before, of possible negative values for Horvitz-Thompson and Yates-Grundy variance estimates. This problem occurs even in the simplest case of simple random sampling without replacement. Non-negative simplified variance estimators are therefore introduced. Conditions for their approximate unbiasedness are given under a design-based and a model-based approach. The properties of our variance estimators are evaluated through a small but realistic simulation study when estimating totals and ratios. Finally, an application to the ELFE data is detailed.
§ =20PT CROSS-CLASSIFIED SAMPLING DESIGN
§.§ =20pt Notations and Horvitz-Thompson estimation
Keeping in mind the ELFE survey, we consider a population $U_M$ of $N_M$ maternities and a population $U_D$ of $N_D$ days. However, the developments below are completely general and may be applied to any populations $U_M$ and $U_D$. We will use the indexes $i$ and $j$ for the maternities, and the indexes $k$ and $l$ for the days. We consider a sampling design $p_M(\cdot)$ on the population $U_M$, leading to a sample $S_M$ of (average) size $n_M$, and a sampling design $p_D(\cdot)$ on the population $U_D$ leading to a sample $S_D$ of (average) size $n_D$. We assume that the two samples are selected independently. The cross-classified sampling design $p(\cdot)$ on the product population $U=U_M \times U_D$ is therefore defined as
\begin{eqnarray*}
p(s)=p_M(s_M) \times p_D(s_D) & \textrm{ for any } & s=s_M \times s_D \subset U_M \times U_D.
\end{eqnarray*}
Let $\pi_i^M$ denote the probability that $i$ is selected in $S_M$, $\pi_{ij}^M$ denote the probability that units $i$ and $j$ are selected jointly in $S_M$, and let $\Delta_{ij}^M=\pi_{ij}^M-\pi_i^M \pi_j^M$. The quantities $\pi_k^D$, $\pi_{kl}^D$ and $\Delta_{kl}^D$ are similarly defined. We assume that the first and second-order inclusion probabilities are non-negative in each population. The probability for the couple $(i,k)$ to be selected in the product sample $S_M \times S_D$ is $\pi_i^M \pi_k^D$, and the probability for the couples $(i,k)$ and $(j,l)$ to be selected jointly in the product sample $S_M \times S_D$ is $\pi_{ij}^M \pi_{kl}^D$.
We are interested in some non-negative variable of interest with value $Y_{ik}$ for the maternity $i$ and the day $k$. The total $t_Y = \sum_{i\in U_M} \sum_{k\in U_D} Y_{ik}$ is then unbiasedly estimated by the Horvitz-Thompson (HT) estimator
\begin{eqnarray} \label{estim:ht:prod}
\hat{t}_{Y} & = & \sum_{i\in S_M} \sum_{k\in S_D} \frac{Y_{ik}}{\pi^M_i \pi^D_k} = \sum_{i\in S_M} \sum_{k\in S_D} \check{Y}_{ik} \;\; \mbox{ where }\;\; \check{Y}_{ik}=\frac{Y_{ik}}{\pi_i^M\pi_k^D}.
\end{eqnarray}
Making use of the independence between $S_M$ and $S_D$, the variance of the HT-estimator is
\begin{eqnarray} \label{eq:var}
V_{CCS}\left(\hat{t}_{Y}\right) & = & \sum_{i,j \in U_M} \sum_{k,l \in U_D} \, \Gamma_{ijkl} \,\check{Y}_{ik}\check{Y}_{jl}
\end{eqnarray}
where $\Gamma_{ijkl}=\pi_{ij}^M \pi_{kl}^D - \pi_{i}^M \pi_{j}^M \pi_{k}^D \pi_{l}^D$. The Sen(1953)-Yates-Grundy(1953) form
\begin{eqnarray} \label{eq:varYG}
V_{CCS}\left(\hat{t}_{Y}\right) & = & -\frac{1}{2}\sum_{(i,k) \neq (j,l) \in U_M \times U_D} \Gamma_{ijkl} \left(\check{Y}_{ik} - \check{Y}_{jl}\right)^2
\end{eqnarray}
can be used alternatively when both sampling designs are of fixed size.
Our set-up can be linked to the usual two-stage framework, by considering $U_M$ as a population of Primary Sampling Units (PSUs) and $U_D$ as a population of Secondary Sampling Units (SSUs), each maternity $i$ being associated to the same population of days. In case of two-stage sampling, denoted by $MD$, a first-stage sample $S_M$ is selected in $U_M$, and some second-stage samples $S_{i}$ are selected independently inside any $i \in S_M$. The variance of the HT-estimator is then
\begin{eqnarray}
V_{MD}\left(\hat{t}_{Y}\right) & = & V^{PSU}_{MD} \left(\hat{t}_Y\right) +V^{SSU}_{MD} \left(\hat{t}_Y\right) \label{var:ht:twost:1}
\end{eqnarray}
\begin{eqnarray}
V^{PSU}_{MD} \left(\hat{t}_Y \right) & = & \sum_{i,j \in U_M} \sum_{k,l \in U_D} \Delta_{ij}^M \pi_k^D \pi_l^D \check{Y}_{ik} \check{Y}_{jl},\label{V:PSU:HT:1} \\
V^{SSU}_{MD} \left(\hat{t}_Y \right) & = & \sum_{i \in U_M} \sum_{k,l \in U_D} \pi_i^M \Delta_{kl}^D \check{Y}_{ik} \check{Y}_{il}. \label{V:SSU:HT:1}
\end{eqnarray}
Alternatively, we could consider $U_D$ as a population of PSUs and $U_M$ as a population of SSUs, each day $k$ being associated to the same population of maternities. In this case, the variance of the HT-estimator under two-stage sampling is
\begin{eqnarray}
V_{DM}\left(\hat{t}_{Y}\right) & = & V^{PSU}_{DM} \left(\hat{t}_Y\right) +V^{SSU}_{DM} \left(\hat{t}_Y\right) \label{var:ht:twost:2}
\end{eqnarray}
\begin{eqnarray}
V^{PSU}_{DM} \left(\hat{t}_Y \right) & = & \sum_{k,l \in U_D} \sum_{i,j \in U_M} \Delta_{kl}^D \pi_i^M \pi_j^M \check{Y}_{ik} \check{Y}_{jl},\label{V:PSU:HT:2} \\
V^{SSU}_{DM} \left(\hat{t}_Y \right) & = & \sum_{k \in U_D} \sum_{i,j \in U_M} \pi_k^D \Delta_{ij}^M \check{Y}_{ik} \check{Y}_{il}. \label{V:SSU:HT:2}
\end{eqnarray}
The different features of CCS and two-stage sampling on a two-dimension population are illustrated on Figure <ref>.
(1.3,7) node[text=red]Days ;
(-1,4.5) node[text=red,rotate=90]Maternities ;
in 0.4,0.6,...,3
(,6.05cm) – (,5.95cm) ;
in 5.6,5.4,...,3
(0.05cm,) – (-0.05cm,) ;
(1,5.2) circle (2pt) ; (1.4,5.2) circle (2pt) ; (2.4,5.2) circle (2pt) ;
(1,5) circle (2pt) ; (1.4,5) circle (2pt) ; (2.4,5) circle (2pt) ;
(1,4.4) circle (2pt) ; (1.4,4.4) circle (2pt) ; (2.4,4.4) circle (2pt) ;
(1,3.6) circle (2pt) ; (1.4,3.6) circle (2pt) ; (2.4,3.6) circle (2pt) ;
[<-] (1,6.25) – (1,6.5); [<-] (1.4,6.25) – (1.4,6.5); [<-] (2.4,6.25) – (2.4,6.5);
[<-] (-0.25,5.2) – (-0.5,5.2); [<-] (-0.25,5) – (-0.5,5); [<-] (-0.25,4.4) – (-0.5,4.4); [<-] (-0.25,3.6) – (-0.5,3.6);
[dotted,draw=red] (0.2,5.2)–(3,5.2);
[dotted,draw=red] (0.2,5)–(3,5);
[dotted,draw=red] (0.2,4.4)–(3,4.4);
[dotted,draw=red] (0.2,3.6)–(3,3.6);
[dotted,draw=red] (1,5.8)–(1,3);
[dotted,draw=red] (1.4,5.8)–(1.4,3);
[dotted,draw=red] (2.4,5.8)–(2.4,3);
(1.6,2) node[text=red]1st stage: Days ;
(-0.5,-0.5) node[text=,rotate=90]Maternities ;
(-1,-0.5) node[text=,rotate=90]2nd stage: ;
in 0.4,0.6,...,3
(,1.05cm) – (,0.95cm) ;
in 0.6,0.4,...,-2
(0.05cm,) – (-0.05cm,) ;
(1,0.2) circle (2pt) ; (1.4,0.8) circle (2pt) ; (2.4,0.6) circle (2pt) ;
(1,-0.4) circle (2pt) ; (1.4,0) circle (2pt) ; (2.4,-1.6) circle (2pt) ;
(1,-0.6) circle (2pt) ; (1.4,-0.6) circle (2pt) ; (2.4,-0.6) circle (2pt) ;
(1,-1.4) circle (2pt) ; (1.4,-1.8) circle (2pt) ; (2.4,-1.4) circle (2pt) ;
[<-] (1,1.25) – (1,1.5); [<-] (1.4,1.25) – (1.4,1.5); [<-] (2.4,1.25) – (2.4,1.5);
[dotted,draw=red] (1,0.8)–(1,-2);
[dotted,draw=red] (1.4,0.8)–(1.4,-2);
[dotted,draw=red] (2.4,0.8)–(2.4,-2);
(1.6,4) node[text=]2nd stage: Days ;
(-1,1.5) node[text=red,rotate=90]Maternities ;
(-1.5,1.5) node[text=red,rotate=90]1st stage: ;
in 0.4,0.6,...,3
(,3.05cm) – (,2.95cm) ;
in 2.6,2.4,...,0
(0.05cm,) – (-0.05cm,) ;
(1,2.2) circle (2pt) ; (1.6,2.2) circle (2pt) ; (1.8,2.2) circle (2pt) ;
(0.6,2) circle (2pt) ; (0.4,2) circle (2pt) ; (2.4,2) circle (2pt) ;
(0.8,1.4) circle (2pt) ; (2,1.4) circle (2pt) ; (2.6,1.4) circle (2pt) ;
(1,0.6) circle (2pt) ; (1.4,0.6) circle (2pt) ; (2.8,0.6) circle (2pt) ;
[<-] (-0.25,2.2) – (-0.5,2.2); [<-] (-0.25,2) – (-0.5,2); [<-] (-0.25,1.4) – (-0.5,1.4); [<-] (-0.25,0.6) – (-0.5,0.6);
[dotted,draw=red] (0.2,2.2)–(3,2.2);
[dotted,draw=red] (0.2,2)–(3,2);
[dotted,draw=red] (0.2,1.4)–(3,1.4);
[dotted,draw=red] (0.2,0.6)–(3,0.6);
Cross-classified sampling (left panel), two-stage sampling $DM$ with primary units in $U_D$ (central panel), two-stage sampling $MD$ with primary units in $U_M$ (right panel)
§.§ =20pt Variance decomposition for cross-classified sampling
The covariance $\Gamma_{ijkl}$ may be written in several ways, leading to alternative variance decompositions. Plugging $\Gamma_{ijkl} = \pi_{kl}^D \Delta_{ij}^M + \pi_{ij}^M \Delta_{kl}^D - \Delta_{ij}^M \Delta_{kl}^D$ into (<ref>) gives
\begin{eqnarray}
V_{CCS}\left(\hat{t}_{Y}\right) & = & V_{1} \left(\hat{t}_Y\right) +V_{2} \left(\hat{t}_Y\right) - V_{3} \left(\hat{t}_Y\right)\label{eq:var1-3}
\end{eqnarray}
\begin{eqnarray}
V_{1} \left(\hat{t}_Y\right) & = & \sum_{k,l \in U_D} \sum_{i,j \in U_M} \pi_{kl}^D \Delta_{ij}^M\, \check{Y}_{ik}\check{Y}_{jl}, \label{eq:var1MD}\\
V_{2} \left(\hat{t}_Y\right) & = & \sum_{i,j \in U_M} \sum_{k,l \in U_D} \pi_{ij}^M \Delta_{kl}^D\, \check{Y}_{ik}\check{Y}_{jl}, \label{eq:var1DM}\\
V_{3} \left(\hat{t}_Y\right) & = & \sum_{i,j \in U_M} \sum_{k,l \in U_D} \Delta_{ij}^M \Delta_{kl}^D \check{Y}_{ik}\check{Y}_{jl}.\label{eq:var3}
\end{eqnarray}
Plugging $\Gamma_{ijkl} = \Delta_{ij}^M \pi_k^D \pi_l^D + \Delta_{kl}^D \pi_i^M \pi_j^M + \Delta_{ij}^M \Delta_{kl}^D$ into (<ref>) gives
\begin{eqnarray}
V_{CCS}\left(\hat{t}_{Y}\right) & = & V^{PSU}_{MD} \left(\hat{t}_Y\right) +V^{PSU}_{DM} \left(\hat{t}_Y\right) +V_{3}^{} \left(\hat{t}_Y\right) \label{eq:var2+3}
\end{eqnarray}
and we have $V_{1} \left(\hat{t}_Y\right)=V^{PSU}_{MD} \left(\hat{t}_Y\right) +V_{3}^{} \left(\hat{t}_Y\right)$
and $V_{2} \left(\hat{t}_Y\right)=V^{PSU}_{DM} \left(\hat{t}_Y\right) +V_{3}^{} \left(\hat{t}_Y\right)$.
This second decomposition was originally derived by Ohlsson (1996). Other decompositions are possible, e.g. through an analysis of variance decomposition as for two-stage sampling.
§.§ =20pt Comparison with two-stage sampling
From expressions (<ref>) and (<ref>), we obtain after some algebra that
\begin{eqnarray} \label{comp:prod:twost:1}
= \sum_{i,j \in U_M} \Delta_{ij}^M \sum_{k \neq l \in U_D} \pi_{kl}^D \check{Y}_{ik} \check{Y}_{jl}.
\end{eqnarray}
In case of Poisson sampling (PO) inside $U_M$, the right-hand side in (<ref>) is non-negative and CCS is thus less efficient than two-stage sampling. In case of fixed-size sampling inside $U_M$, equation (<ref>) may be alternatively written as
\begin{eqnarray} \label{comp:prod:twost:2}
= \sum_{i \neq j \in U_M} \frac{(-\Delta_{ij}^M)}{2} \sum_{k \neq l \in U_D} \frac{\pi_{kl}^D}{\pi_{k}^D \pi_{l}^D} \left(\frac{Y_{ik}}{\pi_i^M}-\frac{Y_{jk}}{\pi_j^M}\right)\left(\frac{Y_{il}}{\pi_i^M}-\frac{Y_{jl}}{\pi_j^M}\right).
\end{eqnarray}
If the so-called Sen-Yates-Grundy conditions are respected for $p_M$, the quantities $(-\Delta_{ij}^M)$ are non-negative. If $Y_{ik}$ is roughly proportional to the size of the maternity unit $i$, as can be expected for count variables, the quantities
$$\displaystyle \left(\frac{Y_{ik}}{\pi_i^M}-\frac{Y_{jk}}{\pi_j^M}\right)\left(\frac{Y_{il}}{\pi_i^M}-\frac{Y_{jl}}{\pi_j^M}\right)$$
will tend to be positive unless the inclusion probabilities $\pi_i^M$ are defined proportionally to some measure of size. CCS sampling would then be less efficient than two-stage sampling. This result is illustrated in section <ref> on some simulated populations when both $p_M$ and $p_D$ are simple random sampling without replacement (SI) designs, and for different sample sizes.
§ =20PT VARIANCE ESTIMATION
§.§ =20pt Design-unbiased variance estimation
The HT variance estimator for $V_{CCS}\left(\hat{t}_{Y}\right)$ is
\begin{eqnarray} \label{eq:HT:var}
\hat{{V}}_{HT}\left(\hat{t}_{Y}\right) & = & \sum_{i,j \in S_M} \sum_{k,l \in S_D} \frac{\Gamma_{ijkl}}{\pi_{ij}^M \pi_{kl}^D} \,\check{Y}_{ik}\check{Y}_{jl}.
\end{eqnarray}
It may be also derived from (<ref>), leading to the alternative writing
\begin{eqnarray}
\hat{{V}}_{HT}\left(\hat{t}_{Y}\right)
& = & \hat{V}_{1,HT}^{} \left(\hat{t}_Y\right) +\hat{V}_{2,HT}^{} \left(\hat{t}_Y\right) -\hat{V}_{3,HT}^{} \left(\hat{t}_Y\right) \label{eq:HT:var1-3}
\end{eqnarray}
\begin{eqnarray}
\hat{V}_{1,HT}^{} \left(\hat{t}_Y\right) &=& \sum_{i,j\in S_M} \sum_{k,l \in S_D} \frac{\Delta_{ij}^M}{\pi_{ij}^M}\, \check{Y}_{ik}\check{Y}_{jl}, \label{eq:estHT:var1MD} \\
\hat{V}_{2,HT}^{} \left(\hat{t}_Y\right) &=& \sum_{i,j\in S_M} \sum_{k,l \in S_D} \frac{\Delta_{kl}^D}{\pi_{kl}^D}\, \check{Y}_{ik}\check{Y}_{jl}, \label{eq:estHT:var1DM} \\
\hat{V}_{3,HT}^{} \left(\hat{t}_Y\right) &=& \sum_{i,j \in S_M} \sum_{k,l \in S_D} \frac{\Delta_{ij}^M}{\pi_{ij}^M} \frac{\Delta_{kl}^D}{\pi_{kl}^D} \,\check{Y}_{ik}\check{Y}_{jl}.\label{eq:estHT:var3}
\end{eqnarray}
If $p_M$ and $p_D$ are both Poisson sampling designs, this variance estimator is always non-negative. Otherwise, it may take negative values even if $p_M$ and $p_D$ are both
SI designs (denoted by SI$^2$) as illustrated in section <ref>. When $p_M$ and $p_D$ are both fixed-size sampling designs, we may alternatively consider the Yates-Grundy like variance estimator:
\begin{eqnarray}
\hat{V}_{YG}\left(\hat{t}_{Y}\right)
& = & \hat{V}_{1,YG}^{} \left(\hat{t}_Y\right) +\hat{V}_{2,YG}^{} \left(\hat{t}_Y\right) -\hat{V}_{3,YG}^{} \left(\hat{t}_Y\right)\label{eq:YG:var1-3}
\end{eqnarray}
\begin{eqnarray}
\hat{V}_{1,YG}^{} \left(\hat{t}_Y\right) &=& -\frac{1}{2} \sum_{i \neq j \in S_M} \frac{\Delta_{ij}^M}{\pi_{ij}^M} \left(\frac{\hat{Y}_{i \bullet}}{\pi_i^M}-\frac{\hat{Y}_{j \bullet}}{\pi_j^M}\right)^2 \\
\hat{V}_{2,YG}^{} \left(\hat{t}_Y\right) &=& -\frac{1}{2} \sum_{k \neq l \in S_D} \frac{\Delta_{kl}^D}{\pi_{kl}^D} \left(\frac{\hat{Y}_{\bullet k}}{\pi_k^D}-\frac{\hat{Y}_{\bullet l}}{\pi_l^D}\right)^2 \\
\hat{V}_{3,YG}^{} \left(\hat{t}_Y\right) &=& - \frac{1}{2}\sum_{(i,k) \neq (j,l) \in S_M \times S_D} \frac{\Delta_{ij}^M \Delta_{kl}^D}{\pi_{ij}^M \pi_{kl}^D} \left(\check{Y}_{ik} - \check{Y}_{jl}\right)^2
\end{eqnarray}
with $\hat{Y}_{\bullet k} = \sum_{i \in S_M} Y_{ik} / \pi_i^M$ is the estimated sub-total for the day $k$ and $\hat{Y}_{i \bullet} = \sum_{k \in S_D} Y_{ik} / \pi_k^D$ is the estimated sub-total for the maternity $i$.
It can be proved that $\hat{V}_{HT}\left(\hat{t}_{Y}\right)$ in (<ref>) and $\hat{V}_{YG}\left(\hat{t}_{Y}\right)$ in (<ref>) match term by term, when $p_M$ and $p_D$ are stratified simple random sampling designs. If both sampling designs satisfy the Sen-Yates-Grundy conditions (SYG), the terms $\hat{V}_{1,YG}^{} \left(\hat{t}_Y\right)$ and $\hat{V}_{2,YG}^{} \left(\hat{t}_Y\right)$ are non-negative. However, the term $\hat{V}_{3,YG}^{} \left(\hat{t}_Y\right)$ is usually non-negative, which may lead to negative values for $\hat{V}_{YG}^{} \left(\hat{t}_Y\right)$ as illustrated in the simulations of section <ref>. It is thus desirable to exhibit non-negative variance estimators with limited bias.
§.§ =20pt Non-negative variance estimators
We consider the variance decomposition in (<ref>), and study the relative order of magnitude of the components. We make the following assumptions:
H1: There exist some constants $\alpha_1$ and $\alpha_2$ such that
\begin{eqnarray*}
\forall k \in U_D, ~~ \frac{1}{N_M} \sum_{i \in U_M} Y_{ik}^2 \leq \alpha_1, & \textrm{ and } & \forall i \in U_M, ~~ \frac{1}{N_D} \sum_{k \in U_D} Y_{ik}^2 \leq \alpha_2.
\end{eqnarray*}
H2: There exists some constants $\lambda_1>0$ and $\lambda_2>0$ such that
\begin{eqnarray*}
\forall k \in U_D, ~~ \pi_{k}^D \geq \lambda_1 \frac{n_D}{N_D}, & \textrm{ and } & \forall i \in U_M, ~~ \pi_{i}^M \geq \lambda_2 \frac{n_M}{N_M}.
\end{eqnarray*}
H3: There exist some constants $\gamma_1$ and $\gamma_2$ such that
\begin{eqnarray*}
\forall k \neq l \in U_D, ~~ \frac{N_D^2}{n_D} \sup_{k \neq l \in U_D} \left|\Delta_{kl}^D\right| \leq \gamma_1, & \textrm{ and } &
\forall i \neq j \in U_M, ~~ \frac{N_M^2}{n_M} \sup_{i \neq j \in U_M} \left|\Delta_{ij}^M\right| \leq \gamma_2.
\end{eqnarray*}
H4: There exists some constant $\delta>0$ such that
\begin{eqnarray*}
V_{CCS}\left(\hat{t}_{Y}\right) & \geq & \delta N_M^2 N_D^2 \left( \frac{1}{n_M} + \frac{1}{n_D} \right).
\end{eqnarray*}
It is assumed in (H1) that the variable $y$ has bounded moments of order 2 for each maternity $i$ and for each day $k$. Assumptions (H2) and (H3) are classical in survey sampling and are satistified for many sampling designs, see for example Cardot et al. (2013). It is assumed in (H4) that the variance of the HT-estimator under CCS sampling has the order $N_M^2 N_D^2 (n_M^{-1}+n_D^{-1})$. From assumptions (H1-H4), there exist some constants $C_1$, $C_2$ and $C_3$ such that
\begin{eqnarray}
\frac{V_{1}^{} \left(\hat{t}_Y\right)}{V_{CCS}\left(\hat{t}_{Y}\right)} & \leq & C_1 \,\frac{1}{1+n_M n_D^{-1}}, \label{order:V1:DM}\\
\frac{V_{2}^{} \left(\hat{t}_Y\right)}{V_{CCS}\left(\hat{t}_{Y}\right)} & \leq & C_2 \,\frac{1}{1+n_D n_M^{-1}}, \label{order:V1:MD}\\
\frac{V_{3} \left(\hat{t}_Y\right)}{V_{CCS}\left(\hat{t}_{Y}\right)} & \leq & C_3 \,\frac{1}{n_D n_M^{-1}+n_M n_D^{-1}} \label{order:V3}
\end{eqnarray}
The proof is given in Appendix <ref>. It follows from (<ref>)-(<ref>) that if $n_D$ is large and $n_M$ is bounded, both $V_{2}^{} \left(\hat{t}_Y\right)$ and $V_{3} \left(\hat{t}_Y\right)$ are negligible and a non-negative simplified variance estimator can be derived by focusing on $V_{1}^{} \left(\hat{t}_Y\right)$ only. This leads to
\begin{eqnarray} \label{vsimp:1}
\hat{V}_{\text{SIMP1}} \left(\hat{t}_Y\right) & = & \hat{V}_{1,YG}^{} \left(\hat{t}_Y\right).
\end{eqnarray}
If the sampling design $p_D$ satisfies the SYG conditions, this simplified estimator is always non-negative. In the particular $\mbox{SI}^2$ case, we obtain
\begin{eqnarray} \label{vsimp:1:SI2}
\hat{V}_{\text{SIMP1}} \left(\hat{t}_Y\right) & = & N_{M}^2 \left(\frac{1}{n_{M}}-\frac{1}{N_{M}}\right) s_{\hat{Y}_{\circ \bullet}}^2
\end{eqnarray}
\begin{eqnarray}
s_{\hat{Y}_{\circ \bullet}}^2 &=& \frac{1}{n_{M}-1} \sum_{i \in S_{M}} \left( \hat{Y}_{i \bullet} - \frac{1}{n_{M}} \sum_{j \in S_{M}} \hat{Y}_{j \bullet}\right)^2 .
\end{eqnarray}
Symmetrically, both $V_{1}^{} \left(\hat{t}_Y\right)$ and $V_{3} \left(\hat{t}_Y\right)$ may be seen as negligible if $n_M$ is large and $n_D$ is bounded. Another simplified variance estimator is thus
\begin{eqnarray} \label{vsimp:2}
\hat{V}_{\text{SIMP2}} \left(\hat{t}_Y\right) & = & \hat{V}_{2,YG}^{} \left(\hat{t}_Y\right).
\end{eqnarray}
If the sampling design $p_M$ satisfies the SYG conditions, this estimator is non-negative. In the particular $\mbox{SI}^2$ case, we have
\begin{eqnarray} \label{vsimp:2:SI2}
\hat{V}_{\text{SIMP2}} \left(\hat{t}_Y\right) & = & N_D^2 \left(\frac{1}{n_D}-\frac{1}{N_D}\right)s^2_{\hat{Y}_{\bullet\circ}}
\end{eqnarray}
\begin{eqnarray}
s^2_{\hat{Y}_{\bullet\circ}} & = & \frac{1}{n_{D}-1} \sum_{k \in S_{D}} \left( \hat{Y}_{\bullet k} - \frac{1}{n_{D}} \sum_{l \in S_{D}} \hat{Y}_{\bullet l}\right)^2 .
\end{eqnarray}
A third possible simplified variance estimator is
\begin{eqnarray} \label{vsimp:3}
\hat{V}_{\text{SIMP3}} \left(\hat{t}_Y\right) &=& \hat{V}_{\text{SIMP1}} + \hat{V}_{\text{SIMP2}} \nonumber \\
& = & \hat{V}_{1,YG}^{} \left(\hat{t}_Y\right)+\hat{V}_{2,YG}^{} \left(\hat{t}_Y\right).
\end{eqnarray}
This estimator is non-negative if both $p_D$ and $p_M$ satisfy the SYG conditions. It is approximately unbiased for $V_{CCS}\left(\hat{t}_{Y}\right)$ if $n_D$ is large and $n_M$ is bounded, or if $n_M$ is large and $n_D$ is bounded. In the particular $\mbox{SI}^2$ case
\begin{eqnarray} \label{vsimp:3:SI2}
\hat{V}_{\text{SIMP3}} \left(\hat{t}_Y\right) = N_{M}^2 \left(\frac{1}{n_{M}}-\frac{1}{N_{M}}\right) s_{\hat{Y}_{\circ \bullet}}^2 + N_D^2 \left(\frac{1}{n_D}-\frac{1}{N_D}\right)s^2_{\hat{Y}_{\bullet\circ}}.
\end{eqnarray}
Similar formula can be easily derived in the case of stratified simple random sampling without replacement and will be used in Section 5.
§.§ =20pt Relative bias under a superpopulation model
We consider the following superpopulation model
\begin{eqnarray} \label{model}
Y_{ik} = \mu + \sigma_M U_i + \sigma_D V_k + \sigma_E W_{ik}
\end{eqnarray}
where $U_i$, $V_k$ and $W_{ik}$ are independently generated according to a standard normal distribution. This is an analysis of variance model with two crossed random factors and without repetition. Let “$E_m$" denote the expectation with respect to the model (<ref>) and “$E_{p}$" denote the expectation with respect to the CCS design. For each simplified variance estimator
$\hat{{V}}_{\textrm{SIMP}i}$, $i=1,2,3$, the relative bias RB under the model and under the sampling design is defined by
\begin{eqnarray}
\textrm{RB}_{m,p} \left[\hat{{V}}_{\textrm{SIMP}i} \left(\hat{t}_Y\right) \right] = \frac{{E}_m \left\{{E}_p \left[\hat{{V}}_{\textrm{SIMPi}}\left(\hat{t}_Y\right)\right] - V_{CCS} \left(\hat{t}_Y\right)\right\}}{{E}_m \left[ V_{CCS}\left(\hat{t}_Y\right) \right]}.
\end{eqnarray}
In the $\mbox{SI}^2$ case, these relative biases are of the form
\begin{eqnarray} \label{rb:mp:vsimp12}
\textrm{RB}_{m,p} \left[\hat{{V}}_{\textrm{SIMP}i} \left(\hat{t}_Y\right) \right] &=& - 1/ (1 + A_i)
\end{eqnarray}
for $i=1$ and 2 and
\begin{eqnarray} \label{rb:mp:vsimp3}
\textrm{RB}_{m,p} \left[\hat{{V}}_{\textrm{SIMP}3} \left(\hat{t}_Y\right) \right] &=& 1/ (1 + A_3)
\end{eqnarray}
for some positive constant $A_i$, $i=1,2,3,$ depending on $\sigma_M$, $\sigma_D$, $\sigma_E$ and $n_M$, $N_M$, $n_D$ and $N_D$, see equations (<ref>)-(<ref>). Equations (<ref>) and (<ref>) imply that the two first simplified variance estimators are negatively biased while the third one is positively biased. Using the notations $r_M=\sigma^2_M/\sigma^2_E$, $r_D=\sigma^2_D/\sigma^2_E$, $f_M=n_M/N_M$ and $f_D=n_D/N_D$, we have
\begin{eqnarray}
A_1 & = & \frac{1 - f_M }{1 - f_D} \,\frac{n_D r_M +1 }{n_M r_D + f_M }, \label{A1}\\
A_2 & = & \frac{1 - f_D }{1 - f_M}\, \frac{n_M r_D +1 }{n_D r_M + f_D }, \label{A2}\\
A_3 & = & \frac{n_D r_M + f_D }{1 - f_D} + \frac{n_M r_D + f_M }{1 - f_M}. \label{A3}
\end{eqnarray}
The bias of $\hat{{V}}_{\textrm{SIMP}1}$ increases from $-1$ to $0$ when $A_1$ increases, which occurs in particular when the ratio $r_M$ or the sample size $n_D$ increases. In other words, $\hat{{V}}_{\textrm{SIMP}1}$ will have a small bias under model (<ref>) if the variable of interest contains some maternity effect or if the number of sampled days is large enough. Symmetrically, $\hat{{V}}_{\textrm{SIMP}2}$ will have a small bias under model (<ref>) if the variable of interest contains some day effect or if the number of sampled maternities is large enough. The bias of $\hat{{V}}_{\textrm{SIMP}3}$ decreases from $1$ to $0$ when $A_3$ increases, which occurs in particular when $r_M$ or $r_D$ increases, or when $n_M$ or $n_D$ increases. In other words, $\hat{{V}}_{\textrm{SIMP}3}$ will have a small bias under model (<ref>) if the variable of interest contains some maternity or some day effect, or if the number of sampled days or the number of sampled maternities is large enough. The simulation study in section <ref> supports these results, and confirm that the variance tends to be underestimated with $\hat{{V}}_{\textrm{SIMP}1}$ or $\hat{{V}}_{\textrm{SIMP}2}$, and overestimated with $\hat{{V}}_{\textrm{SIMP}3}$.
§.§ A central-limit theorem
To produce confidence intervals with appropriate asymptotic coverage, it is of interest to state a central-limit theorem (CLT) for CCS. Roughly speaking, Theorem <ref> below states that if the HT-estimator follows a CLT under both sampling designs $p_D$ and $p_M$, then the HT-estimator also follows a CLT under CCS. It is derived almost directly from Theorem 2 in Chen and Rao (2007), and the proof is therefore omitted.
Suppose that assumptions (H1)-(H4) hold. Suppose that
H5 $\sigma_{1}^{-1} V_1 \rightarrow_{\mathcal{L}} \mathcal{N}(0,1)$, where $\rightarrow_{\mathcal{L}}$ stands for the convergence in distribution under the sampling-design, with
\begin{eqnarray} \label{theo:clt:eq1}
V_1 = \frac{1}{N} \left(\sum_{i \in S_M} \frac{Y_{i\bullet}}{\pi_i^M} - \sum_{i \in U_M} Y_{i\bullet} \right) & \textrm{ and } & \sigma_{1}^2=V(V_1).
\end{eqnarray}
H6 $\sup_t |P(\sigma_{2}^{-1} U_1 \leq t|S_M)-\Phi(t)|=o_p(1)$, where $\Phi$ is the cumulative distribution function of the standard normal distribution, and where
\begin{eqnarray} \label{theo:clt:eq2}
U_1 = \frac{1}{N} \sum_{i \in S_M} \frac{1}{\pi_i^M} (\hat{Y}_{i\bullet}-Y_{i\bullet}) & \textrm{ and } & \sigma_{2}^2=V(U_1|S_M).
\end{eqnarray}
H7 $\sigma_{1}^2/\sigma_{2}^2 \rightarrow_{P} \gamma^2$, where $\rightarrow_{P}$ stands for the convergence in probability under the sampling-design.
\begin{eqnarray} \label{theo:clt:eq3}
\frac{N^{-1}(\hat{t}_Y-t_Y)}{\sqrt{\sigma_{1n}^2+\sigma_{2n}^2}} & \rightarrow_{\mathcal{L}} & \mathcal{N}(0,1).
\end{eqnarray}
For illustration, we consider the particular case when $p_D$ and $p_M$ are both SI designs. Suppose that (H2)-(H4) hold, and that (H1) is strengthened to
H1b: There exists $\delta>0$ and some constants $\alpha_1$ and $\alpha_2$ such that
\begin{eqnarray*}
\forall k \in U_D ~~ \frac{1}{N_M} \sum_{i \in U_M} Y_{ik}^{2+\delta} \leq \alpha_1, & \textrm{ and } & \forall i \in U_M ~~ \frac{1}{N_D} \sum_{k \in U_D} Y_{ik}^{2+\delta} \leq \alpha_2.
\end{eqnarray*}
Then by using the CLT in Hajek (1961), the assumption (H5) can be shown to hold. By mimicking the proof of Lemma 2 in Chen and Rao (1997), the assumption (H6) can be shown to hold as well.
§ =20PT SIMULATIONS
In this Section, two artificial populations are first generated using the superpopulation model (<ref>). In Section <ref>, CCS is compared with two stage sampling in terms of variance, which illustrates the results in Section <ref>. A Monte Carlo experiment is then presented in Section <ref>, and the variance estimators introduced in Section <ref> are compared for the estimation of a total. Some attention is paid to the issue of negative values for the unbiased variance estimator. In Section <ref>, two other populations with two variables of interest for each are generated. We focus on variance estimation for a ratio, making use of the variance estimators introduced in Section <ref> with estimated linearized variables instead of the variable of interest. The results from Tables <ref> and <ref> are readily reproducible using the R code provided in the supplementary materials of the present paper.
§.§ =20pt Comparison with two-stage sampling
Two populations are generated according to model (<ref>), with $N_M=1000$ maternities and $N_D=1000$ days for each population, and with $\mu=200$ and $\sigma_E=5$. Equal random effects $\sigma_M=\sigma_D=5$ are used for population 1, while we use $\sigma_M=0.5$ and $\sigma_D=5$ for population 2. For each population, the $\mbox{SI}^2$ sampling design is used, with sample sizes equal to $5$, $10$, $100$ and $500$. The ratios $V_{MD} \slash V_{CCS}$ between the variance under two-stage sampling and the variance under CCS are computed, and plotted in percentage on Figure <ref>. A ratio smaller than $100$ indicates that two-stage sampling is more accurate than CCS, which holds true in all cases considered in our experiment.
The ratio increases with $n_D$ and decreases when $n_M$ increases. Also, it can be observed that the ratio decreases with $\sigma_M$. This impact of the maternity effect is noticeable, and illustrates the substantial loss in accuracy induced by using a CCS instead of a two-stage sampling design if the maternity effect is small. Similar conclusions could be derived when computing the ratio $V_{DM} \slash V_{CCS}$.
Population 1 Population 2
$\sigma_M=5$, $\sigma_D=5$ $\sigma_M=0.5$, $\sigma_D=5$
$V_{MD} \slash V_{CCS}$ ( % ) for population 1 (left panel) and population 2 (right panel)
§.§ =20pt Variance estimation for a total
We consider the two artificial populations generated as described in Section <ref>. For each population, the $\mbox{SI}^2$ sampling design is used, with sample sizes equal to $5$, $10$, $100$ and $500$, and the sample selection is repeated $B=10,000$ times. For each sample $b=1,\ldots,B$, we compute the estimate $\hat{t}_Y^{(b)}$ of the total $t_Y$. The unbiased variance estimator $\hat{V}_{}^{(b)}$ and
the simplified variance estimators $\hat{V}_{\text{SIMP1}}^{(b)}$, $\hat{V}_{\text{SIMP2}}^{(b)}$, $\hat{V}_{\text{SIMP3}}^{(b)}$ are also computed for
For each variance estimator $\hat{V}$, we compute the Monte Carlo Percent Relative Bias
\begin{eqnarray*}
\mbox{RB}_{\mbox{\sc mc}}(\hat{V}) = 100 \times \frac{B^{-1} \sum_{b=1}^B \hat{V}^{(b)}-V}{V},
\end{eqnarray*}
where the true variance $V$ was approximated through an independent set of $50,000$ simulations. The number (#NEG) of negative variance estimators $\hat{V}_{}^{(b)}$ is also computed.
The results are reported in Table <ref>. The variance estimator $\hat{{V}}$ is almost unbiased in all situations, as expected. However,
this variance estimator is prone to negative values with small sample sizes when the value of $\sigma_M$ and/or the value of $\sigma_D$ is small as compared to
$\sigma_E$. The problem vanishes when the sample sizes increase. We now turn to the simplified variance estimators. The relative bias of
$\hat{{V}}_{\text{SIMP1}}$ decreases when $n_D$ increases or when $n_M$ decreases, and when $\sigma_M$ increases or when $\sigma_D$ decreases. This supports
the findings in Section <ref>. Symmetrical conclusions are drawn for the relative bias of $\hat{{V}}_{\text{SIMP2}}$. Turning to
$\hat{{V}}_{\text{SIMP3}}$, we note that the relative bias decreases when either $\sigma_M$ or $\sigma_D$ increases. This variance estimator is therefore
advisable in all cases but those where there is no maternity nor day effect.
$n_M$ 5 10 10 100 500 5 10 10 100 500
$n_D$ 5 10 100 100 500 5 10 100 100 500
${\sigma}_M$ 5c|5 5c50
${\sigma}_D$ 5c|5 5c5
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}\right)$ 1 -1 2 0 -0 1 -1 1 0 0
#NEG 6 0 0 0 0 0 0 0 0 0
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}_{SIMP1}\right)$ -43 -47 -6 -49 -49 - -2 1 -1 -1
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}_{SIMP2}\right)$ -46 -50 -91 -51 -51 -99 -99 -100 -99 -99
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}_{SIMP3}\right)$ 11 3 2 1 -0 1 -0 1 0 0
${\sigma}_M$ 5c|0.5 5c0.5
${\sigma}_D$ 5c|5 5c0.5
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}\right)$ 1 -1 0 1 -1 1 -1 2 -0 -0
#NEG 91 0 0 0 0 1393 298 0 0 0
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}_{SIMP1}\right)$ -82 -90 -81 -98 -99 -4 -9 -3 -34 -47
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}_{SIMP2}\right)$ -1 -2 -10 -0 -2 -5 -10 -52 -36 -49
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}_{SIMP3}\right)$ 18 8 9 2 -0 90 81 45 29 4
Comparison between variance estimators for a total
§.§ =20pt Variance estimation for a ratio
We now consider variance estimation for a ratio. Two populations are generated with $N_M=1000$ maternities and $N_D=1000$ days. In each population, two count variables are generated so as to mimic the data encountered in the ELFE survey. More precisely, we first generate an auxiliary variable $Z_{ik}$
according to model (<ref>) with $\mu=200$, $\sigma_E=\sigma_D=5$, and $\sigma_M=5$ or $50$. The first variable of interest $X_{ik}$ is generated
according to a Poisson distribution with parameter $Z_{ik}$. The second variable of interest $Y_{ik}$ is generated according to a binomial distribution with
parameters $X_{ik}$ and $p_{ik}$. We consider two cases: (i) equal probabilities with $p_{ik}=0.3$; (ii) unequal probabilities with $\logit(p_{ik})=\beta Z_{ik}$, where $\beta$ was chosen so that the average probability is approximately 0.3. Note that $Y_{ik}$ follows a Poisson distribution with parameter $p_{ik}Z_{ik}$.
The reason for this generating process is that some variable of interest $X_{ik}$, like the number of births in the ELFE survey, may contain some
maternity and/or day effect which is reflected in the way $Z_{ik}$ is generated. On the other hand, some maternity and/or day effect may also be contained in some other variable of interest $Y_{ik}$, like the number of births per caesarean. Such effects may be either similar to those for $X_{ik}$ like with pattern (i), or may occur differently like with pattern (ii).
For each population, the $\mbox{SI}^2$ sampling design is used, with sample sizes equal to $5$, $10$, $100$ and $500$, and the sample selection is
repeated $B=10,000$ times. For each sample $b=1,\ldots,B$, we compute the substitution estimator $\hat{R}^{(b)}={\hat{t}_Y^{(b)}} \slash {\hat{t}_X^{(b)}}$
of the ratio $R={t_Y} \slash {t_X}$. The variance estimator $\hat{V}_{}^{(b)}$ and the simplified variance estimators $\hat{V}_{\text{SIMP1}}^{(b)}$,
$\hat{V}_{\text{SIMP2}}^{(b)}$, $\hat{V}_{\text{SIMP3}}^{(b)}$ are also computed for $\hat{t}_Y^{(b)}$, where the variable of interest $Y_{ik}$ is replaced with
the estimated linearized variable of the ratio.
The results are reported in Table <ref>. The variance estimator $\hat{{V}}$ is almost unbiased in all situations, as expected,
but is prone to negative values even when the maternity or day effect is small. We now turn to the relative bias for the simplified variance estimators. With
pattern (i), the situation is much different to that when a total is estimated, since the relative bias of $\hat{{V}}_{\text{SIMP3}}$ is much
larger than for the other two simplified estimators. This can be explained as follows: when the probabilities $p_{ik}$ are uniform, both $Y_{ik}$ and $X_{ik}$
contain the same maternity and day effect, but these effects wear off in the linearized variable. Whatever the values of $\sigma_M$ and $\sigma_D$ are, the situation is therefore comparable to that observed in the bottom right cell of Table <ref>. With pattern (ii), the probabilities $p_{ik}$ depend on
$i$ and $k$, leading potentially to some remaining maternity and/or day effect in the linearized variable. In such situation, which seems more realistic in
practice, the relative bias of $\hat{{V}}_{\text{SIMP1}}$ and $\hat{{V}}_{\text{SIMP2}}$ increase when $\sigma_M$ or $\sigma_D$ increase, while the
relative bias of $\hat{{V}}_{\text{SIMP3}}$ decreases.
$n_M$ 5 10 10 100 500 5 10 10 100 500
$n_D$ 5 10 100 100 500 5 10 100 100 500
${\sigma}_M$ 5c|5 5c50
${\sigma}_D$ 5c|5 5c5
Case (i) $\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}\right)$ -0 -1 -1 0 -0 -2 -1 -1 0 1
#NEG 1645 484 14 0 0 1656 499 12 0 0
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}_{SIMP1}\right)$ -1 -1 -2 -10 -37 -1 -1 -1 -8 -32
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}_{SIMP2}\right)$ 0 -2 -10 -8 -30 -2 -1 -9 -8 -31
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}_{SIMP3}\right)$ 99 96 89 82 33 97 98 90 84 37
Case (ii) $\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}\right)$ 0 -1 2 0 -0 -4 -3 -1 -0 0
4*$p_{ik} = \frac{e^{\beta Z_{ik}}}{1 + e^{\beta Z_{ik}}}$
#NEG 1351 235 0 0 0 67 0 0 0 0
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}_{SIMP1}\right)$ -7 -13 -4 -39 -48 -5 -4 -1 -1 -1
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}_{SIMP2}\right)$ -6 -14 -61 -40 -49 -87 -93 -99 -98 -99
$\mbox{RB}_{\mbox{\sc mc}}\left(\hat{V}_{SIMP3}\right)$ 87 73 35 22 3 8 3 -0 0 0
Comparison between variance estimators for a ratio
§ =20PT APPLICATION TO THE ELFE SURVEY
ELFE is the first longitudinal study of its kind in France, tracking children from birth to adulthood. This cohort comprises more than 18,000 children
whose parents consented to their inclusion. The population of inference consists of babies born during 2011 in French maternities, excluding very premature
infants. It is a two-dimensional population with 544 maternities as spatial units and 365 days as time units. The crossing of one day and one maternity represents a cluster of infants.
The sample is obtained by CCS, where days and maternities are selected independently with selected families surveyed shortly after birth in $320$ metropolitan maternities and during $25$ days for one year. The sample selection for maternities may be modeled as stratified simple random sampling (STSI), the
population of maternities being divided into five strata of equal size. The allocation per stratum is proportional to the number of deliveries recorded in 2008.
The sample selection for days may be modeled as STSI, with four strata associated to the four seasons of 2011. The sample sizes inside strata are provided in
Tables <ref> and <ref>.
Strata Strata size Sample size
$g$ $N_{Mg}$ $n_{Mg}$
1 108 21
2 108 41
3 109 55
4 108 80
5 111 90
Total 544 287
Population and sample strata sizes for the maternities design $p_M$.
Strata Strata size Sample size
$h$ $N_{Dh}$ $n_{Dh}$
1 91 4
2 91 6
3 91 7
4 92 8
Total 365 25
Population and sample strata sizes for the days design $p_D$.
In this Section, we aim at illustrating the results previously obtained on a real data set. Some aspects of the ELFE survey, like the non-response
issue or the calibration step, deserve a specific attention but are beyond the scope of the present paper and are therefore not considered. In particular, the
ELFE survey is prone to several levels of non-response, since some sampled maternities and some families refused to participate either for some specific days
or for the whole period. In the present study, the sample of respondents is viewed as the original sample and in particular, we consider only the 287
maternities that participate during the 25 days of survey. The calibration step is not taken into account. The results below are meant to illustrate our
theoretical results, but are not intended for use in other contexts.
We consider seven count variables from the ELFE survey. Some of them depend on the characteristics of the maternities (e.g., the spatial location), like the variable indicating whether the mother is followed by a midwife. Others are related to the days of the survey, like the variable indicating whether the birth occurred by caesarean. For each variable, the estimated total $\hat{t}_{Y}$ from equation (<ref>), the estimated variance
$\hat{\mathbf V} \left(\hat{t}_{Y}\right)$ from equation (<ref>) and the three simplified estimators are given in the upper part of Table
<ref>. Similar indicators are given in the bottom part of Table <ref> for ratios, when the totals of the variables of interest are divided by
the total number of births.
Birth Born by Twins Born To have To have a To have a To have an
Caesarean within a mother mother aged primiparous immigrant
marriage followed by between 18 mother mother
a midwife and 25 years
$\hat{t}_Y$ 362924 33873 10187 160283 42337 43238 162316 44169
$\hat{\mathbf{V}} \left(\hat{t}_Y\right)$ 7,6E+07 1,5E+07 5,3E+05 2,0E+07 3,9E+06 2,6E+06 1,5E+07 3,6E+06
$\mbox{RD}\left(\hat{\mathbf{V}}_{\text{SIMP1}} \right)$ -63,7 % -95,5 % -63,5 % -64,6 % -13,2 % -49,7 % -46,5 % -58,2 %
$\mbox{RD}\left(\hat{\mathbf{V}}_{\text{SIMP2}} \right)$ -31,1 % -1,9 % -13,3 % -29,7 % -76,3 % -35,2 % -41,4 % -33,4 %
$\mbox{RD}\left(\hat{\mathbf{V}}_{\text{SIMP3}} \right)$ 5,2 % 2,6 % 23,2 % 5,7 % 10,5 % 15,1 % 12,2 % 8,4 %
$\hat{R}$ 1,00 0,09 0,03 0,44 0,12 0,12 0,45 0,12
$\hat{\mathbf{V}} \left(\hat{R}\right)$ 7,9E-05 2,8E-06 2,4E-05 2,5E-05 1,2E-05 3,0E-05 1,6E-05
$\mbox{RD}\left(\hat{\mathbf{V}}_{\text{SIMP1}} \right)$ -96,2 % -51,0 % -31,0 % -7,9 % -40,2 % -69,3 % -49,2 %
$\mbox{RD}\left(\hat{\mathbf{V}}_{\text{SIMP2}} \right)$ -0,4 % -17,0 % -44,7 % -80,5 % -35,5 % -5,0 % -37,5 %
$\mbox{RD}\left(\hat{\mathbf{V}}_{\text{SIMP3}} \right)$ 3,4 % 31,9 % 24,3 % 11,5 % 24,3 % 25,7 % 13,3 %
Variance estimates of estimated total and ratio on some ELFE variables
The relative difference $RD$ between $\hat{V}_{\text{SIMP}}$ and the unbiased estimator $\hat{V}$ is
\begin{eqnarray*}
RD = \frac{\hat{V}_{\text{SIMP}} \left(\hat{t}_{Y\star}\right) - \hat{V} \left(\hat{t}_{Y\star}\right)}{\hat{V} \left(\hat{t}_{Y\star}\right)}.
\end{eqnarray*}
Different behaviours may be observed for the variables of interest, depending on the maternity/day effect. For instance, the variable indicating whether the birth
occurred by caesarean contains an important day effect, and the RD of $\hat{{V}}_{\text{SIMP2}}$ is therefore small while that of
$\hat{{V}}_{\text{SIMP1}}$ is large. Symmetrically, the variable indicating whether the mother is followed by a midwife contains a small day effect as
compared to the maternity effect, and the RD of $\hat{{V}}_{\text{SIMP2}}$ is therefore large while that of
$\hat{{V}}_{\text{SIMP1}}$ is small. Also, we note that the RD of $\hat{{V}}_{\text{SIMP3}}$ is relatively stable for all variables when estimating a
total, which is an important feature in favour of this third simplified estimator. We note however that the absolute RD of $\hat{{V}}_{\text{SIMP3}}$ can
be large when estimating a ratio, which confirms the simulation results.
§ CONCLUSION
The present paper derives some general estimation theory for the cross-classified sampling design which was used in the recent ELFE survey on childhood. The issue of possibly negative variance estimates arised even in case of simple random sampling without replacement. Alternative estimators to the usual Horvitz-Thompson and Yates-Grundy variance estimators are thus proposed, and proved to be non-negative under the usual Sen-Yates-Grundy conditions. The relative bias of the proposed variance estimators is derived for a superpopulation model. The behavior of these estimators is also investigated for totals and ratios on simulated data and on data extracted from the ELFE survey. Among the proposals, one variance estimator that leads to a slight overestimation of the variance in many cases, appears to be advisable.
Despite the present results and the recent paper by Skinner (2015), the cross-classified sampling design still deserves some attention. In particular, the treatment of non-response and the calibration problem should also be taken into account, and is currently under investigation.
§ SUPPLEMENTARY MATERIALS
The three supplemental files are contained in a single archive (and can be obtained via a single download).
readme: description of the supplemental files. (txt file)
CodeR_functions: basic functions required to calculate estimators. (R file)
CodeR_Tables: commands that calculate and display the results in Table 1 and Table 2 (call the CodeR_functions). (R file)
§ BIBLIOGRAPHY
Cardot, H., and Goga, C. and Lardin, P. (2013). Uniform convergence and asymptotic confidence bands for model-assisted estimators of the
mean of sampled functional data. Electronic Journal of Statistics, 7, 562-596.
Chen, J. and Rao, J.N.K. (2007). Asymptotic normality under two-phase sampling designs. Statistica Sinica, 17, 1047-1064.
Dalén, J. and Ohlsson, E. (1995). Variance Estimation in the Swedish Consumer Price Index. Journal of Business & Economic Statistics, 13, No.3, 347-356.
Hajek, J. (1961). Some extensions of the Wald-Wolfowitz-Noether theorem. The Annals of Mathematical Statistics, 32, 506-523.
Ohlsson, E. (1996). Cross-Classified Sampling. Journal of Official Statistics, 12, No.3, 241-251.
Särndal, C.-E., Swensson, B. and Wretman, J.H. (1992). Model Assisted Survey Sampling. New-York, Springer-Verlag.
Sen, A.R. (1953). On the estimate of the variance in sampling with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 5, 119-127.
Skinner, C.J. (2015). Cross-classified sampling: some estimation theory. Statistics and Probability Letters, 104, 163-168.
Vos, J. W. E. (1964). Sampling in space and time. Review of the International Statistical Institute, 32, No. 3, 226-241.
Yates, F. and Grundy, P.M. (1953). Selection without replacement from within strata with probability proportional to size. Journal of the Royal Statistical Society B, 15, 235-261.
§ APPENDIX
§.§ Proof of equations (<ref>)-(<ref>)
We can rewrite
\begin{eqnarray} \label{proof1:eq1}
V_{1} \left(\hat{t}_Y\right) & = & \sum_{k \in U_D} \frac{V(\hat{Y}_{\bullet k})}{\pi_k^D} + \sum_{k \neq l \in U_D} \frac{\pi_{kl}^D}{\pi_{k}^D\pi_{l}^D} Cov(\hat{Y}_{\bullet k},\hat{Y}_{\bullet l}).
\end{eqnarray}
We have
\begin{eqnarray} \label{proof1:eq2}
V(\hat{Y}_{\bullet k}) & = & \sum_{i \in U_M} (1-\pi_i^M) \frac{(Y_{ik})^2}{\pi_i^M}+ \sum_{i \neq j \in U_M} \frac{\pi_{ij}^M-\pi_i^M \pi_j^M}{\pi_i^M \pi_j^M} Y_{ik} Y_{jk}.
\end{eqnarray}
From assumptions (H1), (H2) (H3) and Cauchy-Schwarz inequality, there exists some constant $C$ such that for any $k \in U_D$,
\begin{eqnarray} \label{proof1:eq3}
V(\hat{Y}_{\bullet k}) & \leq & C \frac{N_M^2}{n_M}.
\end{eqnarray}
Also, from the Cauchy-Schwarz inequality, there exists some constant $C$ such that for any $k \neq l \in U_D$:
\begin{eqnarray} \label{proof1:eq4}
Cov(\hat{Y}_{\bullet k},\hat{Y}_{\bullet l}) & \leq & C \frac{N_M^2}{n_M}.
\end{eqnarray}
From equation (<ref>) and assumption (H2), the first term in the right hand sum of (<ref>) is $O(N_D^2 N_M^2 n_M^{-1} n_D^{-1})$. From equation (<ref>) and assumptions (H2) and (H3), the absolute value of the second term in the RHS of (<ref>) is $O(N_D^2 N_M^2 n_M^{-1})$. Therefore, there exists some constant $C$ such that
\begin{eqnarray} \label{proof1:eq5}
V_{1} \left(\hat{t}_Y\right) & \leq & C \frac{N_D^2 N_M^2}{n_M}.
\end{eqnarray}
We can prove similarly that there exists some constant $C$ such that
\begin{eqnarray} \label{proof1:eq6}
V_{2} \left(\hat{t}_Y\right) & \leq & C \frac{N_D^2 N_M^2}{n_D}.
\end{eqnarray}
From equation (<ref>), the term $V_{3} \left(\hat{t}_Y\right)$ may be split into four terms according to the intersection of $\{i,j\}$ and $\{k,l\}$. From assumptions (H1)-(H3), it is easily shown that the absolute value of each of these four terms is $O(N_D^2 N_M^2 n_M^{-1} n_D^{-1})$. Therefore, there exists some constant $C$ such that
\begin{eqnarray} \label{proof1:eq7}
V_{3} \left(\hat{t}_Y\right) & \leq & C \frac{N_D^2 N_M^2}{n_M n_D}.
\end{eqnarray}
Equations (<ref>)-(<ref>) follow immediately from equations (<ref>)-(<ref>) and assumption (H4).
|
1511.00110
|
$^{1}$Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan
$^{2}$RECCS, Akita Prefectural University, Akita 015-0055, Japan
The center-of-mass energy of two particles can become arbitrarily large
if they collide near the event horizon of an extremal Kerr black hole,
which is called the Ba$\rm \t n$ados-Silk-West (BSW) effect.
We consider such a high-energy collision of two particles
which started from infinity and follow geodesics in the equatorial plane
and investigate the energy extraction from such a high-energy
particle collision and the production of particles in the equatorial plane.
We analytically
show that, on the one hand, if the produced particles are
as massive as the colliding
particles, the energy-extraction efficiency is bounded by
$2.19$ approximately. On the other hand, if a very massive particle is produced as a result of the high-energy collision, which has negative energy and
necessarily falls into the black hole, the upper limit of the
energy-extraction efficiency is increased to $(2+\sqrt{3})^2 \simeq 13.9$.
Thus, higher efficiency of the energy extraction, which is
typically as large as 10,
provides strong evidence for the production of a heavy particle.
§ INTRODUCTION
Ba$\rm \t n$ados, Silk, and West (BSW) pointed out that the
center-of-mass (CM) energy of two colliding particles can be arbitrarily
large, if the collision occurs near the event horizon of an extremal Kerr
black hole and the angular momentum of either of the colliding particles is
finetuned to the critical value <cit.>.
This is now called the BSW effect. See Harada and
Kimura <cit.>
for a brief review and references therein for further details.
Particle collision with high CM energy had already been noticed by
Piran, Shaham, and Katz in the study of energy extraction from
collisional events in the ergoregion, which is called collisional
Penrose process <cit.>. This process is
typically as follows. We consider the reaction of particles 1 and 2 into
particles 3 and 4 in the ergoregion, where particle 3 escapes to the
infinity after the collision, while particle 4 falls into the black hole
possibly with negative energy due to the existence of the ergosphere.
If one defines an energy-extraction efficiency as
\begin{eqnarray}
\eta:=\frac{ {\rm energy~of~the~escaping~particle} }{ {\rm total~energy~of~the~injected~particles} }=\frac{E_3}{E_1+E_2},
\end{eqnarray}
the energy extraction ($\eta>1$) from the black hole is possible provided $E_4<0$.
The high-CM-energy collision can produce a very massive and/or energetic
particle. This means that Kerr black holes act as natural particle
accelerators, which can accelerate even neutral particles. Recently, the
interplay between such particle acceleration and energy extraction
have been intensively investigated <cit.>. In particular, Schnittman <cit.> numerically showed that the upper limit of energy-extraction efficiency in the collisional Penrose process can reach about 13.9.
Berti, Brito, and Cardoso confirmed the result of
Ref. <cit.>, and also showed that an arbitrarily high efficiency is
possible by more general processes. Namely, they considered a head-on
collision of two subcritical particles, which is called
super-Penrose process. This process needs an outgoing particle
which must be generated in the ergoregion by some preceding
process <cit.>.
In this paper, we study particle collision near the horizon of an
extremal Kerr black hole and the resultant energy extraction.
We present an analytic formulation to investigate collisional Penrose
process under the assumption that particles follow geodesics in the
equatorial plane and two particles collide near the horizon to produce
two particles. We find that if the
produced particles are as massive as the colliding particles,
the energy-extraction efficiency is bounded by $2.19$ approximately.
However, if a very massive particle is allowed to be produced, which
has negative energy and necessarily falls into the black hole,
the upper limit is increased to $(2+\sqrt{3})^2 \simeq 13.9$, confirming the numerical result in Ref. <cit.>.
The organization of this paper is as follows. In Sec. <ref>, we
prepare for the analysis of collisional Penrose process, reviewing
geodesic motions and near-horizon collision in the Kerr black hole. In
Sec. <ref>, we investigate the upper limits of
the energy of escaping particle and of the energy-extraction efficiency
in the case of produced particles as massive as the colliding particles.
In Sec. <ref>, we will see that how the upper limits will be
significantly increased if we take the production of a very massive
particles due to the BSW effect into account.
Section <ref> is devoted to conclusion.
We adopt the geometrized unit in which $c=G=1$.
§ PRELIMINARIES
§.§ Geodesics in the Kerr black hole
The spacetime metric of the Kerr black hole is given by
\begin{eqnarray}
&& g_{\mu\nu} dx^\mu dx^\nu =
\frac{4Mar\sin^2\theta}{\rho^2}dtd\varphi
+\rho^2 d\theta^2
\nonumber\\
&& \hspace{6cm} +\left(r^2+a^2+\frac{2Ma^2r\sin^2\theta}{\rho^2}\right)\sin^2\theta d\varphi^2,
\end{eqnarray}
where $\rho^2(r,\theta) := r^2+a^2\cos^2\theta$,
$\Delta(r):=r^2-2Mr+a^2$ and $M$ and $a \; (0 \leq a \leq M)$ are the mass and spin parameters, respectively. $\Delta(r)$ vanishes at $r_\pm:=M\pm\sqrt{M^2-a^2}$, and $r=r_+$ and $r=r_-$ correspond to the event horizon and Cauchy horizon, respectively.
This spacetime is stationary and axisymmetric with Killing vectors
$\partial_t$ and $\partial_\varphi$. The conserved energy $E$ and
angular momentum $L$ of a particle with the four-momentum $p^\mu$ are given
by $E=-g_{\mu\nu}(\partial_t)^\mu p^\nu=-g_{t\mu}p^\mu$ and $L=g_{\mu\nu}(\partial_\varphi)^\mu p^\nu=g_{\varphi\mu}p^\mu$,
respectively. The components of the four-momentum are given in terms of
these conserved charges as (e.g. <cit.>)
\begin{eqnarray}
\;\;\;
\label{p^phi}
\\
\;\;\;
\nonumber \\
\label{V}
\end{eqnarray}
where $m \; (\geq 0)$ denotes the mass of the particle, and the motion
is assumed to be confined in the equatorial plane, where $\theta=\pi/2$.
The forward-in-time condition $p^t>0$ near the horizon $r\rightarrow r_+ + 0$ reduces to
\begin{eqnarray}
\label{fit}
\end{eqnarray}
where $\Omega_H := a/(r^2_+ +a^2)$ is the angular velocity of the horizon.
We call a particle a critical particle if it has a critical angular
momentum $E/\Omega_H$, for which the equality in Eq. (<ref>) holds. Accordingly, we call a particle with $L<E/\Omega_H $ ($L>E/\Omega_H$) a subcritical (supercritical) particle.
In the rest of this paper, we only consider the extremal black hole $a=M$. In this case, the forward-in-time condition for general position $r>r_+$ is written as
\begin{eqnarray}
\frac{1}{2}\left[\left(\frac{r}{M}\right)^3+\frac{r}{M}+2\right]E>\t L,
\label{froward in time condition}
\end{eqnarray}
where $\t L:=L/M$ is a reduced angular momentum.
(a) (b)
(a) The radial turning points are plotted
for a massless particle with $b=b_\pm
(r)$. (b) The radial turning points are plotted for a massive particle
with $\tilde{L}=\tilde{L}_\pm (r,E,m)$, where we set $E=m=1$. The negative
energy particles are confined to the gray regions.
§.§ Radial turning points of a geodesic particle
Here, we are concerned with a particle that comes from or escapes to the
infinity, which requires the effective potential $V$ in Eq. (<ref>) is
non-positive for large $r$. This requires $E \geq m$.
For a massless particle ($m=0$) , solving $V=0$ for the impact parameter $b:=L/E$, we obtain $b=b_\pm(r)$, where
\begin{eqnarray}
b_+(r) := r+M, \;\;\; b_-(r) := -\left(r+M+\frac{4M^2}{r-2M}\right).
\label{bpm}
\end{eqnarray}
This means that a particle of which impact parameter $ b= b_\pm(r)$ has
a turning point at $r$. The numerical plot of $b=b_\pm (r)$ is given in
Fig. <ref>(a). As $r$ increases from $M$ to
infinity, $b_+(r)$ begins with $2M$ and monotonically increases to
infinity. As $r$ increases from $M$ to $2M$, $b_-(r)$ begins with $2M$
and monotonically increases to infinity. As $r$ increases from $2M$ to
infinity, $b_-(r)$ begins with negative infinity, monotonically
increases to a local maximum $-7M$ at $r=4M$ and monotonically decreases to negative infinity. Therefore, for
$2M<b<b_+(r_*)$, the particle can escape to the infinity irrespective of
the sign of the initial velocity, which is shown by the yellow region in
Fig. <ref>(a) and where we denote the radial coordinate of
collision by $r_*$. On the other hand, for $M<r_{*}<4M$
and $-7M<b\leq 2M$, the
particle can escape to the infinity only if it moves initially
outwardly, which is shown by the blue region of Fig. <ref>(a).
For a massive particle ($m>0$),
solving $V(r)=0$ for $\t L$, we obtain $\t L = \t L_\pm(r,E,m)$, where
\begin{eqnarray}
\t L_\pm(r,E,m) := \frac{-2M^2E\pm r(r-M)\sqrt{E^2-m^2+ 2Mm^2/r}}{M(r-2M)}.
\label{Lpm}
\end{eqnarray}
This means that a particle with $E$, $m$, and $\t L=\t L(r,E,m)$ has a
turning point at $r$.
The numerical plot of $\t L = \t L_\pm(r,E,m) $ is given in
Fig. <ref>(b).
As $r$ increases from $M$ to infinity, $\t
L_+(r,E,m)$ begins with $2E$ and monotonically increases to
As $r$ increases from $M$ to $2M$,
$\t L_-(r,E,m)$ begins with $2E$ and monotonically increases to
infinity. As $r$ increases from $2M$ to infinity, $\t L_-(r,E,m)$ begins
with negative infinity, monotonically increases to a local maximum
$\t L_{\rm max}(E,m) \; (<0)$ at $r=r_{\rm max}$ and monotonically
decreases to negative infinity. Therefore, for $2E<\t L<\t
L_+(r_*,E,m)$, the particle can escape to infinity irrespective of the
sign of the initial velocity, which is shown by the yellow region of
Fig. <ref>(b). On the other hand, for $M<r_{*}<r_{\rm
max}$ and $\t L_{\rm max}(E,m)<\t L\leq2E$, the particle can escape to
infinity only if it moves
initially outwardly, which is shown by the blue region of Fig. <ref>(b).
§.§ Particle collision on the horizon
Let us consider the reaction of two colliding particles, named particles 1 and 2, to
two product particles, 3 and 4. The local conservation of four-momenta can be written as
\begin{eqnarray}
\label{p^mu,conservation}
\end{eqnarray}
The $t$- and $\varphi$-components of Eq. (<ref>) represent the conservations of energy and angular momentum
\begin{eqnarray}
\quad \mbox{and}\quad
\;\;\;
\t L_1+\t L_2=\t L_3+\t L_4,
\label{L,conservation}
\end{eqnarray}
The $r$-component represents the conservation of radial momentum
\begin{eqnarray}
\sigma_1|p^r_1|+\sigma_2|p^r_2|=\sigma_3|p^r_3|+\sigma_4|p^r_4|,
\label{p^r,conservation}
\end{eqnarray}
where $\sigma_i = {\rm sgn}(p_i^r)$ for $i=1,2,3,4$.
Note that the mass and four-momentum of particle 4 can be written in terms of those of the other particles using the momentum conservations and identity $m^2_4=-p^\mu_4p_{4\mu}$. From Eq. (<ref>), we obtain
\begin{eqnarray}
|p^r_i|=2E_i-\t L_i,
\end{eqnarray}
where we have used the forward-in-time condition to open the square root.
In the rest of this paper, we assume particle 1 to be critical ($\t L_1=2E_1$), particle 2 to be subcritical ($ \t L_2 < 2E_2 $), particle 3 to escape to infinity, and particle 4 to fall into the black hole with negative energy ($E_4<0$). Then, since particle 1 is critical, Eq. (<ref>) is written as
\begin{eqnarray}
\sigma_2(2E_2-\t L_2)
\begin{cases}
\sigma_3(2E_2-\t L_2) & (\mbox{for $\sigma_3=\sigma_4$}) \\
\sigma_3\left[2(2E_3-\t L_3)-(2E_2-\t L_2)\right] & (\mbox{for $\sigma_3=-\sigma_4$}) \\
\end{cases}.
\label{sigma_2(2E-L)}
\label{pr-cons}
\end{eqnarray}
When we choose $\sigma_2=-1 $, several situations are possible depending
on the values of $\sigma_3$ and $\sigma_4$. We will see, however, that only a few situations among them are interesting for our considerations. If $\sigma_3=\sigma_4=1$, from Eq. (<ref>), we obtain $2E_2-\t L_2=0$, which contradicts our assumption. If $\sigma_3=\sigma_4=-1$, we obtain $2E_2-\t L_2=2E_3-\t L_3+2E_4-\t L_4$, which implies that particle 3 can be either critical ($\t E_3=2E_3$) or subcritical ($\t E_3<2E_3$). Nevertheless, only the critical case is interesting since a subcritical ingoing particle cannot escape to infinity. If $\sigma_3=- \sigma_4= 1$, we obtain $2E_3-\t L_3=0$ (particle 3 is critical). If $\sigma_3 = - \sigma_4= -1$, we obtain $2E_2-\t L_2=2E_3-\t L_3$, which implies particle 3 is subcritical, it is not interesting since a subcritical ingoing particle cannot escape to infinity again.
When we choose $\sigma_2=1$, only a few situations are interesting
If $\sigma_3=\sigma_4=1$, we obtain $2E_2-\t L_2= (2E_3-\t L_3) + (2E_4-\t L_4) $, which implies that particle 3 can be either critical or subcritical.
Since $\sigma_3=1$ in the present case, particle 3 can escape to infinity even if it is subcritical and outgoing, provided $b_3$ or $\t L_3$ satisfy $b_{\rm max, 3}<b_3$ (massless case) or $\t L_{\rm max, 3}(E_3,m_3)<\t L_3$ (massive case). If $\sigma_3=\sigma_4=-1$, we obtain $2E_2-\t L_2=0$, which contradicts our assumption. If $\sigma_3=-\sigma_4=1$, we obtain $2E_2-\t L_2=2E_3-\t L_3$, which implies that particle 3 is subcritical. If $\sigma_3=-\sigma_4=-1$, we obtain $2E_3-\t L_3=0$.
From the above considerations, we see the following four situations are
interesting for the energy efficiency.
Case A: $\sigma_2=-1$ and particle 2 is subcritical. In this case, particles 1 and 2 come from infinity and particle 3 is critical at the horizon.
Case B: $\sigma_2=1$ and particle 2 is subcritical. In this case,
particle 2 must be created inside the ergoregion by some preceding process.
In the Appendix, we will see that the upper limit of the energy
extraction efficiency is in case B is the same as in case A.
Hence, we will focus on case A in Secs. <ref> and <ref>.
§.§ Near-horizon and near-critical behaviors of a particle
We parameterize the radial position of near-horizon collision $r_\ast$ as
\begin{eqnarray}
r_\ast = \frac{M}{1-\epsilon},
\;\;\;
0<\epsilon \ll 1,
\label{r_ast}
\end{eqnarray}
and the near-critical angular momentum as
\begin{eqnarray}
\t L&=&2E(1+\delta),
\;\;\;
|\delta|\ll1 .
\end{eqnarray}
Then, we assume that $\delta$ can be expanded in powers of $\epsilon$ as
\begin{eqnarray}
\delta
\delta_{(1)}\epsilon+\delta_{(2)}\epsilon^2+ O(\epsilon^3).
\end{eqnarray}
Under the assumption that particle 3 escapes to the infinity, $\t
L_3\leq\t L_{+}(r_*,E_3,m_3)$ has to hold, which implies
\begin{eqnarray}
\delta\leq
\left(\frac{2E_3-\sqrt{E^2_3+m^2_3}}{2E_3}\right)\epsilon
\end{eqnarray}
The forward-in-time condition for the near-horizon and near-critical particle implies
\begin{eqnarray}
\delta<\epsilon+\frac{7}{4}\epsilon^2+O(\epsilon^3).
\end{eqnarray}
Therefore, the forward-in-time condition is always satisfied.
§.§ Expansion of $|p^r_i|$ by $\epsilon$
Let us consider the series expansion of the radial momentum in powers of
$\epsilon$ for each particle.
Since we have assumed particle 1 to be critical and particle 2 to be subcritical, $|p^r_1|$ and $|p^r_2|$ are expanded as
\begin{eqnarray}
\sqrt{3E^2_1-m^2_1}\epsilon
\label{p1r}\\
(2E_2-\tilde L_2)
-2(E_2-\tilde L_2)\epsilon
+\frac{(3E_2-\tilde L_2)(E_2-\tilde L_2)-m^2_2}{2(2E_2-\tilde L_2)}\epsilon^2
\label{p2r}
\end{eqnarray}
If particle 3 is near critical and particle 4 is subcritical, the expansions of $|p^r_3|$ and $|p^r_4|$ are given by
\begin{eqnarray}
\sqrt{E^2_3\left[4(1-\delta_{(1)})^2-1\right]-m^2_3}\epsilon
-\frac{E^2_3\left[1-4(2\delta_{(1)}-\delta_{(2)})(1-\delta_{(1)})\right]}{\sqrt{E^2_3\left[4(1-\delta_{(1)})^2-1\right]-m^2_3}}\epsilon^2 + O(\epsilon^3), \nonumber \\
\label{p3r.critical}\\
(2E_2-\tilde L_2)
-\left[2(E_2-\tilde L_2)+2E_3(1-\delta_{(1)})-2E_1\right]\epsilon
\nonumber\\&&~
\frac{(2E_2-\tilde L_2)}{2}-2E_3(2\delta_{(1)}-\delta_{(2)})-\frac{(E_1+E_2-E_3)^2+m^2_4}{2(2E_2-\tilde L_2)}
\right]\epsilon^2
\label{p4rcritical}
\end{eqnarray}
If particle 3 is subcritical and particle 4 is near critical, the expansion of $|p^r_3|$ and $|p^r_4|$ are obtained by exchanging subscripts 3 and 4 in Eqs. (<ref>) and (<ref>).
If both particles 3 and 4 are subcritical, the expansion of $|p^r_3|$ and $|p^r_4|$ are given by
\begin{eqnarray}
(2E_3-\tilde L_3)
-2(E_3-\tilde L_3)\epsilon
+\frac{(3E_3-\tilde L_3)(E_3-\tilde L_3)-m^2_3}{2(2E_3-\tilde L_3)}\epsilon^2
\label{p3rsubcritical}\\
(2E_2-\tilde L_2)-(2E_3-\tilde L_3)
+2(E_1-E_2+E_3+\t L_2-\t L_3)\epsilon
\nonumber\\
\frac{(E_1-E_2+E_3+\t L_2-\t L_3)(E_1+3E_2-3E_3-\t L_2+\t L_3)+m^2_4}{2[(2E_2-\tilde L_2)-(2E_3-\tilde L_3)]}\epsilon^2
\label{p4rsubcritical}
\end{eqnarray}
§ ENERGY-EXTRACTION EFFICIENCY
§.§ Case for $m_{4}=O(\epsilon^{0})$
We focus on case A, where
$\sigma_2=\sigma_4=-1$ and particle 3 is near critical. $\sigma_2=-1$
implies that we can assume that particles 1 and 2 come from the
infinity. The $O(\epsilon)$ and $O(\epsilon^2)$ terms of radial momentum conservation Eq. (<ref>) yield
\begin{eqnarray}
\sigma_1\sqrt{3E^2_1-m^2_1}
=\sigma_3\sqrt{E^2_3 [4(1-\delta_{(1)})^2-1 ]-m^2_3},
\label{r.1.conservation.case(A)}
\end{eqnarray}
\begin{eqnarray}
\sigma_1\frac{E^2_1}{\sqrt{3E^2_1-m^2_1}}
+\frac{(3E_2-\tilde L_2)(E_2-\tilde L_2)-m^2_2}{2(2E_2-\tilde L_2)}
\sigma_3\frac{E^2_3 [1-4(2\delta_{(1)}-\delta_{(2)})(1-\delta_{(1)}) ]}{\sqrt{E^2_3 [4(1-\delta_{(1)})^2-1 ]-m^2_3}}
&&\nonumber\\ +
\frac{(2E_2-\tilde L_2)}{2}-2E_3(2\delta_{(1)}-\delta_{(2)})-\frac{(E_1+E_2-E_3)^2+m^2_4}{2(2E_2-\tilde L_2)},&&
\label{r.2.conservation.case(A)}
\end{eqnarray}
When we choose $\sigma_1=1$, Eq. (<ref>) implies
\begin{eqnarray}
=\sigma_3\sqrt{E^2_3 [ 4(1-\delta_{(1)})^2-1 ]-m^2_3},
\label{conserve1,+1,-1,-1}
\end{eqnarray}
where $B_1:=2E_1+\sqrt{3E^2_1-m^2_1} \; (>0)$.
Squaring the both sides of Eq. (<ref>), we obtain
\begin{eqnarray}
\label{1-delta,+1,-1,-1}
\end{eqnarray}
which implies
\begin{eqnarray}
\delta_{(1),\rm max}-\delta_{(1)}=\frac{(B_1-\sqrt{E^2_3+m^2_3})^2}{4B_1E_3}\geq0.
\end{eqnarray}
Substituting Eq. (<ref>) into the left-hand side of Eq. (<ref>), we obtain
\begin{eqnarray}
B_1-\frac{E^2_3+m^2_3}{B_1}=2\sigma_3\sqrt{E^2_3 [4(\delta_{(1)}-1)^2-1 ]-m^2_3}.
\label{conserve1,B,+1,-1,-1}
\end{eqnarray}
This implies $E_3 \leq \tilde\lambda_0 := \sqrt{B^2_1-m^2_3}$ ($E_3 \geq \tilde\lambda_0$) for $\sigma_3=1$ ($\sigma_3=-1$).
If we choose $\sigma_3=-1$, we need $\delta_{(1)}\geq0$ since particle 3 has to be scattered by the potential barrier.
Supposing $\delta_{(1)}\geq0$ in Eq. (<ref>), we have
\begin{eqnarray}
\label{E2ineq}
\end{eqnarray}
Inequality (<ref>) is satisfied by
\begin{eqnarray}
\t\lambda_-\leq E_3\leq\t\lambda_+,
\;\;\;
\t\lambda_\pm:=2B_1\pm\sqrt{3B^2_1-m^2_3},
\label{t,lambda_-,E,t,lambda_+}
\end{eqnarray}
where the discriminant $D$ of Eq. (<ref>) has to satisfy $D/4=3B^2_1-m^2_3\geq0$. Since particle 3 escapes to infinity, it is marginally bound or unbound ($E_3\geq m_3$). $m_3\leq\t\lambda_+$ has to be satisfied so that Eq. (<ref>) and $E_3\geq m_3$ have an intersection.
The relation $m_3\leq\t\lambda_+$ is satisfied if $m_3\leq\sqrt{3}B_1$, which is equivalent to $D/4\geq0$.
Therefore, if $D/4\geq0$ is satisfied, Eq. (<ref>) and $E_3\geq m_3$ always have an intersection.
$\t\lambda_0$ and $\t\lambda_+$ are the upper limits on $E_3$ for $\sigma_3=1$ and $\sigma_3=-1$, respectively.
Since $\t\lambda_+$ is larger than $\t\lambda_0$, we concentrate on the case of $\sigma_3=-1$.
The maximum of $\t\lambda_+$ is given by
\begin{eqnarray}
\t\lambda_{+,\rm max}=(2+\sqrt{3})^2E_1,
\end{eqnarray}
where we have assumed $m_1=m_3=0$.
Next, we consider the $O(\epsilon^2)$ terms in the radial momentum conservation. $E_3=\t\lambda_+$ can be realized when $\delta_{(1)}=0$.
Substituting $\delta_{(1)}=0$, $\sigma_1=1$, and $\sigma_3=-1$ into Eq. (<ref>), and then solving it for $m^2_4$, we obtain
\begin{eqnarray}
-2 (2E_2-\t L_2 )\Big[
\frac{E^2_1}{\sqrt{3E^2_1-m^2_1}}
\Big]\nonumber\\
&& \hspace{8cm}
\label{m^2_4,+1,-1,-1,-1}
\end{eqnarray}
We need $\delta>0$ for particle 3 to escape to the infinity. Since we choose $\delta_{(1)}=0$, we need $\delta_{(2)}\geq0$. This implies the first term on the right-hand side of Eq. (<ref>) is negative.
The lower limit of $E_2$ is then given by
\begin{eqnarray}
E_2 \geq \t\kappa := \frac{1}{2}\left[(\t\lambda_+-E_1)-\frac{m^2_2}{(\t\lambda_+-E_1)}\right].
\label{E_2,min}
\end{eqnarray}
Since we assume that particle 2 comes from infinity, it must be
marginally bound or unbound ($E_2\geq m_2$). Therefore, we have to
compare $\t\kappa$ with $m_2$. If $\t\kappa\geq m_2$, i.e.,
$(\sqrt{2}-1)(\t\lambda_+-E_1)\geq m_2$, the lower limit of $E_2$ is $\t\kappa$. Thus, we find
\begin{eqnarray}
\eta\leq\frac{\t\lambda_+}{E_1+\t\kappa}
\end{eqnarray}
One can see that $f(m_2)$ defined above begins with $2\t\lambda_+/(\t\lambda_++E_1)$ and monotonically increases to
\begin{eqnarray}
\frac{(3+2\sqrt{2})\t\lambda_+}{(2+\sqrt{2})E_1+(1+\sqrt{2})\t\lambda_+}=:g(\t\lambda_+),
\end{eqnarray}
as $m_2$ increase from $0$ to $(\sqrt{2}-1)(\t\lambda_+-E_1)$.
Since $g(\t\lambda_+)$ is a monotonically increasing function of $\t\lambda_+$, the maximum of $g(\t\lambda_+)$ is given by
\begin{eqnarray}
g(\t\lambda_{+,\rm max})=
\frac{179+186\sqrt{2}+88\sqrt{3}+100\sqrt{6}}{383}
\simeq2.19.
\end{eqnarray}
If $\t\kappa\leq m_2$, i.e., $(\sqrt{2}-1)(\t\lambda_+-E_1)\leq m_2$, the lower limit of $E_2$ is $m_2$. Thus, we find
\begin{eqnarray}
\eta\leq\frac{\t\lambda_+}{E_1+m_2},
\end{eqnarray}
and easily notice that the right-hand side monotonically decreases as $m_2$ increase. For the above reason, the maximum of the right-hand side is about $2.19$.
In summary, the upper limit of the energy extraction efficiency is $\eta_{\rm max}\simeq2.19$, which is realized $\sigma_1=1$, $\sigma_2=\sigma_3=\sigma_4=-1$, $m_1=m_3=0$, and $\delta_{(1)}=0$.
Next, let us consider the case of $\sigma_1=-1$. Equation (<ref>) implies
\begin{eqnarray}
\label{conserve-1,+1,-1,-1}
\end{eqnarray}
where $A_1:=2E_1-\sqrt{3E^2_1-m^2_1} \; (>0)$.
Performing the following replacement, an argument similar to that in the
$\sigma_1=1$ case applies.
\begin{eqnarray*}
\t\lambda_0&\rightarrow&\lambda_0:=\sqrt{A^2_1-m^2_3},\\
\t\lambda_\pm&\rightarrow&\lambda_\pm:=2A_1\pm\sqrt{3A^2_1-m^2_3},\\
\t\kappa&\rightarrow&\kappa:=\frac{1}{2}\left[(\lambda_+-E_1)-\frac{m^2_2}{(\lambda_+-E_1)}\right].
\end{eqnarray*}
However, the value of the upper limit is different. Since $\lambda_+$ is larger than $\lambda_0$, we concentrate on the case $\sigma_3=-1$.
The maximum of $\lambda_+$ is given by
\begin{eqnarray}
\lambda_{+, \rm max}=(2+\sqrt{3})(2-\sqrt{2})E_1,
\end{eqnarray}
where we have assumed $m_1=E_1$ and $m_3=0$.
We have seen that $E_3=\lambda_+$ can be realized $\delta_{(1)}=0$.
Substituting $\delta_{(1)}=0$ and $\sigma_1=\sigma_3=-1$ into Eq. (<ref>), and then solving it for $m^2_4$, we obtain
\begin{eqnarray}
-2\left(2E_2-\t L_2\right) \Big[
\Big]\nonumber\\
&& \hspace{8cm}
\label{m^2_4,-1,-1,-1,-1}
\end{eqnarray}
Again, we need $\delta>0$ for particle 3 to escape to infinity. Since we choose $\delta_{(1)}=0$, we need $\delta_{(2)}\geq0$.
In this case, we can prove the first term on the right-hand side of Eq. (<ref>) is negative (See Ref. <cit.>).
The lower limit of $E_2$ is given by
\begin{eqnarray}
E_2\geq \kappa := \frac{1}{2}\left[(\lambda_+-E_1)-\frac{m^2_2}{(\lambda_+-E_1)}\right].
\end{eqnarray}
From the discussion similar to that in the case of $\sigma_1=1$, the upper limit of the energy extraction efficiency is also obtained as
\begin{eqnarray}
\eta_{\rm max}=\frac{2+\sqrt{2}+\sqrt{6}}{4}\simeq1.47,
\end{eqnarray}
which is realized $\sigma_1=\sigma_2=\sigma_3=\sigma_4=-1$, $E_1=m_1$, $m_3=0$, $\delta_{(1)}=0$, and $m_2/m_1\simeq0.491$.
This reproduces the result of Ref. <cit.>.
§.§ Case for $m_{4}=O(\epsilon^{-1/2})$
In general, the CM energy of particles 1 and 2 is given by
\begin{eqnarray}
E^2_{\rm cm}
\end{eqnarray}
For example, in the original BSW process <cit.>, in which $\tilde{L}_1=2E_{1}$, $\tilde{L}_2<2E_{2}$ and $\sigma_1=\sigma_2 =-1$, the leading term of the CM energy is
\begin{eqnarray}
E^2_{\rm cm}
\simeq
\frac{2(2E_2-\t L_2)(2E_1-\sqrt{3E^2_1-m^2_1})}{\epsilon}.
\end{eqnarray}
In such a high-energy collision, the masses of the product particles (particles 3 and 4) can become large with the following restriction
\begin{eqnarray}
m_3+m_4\leq E_{\rm cm}.
\end{eqnarray}
Since $E_{\rm cm}$ for the collision between a critical particle and a subcritical particle is proportional to $\epsilon^{-1/2}$ for both a rear-end collision and head-on collision, we assume
\begin{eqnarray}
m_4 = O(\epsilon^{-1/2}).
\end{eqnarray}
Then, we assume that particle 4 is very massive as
\begin{eqnarray}
\label{heavy,m_4}
\end{eqnarray}
where $\mu_4 \; (>0)$ and $\nu_4$ are constants.
In fact, we can write $m^2_4$ in terms of the quantities of the other particles, using the momentum conservation, as
\begin{eqnarray}
E^2_{\rm cm}+m^2_3+2(p^\mu_1+p^\mu_2)p_{3\mu}
\end{eqnarray}
where we have assumed $\t L_3=2E_3(1+\delta)$. This implies
\begin{eqnarray}
\mu_4 = 2(2E_2-\t L_2) \Big[2E_1-2E_3(1-\delta_{(1)})+\sigma_1\sqrt{3E^2_1-m^2_1}-\sigma_3\sqrt{E^2_3 [4(1-\delta_{(1)})^2-1]-m^2_3} \Big].
\nonumber\\
\label{mu_4}
\end{eqnarray}
The expansion of radial momentum of particle 4 is given by
\begin{eqnarray}
(2E_2-\t L_2)
-\left[2(E_2-\t L_2)+2E_3(1-\delta_{(1)})-2E_1+\frac{\mu_4}{2(2E_2-\t
L_2)}\right]\epsilon \nonumber \\
\end{eqnarray}
\begin{eqnarray}
\frac{(2E_2-\tilde L_2)}{2}-2E_3(2\delta_{(1)}-\delta_{(2)})-\frac{(E_1+E_2-E_3)^2+\nu_4}{2(2E_2-\tilde L_2)}
\nonumber\\
+\frac{[E_1-E_2+\t L_2-E_3(1-\delta_{(1)})]\mu_4}{(2E_2-\tilde L_2)^2}
-\frac{\mu^2_4}{2(2E_2-\tilde L_2)^3}.
\end{eqnarray}
This implies that the $O(\epsilon)$ and $O(\epsilon^2)$ terms of the radial momentum conservation yield Eq. (<ref>) and
\begin{eqnarray}
\sigma_1\frac{E^2_1}{\sqrt{3E^2_1-m^2_1}}
+\frac{(3E_2-\tilde L_2)(E_2-\tilde L_2)-m^2_2}{2(2E_2-\tilde L_2)}-F_{4,\epsilon^2}
\sigma_3
\frac{E^2_3 [1+4(2\delta_{(1)}-\delta_{(2)})(\delta_{(1)}-1) ]}{\sqrt{E^2_3 [4(\delta_{(1)}-1)^2-1 ]-m^2_3}},
\nonumber\\
\label{p.2.momentum.BSW}
\end{eqnarray}
If we define
\begin{eqnarray}
C:=2E_1+\sigma_1\sqrt{3E^2_1-m^2_1}-\frac{\mu_4}{2(2E_2-\t L_2)}>0,
\end{eqnarray}
we can discuss the upper limit of $E_3$ in the way similar to that in Sec. <ref>.
As we have already seen, we need $\sigma_3=-1$ and $\delta_{(1)}\geq0$ to obtain the maximum of $E_3$.
Setting $\delta_{(1)}\geq0$ in Eq. (<ref>), we have
\begin{eqnarray}
\end{eqnarray}
This inequality is satisfied by
\begin{eqnarray}
\b\lambda_-\leq E_3\leq\b\lambda_+,
\;\;\;
\b\lambda_\pm:=2C\pm\sqrt{3C^2-m^2_3}.
\end{eqnarray}
The maximum value of $\b\lambda_+$ is given by
\begin{eqnarray}
\b\lambda_{+,\rm max}=(2+\sqrt{3})^2E_1-\frac{(2+\sqrt{3})\mu_4}{2(2E_2-\t L_2)},
\label{lambda+max}
\end{eqnarray}
where we have assume $m_1=m_3=0$ and $\sigma_1=1$.
$E_3=\b\lambda_+$ is realized when $\delta_{(1)}=0$.
Substituting $\delta_{(1)}=0$, $\sigma_1=1$, and $\sigma_3=-1$ into Eq. (<ref>), and then solving it for $\nu_4$, we obtain
\begin{eqnarray}
\nu_4&=&
-2 (2E_2-\t L_2 ) \Big[
\frac{E^2_1}{\sqrt{3E^2_1-m^2_1}}
+\frac{2\delta_{(2)}\b\lambda_+}{\sqrt{3\b\lambda^2_+-m^2_3}} \big( 2\b\lambda_+-\sqrt{3\b\lambda^2_+-m^2_3} \big)
\Big]\nonumber\\&&~
+(E^2_2+m^2_2)-(E_1+E_2-\b\lambda_+)^2+\frac{2(E_1-E_2+\t L_2-\b\lambda_+)\mu_4}{(2E_2-\t L_2)}-\frac{\mu^2_4}{(2E_2-\t L_2)^2}.
\label{nu_4=}
\end{eqnarray}
In Sec. <ref>, we have used the above equation
with $\mu_{4}=0$ and $\nu_{4}=m_{4}^{2}\ge 0$
to estimate the lower limit of $E_2$.
However, in the present case, the sign of $\nu_4$ is not restricted
to be positive if $\mu_{4}>0$. Hence,
we use Eq. (<ref>) not for the estimate of the lower
limit of $E_2$ but for the determination of $\nu_4$.
Therefore, the upper limit of the energy-extraction efficiency is given by
\begin{eqnarray}
\eta_{\rm max}=\frac{\b\lambda_{+,\rm max}}{E_1+E_2}
=\frac{(2+\sqrt{3})^2E_1-\frac{(2+\sqrt{3})\mu_4}{2(2E_2-\t L_2)}}{E_1+E_2}.
\end{eqnarray}
From the above equation, we see that if $E_1E_2\gg\mu_4$, $\eta_{\max}$ is approximately given by $(2+\sqrt{3})^2E_1/(E_1+E_2)$.
Moreover, for $E_2/E_1 \ll 1$, we find that the escaping
massless particle has the energy that is nearly equal to
$(2+\sqrt{3})^{2}\simeq 13.9$ times the total energy of
the incoming massless particles.
However, $E_3$ cannot be exactly equal to $(2+\sqrt{3})^2E_1$, for the
following reason.
From Eq. (<ref>), $E_3=(2+\sqrt{3})^2E_1$ is realized when $m_1=m_3=0$, $\delta_{(1)}=0$ and $\mu_4=0$.
However, $\mu_4=0$ implies $m_4^{2}=O(\epsilon^{0})$ and hence
$\nu_4$ must be positive. Substituting
$\mu_4=0$ into Eq. (<ref>) and requiring $\nu_4>0$, we obtain the
lower limit of $E_2$, which is given by Eq. (<ref>). In this
case, we have already seen that the efficiency cannot reach
$(2+\sqrt{3})^2\simeq 13.9$ but only 2.19.
While there are other cases in which a very massive and/or energetic
particle is produced, one can conclude that a large efficiency is
possible only in the above case.
In any other cases, $p^r_3$ and $p^r_4$ have $O(\epsilon^{1/2})$
terms. Since there is no half-integer order terms on the left-hand side
of Eq. (<ref>), $O(\epsilon^{1/2})$ term in the sum of
$p^r_3$ and $p^r_4$ has to be zero. In fact, if one assumes particle 3
to be very energetic so that $E^2_3={\rho_3}/{\epsilon}+\phi_3$, where $\rho_3 \; (>0)$ and $\phi_3$ are constants, the $O(\epsilon^{1/2})$ terms in the radial momentum conservation yield
\begin{eqnarray}
\sigma_3\sqrt{\rho_3 [4(1-\delta_{(1)})^2-1 ]}-2(1-\delta_{(1)})\sqrt{\rho_3}=0.
\end{eqnarray}
There is no solution for $\rho_3$ under the assumption of $\rho_3>0$.
By similar arguments, one can conclude that there is no energy
extraction, expect for the case where particle 4 is very massive.
§ CONCLUSION
We have studied particle collision and energy-extraction efficiency,
where a critical particle (particle 1) and
subcritical particle (particle 2) collide near the event horizon of an
extremal Kerr black hole and then two particles are produced, one of which
escapes to infinity (particle 3) and another falls into the black hole
(particle 4).
There is an upper limit of the energy of particle 3, which is given by
$E_{3,{\rm max}}=(2+\sqrt{3})^2E_1$.
This is realized in the situation where particles 1 and 3 are massless,
particle 3 is near critical,
particle 1 is temporary outgoing and particles 2, 3 and 4
are temporarily ingoing.
The energy-extraction efficiency,
however, is bounded by $ 2.19$ approximately, under the assumption
that particle 4 has a mass of $O(\epsilon^{0})$, where $\epsilon$
parametrizes the distance from the collision point to the horizon.
Since the CM energy of the near-horizon collision can be arbitrarily
large, the collision can produce a heavy particle. From this
viewpoint, we have considered the case in which the mass of particle 4
is of $O(\epsilon^{-1/2})$.
In this case, we found that
the upper limit of
the efficiency can indeed reach $(2+\sqrt{3})^{2}\simeq 13.9$.
Thus, if the efficiency as large as 10 is observed for
a collisional Penrose process, this strongly suggests
the production of a heavy particle as a result of the collision of
high CM energy.
Finally, let us give an example of collision with a high energy-extraction
efficiency. We assume that particles 1 and 2 are protons, 3 is a
photon, and 4 is some heavy particle and that
the collision happens in the vicinity of the horizon with $\epsilon=10^{-8}$.
Using $m_{1}=m_{2} \sim $ 1 GeV, we have $E_{\rm
cm}\sim\sqrt{m_1m_2/\epsilon}\sim10^4$ GeV. If $m_4 \sim 10^3$ GeV,
we have $\mu_4\sim
m^2_4\epsilon\sim10^{-2}$ GeV$^2$ which is much smaller than $E_{1}E_{2}$.
In this case, the efficiency is typically as large as 10.
Note added: While completing the current paper, the authors found
that a paper <cit.> studying a similar problem
appeared in arXiv. It will be interesting to compare the result in
Ref. <cit.> with that in the current paper.
The authors would like to thank T. Igata, M. Kimura, T. Kobayashi, K. Nakao, M. Patil
and S. Yokoyama.
This work was supported by JSPS KAKENHI Grant Numbers 26400282 (TH) and
15K05086 (UM).
§ COLLISION BETWEEN A CRITICAL PARTICLE AND AN OUTGOING
SUBCRITICAL PARTICLE
Here we consider case B, where $\sigma_2=1$ and particle 2 is
subcritical. We can divide this case into the following three subcases:
B1: $\sigma_2=\sigma_4=1$ ($\sigma_1$ and $\sigma_3$ can be either $1$ or $-1$). Particle 3 is critical.
B2: $\sigma_2=\sigma_3=\sigma_4=1$.
Both particles 3 and 4 are subcritical, and $2E_2-\t L_2 = (2E_3-\t L_3)
+ (2E_4-\t L_4)$.
B3: $\sigma_2=\sigma_3=1$, $\sigma_4=-1$.
Particle 3 is subcritical, and $2E_2-\t L_2=2E_3-\t L_3$, which implies particle 4 to be critical.
For case B1, the $O(\epsilon)$ terms in the radial momentum conservation yield
\begin{eqnarray}
\sigma_1\sqrt{3E^2_1-m^2_1}-2E_1+2E_3(1-\delta_{(1)})
=\sigma_3\sqrt{E^2_3 [4(\delta_{(1)}-1)^2-1 ]-m^2_3}.
\end{eqnarray}
This is obtained from Eq. (<ref>) after replacing $\sigma_1$ and $\sigma_3$ with $-\sigma_1$ and $-\sigma_3$, respectively.
The equation obtained from the $O(\epsilon^2)$ terms of the radial
momentum conservation is also the same as
Eq. (<ref>) after the above
replacement. Therefore, $\eta_{\rm max}$ in this case is given by that
in Sec. <ref> after changing the signs $\sigma_i$
($i=1,2,3,4$). Note that the change of $\sigma_3$ requires the argument
on the turning point, but one can finally see that
this prescription is valid.
For case B2, the $O(\epsilon)$ terms in the radial momentum conservation equations yield
\begin{eqnarray}
\sigma_1\sqrt{3E^2_1-m^2_1}-2E_1=0.
\label{r.1.conservation.p3rsubcritical}
\end{eqnarray}
This equation has no solution for $E_1$ under the assumption $E_1>0$ and $m_1^2\geq0$. Thus, there is no energy extraction in this case.
For case B3, the $O(\epsilon)$ terms of the radial momentum conservation equations yield
\begin{eqnarray}
2E_1-\sigma_1\sqrt{3E^2_1-m^2_1}-2E_4(1-\delta_{(1)})=\sqrt{E^2_4 [4(\delta_{(1)}-1)^2-1 ]-m^2_4}.
\label{r.1.momentum.case(d)}
\end{eqnarray}
Equation (<ref>) implies
\begin{eqnarray}
B_1-2E_4(1-\delta_{(1)})=\sqrt{E^2_4 [4(\delta_{(1)}-1)^2-1 ]-m^2_4}
\label{pr1,4near}
\end{eqnarray}
for $\sigma_1=-1$ and $-\sqrt{B^2_1-m^2_4}\leq E_4<0$.
From the forward-in-time condition and the argument on the turning point for a near-critical particle with negative energy, the restriction on $\delta_{(1)}$ is obtained as
\begin{eqnarray}
\delta_{(1)}\geq1-\frac{\sqrt{E^2_4+m^2_4}}{2E_4}.
\end{eqnarray}
Squaring the both sides of Eq. (<ref>) and then solving it for $E_4$, we obtain
\begin{eqnarray}
E_4=2(1-\delta_{(1)})B_1+\sqrt{B^2_1 [4(\delta_{(1)}-1)^2-1 ]-m^2_4}.
\end{eqnarray}
This implies the lower limit of $E_4$ is given by
\begin{eqnarray}
E_{4,\rm min}=-(2+\sqrt{3})E_1,
\end{eqnarray}
where $m_1=m_4=0$ and $\delta_{(1)}=3/2$ hold.
The $O(\epsilon^2)$ terms of the radial momentum conservation equation yield
\begin{eqnarray}
+\frac{(3E_2-\tilde L_2)(E_2-\tilde L_2)-m^2_2}{2(2E_2-\tilde L_2)}
\frac{E^2_4 [1-4(2\delta_{(1)}-\delta_{(2)})(1-\delta_{(1)}) ]}{\sqrt{E^2_4 [4(\delta_{(1)}-1)^2-1 ]-m^2_4}}
\nonumber \\
+\frac{ 2E_2-\tilde L_2 }{2}-2E_4(2\delta_{(1)}-\delta_{(2)})-\frac{(E_1+E_2-E_4)^2+m^2_3}{2(2E_2-\tilde L_2)} .
\label{B3O2}
\end{eqnarray}
The lower limit of $E_4$ can be realized only for
$m_1=m_4=0$ and $\delta_{(1)}=3/2$.
Because the denominator of the first term on the right-hand side of Eq. (<ref>) becomes zero when $m_4=0$ and $\delta_{(1)}=3/2$, we need $1-4(2\delta_{(1)}-\delta_{(2)})(1-\delta_{(1)})=0$, which is possible when $\delta_{(2)}=7/2$.
Therefore, the upper limit of the energy-extraction efficiency in this case is given by
\begin{eqnarray}
\eta_{\rm
\end{eqnarray}
where we have assumed $m_1=m_4=0$, $\delta_{(1)}=3/2$, and $\delta_{(2)}=7/2$.
If we assume $m_{4}=O(\epsilon^{-1/2})$, the energy-extraction
efficiency can be larger as in case A.
For case B1, the result is the same as for case A after replacing $\sigma_i$ with
$-\sigma_i$ ($i=1,2,3,4$). Therefore, we can obtain the upper limit
$\simeq 13.9$, only if a very massive particle is produced to fall into the black hole.
For cases B2 and B3, the energy extraction is quite modest.
Namely, the efficient energy extraction is not realized even when
an subcritical outgoing particles is considered, as long as
its counterpart is critical.
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|
1511.00321
|
C. Ding's research was supported by
The Hong Kong Research Grants Council, under Proj. No. 16300415.
Department of Computer Science and Engineering,
The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Boolean functions have important applications in cryptography and coding theory. Two famous classes of binary codes derived
from Boolean functions are the Reed-Muller codes and Kerdock codes. In the past two decades, a lot of progress on the
study of applications of Boolean functions in coding theory has been made. Two generic constructions of binary linear
codes with Boolean functions have been well investigated in the literature. The objective of this paper is twofold. The
first is to provide a survey on recent results, and the other is to propose open problems on one of the two generic
constructions of binary linear codes with Boolean functions. These open problems are expected to stimulate further research
on binary linear codes from Boolean functions.
Almost bent functions, bent functions, difference sets, linear codes, semibent functions, o-polynomials.
§ INTRODUCTION
Let $p$ be a prime and let $q=p^m$ for some positive integer $m$.
An $[n,\, k,\, d]$ code $\C$ over $\gf(p)$ is a $k$-dimensional subspace of $\gf(p)^n$ with minimum
(Hamming) distance $d$.
Let $A_i$ denote the number of codewords with Hamming weight $i$ in a code
$\C$ of length $n$. The weight enumerator of $\C$ is defined by
1+A_1z+A_2z^2+ \cdots + A_nz^n.
The sequence $(1, A_1, A_2, \cdots, A_n)$ is called the weight distribution of the code $\C$.
A code $\C$ is said to be a $t$-weight code if the number of nonzero
$A_i$ in the sequence $(A_1, A_2, \cdots, A_n)$ is equal to $t$.
Boolean functions are functions from $\gf(2^m)$ or $\gf(2)^m$ to $\gf(2)$. They are important building blocks
for certain types of stream ciphers, and can also be employed to construct binary codes. Two famous families of
binary codes are the Reed-Muller codes <cit.> and Kerdock codes <cit.>.
In the literature two generic constructions of binary
linear codes from Boolean functions have been well investigated. A lot of progress on the study of one of the two
constructions has been made in the past decade. The objective of this paper is twofold. The first one is to provide
a survey on recent development on this construction, and the other is to propose open problems on this generic
constructions of binary linear codes with Boolean functions.
These open problems are expected to stimulate further research on binary linear codes from Boolean functions.
§ MATHEMATICAL FOUNDATIONS
§.§ Difference sets
For convenience later, we define the difference functiondifference function of a subset
$D$ of an abelian group $(A,\,+)$ as
\begin{eqnarray}\label{eqn-DifferenceFunction}
\diff_D(x)=|D \cap (D+x)|,
\end{eqnarray}
where $D+x=\{y+x: y \in D\}$.
A subset $D$ of size $k$ in an abelian group $(A, \, +)$ with order $v$ is called
a $(v, \,k, \,\lambda)$ difference setdifference set in $(A,\,+)$ if the difference function
$\diff_D(x)=\lambda$ for every nonzero $x \in A$. A difference set $D$ in $(A,\,+)$ is called cyclic if
the abelian group $A$ is so.
Difference sets could be employed to construct linear codes in different ways. The reader is referred to
<cit.> for detailed information. Some of the codes presented in this survey paper
are also defined by difference sets.
§.§ Group characters in $\gf(q)$
An additive character of $\gf(q)$ is a nonzero function $\chi$
from $\gf(q)$ to the set of nonzero complex numbers such that
$\chi(x+y)=\chi(x) \chi(y)$ for any pair $(x, y) \in \gf(q)^2$.
For each $b\in \gf(q)$, the function
\begin{eqnarray}\label{dfn-add}
\chi_b(c)=\epsilon_p^{\tr(bc)} \ \ \mbox{ for all }
\end{eqnarray}
defines an additive character of $\gf(q)$, where and whereafter $\epsilon_p=e^{2\pi \sqrt{-1}/p}$ is
a primitive complex $p$th root of unity and $\tr$ is the absolute trace function. When $b=0$,
$\chi_0(c)=1 \mbox{ for all } c\in\gf(q),
and is called the trivial additive character of
$\gf(q)$. The character $\chi_1$ in (<ref>) is called the
canonical additive character of $\gf(q)$.
It is known that every additive character of $\gf(q)$ can be
written as $\chi_b(x)=\chi_1(bx)$ <cit.>.
§.§ Special types of polynomials over $\gf(q)$
It is well known that every function from $\gf(q)$ to $\gf(q)$ can be expressed as a polynomial over $\gf(q)$.
A polynomial $f \in \gf(q)[x]$ is called a permutation polynomialpermutation polynomial
if the associated polynomial function $f: a \mapsto f(a)$ from $\gf(q)$ to $\gf(q)$ is a permutation of
Dickson polynomials of the first kind over $\gf(q)$ are defined by
\begin{eqnarray}\label{eqn-1stDP}
D_h(x, a)=\sum_{i=0}^{\lfloor \frac{h}{2} \rfloor} \frac{h}{h-i} \binom{h-i}{i} (-a)^i x^{h-2i},
\end{eqnarray}
where $a \in \gf(q)$ and $h$ is called the order of the polynomial. Some of the linear codes that
will be presented in this paper are defined by Dickson permutation polynomials of order 5 over $\gf(2^m)$.
A polynomial $f \in \gf(q)[x]$ is said to be $e$-to-$1$ if the equation $f(x)=b$ over $\gf(q)$
has either $e$ solutions $x \in \gf(q)$ or no solution for every $b \in \gf(q)$, where $e \geq 1$ is an
integer, and $e$ divides $q$. By definition, permutation polynomials are 1-to-1. In this survey paper, we
need $e$-to-$1$ polynomials over $\gf(2^m)$ for the construction of binary linear codes.
§.§ Boolean functions and their expressions
A function $f$ from $\gf(2^m)$ or $\gf(2)^m$ to $\gf(2)$ is called a Boolean function.
A function $f$ from $\gf(2^m)$ to $\gf(2)$ is called linear if $f(x+y)=f(x)+f(y)$ for all $(x, y) \in \gf(2^m)^2$.
A function $f$ from $\gf(2^m)$ to $\gf(2)$ is called affine if $f$ or $f-1$ is linear.
The Walsh transform of $f: \gf(2^m) \to \gf(2)$ is defined by
\begin{eqnarray}\label{eqn-WalshTransform2}
\hat{f}(w)=\sum_{x \in \gf(2^m)} (-1)^{f(x)+\tr(wx)}
\end{eqnarray}
where $w \in \gf(2^m)$. The Walsh spectrum of $f$ is the following multiset
\left\{\left\{ \hat{f}(w): w \in \gf(2^m) \right\}\right\}.
Let $f$ be a Boolean function from $\gf(2^m)$ to $\gf(2)$. The support of $f$ is defined to be
\begin{eqnarray}\label{eqn-Booleanfsupport}
D_f=\{x \in\gf(2^m) : f(x)=1\} \subseteq \gf(2^m).
\end{eqnarray}
Clearly, $f \mapsto D_f$ is a one-to-one correspondence between the set of Boolean functions from
$\gf(2^m)$ to $\gf(2)$ and the power set of $\gf(2^m)$.
§ THE FIRST GENERIC CONSTRUCTION OF LINEAR CODES FROM FUNCTIONS
Let $f$ be any polynomial from $\gf(q)$ to $\gf(q)$, where $q=p^m$. A code over $\gf(p)$ is defined by
\C(f)=\{ \bc=(\tr(af(x)+bx))_{x \in \gf(q)}: a \in \gf(q), \ b \in \gf(q)\},
where $\tr$ is the absolute trace function.
Its length is $q$, and its dimension is at most $2m$ and is equal to $2m$ in many cases.
The dual of $\C(f)$ has dimension at least $q-2m$.
Let $f$ be any polynomial from $\gf(q)$ to $\gf(q)$ such that $f(0)=0$. A code over $\gf(p)$ is defined by
\C^*(f)=\{ \bc=(\tr(af(x)+bx))_{x \in \gf(q)^*}: a \in \gf(q), \ b \in \gf(q)\}.
Its length is $q-1$, and its dimension is at most $2m$ and is equal to $2m$ in many cases.
The dual of $\C^*(f)$ has dimension at least $q-1-2m$.
This is a generic construction of linear codes, which has a long history and its importance is supported by
Delsarte's Theorem <cit.>. It gives a coding-theory characterisation of APN monomials, almost bent
functions, and semibent functions (see, for examples, <cit.>, <cit.> and <cit.>) when $q=2$.
We will not deal with this construction in this paper.
§ THE SECOND GENERIC CONSTRUCTION OF LINEAR CODES FROM FUNCTIONS
In this section, we present the second generic construction of linear codes over $\gf(p)$ with
any subset $D$ of $\gf(p^m)$, and introduce basic results about the linear codes.
In Section <ref>, we will consider specific families of binary linear codes
from Boolean functions obtained with this generic construction.
§.§ The description of the construction of linear codes
Let $D=\{d_1, \,d_2, \,\ldots, \,d_n\} \subseteq \gf(q)$, where again $q=p^m$.
Recall that $\tr$ denotes the trace function from $\gf(q)$ onto $\gf(p)$ throughout
this paper. We define a linear code of
length $n$ over $\gf(p)$ by
\begin{eqnarray}\label{eqn-maincode}
\C_{D}=\{(\tr(xd_1), \tr(xd_2), \ldots, \tr(xd_n)): x \in \gf(q)\},
\end{eqnarray}
and call $D$ the defining set of this code $\C_{D}$. By definition, the
dimension of the code $\C_D$ is at most $m$.
This construction is generic in the sense that many classes of known codes could be produced
by selecting the defining set $D \subseteq \gf(q)$ properly. This construction technique was
employed in <cit.>, <cit.>, <cit.>, <cit.> and other papers for obtaining
linear codes with a few weights. If the set $D$ is properly chosen, the code $\C_D$ may have
good or optimal parameters. Otherwise, the code $\C_D$ could have bad parameters.
§.§ The weights in the linear codes $\C_D$
It is convenient to define for each $x \in \gf(q)$,
\begin{eqnarray}\label{eqn-mcodeword}
\bc_{x}=(\tr(xd_1), \,\tr(xd_2), \,\ldots, \,\tr(xd_n)).
\end{eqnarray}
The Hamming weight $\wt(\bc_x)$ of $\bc_x$ is $n-N_x(0)$, where
N_x(0)=\left|\{1 \le i \le n: \tr(xd_i)=0\}\right|
for each $x \in \gf(q)$.
It is easily seen that for any $D=\{d_1,\,d_2,\,\ldots, \,d_n\} \subseteq \gf(q)$
we have
\begin{eqnarray}\label{eqn-hn3}
= \sum_{i=1}^n \sum_{y \in \gf(p)} e^{2\pi \sqrt{-1} y\tr(xd_i)/p} \nonumber
= \sum_{i=1}^n \sum_{y \in \gf(p)} \chi_1(yxd_i) \nonumber
%&=& n + \sum_{i=1}^n \sum_{y \in \gf(p)^*} \chi_1(yxd_i) \nonumber \\
= n + \sum_{y \in \gf(p)^*} \chi_1(yxD)
\end{eqnarray}
where $\chi_1$ is the canonical additive character of $\gf(q)$, $aD$ denotes the set
$\{ad: d \in D\}$, and $\chi_1(S):=\sum_{x \in S} \chi_1(x)$ for any subset $S$ of $\gf(q)$.
\begin{eqnarray}\label{eqn-weight}
\wt(\bc_x)=n-N_x(0)=\frac{(p-1)n-\sum_{y \in \gf(p)^*} \chi_1(yxD)}{p}.
\end{eqnarray}
§.§ Differences between the first and second generic constructions
The second generic construction of this section is different from the first generic construction of Section
<ref> in the following aspects:
* While the length of the codes in the first generic construction in Section <ref> is either
$q$ or $q-1$, that of the codes in the second generic construction could be any integer between 1 and $q$, depending on
the underlying defining set $D$.
* While the dimension of the codes in the first construction in Section <ref> is usually $2m$,
that of the codes in the second construction is usually $m$ and is at most $m$.
§ BINARY CODES FROM THE PREIMAGE $F^{-1}(B)$ OF BOOLEAN FUNCTIONS $F$
Let $f$ be a function from $\gf(2^m)$ to $\gf(2)$, and let $D$ be any subset of the preimage $f^{-1}(b)$ for any
$b \in \gf(2)$. In general, it is very hard to determine the parameters of the code $\C_{D}$.
Recall the support $D_f$ of $f$ defined in (<ref>). Let $n_f=|D_f|$.
In this section, we deal with the binary code $\C_{D_f}$ with length $n_f$ and dimension at most $m$, and will
focus on the weight distribution of the code $\C_{D_f}$ for several classes of Boolean functions $f$.
The following theorem plays a major role in this section whose proof can be found in <cit.>.
Let $f$ be a function from $\gf(2^m)$ to $\gf(2)$, and let $D_f$ be the support of $f$. If $2n_f + \hat{f}(w) \neq 0$
for all $w \in \gf(2^m)^*$, then $\C_{D_f}$ is a binary linear code with
length $n_f$ and dimension $m$, and its weight distribution is given by the following multiset:
\begin{eqnarray}\label{eqn-WTBcodes}
\left\{\left\{ \frac{2n_f+\hat{f}(w)}{4}: w \in \gf(2^m)^*\right\}\right\} \cup \left\{\left\{ 0 \right\}\right\}.
\end{eqnarray}
Theorem <ref> establishes a connection between the set of Boolean functions $f$ such that
$2n_f + \hat{f}(w) \neq 0$ for all $w \in \gf(2^m)^*$ and a class of binary linear codes.
The determination of the weight distribution of the binary linear code $\C_{D_f}$ is equivalent to that of the Walsh
spectrum of the Boolean function $f$ satisfying $2n_f + \hat{f}(w) \neq 0$ for all $w \in \gf(2^m)^*$.
When the Boolean function $f$ is selected properly, the code $\C_{D_f}$ has
only a few weights and may have good parameters. We will demonstrate this in the remainder of this section.
We point out that Theorem <ref> can be generalized into the following whose proof is the same
as that of Theorem <ref>.
Let $f$ be a function from $\gf(2^m)$ to $\gf(2)$, and let $D_f$ be the support of $f$. Let $e_w$ denote the
multiplicity of the element $\frac{2n_f+\hat{f}(w)}{4}$ and $e$ the multiplicity of 0 in the following multiset
of (<ref>).
Then $\C_{D_f}$ is a binary linear code with length $n_f$ and dimension $m-\log_2 e$, and
the weight distribution of the code is given by
\begin{eqnarray*}
\frac{2n_f+\hat{f}(w)}{4} \mbox{ with frequency } \frac{e_w}{e}
\end{eqnarray*}
for all $\frac{2n_f+\hat{f}(w)}{4}$ in the multiset of (<ref>).
§.§ Linear codes from bent functions
A function from $\gf(2^m)$ to $\gf(2)$ is called bentbent if $|\hat{f}(w)|=
2^{m/2}$ for every $w \in \gf(2^m)$. Bent functions exist only for even $m$ <cit.>.
It is well known that
a function $f$ from $\gf(2^m)$ to $\gf(2)$ is bent if and only if $D_f$ is
a difference set in $(\gf(2^m),\,+)$ with the following parameters
\begin{eqnarray}\label{eqn-MenonHadamardPara}
(2^m, \, 2^{m-1} \pm 2^{(m-2)/2}, \, 2^{m-2} \pm 2^{(m-2)/2}).
\end{eqnarray}
Let $f$ be bent. Then by definition $\hat{f}(0)=\pm 2^{m/2}$. It then follows that
\begin{eqnarray}\label{eqn-bentfuncsupportsize}
n_f=|D_f|=2^{m-1} \pm 2^{(m-2)/2}
\end{eqnarray}
The weight distribution of the codes of Corollary <ref>
Weight $w$ Multiplicity $A_w$
$0$ $1$
$\frac{n_f}{2}-2^{\frac{m-4}{2}}$ $\frac{2^m-1-n_f2^{-\frac{m-2}{2}}}{2}$
$\frac{n_f}{2}+2^{\frac{m-4}{2}}$ $\frac{2^m-1+n_f2^{-\frac{m-2}{2}}}{2}$
As a corollary of Theorem <ref>, we have the following <cit.>.
Let $f$ be a Boolean function from $\gf(2^m)$ to $\gf(2)$ with $f(0)=0$, where $m \geq 4$ and is even.
Then $\C_{D_f}$ is an $[n_f, \,m, \,(n_f-2^{(m-2)/2})/2]$
two-weight binary code with the weight distribution in Table <ref>, where $n_f$ is defined in (<ref>), if and only if $f$ is bent.
There are many constructions of bent functions and thus Hadamard difference sets. We refer the reader to
<cit.>, <cit.>, <cit.>, the book chapter <cit.> and the references
therein for details. Any bent function can be plugged into Corollary <ref> to obtain a two-weight
binary linear code.
The construction of binary codes with bent functions above can be generalized as follows.
Let $D$ be a $(2^m, n, \, \lambda)$ difference sets in $(\gf(2^m), +)$.
Then $\C_D$ is a two-weight binary code with parameters $[n,\, m]$
and weight enumerator
\begin{eqnarray*}
1+ \frac{(2^m-1)\sqrt{n-\lambda} - n}{2\sqrt{n-\lambda}} z^{\frac{n-\sqrt{n-\lambda}}{2} } +
\frac{(2^m-1)\sqrt{n-\lambda} + n}{2\sqrt{n-\lambda}} z^{\frac{n + \sqrt{n-\lambda}}{2}}.
\end{eqnarray*}
§.§ Linear codes from semibent functions
Let $m$ be odd. Then there is no bent Boolean function on $\gf(2^m)$.
A function $f$ from $\gf(2^m)$ to $\gf(2)$ is called semibent if $\hat{f}(w) \in \{0, \,
\pm 2^{(m+1)/2}\}$ for every $w \in \gf(2^m)$.
Let $f$ be a semibent function from $\gf(2^m)$ to $\gf(2)$. It then follows from the definition of semibent functions
\begin{eqnarray}\label{eqn-semibf}
n_f=|D_f| = \left\{ \begin{array}{ll}
2^{m-1}-2^{(m-1)/2} & \mbox{ if } \hat{f}(0)=2^{(m+1)/2}, \\
2^{m-1}+2^{(m-1)/2} & \mbox{ if } \hat{f}(0)=-2^{(m+1)/2}, \\
2^{m-1} & \mbox{ if } \hat{f}(0)=0.
\end{array}
\right.
\end{eqnarray}
The weight distribution of the codes of Corollary <ref>
Weight $w$ Multiplicity $A_w$
$0$ $1$
$\frac{n_f-2^{(m-1)/2}}{2}$ $n_f(2^m-n_f)2^{-m} - n_f2^{-(m+1)/2}$
$\frac{n_f}{2}$ $2^m-1-n_f(2^m-n_f)2^{-(m-1)}$
$\frac{n_f+2^{(m-1)/2}}{2}$ $n_f(2^m-n_f)2^{-m} + n_f2^{-(m+1)/2}$
As a corollary of Theorem <ref>, we have the following <cit.>.
Let $f$ be a Boolean function from $\gf(2^m)$ to $\gf(2)$ with $f(0)=0$, where $m$ is odd.
Then $\C_{D_f}$ is an $[n_f, \,m, \,(n_f-2^{(m-1)/2})/2]$
three-weight binary code with the weight distribution in Table <ref>, where $n_f$ is defined in (<ref>), if and only if $f$ is semibent.
There are a lot of constructions of semibent functions from $\gf(2^m)$ to $\gf(2)$. We refer the reader to
<cit.> for detailed constructions. All semibent functions can be plugged into Corollary <ref> to obtain three-weight binary linear codes.
§.§ Linear codes from almost bent functions
For any function $g$ from $\gf(2^m)$ to $\gf(2^m)$, we define
\lambda_g(a, b) = \sum_{x \in \gf(2^m)} (-1)^{\tr(ag(x)+bx)}, \ a,\, b \in \gf(2^m).
A function $g$ from $\gf(2^m)$ to $\gf(2^m)$ is called almost bent (AB) if
$\lambda_g(a, b) = 0, \mbox{ or } \pm 2^{(m+1)/2}$ for every pair $(a, b)$ with $a \neq 0$.
By definition, almost bent functions over $\gf(2^m)$ exist only for odd $m$.
Specific almost bent functions are available in <cit.>.
By definition, $\lambda_g(1, 0) \in \{0, \,\pm 2^{(m+1)/2}\}$ for any almost bent function $g$ on $\gf(2^m)$. It is straightforward
to deduce the following lemma.
For any almost bent function $g$ from $\gf(2^m)$ to $\gf(2^m)$, define $f=\tr(g)$. Then we have
\begin{eqnarray}\label{eqn-newnf2}
n_{f}=|D_{\tr(g)}| &=& \left\{ \begin{array}{ll}
2^{m-1}+2^{(m-1)/2} & \mbox{ if } \lambda_g(1, 0)=-2^{(m+1)/2}, \\
2^{m-1}-2^{(m-1)/2} & \mbox{ if } \lambda_g(1, 0)=2^{(m+1)/2}, \\
2^{m-1} & \mbox{ if } \lambda_g(1, 0)=0.
\end{array}
\right.
\end{eqnarray}
As a corollary of Theorem <ref>, we have the following <cit.>.
Let $g$ be an almost bent function from $\gf(2^m)$ to $\gf(2^m)$ with $\tr(g(0))=0$, where $m$ is odd. Define $f=\tr(g)$.
Then $\C_{D_{f}}$ is an $[n_f, \,m, \,(n_f-2^{(m-1)/2})/2]$
three-weight binary code with the weight distribution in Table <ref>, where $n_f$ is given in (<ref>).
The following is a list of almost bent functions $g(x)=x^d$ on $\gf(2^m)$ for odd $m$:
* $d=2^h+1$, where $\gcd(m, h)=1$ is odd <cit.>.
* $d=2^{2h}-2^h+1$, where $h \geq 2$ and $\gcd(m, h)=1$ is odd <cit.>.
* $d=2^{(m-1)/2}+3$ <cit.>.
* $d=2^{(m-1)/2}+2^{(m-1)/4}-1$, where $m \equiv 1 \pmod{4}$ <cit.>.
* $d=2^{(m-1)/2}+2^{(3m-1)/4}-1$, where $m \equiv 3 \pmod{4}$ <cit.>.
This list of almost bent monomials $g(x)$ are permutation polynomials on $\gf(2^m)$. Hence,
the length of the code $\C_{D_{f}}$ is equal to $2^{m-1}$, and the weight distribution of the
code is given in Table <ref>.
§.§ Linear codes from quadratic Boolean functions
\begin{eqnarray}\label{eqn-QBFs}
f(x)=\tr_{2^m/2} \left( \sum_{i=0}^{\lfloor m/2 \rfloor} f_i x^{2^i +1} \right)
\end{eqnarray}
be a quadratic Boolean function from $\gf(2^m)$ to $\gf(2)$, where $f_i \in \gf(2^m)$. The rank of
$f$, denoted by $r_f$, is defined to be the codimension of the $\gf(2)$-vector space
V_f=\{x \in \gf(2^m): f(x+z)-f(x)-f(z)=0 \ \forall \ z \in \gf(2^m)\}.
The Walsh spectrum of $f$ is known <cit.> and given in Table <ref>.
The Walsh spectrum of quadratic Boolean functions
$\hat{f}(w)$ the number of $w$'s
$0$ $2^m-2^{r_f}$
$2^{m-r_f/2}$ $2^{r_f-1}+2^{(r_f-2)/2}$
$-2^{m-r_f/2}$ $2^{r_f-1}-2^{(r_f-2)/2}$
Let $D_f$ be the support of $f$. By definition, we have
\begin{eqnarray}\label{eqn-QBFcodeL}
\begin{array}{ll}
2^{m-1} & \mbox{ if } \hat{f}(0)=0, \\
2^{m-1}-2^{m-1-r_f/2} & \mbox{ if } \hat{f}(0)=2^{m-r_f/2}, \\
2^{m-1}+2^{m-1-r_f/2} & \mbox{ if } \hat{f}(0)=-2^{m-r_f/2}.
\end{array}
\right.
\end{eqnarray}
The following theorem then follows from Theorem <ref> and Table <ref>.
Let $f$ be a quadratic Boolean function of the form in (<ref>) such that $r_f > 2$.
Then $\C_{D_f}$ is a binary code with length $n_f$ given in (<ref>),
dimension $m$, and the weight distribution in Table <ref>, where
\begin{eqnarray}
(\epsilon_1, \epsilon_1, \epsilon_3)=
\left\{
\begin{array}{ll}
(1,0,0) & \mbox{ if } \hat{f}(0)=0, \\
(0,1,0) & \mbox{ if } \hat{f}(0)=2^{m-1-r_f/2}, \\
(0,0,1) & \mbox{ if } \hat{f}(0)=-2^{m-1-r_f/2}.
\end{array}
\right.
\end{eqnarray}
The weight distribution of the code $\C_{D_{f}}$ in Theorem <ref>
Weight $w$ $A_w$
$0$ $1$
$\frac{n_f}{2}$ $2^m-2^{r_f}-\epsilon_1$
$\frac{n_f+2^{m-1-r_f/2}}{2}$ $2^{r_f-1}+2^{(r_f-2)/2}-\epsilon_2$
$\frac{n_f-2^{m-1-r_f/2}}{2}$ $2^{r_f-1}-2^{(r_f-2)/2}-\epsilon_3$
Note that the code $\C_{D_f}$ in Theorem <ref> defined by any quadratic Boolean function $f$ is
different from any subcode of the second-order Reed-muller code, due to the difference in their lengths. The weight
distributions of the two codes are also different.
§.§ Some binary codes $\C_{D_f}$ with three weights
Boolean functions with three-valued Walsh spectrum
$\hat{f}(w)$ the number of $w$'s
$0$ $2^m-2^{m-e}$
$2^{(m+e)/2}$ $2^{m-e-1}+2^{(m-e-2)/2}$
$-2^{(m+e)/2}$ $2^{m-e-1}-2^{(m-e-2)/2}$
Let $m \geq 4$ be even. Then the code $\C_{D_f}$ has parameters $[2^{m-1},\, m,\, 2^{m-2}-2^{(m-2)/2}]$
and the weight distribution of Table <ref>, where $e=2$, for $f(x)=\tr(x^d)$ for the following $d$:
* $d=2^h+1$, where $\gcd(m, h)$ is odd and $1 \leq h \leq m/2$ <cit.>.
* $d=2^{2h}-2^h+1$, where $\gcd(m, h)$ is odd and $1 \leq h \leq m/2$ <cit.>.
* $d=2^{m/2}+2^{(m+2)/4}+1$, where $m \equiv 2 \pmod{4}$ <cit.>.
* $d=2^{(m+2)/2}+3$, where $m \equiv 2 \pmod{4}$ <cit.>.
It can be verified that $\gcd(d, 2^m-1)=1$ for all the $d$ listed above. Hence $n_f=|D_f|=2^{m-1}$.
The Walsh spectrum of the functions $f$ above is given in Table <ref> according
to the references given in this theorem. The desired conclusions on the parameters and the weight
distribution of the code $\C_{D_f}$ then follow from Theorem <ref>.
The weight distribution of some three-weight codes
Weight $w$ Multiplicity $A_w$
$0$ $1$
$2^{m-2}$ $2^m-2^{m-e}-1$
$2^{m-2}+2^{(m+e-4)/2}$ $2^{m-e-1}+2^{(m-e-2)/2}$
$2^{m-2}-2^{(m+e-4)/2}$ $2^{m-e-1}-2^{(m-e-2)/2}$
§.§ Binary codes $\C_{D_f}$ with four weights
Boolean functions with four-valued Walsh spectrum: Case I
$\hat{f}(w)$ the number of $w$'s
$-2^{m/2}$ $(2^{m}-2^{m/2})/3$
$0$ $2^{m-1}-2^{(m-2)/2}$
$2^{m/2}$ $2^{m/2}$
$2^{(m+2)/2}$ $(2^{m-1}-2^{(m-2)/2})/3$
Boolean functions with four-valued Walsh spectrum: Case 2
$\hat{f}(w)$ the number of $w$'s
$-2^{m/2}$ $2^{m-1}-2^{(3m-4)/4}$
$0$ $2^{3m/4}-2^{m/4}$
$2^{m/2}$ $2^{m-1}-2^{(3m-4)/4}$
$2^{3m/4}$ $2^{m/4}$
Boolean functions with four-valued Walsh spectrum: Case 3
$\hat{f}(w)$ the number of $w$'s
$-2^{m/2}$ $(2^{m}-2^{m/2}-2)/3$
$0$ $2^{m-1}-2^{(m-2)/2}+2$
$2^{m/2}$ $2^{m/2}-2$
$2^{(m+2)/2}$ $(2^{m-1}-2^{(m-2)/2}+2)/3$
The code $\C_{D_f}$ has four weights and its weight distribution is known when $f(x)=\tr(x^d)$ and $d$ is given
in the following list.
* When $d=2^{(m+2)/2}-1$ and $m \equiv 0 \pmod{4}$,
the code $\C_{D_f}$ has length $2^{m-1}$ and dimension $m$,
and the weight distribution of $\C_{D_f}$ is deduced from Theorem <ref> and
Table <ref>, where $h=1$ <cit.>.
* When $d=2^{(m+2)/2}-1$ and $m \equiv 2 \pmod{4}$,
the code $\C_{D_f}$ has length $2^{m-1}-2^{m/2}$ and dimension $m$,
and the weight distribution of $\C_{D_f}$ is deduced from Theorem <ref> and
Table <ref> <cit.>. Note that in this case,
$\gcd(d, 2^m-1)=3$.
* When $d=(2^{m/2}+1)(2^{m/4}-1)+2$ and $m \equiv 0 \pmod{4}$,
the code $\C_{D_f}$ has length $2^{m-1}$ and dimension $m$,
and the weight distribution of $\C_{D_f}$ is deduced from Theorem <ref> and
Table <ref> <cit.>.
* When $d=\frac{2^{(m+2)h/2}-1}{2^h-1}$ and $m \equiv 0 \pmod{4}$, where $1 \leq h < m$ and $\gcd(h, m)=1$,
the code $\C_{D_f}$ has length $2^{m-1}$ and dimension $m$,
and the weight distribution of $\C_{D_f}$ is deduced from Theorem <ref> and
Table <ref>, where $h=1$ <cit.>.
* When $d=\frac{2^{(m+2)h/2}-1}{2^h-1}$ and $m \equiv 2 \pmod{4}$, where $1 \leq h < m$ and $\gcd(h, m)=1$,
the code $\C_{D_f}$ has length $2^{m-1}-2^{m/2}$ and dimension $m$,
and the weight distribution of $\C_{D_f}$ is deduced from Theorem <ref> and
Table <ref> <cit.>. Note that in this case,
$\gcd(d, 2^m-1)=3$.
* When $d=\frac{2^m+2^{h+1}-2^{m/2+1}-1}{2^h-1}$, where $2h$ divides $m/2$ and
$m \equiv 0 \pmod{4}$,
the code $\C_{D_f}$ has length $2^{m-1}$ and dimension $m$,
and the weight distribution of $\C_{D_f}$ is deduced from Theorem <ref> and
Table <ref> <cit.>.
* When $d=(2^{m/2}-1)s+1$ with $s=2^h(2^h \pm 1)^{-1} \pmod{2^{m/2}+1}$, where $e_2(h) < e_2(m/2)$
and $e_2(h)$ denotes the highest power of $2$ dividing $h$, the parameters and the weight distribution of the
code $\C_{D_f}$ can be deduced from Theorem <ref> and the results in <cit.>.
* Let $d$ be any integer such that $1 \leq d \leq 2^m-2$ and $d(2^\ell +1) \equiv 2^h \pmod{2^m-1}$ for some
positive integers $\ell$ and $h$. Then the parameters and the weight distribution of the code $\C_{D_f}$ can be
deduced from Theorem <ref> and the results in <cit.>.
All these cases of $d$ above are derived from the cross-correlation of a binary maximum-length
sequence with its $d$-decimation version.
§.§ Other binary codes $\C_{D_f}$ with at most five weights
The code $\C_{D_f}$ has at most five weights for the following $f$:
* When $f(x)=\tr(x^{2^{m/2}+3})$, where $m \geq 6$ and is even, $\C_{D_f}$ is a five-weight code with
length $2^{m-1}$ and dimension $m$, and its weight distribution can be derived from <cit.>.
* When $f(x)=\tr(ax^{(2^m-1)/3})$ with $\tr_{4}^{2^m}(a) \ne 0$, where $m$ is even, $\C_{D_f}$ is a
two-weight code with length $(2^{m+2}-4)/6$ and dimension $m$, and its weight distribution can be derived from
* When $f(x)=\tr(\lambda x^{2^{m/2}+1}) +\tr(x)\tr(\mu x^{2^{m/2}-1})$,
where $m$ is even, $\mu \in \gf(2^{m/2})^*$, and $\lambda \in \gf(2^m)$
with $\lambda + \lambda^{2^m}=1$, $\C_{D_f}$ is a five-weight code <cit.>.
* When $f(x)=(1+\tr(x))\tr(\lambda x^{2^{m/2}+1}) +\tr(x)\tr(\mu x^{2^{m/2}-1})$,
where $m$ is even, $\mu \in \gf(2^{\frac{m}{2}})^*$, and $\lambda \in \gf(2^m)$
with $\lambda + \lambda^{2^m}=1$, $\C_{D_f}$ is a five-weight code <cit.>.
Some Boolean functions $f$ documented in <cit.> gives also binary linear code $\C_{D_f}$ with five
§.§ A class of two-weight codes from the preimage of a type of Boolean functions
Let $m$ be a positive integer and let $r$ be a prime such that $2$ is a primitive root modulo
$r^m$. Let $q=2^{\phi(r^m)}$, where $\phi$ is the Euler function.
\begin{eqnarray}\label{eqn-wdxD}
D=\left\{ x \in \gf(q)^*: \tr\left( x^{\frac{q-1}{r^m}} \right) =0 \right\}.
\end{eqnarray}
The following theorem was proved in <cit.>.
Let $r^m \geq 9$ and let $D$ be defined in (<ref>). Then the set $\C_D$ of (<ref>)
is a binary code with length $(q-1)(r^m-r+1)/r^m$, dimension $(r-1)r^{m-1}$ and the weight distribution
in Table <ref>.
The weight distribution of the codes of Theorem <ref>
weight $w$ Multiplicity $A_{w}$
0 1
$\frac{q-\sqrt{q}}{4}+\frac{q+\sqrt{q}}{4r^{m}}(r^m-2r+2)$ $\frac{(q-1)(r^{m}-r+1)}{r^{m}}$
$\frac{q+\sqrt{q}}{4}+\frac{q+\sqrt{q}}{4r^{m}}(r^m-2r+2)$ $\frac{(q-1)(r-1)}{r^{m}}$
§.§ Binary codes from Boolean functions whose supports are relative difference sets
Let $(A, +)$ be an abelian group of order $m\ell$ and $(N, +)$ a subgroup of $A$ of order $\ell$.
A $n$-subset $D$ of $A$ is called an $(m, \ell, n, \lambda)$ relative difference set, if the
multiset $\{\{ d_1-d_2: d_1, \, d_2 \in D,\ d_1 \neq d_2 \}\}$ does not contain all elements in
$N$, but every element in $A \setminus N$ exactly $\lambda$ times.
It well known in combinatorics that
|\chi(D)|^2 \in \{n, n-\lambda \ell\}
for any nontrivial group character $\chi$. Hence, any relative difference set $D$ in $(\gf(2^m), +)$ defines
a binary code $\C_D$ with at most the following four weights:
\frac{n \pm \sqrt{n}}{2}, \ \frac{n \pm \sqrt{n-\lambda \ell}}{2}.
Obviously, $D$ is the support of a Boolean function on $\gf(2^m)$.
§ BINARY CODES FROM THE IMAGES OF CERTAIN FUNCTIONS ON $\GF(2^M)$
Let $f(x)$ be a function from $\gf(2^m)$ to $\gf(2^m)$. We define
D(f) =\{f(x): x \in \gf(2^m)\} \mbox{ and } D(f)^* =\{f(x): x \in \gf(2^m)\} \setminus \{0\}.
In this section, we consider the code $\C_{D(f)}$. In general, it is difficult to determine the length $n_f:=|D(f)|$ of this code,
not to mention its weight distribution. However, in certain special cases, the parameters and the weight distribution of $\C_{D(f)}$
can be settled.
If $0 \not\in D(f)$, then the two codes $\C_{D(f)}$ and $\C_{D(f)^*}$ are the same. Otherwise, the length of
the code $\C_{D(f)^*}$ is one less than that of the code $\C_{D(f)}$, but the two codes have the same weight
distribution. Thus, we will not give information about the codes $\C_{D(f)^*}$ in this section.
Let $D$ be any subset of $\gf(2^m)$. The characteristic function, denoted by $f_D(x)$, of $D$ is defined by
f_D(x)= \left\{ \begin{array}{ll}
1 & \mbox{ if } x \in D, \\
0 & \mbox{ otherwise.}
\end{array}
\right.
Hence, the Boolean function $f_D$ has support $D$. Thus, the code $\C_{D(f)}$ is in fact defined by the support
of the characteristic function (Boolean function) of the set $D(f)$. Therefore, the construction method of this section
is actually equivalent to that of Section <ref>.
§.§ The codes $\C_{D(f)}$ from o-polynomials on $\gf(2^m)$
A permutation polynomial $f$ on $\gf(2^m)$ is called an o-polynomialo-polynomial if
$f(0)=0$, and for each $s \in \gf(2^m)$,
\begin{eqnarray}
\end{eqnarray}
is also a permutation polynomial. O-polynomial can be used to construct hyperovals in finite geometry.
In the original definition of o-polynomials, it is required that $f(1)=1$. However, this is not essential, as
one can always normalise $f(x)$ by using $f(1)^{-1} f(x)$ due to that $f(1) \neq 0$.
In this section, we consider binary codes $\C_{D(f)}$, where $f$ is defined by an o-polynomial in some way.
§.§.§ O-polynomials and their binary codes $\C_{D(f_u)}$
For any permutation polynomial $f(x)$ over $\gf(2^m)$, we define $\overline{f}(x)=xf(x^{2^m-2})$, and use $f^{-1}$ to denote the compositional inverse of $f$, i.e., $f^{-1}(f(x))=x$ for all $x \in \gf(2^m)$.
The following two theorems introduce basic properties of o-polynomials whose proofs can be
found in references about hyperovals. .
Let $f$ be an o-polynomial on $\gf(2^m)$. Then the following statements hold:
* $f^{-1}$ is also an o-polynomial;
* $f(x^{2^{j}})^{2^{m-j}}$ is also an o-polynomial for any $1 \leq j \leq m-1$;
* $\overline{f}$ is also an o-polynomial; and
* $f(x+1)+f(1)$ is also an o-polynomial.
Let $x^k$ be an o-polynomial on $\gf(2^m)$. Then every polynomial in
$$\left\{x^{\frac{1}{k}},\, x^{1-k},\, x^{\frac{1}{1-k}},\, x^{\frac{k}{k-1}},\, x^{\frac{k-1}{k}}\right\}$$
is also an o-polynomial, where $1/k$ denotes the multiplicative inverse of $k$ modulo $2^m-1$.
The following property of o-polynomials plays an important role in our construction of binary linear codes with
A polynomial $f$ from $\gf(2^m)$ to $\gf(2^m)$ with $f(0)=0$ is an o-polynomial if and only if $f_u:=f(x)+ux$
is $2$-to-$1$ for every $u \in \gf(2^m)^*$.
Let $f$ be any o-polynomial over $\gf(2^m)$. Define
where $u \in \gf(2^m)^*$. It follows from Theorem <ref> that $f_u$ is 2-to-1 for every $u \in \gf(2^m)^*$.
In the rest of Section <ref>, we consider the codes $\C_{D(f_u)}$ defined by o-polynomials. By Theorem
<ref>, the o-polynomial property of $f$ guarantees that the length of the code $\C_{D(f_u)}$ is equal to $2^{m-1}$ for any $u \in \gf(2^m)^*$.
The dimension of $\C_{D(f_u)}$ usually equals $m$, but may be less than $m$. The minimum weight
and the weight distribution of $\C_{D(f_u)}$ cannot be determined by the o-polynomial property alone,
and differ from case to case.
§.§.§ Binary codes from the translation o-polynomials
The translation o-polynomials are described in the following theorem <cit.>.
$\Trans(x)=x^{2^h}$ is an o-polynomial on $\gf(2^m)$, where $\gcd(h, m)=1$.
The following is a list of known properties of translation o-polynomials.
* $\Trans^{-1}(x)=x^{2^{m-h}}$ and
* $\overline{\Trans}(x)= xf(x^{2^m-2})=x^{2^m-2^{m-h}}$.
The proof of the following theorem is straightforward.
Let $f(x)=x^{2^h}$, where $\gcd(h,m)=1$. Then for any $u \in \gf(2^m)^*$, the code $\C_{D(f_u)}$
has parameters $[2^{m-1},\, m-1,\, 2^{m-2}]$ and is a one-weight code.
The codes $\C_{D(f_u)}$ in Theorem <ref> have the same parameters as a subcode of
the first order binary Reed-Muller code.
§.§.§ Binary codes from the Segre and Glynn o-polynomials
The following theorem describes a class of o-polynomials, which are an extension of the original Segre
Let $m$ be odd. Then
$\Segre_a(x)=x^{6}+ax^4+a^2x^2$ is an o-polynomial on $\gf(2^m)$ for every $a \in \gf(2^m)$.
The conclusion follows from
\Segre_a(x)=(x+\sqrt{a})^6 + \sqrt{a}^3.
We have the following remarks on this family of o-polynomials.
* $\Segre_0(x)=x^6$ is the original Segre o-polynomial <cit.>. So this is an extended family.
* $\Segre_a(x)=xD_5(x, a)=a^2D_5(x^{2^m-2}, a^{2^m-2})x^7$, where $D_5(x, a)=x^5+ax^3+a^2x$,
which is the Dickson polynomial of the first kind of order 5.
* $\overline{\Segre}_a=D_5(x^{2^m-2}, a)=a^2x^{2^m-2}+ax^{2^m-4}+x^{2^m-6}$.
* $\Segre_a^{-1}(x)=(x+\sqrt{a}^3)^{\frac{5\times 2^{m-1}-2}{3}} +\sqrt{a}$.
Let $m$ be odd. Then
\begin{eqnarray}\label{eqn-PayneInverse2}
\overline{\Segre}_1^{-1}(x)=\left( D_{\frac{3 \times 2^{2m}-2}{5}}(x, 1)\right)^{2^m-2}.
\end{eqnarray}
Glynn discovered two families of o-polynomials <cit.>. The first is described as follows.
Let $m$ be odd. Then $\Glynnone(x)=x^{3\times 2^{(m+1)/2}+4}$ is an o-polynomial.
Let $m \ge 3$ be odd, and let $f(x)=x^{3\times 2^{(m+1)/2}+4}$ be the Glynn o-polynomial.
When $m \in \{5, 7\}$, $\C_{D(f_u)}$ is a $[2^{m-1},\, m]$ code with the weight distribution
of Table <ref>. When $m \ge 9$, $\C_{D(f_u)}$ is a $[2^{m-1},\, m]$
code with five weights.
An extension of the second family of o-polynomials discovered by Glynn is documented in the following theorem.
Let $m$ be odd. Then
\begin{eqnarray*}
\Glynntwo_a(x)= \left\{
\begin{array}{ll}
x^{2^{(m+1)/2}+2^{(3m+1)/4}} + a x^{2^{(m+1)/2}} + (ax)^{2^{(3m+1)/4}} & \mbox{ if } m \equiv 1 \pmod{4}, \\
x^{2^{(m+1)/2}+2^{(m+1)/4}} + a x^{2^{(m+1)/2}} + (ax)^{2^{(m+1)/4}} & \mbox{ if } m \equiv 3 \pmod{4}.
\end{array}
\right.
\end{eqnarray*}
is an o-polynomial for all $a \in \gf(q)$.
Let $m \equiv 1 \pmod{4}$. Then
\Glynntwo_a(x)=(x+a^{(m-1)/4})^{2^{(m+1)/2}+2^{(3m+1)/4}} + a^{2^{(m+1)/2}+2^{(3m+1)/4}}.
The desired conclusion for the case $m \equiv 1 \pmod{4}$ can be similarly proved.
Note that $\Glynntwo_0(x)$ is the original Glynn o-polynomial. So this is an extended family. For some applications,
the extended family may be useful.
For certain quadratic Boolean functions $f$, the code $\C_{D(f_u)}$ has good parameters and its weight distribution
is known. The following result is an extended version of a result proved in <cit.>.
The weight distribution of the codes of Theorem <ref>
Weight $w$ Multiplicity $A_w$
$0$ $1$
$2^{m-2}-2^{(m-3)/2}$ $2^{m-2}+2^{(m-3)/2}$
$2^{m-2}$ $2^{m-1}-1$
$2^{m-2}+2^{(m-3)/2}$ $2^{m-2}-2^{(m-3)/2}$
Let $\rho=2^i + 2^j$ and define
f_{u}(x)=x^\rho+ux, \, u \in \gf(2^m)^*.
If $f_u(x)$ is 2-to-1 on $\gf(2^m)$ and $\gcd(2^\kappa+1, \,2^m-1)=1$, where $\kappa=j-i$,
then the binary code $\C_{D(f_u)}$ has parameters $[2^{m-1}, \,m, \,2^{m-2} - 2^{(m-3)/2}]$ and the weight
distribution of Table <ref> for any $u \in \gf(2^m)^*$.
The following $\rho$ satisfies the conditions of Theorem <ref>:
* $\rho=6$ (Segre case).
* $\rho=2^\sigma + 2^\pi$ with $\sigma=(m+1)/2$ and $4\pi \equiv 1 \bmod{m}$ (Glynn I case).
Let $f(x)=xD_5(x, a)$, where $a \in \gf(2^m)$, and let $m$ be odd. Then the code $\C_{D(f_u)}$ has parameters
$[2^{m-1},\, m]$ and the weight distribution of Table <ref> for any $u \in \gf(2^m)^*$.
It is easily verified that $f(x)=(x+\sqrt{a})^6 + \sqrt{a}^3$. The desired conclusions then follow from Theorem
<ref> in the Segre case, i.e., $\rho=6$.
Let $F(x)=x^{(5 \times 2^{m-1}-2)/3}$, and let $m$ be odd. Then code $\C_{D(f_u)}$ has parameters
$[2^{m-1},\, m]$ and the weight distribution of Table <ref> for any $u \in \gf(2^m)$.
Note that the multiplicative inverse of $6$ modulo $2^m-1$ is equal to $(5 \times 2^{m-1}-2)/3$. The desired
conclusion then follows from Theorem <ref>.
§.§.§ Binary codes from the Cherowitzo o-polynomials
The following describes another conjectured class of o-polynomials.
Let $m$ be odd and $e=(m+1)/2$. Then
\Cherowitzo_a(x)=x^{2^e}+ax^{2^e+2}+a^{2^e +2}x^{3 \times 2^e +4}
is an o-polynomial on $\gf(2^m)$ for every $a \in \gf(2^m)$.
We have the following remarks on this family.
* $\Cherowitzo_1(x)$ is the original Cherowitzo o-polynomial <cit.>. So this is an extended family.
* No proof of the o-polynomial property is known in the literature.
* $\overline{\Cherowitzo}(x)=x^{2^m-2^e}+ax^{2^m-2^e-2}+a^{2^e +2}x^{2^m-3 \times 2^e -4}$.
* Carlet and Mesnager showed that $\Cherowitzo_1^{-1}(x)=x(x^{2^e+1}+x^3+x)^{2^{e-1}-1}$.
We can prove the following.
\Cherowitzo_a^{-1}(x)=x(ax^{2^e+1}+a^{2^e}x^3+x)^{2^{e-1}-1}.
\overline{\Cherowitzo}=(ax^{2^m-2^e-2}+a^{2^e}x^{2^m-4}+x^{2^m-2})^{2^{e-1}-1}.
Let $m$ be odd, and let
$$f(x)=b^{2^{(m+1)/2}+2}x^{2^{(m+1)/2}}+b^{2^{(m+1)/2}+1}x^{2^{(m+1)/2} +2}+ x^{3 \times 2^{(m+1)/2}+4},$$
where $b \in \gf(2^m)$.
If $m \in \{5, 7\}$,
$\C_{D(f_u)}$ is a $[2^{m-1},\, m]$ code with at most five weights for every $u \in \gf(2^m)^*$.
If $m \geq 9$, $\C_{D(f_u)}$ is a five-weight code with length
$2^{m-1}$ and dimension $m$ for every $u \in \gf(2^m)^*$.
§.§.§ Binary codes from the Payne o-polynomials
The following documents a conjectured family of o-trinomials.
Let $m$ be odd. Then
$\Payne_a(x)=x^{\frac{5}{6}}+ax^{\frac{3}{6}}+a^2x^{\frac{1}{6}}$ is an o-polynomial on $\gf(2^m)$ for every $a \in \gf(2^m)$.
We have the following remarks on this family.
* $\Payne_1(x)$ is the original Payne o-polynomial <cit.>. So this is an extended family.
* $\Payne_a(x)=xD_5(x^{\frac{1}{6}}, a)$.
* $\overline{\Payne}_a(x)=a^{2^m-3}\Payne_{a^{2^m-2}}(x)$.
* Note that
\frac{1}{6}=\frac{5 \times 2^{m-1}-2}{3}.
We have then
\Payne_a(x)=x^{\frac{2^{m-1}+2}{3}} + ax^{2^{m-1}} + a^2x^{\frac{5 \times 2^{m-1}-2}{3}}.
Let $m$ be odd. Then
\begin{eqnarray}\label{eqn-PayneInverse}
\Payne_1^{-1}(x)=\left( D_{\frac{3 \times 2^{2m}-2}{5}}(x, 1)\right)^6
\end{eqnarray}
and $\overline{\Payne}_1^{-1}(x)$ are an o-polynomial.
Note that the multiplicative inverse of $5$ modulo is $\frac{3 \times 2^{2m}-2}{5}$. The conclusion then
follows from the definition of the Payne polynomial and the fact that
D_5(x, 1)^{-1}=D_{\frac{3 \times 2^{2m}-2}{5}}(x, 1).
Let $m$ be odd, and let
$$f(x)=x^{\frac{5}{6}}+bx^{\frac{3}{6}}+b^2x^{\frac{1}{6}}=D_5\left(x^{\frac{5 \times 2^{m-1}-2}{3}}, \, b\right),$$
where $b \in \gf(2^m)$. If $m \geq 7$, $\C_{D(f_u)}$ is a three-weight or five-weight code with length $2^{m-1}$ and dimension $m$ for
all $u \in \gf(2^m)^*$.
§.§.§ Binary codes from the Subiaco o-polynomials
The Subiaco o-polynomials are given in the following theorem <cit.>.
\Subiaco_a(x)=((a^2(x^4+x)+a^2(1+a+a^2)(x^3+x^2)) (x^4 + a^2 x^2+1)^{2^m-2}+x^{2^{m-1}},
where $\tr(1/a)=1$ and $d \not\in \gf(4)$ if $m \equiv 2 \bmod{4}$. Then $\Subiaco_a(x)$ is an
o-polynomial on $\gf(2^m)$.
As a corollary of Theorem <ref>, we have the following.
Let $m$ be odd. Then
\begin{eqnarray}\label{cor-Subiaco}
\Subiaco_1(x)=(x+x^2+x^3+x^4) (x^4 + x^2+1)^{2^m-2}+x^{2^{m-1}}
\end{eqnarray}
is an o-polynomial over $\gf(2^m)$.
Experimental data shows that the binary codes $\C_{D(f_u)}$ from the Subiaco o-polynomials have many weights
and have smaller minimum weights compared with binary codes from other o-polynomials described in the previous
subsections. Hence, the binary code $\C_{D(f_u)}$ from an o-polynomial could be very good and bad, depending on
the specific o-polynomial $f$.
§.§ Binary codes $\C_{D(f)}$ from functions on $\gf(2^m)$ of the form $f(x)=F(x)+F(x+1)+1$
A function $F(x)$ over $\gf(2^m)$ is called almost perfect nonlinear (APN) if
\max_{a \in \gf(2^m)^*} \max_{b \in \gf(2^m)} \left|\{x \in \gf(2^m): F(x+a)-F(x)=b\}\right| =2.
Let $F$ be any function on $\gf(2^m)$. Define
For certain APN functions $F(x)$ over $\gf(2^m)$, it is known that $f$ is 2-to-1.
Let $F(x)=x^{2^{(m-1)/2} + 3}$ and $m$ be odd. It is known that $F$ is both APN and AB.
If $m \in \{5, 7\}$, $\C_{D(f)}$ is a three-weight code
with length $2^{m-1}$ and dimension $m$. If $m \geq 9$, $\C_{D(f)}$ is a five-weight code with length
$2^{m-1}$ and dimension $m$.
Let $F(x)=x^{2^{2h}-2^h+1}$, and let $\gcd(h, m)=1$. It is known that $F$ is both APN and AB.
When $h=1$, $\C_{D(f)}$ is a $[2^{m-1}, m-1, 2^{m-2}]$ one-weight code.
When $h \geq 2$ and $m$ is odd, $\C_{D(f)}$ is a three-weight or
five-weight code with length $2^{m-1}$ and dimension $m$.
In particular, when $h=3$ and $m$ is odd, $\C_{D(f)}$ is a three-weight code with length
$2^{m-1}$ and dimension $m$ for every odd $m \geq 5$ and $m \not\equiv 0 \pmod{3}$.
In this case, $d=57$ and the weight distribution of the code $\C_{D(f)}$ is given in Table
It is known that $f(x)=x^{2^{2h}-2^h+1} + (x+1)^{2^{2h}-2^h+1}$ is $2^s$-to-1, where $s=\gcd(h, m)$ <cit.>.
Let $F(x)=x^{2^h+1}$, and let $\gcd(h, m)=1$. Then $\C_{D(f)}$ is a one-weight code with
parameters $[2^{m-1},\, m-1,\, 2^{m-2}]$.
The proof is straightforward and omitted, as $f(x)$ is linear.
We have the following comments on other APN monomials.
* Let $F(x)=x^{2^m-2}$. Then $\C_{D(f)}$ is a binary code with length $2^{m-1}$ and
dimension $m$, and has at most $m$ weights. The weights are determined by the Kloosterman
* For the Niho function $F(x)=x^{2^{(m-1)/2}+2^{(m-1)/4}-1}$, where $m \equiv 1 \pmod{4}$,
the code $\C_{D(f)}$ has
length $2^{m-1}$ and dimension $m$, but many weights.
* For the Niho function $F(x)=x^{2^{(m-1)/2}+2^{(3m-1)/4}-1}$, where $m \equiv 3 \pmod{4}$,
the code $\C_{D(f)}$ has
length $2^{m-1}$ and dimension $m$, but many weights.
It would be extremely difficult to determine the weight distribution of the code $\C_{D(f)}$ for
these three classes of APN monomials.
§.§ Binary linear codes from some trinomials
A lot of constructions of cyclic difference sets in $(\gf(2^m)^*, \,\times)$ with the Singer parameters
$(2^m-1,\, 2^{m-1},\, 2^{m-2})$ or $(2^m-1,\, 2^{m-1}-1,\, 2^{m-2}-1)$ are proposed in the literature
<cit.>. These difference sets can certainly be plugged into the second generic
construction of this paper and obtain binary linear codes with good parameters. But determining the parameters
of the binary linear codes may be difficult in general.
There are also a number of conjectured cyclic difference sets in $(\gf(2^m)^*, \,\times)$. They give naturally
binary linear codes with this construction.
The following is a list of conjectured cyclic difference sets in $(\gf(2^m)^*, \,\times)$ with Singer parameters
(see Chapter 4 of <cit.>).
For any $f \in \gf(2^m)[x]$, we define
D(f)^*=\{f(x): \,x \in \gf(2^m)\} \setminus \{0\}.
Let $m \geq 5$ be odd. Then $D(f)^*$ is
a difference set in $(\gf(2^m)^*, \,\times)$ with Singer parameters $(2^{m}-1, \,2^{m-1}, \,2^{m-2})$ for the following trinomials $f \in \gf(2^m)[x]$:
* $f(x)=x^{2^m-17} + x^{(2^m +19)/3} + x$.
* $f(x)=x^{2^m - 2^{m-4} -1} + x^{2^m - (2^{m-2}+4)/3} + x$.
* $f(x)=x^{2^m-3} + x^{2^{(m+3)/2} + 2^{(m+1)/2} +4} + x$.
* $f(x)=x^{2^m - 2^{(m-1)/2} -1} + x^{2^{m-1} - 2^{(m-1)/2}} + x$.
* $f(x)=x^{2^m -2 - (2^{m-1} -2^2)/3} + x^{2^m -2^2 - (2^m -2^3)/3} + x$.
* $f(x)=x^{2^m - 2^{(m+1)/2} + 2^{(m-1)/2}} + x^{2^m - 2^{(m+1)/2} -1} + x$.
* $f(x)=x^{2^m-3(2^{(m+1)/2}-1)} + x^{2^{(m+1)/2} + 2^{(m-1)/2}-2} +x$.
* $f(x)=x^{2^m-2^{m-2}-1} + x^{2^{m-1}-2}+x$.
* $f(x)=x^{2^m-2^{(m+3)/2}-3} + x^{2^{(m+1)/2} +2} + x$.
* $f(x)=x^{2^m-3(2^{(m-1)/2}+1)} + x^{2^{m-1}-1} + x$.
* $f(x)=x^{2^m-5} + x^6 + x$.
For the linear codes $\C_{D(f)^*}$ of the conjectured difference sets $D(f)^*$ in Conjecture <ref>, we have the
following conjectured parameters.
Let $m \geq 5$ and let $D(f)^*$ be defined as in Conjecture <ref>. Then for every $f$ given in Conjecture <ref>,
the binary linear code $\C_{D(f)^*}$ has parameters $[2^{m-1}, \,m, \,2^{m-2}-2^{(m-3)/2}]$ and weight enumerator
1+(2^{m-2}-2^{(m-3)/2}) z^{2^{m-2}-2^{(m-3)/2}} + (2^{m-1}-1) z^{2^{m-2}} + (2^{m-2}+2^{(m-3)/2}) z^{2^{m-2}+2^{(m-3)/2}} .
The dual code of $\C_{D(f)^*}$ has parameters $[2^{m-1},\, 2^{m-1}-m,\, 3]$.
Conjecture <ref> describes binary three-weight codes for the case that $m$ is odd. The next one is about
binary three-weight codes for the case that $m$ is even.
Let $f(x)=x+x^{2^{(m-2)/2} + 2^{m-1}} + x^{2^{(m-2)/2} + 2^{m-1}+1} \in \gf(2^m)[x]$, where $m \equiv 2 \bmod{4}$
and $m \geq 6$. Define
D(f)=\{f(x): \,x \in \gf(2^m)\}.
Then the binary code $\C_{D(f)}$ has parameters $[2^{m-1}, \,m, \,2^{m-2}-2^{(m-2)/2}]$ and
weight enumerator
1+(2^{m-3}+2^{(m-4)/2})z^{2^{m-2}-2^{(m-2)/2}} + (3 \times 2^{m-2}-1) z^{2^{m-2}} +
It was conjectured in <cit.> that $D(f)^*$ is
a difference set in $(\gf(2^m)^*, \,\times)$ with the parameters $(2^{m}-1, \,2^{m-1}-1, \,2^{m-2}-1)$.
The weight distribution of the codes of Conjecture <ref>
Weight $w$ Multiplicity $A_w$
$0$ $1$
$2^{m-2}-2^{(m-2)/2}$ $2^{(m-2)/2}$
$2^{m-2}-2^{(m-4)/2}$ $2^{m-1}-2^{m/2}$
$2^{m-2}$ $2^{m/2}+2^{(m-2)/2}-1$
$2^{m-2}+2^{(m-4)/2}$ $2^{m-1}-2^{m/2}$
Binary four-weight codes may also be produced with difference sets in $(\gf(2^m)^*, \,\times)$ as follows.
Let $f(x)=x+x^2+x^{2^m-2^{m/2} + 1} \in \gf(2^m)[x]$, where $m \geq 4$ and $m$ is even. Define
D(f)=\{f(x): \,x \in \gf(2^m)\}.
Then the binary linear code $\C_{D(f)}$ has parameters $[2^{m-1}, \,m, \,2^{m-2}-2^{(m-2)/2}]$ and
the weight distribution of Table <ref>.
It was conjectured in <cit.> that $D(f)^*$ is
a difference set in $(\gf(2^m)^*, \,\times)$ with parameters $(2^{m}-1, \,2^{m-1}-1, \,2^{m-2}-1)$.
The following is an another list of conjectured cyclic difference sets in $(\gf(2^m)^*, \,\times)$ with Singer parameters
(see Chapter 4 of <cit.>).
For any $f \in \gf(2^m)[x]$, we define
D(f)=\{f(x(x+1)): x \in \gf(2^m)\}.
Let $m \geq 4$. Then $D(f)^*$ is
a difference set in $(\gf(2^m)^*, \,\times)$ with Singer parameters $(2^{m}-1, \,2^{m-1}-1, \,2^{m-2}-1)$ for the following polynomials $f \in \gf(2^m)[x]$:
* $f(x)=x + x^{2^{(m+1)/2}-1} + x^{2^m-2^{(m+1)/2}+1} $, where $m$ is odd.
* $f(x)=x + x^{(2^m +1)/3} + x^{(2^{m+1} -1)/3}$, where $m$ is odd.
* $f(x)=x + x^{2^{(m+2)/2}-1} + x^{2^m-2^{m/2}+1}$, where $m$ is even.
For the linear codes $\C_{D(f)^*}$ of the conjectured difference sets $D(f)^*$ in Conjecture <ref>, we have the
following conjectured parameters.
Let $m \geq 4$ and let $D(f)$ be defined as in Conjecture <ref>. Then for every $f$ given in Conjecture <ref>,
the binary linear code $\C_{D(f)^*}$ has parameters $[2^{m-1}-1, \,m-1, \, 2^{m-2}]$ and is a one-weight code.
To determine the weight distribution of the code $\C_{D(f)}$ or $\C_{D(f)}^*$ of the conjectured difference sets
listed in this section, one does not have to prove the difference set property of the set $D(f)$ or $D(f)^*$.
§ AN EXPANSION OF THE BINARY CODES
The weight distribution of the codes of Theorem <ref>
Weight $w$ Multiplicity $A_w$
$0$ $1$
$2^{m-2}-2^{(m-3)/2}$ $2^{m-1}+2^{(m-1)/2}$
$2^{m-2}$ $2^{m}-2$
$2^{m-2}+2^{(m-3)/2}$ $2^{m-1}-2^{(m-1)/2}$
$2^{m-1}$ 1
Let $\bone$ denote the all-one vector, that is, $(1,1,\ldots,1)$, of any length. The complement of any
vector $\bc \in \gf(2)^n$ is defined to be $\bc+\bone$. For any binary code $\C$, we define
\overline{\C}=\C \cup \{\bc+\bone: \bc \in \C\}.
Then $\overline{\C}$ is a binary linear code, which has the same length as $\C$. For most of the binary codes
$\C$ presented in this paper, the dimension of $\overline{\C}$ is one more than that of $\C$. In many cases,
the weight distribution of $\overline{\C}$ can be deduced from that of $\C$. As an example, we have the
Let $m$ be odd and let $\C$ be any binary linear code with parameters $[2^{m-1}, m]$ and the weight
distribution of Table <ref>. Then $\overline{\C}$ is binary linear code with parameters
$[2^{m-1}, m+1]$ and the weight distribution of Table <ref>.
When $m=5$, the code $\overline{\C}$ of Theorem <ref> has parameters $[16, 6, 6]$
and is optimal.
When $m=7$, the code $\overline{\C}$ of Theorem <ref> has parameters $[64, 8, 28]$
and is almost optimal.
§ CONCLUDING REMARKS
In this paper, we surveyed binary linear codes from Boolean functions and functions on $\gf(2^m)$ obtained
from the second generic
construction. Our focus was on such binary linear codes with at most five weights. Many one-weight codes,
two-weight codes, three-weight codes, four-weight codes are presented in this paper. Some of them are
optimal and some are almost optimal. The codes are also quite interesting in the sense that they may have
applications in secret sharing <cit.> and authentication codes <cit.>. The
parameters of some of the binary codes are different from those in <cit.>, <cit.>, <cit.>,
<cit.>, <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>.
A number of conjectures were presented in this paper as open problems. All the conjectures on difference sets,
o-polynomials and the corresponding binary codes were confirmed for sufficiently many integers $m$ by Magma.
The reader is warmly invited to attack these open problems.
Finally, we make it clear that this is by no means a survey of all binary linear codes from Boolean functions,
but a survey of binary linear codes from Boolean functions from the second generic construction described in
Section <ref>.
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Boolean functions with four-valued Walsh spectrum: Case I
$\hat{f}(w)$ the number of $w$'s
$-2^{m/2}$ $(2^{m}-2^{m/2})/3$
$0$ $2^{m-1}-2^{(m-2)/2}$
$2^{m/2}$ $2^{m/2}$
$2^{(m+2)/2}$ $(2^{m-1}-2^{(m-2)/2})/3$
§.§.§ Binary codes $\C_{D_f}$ with three or five weights from some polynomials
For certain Boolean functions $f$, the Walsh
transform of $f$ takes on only the following values
\{0, \pm 2^{(m+1)/2}\}
when $m$ is odd, and
\{0, \pm 2^{m/2}, \pm 2^{(m+2)/2}\}
when $m$ is even. Then $\C_{D_f}$ is a three-weight code when $m$ is odd, and a five-weight code when $m$ is
Below is a list of such functions:
* $f(x)=\tr(x^3+\tr(x^9))$ <cit.>
Let $m=2e$ for any positive integer $e$. Let $\beta$ be a generator of $\gf(2^m)$. Then $\{1, \beta\}$ is a basis of
$\gf(2^m)$ over $\gf(2^e)$, and any element $z \in \gf(2^m)$ can be expressed as
\begin{eqnarray}\label{eqn-basisexpression2}
x=\frac{\beta^{2^e} z+\beta z^{2^e}}{\beta^{2^e}+\beta} \in \gf(2^e), \ \
y=\frac{z+z^{2^e}}{\beta^{2^e}+\beta} \in \gf(2^e).
\end{eqnarray}
|
1511.00583
|
Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems.
Concretely, these are set systems $(X,\Sigma)$, where each element $x \in X$ lies in $t$
randomly selected sets of $\Sigma$, where $t$ is an integer parameter.
We provide new bounds in two regimes of parameters.
We show that when $|\Sigma| \ge |X|$ the hereditary discrepancy of $(X,\Sigma)$ is with high probability $O(\sqrt{t \log t})$;
and when $|X| \gg |\Sigma|^t$ the hereditary discrepancy of $(X,\Sigma)$ is with high probability $O(1)$.
The first bound combines the Lovász Local Lemma with a new argument based on partial matchings;
the second follows from an analysis of the lattice spanned by sparse vectors.
§ INTRODUCTION
Let $(X, \Sigma)$ be a finite set system, with $X$ a finite set and $\Sigma$ a collection of subsets of $X$.
A two-coloring of $X$ is a mapping $\chi : X \rightarrow \{-1 ,+1\}$.
For a subset $S \in \Sigma$ we define $\chi(S) := \sum_{x \in S} \chi(x)$.
The discrepancy of $\Sigma$ is defined as
\disc(\Sigma) := \min_{\chi} \max_{S \in \Sigma} |\chi(S)| .
In other words, the discrepancy of the set system $(X,\Sigma)$ is the minimum over all colorings $\chi$ of the
largest deviation from an even split, over all subsets in $\Sigma$. For background on discrepancy theory, we refer the
reader to the books of Chazelle <cit.> and
Matoušek <cit.>.
In this paper, our interest is in the discrepancy of sparse set systems.
The set system $(X,\Sigma)$ is said to be $t$-sparse if any element $x \in X$ belongs to at most $t$
sets $S \in \Sigma$.
A well-known result of Beck and Fiala <cit.> is that sparse set systems have discrepancy bounded only in terms of their sparsity.
If $(X,\Sigma)$ is $t$-sparse then $\disc(\Sigma) \le 2t-1$.
Beck and Fiala conjectured that in fact, the bound can be improved to $O(\sqrt{t})$,
analogous to Spencer's theorem for non-sparse set systems <cit.>.
This is a long standing open problem in discrepancy theory. The best result to date is by Banaszczyk <cit.>.
If $(X,\Sigma)$ is $t$-sparse with $|X|=n$ then $\disc(\Sigma) \le O(\sqrt{t \log n})$.
Our results.
In this paper, we study random sparse set systems.
To sample a random $t$-sparse set system $(X,\Sigma)$ with $|X|=n,|\Sigma|=m$, for each $x \in X$
choose uniformly and independently a subset $T_x \subset [m]$ of size $|T_x|=t$. Then set $S_i=\{x \in X: i \in T_x\}$ and $\Sigma=\{S_1,\ldots,S_m\}$. Letting $\E[\cdot]$ denote expectation, our main quantity of interest is $\E[\disc(\Sigma)]$. We show that when $m \ge n$, this is close to the conjectured bound of Beck and Fiala.
Specifically, we show $\E[\disc(\Sigma)] = O(\sqrt{t\log{t}})$. In particular, the bound does not depend on $n$.
In fact, we obtain such bound for the hereditary discrepancy of the set system.
For $Y \subset X$ let $\Sigma|_Y=\{S \cap Y: S \in \Sigma\}$ be the set system restricted to $Y$. The
hereditary discrepancy of a set system $(X,\Sigma)$ is defined as
\herdisc(\Sigma) = \max_{Y \subset X} \disc(\Sigma|_Y).
Our main result is the following.
Assume $m \ge n \ge t$. Let $(X,\Sigma)$ be a random $t$-sparse set system with $|X|=n, |\Sigma|=m$.
\E[\disc(\Sigma)] \le \E[\herdisc(\Sigma)] \le O(\sqrt{t\log{t}}) .
In fact, the bound holds with probability $1-\exp(-\Omega(t))$.
We note that our technique can be extended to the case where $m \ge cn$ for any absolute constant $c>0$,
but fails whenever $m \ll n$. The main reason is that in this regime, most sets are large.
Nevertheless, when $n$ is considerably larger than $m$, we use a different approach and show that the discrepancy is small
in this case as well. Specifically, when $n$ is somewhat larger than ${m \choose t}$ we show that the discrepancy
is only $O(1)$.
Fix $m \ge t$ and let $N = {m \choose t}$.
Assume that $n \ge \Omega(N \log{N})$.
Let $(X,\Sigma)$ be a random $t$-sparse set system with $|X|=n, |\Sigma|=m$.
\E[\disc(\Sigma)] = O(1) .
In fact, the bound holds with probability $1-N^{-\Omega(1)}$.
To summarize, the work in this paper was motivated by the elusive Beck-Fiala conjecture. We considered a natural setting of random $t$-sparse set systems,
and showed that in this case, in some regimes of parameters, the conjecture holds (with the bound of $O(\sqrt{t})$ replaced by the slightly weaker
bound of $O(\sqrt{t \log t})$ in our first result). We hope that the techniques developed in this work will be useful for the study of random sparse set systems in the
full spectrum of parameters, as well as for the original Beck-Fiala conjecture.
§ PRELIMINARIES AND PROOF OVERVIEW
The Lovász Local Lemma.
The Lovász Local Lemma <cit.> is a powerful probabilistic tool. In this paper we only need its symmetric version.
Let $E_1, E_2,..., E_k$ be a series of events such that each event occurs with probability at most $p$ and such that each event is independent of all the other events except for at most $d$ of them. If $ep(d+1) \le 1$ then $\Pr[\wedge_{i=1}^m \overline{E_i}]>0$.
Tail bounds.
In our analysis we exploit a few standard tail bounds for the sum of independent random variables
(Chernoff-Hoeffding bounds, see, e.g., <cit.>).
Let $Z_1,\ldots,Z_k \in \{-1,1\}$ be independent random variables and let $Z=Z_1+\ldots+Z_k$. Then for any $\lambda>0$
\Pr\left[|Z- \E[Z]| \ge \lambda \sqrt{k}\right] \le 2 \exp(-2 \lambda^2).
Let $Z_1,\ldots,Z_k \in \{0,1\}$ be independent random variables and let $Z=Z_1+\ldots+Z_k$. Then for any $\lambda>0$
\Pr\left[Z \ge (1+\lambda) \E[Z]\right] \le \exp(-\lambda^2/3 \cdot \E[Z]).
§.§ Proof Overview for Theorem <ref>
We next present an overview of our proof for Theorem <ref>.
For simplicity of exposition, we present the overview only for the derivation of the discrepancy bound.
In Section <ref> we present the actual analysis and show a bound on the hereditary discrepancy.
First, we classify each set as being either “small” if its cardinality is $O(t)$, or “large” otherwise.
Then we proceed in several steps:
* (i) Making large sets pairwise disjoint:
Initially, we show that with high probability over the choice of the set system, it is possible to delete at most one element from each large set, such that
they become pairwise disjoint after the deletion. This property is proved in Lemma <ref>.
* (ii) Partial matching:
For each large set resulting after step (i), we pair its elements, leaving at most, say, two unpaired elements.
Since each pair appears in a unique set, this process results in a partial matching $M=\{(a_1,b_1),\ldots,(a_k,b_k)\}$ on $X$.
We observe that as soon as we have such a matching, we can restrict the two-coloring function $\chi$ on $X$ to assign
alternating signs on each pair of $M$. Since each large set $S$ has at most two unpaired elements, we immediately conclude that
$|\chi(S)| \le 2$.
* (iii) Applying the Lovász Local Lemma on the small sets:
We are thus left to handle the small sets.
In this case, we observe that a random coloring $\chi$, with alternating signs on $M$ as above[That is, each pair in $M$ is assigned $(+1,-1)$ or $(-1,+1)$ independently with probability $1/2$.], satisfies with positive probability that
$|\chi(S)| \le O(\sqrt{t \log t})$ for all small sets $S \in \Sigma$.
This is a consequence of the Lovász Local Lemma, as each small set $S$ contains only $O(t)$ elements, and each of these elements
participates in $t$ sets of $\Sigma$. The fact that some of these elements appear in the partial matching implies that $S$ can “influence”
(w.r.t. the random coloring $\chi$) at most $2|S| t = O(t^2)$ other small sets; see Section <ref> for the details.
We point out that as soon as we have a partial matching $M$ as above, we can “neutralize” the deviation that might be caused by the large sets,
and only need to keep the deviation, caused by the small sets, small. The latter is fairly standard to do, and so the main effort in the analysis
is to show that we can indeed make large sets disjoint as in step (i).
We note that our proof technique is constructive. Our arguments for steps (i) and (ii) (see Lemma <ref>
and our charging scheme in Claim <ref>) give an efficient algorithm to
find an element to delete in each large set, thereby making large sets disjoint, as well as build the partial matching,
or, alternatively, report (with small probability) that a partial matching of the above kind does not exist and halt.
In step (iii) we can apply the algorithmic Lovász Local Lemma of Moser and Tardos <cit.>, since the colors are assigned independently among the pairs in $M$ as well as the unpaired elements.
Thus, we obtain an expected polynomial time algorithm, which, with high probability over the choice of the set system,
constructs a coloring with discrepancy $O(\sqrt{t \log t})$.
§ A LOW HEREDITARY DISCREPANCY BOUND: THE ANALYSIS
We now proceed with the proof of Theorem <ref>.
We classify the sets in $\Sigma$ based on their size. A set $S \in \Sigma$ is said to be large if $|S| \ge 6t$ and small otherwise.
Note that as $m \ge n$, most sets in $\Sigma$ are small. Let $I=\{i: S_i \textrm{ is large}\}$ be a random variable capturing the indices of the large sets.
To construct a coloring, we proceed in several steps. First, we show that with high probability the large sets are nearly disjoint. We will assume throughout that $t$ is sufficiently large (concretely $t \ge 55$).
Fix $t \ge 55$.
Let $E$ denote the following event: “there exists a choice of $x_i \in S_i$ for $i \in I$ such that the sets $\{S_i \setminus \{x_i\}: i \in I\}$ are pairwise disjoint". Then
$\Pr[E] \ge 1-2^{-t}$.
We defer the proof of Lemma <ref> to Section <ref> and prove Theorem <ref> based on it, in the remainder of this section.
\begin{align*}
\E[\herdisc(\Sigma)] &= \E[\herdisc(\Sigma) | E] \Pr[E] + \E[\herdisc(\Sigma) | \overline{E}] \Pr[\overline{E}] \\
& \le \E[\herdisc(\Sigma) | E]+(2t-1) \Pr[\overline{E}]\\
& \le \E[\herdisc(\Sigma) | E]+1
\end{align*}
where we bounded $\E[\herdisc(\Sigma) | \overline{E}]$ by the Beck-Fiala theorem (Theorem <ref>) which holds for any $t$-sparse set system, and bounded $\Pr[\overline{E}]$ by $2^{-t}$ according to Lemma <ref>.
To conclude the proof we will show that when $E$ holds then $\herdisc(\Sigma) \le O(\sqrt{t \log t})$.
Thus, we assume from now on that the event $E$ holds. Fix a subset $Y \subset X$, where we will construct a two-coloring for $\Sigma'=\Sigma|_Y$ of low discrepancy.
Partition each $S_i \cap Y = A_i \cup B_i$ for $i \in I$, where $|A_i|$ is even, $|B_i| \le 2$ and the sets $\{A_i: i \in I\}$ are pairwise disjoint.
Partition each $A_i$ arbitrarily into $|A_i|/2$ pairs, and let $M$ be the union of these pairs.
That is, $M$ is a partial matching on $Y$ given by $M=\{(a_1,b_1),\ldots,(a_k,b_k)\}$ where $a_1,b_1,\ldots,a_k,b_k \in Y$ are distinct,
and each $A_i$ is a union of a subset of $M$, and each pair $a_j, b_j$ appears in a unique set $A_i$ due to the fact that these sets are pairwise
disjoint (they thus form a partition of $M$).
We say that a coloring $\chi:Y \to \{-1,+1\}$ is consistent with $M$ if $\chi(a_j)=-\chi(b_j)$ for all $j \in [k]$. Note that
if $S_i$ is a large set, then for any coloring $\chi$ consistent with $M$,
|\chi(S_i \cap Y)|=|\chi(A_i)+\chi(B_i)|=|0+\chi(B_i)| \le |B_i| \le 2.
Thus, we only need to minimize the discrepancy of $\chi$ over the small sets in $\Sigma$.
To do so, we choose $\chi$ uniformly from all two-colorings consistent with $M$. These are given
by choosing uniformly and independently $\chi(a_i) \in \{-1,+1\}$ for $i \in [k]$, setting $\chi(b_i)=-\chi(a_i)$ and choosing $\chi(x) \in \{-1,+1\}$ uniformly and independently
for all $x \notin \{a_1,b_1,\ldots,a_k,b_k\}$.
Let $S_i$ be a small set, that is $|S_i| \le 6t$. Let $E_i$ denote the event
E_i := \left[|\chi(S_i \cap Y)| \ge c \sqrt{t \log t}\right] .
Each pair $\{a_{j},b_{j}\}$ contained in $S_i$ contributes $0$ to the discrepancy, and all other elements obtain independent colors. Hence
$\chi(S_i)$ is the sum of $t' \le 6t$ independent signs. By Lemma <ref>, for an appropriate constant $c$ we have
\Pr[E_i] \le 1/100 t^2.
We next claim that each event $E_i$ depends on at most $d=12 t^2$ other events $\{E_j: j \ne i\}$.
Indeed, let $S'_i = S_i \cup \{a_j: b_j \in S_i\} \cup \{b_j: a_j \in S_i\}$. Then
$|S'_i| \le 2 |S_i| \le 12t$ and $\chi(S_i)$ is independent of $\chi(x)$ for all $x \notin S'_i$. So, if $E_i$ depends on $E_j$, it must be the case that $S_j$ intersects $S'_i$.
However, as each $x \in S'_i$ is contained in $t$ sets, there are at most $12 t^2$ such events $E_j$.
We are now in a position to apply the Lovász Local Lemma (Theorem <ref>).
Its condition are satisfied as we have $p=1/100 t^2$ and $d=12 t^2$.
Hence $\Pr[\wedge \overline{E_i}]>0$, that is, there exists a coloring $\chi$ consistent with $M$ for which $|\chi(S_i)| \le c \sqrt{t \log t}$ for all small sets $S_i$. This coloring shows that $\disc(\Sigma') \le \max(c \sqrt{t \log t},2)$ as claimed.
§ PROOF OF LEMMA <REF>
Let $(X,\Sigma)$ be a $t$-sparse set system with $|X|=n,|\Sigma|=m$. It will
be convenient to identify it with a bi-partite graph $G=(X,V,E)$ where $|V|=m$ and $E=\{(x,i): x \in S_i\}$. Then, a random $t$-sparse set system
is the same as a random left $t$-regular bi-partite graph. That is, a uniform graph satisfying $\deg(x)=t$ for all $x \in X$.
Large sets in $\Sigma$ correspond to the subset of the vertices $V'=\{v \in V: \deg(v) \ge 6t\}$.
For a vertex $v \in V$ let $\Gamma(v) \subset X$ denote its neighbors.
Lemma <ref> is equivalent to the following lemma, which we prove in this section.
Fix $t \ge 55$. With probability at least $1-2^{-t}$ over the choice of $G$, there exists a choice of $x_v \in \Gamma(v)$ such that
the sets $\{\Gamma(v) \setminus \{x_v\}: v \in V'\}$ are pairwise disjoint.
Let $G'$ be the induced (bi-partite) sub-graph on $(X,V')$.
We will show that with high probability $G'$ has no cycles.
In such a case Lemma <ref> follows from the straightforward scheme
described below:
Assume that $G'$ has no cycles. Then there exists a choice of $x_v \in \Gamma(v)$ such that
the sets $\{\Gamma(v) \setminus \{x_v\}: v \in V'\}$ are pairwise disjoint.
We present a charging scheme of the vertices $x_v \in \Gamma(v)$, for each $v \in V'$.
If $G'$ has no cycles then it is a forest. Fix a tree $T$ in $G'$ and an arbitrary root $v_T \in V'$ of $T$.
Orient the edges of $T$ from $v_T$ to the leaves.
For each $v \in T$ other than the root, choose $x_v$ to be the parent of $v$ in the tree, and choose $x_{v_T}$ arbitrarily.
Let $A_v = \Gamma(v) \setminus \{x_v\}$ for $v \in V'$.
We claim that $\{A_v: v \in V'\}$ are pairwise disjoint. To see that, assume towards contradiction that $x \in A_{v_1} \cap A_{v_2}$ for some $x \in X, v_1,v_2 \in V'$.
Then $v_1,x,v_2$ is a path in $G'$ and hence $v_1,v_2$ must belong to the same tree $T$. However,
the only case where this can happen (as $T$ is a tree) is that $x$ is the parent of both $v_1,v_2$ in $T$.
However, by construction in this case $x=x_{v_1}=x_{v_2}$ and hence $x \notin A_{v_1}, A_{v_2}$,
from which we conclude that $\{A_v: v \in V'\}$ are pairwise disjoint, as claimed.
In the remainder of the proof we show that with high probability $G'$ has no cycles.
The girth of $G'$, denoted $\girth(G')$, is the minimal length of a cycle in $G'$ if such exists, and otherwise it is $\infty$.
Note that as $G'$ is bipartite, then $\girth(G')$ is (if finite) the minimal $2\ell$ such that there exist a cycle $x_1,v_1,x_2,v_2,\ldots,x_{\ell},v_{\ell},x_1$ in $G'$ with $x_i \in X$ and $v_i \in V'$.
$\Pr[\girth(G')=4] \le t^4 \exp(-t)$.
Fix $x_1,x_2 \in X$ and $v_1,v_2 \in V$. They form a cycle of length $4$ if $v_1,v_2 \in \Gamma(x_1) \cap \Gamma(x_2)$. As each $\Gamma(x_i)$
is a uniformly chosen set of size $t$ we have that
\Pr[v_1,v_2 \in \Gamma(x_1) \cap \Gamma(x_2)] = \left(\frac{{t \choose 2}}{{m \choose 2}}\right)^2 \le (t/m)^4.
Next, conditioned on the event that $v_1,v_2 \in \Gamma(x_1) \cap \Gamma(x_2)$, we still need to have $v_1,v_2 \in V'$ (that is $v_1$, $v_2$ represent
large sets of $\Sigma$).
We will only require that $v_1 \in V'$
for the bound. Note that so far we only fixed $\Gamma(x_1),\Gamma(x_2)$, and hence the neighbors of $\Gamma(x)$ for $x \ne x_1,x_2$ are still uniform.
Then $v_1 \in V'$ if at least $6t-2$ other nodes $x \in X$ have $v_1$ as their neighbor. By Lemma <ref>, the probability for this is bounded by
\Pr[v_1 \in V' | v_1,v_2 \in \Gamma(x_1) \cap \Gamma(x_2)] \le \exp(-((5t-2)/t)^2 /3 \cdot t) \le \exp(-t).
\Pr[v_1,v_2 \in \Gamma(x_1) \cap \Gamma(x_2) \wedge v_1 \in V'] \le (t/m)^4 \cdot \exp(-t).
To bound $\Pr[\girth(G')=4]$ we union bound over all ${n \choose 2} {m \choose 2}$ choices of $x_1,x_2,v_1,v_2$.
Using our assumption that $m \ge n$ we get
\Pr[\girth(G')=4] \le m^4 (t/m)^4 \exp(-t) \le t^4 \exp(-t).
For any $\ell \ge 3$, $\Pr[\girth(G')=2 \ell] \le \exp(-t \ell)$.
Let $x_1,v_1,\ldots,x_{\ell},v_{\ell}$ denote a potential cycle of length $2 \ell$. As it is a minimal cycle and $\ell \ge 3$, the vertices $v_i,v_j$ have no common
neighbors, unless $j=i+1$ in which case $x_i$ is the only common neighbor of $v_i,v_{i+1}$ (where indices are taken modulo $\ell$). Thus there exist
sets $X_i \subset X$ of size $|X_i|=6t-2$ such that $X_i \subset \Gamma(v_i)$ and $X_1,\ldots,X_{\ell},\{x_1,\ldots,x_{\ell}\}$ are pairwise disjoint.
Let $E(x_1,v_1,\ldots,x_{\ell},v_{\ell},X_1,\ldots,X_{\ell})$ denote the event described above, for a fixed choice of $x_1,v_1,\ldots,x_{\ell},v_{\ell},X_1,\ldots,X_{\ell}$.
The event holds if
* $v_i,v_{i+1}$ are neighbors of $x_i$.
* $v_i$ is a neighbor of all $x \in X_i$.
There are independent events, as $\Gamma(x)$ is independently chosen for each $x \in X$. So
\begin{align*}
&\Pr[E(x_1,v_1,\ldots,x_{\ell},v_{\ell},X_1,\ldots,X_{\ell})] \\
&= \prod_{i=1}^{\ell} \Pr[v_i,v_{i+1} \in \Gamma(x_i)] \cdot \prod_{i=1}^{\ell} \prod_{x \in X_i} \Pr[v_i \in \Gamma(x)]\\
&=\left( \frac{{t \choose 2}}{{m \choose 2}} \right)^{\ell} \cdot \left( \frac{t}{m}\right)^{(6t-2)\ell}
\le \left( \frac{t}{m}\right)^{6 t \ell}.
\end{align*}
To bound $\Pr[\girth(G')=2 \ell]$ we union bound over all choices of $x_1,v_1,\ldots,x_{\ell},v_{\ell},X_1,\ldots,X_{\ell}$.
The number of choices is bounded by
n^{\ell} m^{\ell} {n \choose 6t-2}^{\ell} \le \left(\frac{nm \cdot e^{6t-2} \cdot n^{6t-2}}{(6t-2)^{6t-2}}\right)^{\ell}
\le \left(\frac{(em)^{6t}}{(6t-2)^{6t-2}}\right)^{\ell}.
\Pr[\girth(G')=2 \ell] \le \left(\frac{(em)^{6t}}{(6t-2)^{6t-2}}\right)^{\ell} \cdot \left( \frac{t}{m}\right)^{6 t \ell} = \left((6t-2)^2 \left(\frac{et}{6t-2}\right)^{6t} \right)^{\ell}
\le \exp(-t \ell).
Using Claims <ref> and <ref>, the probability that $\girth(G') < \infty$ is bounded by:
\Pr[\girth(G') < \infty] = \sum_{\ell=2}^{\infty} \Pr[\girth(G')=\ell] \le t^4 \exp(-t) + \sum_{\ell=3}^{\infty} \exp(-t \ell) \le 2 t^4 \exp(-t).
For $t \ge 55$, we have that $\Pr[\girth(G') < \infty] \le 2^{-t}$.
§ THE REGIME OF LARGE SETS
We next prove Theorem <ref>.
Let $(X,\Sigma)$ be a $t$-sparse set system with $|X|=n, |\Sigma|=m$. In this setting, we consider the case of fixed $m,t$
and $n \to \infty$. Consider its $m \times n$ incidence matrix. The columns are $t$-sparse vectors in $\{0,1\}^m$,
and hence have $N={m \choose t}$ possible values. When $n \gg N$, there will be many repeated columns. We show that in this case,
the discrepancy of the set system is low. Setting notations, let $v_1,\ldots,v_N \in \{0,1\}^m$ be all the possible $t$-sparse
vectors, and let $r_1,\ldots,r_N$ denote their multiplicity in the set system.
Note that they define the set system uniquely (up to permutation of the columns, which does not effect the discrepancy).
Our main result in this section is the following. We will assume throughout that $m$ is large enough and that $4 \le t \le m-4$. We note that if $t \le 3$ or $t \ge m-3$ then result immediately follows from the Beck-Fiala theorem (Theorem <ref>), for any set systems. The first case follows by a direct application, and the second case by first partitioning the columns to pairs and subtracting one vector from the next in each pair, which gives a $6$-sparse $\{-1,0,1\}$ matrix, to which we apply the Beck-Fiala theorem.
Let $(X,\Sigma)$ be a $t$-sparse set system with $4 \le t \le m-4$ and $m$ large enough.
Assume that $\min(r_1,\ldots,r_N) \ge 7$. Then $\disc(\Sigma) \le 2$.
Note that the statement in Theorem <ref> is somewhat stronger than that in Theorem <ref>,
as it only assumes that all possible $t$-sparse column vectors comprise the incidence matrix of $(X,\Sigma)$,
and their multiplicity is $7$ or higher. In fact, Theorem <ref> follows
from Theorem <ref> using a straightforward coupon-collector argument <cit.>. In this regime,
with high probability (say, with probability at least $1 - 1/N$), a random sample of $\Theta(N \log{N})$ columns
guarantees that each $t$-sparse column appears with multiplicity $7$ (or higher).
Therefore, we obtain:
\E[\disc(\Sigma)] \le 2\left(1 - \frac{1}{N}\right) + \frac{2t-1}{N} = O(1) .
We are thus left to prove Theorem <ref>.
First, we present an overview of the proof.
Proof overview.
Every column $v_i$ is repeated $r_i$ times. As we may choose arbitrary signs for each occurrence of a vector, the aggregate total would be $c_i v_i$,
where $c_i \in \Z$, $|c_i| \le r_i$ and $c_i \equiv r_i \mod 2$. Our goal is to show that such a solution $c_i$ always exists, for which
$\|\sum c_i v_i\|_{\infty}$ is bounded, for any initial settings of $r_1,\ldots,r_N$, as long as they are all large enough.
We show that such a solution always exists, with $|c_i| \le 7$. In order to show it, we first fix some solution with the correct parity, and then
correct it to a low discrepancy solution, by adding an even number of copies of each vector.
In order to do that, we study the integer lattice $\L$ spanned by the vectors $v_1,\ldots,v_N$, as our correction comes from $2 \L$.
We show that $\L=\{x \in \Z^m: \sum x_i=0 \mod t\}$, which was already proved by Wilson <cit.> in a more general scenario.
However, we need an additional property: vectors in $\L$ are efficiently spanned by $v_1,\ldots,v_N$. This allows us to perform the above
correction efficiently, keeping the number of times that each $v_i$ is repeated bounded. Putting that together,
we obtain the result.
§.§ Proof of Theorem <ref>
Initially, we investigate the lattice spanned by the vectors $v_1,\ldots,v_N$.
As the sum of the coordinates of each of them is $t$, they sit within the lattice
\L = \left\{x \in \Z^m: \sum x_i \equiv 0 \mod t\right\}.
We first show that they span this lattice, and moreover, they do so effectively.
For any $w \in \L$ there exist $a_1,\ldots,a_N \in \Z$ such that $\sum a_i v_i =w$. Moreover, $|a_i| \le A$ for all $i \in [N]$ where
$A=\frac{2 \|w\|_1}{{m-2 \choose t-1}}+2$.
Assume first that we have $\sum w_i=0$. We will later show how to reduce to this case.
Pair the positive and negative coordinates of $w$. For $L=\|w\|_1 / 2$ let $(i_1,j_1),\ldots,(i_L,j_L)$ be pairs
of elements of $[N]$ such that: if $(i,j)$ is a pair then $w_i>0, w_j<0$; each $i \in [m]$ with $w_i>0$ appears $w_i$ times
as the first element in a pair; and each $j \in [m]$ with $w_j<0$ appears $-w_j$ times as the second element in a pair.
For any $\ell \in [L]$ choose $S_\ell \subset [m]$ of size $t-1$. Set $I_{\ell}=S_{\ell} \cup \{i_{\ell}\}$ and
$J_{\ell}=S_{\ell} \cup \{j_{\ell}\}$.
Identifying $[N]$ with subsets of $[m]$ of size $t$, we have
w = \sum_{\ell=1}^L v_{I_{\ell}} - v_{J_{\ell}}.
We choose the sets $S_1,\ldots,S_{L}$ to minimize the maximum number of times that each vector from $\{v_1,\ldots,v_N\}$
is repeated in the decomposition. When we
choose $S_{\ell}$, we can choose one of $M={m-2 \choose t-1}$ many choices. There is a choice for $S_{\ell}$ such that both $I_{\ell}$ and $J_{\ell}$
appeared thus far less than $2 \ell / M$ times. Choosing such a set, we maintain the invariant that after choosing $S_1,\ldots,S_{\ell}$, each vector
is repeated at most $2 \ell / M+1$ times. Thus, at the end each vector is repeated at most $2L/M + 1$ times.
In the general case, we have $\sum w_i = st$, where we may assume $s>0$. We
apply the previous argument to $w-(v_{i_1}+\ldots+v_{i_s})$, whose coordinates
sum to zero. We choose $i_1,\ldots,i_s \in [N]$ (potentially with repetitions) so as to minimize the maximum number of times that
each vector participates; this number is $\lceil s / N \rceil \le \|w\|_1/M+1$.
Combining the two estimates, we obtain that at the end each vector is repeated at most $4L/M + 2 = 2\|w\|_1/M + 2$ times.
For any $b_1,\ldots,b_N \in \{0,1\}$ there exist $c_1,\ldots,c_N \in \Z$ such that
(i) $c_i \equiv b_i \mod 2$.
(ii) $\|\sum c_i v_i\|_{\infty} \le 2$.
(iii) $|c_i| \le 7$ for all $i \in [N]$.
As a first step, choose $z_i \in \{-1,0,1\}$ such that $z_i=0$ if $b_i=0$, and $z_i \in \{-1,1\}$ chosen uniformly if $b_i=1$. Let $u=\sum z_i v_i$.
Note that for $j \in [m]$,
if there are $k_j$ indices $i \in [N]$ for which $(v_i)_j=1$ and $b_i=1$, then $\E_z[u_j^2]=k_j$. Thus,
\E_z[\|u\|_2^2] = \sum k_j \le Nt.
Thus, with probability at least $1/2$, $\|u\|_2 \le \sqrt{2 N t}$ and hence $\|u\|_1 \le \sqrt{2 N t m}$. Fix such a $u$.
Next, we choose $w \in \L$ such that $\|u - 2 w\|_{\infty}$ is bounded. If we only wanted that $w \in \Z^m$
we could simply choose $q \in \{0,1\}^m$ with $q_i = u_i \mod 2$ and take $w=(u-q)/2$. In order to guarantee that $w \in \L$,
namely that $\sum w_i=0 \mod t$, we change at most $t$ coordinates in $q$ by adding or subtracting $2$.
Thus, we obtain $q \in \{-2,-1,0,1,2\}^m$ where $q_i \equiv u_i \mod 2$ and set $w=(u-q)/2 \in \L$.
We have $\|u - 2 w\|_{\infty} \le 2$.
Next, we apply Lemma <ref> to $w$. We obtain a decomposition $w=\sum a_i v_i$. This implies that
if we set $c_i =z_i - 2 a_i$ then indeed $c_i \equiv b_i \mod 2$ and $\|\sum c_i v_i\|_{\infty} = \| u - 2w\|_{\infty} \le 2$.
To bound $|c_i|$, note that $\|w\|_1 \le \|u\|_1/2+m$. We have by Lemma <ref> that $|a_i| \le A$ for
A = 2 + \eta \le 3,
\eta = 2 \frac{\|w\|_1}{{m-2 \choose t-1}} \le O \left(\frac{\sqrt{mt {m \choose t}}}{{m-2 \choose t-1}} \right) \le
O \left(\frac{m^{3/2}}{{m \choose t}^{1/2}} \right) \le 1 ,
whenever $4 \le t \le m-4$ and $m$ is large enough, as is easily verified by the fact that the last term is a
decreasing function of $m$.
Assume that $r_1,\ldots,r_N \ge 7$. By Lemma <ref>, there exists $c_i \in \Z$ such that $c_i \equiv r_i \mod 2$, $|c_i| \le 7$ and
$\|\sum c_i v_i\|_{\infty} \le 2$.
For each $i \in [N]$, we color $|c_i|$ of the vectors $v_i$ with $\text{sign}(c_i) \in \{-1,+1\}$ and the remaining $r_i-|c_i|$ vectors with alternating $+1$ and $-1$
colors (so that their contribution cancels, since $r_i-|c_i|$ is even). The total coloring produces exactly the vector $\sum c_i v_i$, which as guaranteed has discrepancy
bounded by $2$.
The authors wish to thank Aravind Srinivasan for presenting this problem during
a discussion at the IMA Workshop on the Power of Randomness in Computation.
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1511.00142
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Department of Chemistry and Biochemistry, Queens College, City University of New York, 65-30 Kissena Boulevard, Queens, NY 11367[mailing address] & PhD programs in Chemistry and Physics, and Initiative for Theoretical Sciences, Graduate Center, City University of New York, 365 Fifth Avenue, New York, NY 10016
This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Caldeira and Leggett, Physica A 121, 587 (1983)] is extended for a nonequilibrium influence functional, which has been obtained for general cases where the ground and the excited electronic state baths can be different. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional with respect to time up to the second order. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation, but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived through short time expansion of the effective action followed by Gaussian integrations in analytically continued complex domain of space. This leads to a compact form of GQFPE with time dependent kernels and additional terms, which make the resulting equation the Dekker form [H. Dekker, Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit.
§ INTRODUCTION
The concept of Brownian motion,<cit.> or more specifically, Langevin equation,<cit.> was originally developed under the premise that the system of interest follows fully deterministic paths if left alone and that environmental effects can be accounted for by random forces and frictional drags satisfying the fluctuation-dissipation relationship. How to extend such description to the quantum mechanical regime governing time evolution of quantum operators had remained difficult in practice<cit.> or was even considered impossible,<cit.> although some advances have been made.<cit.> On the other hand, the Fokker-Planck equation (FPE),<cit.> which considers time evolution of distribution function instead, is more amenable to quantum generalization because the distribution can be obtained naturally from a quantum density operator retaining complete information on the quantum system.[The definition of QFPE seems to be somewhat subjective and can differ for each researcher. Here, we define it loosely as a time evolution equation for quantum particle approaching the FPE in the classical limit. Thus, it can be viewed as a kind of quantum master equation for a continuous quantum degree of freedom.] Indeed, a few well defined and tractable derivations of quantum FPE (QFPE)<cit.> or hierarchical QFPEs<cit.> are available now.
One of the most well-known derivation of QFPE was provided by Caldeira and Leggett (CL)<cit.> based on the Feynman-Vernon influence functional formalism.<cit.> This approach has also been extended to the case of nonadiabatic quantum dynamics by Garg, Onuchic, and Ambegaokar (GOA).<cit.> CL's derivation of QFPE invokes high temperature and the Markovian approximation for the bath dynamics. Although not explicit, an assumption of weak system-bath coupling appears to be implicit in the derivation as well. Indeed, a distinct quantum Smoluchowski equation (QSE)<cit.> is obtained following a similar approach but taking the effect of strong system-bath coupling properly. However, this QSE still assumes that the bath relaxes much faster than the system, and does not account for the non-Markovian effect that can have potentially important effects.
Major applications of QFPE include quantum extension<cit.> of Kramers' barrier crossing problem and proton or electron transfer dynamics.<cit.> In particular, for the latter case, there has been growing interest in the study of fast photo-induced reaction dynamics that can occur during time scales comparable to those of molecular relaxation and dephasing dynamics.<cit.> For these, currently available QFPE or QSE are not well suited. Thus, generalization of QFPE to include non-Markovian and nonequilibrium effects remains an important and interesting theoretical issue to be addressed. Although more general hierarchical equations<cit.> may be used to this end, the benefit of having a single closed form equation, which can account for such non-Markovian and nonequilibrium effects, cannot be overestated.
In addition, the fact that CL's QFPE<cit.> is not positive definite remains a lingering theoretical issue. Although it is true that positive definiteness is not necessary for accurate description of the open system quantum dynamics,<cit.> it is still important to understand the source of its violation and how to fix the problem. Diósi showed that the non-positivity can result from an inconsistent application of the Markovian approximation<cit.> and that it can partially be corrected by including next order terms in the intermediate temperature regime. The present work shows that a a similar consideration can be made in deriving a generalized QFPE (GQFPE) and that the resulting equation is of the Dekker form,<cit.> which has a well-defined condition for positive definiteness. A detailed consideration of this equation for Ohmic spectral density with Drude cutoff shows that the Lindblad's positive definiteness condition<cit.> can indeed be satisfied in the steady state limit under reasonable physical condition.
The paper is organized as follows. Section II presents the main theoretical development based on the standard path integral formulation. Section III provides numerical analysis of major terms of newly derived GQFPE for the case of Ohmic spectral density with Drude cutoff. Section IV concludes the paper by summarizing the main results and their implication.
§ THEORY
Consider the following total Hamiltonian:
Ĥ=Ĥ_g|g⟩⟨g|+Ĥ_e|e⟩⟨e|where $|g\rangle$ is the ground electronic state and $|e\rangle$ is the excited electronic state. $\hat H_g$ and $\hat H_e$ are nuclear Hamiltonians in respective ground and electronic states, and have the following forms:
+∑_α{p̂_α^2/2m_α+m_αω_α^2/2(x̂_α-c_α,e/m_αω_α^2q̂)^2} In the above expression, $\hat q$ and $\hat p$ represent the position and momentum operators of the quantum nuclear degree of freedom of the system, and $\hat x_\alpha$'s and $\hat p_\alpha$'s represent the position and momentum operators of all the bath modes bilinearly coupled to the system. The system nuclear degree of freedom is assumed to be one dimensional here, but extension of the present work for multidimensional situation is straightforward.
It is assumed that there is no coupling between the ground and the excited state in the absence of radiation.
The total density operator at time $t$ is denoted as $\hat \rho_T(t)$. As the initial condition at $t=0$, we consider the situation where the entire system plus bath degrees of freedom are prepared at their canonical equilibrium for the ground electronic state as follows:
ρ̂_T(0)=|g⟩⟨g|ρ̂_g=|g⟩⟨g|e^-βĤ_g/Tr{e^-βĤ_g} .
Given that an impulsive excitation is applied to the system at time zero and under the Condon approximation that the transition dipole is independent of nuclear coordinates, a vertical transition from $|g\rangle$
to $|e\rangle$ occurs while all other degrees of freedom remain frozen. Thus, the total density operator for $t\geq 0+$, following an impulsive excitation at $t=0$,
is given by
ρ̂_T(t)=ρ̂_e(t)|e⟩⟨e|=e^-iĤ_e t/ħρ̂_g e^iĤ_e t/ħ|e⟩⟨e| ,
where $\hat \rho_e (t)$ is the total density operator representing the system nuclear coordinate in the excited electronic state and the bath.
Taking the trace of this over the bath degrees of freedom, we obtain the reduced density operator describing the system nuclear degree of
freedom as follows:
σ̂_e(t)=Tr_b{ρ̂_e(t)}=Tr_b{e^-iĤ_e t/ħρ̂_g e^iĤ_e t/ħ}
where $Tr_b$ represents trace over the bath.
For the derivation of GQFPE governing time evolution of $\hat \sigma_e (t)$, we extend the path integral approach developed by CL,<cit.> which has also been adopted by GOA<cit.> for nonadiabatic quantum dynamical processes.
§.§ Path integral representation and short time expansion
The path integral representation for the reduced system density operator in the excited electronic state, $\hat \sigma_e(t)$ defined by Eq. (<ref>), can be found by utilizing standard expressions for both the real and imaginary time propagators. The major steps are described in Appendix A, where the final expression for $\hat \sigma_e(t)$ is given by Eq. (<ref>). This expression can be simplified by introducing the following three spectral densities of the bath:
η_g(ω)≡π/2∑_αc_α,g^2/m_αω_αδ(ω-ω_α) .
Then, with the definitions of Eqs. (<ref>)-(<ref>), we can introduce the following nonequilibrium influence functional:
J[q'(·),q”(·),q_g(·); t,βħ] =Z_b exp{i/ħ W_e,I(t)
-1/ħW_e,R(t) +i/ħ W_c(t)+1/ħW_g(βħ} } ,
where $Z_b=\prod_\alpha \big(2\sinh(\omega_\alpha\beta\hbar/2)\big)^{-1}$, $q'(\cdot)$ and $q''(\cdot)$ represent real time paths in the excited electronic state, and $q_g(\cdot)$ the imaginary time path in the ground electronic state. Different exponents in the above nonequilibrium influence functional represent effective actions coming from different sources of the system-bath interaction. $W_{e,I}(t)$ and $W_{e,R}(t)$ are imaginary and real components of the contribution from the bath dynamics in the excited state, $W_c(t)$ represents coupling between the two baths in the excited and ground electronic states, and $W_g(\beta\hbar)$ is the imaginary time action due to thermal distribution of the bath in the ground electronic state.
With the definition of Eq. (<ref>), Eq. (<ref>) can now be expressed as
σ̂_e(t)=1/Z_g ∫dq_i' ∫dq_i” ∫_q_i'^q_i”q_g(·)∫dq_f'∫dq_f”
×∫_q_i'^q_f' 𝒟 q'(·)∫_q_i”^q_f” 𝒟q”(·) J[q'(·),q”(·),q_g(·); t,βħ]
×e^i/ħS_e[q'(·);t]-i/ħS_e[q”(·);t]-1/ħS_g^E[q_g(·);βħ] |q_f'⟩⟨q_f”| .
The nonequilibrium influence functional appearing in the above equation, as defined by Eq. (<ref>), is a direct extension of the Feynman and Vernon influence functional<cit.> to the general case where the Hamiltonian for the initial equilibrium distribution can be different from that of the dynamics. Numerical evaluation of this is feasible extending novel computational methods,<cit.> which is not the main focus here.
For the derivation of GQFPE, let us introduce the following time dependent kernels:
η̃_e,I(t)≡1/π∫_0^∞dω η_e(ω)sin(ωt) ,
η̃_e,R(t)≡1/π∫_0^∞dω η_e(ω)(ωβħ/2)cos(ωt) ,
η̃_c (t;q_g(·))≡1/π∫_0^βħ dτ∫_0^∞dω η_c(ω)
×cosh(ω(τ-it-βħ/2))/sinh(ωβħ/2)q_g(τ)Then, the three time dependent exponents in Eq. (<ref>), which are defined by Eqs. (<ref>)-(<ref>),
can be expressed as
W_e,I(t)=∫_0^t dt_1 ∫_0^t_1 dt_2 η̃_e,I(t_2) (q'(t_1)-q”(t_1))
×(q'(t_1-t_2)+q”(t_1-t_2)) ,
W_e,R(t)=∫_0^t dt_1∫_0^t_1 dt_2 η̃_e,R(t_2)(q'(t_1)-q”(t_1))
×(q'(t_1-t_2)-q”(t_1-t_2)) ,
W_c(t;q_g(·))=∫_0^tdt_1 η̃_c(t;q_g(·)) (q'(t_1)-q”(t_1)) .
Equations (<ref>) and (<ref>) above involve double time integrations, which need to be converted to single time integrations<cit.> for the derivation of GQFPE.
Under the assumption that the decay of $\tilde \eta_{e,I}(t)$ is fast enough, $q'(t_1-t_2)$ and $q''(t_1-t_2)$ in Eq. (<ref>) can be approximated with their second order expansions with respect to $t_2$ around $t_1$. The resulting expression can then be converted to single time integration through partial integration. Thus, Eq. (<ref>) can be approximated as
W_e,I(t)≈∫_0^t dt_1 𝒦_I^(0)(t_1)(q'(t_1)^2-q”(t_1)^2)
-∫_0^t dt_1 (𝒦_I^(1)(t_1) +1/2𝒦̇_I^(2)(t))
-1/2∫_0^t dt_1 𝒦_I^(2) (t_1) (q̇'(t_1)^2-q̇”(t_1)^2)
.+1/2𝒦_I^(2)(t_1) (q'(t_1)-q”(t_1))(q̇'(t_1)+q̇”(t_1))|_0^t ,
where the single dot over $q'(t)$ and $q''(t)$ denotes the first derivatives with respect to time and
𝒦_I^(n)(t_1)≡∫_0^t_1 dt_2 η̃_e,I(t_2)t_2^n
Similarly, Eq. (<ref>) can be approximated as
W_e,R(t)≈∫_0^t dt_1 𝒦_R^(0)(t_1)(q'(t_1)-q”(t_1))^2
-∫_0^t dt_1 (𝒦_R^(1)(t_1)+1/2 𝒦̇_R^(2)(t_1))
-1/2∫_0^t dt_1 𝒦_R^(2)(t_1) (q̇'(t_1)-q̇”(t_1))^2
.+1/2𝒦_R^(2)(t_1) (q'(t_1)-q”(t_1))(q̇'(t_1)-q̇”(t_1))|_0^t ,
𝒦_R^(n)(t_1)≡∫_0^t_1 dt_2 η̃_e,R(t_2)t_2^n
With Eqs. (<ref>) and (<ref>), Eq. (<ref>) can be converted to an expression that involves only single time integrations, but with additional boundary terms. Before presenting the final form, let us first collect all the contributions to Eq. (<ref>) from the boundary terms in Eqs. (<ref>) and (<ref>), and define
𝒜(r', r”,ṙ',ṙ”,t)=
exp{-1/2ħ 𝒦_R^(2)(t) (q'-q”)(q̇'-q̇”)
+i/2ħ𝒦_I^(2)(t) (q'-q”)(q̇' +q̇”)} .
Collecting all the contributions from single time integration terms in Eqs. (<ref>) and (<ref>), let us also define
.-i/ħ 𝒞_I[q'(·),q”(·);t,t_0]} ,
𝒞_R[q'(·),q”(·);t,t_0]=∫_t_0^t dt_1{ 𝒦_R^(0)(t_1)(q'(t_1)-q”(t_1))^2
-1/2𝒦_R^(2)(t_1) (q̇'(t_1)-q̇”(t_1))^2} ,
∫_t_0^t dt_1 𝒦̃_I^(1)(t_1)(q'(t_1)-q”(t_1))(q̇'(t_1)+q̇”(t_1)) ,
𝒦̃_R^(1)(t_1)=𝒦_R^(1)(t_1)+1/2𝒦̇_R^(2)(t_1) ,
𝒦̃_I^(1)(t_1)=𝒦_I^(1)(t_1)+1/2𝒦̇_I^(2)(t_1) .
Finally, the following effective time dependent action can be introduced:
=∫_t_0^t dt_1 { m_e(t)/2q̇(t_1)^2 -U_e(q(t_1),q_g(·);t_1) } ,
m_e(t)=m -𝒦_I^(2)(t) ,
U_e(q, q_g(·),t)=V_e(q)+(κ_e/2-𝒦_I^(0)(t)) q^2
- η̃_c(t,q_g(·)) q .
In the above expression, $\kappa_{e}$, which is defined by Eq. (<ref>), is an effective harmonic oscillator spring constant due to the bath in the excited electronic state.
Then, the position space matrix element of the reduced density operator, Eq. (<ref>), can be expressed as follows:
⟨q_f'|σ̂_e(t)|q_f”⟩= ∫dq_i'∫dq_i” ∫_q_i'^q_i” q_g(·) 𝒫[q_g(·);βħ]
×∫_q_i'^q_f'q'(·)∫_q_i”^q_f”q”(·) 𝒜(q_f',q_f”,q̇_f',q̇_f”,t)
×e^i/ħS_eff[q'(·),q_g(·);t,0]-i/ħS_eff[q”(·),q_g(·);t,0] ,
where the fact that ${\mathcal A}(q',q'',\dot q',\dot q'',0)=1$ has been used and ${\mathcal P}[q_g (\cdot);\beta\hbar]$ is the probability density for the imaginary time path with the following expression:
𝒫[q_g(·);βħ]=Z_b/Z_gexp{-1/ħS_g^E[q_g(·);βħ] .
+1/ħ∫_0^βħdτ∫_0^τdτ_1 ∫_0^∞dω/π η_g(ω)
. ×cosh(ω(τ-τ_1-βħ/2))/sinh(ωβħ/2)q_g(τ_1)q_g(τ)} .
Equation (<ref>) is the best form available for deriving a GQFPE. Note the presence of the time dependent prefactor ${\mathcal A}(q_f',q_f'',\dot q_f',\dot q_f'',t)$, which comes from the boundary values of time integrations. This term vanishes for $q_f'=q_f''$ at all time or in the long time limit where ${\mathcal K}_R^{(2)}(t)$ and ${\mathcal K}_I^{(2)}(t)$ decay to zero. Thus, it does not contribute to the calculation of position dependent observables at any time or any observables in the steady state limit where the initial memory of the bath disappears. However, for general situations, it remains as a source of ambiguity in deriving the time evolution equation and has not been considered in previous treatments by CL,<cit.> GOA,<cit.> and Diósi<cit.> who all considered only the Markovian or steady state limit.
§.§ Time evolution equation
Let us define the time dependent part in Eq. (<ref>) except for the prefactor and the ground state influence functional as follows:
σ̃_e(q_f',q_f”;t) ≡∫_q_i'^q_f'q'(·)∫_q_i”^q_f”q”(·)
×e^i/ħS_eff[q'(·),q_g(·);t,0]-i/ħS_eff[q”(·),q_g(·);t,0] .
In the above expression, dependences of $\tilde \sigma_e$ on $q'$, $q''$, and $q_g(\cdot)$ have not been shown explicitly.
A time evolution equation for $\tilde \sigma_e(q_f',q_f'';t)$ can be derived employing the short time expansion of path integral expression as was done by CL<cit.> and GOA.<cit.>
Due to the fact that $J_{eff}[q'(\cdot),q''(\cdot);t,0]$ defined by Eq. (<ref>) now involves single time integration in the exponent, it can be expressed as the product of discretized terms as follows:
J_eff[q'(·),q”(·);t+δt,0] = J_eff[q'_δ(·),q”_δ(·);t+δt,t]
×J_eff[q'_δ(·),q”_δ(·);δt,0] .
Similar expressions can be found for $e^{\frac{i}{\hbar}S_{eff}[q'(\cdot),q_g(\cdot);t,0]}$ and $e^{-\frac{i}{\hbar}S_{eff}[q''(\cdot),q_g(\cdot);t,0]}$ as well. Therefore, approximating the paths $q'(\cdot)$ and $q''(\cdot)$ by a collection of discretized paths $q'_\delta(\cdot)$'s and $q''_\delta (\cdot)$'s with time interval $\delta t$, and assuming that $m_e(t)$ remains virtually constant during each time interval $\delta t$, we can express Eq. (<ref>) as follows:
σ̃_e(q_f',q_f”;t+δt) = m_e/2πħt∫dq'∫dq” σ̃_e(q',q”;t)
×e^i/ħS_eff[q'(·);t+δt, t]-i/ħS_eff[q”(·);t+δt,t] ,
where the standard normalization factor of $\sqrt{m_e/(2\pi \hbar t)}$ was used for the path integral.
Let us introduce $\delta q'=q_f'-q'$, $\delta q''=q_f''-q''$, $\bar q'=(q_f'+q')/2$, and $\bar q''=(q_f''+q'')/2$. Then, assuming that ${\mathcal K}_I^{(n)}(t)$ and ${\mathcal K}_R^{(n)}(t)$ also remain virtually constant during the time interval of $\delta t$ and approximating the trajectories as straight lines, we obtain the following expressions:
S_eff[q'(·);t+δt, t]≈m_e(t)/2δtδq'^2-δt U_e(q̅',t) ,
S_eff[q”(·);t+δt,t]≈m_e(t)/2δtδq”^2-δt U_e(q̅”,t) ,
𝒞_I [q'(·),q”(·);t+δt, t]
≈𝒦̃_I^(1)(t)(q̅'-q̅”)(δq'+δq”) ,
𝒞_R[q'(·),q”(·);t+δt,t]≈𝒦_R^(0)(t)(q̅'-q̅”)^2 δt
-𝒦_R^(2)(t)/2δt (δq'-δq”)^2 .
Inserting these expressions into Eqs. (<ref>) and (<ref>), we find that
σ̃_e(q_f',q_f”;t+δt) ≈m_e(t)/2πħδt ∫dq' ∫dq” σ̃_e(q',q”;t)
×exp{im_e(t)/2ħδt (δq'^2 - δq”^2)-𝒦_R^(2)(t)/2ħδt (δq'-δq”)^2
-i/ħ 𝒦̃_I^(1)(t) (q̅'-q̅”)(δq'+δq”)
+1/ħ𝒦̃_R^(1)(t) (q̅'-q̅”)(δq'-δq”)
-δt/ħ𝒦_R^(0)(t) (q̅'-q̅”)^2
-iδt/ħ (U_e(q̅',t) - U_e(q̅”,t) )} .
The remaining steps in deriving a time evolution equation for $\tilde \sigma_e$ from Eq. (<ref>) are (i) to expand $\tilde \sigma_e(q',q'';t)$ in the integrand around $q_f'$ and $q_f''$, (ii) to perform integrations with respect to
$\delta q'=q_f'-q'$ and $\delta q''=q_f''-q''$, and (iii) to retain terms up to the order of $\delta t$ only. The integrations with respect to $\delta q'$ and $\delta q''$ can be done through analytic continuation of the integrands into the complex domain of space followed by normal mode transformation, which results in standard Gaussian integrations. Appendix B provides detailed description of all the steps (i)-(iii) of calculations listed above.
The resulting expression, Eq. (<ref>) or (<ref>), can be summarized as
∂/∂tσ̃_e(q',q”;t)={ iħ/2m_e(t)(∂/∂q'^2 -∂^2/∂q”^2)
-i/ħ (U_e(q',t)-U_e(q”,t)) -α(t)/ħ(q'-q”)^2
-𝒦̃_I^(1)(t)/m_e(t) (q'-q”) (∂/∂q' -∂/∂q”)
-i(𝒦̃_R^(1)(t)/m_e(t) -2 𝒦_R^(2)(t)/m_e(t)^2 𝒦̃_I^(1)(t))
×(q'-q”) (∂/∂q' +∂/∂q”)
+ħ𝒦_R^(2)(t)/2m_e(t)^2(∂^2/∂q'^2+∂^2/∂q”^2+2∂^2/∂q' ∂q”) }σ̃_e(q',q”;t) .
where $\alpha(t)$ is a real valued function defined by Eq. (<ref>) or (<ref>) and can be expressed as follows:
-1/m_e(t)𝒦_R^(2)(t)K̃_I^(1)(t)) .
Equivalently, we can express $\tilde \sigma_e(q',q'';t)$ in an operator form as follows:
σ̂̃̂_e(q_g(·);t)= ∫dq' ∫dq” |q' ⟩σ̃_e(q',q”,q_g(·);t)⟨q”| ,
where the dependence on $q_g(\cdot)$ has been shown explicitly.
Then, Eq. (<ref>) can be translated into a time evolution equation for this operator as follows:
∂/∂t σ̂̃̂_e (q_g(·);t)=-i/ħ [Ĥ_eff(t),σ̂̃̂_e]-i/ħ𝒦̃_I^(1)(t)/m_e(t)[q̂,{p̂,σ̂̃̂_e }]
+1/ħ (𝒦̃_R^(1)(t)/m_e(t)-2𝒦_R^(2)(t)/m_e(t)^2𝒦̃_I^(1)(t)) [q̂,[p̂,σ̂̃̂_e]]
-α(t)/ħ [q̂,[q̂,σ̂̃̂_e]] -𝒦_R^(2)(t)/2ħm_e(t)^2[p̂,[p̂, σ̂̃̂_e]] ,
Ĥ_eff(t)=p̂^2/2m_e(t)+U_e(q̂,q_g(·),t) ,
with $U_e(\hat q,q_g(\cdot),t)$ defined by Eq. (<ref>). Note that the effects of the ground state bath appear only in the effective time dependent potential $U_e(\hat q,q_g(\cdot),t)$. A phase space representation for Eq. (<ref>), which is a more conventional form of QFPE, can be found by applying the Wigner transformation<cit.> to Eq. (<ref>).
Equation (<ref>) is in the Dekker form<cit.> unlike the original CL's QFPE.<cit.> Thus, the Lindblad's condition of positive definiteness<cit.> can be satisfied for appropriate range of physical variables. To this end, more detailed analysis of each term is necessary for a specific form of the spectral density chosen.
§ RESULTS FOR OHMIC SPECTRAL DENSITY WITH DRUDE CUTOFF
Let us consider the case where the excited state bath spectral density, Eq. (<ref>), is given by the Ohmic spectral density with Drude cutoff as follows:
η_e(ω)=2mγ_e ω_c^2ω/ω^2+ω_c^2=2 mγ_s ω_c^2 ω/ω_c/(ω/ω_c)^2+1 ,
where $\gamma_e$ is the friction constant in the excited electronic state and $\gamma_s=\gamma_e/\omega_c$ is a dimensionless scaled version of the same friction constant.
For the above spectral density, $\kappa_e$, defined by Eq. (<ref>), has the following form:
κ_e=2mγ_eω_c=2mγ_sω_c^2 ,
and the imaginary component of the bath correlation function, Eq. (<ref>), can be expressed as
η̃_e,I(t)=mγ_eω_c^2 e^-ω_c t .
Then, it is straightforward to show that ${\mathcal K}_I^{(n)}(t)$ defined by Eq. (<ref>), for $n=0-2$, can be expressed as
𝒦_I^(0)(t)=mγ_eω_c F_0(t_s) ,
𝒦_I^(1)(t)=mγ_e F_1(t_s) ,
𝒦_I^(2)(t)=2mγ_s F_2(t_s) ,
where $t_s=\omega_c t$ is a scaled time, and
F_0(t_s)=1-e^-t_s ,
F_1(t_s)=1-(1+t_s)e^-t_s ,
F_2(t_s)=1-(1+t_s+t_s^2/2)e^-t_s ,
which all approach $1$ in the limit of $t_s\rightarrow \infty$. All of these three functions are monotonically increasing and positive for $t_s >0$.
Combining Eq. (<ref>) and the time derivative of Eq. (<ref>), we also obtain the expression for $\tilde {\mathcal K}_1^{(1)}(t)$, defined by Eq. (<ref>), as follows:
𝒦̃_I^(1) (t)=mγ_e F̃_1(t_s) ,
F̃_1(t_s)=1-(1+t_s-t_s^2/2)e^-t_s .
Note that $\tilde F_1(t_s)$ also approaches $1$ in the limit of $t_s\rightarrow \infty$ and is positive for $t_s >0$.
Taking the ratio of the effective time dependent mass $m_e(t)$, Eq. (<ref>), to the actual mass $m$, let us introduce
r_m(t)=m_e(t)/m= 1-2γ_s F_2(t_s) ,
which approaches $1-2\gamma_s$ in the limit of $t\rightarrow \infty$.
The real component of the bath correlation function, Eq. (<ref>), can be evaluated employing the well-known Matsubara expansion of $\coth (x)$ and $\cot(x)$, and is expressed as
η̃_e,R(t)=mγ_eω_c^2(βħω_c/2)e^-ω_c t
+4 mγ_eω_c^2/βħ∑_n=1^∞ω_n/(ω_n^2-ω_c^2) e^-ω_n t ,
where $\omega_n=2\pi n/(\beta\hbar)$.
Then, it is straightforward to show that ${\mathcal K}_R^{(n)}(t)$ defined by Eq. (<ref>), for $n=0-2$, can be expressed as
𝒦_R^(0)(t)=mγ_e ω_c G_0(β_s,t_s) ,
𝒦_R^(1)(t)=mγ_e G_1(β_s,t_s) ,
𝒦_R^(2)(t)=mγ_s G_2(β_s,t_s) ,
where $\beta_s=\beta\hbar \omega_c$, a dimensionless and scaled inverse temperature, and
G_0(β_s,t_s)= (β_s/2 )F_0(t_s)
+4/β_s∑_n=1^∞F_0(2πnt_s/β_s)/(2πn/β_s)^2-1 ,
G_1(β_s,t_s)=(β_s/2 )F_1(t_s)
+4/β_s∑_n=1^∞F_1(2πn t_s/β_s)/(2πn/β_s)((2πn/β_s)^2-1) ,
G_2(β_s,t_s)=(β_s/2 )F_2(t_s)
+4/β_s∑_n=1^∞F_2(2πn t_s/β_s)/(2πn/β_s)^2((2πn/β_s)^2-1) .
In addition, combination of Eq. (<ref>) and the time derivative of Eq. (<ref>) leads to the expression for $\tilde {\mathcal K}_R^{(1)}(t)$, defined by Eq. (<ref>), as follows:
𝒦̃_R^(1)(t)=mγ_e G̃_1(β_s;t_s) ,
G̃_1(β_s;t_s)=( β_s/2 )F̃_1 (t_s)
+4/β_s∑_n=1^∞F̃_1 (2πn t_s/β_s)/(2πn/β_s)((2πn/β_s)^2-1) .
Employing the above expressions, Eq. (<ref>) for the present spectral density can be expressed as
∂/∂t σ̂̃̂_e (q_g(·);t)=-i/ħ [Ĥ_eff(t),σ̂̃̂_e]
-iγ_e/ħ Γ(t)[q̂,{p̂,σ̂̃̂_e }]+γ_e/ħ ℛ_pq (t) [q̂,[p̂,σ̂̃̂_e]]
-mγ_e^2/ħ ℛ_pp(t) [q̂,[q̂,σ̂̃̂_e]] -ℛ_qq(t)/m ħ[p̂,[p̂, σ̂̃̂_e]] ,
Γ(t)=F̃_1(t_s)/r_m(t) ,
ℛ_pq(t)=1/r_m(t)G̃_1(β_s,t_s) -2γ_s/r_m^2 G_2(β_s,t_s)F̃_1(t_s) ,
ℛ_qq(t)=γ_s/2 r_m(t)^2 G_2(β_s,t_s) ,
ℛ_pp(t)=1/γ_sG_0(β_s,t_s)- 2 Γ(t)G̃_1(β_s,t_s)
+2γ_s Γ(t)^2 G_2(β_s,t_s) .
As mentioned in the previous section, Eq. (<ref>) is in the Dekker form<cit.> and satisfies the Lindblad's positive definiteness condition<cit.> given that the following inequality holds.
D(t)≡4ℛ_pp (t)ℛ_qq(t)-ℛ_pq(t)^2-Γ(t)^2 >0 .
Values of ${\mathcal R}_{pq}(t)$, ${\mathcal R}_{qq}(t)$, ${\mathcal R}_{pp}(t)$, and $D(t)$ versus $t_s=\omega_c t$ for $\beta_s=0.5$, $1$, and $5$. The value of $\gamma_s=0.1$.
Figure 1 shows calculated results of ${\mathcal R}_{pq}(t)$, ${\mathcal R}_{qq}(t)$, ${\mathcal R}_{pp}(t)$, and $D(t)$ for three cases of $\beta_s=0.5$, $1$, and $5$. A small value of $\gamma_s=0.1$ was chosen, for which the weak damping condition of $\gamma_s G_2(\beta_s,t_s) <1-2\gamma_s F_2(t_s)$ is satisfied throughout the entire time. Except for the case of very low temperature, $\beta_s=5$, all the values of ${\mathcal R}_{qq}(t)$, ${\mathcal R}_{qq}(t)$, and ${\mathcal R}_{pp}(t)$ remain positive. Although $D(t)$ becomes negative transiently in early stage, its steady state limits are positive for $\beta_s=0.1$ and $1$. This shows that the non positivity condition of CL's QFPE<cit.> can be fixed by avoiding inconsistent use of Markovian approximation, confirming the analysis by Diosi.<cit.> On the other hand, for $\beta_s=5$, $D(t)$ becomes negative for all values of $t>0$. The main contribution to this negative value comes from that of ${\mathcal R}_{pp}(t)$, and indicates that the second order approximation for the real part of the bath correlation function is not valid at this temperature due to nonlocality of the quantum dynamics in time.
§ CONCLUSION
The present work has provided a derivation of a GQFPE by extending CL's path integral approach.<cit.>
The only assumption used in this derivation is that the bath correlation functions are short ranged in time so that the second order expansions of trajectories within the integrands in the exponents of the influence functional are well justified. This seems to be the most general assumption one can make in order to convert the double time integrations in the exponents of the influence functional into single time integrations, from which a time evolution equation can be derived.
Thus, the resulting GQFPE, Eq. (<ref>), may serve as a general form that can include various known QFPEs as special cases. In addition, Eq. (<ref>) can also serve as a new and useful means to describe photo-induced quantum relaxation processes beyond typical high temperature and weak coupling limits, and thus will serve as a more satisfactory theoretical tool to study wider range of photoinduced electron and proton transfer processes.
The importance of the general form of the GQFPE, Eq. (<ref>), is that it has not been constructed phenomenologically, but rather derived from a well defined Hamiltonian. Thus, it offers detailed microscopic expressions for all the terms entering the equation in terms of the parameters defining the Hamiltonian. This makes it possible to examine the validity of the assumptions underlying the derivation a posteriori, for a given Hamiltonian and bath spectral density. Most of all, because Eq. (<ref>) is in the Dekker form,<cit.> the condition of positive definiteness can be tested explicitly.
For one of the most well-known spectral densities, the Ohmic spectral density with Drude cutoff, all terms in the GQFPE have been calculated explicitly in Sec. III. The results demonstrated in Fig. 1 show that the GQFPE indeed is well defined and its steady state limit satisfies the Lindblad's condition<cit.> for reasonable physical situation. This confirms that the violation of positive definiteness results from an inconsistent application of the Markovian approximation or the breakdown of time locality in the quantum dynamics as is typical at very low temperature regime.
There remain some subtle issues that need to be clarified in the future. For example, the physical implication of the boundary terms in Eq. (<ref>) should be understood better. In addition, the expressions of terms in Eq. (<ref>) and Eq. (<ref>) show that the detailed manner of the high frequency cutoff, even for the Ohmic spectral density, makes significant contribution to the final form of the equation. This is consistent with a previous analysis based on a fourth order quantum master equation.<cit.> This also shows the possibility that Eq. (<ref>) can serve as a useful theoretical tool to examine the type of spectral density and physical conditions for which a time local QFPE can be established.
The methodology of present work can easily be extended to multidimensional system and nonadiabatic cases. Consideration of these cases will be another subject of future theoretical investigation. Finally, further test of the GQFPE against exactly solvable models<cit.> and virtually exact calculation approaches<cit.> remain as important future tasks. Outcomes of these studies will help provide ultimate validation of the GQFPE of this work and understanding of the extent to which a closed form time local equation can be used to describe nonequilibrium and non-Markovian quantum dynamical processes.
The author acknowledges the support for this research from the National Science Foundation (CHE-1362926), the Office of Basic Energy Sciences, Department of Energy (DE-SC0001393), and the Camille Dreyfus Teacher Scholar Award.
§ NONEQUILIBRIUM INFLUENCE FUNCTIONAL
The path integral expression for Eq. (<ref>) and an appropriate expression
for the influence functional, which are well known,<cit.> are derived here for the sake of completeness.
For the real time propagator, $e^{-iH_et/\hbar}$, the path integral expression is given by
e^-iĤ_e t/ħ= ∫dq_i ∫dq_f ∫dx_i∫dx_f |q_f,x_f⟩⟨q_i,x_i|
×∫_q_i^q_f 𝒟q(·)∫_x_i^x_f 𝒟 x(·) e^i/ħS_e[q(·);t]+i/ħS_eb[x(·),q(·);t]
where $x\equiv (x_1,\cdots,x_\alpha,\cdots)$ and $|q,x\rangle \equiv |q\rangle \prod_\alpha |x_\alpha\rangle$. $S_e$ and $S_{eb}$ are real time actions of the excited state, respectively given by
S_e[q(·);t]=∫_0^t dt_1 (m/2q̇(t_1)^2-V_e(q(t_1))-κ_e/2 q(t_1)^2) with
κ_e=∑_αc_α,e^2/m_αω_α^2=2/π∫_0^∞dωη_e(ω)/ω ,
S_eb[x(·), q(·);t]=∑_α∫_0^t dt_1 (m_α/2ẋ_α(t_1)^2.
. -m_αω_α^2/2x_α(t_1)^2 +c_α,eq(t_1)x_α(t_1)) On the other hand, for $e^{-\beta \hat H_g}$, the path integral expression is given by
e^-βĤ_g = ∫dq' ∫dq” ∫d x' ∫d x” |q',x'⟩⟨q”,x”|
×∫_q'^q” 𝒟q(·) ∫_x'^x”𝒟x(·) e^- 1/ħS_g^E[q(·);βħ]-1/ħS_gb^E[q(·),x(·);βħ]
where $S_g^E$ and $S_{gb}^E$ are Euclidean actions, respectively given by
with $\kappa_g=\sum_\alpha c_{\alpha,g}^2/(m_\alpha \omega_\alpha^2)$, and
S_gb^E[x(·), q(·);βħ]=∑_α∫_0^t dt_1 (m_α/2ẋ_α(t_1)^2.
.+m_αω_α^2/2x_α(t_1)^2 -c_α,gq(t_1)x_α(t_1))
The path integral expression for $\hat \sigma_e(t)$ can be obtained by inserting
Eq. (<ref>), its complex conjugate, and Eq. (<ref>),
into Eq. (<ref>). By performing explicit path integration over the bath degrees of freedom, one
can show that Eq. (<ref>) reduces to
e^-iĤ_e t/ħ=∫dq_i ∫dq_f ∫dx_i ∫dx_f |q_f,x_f⟩⟨q_i,x_i|
×∫_q_i^q_f 𝒟q(·) e^ i/ħS_e[q(·);t]∏_αT_e,α[q(·);x_α,f,x_α,i,t]
×exp{i/ħ[ m_αω_α/2sin(ω_αt) ( x_α,f^2cos(ω_αt).
. +x_α,i^2 cos(ω_αt)-2x_α,f x_α,i )
+c_α,ex_α,f/sin(ω_αt)∫_0^t dt_1 sin(ω_αt_1)q(t_1)
+c_α,ex_α,i/sin(ω_αt)∫_0^t dt_1 sin(ω_α(t- t'))q(t_1)
-c_α,e^2/m_αω_αsin(ω_αt) ∫_0^t dt_1∫_0^t_1 dt_2 sin(ω_α(t-t_1))
×sin(ω_αt_2) q(t_1) q(t_2) ]} Similarly, Eq. (<ref>) can be shown to be
e^-βĤ_g = ∫dq' ∫dq” ∫d x' ∫d x” |q',x'⟩⟨q”,x”|
×∫_r'^r” 𝒟q(·) e^- 1/ħS_g^E[q(·);βħ]∏_αT_g,α^E[q(·);x'_α,x”_α,βħ]
×exp{-1/ħ[ m_αω_α/2sinh(ω_αβħ) ( x_α'^2cosh(ω_αβħ) .
. +x_α”^2 cosh(ω_αβħ)-2x'_αx”_α)
-c_α,gx'_α/sinh(ω_αβħ)∫_0^βħ dτsinh(ω_ατ)q_g(τ)
-c_α,gx_α”/sinh(ω_αβħ)∫_0^βħ dτsinh(ω_α(βħ-τ))q_g(τ)
-c_α,g^2/m_αω_αsinh(ω_αβħ) ∫_0^βħ dτ∫_0^τ dτ_1
×sinh(ω_α(βħ-τ)) sinh(ω_ατ_1)q_g(τ)q_g(τ_1) ] }
With the use of Eq. (<ref>), its complex conjugate, and
Eq. (<ref>) in Eq. (<ref>), the reduced density operator can be expressed as
σ̂_e(t)=1/Z_g∫dq_f'∫dq_f” |q_f'⟩⟨q_f”|
×∫dq_i'∫dq_i”∫dx_f ∫dx_i'∫dx_i”
×∫_q_i'^q_f' 𝒟 q'(·)∫_q_i”^q_f” 𝒟q”(·) ∫_q_i'^q_i” q_g(·) exp{i/ħS_e[q'(·);t]
Performing integrations over $x_f$, $x_i'$, and $x_i''$ leads to Eqs. (<ref>) and (<ref>) with the following definitions of its exponents.
W_e,I(t) = ∫_0^t d t_1∫_0^t_1 dt_2 ∫_0^∞dω/π η_e (ω)sin(ω(t_1-t_2))
×(q'(t_1)-q”(t_1))(q'(t_2)+q”(t_2)) ,
W_e,R(t) = ∫_0^t dt_1∫_0^t_1 dt_2 ∫_0^∞dω/π η_e(ω)
×(q'(t_1)-q”(t_1))(q'(t_2)-q”(t_2)) ,
W_c(t) = ∫_0^tdt_1∫_0^βħ dτ∫_0^∞d ω/π η_c(ω)
×(q'(t_1)-q”(t_1))q_g(τ) ,
W_g(βħ) = ∫_0^βħdτ∫_0^τdτ_1 ∫_0^∞dω/π η_g(ω)
×cosh(ω(τ-τ_1-βħ/2))/sinh(ωβħ/2)q_g(τ_1)q_g(τ) .
§ DERIVATION OF EQ. (<REF>)
In the integrand of Eq. (<ref>), consider the following term:
-1/m_e 𝒦_R^(2)(t) (δq'-δq”)^2 } .
This term can be diagonalized into a sum of two quadratic terms by using complex-valued coordinates, the choice of which depends on the magnitude of $ {\mathcal K}_R^{(2)}(t)/m_e$ as described below. In the above expressions, the time dependence of $m_e(t)$ has not been shown explicitly and will remain so throughout this section.
§.§.§ Case for $0< {\mathcal K}_R^{(2)}(t)/m_e < 1$
For this case, we can introduce $\mu_1 (t)$ such that
sin(μ_1(t))=𝒦_R^(2)(t)/m_e ,
where $0 < \mu_1(t) < \pi/2$.
Solving the eigenvalue problem for the quadratic form, Eq. (<ref>), it is straightforward to find out the following two normal modes defined in the complex domain.
u'=1/√(cosμ_1)( cos(μ_1/2) δq'-isin(μ_1/2)δq”) ,
u”=1/√(cosμ_1) (isin(μ_1/2) δq'+cos(μ_1/2)δq”) .
Equivalently, $\delta q'$ and $\delta q''$ can be expressed in terms of $u'$ and $u''$ as follows:
δq'=1/√(cosμ_1)( cos(μ_1/2) u'+isin(μ_1/2)u”) ,
δq”=1/√(cosμ_1) (-isin(μ_1/2) u'+cos(μ_1/2)u”) .
Inserting these expressions into Eq. (<ref>), we find that
Q(δq',δq”) = m_e/2ħδt{(icosμ_1 -sinμ_1)u'^2
+(-icosμ_1-sinμ_1)u”^2 }
= m_e/2ħδt{ ie^iμ_1u'^2-ie^-iμ_1u”^2} .
Then, Eq. (<ref>) can be expressed as
σ̃_e(q_f',q_f”;t+δt) ≈m_e/2πħδt ∫du' ∫du” σ̃_e(q',q”;t)
×exp{im_e/2ħδt (e^iμ_1u'^2 - e^-iμ_1 u”^2)
-i/ħ 𝒦̃_I^(1)(t) (q̅'-q̅”)1/√(cosμ_1)(e^-iμ_1/2u'+e^iμ_1/2u”)
+1/ħ𝒦̃_R^(1)(t) (q̅'-q̅”)1/√(cosμ_1)(e^iμ_1/2u'-e^-iμ_1/2u”)
-δt/ħ𝒦_R^(0)(t) (q̅'-q̅”)^2
-iδt/ħ (U_e(q̅',t) - U_e(q̅”,t) )} .
Let us introduce
ξ'=u'-iδt e^-iμ_1/m_e√(cosμ_1) (q̅'-q̅”)
×(𝒦̃_R^(1)(t)e^iμ_1/2-i𝒦̃_I^(1)(t)e^-iμ_1/2 ) ,
ξ”=u”-iδt e^iμ_1/m_e√(cosμ_1) (q̅'-q̅”)
×(𝒦̃_R^(1)(t)e^-iμ_1/2+i𝒦̃_I^(1)(t)e^iμ_1/2 ) .
Then, the squares of the exponents in Eq. (<ref>) can be completed as follows:
σ̃_e(q_f',q_f”;t+δt) ≈m_e/2πħδt ∫dξ' ∫dξ” σ̃_e(q',q”;t)
×exp{im_e/2ħδt (e^iμ_1ξ'^2 - e^-iμ_1 ξ”^2)
-δt/ħα(t) (q̅'-q̅”)^2
-iδt/ħ (U_e(q̅',t) - U_e(q̅”,t) ) } ,
α(t) =𝒦_R^(0)(t)-2/m_e𝒦̃_I^(1)(t)(K̃_R^(1)(t)-sinμ_1 K̃_I^(1)(t)) .
In Eq. (<ref>), $q'=q_f'-\delta q'$ and $q''=q_f''-\delta q"$ can be expressed as
q'=q_f'-1/√(cosμ_1) ( cos(μ_1/2)ξ'+isin(μ_1/2)ξ”)
-δt/m_e 𝒦_c(t) (q̅'-q̅”) ,
q”=q_f”-1/√(cosμ_1) ( cos(μ_1/2)ξ”-isin(μ_1/2)ξ')
+δt/m_e𝒦_c^*(t)(q̅'-q̅”) ,
where Eqs. (<ref>), (<ref>), (<ref>), and (<ref>) have been used and
𝒦_c(t)=𝒦̃_I^(1)(t)+i(𝒦̃_R^(1)(t)-2sinμ_1 𝒦̃_I^(1)(t)) .
Inserting the above expressions into the arguments of $\tilde \sigma_e(q',q'';t)$ in Eq. (<ref>) and expanding the integrand up to the second order of $\xi'$ and $\xi''$, we find the following expression:
σ̃_e(q_f',q_f”;t+δt) ≈
σ̃_e(q_f'-δt 𝒦_c(t)/m_ecosμ_1 (q̅'-q̅”),q_f”+δt 𝒦_c^*(t)/m_ecosμ_1 (q̅'-q̅”);t)
×exp{-iδt/ħ (U_e(q̅',t) - U_e(q̅”,t)) -δt/ħ α(t)(q̅'-q̅”)^2}
+m_e/4πħδtcosμ_1 ∫dξ'∫dξ”
×exp{im_e/2ħδt( e^iμ_1ξ'^2-e^-iμ_1ξ”^2)}
×( cos^2(μ_1/2)ξ'^2 -sin^2(μ_1/2)ξ”^2 ) ∂^2/∂q_f'^2 σ̃_e(q_f',q_f”;t)
+m_e/4πħδtcosμ_1 ∫dξ'∫dξ”
×exp{im_e/2ħδt( e^iμ_1ξ'^2-e^-iμ_1ξ”^2)}
×( cos^2(μ_1/2)ξ”^2 -sin^2(μ_1/2)ξ'^2 ) ∂^2/∂q_f”^2 σ̃_e(q_f',q_f”;t)
+im_esinμ_1/4πħδtcosμ_1 ∫dξ'∫dξ”
×exp{im_e/2ħδt ( e^iμ_1ξ'^2-e^-iμ_1ξ”^2)}
×(ξ”^2 -ξ'^2 ) ∂^2/∂q_f'∂q_f” σ̃_e(q_f',q_f”;t) ,
where the terms linear in $\xi'$ and $\xi''$ and the term containing mixed quadratic term $\xi'\xi''$ were dropped because they become zero after the integration.
Performing Gaussian integrations over $\xi'$ and $\xi''$ and expanding all terms up to the first order of $\delta t$, we obtain
σ̃_e(q_f',q_f”;t+δt) ≈{ 1-iδt/ħ (U_e(q̅',t)-U_e(q̅”,t))
- δt/m_e𝒦̃_I^(1)(t)(q̅'-q̅”)(∂/∂q_f'-∂/∂q_f”)
-iδt/m_e (𝒦̃_R^(1)(t)-2sinμ_1 𝒦̃_I^(1)(t))
+iħδt/2m_e (∂^2/∂q_f'^2-∂^2/∂q_f”^2)
. +ħδt/2m_e sinμ_1 (∂^2/∂q_f'^2+∂^2/∂q_f”^2+2∂^2/∂q_f' ∂q_f” )} σ̃_e(q_f',q_f”;t) .
Taking the limit of $\delta t\rightarrow 0$, and making the replacement of $q_f',\bar q' \rightarrow q'$ and $q_f'',\bar q''\rightarrow q''$ in the resulting equation, we obtain the following time evolution equation:
∂/∂tσ̃_e(q',q”;t)={ iħ/2m_e(∂/∂q'^2 -∂^2/∂q”^2)
-i/ħ (U_e(q',t)-U_e(q”,t)) -α(t)/ħ(q'-q”)^2
-𝒦̃_I^(1)(t)/m_e (q'-q”) (∂/∂q' -∂/∂q”)
-i/m_e(𝒦̃_R^(1)(t) -2sinμ_1 𝒦̃_I^(1)(t)) (q'-q”) (∂/∂q' +∂/∂q”)
+ħ/2m_e(t)sinμ_1(∂^2/∂q'^2+∂^2/∂q”^2+2∂^2/∂q' ∂q”) }σ̃_e(q',q”;t) .
where $\mu_1$ and $m_e$, respectively defined by Eqs. (<ref>) and (<ref>), are all time dependent although not shown explicitly.
§.§.§ Case for ${\mathcal K}_R^{(2)}(t)/m_e > 1$
For this case, we can introduce $\mu_2(t)$ such that
sin(μ_2(t))=m_e/ 𝒦_R^(2)(t) .
Solving the eigenvalue problem for the quadratic form, Eq. (<ref>), it is easy to find out the following two normal modes defined in the complex domain.
u'=1/√(2cosμ_2) ( e^iμ_2/2 δq' +e^-iμ_2/2δq”) ,
u”=1/√(2cosμ_2) (e^-iμ_2/2δq'-e^iμ_2/2 δq” ) .
Equivalently, $\delta q'$ and $\delta q''$ can be expressed in terms of $u'$ and $u''$ as follows:
δq'=1/√(2cosμ_2) ( e^iμ_2/2 u' +e^-iμ_2/2 u”) ,
δq”=1/√(2cosμ_2) (e^-iμ_2/2 u'-e^iμ_2/2 u” ) .
Inserting these expressions into Eq. (<ref>), we find that
Q(δq',δq”)=-m_e/2ħδt{tan(μ_2/2) u'^2+(μ_2/2)u”^2} .
In the exponent of the integrand in Eq. (<ref>), squares can be completed introducing the following variables:
ξ'=u'+2iδt/m√(2cosμ_2) (μ_2/2) (q̅'-q̅”)
×(𝒦̃_I^(1)(t)cos(μ_2/2)-𝒦̃_R^(1)(t)sin(μ_2/2)) ,
ξ”=u”+2δt/m√(2cosμ_2) tan(μ_2/2)(q̅'-q̅”)
×(𝒦̃_I^(1)(t)sin(μ_2/2)-𝒦̃_R^(1)(t)cos(μ_2/2)) ,
Thus, Eq. (<ref>) can be expressed as
σ̃_e(q_f',q_f”;t+δt) ≈m_e/2πħδt ∫dξ' ∫dξ” σ̃_e(q',q”;t)
×exp{-m_e/2ħδt (tan(μ_2/2) ξ'^2 +(μ_2/2) ξ”^2)
-iδt/ħ (U_e(q̅',t) - U_e(q̅”,t) ) } ,
α(t) =𝒦_R^(0)(t)-2/m_e𝒦̃_I^(1)(t)(K̃_R^(1)(t)-1/sinμ_2 K̃_I^(1)(t)) .
Comparing the definitions of Eqs. (<ref>) and (<ref>), it is easy to see that the above expression is equivalent to Eq. (<ref>).
In Eq. (<ref>), $q'$ and $q''$ can be expressed as
q'=q_f'-1/√(2cosμ_2) ( e^iμ_2/2 ξ'+e^-iμ_2/2 ξ”)
-δt/m_e 𝒦_c(t)(q̅'-q̅”) ,
q”=q_f”-1/√(2cosμ_2) ( e^-iμ_2/2 ξ'-e^iμ_2/2 ξ”)
+δt/m_e 𝒦_c^*(t)(q̅'-q̅”) ,
where ${\mathcal K}_c(t)$ can be expressed as
𝒦_c(t)=𝒦̃_I^(1)(t)+i(𝒦̃_R^(1)(t)-2/sinμ_2 𝒦̃_I^(1)(t)) .
Inserting Eqs. (<ref>) and (<ref>) into Eq. (<ref>) and following the same procedure as deriving Eq. (<ref>), it is straightforward to show that
∂/∂tσ̃_e(q',q”;t)={ iħ/2m_e(∂/∂q'^2 -∂^2/∂q”^2)
-i/ħ (U_e(q',t)-U_e(q”,t)) -α(t)/ħ(q'-q”)^2
-𝒦̃_I^(1)(t)/m_e (q'-q”) (∂/∂q' -∂/∂q”)
-i/m_e(𝒦̃_R^(1)(t) -2/sinμ_2 𝒦̃_I^(1)(t)) (q'-q”) (∂/∂q' +∂/∂q”)
+ħ/2m_esinμ_2(∂^2/∂q'^2+∂^2/∂q”^2+2∂^2/∂q' ∂q”) }σ̃_e(q',q”;t) ,
Comparing the definitions of Eqs. (<ref>) and (<ref>), it is easy to show that the above equation is equivalent to Eq. (<ref>).
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|
1511.00443
|
$^1$LaserLaB, Department of Physics and Astronomy, Vrije Universiteit Amsterdam,
De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
High precision spectroscopy on the $2 \ ^3 S \rightarrow 2 \ ^1 S$ transition is possible in ultracold optically trapped helium but the accuracy is limited by the ac-Stark shift induced by the optical dipole trap. To overcome this problem, we have built a trapping laser system at the predicted magic wavelength of 319.8 nm. Our system is based on frequency conversion using commercially available components and produces over 2 W of power at this wavelength. With this system, we show trapping of ultracold atoms, both thermal ($\sim0.2 \ \mathrm{\mu K}$) and in a Bose-Einstein condensate, with a trap lifetime of several seconds, mainly limited by off-resonant scattering.
(020.7010) Laser trapping; (300.6210) Spectroscopy, atom; (140.3515) Lasers, frequency doubled; (140.3610) Lasers, ultraviolet
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§ INTRODUCTION
The helium atom has proven to be a productive testing ground for fundamental physics. Frequency metrology has been employed as a sensitive test of QED calculations, and also to probe the influence on level energies of the finite size of the nucleus. Such measurements are currently receiving a lot of attention since a measurement of the proton radius in muonic hydrogen found a $7 \sigma$ discrepancy with the 2010 CODATA value <cit.>. This discrepancy has been named the “proton radius puzzle”, and research is extended to the helium nucleus, with current experiments focussing on the muonic helium ion <cit.>.
In neutral helium, theory and experiment have developed to a level where it is possible to determine the difference in nuclear charge radius between $\mathrm{^3 He}$ and $\mathrm{^4 He}$ with high accuracy <cit.>. One transition very sensitive to the nuclear charge radius is the $2 \ ^3 S \rightarrow 2 \ ^1 S$ transition, which has recently been measured in an ultracold gas trapped in an optical dipole trap (ODT) <cit.>. This was done both in a Bose-Einstein condensate (BEC) of $\mathrm{^4He}$ in the metastable $2 \ ^3 S$ state ($\mathrm{^4He^*}$) and in a degenerate Fermi gas of $\mathrm{^3 He^*}$. The 2.3 kHz experimental accuracy of the isotope shift was limited by the ac-Stark shift induced by the ODT. This problem is also encountered in optical lattice clocks where it is solved by employing so-called magic wavelength traps <cit.>. In a magic wavelength trap, the wavelength of the trapping laser is chosen such that the upper and lower state polarizability are exactly the same, cancelling out the differential ac-Stark shift. For the $2 \ ^3 S$ and $2 \ ^1 S$ levels of helium, we have previously calculated the ac polarizability and compiled a list of magic wavelengths for both isotopes <cit.>. All useful magic wavelengths are in the ultraviolet spectral region with the most promising candidate located at 319.815 nm for $\mathrm{^4 He}$ and 319.830 nm for $\mathrm{^3 He}$.
Trapping atoms at this wavelength is not straightforward. First of all, to achieve a trap depth comparable to <cit.>, where an infrared ODT at 1557 nm was used, an optical power of approximately 1 W is required. Such powers are not readily available at ultraviolet wavelengths. Secondly, the lifetime of atoms trapped at this magic wavelength is intrinsically limited by two mechanisms: off-resonant excitation to the nearby $2 \ ^3 S \rightarrow 4 \ ^3 P$ transition at 318.9 nm, and two-photon ionization. The total loss rate from these processes should not exceed about 1 $s ^{-1}$, to allow sufficient probe time for spectroscopy.
Considerable progress has been made in the production of laser light at the nearby wavelength of 313 nm, for the purpose of laser cooling $\mathrm{Be^+}$ ions <cit.>. High power (750 mW) was generated by sum frequency mixing and subsequent frequency doubling of two fiber lasers, retaining most of their high spatial and spectral mode quality <cit.>. More recently, production of up to 2 W was demonstrated with such a scheme <cit.>.
In section <ref>, we demonstrate a laser system producing over 2 W at 319.8 nm based on a modification of this scheme. Our system is built out of commercially available components which makes it simple, compact, and reliable. In section <ref> we show that our source can be used to trap helium atoms with an acceptable lifetime of a few seconds, such that spectroscopy on the $2 \ ^3 S \rightarrow 2 \ ^1 S$ transition can be performed.
§ LASER SYSTEM
The system can be divided in a sum frequency generation (SFG) part, which generates 639.6 nm light from two infrared lasers, and a second harmonic generation (SHG) part which frequency doubles the SFG light to 319.8 nm. In the following, we will first give a full overview of the optical system before discussing the results and performance of the SFG and SHG parts separately.
§.§ Overview
Figure <ref> shows a schematic overview of the optical setup. The setup is relatively compact with all of the components, except for the lasers and the control electronics, mounted on a single 1000$\times$500 $\mathrm{mm^2}$ optical breadboard. The system starts with two fiber lasers (NKT photonics Koheras Adjustik E15 and Y10) with center wavelengths of 1557.28 nm and 1085.45 nm. The thermal tuning ranges are 1000 pm and 700 pm respectively, covering a spectral range much larger than the uncertainty in the calculated magic wavelength. The lasers seed two 10 W fiber amplifiers (NuFern NUA-1084-PB-0010-C2 and NUA-1550-PB-0010-C2), with isolated polarization maintaining free-space output couplers. The beams are separately focussed to achieve optimal sum frequency generation.
Schematic view of the UV laser system. Two infrared fiber lasers at 1085.5 nm and 1557.3 nm seed two 10 W amplifiers. The infrared beams are independently focussed, and overlapped on a dichroic mirror (DM). The combined beam passes through a temperature stabilized PPLN crystal. More dichroic mirrors filter residual infrared light from the SFG beam at 639.6 nm. This beam is mode-matched and phase modulated by a 2 MHz electro-optic modulator (EOM) and the cavity reflection is monitored using a photodiode (PD) to allow Pound-Drever-Hall locking of the cavity. The final UV output beam is then collimated and ellipticity is compensated by an anamorphic prism pair (APP).
The output beams are to an excellent degree of approximation Gaussian, and can be described completely by two parameters: their (minimum) waist size ($w_0$), which is directly related to their Rayleigh range ($z_R = \pi w_0^2/\lambda$), and the position of their focus. In order to achieve optimal conversion these parameters must be matched both to each other and to the crystal. Achieving this condition is not entirely straightforward because of the coupled nature of the problem.
We first collimate the 1557.3 nm beam with an $f = 300 \ \mathrm{mm}$ lens, and then focus by two lenses with focal distances of -100 mm and 200 mm. The beam waist can now be changed by moving either of the focussing lenses, but doing so will also move the focal point. The 1085.5 nm beam is first passed through a telescope consisting of two lenses with focal distances of 200 mm and 50 mm and is then focussed by an $f = 300 \ \mathrm{mm}$ lens. The focal point can be changed without affecting the waist size by moving the focussing lens. In this way, the foci of the beams are matched by first setting the waist of the 1085.5 nm beam to the desired focussing, secondly matching the 1557.3 nm beam waist to it, and finally overlapping the focal point of the 1085.5 nm beam with that of the 1557.3 nm beam.
With fixed beam parameters, the infrared beams are overlapped on a dichroic mirror and passed through a 40 mm MgO doped periodically poled lithium niobate (PPLN) crystal (Covesion) with a poling period of $12.1\mathrm{\mu m}$. The crystal is mounted in an oven and temperature stabilized at $ \sim 90^\circ$C. The output beam from the crystal contains both the sum frequency and residual infrared light. This residual light is filtered from the beam by two dichroic mirrors and the SFG beam is collimated by a $f = 250 \ \mathrm{mm}$ lens to a waist of $\sim 1 \ \mathrm{mm}$.
The light is then coupled (free-space) into a commercial frequency doubling system (Toptica SHG pro) where it is mode-matched to the cavity and passed through an electro-optical modulator (EOM). The EOM modulates 2 MHz sidebands on the laser carrier frequency to allow Pound-Drever-Hall locking of the cavity. The doubling cavity is similar to <cit.>, the main differences being the locking scheme (Pound-Drever-Hall instead of Hänsch-Couillaud) and the crystal (AR-coated rather than Brewster cut). The UV output is collimated and passed through an anamorphic prism pair to reduce beam ellipticity.
§.§ Sum Frequency Generation
The purpose of the SFG stage is to convert the available infrared laser light into useful SFG light with high efficiency. To achieve this it is necessary to focus the input beams tightly so that a high peak intensity is reached, but not so tightly that the beams diverge too quickly before they reach the end of the crystal. As described by Boyd and Kleinman <cit.>, this process can be optimized with respect to the dimensionless focussing parameter $\xi = l / 2 z_R$, which is the ratio of the crystal length $l$ to the confocal parameter of the beam (twice the Rayleigh length $z_R$). Although optimal at $\xi \approx 2.84$, the efficiency varies quite slowly so that at confocal focussing ($\xi = 1$) it is still approximately 80% of its maximum value. The system produces more power than required and the confocal condition is chosen because of practical considerations such as the available path length and the size of the entrance surface of the crystal.
The infrared beams were set to the confocal condition as described in section <ref>. The waist sizes are measured to be $56(1) \ \mathrm{\mu m}$ ($z_R = 8.9(3) \ \mathrm{mm}$) for the 1085.5 nm beam and $63(1) \ \mathrm{\mu m}$ ($z_R = 7.9(3) \ \mathrm{mm}$) for the 1557.3 nm beam, with their waist positions located within 1 mm of each other. By correcting for the refractive index of the crystals, which can be calculated based on known Sellmeier coefficients <cit.>, the focussing parameter inside the crystal is found to be $\xi = 1.03$ for the 1085.5 nm beam and $\xi = 1.16$ for the 1557.3 nm beam.
Results of sum frequency generation. (a) Sum frequency power as a function of the input power product (IPP) of the infrared lasers. The dashed line is a linear fit at low input powers (slope $0.108(1) \ \mathrm{W^{-1}}$). (b) Contourplot of the sum frequency power as a function of both input powers, based on a linear interpolation of the same dataset as (a). Black diamonds indicate measured datapoints, the dashed lines indicate contours of constant IPP.
Figure <ref>a shows the converted power as a function of the product of the input powers (input power product, IPP). When different combinations of input powers with equal IPP are used they produce almost the same SFG output power. This can be seen more clearly in a contour plot of the output power as a function of input powers (figure <ref>b, based on a linear interpolation of the same dataset) where contours of constant IPP follow constant output power. From this we conclude that the total converted power is a function of IPP only, and does not depend on the exact composition of infrared powers.
In order to achieve the power conversion plotted in Figs. <ref>a and <ref>b, it is necessary to optimize for crystal temperature at each input power product. The reason for this is shown in Figure <ref>, which shows the conversion efficiency as a function of crystal temperature at different input powers. As the input power becomes higher, the optimal temperature shifts to lower temperature and the crystal temperature needs to be adjusted to achieve maximum conversion. A possible explanation for this behaviour is that light is absorbed in the center of the crystal and heats it locally. This causes a slightly elevated temperature, and consequently imperfect phase matching at the center of the crystal (where conversion occurs) compared to the crystal edge (with respect to which the temperature is controlled). This effect is compensated when the crystal temperature is set to a slightly lower temperature.
Sum frequency production as a function of crystal temperature, normalized for measured input power product. The plot shows input power products of $1.4 \ \mathrm{W^2}$ (blue circles), $35.7 \ \mathrm{W^2}$ (red squares), and $78.0 \ \mathrm{W^2}$ (black diamonds). At higher input powers the temperature graph becomes slanted towards lower temperature, indicating thermal effects.
At low input powers the SHG output power scales linearly with a slope of $0.108(1) \ \mathrm{W^{-1}}$, comparable to other experiments using a similar crystal <cit.>. At higher output powers a deviation from the linear behaviour is seen. This may be a left-over thermal effect, or it may be that, because of the high conversion efficiency, pump depletion needs to be taken into account. At an IPP of $80 \ \mathrm{W^2}$, consisting of 8 W at 1557.3 nm and 10 W at 1085.5 nm, a maximum output power of almost 6 W of SFG light is produced, which corresponds to a conversion efficiency of 33%.
§.§ Second Harmonic Generation
With the generated SFG light, the Toptica SHG pro system is pumped. Using a commercial frequency doubling system has the advantages of reliability and stability but a downside is that some system parameters are not disclosed. We will therefore describe the system as a whole with reported in- and output powers measured before and after the full system.
Results from the SHG system. (a) UV output power (319.8 nm) as a function of total input power (639.6 nm) going into the full system. The output increases quadratically at low input powers, but saturates to a linear asymptote at higher powers. (b) System conversion efficiency (ratio of total input and output power) as a function of input power. The efficiency saturates at $\sim$50%.
Figure <ref> shows the system output power and conversion efficiency of the SHG section. At an input power of $\sim$2 W the conversion efficiency saturates at about 50% and at an input power of 4 W a maximum output power of more than 2 W of UV light was achieved. At this point the cavity coupling efficiency is just over 80%. This behaviour is qualitatively similar to what is observed in other SHG systems <cit.>. Although more SFG input power is available, we choose to remain below 4 W input power as the system is already operating close to the (UV induced) damage threshold of the crystal at these powers <cit.>. Peak-to-peak UV intensity fluctuations of 4% were measured over a 4 hour period. This is within the specifications for intensity noise of the fiber amplifiers.
The beam profile of the cavity output is still Gaussian, but shows ellipticity due to crystal walk-off. Directly from the cavity we observe a beam waist of $\sim0.7 \ \mathrm{mm}$ in the vertical and $\sim 0.1 \ \mathrm{mm}$ in the horizontal direction. This beam is passed through a collimating lens and an anamorphic prism pair to produce a more circular beam. We measure the final beam waist of $\sim0.21 \times 0.35 \ \mathrm{mm^2}$. Despite the modest ellipticity these beam parameters still allow tight focussing, an essential requirement for optical dipole trapping.
§ TRAPPING
Now that we are able to produce sufficient power at 319.8 nm, we implement the laser system into our existing setup to demonstrate trapping and to characterize the trap lifetime. We prepare the beam for trapping by enlarging it with a 1:2 telescope and focus it inside the vacuum chamber with an $f=400 \ \mathrm{mm}$ lens to a waist of $w_0^{(1)} \times w_0^{(2)} = 64.3(1.0)\times 55.6(7) \ \mathrm{\mu m^2}$. The focus positions along the horizontal and vertical axes are found to lie 12(2) mm apart which is small compared to the Rayleigh lengths (40.0(6) mm and 30.0(4) mm). We measure 62% total transmission of the two windows of the vacuum chamber. While high, these losses are expected from uncoated sapphire vacuum windows <cit.>.
A transmission per window of $T_1 = \sqrt{0.62} \approx 0.78$ is assumed to estimate the power inside the vacuum chamber. Therefore, at a power ($P$) of 1 W and neglecting astigmatism, the beam has peak intensity
\begin{equation}
I_p = \frac{2 T_1 P}{\pi w_0^{(1)} w_0^{(2)}} = 14 \ \mathrm{kW cm^{-2}}.
\end{equation}
Based on the polarizability (calculated in <cit.>) this translates to a trap depth of $\sim 6.0 \ \mathrm{\mu K \ W^{-1}}$ for a single beam. The trap depth of our ODT is therefore far below the recoil temperature
\begin{equation}
T_{rec} = \frac{\hbar^2 k^2}{k_B m} = 46.7 \ \mathrm{\mu K},
\end{equation}
where $k = 2 \pi / \lambda$ is the photon wavenumber and $m$ is the atomic mass of helium. We can therefore safely assume that each photon scattering event leads to the loss of the scattered atom. Additionally, the excess energy of the scattered atom can heat the other atoms or even kick more atoms out of the trap.
When two- and three-body losses can be neglected, the total loss rate is a combination of three distinct rates: a background loss rate $R_{bg}$ due to the background pressure inside the chamber, a loss rate $R_{sc}$ due to photon scattering, and a loss rate $R_{ion}$ due to two-photon ionization. These mechanisms scale in different ways with ODT power. The total loss rate is
\begin{equation}
R_{tot} = R_{bg} + R_{sc} I_p + R_{ion} I_p^2.
\label{eq:rates}
\end{equation}
The Rayleigh length of the focussed UV beam is comparable to the $\sim$4 cm spacing between the vacuum windows. Therefore, in a single beam ODT trapped atoms are able to collide with the windows and leave the trap. To avoid this some means of axial confinement is necessary. We use two different methods to provide this confinement. The first is to add a magnetic field gradient along the beam direction to create a hybrid trap <cit.>. With this method the loss rate and trap depth are straightforward to interpret. However, peak densities are not high enough to produce a BEC. The other method is to create a two-colour crossed-beam ODT using the UV beam and additionally a focussed 1557 nm beam which was already in place <cit.>. This trap gives a high enough peak density for a BEC to form.
§.§ Hybrid Trap
We prepare an ultracold sample in a way previously described <cit.>. A beam of $\mathrm{He^*}$ atoms is generated from a liquid nitrogen cooled dc-discharge, collimated, slowed in a Zeeman slower, and captured in a magneto-optical trap. Here the atoms are cooled to approximately 0.5 mK. Subsequently they are spin-polarized, loaded into a Ioffe-Pritchard type magnetic trap and Doppler cooled to approximately $130 \ \mathrm{\mu K}$. Finally, the atoms are cooled to $\sim0.2 \ \mathrm{\mu K}$ by forced rf evaporative cooling inside the magnetic trap.
The cloud is transferred to the hybrid trap consisting of the UV beam and a quadrupole magnetic trap (QMT) generated by a set of QMT coils in anti-Helmholtz configuration. Figure <ref> shows a schematic of this trap. The quadrupole field has a strong axis gradient of 0.54 Gauss/cm but gravity acts in a direction perpendicular to this axis. In this direction the gradient is only half the magnitude which is well below the leviation gradient of $m g / \mu = 0.351 \ \mathrm{Gauss/cm}$ <cit.>. Below this gradient gravity is stronger than the confining magnetic force so that atoms are not trapped in the absence of the UV beam. The QMT therefore adds confinement but does not contribute to the trap depth. A homogeneous magnetic field is applied with a set of fine-tune coils to minimize trap oscillations induced by the loading step.
Schematic view of the hybrid trap geometry. Atoms are trapped by the combination of the UV-laser beam and the quadrupole magnetic trap (QMT) coils. A set of fine-tune coils allows precise tuning of the trap center. Because of limited optical access the UV laser beam is at an angle of $\sim 9.5^\circ$ with the magnetic field axis. High-resolution detection is done using a micro-channel plate detector (MCP). Inset: in-situ absorption image of atoms in the UV hybrid trap.
The atoms are detected either by absorption imaging or by a micro-channel plate detector (MCP) located 17 cm below the trap center. MCP time of flight measurements are done as a function of hold time in the hybrid trap. The time of flight signals are fitted with a thermal Bose-Einstein distribution to extract the temperature of the gas as well as the atom number. The decay in atom number is fitted with an exponential (with an oscillating component to account for residual trap oscillations). The first few seconds of the decay are not fitted to neglect two-body loss and thermalization effects.
Hybrid trap one-body loss rate as a function of UV power. The blue line is a linear fit with a slope of 0.16(2) $\mathrm{s^{-1} W ^{-1}}$, corresponding to off-resonant scattering.
Figure <ref> shows the loss rate of a thermal gas inside the hybrid trap. The loss rate varies linearly with power, which is consistent with only background collisions and off-resonant scattering; two-photon ionization would depend quadratically on power. A linear fit gives a slope of about 0.16(2) $\mathrm{s^{-1} W^{-1}}$ ($R_{sc} \approx 1.2 \ \mathrm{s^{-1} W^{-1} m^2}$, see equation <ref>), with a background loss rate of 0.12(2) $\mathrm{s^{-1}}$. This background loss rate is consistent with the loss rate of 0.13(1) $\mathrm{s^{-1}}$ in a hybrid trap of comparable depth using our 1557 nm ODT for which off-resonant scattering is negligible <cit.>. Figure <ref> shows the fitted temperature as a function of hold time in the hybrid trap. This temperature is not constant; after a quick thermalization the cloud starts heating up. This heating may be caused by scattered atoms dumping a small portion of their high recoil energy in the atomic cloud. In principle it is also possible that a scattered atom is able to dump all of its energy in the cloud and thermalize instead of leaving the trap. At the highest observed heating rate $\dot{T} \approx 0.04 \ \mathrm{\mu K \ s^{-1}}$, this contributes no more than $\dot{T}/T_{rec} \approx 0.001 \mathrm{s^{-1}}$ to the loss rate, negligible compared to other processes.
(a) Temperature as a function of hold time in the hybrid trap. After some thermalization time the atomic cloud starts to heat as a result of UV absorption. The blue dotted line is a constant fit to the temperature around the minimum to extract the minimum temperature. The red dashed line is a linear fit to the temperature after a few seconds to extract the heating rate. (b) Determination of the trap temperature based on minimum temperature (blue circles), and an extrapolation to zero hold time of the heating rate (red squares). From linear fits to these data an upper (lower) bound is set on the temperature in absence of heating of $0.62(2) \ \mathrm{\mu K \ W^{-1}}$ ($0.51(2) \ \mathrm{\mu K \ W^{-1}}$).
To give an estimate of the equilibrium temperature inside the trap (in the absence of scattering) we take the minimum achieved temperature as an upper bound, and a linear extrapolation of the heating to zero hold time as a lower bound. In this way we extract temperatures of $0.62(2) \ \mathrm{\mu K \ W^{-1}}$ and $0.51(2) \ \mathrm{\mu K \ W^{-1}}$ respectively. This is approximately a factor 10 lower than the calculated trap depth and corresponds to a truncation parameter $\eta=10$ which is typically found in a thermalized trapped gas <cit.>.
(a) Schematic picture of the two-colour ODT setup. The angle between the beams is 19$^\circ$. (b) MCP time-of-flight measurement of a BEC in a two colour optical dipole trap. The signal is fit with a bimodal distribution indicating a BEC (red dashed line) and a thermal fraction (blue dotted line).
§.§ Two-colour Trap
Because of the low confinement provided by the hybrid trap, no BEC was observed. To provide enough confinement to observe BEC we switch to a two-colour trap consisting of the UV beam and an IR beam (focussed to a waist of $85 \ \mathrm{\mu m}$) which cross at an angle of $19^\circ$. Figure <ref>a shows a schematic of the setup. Figure <ref>b shows the corresponding time-of flight signal on the MCP, fit with a bimodal distribution. Superposed on the thermal distribution is a clear inverted parabola Thomas-Fermi profile demonstrating Bose-Einstein condensation. For this BEC a one-body lifetime is observed of $\sim 4 s$.
§ CONCLUSION
In order to perform magic wavelength trapping of metastable helium atoms we have realized a laser system which produces over 2 W at 319.8 nm, with not yet an indication of a reduction of SHG efficiency at higher pump power. The setup is built from commercially available fiber lasers, amplifiers and SFG/SHG components. The produced UV light is used to trap an ultracold ($\sim 0.2 \ \mathrm{\mu K}$) thermal gas in a hybrid trap and a BEC in a two colour ODT. Trap losses are found to be mainly due to off-resonant scattering with a rate of 0.16(2) $\mathrm{s^{-1} \ W^{-1}}$.
With this system we can make a sufficiently deep dipole trap in the UV while keeping the intrinsic losses at an acceptable level such that spectroscopy is possible. This opens the door to a full magic wavelength ODT and a more precise measurement of the $2 \ ^3 S \rightarrow 2 \ ^1 S$ transition frequency.
§ ACKNOWLEDGMENTS
We would like to thank Christian Rahlff of Covesion for useful comments about the PPLN crystal, Steven Knoop for useful discussions about hybrid trapping and Rob Kortekaas for technical support. This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).
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